.Mmhsg’m a. u) l UDEVQVo’i‘y w PLACE IN RETURN BOX to remove this checkout from your teeord. TO AVOID FINES Mum on or before date due. [ -|_________ __][__ ___JL_ ,4; ”T m NJ. JA '6 p»! Z P“ L I MSU Is An Affirmative Action/Equal Opponunity Institukion MODELING YIELD OF DALBERGIA SISSOO (SHISHAM) FOR IRRIGATBD PLANTATIONB OF THE PUNJAB, PAKISTAN BY Muhammad Jahangir Ghauri A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Forestry 1991 ABSTRACT MODELING YIELD OF DALBERGIA SISSOO (SHISHAM) FOR IRRIGATED PLANTATIONS OF THE PUNJAB, PAKISTAN by MUHAMMAD JAHANGIR GHAURI A yield prediction model for shisham, the main species of irrigated plantations of the Punjab, was developed using data from stand-level permanent sample plots. The plantations were tested for their similarity on the basis of stand volume and basal area using age as a covariate. All contrasts for plantation groups were nonsignificant, therefore, all plantations were pooled for model building. Stand age, basal area per hectare, mean stand dbh, mean stand height, log10 (number of trees per hectare), and indices of irrigation and site quality were used to predict mean stand volume per hectare. Models were developed using all possible regression techniques as well as backward elimination. A subset of seven models were selected using the criteria of adjusted R2, error mean square (MSE) and Mallow's CI, statistic. The final model developed for the Punjab zone included stand age (A), stand height (H), stand basal area (B) and site quality index (8). The model was evaluated for outliers and influential observations using residual plots and Cook's D statistics. Multicollinearity among independent variables was examined using variance inflation factors (VIP) and condition numbers (CN). The regression coefficients were recalculated after deleting 19 outliers from a total of 313 observations. The final yield model was of the form: Y = -768.8 -15.9(A) +24.7(H) +16.2(B) +0.00004(S)2 The regression model was compared with the Schumacher, Buckman, Clutter and Chapman-Richards models using the criteria of adjusted R2, MSE, CP, (VIF),,m and (cmm. The Schumacher model produced better measures of goodness-of-fit. Predicted yield was compared with existing provisional yield tables of shisham for the irrigated.plantations of the Punjab. DEDICATED to Ayatullah Ruhollah Khomeini, the great Islamic scholar and to the innocent Iraqi civilians killed during the gulf war iv ACKNOWLEDGEMENTS I am thankful to God Almighty, the most Merciful, for giving' me ‘the strength. to. complete ‘this ‘tiring' task. of acquiring worldly knowledge. My thanks are to my professor, Dr. Carl W. Ramm, for his guidance and continuous patience over the progress of this dissertation. I also extend my gratitude to the members of the graduate committee, Drs. Charles E. Cress, Donald I. Dickmann and Michael A. Gold, for their helpful comments, suggestions and respect throughout my program. I gratefully thank Dr. Karen W. Potter-Witter and Mr. Rubens Humpherys for their support during the sabbatical period of Dr. Ramm. I wish to acknowledge the time and help of Raja Walayat Hussain, Pakistan Forest Institute who acted as co-supervisor during data collection in Pakistan. Thanks are due to Mr. Muhammad Hafeez, Director, Punjab Forestry Research Institute for his special arrangements in collecting irrigation data used in the study. Financial sponsorship of this study program by FAO of the United Nations through Pakistan Agricultural Research Council is acknowledged. My profound, hearty and sincere gratitude goes to my wife, Amatus Sami, and children who courageously struggled with me all the way , and other family members for their understanding and patience during this academic ‘exile'. May God Almighty reward them in this life and in the Hereafter. vi TABLE OF CONTENTS List of Tables ...... ..... .................. ...... ... ix List of Figures ...... ................... ....... ...... xi CHAPTER 1. INTRODUCTION 1.1 Forestry in Pakistan - an overview ........ ..... . 1 1.2 Environmental factors ..................... ...... 3 1.3 Scope of the problem and study objectives ....... 5 2. LITERATURE REVIEW 2.1 Overview of growth and yield modeling .......... 9 2.2 Stand-level models of yield .................... 11 2.3 Nonlinear regression models . ........... . ...... . 18 3. DESCRIPTION OF STUDY AREA 3.1 Irrigated plantations .............. ........... . 23 3.2 Sample plots and study plantations ........... .. 26 Sampling ........ ....... ..... . ................ 27 Description of sample plantations ............ 29 3.3 Description of Dalbergia sissoo ........ ........ 40 Botanical description ........................ 40 Distribution ................................. 41 Nursery techniques ........................... 42 Planting method .............................. 43 Cultural operations .......................... 44 Injuries to Shisham plantations .............. 47 uses 0...........OOOOOOOOOOOOOOOOOO0.0.0.0.... 49 4. METHODS AND MATERIALS Data 0.00............OOOOOOOOOOOOIOOO...-0...... 50 4 1 4 2 Analysis for plantations grouping ............. . 53 4.3 Univariate analysis ...................... ...... 58 4 4 Bivariate relationships ........................ 61 4 5 Empirical model ........ ......... . .............. 63 Variable selection ........................... 64 All possible regressions ..................... 67 Backward elimination .................... ..... 73 vii 5. RESULTS AND DISCUSSION 5.1M0del evaluation ......OOIOOOOOOOOOOO0.0.0.0.... 76 Residual analysis ..................... ..... .. 76 Influential observations ......... ............ 80 Multicollinearity ............................ 84 5.2 Comparison with conceptual models .............. 87 5.3 Model validation ............................... 92 5. SUMMARY AND CONCLUSION ............................ 99 Future recommendations ........................ 101 APPENDICES APPENDIX A-1: Plantation-wise univariate statistics ......OOOOOOOOOOOOOO ..... 103 APPENDIX A-2: Age-wise univariate statistics ..... 105 APPENDIX B: All possible regressions ............. 111 LITERATURE CITED .0.........OOOOOOOOOOOOOOOOOO0...... 117 viii Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table 10: 11: 12: 13: 14: 15: LIST OF TABLES Land use pattern in Pakistan .. ............. 2 State-owned forest land by forest type in Pakistan ......... ...... ...... ........... 5 Regression statistics for shisham volume tables ..................................... 17 Forest area under the control of Provincial Forest Departments OOOOOOOOOOOOOOO0.00...... 25 Sample versus population plantations by category O....0......OOOOOOOOOOOOOOOOOOIO... 28 Distribution of plots by plantation ........ 51 Distribution of plots by age class ......... 51 Analysis of covariance for the dependent variable, volume per ha, with age as covariate .................................. 55 Analysis of covariance for the independent variable, basal area per ha, with age as covariate .............. ............ . ....... 57 Analysis of normality ...................... 58 Analysis of normality for site quality indices ........ .................... 59 Analysis of normality for irrigation indices indices 61 Analysis of normality for the transformed variables of number of trees per ha ........ 63 Summary of backward elimination procedure .. 75 Regression statistics for subset models using all observations ..................... 77 ix Table Table Table Table Table 16: 17: 18: 19: 20: Regression statistics for subset models Without outliers ..........OOOOOOOOOOOOOOOOO Multicollinearity diagnosis ......... ....... Regression statistics and multicollinearity diagnosis for conceptual models ............ Regression statistics for the Chapman- RiChardS madel 0.........OOOOOOOOOOOOOO0.00. Comparison of observed and predicted yield by site index class ........................ 82 87 88 91 98 Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Forest types of Pakistan ................... Layout of irrigation in plantation compartment ......................... ...... . Adjusted szersus number of parameters in themOdel ............OOOOOOOOOOOOO0.0... MSE versus number of parameters in the madel 00............OOOOOOOOIOOOOO0.0... C versus number of parameters in themedel 000............OOOOOOOOOOOOO..0... Studentized residual versus predicted valume 00.0.0.0.........OOOOOOOOOOCOOOO0.... Observed and predicted yields for site index I ............................... Observed and predicted yields for Site index II ..........OOOOOOOOOOOOOOOOO... Observed and predicted yields for site index III ........ ............ .......... xi 45 69 71 72 79 95 96 CHAPTER 1 INTRODUCTION 1.1 FORESTRY IN PAKISTAN - AN OVERVIEW Pakistan occupies an area of 87.978 million hectares (ha) and is one of the most populous countries in the world. According to the 1981 census, its total population was 86.8 million, giving a density of 106 people per square kilometer. With an annual growth rate of 2.8%, Pakistan's population is expected to reach 112 million in 1990. Being an agricultural country, Pakistan has only 5.4 percent of its land under forest use. Of about 4.737 million ha in forests, the productive forests occupy only 1.477 million ha or 1.7 percent of the total area of the country. This scarcity of forest resources is mainly because about 70 percent of the country is arid and semi-arid. The land utilization pattern given in Table 1 reveals the shortage of forest areas in the country. The gap between supply and demand of wood and other forest products is very pronounced. The factors responsible for this gap are the high population growth rate in the country and the steady development of forest industries. Timber consumption during 1986-87 was estimated to be 2.5 million cubic meters (m3) for the estimated population of 104 2 million (Amjad and Khan, 1988). Table 1. Land use pattern in Pakistan Land-use category Million Percent of ha total area 1. cultivation 20.54 23.4 2. Cultivatable waste 11.03 12.5 3. Not available for 51.67 58.7 cultivation (mountains, deserts, rivers, urban) 4. Forests 4.74 5.4 About 15 percent of the wood consumption was supplied by government (state controlled) forests, 41 percent by farmlands and 44 percent was imported. In the case of firewood, the consumption for 1986-87 was estimated to be 20.88 million mt. About 10 percent of this demand was fulfilled by the government owned forests and the rest was supplied by private farmlands and wastelands. Imported wood and wood-products, thus, are a regular feature in the country. The 1986-87 value for imported wood and wood products was 2,399 million rupees ( Rs.), or U.S. $ 139 million, which included round and sawn timber (6 percent), wood-based pulp and paper (90 percent) and wood manufactures and miscellaneous items (4 percent). Pakistan, however, exports a small quantity of forest products, mainly sporting goods. These include hockey sticks, cricket bats, and tennis, squash and badminton 3 rackets. In 1986-87 the total wood export value was Rs. 383 million, out of which about 84 percent were sporting goods (Amjad and Khan, 1988). 1.2 ENVIRONMENTAL FACTORS 1.2.1 TOPOGRAPHY Pakistan lies between 24 degrees and 37 degrees north latitude and 61 degrees and 75 degrees east longitude. It has a continuous mountainous tract in the northwest and southwest. Pakistan is generally arid and semi-arid with small patches of humid and sub-humid areas. The soil is mainly stratified alluvial deposits formed by the Indus river and its tributaries in arid and semi-arid zones. The alluvial deposits contain mixed minerals derived from Himalayan rocks, usually containing calcareous silt loams. In humid and sub-humid tracts, the soils are generally light sandy brown and fertile (Champion et a1. 1972). The water-table is very high in most part of the Indus plains, usually within 5 meters. Salinity and water logging have become major factors contributing to the decline in agriculture as well as forestry. 1.2.2 CLIMATE Although generally dry, Pakistan has an annual rainfall that varies from 50 millimeters (mm) in the west (Baluchistan province) to about 1,500 mm in the north (Himalayan range). About 75 percent of the country is very dry, with less than 4 250 mm annual rainfall. The most important source of rainfall is the monsoon season, July through September, which brings 70 to 80 percent of the annual rainfall. Monsoon rains mostly occur in the eastern, north-eastern and southern areas. The rains during winter occur in the south-western, west and north—western regions of the country. Summers prevail from May to August and are very hot in the plains, with June as the hottest month. The temperature is normally more than 40° C during summer in the plains and drops as low as -12° C in the high mountainous regions during winter. The mean minimum temperature is about 5° to 7° C‘during the month of January in most of the Indus plains. Humidity is generally moderate in most of the country, August being the most humid month. Desiccating and hot winds are common in the plains during April to September; they can blow at a speed of over 80 kilometers (km) per hour. Dust storms are also a regular feature in the desert parts of the country. Climatic and edaphic factors control the distribution of forests. Most of the forest areas contain naturally regenerated conifers and broad—leaved species, while some man- made irrigated.plantations are present. The major forest.types found in the country, from the Himalayas down to the delta of the Indus River near the Arabian Sea (Champion et a1. 