4 int-2s: max-r». ..-.--.:.v.w~- gsfifi’gm '- IfliChigafl gtffiég UniVel‘sitY '..-'."5 This is to certify that the dissertation entitled A COMPUTER-BASED SIMULATION MODEL FOR AGRO-ECOLOGICAL ZONE YIELD ASSESSMENT presented by Nilson Amaral has been accepted towards fulfillment of the requirements for Ph . D . degree in Research Development/ Resource Information Systems“ \ Major professor Date December 10, 1986 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES RETURNING MATERIALS: Piece in book drop to remove this checkout from your record. FINES wiil be charged if book is returned after the date stamped beiow. V7733 A COMPUTER-BASED SIMULATION MODEL FOR AGRO-ECOLOGICAL ZONE YIELD ASSESSMENT BY Nilson Amaral A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Resource Development 1986 4128.8a2 I Copyright by NILSON AMARAL 1986 ABSTRACT A COMPUTER-BASED SIMULATION MODEL FOR AGRO-ECOLOGICAL ZONE YIELD ASSESSMENT By Nilson Amaral Computers and information systems are fundamental tools for decision makers and planners. A computer-based simula- tion model was developed in this study with the objective of providing a practical and useful tool for decision makers and planners to estimate crop yields in relatively large and homogeneous regions, the so-called agro-ecological zones. The simulation model was developed in two modes: First, the deterministic approach was used to analyze yield trends in an a posteriori type of analysis. Second, a stochastic approach, with random environmental inputs, was used to provide decision makers with the yield-distribution parame- ters necessary to make inferences about yield values, as well as crop-yield prediction. The technique used in the simulation model does not rely solely on yield time series but on the process of biomass production, where water deficit is a primary constraint. The Monte Carlo approach was employed to generate random Nilson Amaral variates based on the distribution parameters of the popula- tion data for the exogenous environmental inputs. A variance-partitioning technique, which considers random variation in the input parameter, was used with the Monte Carlo approach. The simulation results, using three regions in Jamaica and three agro-ecological zones in the Dominican Republic as data sources, showed the validity of the model when results were compared with observed-yield data for those locations, as well as with FAO yield guidelines. Results are presented in the form of tables, scattergrams, and histograms to serve as an aid to decision making and planning. Limitations do exist in the simulation model and are presented in the last chapter. Data completeness and preci— sion pose some limitations to the final analysis, which indicates a need to improve data collection. Despite its limitations, the model shows the feasibil- ity of the systems approach to crop-yield estimation and opens new insights into the process of yield prediction for use in decision making and planning, and as a linkage to other models such as economic—optimization models. ACKNOWLEDGMENTS I wish to express my gratitude to: Dr. G. Schultink, who gave me ample freedom to approach the problem of this dissertation with my own ideas, but gave his objective guidance and criticism to direct those ideas. His counseling, friendship, and support during my training at Michigan State University will always be appreciated. Drs. D. E. Chappelle, T. Manetsch, and M. Steinmueller, members of the advisory committee, for their comments and suggestions at different stages of this work. Dr. S. Witter for his help at several stages of the model development, as well as in the preparation of the data set used in the simulation runs. Thanks are also due to my colleagues and friends David Mendez, Dorothy Dunkley, and Sashi Nair for their help in data preparation. A number of friends, especially Eliseu R. A. Alves, Mauro R. Lopes, and several others in Brazil, in many ways helped me in my training process. Mrs. Sue Cooley helped in editing and final typing. Acknowledgment is also made to the CRIBS project at Michigan State University, EMBRAPA, and CNPQ for their financial support throughout the course of the investigation and my training at MSU. Last but not least, my sincere appreciation and admira- tion to my wife, Milza; my son, Nilson; and my daughter, Larissa, for their love, encouragement, and cooperation, which made this undertaking possible. vi TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES O I O I O O I I C O xiii Chapter I. INTRODUCTION 0 O O O O C O 1 Objectives of the Study . . . . . . . . . . 2 Literature Review . . . . . . . . . . . . . 6 Technique Used . . . . . . . . . 11 Organization of the Dissertation . . . . . . 14 II. AGRO-ECOLOGICAL ZONE DETERMINATION . . . . . . 16 Data- Acquisition Process . . . . . . . l7 Computational Tools for AEZ Determination . 19 CRIES- GIS: An Overview . . . . . . . . . . . 22 CRIBS—AIS: An Overview . . . . . . . . . . . 27 III. DETERMINISTIC YIELD SIMULATOR (DYS) . . . . . 31 Model Assumptions . . . . . . . . . . . . . 34 Model Structure . . . . . . . . . . . . . . 36 Mathematical Formulation . . . . . . . . . . 39 Data Requirements . . . . . . . . . . . . . 61 IV. STOCHASTIC YIELD SIMULATOR (SYS) . . . . . . . 65 Stochastic Approach . . . . . . . . . . . . 67 Random—Variates Generation . . . . . . . . . 74 Variance Partitioning and Common Scenario Analysis . . . . . . . 77 Stochastic Yield Simulator (SYS) Structure . 78 Mathematical Formulation . . . . . . . . . . 89 Data Requirements . . . . . . . . . . . . . 93 V. MODEL VALIDATION AND SIMULATION RUNS Deterministic YIELD Simulator Run, Jamaica-~Introduction . . . Deterministic YIELD Simulator Run, Jamaica--Environmental Inputs . Deterministic YIELD Simulator, Jamaica-~Crop Parameters . . . Deterministic YIELD Simulator, 0 Jamaica--Farm-Management-Practice Parameters . . . . . Deterministic YIELD Simulator, Jamaica--Local Parameters . . . Deterministic YIELD Simulator, Jamaica--Simulation Results . . Stochastic YIELD Simulator, Jamaica--Environmental Inputs . Stochastic YIELD Simulator, Jamaica--Simulation Results . . Stochastic YIELD Simulator, Dominican Republic—-Environmental Inputs Stochastic YIELD Simulator, Dominican Republic--Simulation Results . VI. SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FURTHER RESEARCH . . . . . . . . . . Summary . . . . . . . . . . . . . Conclusions .. . . . . . .. . Suggestions for Further Research . . . APPENDICES A. INTERPOLATING FUNCTIONS .. . .. . B. NUMERICAL INTEGRATION AND DIFFERENTIATION C. THE INVERSE TRANSFORMATION METHOD . LIST OF REFERENCES 0 C O I C O C C C O O O 0 viii Page 95 99 100 101 101 103 105 129 139 159 172 186 186 191 195 200 207 210 215 LIST OF TABLES Table Page 1. Percentage of Water that Percolates into the Soil as a Function of Percentage Slope and Soil Textural Classes . . . . . . . . . . 53 2. Crops, Salinity Levels in mmhos/cm, and Percentage Yield Decrease Values . . . . . . . 60 3. Deterministic YIELD Simulator: Jamaica-—Crop Parameters--Sugarcane, Tobacco, and Sorghum for Worthy Park, Caymanas, and Monymusk . . . 102 4. Deterministic YIELD Simulator: Jamaica——Farm- Management—Practice Parameters for Sugarcane, Tobacco, and Sorghum, for Worthy Park, Caymanas, and Monymusk . . . . . . . . . . . . 104 5. Deterministic YIELD Simulator: Jamaica-—Local Parameters for Worthy Park, Caymanas, and Monymusk . . . . . . . . . . . . . . . . . . . 106 6. Deterministic YIELD Simulator: Jamaica—- St. Catherine-~Worthy Park. Sugarcane-- Observed Irrigated Yield and Simulated Irrigated and Rain-fed Yield, 1963—1982 . . . 107 7. Deterministic YIELD Simulator: Jamaica—- St. Catherine-—Caymanas. Sugarcane-- Observed Irrigated Yield and Simulated Irrigated and Rain-fed Yield, 1963-1982 . . . 115 8. Deterministic YIELD Simulator: Jamaica-- Clarendon--Monymusk. Sugarcane-— Observed Irrigated Yield and Simulated Irrigated and Rain-fed Yield, 1963—1982 . . . 121 9. Deterministic YIELD Simulator: Tobacco and Sorghum: Jamaica-—Worthy Park, Caymanas, and Monymusk. Simulated "Average" Yield Results Over the Period From 1963 to 1982 . . 127 ix Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for Jamaica--Worthy Park, for the Years 1963 to 1982 Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for Jamaica--Caymanas, for the Years 1963 to 1982 Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for Jamaica--Monymusk, for the Years 1963 to 1982 Stochastic YIELD Q 0 O o I O o I o o o o I o Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for Jamaica-~Worthy Park for the Years 1963 to 1982 Stochastic YIELD - o o a o I O O I a o a I I Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for Jamaica—-Caymanas for the Years 1963 to 1982 Stochastic YIELD Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for Jamaica--Monymusk for the Years 1963 to 1982 Stochastic YIELD for Sugarcane, the Years 1963 Stochastic YIELD for Sugarcane, the Years 1963 Stochastic YIELD for Sugarcane, the Years 1963 Stochastic YIELD Simulator: Fertilizer Usage Jamaica-—Worthy Park for to 1982 O C O O O O O O O O Simulator: Fertilizer Usage Jamaica—-Caymanas, for to 1982 I O I I O O O O 0 O Simulator: Fertilizer Usage Jamaica-- Monymusk, for to 1982 0 O O O O O O O I O Simulator: Sugarcane Results, Jamaica--Worthy Park, Caymanas, and Monymusk Page 130 131 132 133 134 135 137 138 139 140 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Stochastic YIELD Simulator: Tobacco Results, Jamaica-—Worthy Park, Caymanas, and Monymusk Stochastic YIELD Simulator: Sorghum Results, Jamaica--Worthy Park, Caymanas, and Monymusk Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for the Dominican Republic Ocoa Watershed's AEZ Valdesia, for the Years 1970 to 1984 . . Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for the Dominican Republic Ocoa Watershed's AEZ Ocoa, for the Years 1970 to 1984 . . . . Stochastic YIELD Simulator: Precipitation Probability Density Function Statistics for the Dominican Republic Ocoa Watershed's AEZ Azua, for the Years 1970 to 1984 . . . . Stochastic YIELD Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for the Dominican Republic's Ocoa Watershed AEZ Valdesia, for the Years 1963 to 1982 . . Stochastic YIELD Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for the Dominican Republic's Ocoa Watershed AEZ Ocoa, for the Years 1963 to 1982 . . . . Stochastic YIELD Simulator: Temperature, Relative Humidity, and Wind Velocity Probability Density Function Statistics for the Dominican Republic's Ocoa Watershed AEZ Azua, for the Years 1963 to 1982 . . . . Stochastic YIELD Simulator: Dominican Republic --Ocoa Watershed Crop Parameters-~Rice, Potato, Fresh Pea for Valdesia, Ocoa, and Azua O C 0 O O O O I O O I O I I O 0 I l I 0 Stochastic YIELD Simulator: Dominican Republic -—Ocoa Watershed Crop Parameters--Onion and Cabbage for Valdesia, Ocoa, and Azua . . Page 149 154 160 161 162 163 164 165 166 168 Stochastic YIELD Simulator: Dominican Republic —-Ocoa Watershed Farm—Management-Practice Parameters--Rice, Potato, and Fresh Pea for Valdesia, Ocoa, and Azua . . . . . . . . . Stochastic YIELD Simulator: Dominican Republic --Ocoa Watershed Farm-Management-Practice Parameters--Onion and Cabbage for Valdesia, Ocoa, and Azua . . . . . . . . . . . . . . Stochastic YIELD Simulator: Dominican Republic ——Ocoa Watershed Local Parameters for Valdesia, Ocoa, and Azua . . . . . . . . . Stochastic YIELD Simulator: Probability Density Function Statistics Results for Onion for the Dominican Republic--Ocoa Watershed--Valdesia, Ocoa, and Azua AEZs . Stochastic YIELD Simulator: Probability Density Function Statistics Results for Rice for the Dominican Republic--Ocoa Watershed-—Valdesia, Ocoa, and Azua AEZs . Stochastic YIELD Simulator: Probability Density Function Statistics Results for Fresh Pea for the Dominican Republic--Ocoa Watershed--Valdesia, Ocoa, and Azua AEZs . Stochastic YIELD Simulator: Probability Density Function Statistics Results for Potato for the Dominican Republic--Ocoa Watershed-~Valdesia, Ocoa, and Azua AEZs . Stochastic YIELD Simulator: Probability Density Function Statistics Results for Cabbage for the Dominican Republic—-Ocoa Watershed-~Valdesia, Ocoa, and Azua AEZs . . Page 169 170 171 173 176 179 182 184 Figure 1. 2' LIST OF FIGURES Agro-Ecological Zone YIELD Assessment as a Component Of CRIES-RIS o o a a o o o o o o 0 Geographic Information System-—OVERLAY Analysis I I I I I I I I I I I I I I I I I I Geographic Information System--Character Map of Elevation . . . . . . . . . . . . . . Geographic Information System--Two-Way Cross—Tabulation . . . . . . . . . . . . . Geographic Information System-~Choroline Printer Map I I I I I I I I I I I I I I I I General Input/Output Diagram for the Deterministic YIELD Simulator with Nomenclature . . . . . . . . . . . Deterministic YIELD Simulator-~General Diagram I I I I I I I I I I I I I I I I I I Deterministic YIELD Simulation Model—~Phase 1: General System Diagram . . . . . . . . . . . Deterministic YIELD Simulation Model--Phase 2: General System Diagram . . . . . . . . . . . Deterministic YIELD Simulation Model——Phase 3: System Diagram . . . . . . . . . . . . . . . Deterministic YIELD Simulator--Phase 4: System Diagram . . . . . . . . . . . . . . . Yield Adjustment Based on Generalized Fertilizer Availability for All Crops . . . General Input/Output Diagram for the Stochastic YIELD Simulator with Nomenclature xiii Page 23 24 26 28 29 32 37 44 49 54 58 58 68 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. Stochastic YIELD Simulator--An Example of Histogram Plot . . . Stochastic YIELD Simulator--Probabi1ity Density Function Skewing Factor Variation Stochastic YIELD Simulator--Triangular Probability Density Function . Stochastic YIELD Simulator--Gamma—Variates- Generation Process . Stochastic YIELD Simulator—~General Flowchart of the Simulation Process Stochastic YIELD Simulator-~Phase l Flowchart Stochastic YIELD Simulator-~Phase 2 Flowchart Stochastic YIELD Simulator-—Phase 3 Flowchart Stochastic YIELD Simulator-~Phases 4 and 5 Flowchart . . . . . Stochastic YIELD Simulator—-Phase 6 Flowchart Stochastic YIELD Simulator--Phase 7 Flowchart Deterministic YIELD Simulator: Worthy Park, Observed Irrigated Yield and Simulated Irrigated Yield for Sugarcane for the Period 1963-1982 Deterministic YIELD Simulator: Worthy Park, Sugarcane Irrigated Observed Yield and Simulated Rain-fed Yield From 1963 to 1982 . . . . Deterministic YIELD Simulator: Worthy Park, Sugarcane Irrigated Observed Yield versus Simulated Irrigated Yield Deterministic YIELD Simulator: Worthy Park, Sugarcane Irrigated Observed versus Simulated Rain-fed Yield Page 69 71 73 76 80 81 83 84 86 87 88 108 110 113 113 Page 29. Deterministic YIELD Simulator: Jamaica-- Caymanas, Sugarcane Observed Irrigated Yield and Simulated Irrigated Yield from 1963 to 1982 . . . . . . . . . . . . . . . . . 116 30. Deterministic YIELD Simulator: Jamaica—- Caymanas, Sugarcane Observed Irrigated Yield and Simulated Rain-fed Yield from 1963 to 1982 o o o o o O 0 I o o c n o o O o I O O o o 118 31. Deterministic YIELD Model: Jamaica--Caymanas, Sugarcane Observed Irrigated Yield Versus Simulated Irrigated Yield . .. . .. . .. . 119 32. Deterministic YIELD Model: Jamaica--Caymanas, Sugarcane Observed Yield versus Simulated Rain-fed Yield I I I I I I I I I I I I I I I I 119 33. Deterministic YIELD Simulator: Jamaica-- Monymusk, Sugarcane Observed Irrigated Yield and Simulated Irrigated Yield from 1963 to 198 o o I o I I a o o 122 34. Deterministic YIELD Simulator: Jamaica-— Monymusk, Sugarcane Observed Irrigated Yield and Simulated Rain-fed Yield from 1963 to 19 I I I I I I I I I I I I I O I I I I I I 35. Deterministic YIELD Simulator: Jamaica-— Monymusk, Sugarcane Observed Irrigated Yield versus Simulated Irrigated Yield . . .. . . . 124 36. Deterministic YIELD Simulator: Jamaica-- Monymusk, Sugarcane Observed Irrigated Yield versus Simulated Rain-fed Yield .. .. .. . 125 37. Stochastic YIELD Simulator: Jamaica--Worthy Park. Sugarcane Absolute Frequency Histogram for Potential, Irrigated, and Rain—fed Yield . 145 38. Stochastic YIELD Simulator: Jamaica-~Caymanas. Sugarcane Absolute Frequency Histogram for Potential, Irrigated, and Rain-fed Yield . . . 146 39. Stochastic YIELD Simulator: Jamaica-~Monymusk. Sugarcane Absolute Frequency Histogram for Potential, Irrigated, and Rain-fed Yield . . . 147 XV 40. 41. 42. 43. 44. 45. 46. 47. 48. Stochastic YIELD Simulator: Jamaica—-Worthy Park. Tobacco Absolute Frequency Histogram for Potential, Irrigated, and Rain-fed Stochastic YIELD Simulator: Jamaica—-Caymanas. Tobacco Absolute Frequency Histogram for Potential, Irrigated, and Rain-fed Yield . Stochastic YIELD Simulator: Jamaica-~Monymusk. Tobacco Absolute Frequency Histogram for Irrigated, and Rain-fed Yield . Stochastic YIELD Simulator: Jamaica--Worthy Park. Sorghum Absolute Frequency Histogram for Potential, Irrigated, and Rain-fed Stochastic YIELD Simulator: Jamaica--Caymanas. Sorghum Absolute Frequency Histogram for Irrigated, and Rain-fed Yield . Stochastic YIELD Simulator: Jamaica-~Monymusk. Sorghum Absolute Frequency Histogram for Irrigated, and Rain-fed Yield . Stochastic YIELD Simulator: Republic, Ocoa Watershed--Ocoa--Onion. Absolute Frequency Histogram for Potential, and Rain-fed Yield Stochastic YIELD Simulator: Republic, Ocoa Watershed--Ocoa--Rice. Absolute Frequency Histogram for Potential, and Rain-fed Yield Stochastic YIELD Simulator: Republic, Ocoa Watershed-—Ocoa—-Fresh Pea. Absolute Frequency Histogram for Potential, and Rain-fed Yield Stochastic YIELD Simulator: Republic, Ocoa Watershed--Ocoa--Cabbage. Absolute Frequency Histogram for Potential, and Rain-fed Yield Page 151 152 153 156 157 158 175 178 181 185 Page 50. Tablex: A Algorithm for Functional Interpolation . . . . . . . . . . . . . . . . 205 51. Cumulative Distribution Function . . . . . . . . 210 xvii CHAPTER I INTRODUCTION For most developing countries, agricultural produc- tivity has an important influence on economic development, trade, and foreign exchange earnings. Developing nations are seeking areas best suited for crop production to satisfy internal demand, to keep food prices at affordable levels for the large percentage of the population with low income, and to generate foreign exchange to pay for the importation of technology, goods, and services. Past and predicted crop yields play an important role in the decision-making process in many areas of a country's economy. Information on harvest size is needed for a vari— ety of purposes. Governments require information for admin- istrative and planning purposes, possibly for measures to regulate quantities imported and/or exported, to control prices, and so on. Private firms are interested in approp— riate data for their marketing and storage arrangements. Farmers may use harvest data as the basis for their seasonal purchases to obtain particulary favorable prices. Researchers are interested in optimizing the regional distribution of agricultural production patterns. Heady (1964) used crop yields as a significant variable for a linear optimization model for crop allocation. Vilas (1975), who focused his Ph.D. research on the spatial equi- librium analysis of the rice economy in Brazil, used crop yield as one of the most important variables in the inter- regional analysis. Heiss (1981) discussed the economic benefits of improved crop information on wheat and all cereals for European countries. He developed a model for estimating the economic benefits of cropeyield assessment for the European Community (BC) as well the benefits for producers, consumers, and governmental agencies. :1. . E 1 S 3 Answers are needed to questions asked by scientists, decision makers, and planners regarding effects of agro- climatic conditions and management practices on the agricul- tural productivity of cash crops and basic food staples. Answers are needed to such questions as: 1. How are yield values for the different farming systems estimated? By farming system is meant "the complex arrangement of soils, water sources, crops, livestock, labor, and other resources and characteristics within an environmental setting that the farm family manages in accordance with its preferences, capabilities and available technologies" (Shaner, 1982L 2. What are the tradeoffs between irrigation invest- ment, the cost of farm management practices, and the increase in productivity through other factors? 3. How can a developing nation, in need of foreign exchange, improve land-use planning for agricultural produc— tion? Those questions demand the application of a new method using computational procedures, preferably a computer-based model, for evaluating responses of a broad range of agricul- tural crops to agro-climatic (rainfall, temperature, soil, slope, eth and farming-system parameters (fertilizer usage, management practices, and so onL The objective of the present research is to address these questions in a microcomputer—based simulation model that will aid the decision-making process in a "user friendly" manner by predicting crop yields for a homogeneous region or agro-ecological zone. These yield predictions are most representative of relatively large-scale farming sys— tems in mono cultivation. A computer-based simulation model was developed for use in a deterministic as well as in a stochastic or probabilis- tic mode for farming-system yield assessment that can be run in an interactive manner on a microcomputer. The deterministic mode may help decision makers compare the simulation results with the observed yield on a year-by- year basis-—a posteriori analysis-~and evaluate the current production systems and/or practices and their associated yield response. The stochastic mode may help decision makers deal with the decision process under conditions of uncertainty. The estimated probability-density statistics such as moments (mean, variance, and skewness) and quantiles, computed from the model‘s results, provide decision makers with a wealth of information for assessing any uncertainty present in the system. The simulation model can simulate yield for the most important crops responsible for generating foreign exchange for developing nations, as well as crops responsible for ensuring internal food security. It is also this researcher's objective to use in the simulation model a methodology that does not rely solely on time series of past yield data because data availability and reliability due to, among other things, government intervention are significant constraints in most developing nations. It is hoped that the simulation model can eventually be linked to an economic-optimization model that can be used to optimize land use and help in determining the "best" land— allocation scheme. The YIELD model developed herein is a component of the Comprehensive Resource Inventory and Evalu- ation System--Resource Information System (CRIES-RIS) (Schultink, 1981, 1983, 1984). The CRIES resource inventory and analysis approach to integrated rural development planning and agricultural sec— tor analysis has two major components: the CRIES-GIS (Geo— graphic Information System) and the CRIBS-AIS (Agro—economic Information System). The YIELD simulator is a component of the CRIBS-AIS information system (Schultink, 1986). Policy variables are not explicitly included in the model at this stage, but model results can be analyzed and changes made in the input parameters and variables to reflect various policy scenarios. Alternative policies and climatic, physical, and farming-system characteristics can easily be examined in an interactive manner using a micro- computer. The model is designed for use by those with little or no computer experience. A series of menus and system prompts provides the user interaction. The simulation model developed herein can aid in evaluating national and international strategies for agricultural-production planning and take advantage of pre- vailing agro—climatic conditions. The model could form the basis for evaluating irrigation and pest-management deci- sions during the growth season, evaluating investment decisions, forecasting yields, or predicting the effects of soil erosion and water deficit during the growing season. The model can determine the yield value on a seasonal and spatial basis by Agro-Ecological Zone (AEZ) (the so- called Resource Planning Unit or RPU [Schultink, 1983]) for major agricultural crops, including those termed cash crops, such as bananas, sugar cane, soybean, tobacco, and wheat, and food staples such as potato, rice, corn, tomato, bean, and cabbage. Findings of the study as intended neither as precise descriptions of the real world nor as final predictions. Instead, the model was designed to provide insight into decision—making criteria associated with local, soil, envi- ronmental, and management practices and their associated variables used in estimating crop yields. The user should realize that the model provides yield predictions for high- yielding varieties, adapted to the agro—ecological condi- tions represented. As such, variety-specific yields may change by location and are affected by general crop adapta- bility, incidence of disease, pests, and other factors. I' I B . The role of crop-weather models has become increasingly important in assessing potential crop production based on climate, monitoring crop prospects from current weather data, evaluating the effect of natural or man-induced climatic variability on crop yields, and interpreting the effect of weather on yields. Much of the early modeling research on agricultural production systems used statistical analysis as a modeling technique (Smith, 1914; Buck, 1961; Gibson, 1979). Regres- sion models (Botkin, 1969; Thompson, 1975; Vilas, 1979; Bortoluzi, 1978; Heady, 1964) that are based on past crop- yield values are often expressed in a functional format, which contains linear, logarithmic, quadratic, or a combina- tion of these terms involving price, fertilizer usage, and so on, and data such as rainfall, temperature, and time. The regression approach is one of the most common tech- niques used in yield estimation. It has some serious limi- tations due to a vast number of variables and the complexity of the processes involved in plant growth. Before any attempt is made to model a process, basic research is needed to understand fully the theory involved. Regression modeling was used by Sakamoto (1981) in his paper entitled "Climatic-Crop Regression Yield Model: An Appraisal." He made it clear that despite its limitations, regression analysis is a useful tool. He also indicated that much of the utility of regression analysis is associ- ated with its simplicity of application and the availability of data. In his paper "Methods of Crop Production Forecasting in the EEC; Present and Expected Trends in Crop Production," Thiede (1981) briefly discussed the methods of estimating harvests that are currently in use. According to Thiede, the methods used to estimate harvests in the EC vary widely from one member state to another, partly for historical reasons and partly because of the differing fundamental attitudes of farmers toward statistics. Thiede also pointed out that, concerning the methods used, a distinction must be made between pure estimates, objective measurements, and calculations based on agricultural meteorological data. In the late 19605, researchers turned their attention to understanding better the physical and chemical processes involved in crop growth (DeWit, 1965; Ducan, 1967; Lake, 1967). Those theoretical developments gave rise to rela- tionships that were tested in laboratories, "but they lacked the dynamic properties of the plant systems" (Curry, 1975). Computer modeling and simulation began to have a place in agricultural production systems in the last decade or so. Several simulation models for single crops were developed for corn (Curry, 1971), soybean (Curry, 1975), and alfalfa (Holt, 1978), based on temporal modeling, without any attempt to have a temporal-spatial resolution for crop yields in aiding the decision-making process. Curry (1971) pointed out that serious limitations still may exist: The ultimate computer model for the soybean plant would be flexible enough to simulate growth and development at any location for which climate and cultural informa- tion is available. The expected results would be reasonable yield estimates and understanding of the physiological processes underlying these yields. Simu- lations of this type are not limited by mathematical or computer capabilities, but rather by lack of under— standing of the interaction of the plant with its environment. One major use of crop-yield simulation is to improve assessment of technology-transfer options based on regional characteristics. By using the yield simulator, the agricul- tural researcher can simulate environmental situations and obtain critical information on future research priorities, such as crop adaptation, the potential of introducing new cultivars, and so on. This is only possible if the model is designed to accommodate changes in the simulation process. To permit this, the yield-simulation model must be capable of evaluating the yield response for several sites with different soils and climate characteristics, thereby provid— ing a rapid and effective means of assessing and transfer- ring crop-production technology to developed nations as well as developing countries around the world. The dynamics of the input variables, together with non- availability of time series data and the spatial-dimension requirements that affect crop yields, requires a more elabo— rate procedure that goes beyond regression models. Economic-development studies and policy analysis in agriculture production and land-use planning make the usual 10 regression technique for estimating crop yield less than appropriate. This requires further research involving dynamic simulation models that provide a more realistic crop-yield assessment. Essentially, such models exist for most of the world's major economic crops, as Hayes (1982) indicated. Most of these models take into consideration temporal and/or deter- ministic modeling but do not attempt to assess yield on a spatial or farming-systems basis. Several attempts have been made to show spatial pat- terns of photosynthesis or yield modeling at a regional level for a specific crop (Monteith, 1972, Baier, 1976). Hayes (1982) constructed a numerical crop-yield model for 11 crops. The Hayes model is a deterministic model that computes yields for crops that are grown mainly by developed nations (spring wheat, winter wheat, spring barley, winter barley, and so on). The Hayes model, besides focusing mainly on cash crops for developed nations, has requirements for its operation that.are out of reach forxnost developing nations (large mainframe computer, numerical calculus libraries, expertise in computer programming, and so on). The spatial dimension of the models described above does not consider the yield evaluation for a micro-region such as an agro-ecological zone, a production potential area, or farming systems. However, those characteristics 11 are critically needed for yield assessment for developing nations, considering the size of the farm holdings and the wide variations in and nonuniformity of the farming systems. Iechnisuflsed The deterministic model was developed in two major modules. The first module is the data entry/data edit man- agement phase, which allows the user to enter and make changes in the local and climate data set. This module is totally menu driven and has user-friendly design character- istics. The second module is the simulation model itself; interaction with the user occurs on a conversational basis. The user responds to the model prompts and changes parame— ters according to specific requests. The C programming language is used for the first module (Richie & Kerninghan, 1978), whereas the second module is programmed using Fortran 77 (Microsoft, 1984L The yield-simulation model is designed to run on an IBM—PC XT or compatible microcomputer, with a hard disk (one needs 1.2 MB disk space), math co-processor 8087 to speed up simulation runs, and a printer to obtain a hard copy of yield-simulation results. The option of on-screen reporting is also available. To achieve the proposed objectives, the model is struc- tured into four components, which permit execution on a 12 microcomputer with at least 256KB RAM (Random Access Memory). The YIELD simulator consists of five phases: Phase 1: Calculation of the maximum potential yield Phase 2: Calculation of the maximum evapotranspiration Phase 3: Calculation of the actual evapotranspiration Phase 4: Calculation of the estimated yield Phase 5: Estimated yield adjustment The deterministic simulation model has data require- ments and default values for the inputs to make it possible to run the model for regions where data are not available or where the expected data precision is low. A discrete time— simulation approach (Forrester, 1961; Manetsch & Park, 1984) is used in all four phases. The numerical integration and differentiation technique (Hamming, 1962; Conte, 1980) is used to implement the equations. A computation sequence for continuous—flow simulation models is used in the following format (Manetsch & Park, 1984; Chappelle, 1985): A. I 'I' J' I' E] 1. Assign values to model parameters N o Initialize state or level variables 81 (0), 82 (0), loo, Sn (0) w o Initialize time T = 0 4. Specify characteristics such as length, number, output, etc. 13 B.Exe_cutinn_2hase 1. Compute rate variables for time T: Rj (t) = gj (s1 (T), 52 (T) ,..., sn ) j = 1,2 ,..., m 2. Print rate variables 3. Update time : T = T+DT 4. Compute state variables for time T+DT i = 1,2 ’00., n 5. Print state variables 0‘ 0 Return to (1) if simulation run is not completed \I o Terminate simulation run The model structure is based on the equations developed by Doorenbos and Kassam (1979) and Slabbers et a1. (1979), which focus on the relationships between crop yield and water availability. To accommodate the simulation structure outlined above and to account for soil- and management- practice parameters, modifications and additions were made in the equations and procedures provided, and the results were transferred into a simulation model and converted into the C programming language. The deterministic model relies on environmental vari— ables value distributed annually (daily values or monthly means) for selected target years. 14 The stochastic or probabilistic model relies on the shape of the distribution and moments (mean, variance, skew- ness) of the agro-ecological data and parameters associated with management practices. Those variables are then gen- erated internally by the simulator for each day of the grow— ing period and for each simulation run. An input data preprocessing scheme, which uses statistical analysis to determine the shape of the distribution and moments of the agro-ecological variables, is required. A stochastic model component was developed as a separate and independent module that contains all the functions and equations used in the deterministic model. : . I' E I] E' l' Chapter I was an introductory chapter. Chapter II is mainly concerned with the definition of the regionalization process as a precondition for model execution. Agro- Ecological Zone (AEZ) determination procedures are pre- sented, and model assumptions are stated. The analytical framework for the deterministic yield simulator, by phases, is shown in Chapter III. Model assumptions, mathematical equations, and the data require— ments are also presented. Chapter IV contains the analytical framework for the stochastic or probabilistic yield simulator. The use of Statistical procedures, the random variable generator and 15 procedures, data requirements, and mathematical equations are presented. In Chapter V the simulation results of both approaches are presented and discussed. Results are shown in the form of tables and graphs. Chapter VI contains an analysis of the results in terms of policy decision tools for decision making and development planning. Implications, a summary of findings, conclusions, and suggestions for further research are also discussed. CHAPTER II AGRO-ECOLOGICAL ZONE DETERMINATION A regionalization and an aggregation process must be carried out in the study area to identify agro—ecological zones (AEZs) with common physical characteristics, such as soil, soil textural class, prevailing slope, and climate. Two concepts in AEZ definitions were given by Schultink (1984), which characterize an AEZ. They are: — Resource Planning Unit (RPU), a geographically deline- ated unit of land that is relatively uniform with respect to land forms, soil types and patterns, climate, and natural vegetation. - Production Potential Area (PPA), an aggregate area of individual soil types and associated climates within an RPU, which is sufficiently homogeneous with respect to plant adaptability, management requirements, and potential produc- tivity to be reliably depicted by unique estimates of those parameters to serve as an analytical reference for national or regional analysis and planning. 16 17 To accomplish the task of determining AEZs, a data- acquisition process must be activated to collect the required information from and for the study area. The type of data required for AEZ determination will depend on the study objectives, resources and technology available, and so on. Data can be acquired through three main types of data- collection procedures (Chappelle, 1985; Schultink, 1984): — Primary data, which are data acquired to meet spe- cific information requirements of the project, such as spe- cial aerial surveys for topographic mapping, soil analysis, and so on. - Secondary data, which are existing data with charac— teristics and format suitable to meet specific information requirements with minor modifications, such as area calcula— tion from vegetation maps, climatic data from meteorological stations, and so on. Secondary sources are the most common and the least expensive sources of data and are frequently used in AEZ-determination processes. - Derived data, which are existing data with charac- teristics and format suitable to meet certain information requirements with major modifications, such as reinterpreta- tion of existing soil maps to assess crop-specific produc— tion potential using vegetation indicator species and special vegetation surveys and indicator species. In 18 general, the process involves secondary data collection followed by data transformation or derivation. — Based on the user and project objectives identified, data aggregation and regionalization is used to define and spatially delineate homogeneous areas with respect to major soil, topographic, and climatic characteristics. The pro- cess makes it possible to differentiate relatively large areas for which a specific crop-yield response can be pre— dicted on the basis of homogeneous criteria. The cost of a project is largely determined by its data-collection procedures. Sometimes a tradeoff between cost of data acquisition and resulting precision must be made to accomplish the project objectives with minimum cost and/or within the project budget. The system or project design team must be mindful of potential constraints on data availability, such as adminis— trative obstacles, confidentiality obstacles, time and con- tinuity constraints, cost constraints, and data-precision problems. The accuracy of the digital representation of spatial data is governed by both user requirements and the inherent characteristics of the source document and the instruments used to create it. 19 C I l' 1 T ] E EEZ E | . I' In general, a large amount of data must be collected and manipulated to determine the AEZs. Existing maps and data, survey observations, and applied remote sensing are some of the methods used in the data-acquisition process. The objective of this section is to describe a "state of the art" technique, which uses computers as tools for AEZ deter- mination. Computers play a fundamental role in natural-resources assessment today. Several software programs are being developed with the objectives of processing data acquired from remote sensing and other data-acquisition methods. Those programs or systems are usually called Geographic Information Systems (GIS) and have the capability of pro- cessing large amounts of data in a spatial context. Examples of such systems are the CRIES-GIS (Schultink et al., 1981) and Canada Geographic Information System (CGIS) (Marble & Peuquet, 1982). A GIS represents a system, commonly computer-based, for handling spatial data. A critical and unique property of spatial data is that each entry must be definedixxterms of its location in a two- or three-dimensional space. The GIS is the main tool for handling spatial data. The major objective of a GIS is to support the spatial decision—making process in resource use and management. The most important 20 functions of a GIS are as follows (Marble & Peuquet, 1982; Schultink et al., 1981): Data_input: Normally consists of a combination of manual and automatic digitizing operations, together with associated data cleaning and edit activities. By digitizing is meant a process of data capture for spatial data—handling purposes; the main source of data frequently is maps. Manual digitizing has some advantages in terms of correct data assessment, but it is slow and labor-intensive, and errors may be generated by the digitizer operator. Auto— matic digitizing is now being carried out by a number of methods (Marble & Peuquet, 1982). The most common method is the use of a large drum scanner, such as those employed in graphic arts. Speed and reliability are the main advantages of drum scanners. Data_stgrass_and_retrieyal: Initial creation of the spatial data base, together with subsequent update opera— tions and query handling. A data base is defined as a collection of interrelated data stored together with con— trolled redundance to serve one or more applications in an optimal fashion. The data are stored so that they are independent of programs that use the data (Date, 1977; Martin, 1977). Usually, construction of the data base that contains the spatial indexed information is based on the relational data base theory (Codd, 1970). It is important 21 to keep in mind that the spatial—dimension characteristics of the GIS require a data-base structure different from the usually known data bases for business applications, such as Data Base Management System (DBMS) and Management Informa- tion System (MIS) (Date, 1977). Data_manipulation: Creation of composite variables through processing activities directed toward both spatial and nonspatial attributes of system entities. Any GIS must be capable of performing a series of manipulations on the spatial data held in its files. Each system contains a specific set of these procedures, determined by the require— ments of the users of the system. Analysis: The combination of the various resources' attributes and their associated measurement scales in a set of mathematical operations designed to derive indices relat— ing to optimum-use aspects, given a complex set of physical and socioeconomic criteria, e.g., suitability and effect assessment, economic feasibility analysis, and optimum allo— cation decisions, given distance parameters and infrastruc- ture. WWW: Creation of both tabular (statistical results, tables, and so on) and cartographic reports, maps, and pictures reflecting selec- tivity retrieval and manipulation of entities within the data base. Those functions or computer—aided procedures are 22 designed to delineate AEZs, representing areas with physical characteristics considered relatively homogeneous at a pre— defined level of detail. CBIES:§ISJ__AD.QxeLEieE The Comprehensive Resource Inventory and Evaluation System (CRIES) Resource Information System (RIS) has two major components: the Geographic Information System (GIS) and the Agro—economic Information System (AIS) (Figure 1). The YIELD simulator and the AEZs are combined for the AEZ- yield estimation. The CRIBS-GIS provides the capability to store, edit, and process digital map data and creates the master data base (disk files) for subsequent analysis. The CRIES—GIS (Schultink, 1981) has in its analysis module an important phase, called raster OVERLAY. In this process one raster file is superimposed upon another file web, and the (weighted) concurrence of these two data sets and derived indices are determined. The system can overlay up to ten files in one operation. The OVERLAY analysis of multiple attributes is shown in Figure 2. Its output is a single layer of information with attribute values resulting from a linear combination of the attribute values from the other information layers. The MATCH phase creates new attribute values for user—specified co—occurrences of existing attributes values. The OVERLAY 23 GIS AIS Geographm Agro-Economc InFornu‘tIon Information System System AEZ YIELD E Dverhy l [Smumtorl Agro-Ecologlcol Zone Weui Estimation Ielol Estlmot; Figure 1. Agro-ecological zone YIELD assessment as a component of CRIES-RIS. 24 and MATCH phases ultimately provide the spatially defined regions called AEZs. - Single Grid Cell 50115 or Location Element .. w. .:_ - 2' 9: in Data File ""\'~'\' 'Q'I'I'\'\'§'I.§'I‘\:\:§:t Land Cover Type ‘.§'.“'§'\'Q I 'I'I'I‘I‘!‘§‘I‘I‘Q'I‘\ I I ' ' . . _ """"' ' Prec1pitation Slope Degree Slope Length lst Approximation of Erosion Poten- tial Index Figure 2. Geographic Information System--OVERLAY analysis. Some data-manipulation functions included in the CRIES- GIS are (Schultink et al.,1986): Digitizing Editing Polygon conversion Cutter (outline boundaries) Histograms Tally (windowing) Cross—tabulation E l H . ] I' --E J . Erosion (soil erosion) Grouping Invert Match Normalize Overlay Search Surface (three-dimensional analysis) D I H . J . __ . Character maps Value maps Locate D l H . J l' --E . H Choroline (map print capabilities) E l M . ] . ——fl|']'|' Reformat Mosaic Aggregate Disaggregate The GIS can be used to cross-reference the AEZs with major land use to identify additional areas suitable for agricul— tural expansion. Examples of output (Schultink, 1986) from the CRIES—GIS are a scaled character map of elevation for Choluteca Department, Southern Honduras (Figure 3); a cross-tabulation output portion of a Two—Way Cross-Tabulation Between Rain- fall and Elevation, Choluteca Department, Southern Honduras 26 Prepared bv: CRIES-GIS / Hichigan State University Thursday April 10. 1986 Time 09:16 File name: cholelev.ras 000000000000000000001lIlll11111llllllIl222222222222222222223333333333333 001122334455667786990011223345566778699001122334455667788990011223345566 l6l6272727273838383849494949505050506[6|61617272727283838383949494940505 00' 001 010 vvvvi O l 0 018 vaiv 018 027 XVVVVV 927 03G vavvvli..ii v 036 044 XXvii ..... iiiv.viiivvv ivvvv 044 053 XXva ..... ivvviivvvvv vvivvaXXXX 053 061 vii...ivvvvvvaXv vvvvivvvaXXXXX 06] 070 v1.1.levvvvvvva vilvvvvvvaXXXX O70 O79 ........ ivlvl.1vleX vavvivvvilivaXXX 079 087 .............. IIVXX XXXXXVI‘IIIIVVXVXXXX 087 096 .vili ..... (Iiivavv XXXlellvvvvvvaXXXXXX 096 105 vvvi....1vviivvaXv1vv1 XleivavvvvivaxXXXXX 105 113 .vl...11vavvaXXvav11 vaIlvaXvaXvavaXXXXXX 113 122 ....I1vvvvaXvXXXXva..lvl111..iiivvaXXXXXXXXXXXXXXX 122 131 ....... IIvvvvvvvaXle ...... 1IitvvaXvXXXXXXXXXXXXXXXX 131 139 1....11Ivvavvivava....§ ...... 139 148 ..vv..vvvaIvvv.l|Xl.. ...... .-..- . 148 156 ........ v...1v.11vv1..vvvv1... [56 I65 ...... lvviv.v.1111.1 .......... |65 174 ......... 11.i.l ............ (iii. XX'XY I74 182 ....Ii ......... 1.1 ...... .....i " 182 I91 ............... ii .......... . ....... -.xxxx. 19] 200 ... ............ 1 ...... . ......... ..5... rtx- 200 208 . ..... I ........................... 111VXXXXVVVVXXXXXXX XXXXXXX 206 217 .. ..... i ........................ I...lvvvavavvvvvvv_ XXXXX 217 225 . ........................... IIIvavllvaXvavvvvvli 225 234 ............ _ ............... v1.1vvaXXXva1vvvvvv 234 243 . ........................... lleXvXvaiivvvvv 243 251 ................. . ....... .....ivvvavvvl.IilI 251 260 .......................... ...Ivvivivvllllii . 260 269 . ........................... 1vvvliii.l....i. 269 277' .................... .......II:..II...:::..- 277 _286 ......................... ...ll.....l...... 286 295 ....................... ........I. ..... 295 303 ......... ........ ...l ........ I. ...... 303 312 ................................... 312 320 ................. . ............ . . . 320 329 ... ........ . ........... ..... 329 338 ...... . ..... . ........... . 333 345 ......... . -.- }‘6 355 355 00000000000000000000111111111111111111l 1111313333313 00112233445566778899001I22334556677889900|I223344556677889900T1223345566 161627272727181979 ‘“‘“‘“ ‘6161617272727283838383949494940505 CHOLUTECA ELEVATION RANGES HONDURAS DATA BASE - REGIONAL LEVEL ATTRIBUTE: ELEVATION. ATTRIBUTE VALUES: CONTOUR INTERVALS. VARIABLE HAP SCALE serrcrro n : soo.ooo Figure 3. Geographic Information System--character map of elevation. 27 (Figure 4); and a Choroline, dot—matrix printer map output of land cover/use map derived from Landsat Satellite Data, Choluteca Department, Southern Honduras (Figure 5). Hardware required to run the CRIES-GIS is: — IBM PC--XT or compatible microcomputer with 512KB of RAM - hard disk - MS—DOS operating system — Calcomp map digitizer - dot matrix printer - Techmar color board - monochrome and color-enhanced display - color jet printer - optional point or mouse system The CRIES-AIS is designed to evaluate and derive bene- fits from physical and socioeconomic variables such as yields, input cost, and producer prices; to assess the comparative advantage of land-use types in meeting food and export crop demands, and to conduct related economic analy- ses regarding agricultural policy alternatives. The AEZs are the spatial units of analysis for the AIS system. The AIS has several components that perform different functions. Usually, the output or results accomplished by one component are inputs to another component. The follow— ing are the main components of the AIS system: - Water balance - Yield simulator - Farm budget Input/output model Optimization model (linear programming) - Statistical analysis 28 Preoared bv: CRIES-GIS / Hichigan State University Thursday April 10. 1986 Time 09:22 Crosstabulation Table + ————————— + 1Frequencv1 1 1 Frequencv in hectares Format: 1 Col Pct l Row Pct 1 1 Tot Pct 1 + ————————— + Attribute --------- Description: ELEVATION AND PRECIPITATION CHOLRAlN.ras Attribute CHOLELEV.ras Atr Val Atr Val Atr Val Atr Val Row Totals 1 2 3 4 1 444469 11 89725 1 83275 1 184900 1 86569 : Column 1 11 1 1 1 1 Totals 1 100.00 :1 100.00 1 100.00 1 100.00 1 100.00 1 1 100.00 11 20.19 1 18.74 1 41.61 : 19.48 1 1 100.00 11 20.19 1 18.74 1 41.61 1 19.48 1 1 209056 11 1944 1 9969 1 125856 1 71287 1 Atr Val 1 11 1 1 1 1 ------- 1 47.04 11 2.17 1 11.98 1 68.07 1 82.35 1 1 1 100.00 11 0.93 1 4.77 1 60.21 1 34.10 1 1 47.04 11 0.44 1 2.25 1 28.32 1 16.04 1 1 53113 11 10106 1 12494 1 24069 1 6444 : Atr Val 1 11 1 1 1 1 ------- 1 11.95 11 11.27 1 15.01 1 13.02 1 7.45 1 2 1 100.00 11 19.03 1 23.53 1 45.32 1 12.14 1 1 11.95 11 2.28 1 2.82 1 5.42 1 1.45 1 1 82425 11 25669 1 23531 1 26556 1 6669 1 At? Val 1 11 1 1 1 1 ------- 1 18.55 11 28.61 1 28.26 1 14.37 1 7.71 1 3 1 100.00 11 31.15 1 28.55 1 32.22 1 8.10 1 1 18.55 11 5.78 1 5.30 1 5.98 1 1.51 1 1 98481 11 51512 1 36381 1 8419 1 2169 1 Atr Val 1 11 1 1 1 1 ------- 1 22.16 11 57.42 1 43.69 1 4.56 1 2.51 1 4 1 100.00 11 52.31 1 36.95 1 8.55 1 2.21 1 1 22.16 11 11.59 1 8.19 1 1.90 1 0.49 1 1 1394 :1 494 1 900 1 0 1 O 1 Atr Val 1 11 1 1 1 1 ------- : 0.32 :: 0.56 : 1.09 1 0-00 l 0-00 I 5 1 100.00 11 35.44 1 64.57 1 0.00 1 0.00 : 1 0.32 :1 0.12 1 0.21 1 0.00 1 0.00 1 Figure 4. Geographic Information System--two-way cross-tabulation. 29 ///’/ 7/// [If/l" 7// //,/,, /:/" ,/M// 7/ -///’///’//u I '1 ’/{I 4 :1" Figure 5. Geographic Information System--choroline printer map. 30 The AIS data base is responsible for data manipulations, data storage of agricultural and socioeconomic data, crop requirements, and so on. The main objective of this study is to develop a yield- simulation model, as a component of the AIS system, that will evaluate crop yield as a function of climatic and local information, management practices, and soil information for AEZs. CHAPTER III DETERMINISTIC YIELD SIMULATOR (DYS) The deterministic or nonprobabilistic yield simulator is designed to generate the maximum potential yield, irri— gated yield, and rain-fed yield for different crops. The term "deterministic" is used here to indicate that the model's inputs and parameters have zero variance. This means that they are known with certainty and that their precision is not questionable. Chapter IV considers the case where the variance isruM:zero for some inputs and model parameters. The DYS was developed largely based on equations and procedures from the publication Xield_Besponse_t9_flater by Doorenbos and Kassam (1979). Its main objective is to estimate maximum potential yield, irrigated yield, and rain- fed yield for the crops under study, based on the climatic conditions, soil and slope characteristics, and management practices of a single location or agro-ecological zone (AEZ) under investigation. This yield assessment provides addi- tional guidelines for decision makers in land-use planning. Its secondary objective is to serve as an analytical tool 31 32 for decision makers and planners to evaluate agricultural production systems in terms of yields, agricultural land use, and natural-resource management in a posteriori analysis. The nomenclature and different system-input classes that are part of the simulation model are described in Figure 6. 9(1) (exogenous Inputs) 7 DETERMINISTIC .———s YIELD u('t) SIMULATOR (declslon (DYS) /control mputs) p(‘t) (model parameters) Figure 6. General input/output diagram for the deterministic YIELD simulator with nomenclature. Exogenous environmental inputs are represented by the vector e(t), decision or control inputs are represented by 33 u(t), the model's parameters are represented by p(t), and the model response is denoted by y(tL The model represents a procedure for estimating crop yield based mainly on water availability, the dominant con- straint in tropical environments. Water represents the major variable in crop production, and optimum use of avail- able water must be made for efficient irrigated crop produc- tion to produce high yields. It is generally believed that water, as an input to crop-production systems, represents 75 to 85 percent of the variation in crop yield. Doorenbos and Kassam (1979) pointed out that: the upper limit of crop production is set by the cli- matic conditions and the genetic potential of the crop. The extent to which this limit can be reached will always depend on how finely the engineering aspects of water supply are in tune with the biological needs for water in crop production. Therefore, efficient use of water in crop production can only be attained when the planning, design and operation of the water supply and distribution system is geared toward meeting in quan- tity and time, including the periods of water short- ages, the crop water needs required for optimum growth and high yields. The production relationships between crops, climate, water, and soil are complex, and many biological, physio- logical, physical, and chemical processes are involved. Much research information is available on these processes in relation to water. For practical applications, such knowl- edge must be reduced to a manageable number of major compo- nents to allow a meaningful analysis of crop response to water. 34 Modifications and additions were necessary to transform the theoretical framework as presented by Doorenbos and Kassam into a computer-simulation model that can be useful in assessing potential, irrigated, and rain-fed yields for 30 different crops in a way that is easy and accessible for decision makers in developed countries as well as developing countries to use in planning and policy-analysis processes. The simulation model was designed to minimize requirements in terms of computational tools as well as computational expertise. Modalesumnticns The 1979 FAO publication by Doorenbos and Kassam entitled Xield.Besans£_tQ_flateL, from which this yield simulator was derived, assumes that the relationships between crop, climate, water, and soil are very complex and that they are also affected by other factors, such as crop variety, fertilizer, salinity, pests and disease, and agro- nomic practices. The relationships presented in this model pertain to high- producing varieties, well-adapted to the growing envi- ronment, growing in large fields where optimum agronomic and irrigation practices, including adequate input supply except for water under rain-fed conditions, are present. The pre— dictive accuracy of the model may be increased by adjusting 35 the model parameters for site-specific conditions and validation through adaptive research. Local conditions other than climate, such as soil depth and texture, availability of fertilizer, salinity, soil slope, rooting depth, and management practices, will be used to adjust the potential yield values based on AEZ and cropping-system conditions. It is assumed that no post- harvesting losses occur. However, these management-practice parameters can easily be included in the model for site— specific applications upon availability of data. Crop requirements and crop coefficients are included in the model in the form of tables and model parameters that were derived from experimental crop research. Socioeconomic factors, such as farmers' preference in relation to market demand, storage facilities, and availa- bility of farm machinery and labor, that are known to affect farmer's management decisions such as selecting the crops to be grown and length of growing season, are not considered in the DYS. Pests and diseases, which are also known to influence yield output, are not considered due to lack of knowledge about explicitly mathematical relationships and probability functions. Numerical relationships and func- tional forms are requirements for inclusion in the numerical simulation model. 36 Surface runoff or internal drainage is assumed to be adequate to prevent yield reduction under average climatic conditions. Water logging or excessive water is known to cause crop damage and to reduce yields. The model does not consider damage due to excess water. Yield adjustment due to soil fertility and fertilizer limitations may be used in the model. If restrictions apply, the user is given the option to adjust productivity accordingly. Crop-rotation considerations and cropping schemes are user-selected. Micro relief~induced climatic effects on precipitation, wind, and solar radiation and resulting changes in evapo- transpiration are not assumed in this model. We In this section an overview of the DYS, its structure with modifications and additions, and the computational procedure are provided (Figure 7). Five consecutive phases are needed to estimate the yield value for a crop (Ye). They are: Ehasg_l: Determine the maximum yield (Ym) of the adapted crop variety, dictated by climate, assuming that other growth factors (e.g. farm management, fertilizer, pests and diseases, and so on) are not limiting. 37 INPUTS WWWB mmasnmnmrmv - 1 W5 __ NUT ruw run: 31' mm DRY MATTER m V l _ HASE a s: E 4 HASE s REFERENCE ACTUAL rsrnuizn ESTIMATED _"w__wmm _m_um- y me (ETo) ‘57“) (Ye) : DECISIEN/CDNTRDL Figure 7. Deterministic YIELD simulator--general diagram. In the first phase, six steps must be performed in order to determine (Ym). Step Step Step Step Step Step They are as follows: 1: Computes gross dry matter production of a standard crop (Yo). 2 3 Applies correction for crop species and temperatures. Applies correction for crop development over time and leaf area (CL). 4: Applies correction for net dry matter production (cN). (1' u 0‘ Applies correction for harvested part (on). Computes the maximum potential yield (Ym). 38 Ehase 2: Calculate maximum evapotranspiration (ETm) when crop requirements are fully met by available water supply. In this phase, three steps are needed to compute (ETm). Step 1: Computes reference evapotranspiration (ETo) based on the meteorological and crop data available. Step 2: Computes growing period and length of crop- development stages and selects the crop coefficient kc. Step 3: Computes maximum evapotranspiration (ETm). Bhase_3: Determine actual crop evapotranspiration (ETa) based on factors concerned with available crop water supply. Step 1: Determines total available soil water. Step 2: Computes soil water depletion. Step 3: Computes actual evapotranspiration (ETa). Ehase__: Select the yield response factor (ky) to evaluate relative yield decrease as related to relative evapotranspiration deficit and obtain actual yield (Ye). Bhase 5: Estimate crop-yield adjustment. In this phase, the resulting estimated yield from Phase 4 is adjusted for fertilizer availability, soil salinity, and moisture content. 39 M 1] I' J E J . Bha5e_l: Calculate Gross Dry Matter Production of a Standard Crop (Yo). To compute the gross dry matter production of a stand- ard crop (Yo) for a given location or AEZ, the DeWit (1965) method is used. This method is based on the level of incom- ing active shortwave radiation for standard conditions, modified after Doorenbos and Kassam (1979). Equation 1.1 provides the rate of change in gross dry matter production of a standard crop as a function of time. (1.1) dYo(t) ------ = F(t) * yo(t) + [1.0 - F(t)] * yc(t)] dt where: Yo(t) = total gross dry matter production for a standard crop [kg/ha] F(t) = fraction of daytime the sky is clouded [fraction] yo(t) = gross dry matter production rate of a standard crop for a given location on a completely overcast day [kg/ha/day] yc(t) = gross dry matter production rate of a standard crop for a given location on a clear (cloudless) day [kg/ha/day] t = time index [days] Doorenbos and Kassam (1979) provided tables to deter— mine the values of maximum active income shortwave radiation (Rse in cal/cmZ/day) and gross dry matter production on 40 overcast (yo) and clear days (yc) (in kg/ha/ day) for a standard crop for time (t) and latitude in degrees. Numeri- cal techniques for function interpolation such as Tablex and Spline are presented (Appendix A) and are used to obtain intermediate or interpolated results. The total gross dry matter production for a standard crop (in Kg/ha), from time t = to to t1 is Presented in Equation 1.2: (1.2) t=tl (t — ) = {F(t) * (t) + [1 0 - F(t)] * (t)}dt Yo l to ft t yo . YC = o Numerical—integration techniques are applied to the above equation to obtain an expression that can easily be used in a simulation model (Forrester, 1961; Manetsch & Park,1984L Euler's approximation formula (Hamming, 1962; Conte, 1980) is used to find an approximate numerical solution to Equation 1.2. Euler's method was derived from the Taylor expansion series by setting the parameter k = 1. (Appendix B shows that procedure in more detailJ Its general form is given by: dy(t) t=tl if ————— = f(x,y) then, y(t) = ff(x,y)dt dt t=to and y(n + h) = y(n) + h * f(Xn,Yn) 41 where: h = fixed step size y = f(x,y) is the functional relationship Applying the above numerical approximation to Equation 1-2 and assuming to = 0 and t1 = t, results in Equation 1.3, which gives the cumulative total dry matter production for a standard crop from time t = 0 to time t = t + dt. ' (1.3) Yo(t + dt) = Yo(t) + dt * {F(t) * yo(t) +[1.0 — F(t)] * yc(t)} Equation 1.3 is corrected and adjusted to reflect dif- ferent crop groups, according to De Wit's (1965) concept, resulting in Equation 1.4: (1.4) Yo(t + dt) = Yo(t) + dt * {F(t) * [0.8 + 0.01 * ym(t)] * yo(t) + [1.0 - F(t)] * [0.5 + 0.025 * ym(t)] * yc(t)} for ym(t) > = 20.0kg/ha/hour; and Yo(t + dt) = Yo(t) + dt * {F(t) * [0.5 + 0.025 * ym(t)] * yo(t) + [1.0 - F(t)] * [0.5 * ym(t)] * yc(t)} for ym(t) < 20.0kg/ha/hour where the ym(t) ternlis the production rate for crop>groups and mean temperature, in kg/ha/day. The gross dry matter production is crop-species and temperature dependent. The production rate, ym(t), can be larger or smaller than 20.0 42 kg/ha/hour as assumed for the standard crop. Doorenbos and Kassam (1979) gave the production rates, ym(t), in kg/ha/ hour for crop groups and mean temperatures. Additional corrections are applied to the gross dry matter production computed above; they are: — Crop Development over Time and Leaf Area (cL). The model assumes, for the standard crop, an active leaf area index of five times the ground surface. When leaf area is smaller, a correction must be applied; when greater than five, the effect is small and is not considered in the model. Correction gives the correction values for different leaf area indices, as supplied by Doorenbos and Kassam (1979). - Net Dry Matter Production (cN). Energy is required by the plant to maintain dry matter production for the within-plant growth processes (also called respirationL Only the remaining energy fraction can be used to produce new growth, which is, according to Doorenbos and Kassam, about 0.6 for cool temperatures (mean < 20 degrees Celsius) and 0.5 for warm temperatures ( > 20 degrees Celsius). - Correction for Harvested Part (CH). In most cases, only a part of the total dry matter such as grain, sugar, or oil produced is harvested. Doorenbos and Kassam provided the ratio between net total dry matter production and the 43 harvested yield for high-producing varieties under irriga- tion. Using these correction factors, Equation 1.5 gives the potential yield of a high-producing, climatically adapted variety grown under constraint—free conditions (Doorenbos & Kassam, 1979): (1.5) Ym(t + dt) = Ym(t) + cL * cN * cH * dt * {F(t) * [0.8 + 0.01 * ym(t)] * yo(t) + [1.0 - F(t)] * [0.5 + 0.025 * ym(t)] * yc(t)} for ym(t) > = 20.0kg/ha/hour, and Ym(t + dt) = Ym(t) + cL * cN * CH * dt * {F(t) * [0.5 + 0.025 * ym(t)] * yo(t) + [1.0 — F(t)] * [0.5 * ym(t)] * yc(t)} for ym(t) < 20.0kg/ha/hour A general system diagram, showing phase 1 of the deter- ministic YIELD simulation model, is supplied in Figure 8. Phase_2: Maximum evapotranspiration (ETm). Climate is an important factor in determining the crop water requirements needed for unrestricted growth and opti- mum yield. Crop water requirements are normally expressed by the rate of evapotranspiration (ET), in mm/day. The level of ET is related to the evaporative demand of the air, which can be expressed as the reference evapotranspiration Figure 8. 44 W EQUATION >~ O '5 \ d .C \ O) M Deterministic YIELD simulation mode1--Phase 1: general system diagram. 45 (ETo), which, when computed, predicts the effect of climate on the level of crop evapotranspiration. ETo represents the rate of evapotranspiration of an extended surface of an 8 by 15 cm tall green grass cover, actively growing, completely shading the ground and without water deficit (Doorenbos & Kassam, 1979L Several methods can be used to calculate ETo: the Penman, Radiation, and Pan Evaporative methods shown in Doorenbos and Kassam; the Thornthwaite method (Thornthwaite, 1948); the Hargreaves method (Hargreaves, 1977); and the Priestley and Taylor method (Priestley & Taylor, 1972). Selection of Penman's method (Penman, 1948) for use in the model was based on worldwide validations of the method for the computation of reference evapotranspiration (ETo) (Hayes, 1982; Doorenbos & Pruitt, 1977; Todhunter, 1981; Burt et al., 1980, 1981). The basis for computation of evapotranspiration for this model was Penman's (1948) equation. The equation was successively modified to include the effects ofaivariety of factors, such as crop type, crop growth stage, and site factors (Doorenbos & Pruitt, 1977). These adjustments include the influence of extreme climatic environments, crop coefficients adjusting ET for specific growth stages, and soil moisture budget considerations. 46 Empirically determined crop coefficients (kc) can be used to relate ETo to maximum crop evapotranspiration (ETm) when water supply fully meets the water requirements of the crop. The value of kc varies with crop, crop development stage, and to some extent windspeed and humidity. Values of kc for different crops were given in Doorenbos and Kassam (1979). The methodology has the advantage of applicability and the fact that the mathematical relationships are well defined for many crops applications. For a given climate, crop, and crop development stage, the:maximum evapotranspiration (ETm) is provided by Equation 1.6: (1.6) " ETm(t) = kc(t) * ETo(t) where: ETm(t) = maximum!evapotranspiration [mm/day] kc(t) = crop coefficient [fraction] ETo(t) = reference evapotranspiration [mm/day] The reference evapotranspiration (ETo) is computed by means of the Penman method (Penman, 1948), modified by Doorenbos and Pruitt (1977), which provides Equation 1.7: ETo(t) = C(t) * {W(t) * Rn(t) + [1.0 - W(t)] * f[U(t)] * [ea(t) - ed(t)]} (1.7) 47 where: f(U(t)) = 0.27 * [1 + U(t) / 100.0] Rn(t) = 0.75 * Rs(t) - Rnl(t) Rs(t) = 0.25 + 0.50 * [n(t) / N(t)] * Ra(t) Rnl(t) = f[T(t)] * f(ed(t)] * f[n(t) / N(t)] f(T) = 1.993-09 * Tk4(t) ed(t) = ea(t) * RH(t) / 100.0 f(ed) = 0.34 + 0.044 * [ed(t)]1/2 f(n/N) = 0.1 + 0.9 * n(t) / N(t) and ea(t) = saturation vapor pressure [mbar] ed(t) = actual vapor pressure [mbar] U(t) = wind velocity measured at 2m height [km/day] ‘ n(t) = actual sunshine duration [hour/day] ‘ N(t) =maximum possible sunshine duration [hour/day] Ra(t) = extra-terrestrial radiation at time t [mm/day] RH(t) = relative humidity at time t [percent] Rnl(t) = net longwave radiation [mm/day] W(t) = temperatureandaltitudedependent weighting factor [fraction] C(t) = adjustment factor [fraction] T(t) = temperature in degree Celsius [C] Tk(t) = temperature in degree Kelvins [K] 48 Doorenbos and Kassam (1979) provided tables to deter- mine the values of those parameters. In summary, they permit determination of: 1. Saturation Vapour Pressure (ea) in mbar as a Function of Mean Air Temperature (T) in Degrees Celsius. 2. Extra-terrestrial Radiation (Ra) Expressed in Equivalent Evaporation in mm/day. 3. Mean Daily Duration of Maximum Possible Sunshine Hours (N) for Different Months and Latitudes. 4. Values of Weighting Factors (W) for the Effect of Radiation on Eto at Different Temperatures and Altitudes. 5. Adjustment Factor (c) in Presented Penman Equation. All of these tables are included in the simulation model, and an interpolation process is used to determine interme- diate values. A general system diagram of the second phase of the YIELD simulation model is provided in Figure 9. Bha5341: Actual Evapotranspiration ETa Crop water demand in the root zone is met by available soil moisture. The actual rate of water uptake by the crop from soil moisture in relation to its maximum evapotranspi— ration (ETm) is determined by whether the available water in the soil is adequate or not. If not enough water is avail- able, water-induced crop stress will occur. 49 ‘P LATITUDE deg @— LAT LI\ LA TEMPALT' - TEMPERATURE A W - t / ALTITUDE m V 0 \ ALT ARELATIVE HUMIDITY PENMAN EQUATION percent LCEQUATIDN 1.7) b VIND VELDCIT V ' n75 Kc ACTIVE SOLAR Rg - Rs R V \ HM V, Rs CT, C CRDP TYPE - CT If 1 LACRDP STAGE - CS Figure 9. Deterministic YIELD simulation model--Phase 2: system diagram. —. ~. v-u... g 12‘: "N‘ "\.