1965), are summarized below. Coniferous forests include sub-alpine forests, dry temperate forests, Himalayan moist temperate forests, and sub- 5 tropical pine forests. Scrub forests include dry subtropical broad-leaved forests and dry tropical thorn forests. A small area in north-western region is under mazri crop. Irrigated plantations include compact and linear plantations. The final two forest types are riverain forests and mangrove forests. Figure 1 shows the distribution of above forest types in the country. Table 2 shows a break down of the total area under the control of provincial forest departments by forest type. Table 2. State-owned forest land by forest type in Pakistan Forest Type Area Percent of (1000 ha) total forest Coniferous 1959 41.4 Irrigated plantations 392 8.3 Riverain 296 6.2 Scrub 1702 35.9 Coastal 347 7.3 Mazri lands 24 0.5 Linear Plantations 17 0.4 Total: 4737 100.0 1.3 SCOPE OF THE PROBLEM AND STUDY OBJECTIVES The poor forestry resources and scarcity of timber and fuelwood in Pakistan demand immediate attention. Putting more area under forest management will be quite difficult, if not impossible, because of the increasing population pressure and, thus, the priority for agricultural crops. Increasing unit yield from the existing forest resources is a more practical 1119/; ’ / V/ "'I’ /.‘;.,,Q * 77/724444 ”' 'o In. fi/‘l/ 17mm ~ / [/07 / / , x %? JIWI‘" 7/ / % r ' 111 Figure 1. Forest types of Pakistan 7 alternative. Increasing the unit yield of forest products is beyond the scope of this study, but more accurate growth and yield projections will help forest managers have more sound quantitative estimates of the resource. Such studies will provide the basis for comprehensive and sound future planning of the forest resources of the country. This study focuses on improving yield prediction for making sound management decisions and future policy of the forests. Irrigated forest plantations are a major source of quality timber and fuelwood in Pakistan- These plantations are situated on lands where irrigation water is available -- mostly in the plains. Approximately 148,000 ha of these plantations are in the Punjab. Shisham (Dalbergia sissoo) has been the principal species since the establishment of these plantations. In the past, no stand level growth and yield models have been developed for shisham, nor for any of the other commercially important species such as mulberry, babul, and eucalyptus. Although some authors have created stand tables for irrigated plantations (Hussain and Abbas, 1974; Hussain and Abbas, 1980; Hussain and Glennie, 1978; Hussain, 1983), these tables are no substitute for accurate growth and yield models. In the above studies, the measurement unit was the individual trees and not the stand. Age of the forest crop is an important variable for the future prediction of forest yield, but it was not considered as an independent variable in these 8 studies. Moreover irrigation water, which is a very important commodity in the whole system of irrigated plantations, was also not used in the above studies. It is, therefore, considered important to develop yield models for irrigated plantations of‘economic importance. Such models should include important variables like age, quantity of irrigation water, and site quality to obtain sound information for decision- making and prediction. This study explores alternative stand- level models of yield for the irrigated shisham plantations of the Punjab, Pakistan. The specific focus will be on the following objectives: i. Development of a yield model using multiple linear regression techniques--empirical modeling. ii. Application of commonly used forestry yield models developed on the biological growth concept. iii. Selection of the best model(s). As a precursor to model building, an hypothesis of equal yield (volume per hectare) from each plantation or groups of plantations was tested. This was done in order to decide whether to develop a pooled model for all plantations or separate models for each group of plantations. An analysis of covariance with stand age as covariate was performed before starting the model building procedures. CHAPTER 2 LITERATURE REVIEW 2.1 OVERVIEW OF GROWTH AND YIELD MODELING 2.1.1 MODELING THEORY A model is a physical, pictorial or symbolic (mathematical) representation of a system, a decision process, an organization or an object. Models are used for the purpose of experimentation or prediction (Krumbien and Graybill, 1965). The commonly used models in forestry are mathematical in nature. Mathematical models are used to study the operation of system or process in mathematical form, e.g. simulation models (Brabb 1968). These models act as predictive models while indicating the consequences of alternative strategies. They are used to predict the outcomes of decisions and to answer "what if" types of questions (Johnson et a1. 1974). One or more mathematical terms or equations may be used to improve their fit, although the greater the number of equations to work with, the more complex the mathematical analysis and interpretation becomes. 2.1.2 GROWTH AND YIELD Growth can be defined as an increase over time in tree size, weight or volume, whereas yield is the amount of volume 10 per unit area available for harvest.at a specified.time. Thus, yield can be regarded as the summation of the annual increments of growth. The changes in stand volumes can be predicted through direct as well as indirect methods. Direct methods, such as stand-table projection, require estimates of mortality and past growth. When direct observations are not possible or feasible, forest managers have to depend on indirect methods of predicting stand dynamics. Growth, mortality and other related quantities of a stand are inferred from the study of other stands. These inferences are possible through the use of tables, equations, or computer simulation models. Techniques for forecasting stand dynamics are collectively referred to as growth and yield modeling. Forest yield models have been classified by Munro (1974) into two primary types - individual-tree and stand-level models. In individual-tree models, projections are made on the basis of individual trees characteristics and then aggregated for stand summaries. In stand-level models, stand characteristics such as age, site and density are used to predict stand volume. Avery and Burkhart (1983) listed the factors closely related to growth and yield of forest stands as the point in time of stand development, site quality, and the degree to which the site is occupied. These factors can be expressed through stand age, site index and stand density for even-aged stands. 11 Titus and Morton (1985) have defined a forest yield model as a quantitative description of the physical system in which trees exist, consisting of one or more mathematical relationships. The authors indicate that all forest yield models depict the development of stands over time. They use a normal yield tables as a familiar example of a yield model which can simply be expressed by the relation: Volume = fn (age, site, species) The more complex models deal with these same components but in more detail. 2.2 STAND-LEVEL MODELS OF YIELD Researchers in forestry have investigated relationships between variables such as volume, diameter, height, basal area, and site index fOr individual trees as well as for entire stands. Linear (multiple and polynomial) as well as nonlinear equations have been used for this purpose. MacKinney and Chaiken (1939) were the first to apply multiple linear regression to yield estimation, taking stand density into account. Their prediction model for loblolly pine was: log (Y) = bo + bl/A + b2 S + b3 log(SDI) +b4 C where log (Y) = logarithm (base 10) of yield, A = stand age, S=site index, SDI=stand density index, C = composition index, and b1, b2, b3, b4 are regression coefficients. 12 Schumacher (1939) developed a yield model which conformed to reasonable biological growth principles for even- aged stands. This function 'was expressed as the linear relationship: ln(V) = b0 + ms + b21n(B) + b3(1/A) where ln(V) is the natural logarithm of volume per unit area, the bfs are regression coefficients, 8 is site index, B is average stand basal area per acre, and A is stand age. Buckman (1962) and Clutter (1963) recognized that growth should be the derivative of a yield function with respect to age. They used growth functions for remeasured plots in even- aged stands, Buckman for red pine in Minnesota and Clutter for loblolly pine in Georgia, Virginia and South Carolina. Buckman used a linear equation to estimate basal area growth: dB/dA = b0 + blB + b2B2 + b3A +1941:2 + bSS + 16613 (A). where dB/dA is basal area growth, B is average stand basal area per acre, A is stand age, and S is the site index. Buckman's equation was based on the assumption that for a given age and site, an optimum‘density exists where basal area growth is maximum. Clutter (1963) developed the idea of compatible growth and yield functions, and noted that errors might arise if 13 compatible functions were not used. He developed compatible growth and yield models for loblolly pine. Initially, he used a modified Schumacher equation to predict yield from age, site, and basal area: ln(V) = b0 + bls + b21n(B) + b3(1/A) where ln(V) is the natural logarithm of volume per unit area, the b's are regression coefficients, and S, B, and A are as defined earlier. Clutter tested 3 possible two-variable interactions in the model. An analysis of variance showed that the three independent variables used in original model were highly significant, and that none of the interaction variables reduced the residual sum of squares significantly. Moser and Hall (1969) found that yield functions used by Buckman (1962) as well as Clutter (1963) were not appropriate for uneven-aged stands. They used an allometric relation between basal area and stand volume to quantify growth rate and subsequently to obtain a stand volume function for uneven- aged stand of mixed northern hardwood in Washington County, Wisconsin. Their investigation showed that the allometric function (V =boB“) adequately related stand volume (V) to basal area per acre (B). Since this equation depicts stand volume, the survivor growth rate would be represented by: dV/dt = (b1 v 8*) (dB/dt) 14 The authors gave a methodology for time dependent yield functions from the integration of rate equations which do not have age or time as an independent variable. Sullivan and Clutter (1972) developed simultaneous growth and.yield.models for even-aged loblolly pine stands located in the Hitchiti Camp, Santee, and Westvaco Experimental Forests of the U.S.D.A. Forest Service Southern Forest Experiment Station. They used Schumacher's equation: ln(V) = bO + bl S + b2 A‘1 + b3 ln(B) where V, B, A and S are as explained earlier. The authors enumerated the features of this model as: "i. Mathematical form of the variate implies relationships which agree with our biological concepts of even-aged stand development (Schumacher 1939). ii. The use of ln(V) as the dependent variable rather than V will generally be more compatible with the statistical assumptions customarily made in regression analysis (linearity, normality, additivity and homogeneity of variance). iii. The use of ln(V) as the dependent variate is a convenient way to express mathematically the interaction of the independent variables in their 15 effect on V." Smith (1983) developed basal area growth and yield models for red pine plantations located throughout southern Ontario based on Buckman's (1962) and Clutter's (1963) equations. His model which included younger stands than earlier studies resulted from applying unweighted least squares stepwise regression to Buckman's and Clutter's equations: dB/dA = 190+ blB+ 10,132+ b3B3+ b4A+ b5A2+ b6A3+ b73*A+ b8A*S+ logs and dB/dA + B ln(B)A“ = com-1+ clams-1+ c23A°+ c3BSA'3 The reduced models obtained were: i. Model 1: dB/dA = 16,134+ b25134+ b3A3B“+ b4A'2B"+ bsA‘B" ii. Model 2: dB/dA = b0+ 10,13 + b2A + b3BA + b4AS iii. Model 3: dB/dA = -B 1n(B)A‘1+ blBA"+ szA'3 Smith concluded that model 2 gave the best fit for the data used, and could be used for both growth and yield estimates. Nautiyal and Couto (1984) in their study of timber production functions discussed various algebraic forms to 16 represent biological functions of growth and yield of eucalyptus species in Brazil. The authors used seven alternative specifications for representing the annual growth of forest crops: polynomial, reciprocal, square-root, semi- logarithmic, logarithmic-reciprocal, transcendental and double-logarithmic models. The polynomial model was considered the best fit. Borders and. Bailey (1986) expanded. Clutter's (1963) equations to develop a compatible system of linear growth and yield equations for slash pine. The data covered a broad geographical area from lower coastal plain of southern Virginia to northern Florida. Hussain and Abbas (1974) prepared volume tables for shisham of irrigated plantations in Pakistan. They used seven simple functional relationships combining diameter and height variables using tree level data (Table 3). Of the seven equations, equation number 5 was selected for the estimation of total volume for trees with 53 to 89 centimeters (cm) diameter at breast height and equation number 7 was selected for the trees with dbh 50 cm or less on the basis of better indices of goodness-of-fit. 17 Table 3. Regression statistics for shisham volume tables Model R2 SEE 1. V==b0+leH+b2D2H 0.9567 8.82 2. V=b0+le2+b2D3 0.9571 8.78 3. V=b0+b1D2H+bZDH2 0.9571 8.79 4. V=bo+b1D2+b2H+b3DzH 0.9571 8.79 5. V=bo+b1D+b2H+b3D2H 0.9571 8.79 6. V=D2(bo+b1/H+b2/H2) 0.6882 3.34 7. logV=b0+bllogD+bzlogH 0.9888 0.07 V = volumewnfi), D = dbh(cm) and H = height of tree(m). Cheema and Abbas (1978) used three linear regression equations to prepare local metric volume tables of shisham in the Punjab: 1.1ogV=b0+b,logH+bzlogD 2.v=loo+lo,0+162H+16302 3. V b0+le2H+b2DH2 where V is the volume(m3), H is the height(m) and D is the diameter at breast height (cm) of the tree. The authors used equations 1 and 2 for the estimation of total volume of trees 18 with dbh less than 53 cm and 53 cm and above, respectively. Equation 3 was used for the estimation of timber volume. 2.3 NONLINEAR REGRESSION MODELS 2.3.1 OVERVIEW Nonlinear growth.models have been used frequently for the estimation and prediction of height, site index, basal area and volume of individual trees as well as of stands. Payandeh (1983) pointed out that many biological processes are basically nonlinear in nature. He illustrated several nonlinear models which can be used in forestry research and indicated that, if chosen properly, nonlinear models usually fit most biological growth data as well or better than the more complex linear model. Tesch et al. (1983) used nonlinear regression techniques to predict the yield of even-aged Ponderosa pine plantations on the lower Blackfoot River of Montana. Three separate nonlinear models were used to predict individual tree diameter, individual tree volume and the number of trees per hectare as a function of stand age, initial stand density and site index. The dynamics of second-growth managed stands were simulated using individual, dominant tree data from unmanaged stands. The results gave useful first-approximation yield estimates for intensively managed ponderosa pine plantations up to 80 years old. 19 Martin. and Bk (1984) used. nonlinear regression for fitting several models for survivor tree growth. Permanent plots from red pine plantations in Wisconsin were used to fit simple empirical and semi-empirical or constrained models. The authors concluded that the empirical model was the most accurate for diameter growth projections in managed plantations within the range of data. The semi-empirical model, however, was likely to be more accurate for unmanaged stands and for extrapolating beyond the range of data. Gertner (1984) used a sequential Bayesian procedure to localize a nonlinear diameter increment model for Douglas fir in the Western Oregon region. The model predicted regional annual individual diameter growth at breast height using live crown ratio, dbh, site index, number of trees per acre, and basal area. Arney (1985) suggested a procedure for height growth projection of managed stands of Douglas-fir using data from permanent research plots located at the Pacific Northwest Experiment Station. He used a nonlinear equation using mean diameter at breast height as independent variable for the prediction. of stand. height. He concluded that. empirical modeling for managed stand growth and yield is not usually based on adequate data for any single species. He therefore recommended that improvements be made in accuracy and reliability of forecast by using more biologically sound equations. He further argued.that the strength.of the modeling 20 approach contributes more towards confidence in forecasting rather than ‘fit statistics' of the empirical regression analysis applied. 2.3.2 CHAPMAN-RICHARDS FUNCTION This function is a generalization of von Bertalanffy's (1951) growth model. From many studies of aquatic and terrestial organisms, von IBertalanffy' concluded. that. the allometric relation between surface area and total volume could be expressed.as S = c‘VMfi where S is the surface area, V is the volume of the organism, c is the regression coefficient, and 2/3 is the allometric constant. Von.Bertalanffy expressed.the rate of volume.growth.of an organism as the difference between its anabolic rate (constructive process) and its catabolic rate (destructive process). Although von Bertalanffy's model applied reasonably well to the growth of many organisms, some researchers did not agree about his restrictive 2/3 allometric constant. Richards (1959), while studying plant. growth and Chapman (1961), studying fish growth, both suggested that the allometric constant should not be specified. They proposed a generalized model: dV/dt=nV‘“-1V where the value of m, the allometric constant, is to be estimated for each organism and environment, and n and r are the parameters to be estimated. Richards (1959) pointed out 21 that if the above model applied to V, it was theoretically applicable to all variables that had an allometric relationship with V. Moser (1972) used a system of first-order differential equations to model the components of basal area growth and survival of uneven-aged mixed northern hardwood stands in Washington County, Wisconsin. The derivative of the Chapman- Richards equation was used for survivors' growth rate. In another study, Moser (1974) used the Chapman-Richards function for the rate of change in number of trees, basal area, volume, number of ingrowth trees, basal area ingrowth, volume ingrowth, mortality of trees, total basal area and total volume for six diameter classes. The result was a system.of 66 first order differential equations. The system, when integrated, predicted number of trees, basal area and volume per unit area in each of the six diameter classes at any particular time. The author used data from uneven-aged stands of sugar maple located in the Upper Peninsula Experiment Forests in Michigan. Pienaar and Turnbull (1973) used the Chapman-Richards equation to investigate basal area growth of slash pine, and basal area, height and volume yield of spruce for even-aged stands in South Africa. Their study indicated the adequacy of the.Chapman-Richards equation for various types of tree growth investigations. The Chapman-Richards model provided an adequate description of the basal area development of 22 unthinned even-aged stands. Murphy (1983) used a nonlinear equation system to predict timber yield for natural even-aged stands of loblolly pine in the West Gulf region. After several plausible expressions for the Chapman-Richards function, the author used the following equation for projected basal area: B. = {n/k - [n/k - BOW] exp [-k(1-m)(A.-Ao)]}““““’ where Bt and B0 are stand basal areas at age A, and A0, (A,-Ao) is the elapsed time between the two ages, and k,n,m are the regression parameters to be estimated. He concluded that inclusion of basal area in stand volume equations increased the precision of the estimated volume. The author also concluded that the use of nonlinear regression for estimating the parameters provided the opportunity to overcome correlated residuals. Ito (1988) used a Chapman-Richards function for modeling even-aged stands of sugi species in Oita Prefecture, Japan. The author used data from temporary sample plots, and estimated parameters for the system of models to predict basal area, average height, stand volume and number of trees per unit area. The results of these estimation were applied to data from long-term permanent sample plots, and confirmed the reliability of the earlier estimates. CHAPTER 3 DESCRIPTION OF STUDY AREA 3.1 IRRIGATED PLANTATIONS In Pakistan, irrigated plantations are the outcome of man's efforts on marginal lands where irrigation water was available. The history of such plantations begins in 1866 when the first irrigated plantation, Changa Manga, was started. The objective was to supply fuelwood to a newly created locomotive system in the western part of the sub-continent. This objective was changed.when coal fields were discovered in Dandot, Jhelum district in 1888, and the burning of firewood in the railway steam.engines ended. The objective of irrigated plantations thus shifted to produce timber and firewood for the local people. After World War I, the sub-continent's demand for timber increased many-fold due to post-war reconstruction programs. The existing plantations were extended and new plantations (e.g., Daphar in 1919) were started. All plantations, with the exception of Changa Manga, were established by leasing out land for temporary cultivation. The lessees cleared the vast growth of tropical thorn forest species, uprooted stumps and leveled the area. In return, the lessees were able to grow 23 24 agricultural/cash crops on the land for the limited time of the lease. The Government agency (Forest department) did the planting work and lay-out of irrigation channels. Plantations range from less than 80 ha to as large as 8000 ha. The main species grown in these irrigated plantations include Dalbergia sissoo (shisham), Morus alba (mulberry), Salmalia malabarica (simal), Eucalyptus species, Acacia nilotica (kikar), Populus species (poplars), Melia azedarach (bakain) and Salix species (willows). These plantations are managed for the production of firewood and timber for construction, for furniture and for the sports- goods industry. In addition, the plantations provide vast grazing facilities for domestic cattle and a refuge for wildlife. A.total of 392,000 ha area in Pakistan is under irrigated plantations, out of which 148,000 ha fall in the province of the Punjab. This provincial total is subdivided into two categories, production areas (72000 ha) and protection areas (76000 ha). The production irrigated. plantations of the Punjab, therefore, comprise 11% of Pakistan's total forest area. A subdivision of the total forest area of the country by vegetation type is given in Table 4. The irrigated plantations of the Punjab comprise about 40% of the total irrigated plantations of the country. During the early years of the establishment of forest plantations, irrigation water was supplied through a technique 25 called flood irrigation, i.e. without small water channels. Later on, with the desire to conserve water, trench and trench-cum-flood irrigation systems were adopted. These techniques were developed by trial and error. Table 4. Forest area (1000 ha) under the control of Provincial Forest Departments (1987) Veget. NWFP Punjab Sind Baluch- North Azad Total type istan areas Kashmir Coniferous 2730 235 - 287 704 885 4841 Irrigated plantations 393 366 203 2 5 - 969 Riverain - 131 595 5 - - 731 Scrub 284 798 25 1470 1626 2 4205 Coastal - - 852 5 - - 857 Mazri land 59 - - - - - 59 Linear plantations 5 32 - 5 - - 42 Total: 3471 1562 1675 1774 2335 887 11704 Though theoretically most of the shisham plantations are irrigated using the trench system, in practice most trenches are filled up in first 3-5 years with silt, twigs, and leaf litter and the area is flooded. Irrigation water for the plantations becomes available from 15 April to 15 October, which is known as Kharif season. The other 6 months of the year the forests have to live on sub-soil moisture. The 26 frequency and timing for irrigating plantations depend on the water requirement of the crop and the quantity of water available for irrigation. Irrigation water'iSjprecious.and is, therefore, a limiting factor. Most plantations do not get their due share of water because of the pressing demands for irrigating agricultural crOps. In cases of reduced water supply from the Irrigation Department, younger forest stands are given priority over older stands. The erratic supply of water is affecting the normal development of plantations, and causing a decrease in economic returns from a majority of the Punjab's irrigated plantations (Ghani 1974). 3.2 SAMPLE PLOTS AND STUDY PLANTATIONS Growth data for shisham are regularly collected through quinquennial measurement of permanent and semi-permanent sample plots by the Pakistan Forest Institute (PFI), Peshawar. The PFI is responsible for conducting forestry research.at the national level. More than 200 sample plots of shisham have been laid out by the PFI in 17 plantations of major economic importance distributed across the Punjab. These 17 plantations cover an area of 41997 hectares, or about 58 percent of total planted area of the province (Olander 1984). The number of sample plots in each plantation depends upon the extent of pure stands of shisham and stand age. Usually the number of plots ranges from five to ten plots for every thousand ha. The size of sample plot varies from 0.1 to 27 0.5 ha depending upon the density and age of the crop. For some mature stands the plot size may exceed 0.5 ha, it may be less than 0.1 ha in cases of densely spaced, young plantations. 