—._ u. 50 To compute the actual evapotranspiration (ETa), the level of available soil water must be considered. First, the available soil water index (A51) is computed. This index indicates when available soil water is adequate to meet full crop requirements (ETa = ETm). .A combination of A81 value, maximum evapotranspiration (ETm), and the remain— ing available soil water [(l-p)*Sa(t)*D(t)] provides an estimate of the actual evapotranspiration (ETa) (Doorenbos & Kassam, 1979). The available soil water index (ASI) may be calculated using Equation 1.8: (1.8) In(t) + Pe(t) + Wb(t) - [(1 - p) * Sa * D(t)] ASI(t) = --------------------------------------------- 30 * ETm(t) where: In(t) = net monthly irrigation application [mm] Pe(t) = monthly effective rainfall [mm] Wb(t) = available soil water moisture [mm] p = depletion factor [fraction] Sa(t) = total soil water holding capacity [mm/m] D(t) = root depth [m] when ASI(t) > = 1.0, then ETa = ETm ASI(t) < 1.0, then Eta is computed according to Doorenbos and Kassam (1979) 51 The growth and development of crops depend on water availability. Sources of water include moisture stored in the soil, rainfall, irrigation, and surface runoff. Pre- cipitation and irrigation recharge soil moisture in succes- sive soil layers from the surface downward. Precipitation and irrigation in excess of that required to bring the crop root zone up to water-holding capacity is removed by runoff, which is a function of soil texture, slope, and infiltration rate. For a short dry period, crop growth may not be affected, even in the critical growth period, if there is sufficient soil moisture to support the cropfls demand for water. Soil moisture is difficult to measure in the field. Several methods have been proposed to estimate soil moisture content. Thornthwaite's (1948) model is based on simple water-balance equations for gains and losses within a single soil layer. A more complex, two-layer soil-moisture budget model was developed by Palmer (1965). The soil—moisture model used in this study is a modi— fied version of Thornthwaite‘s model, which includes an evaporation-reduction factor to account for farm management practices such as mulching and tillage, and a water- depletion factor, which is crop specific. Adams (1976) stated that, based on his research findings, it may be inferred that management systems that combine trash mulch 52 tillage and narrow-row spacing should add to the beneficial effects of both plant canopy soil shading and mulch. In addition, he stated that the use of mulch at a rate of 4,000 kg/ha with no soil shading reduces evaporation by as much as 58 percent as compared to potential evaporation measured from a bare plate with no canopy, for first-stage drying. Much more research is needed to determine the effect of an evaporation-reduction factor and to make it practical for direct implementation into a numerical simulation model. One additional side effect of some evapotranspiration- reducing management practices is the increased occurrence of pests and diseases, which in some countries and/or regions may cause a significant reduction in yield. The computation of soil moisture is given at time t by Equation 1.9: (1.9) Wb(t) = Wb(t-l) + [Pe(t) + Ir(t)] * Roff - p * Wb(t-l) - ETa * (100.0 - Mu) / 100.0 where: Wb(t) = soil moisture at time t [mm] Wb(t-1)= soil moisture at time t-l [mm] Pe(t) = precipitation at time t [mm] Ir(t) = irrigation at time t [mm] Roff = runoff coefficient [fraction] p = water depletion factor [fraction] Whfi .... _...«._'« v- 53 Eta(t) actual evapotranspiration at time t [mm] Mu evapotranspiration reduction factor [percent] The runoff coefficient (Roff) is a function of soil slope and soil textural class. According to Beasley et a1. (1984), adjustment in infiltration rate due to soil slope and soil textural class may be accomplished using Table 1. Table 1: Percentage of water that percolates into the soil as a function of percentage slope and soil textural classes Slope Class Soil Texture Coarse Fine Silty, Very Loamy Fine Loamy & Fine Fine 0 - 4% 90% 80% 70% 4 - 8% 70% 60% 50% 8 - 12% 62% 52% 42% 12 - 15% 55% 45% 35% 15 - 20% 50% 40% 30% 20 - 30% 40% 30% 20% 30 - 50% 38% 25% 18% > 50% 37% 27% 17% Source: Beasley et al., 1984. The water-balanced equation, adjusted for soil texture and topology, is used to keep track of the moisture content of the soil from time t to time t + dt. A system diagram of phase 3 of the deterministic YIELD simulation model is provided in Figure 10. 54 AXIHUH EVAPDTRANSPTRATIUN . ETm ' 47 PA I SELECT ‘ KCRDP TYPE ' c, CRUP GRDUP GP ETmfiP"D : ,L VATER Hm DING CAPACITY m/n L "REID?"1 DEPTH EQUATIONS 1.8 AND 1.9 T % k‘PRECIPITATIUN run/no 4) 7 / ET U“ m CUMPUTE ETa salRRIGATmN SUIL WATER V mm —9 WW AVAILABILITY "NW” .____.9 ETn LASUTL MOISTURE EV run/n 10. Deterministic YIELD simulation model--Phase 3: Figure system diagram. 55 Bhase_4: Estimated Yield (Ye) Soil water stress influences crop evapotranspiration and yield. An index of water stress is the ratio of actual to maximum evapotranspiration, ETa/ETm. Similarly, an index of crop yield is the ratio of estimated to maximum possible yield, Ye/Ym. The way the first ratio affects the second (called yield response factor [ky]) varies with crop species and crop—development stages or time. Under sufficient water supply, ETa = ETm. The rate of change of the estimated harvested yield, at time t, is given by Equation 1.10, modified after Doorenbos and Kassam (1979): (1.10) d[Ye(t)] -------- = Ym(t) * [1.0 - ky(t) * [1.0 - ETa(t)/ETm(t)]l dt or t=t Ye(t) = f {Ym](t) * [1 — ky(t) *[1 - ETa(t) / ETm(t)]1dt t=t0 Using Euler's numerical approximation formula and assuming t0 = 0, results in Equation 1.11: (1.11) Ye(t + dt) = Ye(t) + dt * {Ym(t) * [1.0 - ky(t) * [1.0 - ETa(t) / ETm(t)]} 56 where: Ye(t) = estimated harvested yield [kg/ha/day] Ym(t) = potential yield [kg/ha/day] ETa(t) = actual evapotranspiration [mm/day] ETm(t) = maximum evapotranspiration [mm/day] ky(t) = yield response factor [fraction] t = time [days] The deterministic YIELD simulation model system diagram, Phase 4, is provided in Figure 11. Phase;§: Estimated Yield Adjustment The Ye(t) computed from Equation 1.11, above, may be adjusted further if fertilizer (NPK) applications are less than optimum, or for the sensitivity of the crop to saline soil conditions. A simplified assumption is made that requirements are met if composite ratio equals 100 percent. In fact, the amounts of N, P, and K requirements are crop specific, and each crop has a different response curve for nutrient applications. Using Evans (1980) and Hayes (1982), the NPK response curve (Figure 12) was derived to compute the yield decrease factor due to fertilizer availability. This represents the generalized yield adjustment due to general fertilizer availability for all crops considered in the model. 57 MAXIMUM POTENTIAL YIELD Yn , ACTUAL EVAPOTRANSPIRATION ' ETTd b MAXIMUM EVAPOTRANSPIRATIDNA ETm Y0 = Ynlfl-KyQ-ETu/ETMH Yo Kg/hu/doy Ky4 CROP TYPE c 't (7 ctcs CROP ST AGE CS Figure 11. Determinis diagram. tic YIELD simulator-~Phase 4: system 58 100- .. ., .. .. .. .. .. ., - l GENERALIZED FERTILIZER RELATION 90- - ao— - Weld Decrease (%) 3 l I sol l 205 1 1o.- _ o: I ' l T' T V U jiT V I ' T ' 1 ‘ I j ' I 0 1O 20 30 4O 50 60 7O 80 90 1 00 Appficd NPK (Z) Figure 12. Yield adjustment based on generalized fertilizer availability for all crops. (From Evans, 1980 and Hayes, 1982.) Future model refinements will require more elaborate research for specific crops' responses to different levels and types of fertilizer, including natural soil fertility. The decision to use a generalized fertilizer curve for all crops reflects the incomplete and inconclusive research of effects of fertilizer availability and toxicity on varying crops under a wide range of agro-ecological conditions (see also Hayes et al., 1982). Doorenbos and Kassam (1979) provided the optimum ferti- lizer requirements (nitrogen-phosphorus-potassium combined) 59 for all crops included in this simulation model. The user should determine the actual deviation from the optimal fer- tilizer requirements for each crop, considering soil fertil— ity and fertilizer applications. The effect of salinity levels on yields was compiled from Doorenbos and Kassam (1979) for all crops included in the simulation model. The results are summarized in Table 2 and incorporated into the model for interpolation. A final conversion of predicted yield is provided to the user. The option is provided to calculate estimated yield as total harvestable biomass or dry matter. This reflects the need to calculate the total harvest production or the final yield amount as dry matter production. The current model adjustment is made by calculating total biomass from dry matter, based on ratios derived from Doorenbos and Kassam (1979). If needed, crop-variety- specific adjustments may be made via model modifications, and these may be adjusted to the specific site. In summary, the result is Equation 1.12, after incor- porating all the adjustments: (1.12) Ye(t + dt) = Ye(t) + dt * {Ym(t) * [1.0 - ky(t) * [1.0 - ETa(t) / ETm(t)] * (1.0 - ydf) * (1.0 - yds / 100.0) /cf} 6O Crops, salinity levels in mmhos/cm, and Table 2. percentage yield decrease values Percentage Yield Decrease.uu.u.. 0.0 Crop Type 25.0 50.0 100.0 10.0 Banana 55 66 Subtropical Tropical Bean 550005 000000 662726 121 660079 c o o o 0 337764 334011 0 o o o o 9 224344 558655 0 o o o o 0 112923 008752 0 O O O O 0 111713 t u e n n 9n ..0 e aoen ethPu rrbtaO GDaorr CCGG Maize 55 22 77 11 Sweet Grain Onion 5555055000600500500 oooooooooooo 000000 0 7768014804830203580 11111211 1111 .l. 7761929055400630880 4435579175000760849 l 111 l 8833816220050045430 2223357761690549538 l 8852582157350534430 LL12236558340337327 2205703000700550075 LL11L35457L002262L4. h S e an m r ter 6 e ene l M r wmmn.ea_wnu e a n ro ouabcocom fs e gt lherrlctrtlue ye Ipaefgbaafaaeaarv [rapthryggnmeEftl “gueegiaoouuuooahlil PPPRSSSSSSTTWWACO Compiled from Doorenbos and Kassam (1979). Source: 61 where: ydf = yield decrease factor from fertilizer usage [fraction] yds = yield decrease factor from salinity soil levels [percent] of = correction factor for humidity inclusion [fraction] Wants To run the deterministic YIELD simulator successfully, the user must assemble a data base. As indicated before, the model predicts yields for homogeneous agro-ecological conditions based on agro-climatic criteria. The delineation of AEZs involves data aggregation and area delineation. Data aggregation is employed to assemble the data set for g the local area or AEZ being considered. Primary weather station and secondary weather station are the main source of climatic data. Those weather stations should be located inside the AEZ (optimal situation). Data collected outside the AEZ boundaries may be interpolated in a trend surface algorithm to obtain the best possible approximation for the AEZ considered. In the case of wind velocity and solar radiation, extra precautions must be be taken in the data-collecting proce- dure to account for shadow effects from elevation and for air current resulting from systematic air flow. 62 The data base must contain several data sets that provide the information necessary to run the model. They include: A. LOCAL data set is needed to identify the location, region, or AEZ for which the simulation will take place. The local data set contains the following information: 1. Average altitude [m] 2. Average latitude [degrees] 3. Hemisphere (north or south) 4. Slope class specification (Table l) 5 . Soil type and texture and associated moisture- holding capacity (Table l) 6. Soil moisture at sowing date [mm/m] 7. Soil salinity level [mmhos/cm] 8. AEZ parameters identification — code - name B. FARM MANAGEMENT PRACTICES data set is required to identify farming-system techniques. The growth period is divided into five stages called crop stages. The duration of the initial stage (first stage) is defined as the time period, in days, from germination to 10 percent of ground cover. The duration of the crop-development stage (second stage) is defined as the time period, in days, from 10 percent to 80 percent ground cover. The duration of the mid—season stage (third stage) is defined as the time period, in days, from 80 percent ground cover to the start 63 of ripening. The duration of the late season (fourth stage) is defined as the time period, in days, from ripening to harvest. The duration of the harvest stage (fifth stage) is defined as the time period, in days, of the harvest. 1. Crop sowing date and harvesting date - day - month - year 2. Crop first stage duration [days] 3. Crop second stage duration [days] 4. Crop third stage duration [days] 5. Crop fourth stage duration [days] 6. Crop fifth stage duration [days] 7. Fertilizer availability [percent] 9 8. Evaporation reduction factor [percent] 9. Irrigation parameters - by crop development stages C. CROP INFORMATION data set must contain the follow- ing information: 1. Crop type 2. Rooting depth for the first stage [m] 3. Rooting depth for the second stage [m] 4. Rooting depth for the third stage [m] 5. Rooting depth for the fourth stage [m] 6. Rooting depth for the fifth stage [m] 7. Crop production rate group 8. Crop water depletion group 64 D. ENVIRONMENTAL-CLIMATE data set. Values are taken as average for the AEZ considered. 1. Temperature in daily or monthly mean [C] 2. Precipitation in daily or monthly mean [mm] 3. Relative humidity in daily or monthly mean [percent] 4. Solar radiation in daily or monthly mean [hours/day] 5. Wind velocity and wind velocity day/night ratio [m/s] CHAPTER IV STOCHASTIC YIELD SIMULATOR (SYS) The stochastic or probabilistic yield simulator is designed to estimate maximum potential yield, irrigated yield, and rain-fed yield with its statistically derived distribution densities (mean, variance, skewness, and so on) for different crops. The word "stochastic" is used here to indicate that the model's inputs and/or parameters (at least one) have nonzero variance. This means that the model's inputs and/or parameters (at least one) are not known with certainty, but statistics and distribution densities can be estimated from their sample data set. The major objective of the SYS is to provide decision makers and planners with information on potential, irri- gated, and rain-fed yields in the form of descriptive sta- tistics such as mean, variance, skewness, and quantiles, and associated histograms. This information will provide insight into stochastic behavior of the yield model and may serve as an important tool in agricultural and land-use planning, as well as natural resource management. 65 66 Several sources of uncertainty and error are present in models. Modeling error may be present because of uncer-. tainty regarding a particular phenomenon or the difficulty of expressing or modeling real-world behavior in mathemati- cal expressions. Errors may be introduced into the simula- tion model by its parameter-estimation procedures and the data-collection technique used. Exogenous environmental- variable inputs such as precipitation, temperature, and relative humidity are probabilistic in nature. Besides, data-collection methods may introduce variation and error into the input data set that will be used by the simulation model. Stochastic models are useful under conditions in which nonreliable estimates are available for the model's parame- ters and a large amount of money and time is needed to improve parameter estimates (Manetsch, 1986). The stochastic yield simulator has an analytical struc- ture similar to that of the deterministic yield simulator (DYS) discussed in Chapter III. In the DYS, all the inputs and model parameters were assumed to be known with cer— tainty, whereas in the SYS some degree of uncertainty is included in the modeling process. From this point of View, the SYS can enhance the contribution of the DYS model in the decision-making and planning process by accounting for some inherent real-world randomness. 67 The use of a simulation model for planning and policy making in developing nations has frequently been described in the literature (Manetsch et al., 1970; Manetsch, 1971, 1984, 1985; Rossmiller et al., 1978). The experience of those models is sometimes characterized by the expression "structure rich/data poor" (Manetsch, 1986), which means that the structure of the problem under investigation is available but time and money are required to provide good estimates for many of the parameters included. W There are two sources of uncertainty in the SYS model, random exogenous model inputs and uncertainty in the values of the parameters. Appropriate terminology for this mode condition is introduced in Figure 13. Following Manetsch (1986), the nomenclature used is: u(t) = defined as control and/or input vector of variables to the simulation model e(t) = defined as vector of exogenous environmental variables, whose values are given by probabil- ity density functions f1 (e1 (t)), f2 (e2 (t)), ..., fn (en (t)) p(t) = defined as vector of model parameters whose values are given by probability density functions 91 (p1 (t)), 92 (p2 (t)), n., gm (pm (t)) ...T 68 y(t) = defined as the simulation output vector, which is also given in terms of its distribution .- functionsl h (yl (t)), h (y2 (t), ..., bk (Yk (t)), and statistics mean, variance, skewness, eth The sources of randomness for the simulation model are e(t) and p(t). The term "control vector" is used here to indicate variables that are totally under the decision maker's control. e(t) n (exogenous Inpu‘t wl'th density functions fJ(eJ(t)), J = 1.2 ..... n) 7 model output wl'th STDCHASTIC densl'ty Functions a YIELD ..(kaD). k = 1,2,...1) u(t) SIMULATOR y(t) (declslon (SYS) /control 7 {7 Inputs) ‘ p(t) (model parameters m-th densrty Punctlons g}. (p {(t», i = 1.2....» Figure 13. General input/output diagram for the stochastic YIELD simulator with nomenclature. 69 From the environmental data base, which contains infor- mation for the environmental variables used in the model, such as precipitation, temperature, solar radiation, rela- tive humidity, and wind velocity, distribution shape and statistics are determined. To find the "correct" stochastic distribution for use in the model, a frequency histogram is prepared. Such a frequency distribution (Figure 14) will help to determine the shape of the distribution density function that provides a "best" fit with our data set or sample data set. Confinuous Don: 45.0- ‘ 40.0“ In x 3500— "l g ‘ ’ l ‘ g 30.0“ " ’ U’ "’ -L’ 8 I. g} L'— 25.0- I I} a 5' D I‘ ’1 ...; ’ ’1 '5 20-0‘ 9‘ 5* I. If m I. , .0 ‘I" ;’ < 15.0-‘ ’5" I I.’ I ’./. I "’ ’; "’ "( l". I’: 10.0- a,’, I ’./ I .’I’I :’ III. I, I.’ .’ 5.0-4 I." I: ~ . III ,I I.’ ’ . "f‘ H" I o _ A'dzdl ‘14‘ ‘ A A ' 61218243036424854606672 Plant Height (cm) Figure 14: Stochastic YIELD simulator-~an example of histogram plot. 70 Expert assistance in providing information for estimat- ing probability density functions for the most important! parameters plays an important role in estimating the distri- bution of a density function. Appropriate experts can and do provide information from which approximate density func- tions can be derived. A manual method may be used to construct the frequency histogram, but several software packages such as Plotit (1985), SAS (1985), SPSS (1984), MSTAT (1984) do exist that can be and are used to provide the statistics needed to run the simulation model. In determining the moments of the distribution func- tion, such statistics as mean, variance, and skewness are of concern. The skew factor, or skewness, is a descriptive statistic that provides information on the tendency of the deviations to be larger in one direction than in another. The skew factor is computed by: (3/2) skewness = m3 / m2 where: N _ mj = 1:1 (xi - x)1 / N mj = moment of order j (j = 1,2,3) Xi = random variate value (i = 1,2, ..., N) x = mean value N = sample size 71 For negative values of skewness, the distribution tail is to the left; for positive values the tail is to the ‘ right. Values of the skew factor > 0.5 or < -0.5 correspond to distributions with significant positive skewing (Figure 15). The distribution statistics computed are used in the simulation model to generate random variates that are approximated random variables drawn from distribution of the variable's population. 1.0- Probability 0 01 l 0.0+ Ifi'r'T T v V T O 1 2 3 4 5 Gommo Distribution 05 ”— Figure 15: Stochastic YIELD simulator-~probability density function skewing factor variation. 72 The most common distributions used in the simulation process for environmental variables are normal, gamma, tri- angular, and uniform distribution. The choice of the dis- tribution function will depend on the "goodness of fit" test with the available data and/or the shape of histogram plot, and the variable characteristics being modeled. One especially important family of theoretical distri- butions is the normal or Gaussian distribution. A normal distribution is a smooth, symmetric function often referred to as "bell-shapedfl' Its skewness is zero. A normal dis- tribution can be completely specified by only two parame- ters: mean and standard deviation. Approximately 68 percent of the values in a normal distribution are within one stand- ard deviation of the population mean; approximately 95 percent of the values are within two standard deviations of the mean; and about 99.7 percent are within three standard deviations. The gamma distribution (Figure 15) is one of the most useful continuous distributions available to the simulation analyst. If the variables from some random phenomenon can- not assume negative values and generally follow a unimodal distribution, then the chances are excellent that a member of the gamma.fami1y can adequately simulate the phenomenon. The gamma.distribution is defined by two parameters, a and k, where a is the shape parameter and k is the scale 73 parameter. As the two parameters vary, the gamma density can assume a wide variety of shapes, making it one of the‘ most versatile of distributions for simulation purposes (Shannon, 1975). The triangular distribution (Figure 16) is useful when data are very limited. The parameters used for determining the triangular probability density function are as follows (Manetsch, 1985, 1986): ' a lower limit (A1) for the parameter value i — an upper limit (Ci) for the parameter value i - a most 11k91Y value (Bi ) for the parameter value i ‘F‘; (pt. )¢ 2/(c — A) Figure 16: Stochastic YIELD simulator-~triangular probability density function. 74 From this information a probability density function for a triangular distribution can be used over the interval (Air Ci) with mode Bi in thesimulation model. The uniform distribution is a continuous probability density function, which is constant over the interval from, say, A to B, and zero otherwise. The uniform distribution is useful for simulating random phenomena with little or no strong variations. W Once the distribution that fits best in the sample data set is known through its moments or statistics, one is in a position to generate the random variates with the same statistics. The inverse transformation method (Naylor, 1968; Shannon, 1975; Manetsch, 1984) is used to generate random variates from a particular statistical population whose density function is given by f(x). (A more detailed description is provided in Appendix CJ The following formulas will be used in the inverse transformation process to generate variates from gamma dis- tribution f(x) with a given mean and variance (Naylor, 1968): ak * xk-l * e*ax a, k, x > 0 75 mean(x) a = """"""" variance(x) [mean(x)]2 k = --------------- variance(x) One of the problems that limited the use of gamma distribution in the past was the lack of a good generator if k is not an integer. Phillips (1971) developed a two- parameter gamma generator to overcome this problem. Shannon (1975) provided the Fortran code for the Phillips two-parameter gamma generator. Naylor (1948) used a simple alternative method to generate gamma random variates when the gamma distribution parameter k is not an integer. An Erlang gamma distribution 0: is an integer) may be generated by simply reproducing a process on which the Erlang distribution is based. This can be accomplished by taking the sum of exponential variates, x1, x2, ..., xn, with identical expected value l/a. Several probability distributions are related to gamma variables. Two of the more important ones are the chi- square and beta distributions (Naylor, 1968). A view of the gamma-variates-generation process to be used in the simulation model is provided in Figure 17. Figure 17. MAIN PROGRAM V CALL GAMMA (m A X) l SUBRDUTINE GAMMA (K, A, X) GENERATE R TR = TR I R -*ji k = -LDG(TR) / A RETURN MAIN Stochastic YIELD simulator—-gamma-variates- generation process. 