3.2.1 SAMPLING Ten plantations were selected as sample plantations for this study from the 17 plantations in the Province. The sampling procedure adopted to select ten out of seventeen plantations was a two-way stratification. The intent was to distribute the sampled plantations across the Province. The two-way stratification was done on the basis of plantation size, age and climatic-edaphic (ecological) factors as follows: (a) Size,and Age Stratum A. The plantations included in this stratum each have a net planted area of more than 2000 hectares. These are older plantations which have completed two or more rotations. Stratum B. This stratum includes plantations with a net planted area of 2000 hectares and less. They are medium to young in age and have completed or are near completion of their first rotation. (b) Climatic-edaphic zones Stratum i. These plantations are situated in the northern half of the province where the annual rainfall is over 230 millimeters. The soil is a clayey loam to silt loam, and the maximum temperature is 45° C or less. 28 Stratum ii. These plantations in the southern part of the province receive less than 230 millimeters of rainfall annually. The soil is sandy loam to clayey loam. Sand storms are frequent due to hot and dry conditions in the area. Maximum temperature rises to 50° C and more. After categorizing the 17 plantations into four groups on the basis of the two-way stratification, 10 sample plantations were selected proportionally from the strata. The ten plantations, uniformly distributed across the Punjab, cover about 62 percent of the total area of the 17 population plantations. Table:5 shows the number of population and sample plantations and their net area in each category. Table 5. Sample versus population plantations by category Category Number of Area Sample Area plantations (ha) plantations (ha) A-i 6 21315 4 15189 A-ii 3 13255 2 6962 B-i 2 1792 2 1792 B-ii 6 5635 2 1960 Total 17 41997 10 25903 There 'were a 'total of 129 plots in ‘the 10 sample plantations. These plots had 306 periodic measurements which were included in this study. All the measurements have been converted to a per ha basis. The data base and variables will be described in section 4.1. 29 3 .2 . 2 DESCRIPTION OF SAMPLE PLANTATIONS A brief description of each sample plantation is given below: 1. Changa Manga Plantation This plantation, established in 1866, is the world's oldest man-made irrgated plantation. It contains 4917 ha, of which 4472 ha are under shisham.and.mulberry. Shisham is grown for timber and firewood production; the annual yield ranges between 790 to 8501f’for timber and between 56000 to 70000 1113 stacked for firewood. Mulberry timber is used in the Pakistani sports industry for manufacturing hockey sticks, cricket bats and rackets for tennis, badminton and squash. About 1100 to 1400 cubic meters of mulberry timber is produced annually from the plantation. Apart from these two'main species, a number of miscellaneous species like simal, eucalyptus, bakain. and poplars are also‘grown for‘varied.purposes. Their total annual yield varies from 800 to 2000 m3 (Bokhari 1974) . The soil of the plantation is very deep, friable and porous. The top soil is loamy-silt or fine silty-loam and the area is highly suitable for tree growth. The plantation falls in the semi-arid subtropical zone characterized by extremes of temperature, low relative humidity and erratic rainfall. The minimum temperature is about 5° C and maximum is about 44° C. The average annual rainfall is about 355 mm, 60 to 70% of 30 which is received during monsoons from July through September. The main management objective at the time of establishment was to supply fuelwood for the railway steam engines. With the change of locomotive technology, as well as the discovery of the Dandot coal fields in 1888, objectives for the management of this plantation changed. The revised objectives were to maximize economic output from forest crops compatible with the optimum utilization of land and water; to promote better growth through judicious and equitable distribution of irrigation water; to eradicate mesquite (Prosopis juliflora) in the shortest possible time; to develop the plantation into a recreational park without jeopardizing the regional character of the plantation; to promote wildlife; and to meet the requirements of local.population for brushwood and for mulberry leaves for sericulture in the best possible way. Major injuries for shisham in Changa Manga plantation include insects, fungi, wild animals, frost, drought and weeds. The insects cause considerable damage to the standing crop as well as to the harvested timber and firewood. Important insects are shisham defoliator, powder post beetles and white ants. The fungi injurious to shisham are Fomes species. The‘wild.animals includerwild.boar, porcupine, jackal and. wild cat. These animals are mostly harmful to the seedlings and young saplings of shisham. Among the common weeds, bathu (Chenopodium album) is by far the worst weed in 31 the plantation. Other weeds include kana (Saccharum munja), patakha (Abutilo bidervatum), puthkanda (Achyranthes aspera), dhatura (Dhatura indica), bandri buti (Anisomeles indica) and dab (Eragrostis species). 2. Daphar Plantation This plantation is situated about 110 kilometers northwest of Gujrat city. The plantation was started in 1918 with the goal of planting 2670 ha in 20 years (planting 130 to 135 ha annually). The area was opened for temporary cultivation leases in 1917 to clear old coppice growth and prepare the land for afforestation. The annual coupes (workable area for planting) of unequal sizes were recovered from temporary cultivation leases and stocked with shisham through sowing and/or planting. Later on, mulberry was introduced and now grows as an understory of the main crop of shisham where soil and irrigation permit. An area of 2463 ha is planted out of a total of 2890 ha of the plantation. The distribution of area under different species is 2067 ha under shisham and mulberry, 356 ha under poplars and 38 ha under eucalyptus (Khan 1972). The soil of the plantation has been formed from mixed alluvium. It varies from silty clay-loam to silt loam and is usually porous and well structured. Annual rainfall, which is highly erratic, is about 380 to 480 millimeters and almost 65 to 70 percent of rainfall is received from mid-June to mid- 32 September. The rest of rainfall occurs throughout the remainder of the year, thus very dry months are rare. The maximum annual rainfall is 760 millimeters. Hail-storms are rare but do occur in the spring. Heavy frosts occur from mid- December to mid-February. The gradual disappearance of Acacia nilotica from the plantation is partially attributed to the cyclic occurance of heavy frosts. The maximum temperature is about 45° C during summer, June being the hottest month. The minimum temperature is about 4 to 5° C during winter. High water table and high alkalinity are major problems (Khan 1972). The plantation's management objectives are to maximize annual sustained yield of shisham and mulberry timber; to grow forest species like poplars and eucalypts for wood products; and to produce fuelwood for local markets. To achieve these objectives, the plantation is being worked under four different working circles: shisham and mulberry working circle; poplar working circle; coppice working circle; and afforestation and reclamation working circle. The common weeds in the plantation are bathoo (Chenopodum album), bhang (Cannabis sativa), khabal (Cynodon dactylon), dab (Eragrostis bifinnata), and gharam (Panicum antidotale). Since these weeds hinder the regeneration process of the desirable species, cleaning is done as a continuous operation, particularly in the early stages of regeneration. 33 3. Kamalia Plantation This plantation is situated in the Faisalabad district and contains a gross area of 4397 ha. Planting was started in 1946 and completed in 1964. Out of a net plantable area of 3554 ha, pure shisham is growing on 1470 ha (41%). Shisham mixed with mesquite (Prosopis species) occupies about 17% of the plantation, shisham mixed with Acacia species, mulberry and tamarix occupies 16%, pure mesquite occupies 3%, other species have about 1.3%, and areas without any vegetation comprise about 22% of the plantable area. The climate in this region is mainly dry; it is very hot in summer and quite cold in ‘winter. The daily average temperature is 5° C during the winter and 44° C during the summer months (based on previous 9 years record). The area is part of the vast Indo-gangatic plains. The geological formation is alluvial soils deposited by the river Ravi, which now flows 5 to 10 km south of the plantation. The plantation has a salinity problem due to high pH (pH values are greater than 8.5). Soils are generally heavy in texture, clayey loam in the upper layers of the profile and sandy loam in the lower layers. The average annual rainfall is about 230 mm (past 16 years record) with. maximum :rainfall in. July and .August (monsoon season) and minimum rainfall in October and November. The water table is at 4 to 6 meters, and is suitable for both cultivation and human consumption. 34 Frost occurs during November through December and damages young crops of kikar, sirus and semal. During severe frosts, even shisham seedlings and kikar pole crops are damaged. Windstorms also injure crops by bending and uprooting trees and.drying up the soil. Wild animals like pigs, porcupines and rats,damage plantations by uprooting young trees, girdling and debarking trees. The common weeds are Prosopis juliflora, Saccharum munja and Chenopodum album. Management objectives include mesquite eradication through temporary cultivation, maximum use of land and water resources, and maximum volume production of shisham and other tree species. The area has been allocated to two different working circles: the shisham working circle and the mesquite eradication working circle. 4. Machhu-Inayat Plantations These two plantations, which are actually National Parks, run contiguously along both sides of the Mianwali-Muzaffargarh Highway in the form of compact belts. The length of these forest belts is about 30 km and their width varies from 3 to 6 km. These parks contain.a total of 8350 ha out of which 4249 ha (51 percent) is planted. Growing stock consists of shisham mixed with kikar and tamarix. Shisham occupies about 87 percent of the total planted area, kikar about 5 percent, tamarix about 2 percent and other species as 6 percent. Soils are sandy, extremely porous and well aerated and are suitable 35 for plantations. Rainfall is scarce and averages about 180 to 200 mm annually. The area has a typical hot desert climate. In summer, the maximum temperature rises to above 46° C and the minimum goes to 0° C in winter. Severe sand storms are common in the summer, although the climate has been ameliorated to a great extent due to forestry and agricultural practices. Contrary to other plantations in the province, these two plantations receive perennial irrigation with only a short closure during January-February each year. In spite of this perennial supply, irrigation water is quite deficient. About 15.8 cusecs (water discharge is measured as cubic feet per second) per 1000 hectares are supplied against the required quantity of 30 cusecs per 1000 hectares. The most noxious weeds in these plantations include Saccharum munja and Cymbopogon jawarancusa. The former chokes up water courses and the latter suppresses the young plants. The management objectives for these plantations are to obtain a sustained yield of forest products, to produce shisham and industrial wood, to plant unstocked areas and provide vegetation cover for sand dunes, and to fulfil the needs of local people for firewood, grazing, grass cutting, and small timber. To achieve these objectives, these plantations are being managed under one plantation working circle. Planting is being done by the method of root-shoot cuttings with the retention of standards. 36 5. Chichawatni Plantation The plantation is situated in the Sahiwal district along the northern side of the Lahore-Karachi railway. It contains 4,666 ha, out of which 3,722 ha (80%) are planted. Soils in the area are mostly clayey loam with about 30 cm of silt and humus at the surface. They are well suited to tree growth if irrigated. Mean annual rainfall is about 200 to 230 mm, mostly during the monsoon season of July and August. It is extremely hot in summer and severely cold in winter. Temperatures range from -1° C to 49° C during the year. Dust and sand storms are frequent, especially in April and May (Khan 1976). Growing stock consists of pure shisham, shisham mixed with mulberry and other species like eucalyptus, kikar, simal, and patches of pure mesquite. Shisham predominates over about 58 percent of the planted area, mulberry in 23 percent and mesquite in 19 percent. Mesquite is spreading fast and dominating the principal species, mainly due to the irrigation shortage. The plantation is being managed for the sustained yield of shisham ‘timber, industrial softwood, sports wood, and firewood; to eradicate mesquite and replace it with shisham and other fast growing timber species; and to meet local requirements for grass cutting, brushwood and chips, and mulberry leaves for sericulture of twigs for basket making. 37 6. Irrigated Plantations of the Bahawalpur Forest Circle: The irrigated plantations of Lal Sohanra, Abbasia and Chak-Katora are covered by the same working plan. The land is generally flat and interspersed with sand dunes varying from less than a ha to 40 ha and with heights from 1 to 3 meters. Occasional sand dunes of 1,200 ha are found in the Lalsohnura and Abbasia areas. Soils are alluvial deposits, generally sandy loam. The layers of soil formation are uniform, varying in depth from 1.5 to 4.5 meters after which.pure sand appears. Rainfall is very low, the annual average is 180 to 200 mm, mostly received in July through August. Summers are very hot and winters are comparatively hot during the day but very cold during the night. Temperatures range from10° C to 49° C during the year. Wind storms are common from April to October. Growing stock consists of pure or mixed shisham or kikar. Mesquite is invading in the Lalsohanra and Abbasia plantations, and presently occupies about 10 to 15 percent of the plantations area, along with jand (Prosopis specigera), karir (Caparis aphylla), frash (Tamarix aphylla), and phog (Calligonum polygonides). The total area of the Ialsohanra plantation is 8488 ha, out of which 3241 ha (38%) is planted. The Abbasia plantation has a total area of 2865 ha, out of which 1490 ha (52%) is planted. The Chak-Katora plantation is smaller, only 536 ha, out of which 470 ha (88%) is planted. These plantations have not been managed under any regular 38 working plan in the past. The present working plan (initiated in 1965) has the following objectives: to achieve maximum economic return from land and water resources; to afforest the unplanted but commandable areas; to meet the requirements of public in general and local people of Bahawalpur Division in particular, for firewood, timber, and minor forest produce; to stabilize uncommanded dunes, arrest their advancing and develop them into productive rangeland. The common weeds found in these plantations are,Saccharum munja, Eragrostis cyonsuroides and crotalaria burbia. Among insects, shisham defoliator, powderpost beetle and white ant are common in these plantations. Wild boar, rat, deer and jackal are harmful for shisham young and mature crops. 7. Shorkot Plantation This plantation lies in Jhang district and has a total area of 4,079 ha, of which 1,228 ha is forested. Most of the plantation faces the problem of salinity and water-logging. Water-logging is due to the seepage of water from the link canal (canal linking two rivers for transporting water during winter). The climate is hot in summer ‘with an average temperature of 43° C, and very cold during winter with an average temperature of less than 4.5° C. Annual rainfall is more than 250 mm with the maximum in the monsoon season. The soils have a high salinity problem with pH value higher than 9 (Ashraf 1965). 39 The main species are shisham, kikar, eucalyptus, and tamarix with a profuse weedy growth of mesquite. More than half of the plantation has been severely affected by water logging and salinity. The water table is at 4.5 to 7.5 meters and the water is not very suitable for drinking. The management objectives are: to reclaim water logged areas by planting suitable species like eucalyptus; to maximize volume production of timber and firewood; and to eradicate mesquite from the plantation. The common weeds found in the plantation are Prosopis juliflora, Chenopodium album, Cannabis sativa, Saccharum munja, S. spontaneum, Panicum antidotale and Xanthium strumarium. The plantation is being worked under rehabilitation and plantation working circles. Small-scale lift pumps have been installed in the water logged areas to drain out water, and resistant species like eucalyptus and 'tamarix are being planted in the rehabilitation working circle. The areas still suitable for normal plantation work have been put under the plantation working circle, and are planted with shisham, eucalyptus, kikar and tamarix. 8. Arifwala Plantation This plantation, situated in the Sahiwal district, was established in 1938. It has a total area of 614 ha, out of which 482 ha (79%) is under vegetation cover. The species 40 composition is 33% pure shisham, 26% shisham mixed with mesquite, 26% pure mesquite, 4% mulberry and other species, and 11% under temporary cultivation. The subsoil water is generally fit for human consumption and cultivation. The soil is alluvium deposited by the river Bias. The climate is severely hot in summer with an average temperature of 45° C and during winter the average temperature is 2° C. Windstorms are common from April to June. Average annual rainfall is about 200 to 225 mm, most of which falls during July and August. In this plantation, there is not much variety of weeds but mesquite is growing on many locations. The plantation is being managed to maximize timber and firewood production, and to eradicate mesquite and replace it with shisham, tamarix and mulberry. The mesquite- infested area is leased out for temporary cultivation for three years. After resumption from temporary cultivation, an irrigation system will be laid out and the area planted up with shisham. As a safeguard against unforeseen shortfalls for pure cropping, a 20% mixture of frash is also maintained. Mulberry is interplanted as an understory with shisham where soil is suitable. 3.3 DESCIRIPTION OF DALBERGIA SISSOO 3.3.1 BOTANICAL DESCRIPTION Dalbergia sissoo belongs to the plant family Leguminosae, sub-family Papilionaceae, and is known as shisham and tali in 41 Urdu and local languages. It is a mid-size to large deciduous tree with a medium crown and light brown to grey bark. Pearson and Brown (1932) reported that this species can reach a maximum dimension of 75 cm in diameter and more than 30 meters in height, with a 10 meter clear cylindrical bole in favorable conditions. But in localities which are dry or do not have suitable conditions, shisham remains a small to medium-size tree. The roots of shisham are dimorphous with vertical roots without buds, and long horizontal roots containing buds from which root suckers are produced. The seedling produces a long taproot which gets stronger as the plant approaches sapling stage. Lateral roots are also developed which spread from a few centimeters to 1 meter below ground level, depending upon frequency of irrigation. Shisham is a nitrogen fixing tree (Troup 1921) and could be taken as a suitable species for agroforestry practices. The probable reasons shisham is not being treated as one of the major choices for agroforestry, might be its slow growth and high water requirements. 3.3.2 DISTRIBUTION Dalbergia sissoo is found throughout the sub-Himalayan tract of the Indo-Pakistan sub—continent at elevations up to 900 meters. It may go even higher on better (hotter) aspects. It is also found on the plains along the rivers. Shisham is a typical species for alluvial grounds in the old beds of rivers, but can also be found on hillsides, landslips and 42 along water channels. In its natural habitat, shisham is found in the Rawalpindi, Gujrat and. Jhelum. districts clinging to ‘the sandstone cliffs and.spreading by means of:root-suckers, along nullahs (water-course) and on fresh alluvium. Globally, shisham is found in its natural habitat in the Kangra valley of India, Burma, Nepal, Sikkim and Bhutan (Troup 1921). Natural forests have also been reported in the Kalmord range of Persia (Tabai 1964), in Afaganistan and in Iraq (Nasir and Ali 1977). Shisham is not an arid zone species but was introduced in the plains of Pakistan during the establishment of irrigated plantations. 3.3.3 NURSERY TECHNIQUES In the past, shisham seed was sown in patches or lines in nurseries. Later on, with the experience gained by practicing foresters, the technique changed to grow shisham on raised nursery beds. The width of the bed has been 3 meters because of the distance between the trenches in the plantations. In recent years, however, the nursery trenches have been dug 1.5 meters apart in order to get maximum utilization of space and to produce higher yield of shisham stumps. Sowing of seed is done on the berms of trenches all along the beds. Seed collection is done‘during late January and February when trees are leafless. The selected seed.pods are stored in dry places, and can remain viable for one year. The nursery site should be fairly level, free of noxious plants and weeds, and have 43 sufficient water and labor supply (Champion et al. 1965). 3.3.4 PLANTING METHOD Shisham is raised by stump (root-shoot cutting) planting during the start of the irrigation season in mid-April. Stumps are planted on the berms of trenches in slots 1.8 meters apart, about 3-4 days after the initial irrigation. The trenches, which are 3 meters apart, are irrigated immediately after planting. Subsequent watering is carried out at 3 to 4 days intervals till sprouting takes place (Champion et al. 1965). Canal water is supplied to irrigated plantations from mid-April to mid-October. In an ideal situation, the optimum supply of water for shisham plantations is about 30 cusecs for 1000 ha. When spread over the six month period of water supply, this becomes 1.37 meters of delta. Delta is the quantity of water measured in meters, that is, a delta of 1 meter is the quantity of water sufficient to acquire a height of 1 meter over an area of 1 ha of land. The data collected for this study show that this quantity of water is never received, causing poor growth of these plantations. Due to this shortage of water, the frequency of irrigation for each month is also reduced. Theoretically, the newly regenerated or afforested areas in the plantations should be irrigated 7 to 8 times each month up to the sprouting of root-shoot cuttings and stumps. Irrigation intensity is then reduced according to the age of the crop in subsequent months. Haq (1985) has reported average delta figures over 15 to 25 years for eight 44 major plantations as ranging from 0.85 to 1.59 meters against the required 1.37 meters delta. This fluctuation shows that the frequency of irrigation is not consistent. The root-shoot cuttings of shisham are obtained from one- year—old nursery stock and are planted out in the field at a conventional spacing of 3.0 by 1.8 meters, as recommended by Troup (1921). The same 3.0 by 1.8 meters spacing has been prescribed in almost all of the latest working plans of irrigated plantations (Ashraf 1965, Bokhari 1974, Hassan 1970, Khan 1976, and Shameem 1980). The standard compartment size for irrigated plantations is 20.2 ha, bounded by a 6.1 meters wide compartment road. The normal layout of a compartment is across the general slope, which means that length of compartment runs along the general contour of the area. The main water course (known as the ‘main') runs on the ridge side along the contour and is connected to small channels known as ‘khals', which run perpendicular to the main. These khals supply irrigation water to the trenches through.‘pasels' which run parallel to and on either sides of the khals. A diagram showing the layout of a plantation compartment and water channels' dimensions is shown in Figure 2. 3.3.5 CULTURAL OPERATIONS Shisham has a rotation length of 22 years, with intermediate thinnings at an interval of 5 to 6 years starting at the age of 6 years. The silvicultural system adopted in irrigated plantations is clear-cutting with the retention of 45 IRRIG ON 0 CO TME if”Forest Distributary METRES 22.7 -——-4.-———- 4 ——-1 a -4JOF——217 7 comm.