77 ll . E !.|. . 3 C S . E J . This section, compiled from Manetsch (1986), deals with different combinations of sources of uncertainty in simula- tion runs. As previously indicated, two sources of uncer- tainty exist for any model, the exogenous inputs and uncertainty in the values to be assigned to model parame- ters. From the results of the simulation runs, that is from the histogram plot generated from the potential, irrigated, and rain-fed yield values, the decision maker can determine how much of the random variation observed in the final results is due to poor data and how much is due to input- parameters estimates. Random variation can be reduced by further data collection, and associated time and cost can be optimized based on that information. To obtain that information, the simulation model must be run with and without randomness in the model parameters. When the simulation is run with model parameters set at most likely values (parameters value assumed to be known with certainty), the procedure is called "variance partitioning." Variance partitioning is a valuable procedure in helping decision makers evaluate the importance of poor model data and the need for further data collection. It may be necessary and/or desirable to carry out this variance partitioning for individual parameters or subsets 78 of parameters to provide better—defined data-collection priorities. A different but very useful simulation technique for facilitating decision making under uncertainty is the "com- mon scenario analysis." Manetsch (1986) provided examples of this technique. Common scenario analysis is very useful when different policy-input alternatives have to be com- pared. Alternative policies are analyzed with the same sequence of random numbers, which then specify a common scenario for random exogenous inputs and parameter varia- tions. In this manner, alternative policies are compared in such a way that the only difference in the comparison is the differing policy specifications. The stochastic yield simulator developed in this study deals with variance partitioning for selected environmental variables. The model's parameters such as crop coefficient (kc), yield response factor to evapotranspiration deficit (ky), and so on, were given by Doorenbos and Kassam (1979) as fixed values. The environmental variables used in the model are the main source of randomness in the stochastic yield simulator. SI 1 I . 1:. 1 3 S' ] | (51:52 SI | In this section, the simulation model structure for the SYS is presented. The basic structure for the SYS is the 79 same as that used in the deterministic yield simulator (DYS), discussed in Chapter III. The Monte-Carlo method is the basis for the SYS simu- lator. In the Monte-Carlo technique, artificial experiences or data are generated by the use of a random—number gen- erator, resulting in the cumulative probability distribution of interest. The random-number generator may be a table of random digits, a computer subroutine or function, or any source of uniformly distributed random digits. The proba- bility distribution to be sampled may be based on empirical data derived from past records, may result from a recent experiment, or may be a known theoretical distribution such as gamma distribution. The random-number generator, as seen in the last section, is used to produce a randomized stream of variates that will duplicate the expected experience, based on the probability distribution being sampled. Some changes in the deterministic yield simulator had to be made to accommodate the stochastic characteristics of the input variables and parameters. A general flowchart for the simulation run with exoge- nous input and randomness in the environmental inputs and/or parameters is provided in Figure 18. The stochastic yield simulator (SYS) structure follows, with some modifications and additions, the equations and procedures in WM (Doorenbos & 80 SET INPUTS’ DISTRIBUTION STATISTICS SET RANDOM VARIATES GENERATOR SET NUMBER OF SIMULATIDN RUNS 7 INITIALIZE STATE VARIABLES INITIALIZE PARAMETERS 4‘ VALUE J EXECUTE SIMULATION RUN PHASES 2 - 6 DMPUTE STATISTICS AND HISTDGRAMS I: | Stochastic YIELD simulator--general flowchart of Figure 18. the simulation process. 81 Kassam, 1979), used in the deterministic yield simulator (DYS). Mainly seven consecutive phases are needed to estimate the probability density function statistics for the crop yields. They are: Ehase_l: Determination of the probability density function and its statistics for the parameters and environ- mental inputs. Three steps are to be followed in this phase (see Figure 19): ( BEGIN SIMULATION) PHASE h mph FHSTDGRAM PLUT ‘STEP 3' Gunnnzss or FFT tumuL______. RANDOM VARIATES ‘___ENEBAIDE_—— Figure 19. Stochastic YIELD simulator--Phase 1 flowchart. 82 Step 1: Identify the probability density function using the histogram plot. Step 2: Conduct a "goodness of fit" test to fit a distribution density function and compute the distribution statistics such as mean, variance, skewness, and so on. Step 3: Implement the random variates generator, using the inverse transformation method. Eha52_2: Determination of the maximum yield (Ym) of the adapted crop variety, dictated by climate, assuming that other growth factors «Lg. farm management, fertilizer, pests and diseases, and so on) are not limiting. In the second phase, six steps are needed to determine (Ym) (see Figure 20): Step 1: Computation of the gross dry matter production of a standard crop (Yo). Step 2: Application of the correction factor for crop species and temperatures. Step 3: Application of the correction factor for crop development over time and leaf area (cL). Step 4: Application of the correction factor for net dry matter production (cN). Step 5: Application of the correction factor for harvested part (cH) . Step 6: Computation of the maximum potential yield (Ym). Figure 20. 83 PHASE 2: STEP h COMPUTE DRY HATTER PRODUCTION (Yo) STEP 2| CORRE CTION PECIES AND TEMP. (ym) :uEuL______. CDRRECTIDN LEAF AREA (CL) ”nap 4. CORRECTION NET DRY MATTER (cN) ‘ TEP 5| CORRECTION HARVESTED PART (CH) ‘TEP 6' CDMPUTE MAXIMUM POTENTIAL YIELD (Yn) Stochastic YIELD simulator--Phase 2 flowchart. 84 £hase_3: Calculation of the maximum evapotranspiration (ETm) when crOp requirements are fully met by available water supply. In this phase, three steps are needed to compute BTm (see Figure 21): Step 1: Computation of the reference evapo- transpiration (ETo) based on the meteorological and crop data available. Step 2: Computation of the growing period and length of crop-development stages and selection of the crop coefficient kc. Step>3: Computation of maximum evapotranspiration (ETm). DHASE 3: TEPI' COMPUTE ETo ‘TEP 2| SELECT TEEL&_____—fi COMPUTE ETm Figure 21. Stochastic YIELD simulator-~Phase 3 flowchart. 85 Ebase 4: Determination of the actual crop evapo- transpiration (ETa) based on factors concerned with the available water supply to the crop (see Figure 22). This includes: Step 1: Determination of the total available soil water. Step 2: Computation of the soil water depletion. Step 3: Computation of the actual evapotranspira— tion (ETa). Ebase 5: Selection of the yield response factor (ky) to evaluate relative yield decrease as related to relative evapotranspiration deficit, and calculation of the actual yield (Ye) (see Figure 22). Ehase 6: Estimation of crop-yield adjustment. In this phase, the resulting estimated yield from Phase 5 is adjusted for fertilizer usage, soil salinity, and moisture content (Figure 23). Three steps are needed in this phase: Step 1: Adjustment of the estimated yield for ferti- lizer availability. Step 2: Adjustment of the estimated yield for salinity levels. Step 3: Adjustment of the estimated yield for moisture content. 86 DHASE 4: TEP 1| CDMPUTE ASI STEP 2| COMPUTE FACTOR p TEP 3| COMPUTE To PHASE 5: STEP II SELECT Ky Figure 22. Stochastic YIELD flowchart. simulator--Phases 4 and 5 87 PHASE 6: P in ADJUSTMENT FERTHJZER P L ADJUSTMENT SALINITY LEVELS fflEUL______fi ADJUSTMENT MOISTURE CONTENT Figure 23. Stochastic YIELD simulator-~Phase 6 flowchart. Rhase_l: Determination of the probability density functions and computation of the final yield statistics. The following three steps are needed (see Figure 24): Step 1: Identify the probability density function using the histogram frequency plot. Step 2: Conduct "goodness of fit" test for selection of the probability density function. Step 3: Compute the distribution density function statistics for potential, irrigated, and rain-fed yields. PHASE 7: TEP It )HSTOGRAM PLOT STEP 2: GUUDNESS OF FIT' __r_ TEP 3| DISTRIBUTION STATISTICS FOR YIELDS (END SIMULATION) Figure 24. Stochastic YIELD simulator--Phase 7 flowchart. M I] I' J’E ] l' The mathematical formulation of the stochastic yield .o simulator follows the procedures used in the deterministic yield simulator. Repetition of the phases, equations, and procedures involved in the deterministic yield simulator is necessary for completeness, consistency, and clear identifi- cation of the random factors included in the model and its relationships with the other components. Bhase_l. Determination of the probability density function and its statistics for the parameters and environ- mental inputs. The mathematical and statistical procedures were discussed in the section on random-variates generation. Ehase_2: Determination of the maximum yield (Ym) of the adapted variety, dictated by climate, assuming that other growth factors “Lg. farm management, fertilizer, pests and diseases, and so on) are not limiting. Refer to Equation 1.1, Chapter III, for the relation— ship of dry-matter production for a standard crop. F(t), fraction of daytime the sky is clouded, is deter- mined from the following formulas (Equation 1.7, Chapter III): [Rse(t) — 0.5 * Rs(t)] / [0.8 * Rse(t)] F(t) Rs(t) 0.25 + 0.50 * [n(t) / N(t)] * Ra(t) 90 Randomness is entered, in Phase 2, through the environ- mental variable n(t), number of sunshine hours a day. The value of n(t) variable is generated for every time (t) from the probability density function statistics derived from the sample data set "number of sunshine hours a dayJ' The parameters yo(t), yc(t) (Equation 1.1, Chapter III), Rse(t), Ra(t), and N(t) are entered into the model by means of table look-up function (tablex, spline) and are assumed to have zero variance: that is, they are values known with cer- tainty. The variable ynflt) (Equation 1.4, Chapter III) depends on the environmental variable temperature, which is a random variable generated by a random variate generator. In summary, Phase 2 of the stochastic yield simulator has two different random variates: an environmental input, number of sunshine hours a day--n(t), and dry matter pro- duction rate-~ym(t), which is a function of the environmen- tal input temperature. A third stochastic variable, which is assumed to vary within a predefined range, is crop-growth duration. In some cases, the sowing or planting date is not known with certainty, and it changes from year to year. If this is the case, a random variate can be defined in a planting interval in days and generated in the model by means of the random-variate generator. 91 Computation of maximum evapotranspiration Bha§s_3: The stochastic environmental variables of Phase 3 of (ETm) . the simulation model are as follows (refer to Equations 1J5 and 1.7, Chapter III): Wind velocity U(t), an exogenous environmental variable generated by a theoretical probability density function whose statistics and shape are determined from the wind velocity sample data set. The number of sunshine hours a day n(t), an exogenous environmental variable generated by a theoretical probabil- ity density function whose statistics and shape are deter- mined from the number of sunshine hours a day data set. Mean daily temperature value T(t), an exogenous envi- ronmental variable generated by a theoretical probability density function whose statistics and shape are determined from the temperature data set. Relative humidity value RH(t), an exogenous environmen— tal variable generated by a theoretical probability density function whose statistics and shape are determined from the relative humidity data set. All other parameters in this phase are deterministic; that is, their values are assumed to be known with cer- tainty. Bhassiiz Computation of actual evapotranspiration In this phase (refer to Equations 1.8 and 1.9, (ETa). 92 Chapter III), precipitation Pe(t) is an exogenous environ- mental variable that is generated from a theoretical proba- bility density function whose statistics are determined from the precipitation data set. Other parameters considered in this phase are assumed to be nonprobabilistic. Phase 5: Computation of estimated yield (Ye). Equa- tionslulO and 1JJ.(Chapter III)are usediJIthis phase. The model parameter ky(t) is assumed to be known with cer- tainty, making it a deterministic parameter. Phase_§: Estimated yield adjustment. The model input factors (refer to Equation 1.12, Chapter III) ydf and yds and the model parameter cf are assumed to be known with certainty, and their values are selected by the user. Bhase_l: Yield statistics generation. Using the results of Phase 6 after several simulation runs, the data set is statistically analyzed to determine the probability density function statistics and the histograms for the potential, irrigated, and rain—fed yield. In this phase, statistical—analysis software is used to provide the fre— quency histogram shape needed to determine the probability density function. The next step is to compute the distribu— tion moments such as mean, variance, skewness, and so on, to aid in the process of planning and decision making in eco- nomic analysis, land-use planning, and natural resource management. 93 W5 To run the stochastic YIELD simulator successfully, the user must assemble and analyze a data base. The data base must contain several data sets, which will provide the information necessary to estimate the probability density function statistics to run the model. For more detailed information on data requirements, refer to the section on data requirements in Chapter III. A. A LOCAL data set is required to identify the loca- tion, the region, or the AEZ parameters where the simulation will be done. All variables in the local data set are deterministic. B. A FARM MANAGEMENT PRACTICES data set is required to identify farming-system techniques. 1. Crop planting data interval [days] - a stochastic input variable 2. Crop first stage duration [days] - a stochastic input variable 3. Crop second stage duration [days] - a stochastic input variable 4. Crop third stage duration [days] - a stochastic input variable 5. Crop fourth stage duration [days] - a stochastic input variable 6. Crop fifth stage duration [days] - a stochastic input variable 7. Fertilizer availability [percent] - a stochastic input variable 94 8. Evaporation reduction factor - a deterministic input variable 9. Irrigation parameters - by crop development stages, a deterministic input variable C. A CROP INFORMATION data set must contain ministic exogenous variables described in Chapter D. An ENVIRONMENTAL-CLIMATE data set, which the stochastic exogenous environmental variables: 1. Temperature in daily or monthly mean 2. Precipitation in daily or monthly mean 3. Relative humidity in daily or monthly mean 4. Solar radiation in daily or monthly mean of sunshine hours 5. Wind velocity and wind velocity day/night ratio [percent] the deter- III. contains [C] [mm] [percent] [hours/day] [m/s] CHAPTER V MODEL VALIDATION AND SIMULATION RUNS The objective of this chapter is to validate the deter- ministic YIELD simulator (DYS) and the stochastic YIELD simulator (SYS). According to Shannon (1975), model valida- tion is a process of bringing to an acceptable level the user's confidence that any inference about a system derived from the simulation is correct. It is not possible to show that a model is the exact representation of the system being modeled. In the modeling process, one is, in general, not concerned with the "truth" of the model, but how it provides insights with a certain confidence in the results of the simulation. In general, one can say that it is the opera- tional utility of the model and its structure and not the "truth" of its structure that is usually of concern. To validate a model, Shannon (1975) indicated that it has to pass three tests, called the "test of validation." First, one must ascertain that the model has face validity; iJe., one must ask if the model results appear to be reason- able. This can be done by comparing the model's results with the system's results--that is, the real-world results. 95 96 Often, an expert opinion is needed to help analyze the model's results. The second method of validation is testing the model assumptions. The third test for model validation involves testing input—output transformations. Kaplan (1964) outlined model validation in terms of norms of validation. He indicated that, to be considered valid, the model must pass the norms of validation tests, which he defined as the correspondence, coherence, and prag- matic norms of validation. Fisherman and Kiviat (1967) divided the evaluation of simulations into three categories: verification, insuring that the model behaves the way the experimenter intends; validation, testing the agreement between the behavior of the model and that of the system; and problem analysis, drawing statistically significant inferences from the data generated by the computer simulation. Schrank and Holt (1967) proposed that "the criterion of the usefulness of the model be adopted as the key to its validation, thereby shifting the emphasis from a conception of its abstract truth or falsity to the question of whether the errors in the model render it too weak to serve the intended pur- poses." The validation process used in this study represents a combination of all the above. The following means of vali— dating the yield model are performed: First, for Jamaica, 97 20 runs will be conducted on a yearly basis, for sugarcane in the regions of Worthy Park, Caymanas, and Monymusk, using the deterministic approach. These represent traditionally rich sugarcane—producing regions. Observed yields are available for a period of 20 years or 1963-1982, to evaluate the model's performance. Second, simulations will be conducted for tobacco and sorghum for the same locationsm Observed yields are not available on a year-by-year basis, but some statistics, such as average tobacco and sorghum yield, are available for Jamaica. The simulation runs will be made in the "average" mode, with the deterministic model using average values of the environmental variables precipitation, temperature, relative humidity, solar radiation, and wind velocity. The simulation results will be compared to the actual average yield for tobacco and sorghum for the Worthy Park, Caymanas, and Monymusk regions in Jamaica. Third, 500 simulation runs will be made with the sto- chastic YIELD simulator for sugarcane, tobacco, and sorghum for the same location in Jamaica. ‘The probability density functions for the environmental variables precipitation, temperature, relative humidity, solar radiation, and wind velocity will be computed and used to generate, in a Monte Carlo simulation approach, the yield values. (The Monte Carlo approach is a technique for generating random variates 98 as input and/or parameters from a population described by some probability function to be used in the simulation ~ processJ Yields resulting from those simulations will be given in terms of yield probability density function statis- tics. Comparisons will be made with observed yield data and average yield data for Jamaica's Worthy Park, Caymanas, and Monymusk regions to infer and measure the yield model's performance. Results will be shown in tables, graphs, and histograms and in the form of statistics such as means, standard deviation, and quantiles. Graphics and summary statistics such as histograms are valuable to decision makers and planners in providing a better understanding of the simulation results. Fourth, 500 simulation runs will be made with the stochastic YIELD model for rice, potato, fresh pea, onion, and cabbage for the Agro-Ecological Zones (AEZs) in the Dominican Republicfls Ocoa Watershed, which are called Valdesia, Ocoa, and Azua. The AEZs were determined by the use of the CRIBS-GIS Geographic Information System using the OVERLAY and MATCH procedures (Schultink, 1986) and the spa- tially referenced information on soil, slope, evapotranspi- ration, temperature, and precipitation. In that simulation process, the probability density function statistics for the environmental variables precipitation, temperature, relative humidity, solar radiation, and wind velocity will be 99 computed using the Statistical Analysis System (1985) to generate, in a Monte Carlo simulation approach, the values of those environmental variables for each year. Yields resulting from those simulations will be given in terms of yield probability density function statistics. Comparison will be made with observed yield data and average yield data for the Dominican Republic's'Valdesia, Ocoa, and Azua AEZs to evaluate the model's performance. Also, the simulation results will be compared with the yield results and guide- lines given by Doorenbos and Kassam (1979). E I . . I' XIEIL S' J l E Jamaicazlntmduction For Jamaica, the DYS model was used to simulate crop productivity for some of the most important "cash“ crops, which are responsible for a large part of Jamaica's foreign exchange earnings, such as sugarcane and tobacco. The model was used to predict yields for sugarcane, tobacco, and sorghum. Sugarcane simulation was done for every year from 1963 to 1982 for three known producing regions: Worthy Park and Caymanas in the parish of St. Catherine and Monymusk in the parish of Clarendon. Twenty years of observed yield data are available for sugarcane, which is considered to be a relatively well organized and primarily state-controlled industry. For tobacco and sorghum, also important crops in 100 the Jamaican economy, only average, no yield data were available. The model was run for an “average" year. 1:! "I'IEIEIES'JI F. W The Jamaican national data base, put together by the Comprehensive Resource Inventory and Evaluation System (CRIES) at Michigan State University, was used to input the environmental data into the simulation model. The required environmental data used to run the simulations are: 1. monthly mean temperature values . monthly mean precipitation values . monthly mean relative humidity values monthly mean solar radiation values . monthly mean wind velocity values and day/night wind ratio thN Environmental and local data set, part of the CRIES— Jamaica national data base, were used on a yearly basis for AEZ conditions associated with Worthy Park, Caymanas, and Monymusk for sugarcane, tobacco, and sorghum. The model was designed to accept as input daily or monthly mean values for the environmental variables described above. It is difficult, if not impossible, to obtain daily measurements for those variables for under— developed or developing nations. However, the daily option is included because it is more realistic to use daily values when available. Parameters for sugarcane, tobacco, and sorghum required to run the model were supplied by the Jamaica Ministry of Agriculture, Rural and Physical Planning Division (RPPD) and complemented, when necessary, by the data set contained in Doorenbos and Kassam (1979). The crop parameters used are: 1. average root size for each phase of the growing period . leaf area index (LAI) . water depletion factor (p) . production rate (ym) . crop coefficient (kc) . yield response factor (ky) OfiUlubWN Crop parameters used in the simulation runs for the Worthy Park, Caymanas, and Monymusk regions of Jamaica are provided in Table 3. In a year-by-year simulation run of the deterministic YIELD simulator, the user enters values of crop parameters at specific prompts during the simulation process. The values for crop parameters (Table 3) were used for each of the three regions where the simulation took place. Being crop specific, they do not vary with location or time. They do vary, however, from crop to crop. E . . !' XIEIE S' 1 l I . __ WW Several farm-management-practice parameters must be available to the user to run the simulation model. The parameter values used were provided by the Rural Physical 102 Table 3. Deterministic YIELD simulator: Jamaica--crop parameters-~sugarcane, tobacco, and sorghum for Worthy Park, Caymanas, Monymusk Crop Parameter Type Sugarcane Tobacco Sorghum Root Size Variation (cm)a 20—120 10-150 10—175 Leaf Area Index (LAI) 3 3 2 Water Depletion Factor Variation (p)b .400-3875 .400-3875 .400-.875 Production rate (ym in kg/ha/day)C 0—65 0-35 0-65 Crop coefficient (kc)d .40-1.3o .30-1.2o .30-1.15 Yield response factor (ky)e .lO—.75 .20—1.00 .20—.55 Source: Doorenbos and Kassam (1979). Note: The values separated by "-" represent ranges of variation to be used by the numerical interpolation function in the model. Crop stage was defined in the data require- ments in Chapter III. aAdjusted for local conditions according to RPPD data. bETm--Maximum evapotranspiration dependent factor. CTemperature dependent factor. dDepend on crop stage, wind velocity, and relative humidity. eDepend on crop stage. 