‘ : c sum: mu mm PASELS cemne Icnoss now ' noao ' ' wmc PASELS flask—4 F—Qgfl-Sl 33H H3{—‘1—%_;j"— ---d ----‘- -m _i£crto~ on AJL_ Figure 2. Layout of irrigation in plantation compartment 46 standards. Artificial regeneration of shisham is carried out by planting the root shoot cuttings, although a considerable part of the new crop is obtained by root-suckers with some stump coppice. Weeding is carried out 3 times during the first year of planting, but. only' occasionally' during' subsequent. years. Cleaning operations (removing weak stems of main crop before thinning stage) are done in the sapling crop, usually before the first thinning, to reduce competition. Cleaning is also done to space out the dense and profuse growth from root- suckers. It is carried out from the early stage of plantation establishment and continued subsequently' with decreasing intensity until the first thinning. Pruning is usually not carried out as a regular operation in the plantations because of its demanding labor cost, and the danger of fungal attack through wounds. Thinning is one of the most important silvicultural operations in the management of the plantations. Although different thinning grades have been suggested for shisham plantations, there has been no hard and fast rule prescribed. Thinning grades range from A (very light) to D (very heavy) depending upon the density and age of crop (Qazi 1973). The first thinning is applied to the crop at age 5 to 7 years, while the second thinning is carried between the age of 11 to 13 years. A. recent, trend is to do only’ the first two thinnings, and shorten the rotation from 22 to 20 years. Qazi 47 (1973) recommended a C-grade thinning for shisham. This is a heavy thinning, removing about.30 percent of total basal area. A number of factors, however, influence thinning practices. These include the influence of site, species, origin of plant, introduction of understory, and the skill and supervision of staff and labor (Champion at al, 1972). Khan (1972) recommended that thinning in shisham should lead the crop to close up after three years and then trees should grow in competition for the next two years. 3.3.6 INJURIES TO SHISHAM PLANTATIONS Shisham is liable to a number of biotic and climate injuries which can be grouped as biotic factors, physiological abnormalities, and climatic and edaphic factors. 3.3.6.1 Biotic Factors The biotic factors injurious to shisham crop include man, animals, birds, weeds, insects and fungi. Man's injurious role is through illicit cutting of trees, lopping of branches, and through setting intentional fires to obtain new grass for cattle feeding. Injury by animals includes cattle grazing and damage by porcupines, wild boars, rodents and birds. Weeds have become a very serious problem in the irrigated plantations. Weeds are hardy competition to the main crop as well as to nursery stock. A number of insect pests have been recorded as causing considerable damage. to shisham. crops (Khan 1983). These 48 include Plecoptera reflexa (shisham defoliator), Dichomeris eridantis (shisham leaf roller), Ascotis selenaria imparata (leaf defoliator),.Apodarus.sissu (leaf roller) and Leucoptera sphenograpta (leaf miner). All attack the standing crop and nursery stock. Those pests which damage the harvested material include Stromatium barbatum (wood borer), Lyctus africanus (powder post beetle), and Perissus dalbergiae (wood borer). Studies have been carried out on the disorders of shisham plantations through fungal attack by Chaudhry and Gul (1984) and Zakaullah (1984). Fungi-causing diseases include Fusarium solani (wilt disease), Ganoderma.lucidum (root.rot), Polyporus gilvus (root and butt rot), leaf spots, leaf rusts, and fungi on felled shisham wood like Irpex flayus and Polystictus proteus. In case of heavy fungal attack, tree growth is affected and eventually the tree dies. 3.3.6.2 Physiological Abnormalities These abnormalities occur mainly due to the shortage of irrigation water. Stressed trees exhibit pale foliage, die- back and finally die. Trees growing in water-logged areas also die because of the poor aeration to the root system. 3.3.6.3 Climatic and edaphic factors Shisham is very susceptible to frost, drought, windstorms, and hailstorms. The main edaphic factors for crop injury are alkaline soils and high water table. Water-logging affects the soil medium for root growth and also brings up the salts to the surface, making the soil more alkaline and unfit 49 for tree growth. 3.3.7 USES Shisham is used for high quality furniture as well as for paneling. Its wood is excellent fuelwood and is suitable for charcoal due to its high calorific value (8,800 to 9,300 BTU). Shisham timber is used in building construction because of its durability and strength. Because of its beautiful grain and dimensional stability, shisham timber is also used in parqueting and flooring. Shisham leaves are used as fodder because of their high fat, protein, carbohydrate and moisture contents. Leaf decoction is believed to be useful in gonorrhea, and leaf mucilage is applied in excoriations. The roots are used as astringents, and the wood is useful in skin infections (Haq 1985) . In fact, the whole tree is very useful. CHAPTER 4 MATERIALS AND METHODS 4.1 DATA Ten sample plantations were selected after stratifying 17 population plantations on the basis of climate, age and plantation size. A total of 313 measurements over time came from 129 sample plots in 10 selected sample plantations. Tables 6 and 7 show the distribution of measurements by plantation and age class. The measurements include age, diameter at breast height (dbh), average stand height, height of dominant/codominant trees, basal area per ha, number of trees per ha, volume per ha, and irrigation level. Univariate statistics were calculated for each plantation as well as each age group (Appendix A). Since 'the jplantations are even-aged. by’ regeneration coupes (working area), sample plot age was determined from each compartment history file. Plot ages ranged from 4 to 33 years, with a mean age of 16 years. Diameter in centimeters of all trees in each sample plot was measured at breast height (1.37 meters) using a diameter tape. Diameters were averaged to determine average stand dbh. 50 51 Table 6. Distribution of plots by plantation Plantation name Number of Period measurements Initial Last Changa Manga 16 1972 1987 Daphar 13 1973 1983 Kamalia 52 1973 1983 Machhu-Inayat 64 1966 1986 Chichawatni 39 1965 1985 Lalsohanra 34 1967 1987 Arifwala 5 1984 1984 Abbasia 19 1972 1982 Shorkot 58 1961 1971 Chak Katora 13 1972 1982 Total 313 Age Number of Age Number of (years) plots (years) plots measured measured 4 2 19 27 5 16 20 12 6 7 21 8 7 6 22 9 8 4 23 9 9 17 24 16 10 29 25 8 11 14 26 4 12 15 27 5 13 13 28 5 14 25 29 6 15 14 30 3 16 13 31 2 17 9 32 2 18 12 33 1 52 Dbh ranged from 6.6 to 46.0 cm with an average of 23.4 cm. In each sample plot a sub-sample of trees across the range of plot diameters was selected for height measurements. Height, defined as the total height from ground level to the tree's tip, was measured by climbing each tree. The range in stand height was from 6.1 to 25.9 meters, the average was 15.85 meters. The heights of at least five dominant/codominant trees were taken in each sample plot to estimate site quality. The range of dominant/codominant height was from 8.84 to 25.9 meters, with an average value of 17.37 meters. The number of trees present in each.plot were counted and converted to a unit area basis (per hectare). The data ranged from 89 to 1,737 trees per ha with a mean of 388 trees per ha, depending on the stand age. The total basal area (square meters) of each plot was calculated by summing individual tree basal areas. Basal area per unit area was calculated by dividing the total basal area of the plot by the plot area. The data ranged from 2.89 to 24.57 square meters per hectare:(nF[ha) with an average value of 12.24 mz/ha. The volume of individual trees was calculated through sectional measurements. Since there is generally no butt swell in shisham, especially young shisham, sectional cubic volume was calculated using Huber's formula for all sectional logs. Volumes were calculated to a 5.1 cm top diameter, as beyond 53 that point the portion of tree is taken.as brushwood (firewood cotaining small branches and twigs). Volume included branchwood, which had a thin end diameter of 5.1 cm or more. Total volume included timber as well as smallwood volume. Timber volume comprised stem and branches having thin end diameter of 20.3 cm and above, and smallwood comprised the rest of the tree and branches with thin end diameter of 5.1 cm and above. As pointed. out earlier, the requirement. of shisham plantations for irrigation water was traditionally set at 30 cusecs for each 1000 ha. This is equivalent to a delta of 1.37 meters on the basis of a six month water supply (15 April to 15 October). Irrigation ranged from 0.34 to 1.43 meters delta of water, with an average of 0.77 meters. 4.2 ANALYSIS OF PLANTATIONS GROUPING Before developing yield models for these plantations, it seemed appropriate to examine the hypothesis that all plantations measured came from the same population. The null hypothesis was that plantations are not significantly different on the basis of any single variable of yield. In this study stand volume per ha, being the dependent variable for yield modeling, was used to test the hypothesis of plantation difference. Basal area per ha was also used to test the hypothesis because, according to Husch et al. (1972), basal area has been found to be the most satisfactory basis 54 for expressing relative stocking. An analysis of covariance (ANCOVA) was used to test for differences between plantation volume with age as the covariate. The results of the ANCOVA are summarized in Table 8. The analysis of covariance results indicated an overall significant F value (33.36) at.a = 0.01 level, meaning that at least one plantation was significantly different from others on the basis of volume per ha. When the effect of age was adjusted, the F value was still significant (11.40) at a = 0.01, indicating that even after adjusting sum of squares for the effect of age as a covariate, there existed a significant difference between the plantation volumes. In order to find out which plantation volumes were significantly different from others, multiple comparison tests were applied. Tukey's and SNK tests were selected for this purpose as these tests have proved superior to the LSD test (Ott 1988). Tukey's test showed that one plantation (Chichawatni) was significantly different at a=0.05 level from other plantations. The SNK test revealed that two plantations (Chichawatni and Machhu-Inayat) were significantly different from the rest of the plantations. Since these two plantations did not form the predetermined groups of plantations described earlier in Section 3.2.1, nor did they constitute a distinct group based on any logical criterion like administrative or management, it was decided to test the predetermined groups for their similiarity. The F statistics for the two contrasts 55 Table 8. Analysis of covariance for the dependent variable, volume per ha, with age as covariate Source DF Sum of Mean F Value Squares Square Model 10 39545769.4 3954576.9 33.36“ Error 302 35796695.2 118532.1 Total 312 75342464.6 Plantations 9 12164267.3 11.40" Age 1 27331502.1 231.00“ Contrast Bigger vs smaller 1 34276.1 0.29 Northern vs southern 1 160140.0 1.35 56 were calculated. The F ‘values of 0.29 and 1.35 in 'the contrasts indicated no significant difference at a = 0.05 level on the basis of criterion of yield. Basal area per ha was picked to be used as the second criterion variable to test the plantation grouping hypothesis, using age as a covariate. The results of the analysis, given in Table 9, revealed that the overall F value (26.0) was significant at a = 0.01 and the F value adjusted for age (7.66) was significant at a = 0.01. It indicated that there existed at least one plantation significantly different from others on the basis of stand basal area per ha adjusted for stand age. Tukey's test revealed that one plantation (Lalsohanra) was significantly different from other plantations at a=0.05 level. The SNK test gave the same results. On the basis of same argument given for the criterion of volume, it was decided to test the plantations similiarity for the predetermined groups. In case of the two predecided contrasts, the F value came down to 0.01 and 0.02, indicating that neither the bigger plantations group was significantly different from smaller plantations group, nor was the northern plantations group significantly different from the southern plantations group at a=0.05 level. On the basis of both the criteria of volume and basal area per ha, it was concluded that plantations would be treated as belonging to the same population for the purpose of model building. 57 Table 9. Analysis of covariance for the independent variable, basal area per ha, with age as covariate Source DF Sum of Mean F Value Squares Square Model 10 31480.8 3148.1 26.00" Error 302 36561.2 121.1 Total 312 68042.0 Plantations 9 8343.5 7.66” Age 1 23137.3 191.12” Contrast Bigger vs smaller 1 1.5 0.01 Northern vs southern 1 2.7 0.02 58 4.3 UNIVARIATE ANALYSIS The assumption of normality is standard throughout regression analysis. In order to have some idea about the distribution of data, all variables were analyzed to examine their skewness and kurtosis. A test of normality (Kolmogorov one-sample test) was used to test the null hypothesis that sample values were a random sample from a normal distribution (Steel and.Torrie, 1980). Results are shown in Table 10. Stand age, dbh, height, basal area and volume per ha did not show any departure from normality. The values of skewness and kurtosis were less than 1 for these variables and the Kolmogorov-D test indicated that the hypothesis of normality for the above mentioned variables could not be rejected at a = 0.05. Number of trees per acre and irrigation quantity (delta) had significant D-statistics at a = 0.05. The values of skewness and kurtosis were high, indicating the normality Table 10. Analysis of normality Variable Skewness Kurtosis Kolmogorov-D Stand age 0.318 -0.654 0.088 Mean stand DBH 0.369 -0.607 0.077 Mean stand height 0.159 -0.518 0.045 Mean No. of trees per hectare 1.843 4.252 0.142* Mean basal area per hectare 0.266 0.907 0.042 Mean irrigation 0.193 -0.814 0.149* Stand volume per hectare 0.398 0.075 0.045 59 assumption was not satisfactory for these two variables. Transformation of these variables is discussed in Section 4.4. Additional variables were constructed to represent specific stand conditions. Five indices of site quality were evaluated: dominant height/stand age (SQ stand basal area/stand age (SQ basal area * stand age (SQ basal area/(35-age) (SJ and (no. of trees per ha) * (stand av. DBH)/stand age (89. A denominator of 35 minus stand age was used in S4 to see the role of the time period remaining for the stand to reach rotation age in representing the site quality. Since the maximum age was 33 years, 35 years was substituted for the rotation age so as to get a positive value in the denominator. The univariate analyses for the site indices are presented in Table 11. Table 11. Analysis of normality for site quality indices. Site quality Skewness Kurtosis Kolmogorov-D 31 1.322 1.666 0.104‘ 52 1.252 1.679 0.129' 53 1.060 1.500 0.088 34 5.947 56.566 0.215‘ 55 1.974 4.077 0.155’ * indicates significance at a=0.05 60 S3, basal area per ha * age, had the minimum values for skewness, kurtosis and Kolmogrov-D. S3 will, therefore, be used in the overall model. The range of theiquantity of irrigation*water (delta) was very narrow. The biplot for this variable, therefore, indicated no specific pattern, rather an irregular trend was found. Most of the delta values were concentrated around 0.61 and 0.91 meters. Correlation. with stand. volume ‘was not significant (r= -0.06) at a=0.05. The reason for this poor correlation seems to be that the variable had been measured as an average quantity of water for the whole plantation for a particular year, and not for the individual sample plots. It was realized that relating quantity of water supplied to the stand with the stand volume was not appropriate. The effect of irrigation on stand volume may not be understood unless the age of the crop is also considered. To compare the yield of an old stand with that of a younger stand on the basis of effect of irrigation, it was felt appropriate to relate mean annual increment to the quantity of water supplied instead of total volume, i.e. quantity of irrigation as proportional to stand volume/stand age. The correlation of stand volume to the combined variable of quantity of irrigation and age improved from -0.06 to 0.47 (significant at a:= 0.05). Two more indices for irrigation were constructed on the basis of stand age and quantity of irrigation. All three indices were examined for their skewness, kurtosis and the 61 Kolmogorov-D statistic (Table 12). Table 12. Analysis of normality for irrigation indices Il (I*A) 1.467 2.819 0.136’ I2 (I/A) 1.334 1.702 0.152‘ I3 (A/I) 1.229 1.382 0.133‘ indicates significance at a=0.05 age irrigation delta The results (Table 12) indicated that the irrigation index I3 had better indices of skewness and kurtosis as well as a lower Kolmogorov-D statistic. Therefore, 13 was selected as the index of irrigation level for the plantations. 4 . 4 BIVARIATE RELATIONSHIPS All variables, of interest. were individually' plotted against stand volume in order to examine bivariate relationships. The results are summarized below. Stand age had a significant correlation coefficient (a = 0.05) of 0.65 with the stand volume. The bivariate plot of stand age with the stand volume showed somewhat a linear trend. Thus, a first degree term for age was used in the linear model. 62 Average stand dbh was significantly correlated with stand volume at the 0.05 significance level with r = 0.80. The bivariate plot showed a linear trend for dbh also. Stand height was significantly correlated with stand volume (r=0.81) at a=0.05. A linear trend was observed in the biplot, thus suggesting this variable as a first degree term in the model. The number of trees per ha had a negative correlation coefficient with the stand volume (r = -0.60) which was significant at a = 0.05 level. The negative correlation indicates that as the number of trees per ha increase, the stand volume decreases. This is the expected trend for stand growth because in young stands, the number of trees is high but the diameters are small. The biplot indicated this phenomenon as well, showing an inverse J-shape between trees per ha and stand volume. In order to linearize the relationship, the trees per ha variable was transformed. For an inverse J—shape curvature, Ott (1988) recommended the transformations of log(X) and (-1/X). These alternative transformations were tested for normality as before, the results are given in Table 13. The log transformation was selected for use in model building based on skewness, kurtosis, and Kolmogorov's D statistic. Basal area per ha showed a significant correlation with stand volume (r=0.79) at a=0.05. The biplot of basal area per ha with the stand volume indicated a linear trend. 63 The selection of the site quality index, 3w was discussed earlier. The biplot of this variable against stand Table 13. Analysis of normality for the transformed variables of number of trees/ha Variable Skewness Kurtosis Kolmogorov-D log (X) 0.207 -o.421 0.037 - (l/X) -l.081 0.778 0.117‘ volume revealed a somewhat curvilinear trend. This suggested the use of a second degree term for this variable in the overall model. The biplot of irrigation index, L“ against stand volume indicated a linear trend and was used as first degree term in the regression model building. 4.5 EMPIRICAL MODELS In reviewing growth and yield modeling, Burkhart et a1. (1981) concluded that most stand-level models were highly empirical "best fit of the data" models based on multiple regression. Titus and Morton (1985) defined models as quantitative descriptions of the physical system with one or more mathematical relationships. As already pointed out, a 64 number of researchers have developed empirical models using three basic variables - age, site quality and density. The objective of the present study was to estimate present and to predict future stand volume. Multiple regression was used to develop an empirical yield model. The variables used in the full model included mean stand age, mean stand dbh, mean stand height, mean basal area per ha, transformed mean number of trees per ha, and the constructed variables of average quantity of irrigation and site quality. 4.5.1 VARIABLE SELECTION The rationale for selecting variables for a model is to strike:a compromise between two opposing views given by Prodan (1968) and Neter et a1. (1985). Prodan (1968) suggested the use of polynomials of higher degrees for growth curves to facilitate interpolation, whereas Neter et al. (1985) cautioned about the use of higher degrees polynomials involving more than three degrees of independent variables. They gave two reasons: first, that the interpretation of the coefficients becomes difficult; and second, that such models end up with highly erratic predictions even on slight extrapolation. The second degree term for site quality index was, therefore, used in the overall model. The full multiple regression model will be of the form: Y = 130 + blA + 1620 + b3H + b4log(N) + 1358 + b6I + 1373 + 16852 (4.1) 65 where Y = mean stand volume (m3/ha) A = mean stand age (years) D = mean stand dbh (cm) H = mean stand height (meter) log(N) = log10 (mean number of trees per ha) B = mean basal area per ha (mZ/ha) I = delta index (quantity of irrigation x stand age) S = site quality (basal area per ha x stand age) There are a number of procedures available to help select independent variables for:regression.analysis. The goal of the study is to predict volume per ha through a set of predictor variables. In the selection of a set of variables from an available pool, the aim is the selection of the minimum number of variables that can account for as much of the variance as possible. The main reasons for reducing the number of predictors have been practical considerations like relative cost, specific needs and circumstances for study, and simplicity. This simplicity in description is usually referred as the principle of parsimony. Chatterjee and Price (1977) have described two advantages for this economy in description. First, the most important variables can be isolated from others, and second, it provides the researchers with a simpler description of the process being studied. There are a number of selection methods available: all possible regressions, forward selection, backward elimination, stepwise selection, and blockwise selection. In the present study, variables selection was done using all possible regressions and.backward.elimination- More than one method was 66 used to guard against screening out important variables. Draper and Smith (1981) pointed out that screening of variables should never be left to the sole discretion of any statistical procedure. The primary advantage of stepwise procedures is that they are readily available on computer systems, easy to implement and fast. Stepwise procedures have been used by some researchers like Lewis (1990), for the selection.of variables. Their main disadvantage is that each of these methods come up with a single final equation, and any inexperienced investigator may believe the equation as the best and optimal. Some researchers like Rencher and Pun (1980), Miller (1984) and.Tsai and.Hurvich (1990), indicated.that models selected by various data-driven methods can produced biased estimates. On the other hand, the all possible regressions technique identifies the subset.mode1s that may be the best on the basis of any criterion set by the investigator. Allen and Cady (1982) point out that all possible regression is more reasonable for model selection if the number of varibles is limited. The basic disadvantage for this technique is the cost and time for computation when the number of variables is large. However, this may not be any problem in this study where the number of regressors is not more than 8. Therefore, all possible selection and backwards elimination procedures were selected for the present study. 67 4.5.1.1 All possible regressions In the all possible regressions method, all regression equations are fit using one predictor variable, two predictor variables, and so on. If there are k predictor variables, there will be 2" total equations to be examined (including an intercept term). In this study, k = 8, so the number of equations examined was 256. These 256 possible models for the 8 variables are given in Appendix B. All 256 regression models were evaluated by the criteria of adjusted R2, MSE and Mallows' Cp statistic (Mallows 1973). Montgomery and Peck (1982) defined adjusted R2 as R’..,- = 1-[ (n-1)/(n-p> 1 (1-122) where n and p denote the number of observations and parameters in the model, respectively, and R2 is the coefficient of determination of the model. The base for the selection of p was the maximum R2,”. The disadvantage of using R2, the multiple coefficient of determination, as a criterion for the best- fitting regression equation is that R2 shall always increase for each independent variable even when the new variable has very little predictive power (Ott, 1988) . The adjusted R2 does not necessarily increase as additional regressors are introduced into the model. Montgomery and.Peck (1982) selected an optimum subset model by choosing the model which has a maximum adjusted R2 . 