103 Planning Division of the Jamaica Ministry of Agriculture. .0 Additional parameter values were derived from Doorenbos and Kassam (1979). Farm-management-practice parameters vary from crop to crop, from region to region, and from year to year. The following farm-management-practice parameters.are considered in the simulation model: O‘U‘lubUONl-J sowing or planting date harvesting date duration of each stage of the growth period irrigation parameter and/or values evaporation reduction factor fertilizer usage The values of farm-management-practice parameters used in the simulation runs for sugarcane, tobacco, and sorghum for the Worthy Park, Caymanas, and Monymusk regions of Jamaica are listed in Table 4. L I . . I' YIEIL S' 1 I I . __ W Parameters that identify the location, region, AEZ, or production potential unit are necessary to run the model. Those parameters give very detailed and specific spatial information for the simulation model. They are spatially referenced parameters, which means that for each location a specific set of parameters is used. Information on the following specific local parameters was collected from the CRIBS-Jamaica national data base. Local parameters, by being local specific, do not vary from 104 Table 4. Deterministic YIELD simulator: Jamaica--farm- management-practice parameters for sugarcane, « tobacco, and sorghum, for Worthy Park, Caymanas, and Monymusk Farm-Mama gem ent- Practice Parameters Sugarcane Tobacco Sorghum Sowing or Planting Date MM/DD/YY 02/15/YY 02/15/YY 02/15/YY Harvesting Date MM—MM/YY+a 01—04/YY+1 05—06/YY 06-07/YY Duration of Growth Stages in Daysb stage 1 30-80 10-15 15-20 stage 2 80—120 20-30 20-30 stage 3 100-220 30-35 15—20 stage 4 30—80 30—40 35-40 stage 5 30~60 10—20 10-15 Irrigation Parameter or ValueC F F F Evaporation Reduction Factord N N N Fertilizer Usagee 80—100 80-100 80-100 Source: Jamaica Ministry of Agriculture, Rural and Physical Planning Division. a+1 means following year. bCompiled from Doorenbos and Kassam, (1979) and adjusted with RPPD data. The deterministic model uses the average value of the range identified. f CFull irrigation was used. No data available on irri- gation scheme or amount of water used in irrigated crops. dNo evaporation reduction factor was used; information not available. eFertilizer usage expressed in percent relative to the crop-requirement guidelines as defined by Doorenbos and Kassam's (1979) crOp—requirement guidelines. 105 crop to crop or from time to time. The average local- specific parameters necessary to run the model are as fol- lows: 1. altitude 2. latitude 3. location (northern or southern hemisphere) 4. slope class (Table 1, Chapter III) 5. soil type (Table 1, Chapter III) 6. soil textural class 7. soil moisture availability 8. soil salinity level The local parameter values for Worthy Park, Caymanas, and Monymusk to be used in simulation runs are provided in Table 5. In spite of a possible variation in time of the local parameters, no time variation was considered from run to run because the local parameters do not change very much in the short or medium time period and because no data are available about changes in those local parameters, such as variations in soil salinity levels, soil-moisture availabil- ity at sowing date, and so on, from year to year. . . . . . __ Dete1minisglc_¥lfLD_S§mnl?fQ£4.Jamalga__ Twenty simulation runs were conducted for sugarcane from 1963 to 1982 for two areas in the parish of St. Catherine and one area in the Clarendon parish. For tobacco and sorghum, an ”average" year was simulated for each of those locations, to permit a comparison with variations in average observed yield for the country. 106 Table 5. Deterministic YIELD simulator: Jamaica--local parameters for Worthy Park, Caymanas, and Monymusk Crop Worthy Parameter Park Caymanas Monymusk Average Altitude in Meters 381.00 27.75 9.15 Average Latitude in Degrees 18.09 17.58 17.48 Location 1 1 1 Average Slope Classa 0—4 0-4 0-4 Average Soil Typea fine silty fine silty fine silty Average Soil Textural Class mm/mb 115 125 125 Soil Moisture mm/mC 90 90 90 Soil Salinity Leve1d N/A N/A N/A Source: Jamaica Ministry of Agriculture, Rural and Physical Planning Division. aTable 1, Chapter III. bMoisture holding capacity. CInitial soil moisture at sowing date. dN/A--Data not available. 107 Results of the yearly simulation runs for sugarcane for irrigated yield and rain-fed yield are presented for the‘ Worthy Park region in Table 6. Table 6. Deterministic YIELD simulator: Jamaica-- St. Catherine--Worthy Park. Sugarcane--observed irrigated yield and simulated irrigated and rain- fed yield, 1963-1982 (tons/ha)a Observed Simulated Simulated Year Yieldb Irrigated Yield Rain-fed Yield 1963 80.82 86.94 81.90 1964 76.00 85.39 82.67 1965 79.95 94.67 89.71 1966 81.06 97.59 80.89 1967 71.58 79.16 72.12 1968 86.55 102.95 95.17 1969 76.99 82.60 74.05 1970 89.31 94.48 86.30 1971 78.87 86.97 79.63 1972 90.97 93.98 83.93 1973 85.64 89.66 85.49 1974 82.50 82.15 71.78 1975 79.34 90.10 85.19 1976 85.31 91.11 76.30 1977 63.77 95.04 91.93 1978 95.49 93.86 90.75 1979 90.72 89.04 86.25 1980 79.31 100.43 99.00 1981 85.58 92.21 81.98 1982 77.73 91.66 85.46 aCompiled from simulation results. bSource: Jamaica Ministry of Agriculture, Rural and Physical Planning Division. 108 A time plot of the observed sugarcane yield and simu: lated irrigated yield for Worthy_Park is presented in Figure 25. Yields for sugarcane from 1963 to 1982 are shown, and comparisons can be made between them. ‘20 T V T I T V T 1 V V 1 I T I Y I V V I I V I j d H Observed Sugarcane Yie 4 110 H Irrigated Sugarcane Yield 3 m . 2 3 $13 '00 1 5 9° 1 0 I L J 8, 5° '1 (,3, : 70 € 60 ‘ , .... 1 . . , . . ., . . ., . . .1 . -111 1960 1964 1968 1972 1976 1980 1984 Harvest Year Figure 25. Deterministic YIELD simulator: Jamaica-~Worthy Park, observed irrigated yield and simulated irrigated yield for sugarcane for the period 1963-1982(t/ha). Simulated results represent the trend in observed yield from 1963 to 1974. In those years (see Figure 25), an almost cyclical yield trend occurred for observed irrigated 109 yield and predicted irrigated yield. From 1975 to 1982,_ there was no clear relationship between simulated irrigated yield and observed yield. This may be explained by several reported changes in the government of Jamaica, affecting the sugarcane industry. It has been reported that sugar mills and sugar states were going through a process of nationali- zation, and that process had a strong influence on sugarcane yield. In addition, strikes took place during that period, affecting the reliability of observed yield data for any meaningful analysis. The observed yield values are usually below the irri- gated simulated values (Figure 25 and Table 6). That may be an indication that, holding other factors constant, improved water-management procedures may improve yields. The rela- tionship between the sugarcane observed yield and simulated rain-fed yield is also presented in Figure 26. In this case, rain-fed production is simulated. It can be observed (Figure 26) that the simulated rain-fed yield is much closer to the observed yield as compared to irrigated yield for the years from 1963 to 1974 (Figure 25). Again, the effects of reported political and socioeconomic conditions related to the sugarcane industry in Jamaica may account for a strong variation in the observed-yield trend from 1975 to 1982. Some simulated rain-fed yields (1964, 1965, 1968, and so on) are larger than the observed yields, suggesting that other 110 yield-reducing factors not included in the model, such as pest and disease, or a major weather phenomenon such as ‘ storms, affected yields. Government policies in terms of agricultural and economic policies and/or external market conditions may affect variations, as well. ‘20 f T——T T j —r Y7 I ‘f j —Y ‘I V r 7 T W T 1 fl ‘T—f H Observed Sugarcane Yield < 110 H Rainfed Sugarcane Yleld : m 1 32 i Q) 1 '>"- ‘°° 1 3 90 3 o 1 ° 1 B 30 . a» 1 (B I 70 '3 so 1 I ' ' V V ‘ V 1' 'TT frr Tirfifrfi 'r T 1960 1964 1968 1972 1976 1980 1984 Harvest Year Figure 26. Deterministic YIELD simulator: Jamaica-~Worthy Park, sugarcane irrigated observed yield and simulated rain-fed yield from 1963 to 1982 (t/ha). For the period from 1975 to 1982, random variations are present. It has been reported that political unrest and government policy in the sugarcane industry were major 111 factors responsible for the random behavior of observed yields for the 1975 to 1982 period. Correlation coefficients were computed from the results presented in Table 6 for a simple linear regression equation and a logarithmic-inverse regression equation. For observed yield and simulated rain—fed yield, using the total data set (20 years), the coefficients of correlation were very low (R2 = .247 for the linear regression and .283 for the logarithmic-inverse regression). Observed yield and simu- lated irrigated yield correlation coefficients were also very low (R2 = .153 and .243 for the linear and logarithmic cases, respectively). When the outliers were removed from the data set for the regression analysis, the correlation coefficient for the linear case changed to .436 and for the logarithmic—inverse case to .497. The significance level of the F—test for the linear case was .0398; for the logarithmic-inverse case, it was .0268. For the simulated irrigated yield the correla- tion coefficient for the linear form was .578 and u629 for the logarithmic-inverse form, with the significance levels of the F-test .0432 and .0346, respectively. In a multiple regression approach in which the depend- ent variable is observed yield, and considering fertilizer usage (N, P, K) (presented in Table 16) and simulated rain- fed yield as independent variables, the correlation 112 coefficient for the linear functional form (taking out the outliers) is .955, with significance of the F-test equal to .1024. For the logarithmic-inverse functional form (taking out the outliers), the correlation coefficient is 0.92, with significance of the F-test equal to .1062. When the simulated rain-fed yield was replaced by simu- lated irrigated yield in the multiple regression equation, the correlation coefficient changed to 0.85 for the linear case and to 0.92 for the logarithmic-inverse case, with significance levels of the F-test at .0873 and .0745, respectively. The degree of association between the observed yield and the simulated rain-fed yield is, of course, increased by entering the fertilizer variable in the regression equa- tion. A different perspective on the relationship between the simulated sugarcane yield results and the observed sugarcane yield is presented in the form of a scattergram in Figures 27 and 28. The sugarcane observed yield and sugarcane irrigated yield results are clustered above the 45-degree line (Figure 27). The 45—degree line indicates equal values for both variables. The values of simulated irrigated yield are in most cases greater than the values of the observed yield. 113 ‘20? 1 ffi v Tfi v v v I 1 Y T 1 ! v v T 1 1 V v v 1 I W j v v ' 1 I Observed Yneld vs [mauled Yueld _ . 5, . < ~° a 110- .. :9 1 J 2 i i x . . y 1: 100- ' - 0 u: . ‘6 q . I - ‘ .9 j . I ' I I : E 903 I u . .. v I I C «I ' 1 U 1 ' 1 8 80" I " D 1 ‘ 0‘ ‘ .l 3 "‘ II (I) « . 7o- - 4 4 ll «1 60- . I—'TT'W'W1‘*"ff"fij‘fi‘l“"1 BO 70 BO 90 100 110 120 Sugarcane Observed Yield Figure 27. Deterministic YIELD simulator: Jamaica--Worthy Park, sugarcane observed irrigated yield versus simulated irrigated yield (t/ha). 1201 .fi- fifijfifirvy+vvv1+vvfiyfirv : I Observed Yoeld vs Rainfed Yield : : a” I no- ” ~ ‘0 j " E2 . i > . . .2 . . .5 1 ' : O ‘ I . 4 g i g. .. 1 o 1 ' l o 'l I I ‘ L. 80" ‘ O 1 ‘l O, J I a 3 II I I U) 4 I '4 70~ "1 «1 '4 4 503 rV'VV'va fivf VTV’V Yfi va so 70 so do 100 ETio T50 Sugarcane Observed Yield Figure 28. Deterministic YIELD simulator: Jamaica-~Worthy Park, sugarcane irrigated observed yield versus simulated rain-fed yield (t/ha). 114 The sugarcane observed yield is also plotted against the rain-fed observed sugarcane yield (Figure 28). Here the points are clustered along the 45-degree line. This is an indication that the results of the rain-fed simulation are closest to the observed sugarcane yield. In both figures, some outlying values can be seen. They represent the years for high and low values and unusual conditions. If one views the cluster center location as one of the performance measures of the model, it can be inferred that the model results reflect, or are similar to, the results of rain-fed yield. The simulation results for the Caymanas region in the parish of St. Catherine in Jamaica are presented in Table 7. The results are from the simulated irrigated yield, the simulated rain-fed yield, and the observed irrigated yield for the 1963 to 1982 period. A plot of the observed sugarcane yield and simulated irrigated yield for Caymanas is shown. Yields for sugarcane from 1963 to 1982 are presented in Figure 29. The modelds performance as a measure of how the simu- lated irrigated yield tracks the observed yield may be considered very good (values are very close and the trend is replicated) for the period from 1963 to 1976. In those years, as one can observe from the data presented in Figure 115 29, the repetition of the observed-yield trend is followed by the irrigated-yield values. Table 7. Deterministic YIELD simulator: Jamaica-- St. Catherine--Caymanas. Sugarcane--observed irrigated yield and simulated irrigated and rain- fed yield, 1963-1982 (tons/ha)a Observed Simulated Simulated Year Yieldb Irrigated Yield Rain-fed Yield 1963 95.39 101.91 72~17 1964 91.46 102.03 82.67 1965 93.46 111.35 80.74 1966 91.29 110.56 75.98 1967 94.18 111.55 82.15 1968 76.62 110.06 64.34 1969 77.48 97.59 57.55 1970 84.62 102.60 80.79 1971 79.63 97.81 75.66 1972 97.39 115.87 74.97 1973 84.18 96.55 75.26 1974 82.20 98.77 63.82 1975 92.28 106.51 73.38 1976 78.72 95.29 60.27 1977 63.75 101.99 56.96 1978 68.98 105.07 67.97 1979 67.67 96.72 68.67 1980 63.72 102.90 82.62 1981 61.91 104.51 81.02 1982 65.75 101.02 77.95 aCompiled from simulation results. bSource: Jamaica Ministry of Agriculture, Rural and Physical Planning Division. 116 ‘60 ' r ' I Tfi 'j T T Y I T V T I 1 v—T I v r- 1 ‘50 e-e Observed Sugarcane Yield ‘40 l—i kfigdmdlmxxmauu‘fldd 130 120 110 100 90 80 70 60 Sugarcane Welds 4O Y j'v—V—V— r ’fi TV '"I F 1964 1968 r T T Tr W 1 fi‘ —1 j I—fi'fij 1960 1972 1976 1980 1984 Harvest Year Figure 29. Deterministic YIELD simulator: Jamaica-- Caymanas, sugarcane observed irrigated yield and simulated irrigated yield from 1963 to 1982 (t/ha). For the period from 1977 to 1982, there is no clear relationship between the simulated irrigated yield and the observed yield. As pointed out before, for the Worthy Park region, during that period several changes occurred in the government of Jamaica in relation to the sugarcane industry. Also, sugar mills and sugar farms were going through a nationalization process, and that process had a great influ- ence on sugarcane yield. Strikes, political unrest, crop destruction by the farm's labor force, farm employees' 117 refusal to harvest the crop, losses in harvesting and trans- portation, and, in some cases, failure to record proper ‘ yield data left the sugarcane industry in chaos. - Also, the observed yield values are below the irrigated simulated values (Figure 29 and Table 7). Again, that may be an indication that, holding everything else constant, a better water-management procedure may improve sugarcane yields. The relationship between sugarcane observed yield and simulated rain-fed yield is plotted in Figure 30. It can be observed that the simulated rain-fed yield is closest to the observed yield for the years considered. Observed sugarcane yield values are in most cases above those for simulated sugarcane rain-fed yield. This fact may suggest that some irrigation was used, but not effectively, because crop-water stress still occurred. The results for Caymanas may also indicate that, every- thing else being equal, better water-management procedures may generate yield increases in sugarcane. Again, the effects of political problems related to the sugarcane industry in Jamaica are shown by a strong random variation in the observed yield trend from 1975 to 1982. There was no reported major disease outbreak or pest infestation on sugarcane crop fields during the period studied, which leaves the government interventions on the 118 sugarcane farms and mills and associated factors as major causes of the random behavior of observed yields for the“- period from 1977 to 1982. ‘60 fit Y— 1 f j V 1 fi— T—l V—f W l 1 i T h V fi 150 o—e Observed Sugarcane field 140 H Ralnfed Sugarcane Yield 1 30 1 20 1 10 100 90 80 7O 60 50 40 Sugarcane Welds fi' 1 —T V— V V V V r"'—T—T TTTTT '—l T—T *1 1960 1964 1968 1972 1976 1980 1984 Harvest Year Figure 30. Deterministic YIELD simulator: Jamaica-- Caymanas, sugarcane observed irrigated yield and simulated rain-fed yield from 1963 to 1982 (t/ha). A scattergram of the simulated sugarcane yield results and the observed sugarcane yield is provided in Figures 31 and 32, which provides a different perspective for analysis of the simulated results. 119 s Observed Yodd vs Irrigated Yield Sugarcane Irrigated field 8 "I. 38838 W 40 50 60 70 80 90100110120130140150160 Sugarcane Observed Yield Figure 31. Deterministic YIELD model: Jamaica-- Caymanas, sugarcane observed irrigated yield versus simulated irrigated yield (t/ha). I Observed Yield vs Rainfed field 120 110 so so f 4' Sugarcane Roinfed Yield 8 I.- 8886‘ W 40 50 60 7O 80 9010011012013014-0150160 Sugarcane Observed Yield Figure 32. Deterministic YIELD model: Jamaica-- Caymanas, sugarcane observed irrigated yield versus simulated rain-fed yield (t/ha). 120 Sugarcane observed yield and sugarcane irrigated yield results are clustered above the 45-degree line (Figure 31). The values for both variables show, as seen before, that the values of simulated irrigated yield are greater for all studied years than the values of the observed yield. Sugar- cane observed yields are also plotted against the rain-fed observed sugarcane yields (Figure 32). Essentially, three clusters can be seen (Figure 31). It can be noted that, for this case, the model predicts yields better for higher values than for lower values (clus- ters of high values are closest to the 45-degree line). Values are clustered mainly below the 45-degree line, an indication that the yields of the rain-fed simulation are smaller than the observed sugarcane yield. If the location of the cluster center is taken as one of the performance indicators of the model, one can infer that the model reflects more the results of rain-fed yield than those of irrigated yield. The result may also suggest that some irrigation scheme was used. Simulated irrigated, rain-fed, and observed sugarcane yields for the Monymusk region in Clarendon parish for the period from 1963 to 1982 are presented in Table 8. 121 Table 8. Deterministic YIELD simulator: Jamaica--C1arendon --Monymusk. Sugarcane--observed irrigated yield and simulated irrigated and rain-fed yield, 1963- 1982 (tons/ha)a Observed Simulated Simulated Year Yieldb Irrigated Yield Rain-fed Yield 1963 92.35 118.66 87.29 1964 86.94 111.05 87.75 1965 83.76 108.25 80.39 1966 90.97 126.55 92.77 1967 76.68 108.28 81.63 1968 70.62 106.53 58.78 1969 78.03 115.05 77.65 1970 82.97 121.92 102.62 1971 71.21 106.28 83.29 1972 78.57 122.41 105.05 1973 62.15 104.16 81.39 1974 70.47 109.70 81.13 1975 71.18 107.99 77.61 1976 72.37 117.84 68.69 1977 54.43 110.33 68.84 1978 74.00 129.05 99.12 1979 67.23 116.88 85.06 1980 61.50 127.45 103.86 1981 55.55 120.31 93.31 1982 53.92 131.90 116.33 aCompiled from simulation results. bJamaica Ministry of Agriculture, Rural and Physical Planning Division. A graphic representation of observed yields and simu- lated irrigated yields for Monymusk from 1963 to 1982 is shown in Figure 33. Again, the model simulates fairly well the trend in observed yield from 1963 to 1974. In those years, the data 122 _ show a good association of the vacillations in the simu- lated values when compared to the irrigated yield trend.“ ‘60 r 1 v 1 I t 1 T v —y ' T 1 1. _' 150 e—e Observed Sugarcane Yield 140 H Wanted Sugarcane Yield 130 120 110 100 90 80 7O 60 50 40 i Y 1+T I f V Sugarcane Yields T ' ' ‘T T— 1 V j r T Tr: 1960 1964 1968 1972 T F I—‘ l 1976 1980 1984 Harvest Year Figure 33. Deterministic YIELD simulator: Jamaica--Monymusk, sugarcane observed irrigated yield and simulated irrigated yield from 1963 to 1982 (t/ha). From 1975 to 1982, there was no clear relationship between simulated irrigated yield and observed yield. 'The specific causes for this were explained before. The fact that the observed yield values are approxi- mately 20 to 40 percent below the simulated irrigated yield values may be an indication that, everything else remaining the same, better water-management procedures are needed to improve sugarcane yields. The relationship between 123 sugarcane observed yield and simulated rain-fed yield is provided in Figure 34. 150 an 1 . T T ' ' l ‘ T T TTT T . I T T ‘ l T T ‘ 150 a—o Observed Sugarcane Yield 140 H Rainfed Sugarcane Yield 130 120 110 100 90 80 7O 60 50 4O Sugarcane Yields 1 1984 V—T ‘7 fl ' . . T . 1968 1972 T I 1' r l r ' v v V 1_fi 1960 1964 1976 1980 Harvest Year Figure 34. Deterministic YIELD simulator: Jamaica-~Monymusk, sugarcane observed irrigated yield and simulated rain-fed yield from 1963 to 1982 (t/ha). From the results presented, it may be observed that the simulated rain-fed yield is closer to the observed yield, as compared to the simulated irrigated yield, for the years from 1963 to 1977. Observed sugarcane yield values are in most cases below the simulated sugarcane rain-fed yield. This fact may suggest that some other factor influenced the decrease in sugarcane yield, such as pest and disease, storms, and/or labor and market conditions. 124 The observed sugarcane yield trend for Monymusk is -~ repeated by the simulated rain-fed sugarcane yield for the years 1963 to 1973. Again, effects of a socio-political nature may have caused strong variations in the observed yield trend from 1974 to 1982, especially because there were no reported cases of major disease outbreaks or pests on sugarcane crop fields during the period studied. i Scattergrams of the simulated sugarcane yield results and the observed irrigated sugarcane yield are presented in Figures 35 and 36. 160 Wu“........,.........,....!....,.........l.........,... I Observed Yield vs lrngated Yield ' 150 _. a z 140 V a I 130 I I I - s- 120 I I s I I T 110 I i. I' .. Sugarcane Irrigated Yield 8 3886' 43' 5'0 6'0 7'0 8'0 '9'0'3'66'1ioiz'dfibiimébiéo Sugarcane Observed Yield Figure 35. Deterministic YIELD simulator: Jamaica--Monymusk, sugarcane irrigated observed yield versus simulated irrigated yield(t/ha). 125 Sugarcane observed yield and sugarcane irrigated yield results (Figure 35) are clustered above the 45-degree line. For all years studied, the values of simulated irrigated yields were larger than the values of the observed yields. Also, sugarcane observed yields were plotted against rain- fed observed sugarcane yields (see Figure 36). 160 I Observed Yield ve Rainfed Yield 150 :3, N' 140 E 93 130 5. .8 120 I. E 110 " I c? 100 ' . ' 3 90 ‘ O l I 8 80 I [I . O I S’ 70 . . (I) 50 I 50 4O WWWMWW 4O 50 60 7O 80 90100110120130140150160 Sugarcane Observed Yield Figure 36. Deterministic YIELD simulator: Jamaica--Monymusk, sugarcane irrigated observed yield versus simulated rain-fed yield (t/ha). The yield data presented in this case are also clus- tered slightly above the 45-degree line. This is an indica- tion that the results of the rain-fed simulation are larger than the observed sugarcane yield. 126 From the location of the cluster center, it can be d inferred that the model reflects more closely the results of rain-fed yield than of irrigated yield when compared to observed yield. The results also suggest that irrigation scheme, in addition to other management practices, must have been used to improve yields. One can also conclude from the results that water deficit affected final yield. The deterministic YIELD simulator model was also used to simulate average yield values of two other crops in Jamaica: tobacco and sorghum. For these crops, only average and no yearly observed yield was available for the study regions. The simulation model was run only once through, to simulate yield for the "average" year; the environmental variables were averaged on a month-by-month basis to generate one year of average values for precipita- tion, temperature, relative humidity, solar radiation, and wind velocity to simulate the irrigated and rain-fed yields. The results of the simulation run for the "average" year for tobacco and sorghum for the Worthy Park, Caymanas, and Monymusk regions are presented in Table 9. The average yields for Jamaica were provided by the Ministry of Agriculture. They are: l. tobacco: 1.0 to 2.0 tons per hectare 2. sorghum: .297 to 1.028 tons per hectare; under good management and cultivar, 1.136 tons per hectare 127 Table 9. Deterministic YIELD simulator: tobacco and sorghum: Jamaica-—Worthy Park, Caymanas, and Monymusk. ’ Simulated ”average" yield results over the period from 1963 to 1982 (tons/ha) Region ......Tobacco...... .......Sorghum..... Irrigated Rain-fed Irrigated Rain-fed Worthy Park 3.148 2.462 3.869 3.803 Caymanas 3.033 2.212 3.308 2.866 Monymusk 4.862 3.589 3.627 3.250 Source: Compiled from simulation results. Doorenbos and Kassam (1979) indicated the following yields of high-producing varieties adapted to the climatic conditions of the available growing season under adequate water supply and high level of agricultural inputs under irrigated farming conditions for sugarcane, tobacco, and sorghum: 1. sugarcane (cane) 100.00 to 150.00 tons/ha 2. tobacco (leaf) 1.50 to 2.50 tons/ha 3. sorghum (grain) 2.00 to 5.00 tons/ha The deterministic YIELD simulator results for sugarcane are within the yield value ranges given by Doorenbos and Kassam (1979). For sorghum, the simulation results are quite good when compared to Doorenbos and Kassam but very high when compared to Jamaica's average, suggesting that 128 there is room for improvement in Jamaicafls sorghum- production system. The simulation results in terms of yield average are very high compared to the average Jamaica tobacco yield and also very high when compared to Doorenbos and Kassanfls yield values. That fact may be an indication that a refinement of the model's parameters or an adjustment of the yield gen- erated by the model may be necessary for tobacco, consider- ing its unique production system. Tobacco is transplanted to the crop field at a certain stage of the growing season, and the harvesting technique is unique. The leaves are harvested during a long period. To summarize the discussion of the deterministic YIELD simulator for agro-ecological conditions represented by the Worthy Park, Caymanas, and Monymusk regions in Jamaica, as a tool in formulating policy decisions regarding the produc- tion of sugarcane, the results indicated that there is room for improvement in the sugarcane—production system; more effective water-management procedures may improve yields. For sorghum, a complete review of Jamaica's production system may be justified to improve yields. For tobacco, some refinements in the model may be needed to improve simulation results and to obtain a more significant rela- tionship between observed yield and simulated results. 129 Improvements in the tobacco-production system seem jus- tified. ‘ The data set used in the deterministic YIELD model was used to run the stochastic YIELD model (SYS). With the stochastic YIELD model, some random variations are intro- duced that are inherent in the exogenous environmental vari- ables. Also, variations will be introduced to some control inputs to the model. It is expected that the SYS will provide insights into the yield variations as some of the exogenous inputs are taken as random variables. 5! l I’ YIEIES' 1| 1 .__ Enviranmentallnauts The CRIBS-Jamaica national data base was used to esti— mate the probability density function statistics for the environmental inputs. The Statistical Analysis System (SAS, 1985) software package was used for the statistical analy- sis and to identify the probability density function and distribution parameters to use in the simulation model. Probability density function statistics for the envi— ronmental variable precipitation for each month of the year were used to run the simulation model. The statistics for Worthy Park, Caymanas, and Monymusk are presented in Tables 10, 11 and 12, respectively. Monthly mean precipitation was used because of the sensitivity of the model to precipitation values. Also, 130 seasonal rainfall distribution is important for agricultural production systems and hence for yield production. No cdr- relation between precipitation in different months was assumed; that is, each month was considered independent of the others. Table 10. Stochastic YIELD simulator: precipitation probability density function statistics for Jamaica--Worthy Park, for the years 1963 to 1982 (mm/month) Month of the Standard Year Mean Deviation Skewness January 88.900 91.766 2.521 February 52.211 39.146 1.554 March 44.027 38.751 1.370 April 97.084 68.330 1.304 May 186.831 116.363 0.377 June 124.178 89.878 1.673 July 111.478 28.973 0.610 August 181.751 116.140 2.107 September 199.249 117.639 1.921 October 197.838 110.303 1.534 November 94.262 60.328 0.838 December 52.776 24.289 0.705 Source: Compiled from CRIBS-Jamaica National Data Base (CRIBS-MSU). 131 Table 11. Stochastic YIELD simulator: precipitation probability density function statistics for Jamaica--Caymanas, for the years 1963 to 1982 (mm/month) Month of the Standard Year Mean Deviation Skewness January 18.473 22.006 2.042 February 16.625 14.255 0.514 March 16.972 134608 0.631 April 41.679 51.519 2.975 May 107.026 108.403 1.296 June 51.954 61.014 1.663 July 44.681 42.362 0.882 August 87.053 77.524 2.211 September 113.492 135.541 3.463 October 166.254 141.858 1.947 November 76.200 66.545 2:737 December 42.603 40.639 1.280 Source: Compiled from CRIBS-Jamaica National Data Base (CRIES-MSU) . Variations in temperature, relative humidity, solar radiation, and wind velocity are not as large as variations in precipitation, and the available data values, mainly solar radiation and wind velocity, allied to precision and the assumption that water availability is the main factor in biomass production, influence the decision to use annual and not monthly distribution as was done with precipitation. Minimum.and the maximum annual values for those variables were taken to run the model. 132 Table 12. Stochastic YIELD simulator: precipitation probability density function statistics for Jamaica--Monymusk, for the years 1963 to 1982“ (mm/month) Month of the Standard Year Mean Deviation Skewness January 43.053 42.226 2.190 February 18.034 13.861 1.248 March 31.115 30.844 1.340 April 40.259 43.107 2.589 May 115.062 84.825 1.103 June 82.042 72.393 0.577 July 45.212 37.797 0.920 August 91.186 64.241 1.754 September 142.240 153.340 2.907 October 167.386 96.305 1.097 November 81.153 47.279 0.044 December 51.689 41.165 1.387 Source: Compiled from CRIBS-Jamaica National Data Base (CRIES—MSU). A uniform random number generator can be used to gen- erate random variates for all environmental variables. The values derived from the CRIBS-Jamaica national data base are presented in Tables 13, 14, and 15. 133 Table 13. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity- probability density function statistics for Jamaica—~Worthy Park for the years 1963 to 1982 Environmental Variable Minimum Maximum Temperaturea 20.44 30:78 Relative Humidityb 75.50 91.50 Solar RadiationC 4.00 7.60 Wind Velocityd 0.00 2.50 Source: Compiled from CRIBS-Jamaica National Data Base (CRIBS-MSU). aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. 134 Table 14. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity’ probability density function statistics for Jamaicar-Caymanas for the years 1963 to 1982 Environmental Variable Minimum Maximum Temperaturea 25.06 29.44 Relative Humidityb 52.00 87.00 Solar RadiationC 1.00 8.00 Wind Velocity d 0.00 2.50 Source: Compiled from CRIBS-Jamaica National Data Base (CRIBS-MSU) . aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. 135 Table 15. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity probability density function statistics for Jamaica--Monymusk for the years 1963 to 1982. Environmental Variable Minimum Maximum Temperaturea 23.89 29.33 Relative Humidityb 61.00 93.00 Solar RadiationC 5.20 9.60 Wind Velocityd 0.00 2.50 Source: Compiled from CRIBS—Jamaica National Data Base (CRIES-MSU). aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. Other stochastic inputs for sugarcane used to run the simulation model are: 1. Sowing or planting date for sugarcane. This is generated in the model by a uniformly distributed random number in the interval from 01/30/YY to 03/25/YY for Worthy Park, Caymanas, and Monymusk. Those values were derived from the sowing or planting scheme given by the RPPD in 136 Jamaica as a result of variations in sugarcane planting dates. 2. Number of days for each growth stage of the growing period for Worthy Park, Caymanas, and Monymusk was generated for the model by a uniformly distributed random-number gen- erator in the interval given by Doorenbos and Kassam (1979). 3. Fertilizer availability, in percentage, was used in the intervals from 36.92 to 100 percent, 40 to 100 percent, and 48468 to 80.69 percent for‘Worthy Park, Caymanas, and Monymusk, respectively, derived from fertilizer-availability data (Table 16, 17 and 18) and crop fertilizer requirements as given by Doorenbos and Kassam (1979). Other stochastic inputs for tobacco and sorghum used to run the simulation model are: 1. The sowing or planting date for tobacco and sor— ghum, derived from uniformly distributed random numbers in the interval from 01/15/YY to 03/15/YY. Those values were derived from the sowing or planting scheme given by the RPPD in Jamaica as a result of variations in tobacco and sorghum planting dates. 2. The number of days for each growth stage of the growing period. This was generated in the model from a uniformly distributed random-number generator in the inter- val provided by Doorenbos and Kassam (1979). 137 3. Fertilizer availability, in percentage. This was assumed to vary in the interval from 50 to 100 percent. 1k) data on fertilizer usage for tobacco and sorghum were avail- able. Table 16. Stochastic YIELD simulator: fertilizer usage for sugarcane, Jamaica-~Worthy Park, for the years 1963 to 1982 (kg/ha) Year ...............Ferti1izer Usage................... Nitrogen(N) Potassium(P) Phosphorus(K) 1963 72.68 24.51 98.98 1964 74.50 26.54 105.96 1965 99.20 28.47 139.54 1966 75.92 27.03 110.05 1967 68.54 24.13 102.65 1968 66.84 22.56 96.06 1969 66.84 22.56 96.06 1970 84.93 23.94 99.73 1971 70.02 17.93 78.20 1972 82.87 27.00 106.67 1973 85.20 275.88 106.77 1974 96.94 41.18 139.44 1975 121.15 40.29 108.54 1976 94.31 24.18 119.17 1977 93.85 30.95 117.57 1978 99.03 31.15 138.88 1979 80.54 a 108.84 1980 89.84 32.29 121.48 1981 92.05 38.03 127.39 1982 95.69 18.43 143.57 Source: Jamaica, Ministry of Agriculture, Rural and Physi- cal Planning Division. aMissing data. 138 Table 17. Stochastic YIELD simulator: fertilizer usage for sugarcane, Jamaica--Caymanas, for the years 1963 to 1976 (kg/ha) Year .................Fertilizer Usage................. Nitrogen(N) Potassium(P) Phosphorus 1963 95.11 94.37 128.22 1964 95.29 106.69 134.38 1965 94.71 69.05 163.88 1966 94.58 91.84 164.65 1967 94.61 89.19 162.01 1968 94.12 87.98 167.91 1969 94.12 87.98 168.44 1970 41.69 20.20 24.67 1971 43.15 20.04 22.67 1972 103.35 86.96 13.00 1973 46.28 19.36 18.55 1974 107.53 80.00 138.71 1975 101.20 81.78 129579 1976 110.42 76.03 137.80 Source: Jamaica, Ministry of Agriculture, Rural and Physi- cal Planning Division. 139 Table 18. Stochastic YIELD simulator: fertilizer usage for sugarcane, Jamaica--Monymusk, for the years 1963 to 1976 (kg/ha) ‘ Year .................Ferti1izer Usage................. Nitrogen(N) Potassium(P) Phosphorus(K) 1963 94.96 31.50 67.37 1964 93.91 38.23 70.69 1965 95.21 39.35 66.93 1966 95.35 42.14 66.57 1967 95.71 43.62 72.45 1968 98.79 51.23 71.75 1969 98.79 51.23 71.75 1970 99.19 53.69 88.17 1971 99.98 54.62 95.96 1972 99.97 54.33 110.77 1973 93.82 a 125.81 1974 94.34 a 45.13 1975 94.69 a a 1976 94.25 a a Source: Jamaica, Ministry of Agriculture, Rural and Physi- cal Planning Division. aMissing data. 5! l l' 1115111 3' 1| I . __ S' ] I' B 1| Five hundred simulation runs for sugarcane, tobacco, and sorghum were made for Worthy Park, Caymanas, and Monymusk. The resulting yield statistics were computed using the Statistical Analysis System (SAS, 1985) software package for microcomputer. Descriptive statistics for sugarcane potential yield, sugarcane irrigated yield, and sugarcane rain-fed yield are presented in Table 19. Table 19. Stochastic YIELD simulator: 140 sugarcane results, Jamaica--Worthy Park, Caymanas, and Monymusk (tons/ha) Worthy Statistics Park Caymanas Monymusk Potential Yield Mean 102.311 104.035 124.171 Std Dev 8.554 10.886 12.958 Skewness 0.005 0.141 0.068 Quantiles 100% 127.968 135.145 157.732 75% 107.931 111.700 133.765 50% 102.679 103.671 124.053 25% 96.181 95.978 115.044 0% 78.217 78.534 88.981 Irrigated Yield Mean 84.511 86.969 99.724 Std Dev 7.058 9.107 10.451 Skewness -0.004 0.130 0.063 Quantiles 100% 106.044 112:709 127.279 75% 89.136 93.505 107.156 50% 84.782 86.760 99.621 25% 79.390 80.274 92.384 0% 64.736 65.197 71.370 Rain-fed Yield Mean 75.984 59.519 68.574 Std Dev 6.846 6.099 6.846 Skewness -0.049 0.116 0.015 Quantiles 100% 97.210 77.518 87.363 75% 80.576 63.881 73.213 50% 76.206 59.444 68.795 25% 71.342 55.039 63.957 0% 57.001 43.088 49.317 Source: Compiled from stochastic YIELD simulator results. 141 For each simulation representing a crop year, random variates were generated by random-number generator for that particular simulation.’ The exogenous environmental inputs were generated and entered into the simulation model as a random number derived from the probability density func- tions. The planting date and duration of each stage were also random variables entered into the simulation run and were assumed to be uniformly distributed (equal chance of being selected, replacing sample) in a given interval. Fertilizer availability was also assumed to be uniformly distributed in an interval because data availability and precision were very poor. The model parameters were assumed to be known with certainty, implying the use of the variance partitioning approach discussed in Chapter IV. The stochastic YIELD simulation results for sugarcane in the Worthy Park, Caymanas, and Monymusk regions are provided in Table 19. Probability density function statis- tics for potential, irrigated, and rain-fed yield for those regions, such as mean, standard deviation, skewness, and quantile values, were computed and are included in the discussion. Once again, potential, irrigated, and rain-fed results were within the FAO yield results presented in Doorenbos and Kassam (1979). 142 The standard deviation was somewhat expected. In general, it was 8 to 10 percent of the mean value. Some" skewness to the right, a positive but not very significant skew factor, was noted in the results for Caymanas (Figure 38 a, b and c). No significant skew factor (all skew factors computed were close to zero) was computed for Worthy Park and Monymusk, which indicates a tendency to cluster around the mean value by the simulated yield. Quantiles also are presented in Table 19, which includes the minimum (0 per- cent) and the maximum (100 percent) values for the simulated yield. In general, the shape of the distribution function generated from a simulation model resembles the Gaussian or normal distribution (where the skew factor is zero). Con- sidering that, it can be said (SAS, 1985) that in 68 percent of the cases, the yield estimated is within one standard deviation (std) of the population mean; that is: P (mean+std <= yield <= mean+std) = 0.68 and that in 95 percent of the cases, the yield estimated is within two standard deviations of the population mean; that is: P (mean+2*std <= yield <= mean+2*std) = 0.95 143 and that in 99.7 percent of the cases, the yield estimated is within three standard deviations of the population mean; that is: P (mean+3*std <== yield <= mean+3*std) = 0.99 where: P(a <= X <= b) = p is the probability (p) of the random variable X being greater than or equal to a and less than or equal to b. The results for Worthy Park indicated that: For the irrigated sugarcane yield P (77.453 <= yield <= 91.569) = .68 P (70.395 <= yield <= 98.627) = .95 P (63.337 <= yield <= 105.685) = .997 For the rain-fed sugarcane yield P (69.138 <= yield <= 82.830) = .68 P (62.292 <= yield <= 89.676) = .95 P (55.446 <== yield <= 96.522) = .997 In a more general form, the probability of any yield value can be computed from: b P (a <= X <= b) = f f(x)dx a where: x - mean 2 1 —1/2( ---------- ) f(x) = ---------- e std , 1/2 ‘lstd*[2pi] or with the use of normal distribution tables. 144 Histograms are provided for the three simulation results--that is, potential, irrigated, and rain-fed yield—- for sugarcane, for Worthy Park, Caymanas, and Monymusk (Figures 37, 38 and 39, a, b, and c, respectively). The histograms were plotted using the results of the 500 simula— tion runs of the stochastic YIELD simulator, each with a different scenario of 500 years of crop-growth simulation. In the histogram plot, the horizontal axis shows, in tons per hectare, the variation in yield results. The vertical axis of the histogram plot represents the number of occurrences for each yield value or group of values of the simulation results and is called absolute frequency. Clearly, the information displayed in Figure 37 can be generated by a number of alternatives of interest to deci- sion makers and policy planners by making changes in con— trollable inputs such as planting date, fertilizer availability, evaporation reduction factor, soil salinity levels, and so on. In addition to giving numerical results such as means, standard deviation, skewness, and quantiles, histograms facilitate assessment of the degree of risk involved or associated with highly unfavorable outcomes. If the skew- ness is toward the right, the probability of obtaining large yield values is higher than if there is a negative skew factor, with skewness to the left. 'Also, histograms and/or § armmwmmm Q Q ', 102.311 ~ mean: 84.511 dam: 5.555 std. 6a).: 7.058 ,9 . 0.005 ,. skew: -o.004 g... g. I: so u. so 3 3 40 40 2 .. 3 . :0 20 $0 10 0 O 50 65 1200001141: 58054:”“11411 smkmm§§'m v n "r Jomaioau- St. fig; .3“ ($4110) Pcrk mg??? 'SCogheio)rine ed .1” N(odhvha)Park O in DMMVMMM N O a n i g Q h 50 *3 .. 3 .. IO 10 o M 06 4 I2 00 I. l 14! 1 Simulated Warsaw...) co - 3. Wk)” - rthy Perk Figure 37. Stochastic YIELD simulator: Jamaica-~Worthy Park. 145 § ammnwmmm Sugarcane absolute frequency histogram for potential, irrigated, and rain-fed yield. 146 Abeolute Frequency 100 100 armmmsoomm :WWMSNWM ’0 90 .0 mean: 104.055 .0 mean: 85.969 ltd. dew: 10.888 dd. dam: 9.107 70 skew: 0.142 3‘ 7° ekew: 0.130 C w a w w E a) D p o N a: ‘° L: EL 5 ‘° “EL" EIHLL a 8 ii: .4! " 30 i .1 a 30 uiujmk Fl p1 u hidi- i , 2° HIHKJN:: HHL m :fiEEJEJL 10 10 ' ' h ' ”a WWW FW"W“W‘"TWIW'W'T'"T"T'I 4050607080901m110120130140 4050507000901m101201 140 Simulated Po entiafil r'led t/ha)“ Simdated Irrigated". Yield t/ho) Jamaica - 81. Jamaica - t. no: a 100 :23 "um W for sun W m 90 ’0 "10011: $9.519 r std. dam: 6.099 6‘ 7° 1 EL skew: 0.116 3 W F “ 5° :1'11 § L ' 2 “’ mfiE: but". 1 3° I:H:I;:l 20 7" [diff ‘0 .1 HH! Hp i ll 0 W 40 50 BO 70 U 901W11012013014O SimulatedRa Romain. Yield 1 Jamaica - $1. crim: - gee-fines Figure 38. Stochastic YIELD simulator: Jamaica--Caymanas. Sugarcane absolute frequency histogram for potential, irrigated, and rain-fed yield. 147 m Gamma-venom». too ash-mushroom..- Q U n mam: 124.170 I) mean: 99.726 ltd. dam: 12.958 ltd. dew: 10.451 79 “I 0.068 m skew: 0.053 “ ’° “' '° :5 Eh 3 e 2 ‘° 5 i ‘° 155:: In ' E Iii-Ii n- H 3° CHH~H::HHF 3° huile:, in »: N 4: .. N P1 HM H a F‘ :‘JI’IfiCHJE m m 1:.ETLEL 0 O "-1 l D so 40 $ fl % n fl1glio1g1g1mlgllo g 40 50 00 x & 001E1101E1E1E15110 arcane S rcane Simulated Fs’ggential Yield (t/ha) Simulated lrrlu’ggted Yield (t/ha) Jamaica - ClaTgon - Monymusk Jamaica - Morgan — Monymusk a b 1” Dmmwmmm no u mean: 65.574 : std. dev.: 8.846 g 75 “H: skew: 0.015 3 an m: E Fa la. a uni e E5 ‘° .Fl :.1 3 . Iriih 10 w ‘0 1* :crl qthl 0 ”NDNNOO‘i 101 1 1 11 Sugarcane Simulated Ra nled Yield (t/ha) la'en n - Monymusk Jamaica - C (a Figure 39. Stochastic YIELD simulator: Jamaica--Monymusk. Sugarcane absolute frequency histogram for potential, irrigated, and rain-fed yield. 148 graphic representations are more readily understood and - evaluated by decision makers than are such abstract concepts as "standard deviation." Comparisons can be made between the potential, irri- gated, and rain-fed yields, and these comparisons can be used as input to a feasibility study on investment in irri- gation projects. Results of the stochastic simulator for tobacco in the Worthy Park, Caymanas, and Monymusk regions are provided in Table 20. Potential, irrigated, and rain-fed yield statis— tics such as means, standard deviation, skewness, and quan- tiles are shown. Yield for tobacco is quite high as compared to Jamaica's average tobacco yield. As discussed before, some model refinements may be justified, together with some more reliable data-collection techniques, which would lead to more precise data for inputs and also improve model results. Also, the introduction of randomness into the model's parameters might indicate which parameters are sensitive and aid in the data-collection plan. The quantile statistics indicate the minimum value and the maximum simulated value of tobacco yield for potential, irrigated, and rain-fed con- ditions. 149 Table 20. Stochastic YIELD simulator: tobacco results, Jamaica--Worthy Park, Caymanas, and Monymusk (tons/ha) Worthy Statistics Park Caymanas Monymusk Potential Yield Mean 4.178 3.974 4.577 Std Dev 0.370 0.378 0.410 Skewness 0.015 0.069 0.002 Quantiles 100% 5.133 4.927 5.579 75% 4.443 4.236 4.862 50% 4.189 3.978 4.598 25% 3.913 2.969 4.283 0% 3.245 1.958 3.512 Irrigated Yield Mean 3.625 3.325 3.966 Std Dev 0.326 0.317 0.356 Skewness 0.006 0.070 -0.008 Quantiles 100% 4.448 4.114 4.827 75% 3.868 3.537 4.216 50% 3.622 3.330 3.976 25% 3.395 3.080 3.712 0% 2.803 2.433 3.045 Rain-fed Yield Mean 2.483 2.188 2.375 Std Dev 0.279 0.243 0.259 Skewness 0.107 0.112 -0.043 Quantiles 100% 3.317 2.831 2.956 75% 2.679 2.361 2.580 50% 2.477 2.187 2.371 ~25% 2.278 2.006 2.173 0% 1.800 1.609 1.690 Source: Compiled from stochastic YIELD simulator results. 150 The skew factor was not significant for tobacco for any of the regions considered. For Caymanas, the skew factor for tobacco was 0.112, which indicates some but not very significant skewing to the right. The variation in tobacco yield was quite large if one considers the minimum value simulated, which is 1.609 t/ha for Caymanas rain—fed mini- mum, and the maximum value simulated, which is 5.579 t/ha for Monymusk potential yield. Histogram plots for tobacco are shown for Worthy Park, Caymanas, and Monymusk in Figures 40, 41, and 42, respec- tively, which illustrate the potential, irrigated, and rain- fed yield. The horizontal axes plot the simulated yield values in tons per hectare, and the vertical axes show the absolute frequency of each value. Histograms give decision makers and planners a good perspective on yield variations. The statistical results for sorghum, generated by the stochastic YIELD simulator for Worthy Park, Caymanas, and Monymusk, are presented in Table 21. Mean, standard devia- tion, skewness, and quantiles are included. Sorghum yield varied from a minimum value under rain—fed conditions in Caymanas, which was 2.346 t/ha, to a maximum value of poten- tial yield for Monymusk, which was 5.321 t/ha. The skewness factor was very Small for all regions and under all condi- tions. i———_ __--, 151 N m Flee-I, nae-en u an Ma eun- a mean: 4.17! 31d. dew: 0.370 04 skew: 0.015 04 8 K Abeohte Frequency - I O 0.0 05101.5 2.0 2.5 10 35 4.0 4.5 5.0 5.5 to 0.0 051.01.: 2.0 2.5 3.0 .15 4.0 4.5 50 5.5 0.0 _ Tobacco Tobacco Simulated Potential Yield (t/ha) Simulated lm' ated Yield (t/ha) S rme-WorthyPark Jamaica-St “gnome-Worm): Park Jamaica - 1. Cali») 8 3 2 8 l Abeolute Frequency .- O O 11.11111111111411111 0.0 0.5 1.9 1.5 2.0 2.5 3.0 35 4.0 4.5 5.0 5.5 ‘0 Tobacco Simulated Ralnfed Yielth/ha) 'w-RCatzniine- orthyPak c Figure 40. Stochastic YIELD simulator: Jamaica--Worthy Park. Tobacco absolute frequency histogram for potential, irrigated, and rain—fed yield. Absolute Frequency 152 ” unmade-summons...» es mmwwumm n 0 mean: 3.974 mean: 3.326 04 std. dev.: 0.378 04 eid. (101.: 0.317 skew: 0.069 skew: 0.070 40 h- 40 3 .2 32 £3: 10 is 0 0 W W 0.00.5 1.0 I310”3.03.54.04.55.05.5l0 “05181523253.0354.045505.55.0 _ Tobacco Tobacco Simulated Potential Yield t/ho) Simulated Irrigated Yield ¢{who} Jamaica - St. Catherine - Jamaica - St. - oymonae d ”- 1 m: m m w :00 M has a: § «J 2 i E i u. as- e -i ‘5 3 .3 1 10-1 l ..i W 0.0 05 1.0 1.5 2.0 2.5 3.0 as 4.0 4.6 5.0 u so Tobacco Simulated Rolnfeleeld 1/h0) Jamaica - St. henna — ones c Figure 41. Stochastic YIELD simulator: Jamaica--Caymanas. Tobacco absolute frequency histogram for potential, irrigated, and rain-fed yield. 153 .0 mmmuumm D:- mmmwumm mean: 4.577 ‘ mean: 3.957 04 std. dev.: 0.410 04-: std. dam: 0.356 skew: 0.002 drew: 41.008 10111-1101; 3 Abeoiute Frequency is- .4 0 ..1 W W 0.0051015202510354045505500 0.005 1.0 1.51.02.53.03540 4.55.05.55.11 Tobacco Tobacco ' Simulated Potential Yield (t/na) Simulated Irrigated Yield (t/na) Jamaica - Okra Son - Monymusk Jamaica - Clarendon - Monymusk (2 (b ”— .. m M m hr 000 M line ' mean: 2.375 04-: std. dev.: 0.259 . skew: 4.043 3 . c. . aa— . er 0‘: .. O n S 32- q 2 . re- 9.. W 0.0 0.5 1.0 1.5 2.0 2.5 5.0 5.5 4.0 4.5 5.0 5.5 so Tobacco Simuiated Related Yield (t/ho) Jamaica - Claengon - Monymusk (c Figure 42. Stochastic YIELD simulator: Jamaica--Monymusk. Tobacco absolute frequency histogram for potential, irrigated, and rain-fed yield. 154 Table 21. Stochastic YIELD simulator: sorghum results, Jamaica--Worthy Park, Caymanas, and Monymusk (tons/ha) ’ Worthy Statistics Park Caymanas Monymusk Potential Yield Mean. 4.632 4.494 5.321 Std Dev 0.430 0.399 0.454 Skewness 0.006 0.028 -0.065 Quantiles 100% 5.691 5.733 6.520 75% 4.957 4.757 5.626 50% 4.622 4.514 5.327 25% 4.319 4.228 5.009 0% 3.368 3.391 3.861 Irrigated Yield Mean 4.016 3.757 4.613 Std Dev 0.375 0.337 0.395 Skewness -0.011 0.013 -0.051 Quantiles 100% 4.967 4.805 5.648 75% 4.295 3.973 4.874 50% 4.010 3.773 4.615 25% 3.743 3.545 4.353 0% 2.909 2:762 3.363 Rain-fed Yield Mean 3.570 3.200 3.814 Std Dev 0.363 0.287 0.329 Skewness -0.059 0.034 0.007 Quantiles 100% 4.470 4.161 45736 75% 3.827 3.386 4.022 50% 3.575 3.211 3.818 25% 3.324 3.009 3.596 0% 2.533 2.346 2.759 Source: Compiled from stochastic YIELD simulator results. 155 Some skew factors had negative values. 'This is an indication that there was a tendency to skew toward the- left, which shows a high probability of low yield values. Observed yield values for Jamaica were very low, in terms of average, as compared to the stochastic yield results and the FAO yield values given in Doorenbos and Kassam (1979). Again, although the results were within the yield range given by Doorenbos and Kassam (1979), a suggestion for further research would be to introduce some variance into the model's parameters. This would permit evaluation of the parameters' sensitivity and also guide future model refine- ment as well as data-collection procedures. Histogram plots for sorghum in the Worthy Park region of Jamaica are presented in Figure 43. Three histograms are shown (Figure 43 a, b, and c): potential sorghum yield, irrigated sorghum yield, and rain-fed sorghum yield. The histograms for stochastic yields resulted from the simulator for the Caymanas region of Jamaica. Potential, irrigated, and rain-fed yield are shown in Figure 44 a, b, and c histograms, respectively. Histogram plots for the Monymusk region of Jamaica, derived from the stochastic YIELD simulator results (Figure 45 a, b, and c) represent the histogram for potential 156 a: an M m 1w“ We be I mean: 4.016 “_ std. dew: 0.375 .. drew: -0.012 E “‘ .. I O a i u‘ l i i ‘ C @- W W 0.00.51.0151025”30.045555510057075” 0.00.51.01529155015404550555005707500 So hum Sor bum Simulated Prague! Yield (t/ha) Simulated lrri 9ted Yield (Una) Jomaca-StCothenre-WorthyPark ca-StCauei-ine- (a) (b) Q q mmmwumm “-1 a. .. 0 4 Exp 3 1 is- ..1 WWW 0.00.51.015152550554.045505500057075“ Sor um smudge Renata Yield'St/ha) ca - St. Gather» - orthy Park c Figure 43. Stochastic YIELD simulator: Jamaica--Worthy Park. Sorghum absolute frequency histogram for potential, irrigated, and rain-fed yield. 157 anihmmqllhlmlwllluummI-n ..j alflhumqlntmusrllinubnhu fl ”.3 mean: 4.494 ‘ std. dev.: 0.399 _‘: g“ skew: 0.029 E“: . he an; O 3 2 10 is- 1 0 ol W W 090510152025305§£0¢55035l005707500 00051.0152025333540455055595.!101500 So hum Simulated Patergtiol Yield t/ho) Simulated lrrigotedm Yield (St/ha) Jamaica - St. ) once Jamaica - aymonoe a 08 .1 m m M er 000 M e... a; .i " mean: 3.200 g 04-: std. dew: 0.287 g . skew: 0.034 g ‘ IL 0.. . d E . 3 ”- 1 u-I .: l 030$1W0132£2£3035404£60550006707509 Jamaica - St. Simulated Ragga; Yield “1031/ )nas 0 Figure 44. Stochastic YIELD simulator: Jamaica--Caymanas. Sorghum absolute frequency histogram for potential, irrigated, and rain-fed yield. 158 n- .- . mmmwumm ”l Gamma-rename:- " «team 5.322 ‘ meam 4.613 ...: dd. don: 0.454 a; std. 4m 0.305 . skew: —0.065 .. skew: -0.051 § . § . ss- se- i . g . k d In. a 9 d 8 J 32— 32- 1 3 : ' 1 10- u- I 1 OJ oJ 0.005101510253015 t.04.55.0$.58.00.57.07.5l0 0.00.510 15201510150045 5.055690571575050 Sor hum Simulated lrriggted Yield (t/ha) Sorghum Simulated Potential YlOld (t/ha) n - Monymusk Jamaica - 0.723.». - Monymusk Jamaica - Gama n: an m M m sea stuns-a su- ‘ mean: 3.815 a _: sid. dew: 0.329 . skew: 0.007 Abealute Frequency 1L1 0.00.51.011 2.025 3.0 31 £0 4.6 5.05.5 6.00.6 7.0 7.5 0.0 smutd§7igiudmmn a n Jemima: Claren - Monymusk c Stochastic YIELD simulator: Jamaica--Monymusk. Sorghum absolute frequency histogram for potential, irrigated. and rain-fed yield. Figure 45. 159 sorghum yield, for irrigated sorghum yield, and for rainffed yield. Stccbasliz‘XIBID S' J l D . . E 11' __ W The CRIES-Dominican Republic (DR) national data base was used to estimate the probability density function sta- tistics for the environmental inputs. The Statistical Analysis System (SAS, 1985) software package was used for the computation and statistical analysis to identify the distri— bution parameters as inputs to the simulation model. Probability density function statistics for the envi— ronmental variable precipitation, for the Valdesia, Ocoa, and Azua Ocoa Watershed AEZs, used to run the simulation model, are provided in Tables 22, 23 and 24, respectively. Monthly mean precipitation values were used because the model is very sensitive to precipitation values. Also, the seasonal rainfall pattern is important for agricultural production systems, and large variation occurs with impor- tant effects on yield response. Variations in temperature, relative humidity, solar radiation, and wind velocity were not as strong as varia- tions in precipitation (Tables 25, 26, and 27). The availa- bility of data for solar radiation and wind velocity is very poor in comparison to the availability of data for precipi- tation. The model's sensitivity to solar radiation and wind 160 velocity inputs is not as significant as for precipitation, and there is no important seasonal variation. These con- siderations are taken into account, and the interval of variation in those environmental variables is taken on an annual basis and not on a monthly basis, as was done for the precipitation inputs. Table 22. Stochastic YIELD simulator: precipitation probability density function statistics for the Dominican Republic Ocoa Watershed's AEZ Valdesia, for the years 1970 to 1984 (mm/month) Month of the Standard Year Mean Deviation Skewness January 56.138 34.076 0.923 February 60.300 29.619 -0.384 March 58.100 50.932 1.950 April 90.946 71.414 1.442 May' . 219.708 207.483 2.006 June 233.085 165.796 0.368 July 142.561 83.638 0.733 August 199.531 96.655 0.653 September 186.354 66.737 0.084 October 192.577 76.736 0.581 November 84.169 45.014 1.076 December 68.162 43.388 1.141 Source: Compiled from CRIBS-Dominican Republic National Data Base (CRIEs-MSU). 161 Table 23. Stochastic YIELD simulator: precipitation probability density function statistics for the Dominican Republic Ocoa Watershed's AEZ Ocoa, for the years 1970 to 1982 Cmm/month) Month of the Standard Year Mean Deviation Skewness January 14.761 19.755 2.786 February 23.523 17.934 0.245 March 23.346 19.935 0.612 April 51.577 61.848 2.455 May 145.754 105.729 1.083 June 80.954 67.267 0.404 July 67.961 60.386 2.442 August 104.131 111.028 1.836 September 140.015 151.684 2.337 October 106.369 62.610 0.457 November 37.780 22.789 0.100 December 25.108 23.586 1.256 Source: Compiled from CRIBS-Dominican Republic National (CRIES—MSU). Data Base 162 Table 24. Stochastic YIELD simulator: precipitation probability density function statistics for the Dominican Republic Ocoa Watershed's AEZ Azua, for the years 1970 to 1984 (mm/month) Month of the Standard Year Mean Deviation Skewness January 11.707 28.495 3.615 February 7.313 10.912 1.613 March 16.320 26.873 2.946 April 14.880 15.229 1.554 May 78.926 76.947 0.867 June 55.193 71.649 1.395 July 28.400 35.197 1.191 August 55.087 37.823 0.111 September 84.507 124.418 3.056 October 88.247 62.215 0.762 November 19.547 26.655 1.754 December 17.647 38.000 3.215 Source: Compiled from CRIBS-Dominican Republic National Data Base (CRIEs-MSU). 163 Table 25. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity probability density function statistics for the Dominican Republic's Ocoa Watershed AEZ Valdesia, for the years 1970 to 1984 Environmental Variable Minimum Maximum Temperaturea 23.90 27.80 Relative Humidityb 64.90 34.70 Solar RadiationC 7.30 9.60 Wind Velocityd 0.70 5.50 Source: Compiled from CRIBS-Dominican Republic National Data Base (CRIBS-MSU). aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. 164 Table 26. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity. probability density function statistics for the Dominican Republicis Ocoa Watershed AEZ Ocoa for the years 1970 to 1984 Environmental Variable Minimum Maximum Temperaturea 21.40 24~80 Relative Humidityb 65.00 87.90 Solar RadiationC 5.00 9.00 Wind Velocityd 0.10 4.50 Source: Compiled from CRIES-Dominican.Republic National Data Base (CRIES-MSU). aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. 165 Table 27. Stochastic YIELD simulator: temperature, relative humidity, solar radiation, and wind velocity- probability density function statistics for the Dominican Republic's Ocoa Watershed AEZ Azua for the years 1970 to 1984 Environmental Variable Minimum Maximum Temperaturea 23.10 27.90 Relative Humidityb 64.10 87.50 Solar RadiationC 1.80 8.30 Wind Velocityd 0.00 3.50 Source: Compiled from CRIBS-Dominican Republic National Data Base (CRIES—MSU). aTemperature units are in degrees Celsius. bRelative humidity units are in percentage. CSolar radiation units are in hours/day. dWind velocity units are in meters/second, day/night wind ratio assumed to be 1. Crop parameters for the simulation of rice, potato, and fresh pea yields for the Ocoa Watershed's Valdesia, Ocoa, and Azua AEZs are given in Table 28. It was assumed that there were no variations from region to region because those regions are close together, and no detailed information on differences among them is available. Values were derived 166 Table 28. Deterministic YIELD simulator: Dominican Republic --Ocoa Watershed crop parameters--rice, potato, fresh pea for Valdesia, Ocoa, and Azua (Doorenbos & Kassam, 1979) ~ Crop Parameter Fresh Type Rice Potato Pea Root Size Variation (cm) 0.0 -100.0 0.0 -50.0 0.0 -150.0 Leaf Area Index (LAI) 3 4 3 Water Depletion Factor (p) Variationa- .300-.800 .175-.500 .225-«675 Production Rate (ym in kg/ha/ day)b .0-35.0 .o-2o.o .o-2o.0 Crop Coefficient (kc) C .95 -1.15 .40 —1.20 .40 -1.20 Yield Response Factor .99e+30) { y2[0] = 0.0; uiO] = 0.0; } else 203 { y2[01 = -0.5; u[0] = (3.0 / (xill - x[01)) * ((y[l] - yi01) / (xill - xiOi) - ypl); } for(i = l; i <= n-2; i++) { sig = (xiii - xii-1]) / (x[i+l] - xii-1]); p = sig * y2[i—l] + 2.0; y2[i] = (sig - 1.0) / p; uli] = (6.0 * ((y[i+l] - yiil) / (x[i+l] - xiii) - (yiii - yii-li) / (xiii - xii-11)) / (xii+l] - xii-1]) - sig * uii—li) / p; } if(ypn > .99e+30) { qn = 0.0; un = 0.0; } else { qn = -0.5; un = (3.0 /(x[n-1]- xin-2])) * (ypn - (yin-1] - yin-2]) / (xin-l] - xin-2])); } y2[n-1] = (un - qn * uin-21) / (qn * y2[n-2] + 1.0); for(i = n—2; 1 >= 0; i--) { y2[i] = y2[i] * y2[i+1] + uii]; } } /* ** x = array of x values ** y = array of y values of function values ** n = number of data points ** y2 = array of interpolated coefficients ** yp = interpolated results ** xp = point to be interpolated ** */ splint(x, y, y2, n, xp, yp) /* generated point yp */ float xi], yi], y2[], xp, *yp; int n; { int klo, khi, k; float h, a, b; klO 0; khi n-l; 204 while((khi - klo) > 1) i k = (khi+klo)./2; iffxik] > xp) khi = k; } else { klo = k; i } h = xikhi] - xiklo]; if(h == 0.0) { printf("bad xi] input. xil's must be distinct"); exit(); } a = (xikhil - xp)/’h; b = (xp - xikloi) / h; *yp = a*y[klo]+b*y[khi]+((a*a*a -a)* y2[klo] + (b * b * b - b) * y2[khi]) * (h * h ) / 6.0; Also used in the simulation model are the table look-up functions called Tablie and Tablex Manetsch (1984). Tablie for table interpolation and tablex for table extrapolation gave more speed to the simulation model (about five to eight times faster), with some increase in the interpolation error, but not to a point where the simulated results were compromised. Other types of errors, such as data input errors, parameter-estimation error, and so on, overshadow in most cases the error incurred by the choice of not using a more sophisticated polynomial interpolation algorithm, such as the time-consuming cubic spline interpolation function. 205 The tablex table look-up function was derived from the following relationship (Figure 50), where xp is the value to be interpolated. -Xe--:-¥i:l yi ' Yi-l Xi - xi_1 F(x) Y/—\ 7' T x ,- y.-. >4. P _l_ >9 - 2: O 5 >§>1 XE :KC >< Figure 50. Tablex: Algorithm for functional interpolation. 206 The C language code for the tablex table look-up func- tion is given as follows: /* ** 4* ** *4 ** 4* ** ** ** ** ** ** ** ** ** ** */ TABLEX This function does table look-up function through interpolation. Extrapolation is also possible for values of independent variables outside of the given vectors. Input: y = vector of dependent variables x = vector of independent variables n = vector dimension xp = independent variable to be interpolated Output: yp = valor interpolated (or extrapolated) tabCX(x' Y, n, xpl YP) float xi], yil, xp, *YP: } { int i; for(i = 1; i < n; i++) { if(xp <= xiii) { *yp = (xp - xii-1]) * (yiil - yii-ll) / (xiii - xii-1]) + yii—l]; return; } i i = n - 1; ' *yp = (xp - xii-1]) * (yiii - yii-ll) / (xiii - xii—1]) + yii-l]; return; APPENDIX B NUMERICAL INTEGRATION AND DIFFERENTIATION In Appendix A, some techniques for approximating a given function, in a tabular format, by a polynomial by interpolation were presented. Here, a major use of such approximating polynomials is considered: a complicated or a merely tabulated function by an approximating polynomial, so that the fundamental operations of calculus can be performed more easily, or can be performed at all. Techniques of numerical differentiation and integration are the main focus of almost all numerical calculus books (Hamming, 1962; Conte, 1980; Press, 1986. A common tech— nique used in most simulation models to numerically approxi- mate a differential and integral equation is called numeri- cal integration by Taylor series. The Taylor series about the point x = x0 has thefollowing form: (x - x )2 ) * W ) + -----9;-- * ”(x ) + n. y(x) = y0 + (x - xO y x0 2! y o where: Y' = f(x,y), and initial condition y(xo) = Yo 207 208 The function f(x,y) may be linear or nonlinear and is assumed to be differentiable with respect to both x and y. To find an approximate solution to the differential equation y' = f(x,y) y(a) = Yo over the interval la, b], the following steps must be followed: 1. Choose a step h = (b - a)/N 2, Set xn = a + n * h, for n = 0, 1, 2, ..., N 3. Generate approximations yn to y(xn) from the recursion Yn+1 = yn + h * Tk (xn, Yn)' for n = 0,1,...,N-1 where Tk (xn, Yn) is the kth term of the Taylor series. On setting k = l in the above algorithm, the Euler method and its local error is obtained by the formulas + h * f(xn, Yn)’ and the local error term Yn+1 = Yn h2 E = ---- * y"(@) 2 The Euler method is the simplest and most commonly used method in simulation work. Several other techniques can be used with the Euler method to improve the interpolated value at a desired precision level. For instance, the predictor- corrector method is commonly used in simulation (Manetsch, 209 1984) and can greatly improve the results, when needed, of a numerical integration and differentiation numerical approxi- mation. Again, trade off between precision and time is a prob- lem to be approached by the systems analyst and the model user. The predictor-corrector method is time consuming, and a number of interactions must be computed to achieve the desired approximation level. APPENDIX C THE INVERSE TRANSFORMATION METHOD If one wants to generate random variate xi's from a particular population whose density function is given by f(x), the cumulative distribution function F(x) must be obtained (Figure 51). F(x) = rnd A. LO..-___ ______ Phdo ______ X. XX Figure 51. Cumulative distribution function. 210 211 Because F(x) is defined over the range 0 to l, uni-- formly distributed random numbers (rnd) can be generated and F(x) = rnd. It is clear that x is uniquely determined by rnd = F(x). It follows, therefore, that for any particular value 0f rnd, say, rndo, that is generated, it is possible to find the value of x, in this case x0, corresponding to rndo by the inverse function of F if it is known. That 15: X0 = F-l (r1160)! Where 15‘“1 (rnd) is the inverse transformation or mapping of rnd on the unit interval into the domain of x. This method can be summarized.mathematically by saying that if one generated uniform random numbers corresponding to a given F(x), x rnd = F(x) = .ff(t)dt -—00 then, P(X<=x) = F(x) = Pir<=F(x)] = PiF'lirnd)<=x], and consequently F’1(rnd) is a variate that has f(x) as its probability density function. The inversion method was applied for the uniform, gamma, normal, and triangular distributions and was used in the simulation model. The C language code that enables the generation of those random variates is given below: /* *‘k *‘k *1: *‘I: *‘k ** *1: ** *‘k *‘k ** ** ** ** ** */ 212 uniform variates lower limit of the interval a < x < b Input: a = b = upper limit of the interval a < x < b seed= seed for the random number generator Output: x = uniform random variate Functions called from here: none Main function must have: #include #include <1 im its.h> uniform(a, b, seed, x) float a, b, *X; short seedi]; { double rn, erand480; rn = erand48(seed); *x = (float)(a + (b - a) * rn); } /* ** ** gamma variates ** ** a = k parameter of the gamma distribution a > 0 0 e s ** b = alpha parameter of the gamma d1str1but10n b > 0 *‘k ** Inputzmi = mean value ** var = variance value ** seed= seed for the random number generator *1: ** mean = a / b ** variance = a / b**2 ** . ** Output: x = gamma random var1able ** ** Functions called from here: none ** 213 ** Main function must have: #include #include <1imits.h> */ gamma(mi, var, seed, x) float mi, var, *x; short seedi]; { float a, b; int i; double rn, erand48(), lg, log(), tr; a = mi*mi/var+0.5; if(a < 1.0)a = 1.0; b = mi / var; tr = 1.0; for(i = 0; i < (int)a; i++) { rn = erand48(seed); tr = tr * rn; } lg = -log(tr); *x = lg / b; l /* ** ** normal variate ** ** Input: mi = mean value ** sigma = standard deviation ** seed = seed for the random number generator *1: ** Output: x = normal random variable ** ** Functions called from here: none ** ** Main function must have: #include ** #include (limits.h> ** ** */ normal(mi, sigma, seed, x) float mi, sigma, *x; short seedi3]; { int 1; double rn, erand48(), sum; /* *st *7: ** *1: *5:- ** ** *1: ** ** ** *1: ** *1: *4: ** *1: */ tr fl sh 214 sum = 0.0; for(i = 0; i < 12; i++) { rn = erand48(seed); sum = sum + rn; } *x = sigma * (sum — 6.0) + mi; } triangular distribution Input- a = minimum value b = maximum value c = mode seed= seed for the random number generator Output: x = triangular random number Functions called from here: none Main function must have: #include #include <1imits.h> #include #include iang(a, b,cn seed, x) oat a, b, c, *X; ort seedI]; { double sqrt(), erand48(), rn; rn = erand48(seed); if(rn < (double)((c - a) / (b - a))) { *x = a + sqrt(rn * (c - a) * (b — a)); } else i *x = b - sqrt((l - rn) * (b - c) * (b - a)); LIST OF REFERENCES LIST OF REFERENCES Adams, J. 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