68 MSE (error mean square) is defined as: MSE = SSE / (n-p) where SSE is the error sum of squares and n and p are the number of observations and parameters in the model. Ott (1988) suggested to choose the model that has the smallest MSE. The criterion of Mallows' Cp statistic for selecting the number of regressors is related to the mean square error of a fitted value and is given by: Cp = SSEp / {MSE-(n-Zp)} where SSEP is the sum of squares for error from a model with p parameters and MSE is the mean square error from the regression equation with the largest number of independent variables. Ott (1988) indicated that the best-fitting model should have Cp z p. All models were first evaluated by plotting their adjusted R2 against number of variables in the model. This plot is shown in Figure 3. It is clear that there is a gain in terms of R2 from the ‘best' univariate model (R2=0.658) with H as the independent variable to the ‘best' bivariate model (R2=0.802) using H and B. The R2 criterion does not improve much beyond the three variable model (R2=0.805) with A, H and B, and remains relatively constant for the four and more 69 pitta NM. In... in“ 41......d1‘l" b. H0005 0:9 so muoumaouon mo nomad: momHm>~m omumcnofl .m mucon Hmcoz aw addendum> no amass: h m M — « a q q q _ _ 0 CD000 0 0 CDDLIIIID [1330 00] 0 mm 00 {HUME-ICED!!!) 0 mm mil-1000 Nd n.O v.0 0.0 wd \u.O m.O m6 axenbs-u pansntpv 70 variable models. So, the three variable model with A, H, and B, was selected as one of the subset models on the basis of R2 criterion. To include irrigation as one of the variables in the model, the four variable model with A, H, B, and I having the highest R2 (R2=0.804) was also selected for further examination. A plot of MSE versus number of variables in the model is shown in Figure 4. The minimum MSE (82682) for a univariate model was obtained using stand height as the regressor. The minimum MSE decreased for the two variable model to 47823 and continued.toldecrease for all further models. The smallest MSE (46251) was for the five variable model with A, H, B, I, and 82. It was noted that the five variable model include the variables selected through R2 criterion. Thus, the minimum MSE criterion confirmed the results obtained by adjusted R2 criterion. Thus, the five variable model with A, H, B, I and E? was also taken as a candidate for further analysis. A biplot of Mallows' Ch with the number of variables in the model (Figure 5) illustrates how the minimum Cp value drops from 243.9 to 11.8 by changing from a univariate (with Cp value ranging from 243.9 to 959.5) to a bivariate model (with Ch value ranging from 11.8 to 633.5). Increasing the number of variables to 4 decreases the minimum value of Cptx> 3.9, after which Ck again starts rising. The four variable model (minimum Cp=3.9) contains A, H, B, and $2. The minimum Cp value for a three variable model (A, H and B), is 8.4. This is 71 Hmooa we» so mumumaouom mo “mafia: m5mum> mm: H0002 ca amanmaum> mo umnfioz .¢ musmflm m DUDE-liq m H 00 1m m D OIIIJ mum 0-4 0000:) ED[] LLL L ON on 0* on On ON on cm 00— OFF ON— one ovp Omp Omp th on? our (spuesnoqm) sazenbs neon 10113 Hmooa 0:» ca muoumaouom no bones: msmum> mu .m mucosa deco: ca uoaouauo> mo Honesz s n 72 El 1 6 m (JDEJ' 1*Elpfl! 00 # ® a n 0 003] UIHDDIMIJ [fl [30 0 CD dLv- nan N6 16 Q6 Q6 N (spuasnoqm) 311813913-50 .BMOTTPH 73 much higher than 3.9, so the four variable model was selected for further examination. In order to find a model that contained the irrigation variable (I), other models were examined. The model with A, H, B, I, and S2 had a Cp statistic of 4.3 which was close to the minimum value of 3.9; therefore, this model was also included for further evaluation. In summary, five models were selected on the basis of RR. MSE, and Cp values. Regression analysis will be used to examine their residual behavior and the significance of their parameter estimates. The models were: Y = bo+ b1H+ sz 4.1(1) Y = b() + blA+ b2H+ b3B 4.1(ii) Y = b0 + blA+ b2H+ b3B+ b482 4.1(iii) Y = b() + blA+ b2H+ b3B+ b4I 4.1(iv) Y = be + blA+ b2H+ b3B+ b,I+ bss2 4.1(v) where the parameters are the same as defined earlier. 4.5.1.2 Backward elimination Although all the stepwise regression techniques including backward elimination have been critcized on various reasons, the most common being that none of the procedures generally guarantees the identification of the "best" subset regression model (Montgomery and Peck 1982), the purpose of using backward elimination procedure in this study is to compare the - u! ...-“E“! 74 results to those obtained by the all possible regression technique. Backward elimination starts with all variables included in the model. The partial F-statistic is computed for each variable as if it were the last to enter the model. The smallest partial F-statistics is compared with a preselected value of F (F-to-remove), and if the computed F-statistic is smaller, the variable is removed from the model. The partial F-statistics are again calculated for the remaining variables in the model and the procedure is repeated. This procedure terminates when the smallest partial F-value is not less than the value of F-to—remove. Backward elimination is better than forward selection because it gives the chance to see the effect of including all the candidate regressors with the sense that nothing obvious is missed. To minimize the chances of excluding many variables, the significance level for F-test was set at 0.10 instead of the conventionally used level of 0.05, thus getting a smaller value of F-to-remove. The full model (4.1) was tested by the backward elimination procedure. Log(N) was dropped out in the first step, and S, D and I were removed in the subsequent steps. The four variable model produced by backward elimination included A, H, B, and 82. The summary of the analysis is given in Table 14. The analysis showed only a marginal decrease in the adjustedIR’(0.8122 to 0.8106) when the number of variables dropped from 7 to 4. The (1 value decreased from 7.22 to 3.91, which is desirable. The four variable model with A, H, B and 82 selected by this 75 Table 14. Summary of backward elimination procedure. Step Variable Variables Model Cp F value removed remaining R2 for the in model model 1 Log(N) 7 0.8122 7.22 188.4" 2 s 6 0.8121 5.43 220.4” 3 D 5 0.8115 4.32 264.4” 4 I 4 0.8106 3.91 329.4" procedure was the same as model 4.1(iii) produced by all possible regression procedure. W—mm41‘: an mag naifiYP. CHAPTER 5 RESULTS AND DISCUSSION 5.1 MODEL EVALUATION The five models selected from all possible regressions were examined for coefficients and residuals (Table 15). All five models had comparable adjusted R2 values (from 0.802 to 0.808). The values of Ck‘varied from 3.9 to 11.8. MSE ranged from 46252 to 47823, with models 4.1(iii) and 4.1(v) having the smallest MSE. To select a final candidate model, the significance of the regression coefficients was examined. Models 4.1(iv) and 4.1(v) had at least one parameter which was not significant at a = 0.05. Out of the remaining three models, model 4.1(iii) had the lowest CP (3.9) and MSE (46342) as compared with the other two models. Although the most likely yield model for the entire Punjab zone seemed to be model 4.1(iii), the final selection of the model was made after analyzing' the residuals for' possible outliers and influential observations. 5.1.1 RESIDUAL ANALYSIS Residuals are the differences between the observed and the predicted response variable. The residuals are highly useful for studying whether a given regression model is 76 «a ’ 6’ M38" ‘74!)90’. n I 77 Table 15. Regression statistics for subset models using all observations Model F-value MSE adj-R2 (1 Parameter estimates 4.1(i): Y=80+81H+828+e 632.7" 47823 0.802 11.8 80=-74o.9” 8,- 20.6“ 82— 15.7" 4.1(ii): Y=80+81A+82H 429.5“ 47162 0.805 8.4 BO=-784.8” +133 B+6 Bl= 6.7' 62— 22.7“ 83— 16.5“ 4.1(iii): Y=80+81A+828 +B3B+B4SZ+6 329.5” 46342 0.808 3.9 Bo=-671.7" 8,= ~12.2“ 82— 23.3“ 133— 14.4“ 8,= 0.00004‘ 4.1(iv): Y=80+81A+BZH +B3B+B4I+e 321.4" 47281 0.804 10.1 Bo=-783.2” 81: -7.5‘ 52= 22.5“ B3= 16.5“ 8,: 1.8 4.1(v) Y=80+81A+828 +B3B+B4I +8532+e 264.4“ 46252 0.808 4.3 Bo=-650.2” 8,: ~15.1” 82: 23.1“ B3= 14.3“ 8,= 4.9 85= 0.0005‘ * indicates significance at a=0.05 ** indicates significance at a=0.01 78 appropriate for a set of data. Plotting the residuals against predicted values of the response variable or the predictors is helpful in examining whether the variance of the error terms is constant, and locating the outliers. The standardized residuals for the model 4.1(iii) were plotted against the predicted volume (Figure 6), and against the independent variables A, H, B and SK The plots showed no obvious pattern except some of the outliers. No obvious pattern of the residuals indicated that the homogeneity and equal variance assumptions for the regression model were valid. There were 19 residuals with studentized values more than 2 standard deviations, out of which 5 values were more than 3 standard deviations. One predicted value was negative which correspond to a stand age of 5 years. On examining the original data, it was found that 7 of the 19 outliers belonged to one plantation (Machhu-Inayat) while the other 12 observations belonged to six other plantations. The measurement years for these 19 observations were also'different from each other. Although about 33% of the outliers were from one plantation, they did not, however, concentrate at any particular year of measurements. This indicated that these outliers might be due to random measurement mistakes or instrument errors. The predicted values of the dependent variable were obtained using regression model 4.1(iii). The prediction was negative in case of one observation, corresponding to an age 79 mesao> oouofiowum mamuw> Hmsowmou cmnwucmosum .o ousmwm 034‘) Guru -00: ocvu ooo~ s. a 8N — 8. 0°C 0 0°... 0 O .--'-'--"|‘-'----.-----'--‘---.'--'--'---I---.---U'----I---'------'-'0I.-'I---‘-|-'-'-'------‘---'-’-I--'00----------‘0------'. O 0 O . O O O O O . I---'----'I-----U'--I-0’--.l'.-"-'.----.--|----.----Iva----...--‘0---"---'-'-"---'--I--------'--||‘---"-'------‘0.--'-'---'-'-. O O O. O O O O O O O O 00 O O O O O O O O 000 O O 000 O O o o 00 O O 00 O 00 00 O 00 O 00 o O . O . O . ...... ...... O 9. O .00.... 0 .0 C . I-II-IO-II----I-D--IIOII.IIOIIOIIOII-.IO-UI.-..--‘CUOOOUICOII..-...I...|.OO-......II--.I.'..'I-I-‘-IIOUIIIIU-|IOIUIIOII'III'I-. . 0 O O O O . ... O. .. .... 0. . O .. ... . O O 0. O . 0.. O. O . .... O O . .... O O O . O O. O O O. O O 0 9 O 9 O O O O O ... O... O O O O O O 09 O 00 O O O O O O 0 O O Q 9 O O O O O O O O O . .-----0---’--------|'llII--UIUIIIUUU---|"-I.I---'---I'IU-'-I---"-l---'--'|--------------l-'---'I----|-'------U'--‘I-C-------O O . QIIIOOIIIIIIIOIOIIIIltllillIll-IIOIIOOIOIIIIIOIIOIIIIIIIIIIIIOIIIII---IIUOIOIIOIIIOIOIIIIDOOIIOIIOI IIIOIIIOIIOOIIO 9 ”h3008b-INIO IUU~°3CJ 80 of 5 years and dbh of 6.4 cm. There are 15 more observations in this age class, i.e. 5 years, but none had a negative predicted value. This observation was examined and it was noticed that the corresponding measurements for independent variables were very low as compared to the rest of the measurements for that stand age. The observed dbh was smaller than the mean value by 9 standard error of the mean (SEmm). Height was 12 SEM smaller and the volume was 10 SEmcan smaller than their means. These deviations suggest that mismeasurement might be the reason for negative prediction for one value. It was, however, stated.in the future recommendations that use of the model needed special attention for younger plantations by collecting fresh data for validation in the field. Draper and Smith (1981) regard an observation as an outlier which lies more than three or four standard deviations from the mean of the residuals. Neter et al. (1985) define outliers as those observations which lie beyond four or more standard deviations from zero. The authors further add.that an observation may appear to be an outlier, but it may not have a strong influence on regression function fitted to data. The analysis of outliers to detect influential observations is, therefore, regarded as an important step in good regression techniques. 5 . 1 . 2 INFLUENTIAL OBSERVATIONS The tests available for outlier detection are DDFITS proposed by Belsey et a1. (1980), Cook's distance (Cook 1977), 81 and the modified-Cook Statistic suggested by Atkinson (1982). Wiesburg (1985) points out that all influence measures will usually give essentially the same information. Cook's distance, D, was selected as a measure of influence because of its frequent use in the literature. Cook's Di's for the 19 potential outliers revealed that all the observations had values less than 1. This indicated that they' were not influential outliers. These 19 observations, therefore, were deleted for parameter estimation purposes. Recalculation of the criteria statistics, adjusted RH MSE and C%, were done for all possible regressions. It was observed that the independent variable Szlof model 4.1(iii) with A, H, B and 82, was substituted with S on the basis of the better values of the above criteria. This new model with the independent variables of A, H, B and S was taken as model 4.1(vi) in further analysis. The variable, SE in model 4.1(v) was substituted by its first-degree term, S, on the basis of better values of adjusted R2, MSE and Cp. This new model with variables A, H, B, I and S, is termed as model 4.1(vii). Plotting of standardized residuals for all the models against the predicted values of thendependent variable as*well as the independent variables in the models was done. No obvious pattern was noticed. The regression statistics for the seven models after deleting the 19 outliers are given in Table 16. Out of the seven models, the irrigation parameter was nonsignificant for 82 Table 16. Regression statistics for subset models without outliers Model F-value MSE adj-R2 (1 Parameter estimates 4.1(i): Y=80+81H+828+e 1248.6“ 24226 0.895 39.9 130=-807.4“ l= 20.8“ 132: 16.9" 4.1(ii): Y=80+81A+82H 912.0“ 22322 0.903 15.1 Bo=-877.1" +133 B+6 Bl= -10.S°° 32: 24.0“ B3= 18.2“ 4.1(iii): Y=BO+BIA+B§H +B3B+B4S +e 715.5" 21436 0.907 4.1 BO=-768.8" 8,: -15.9” 82: 24.7“ B3= 16.2“ {34: 0.00004“ 4.1(iv): Y=BO+BIA+BZH +B3B+B4I+6 681.6“ 22399 0.903 17.1 Bo=-877.0“ 8,: -10.4” 32: 24.0" B3= 18.2" B4= 0.021 4.1(v): Y=80+81A+82H +83B+8,I +8552+e 573.1” 21420 0.907 5.0 80=-755.0” 81= -17.6“ 82: 24.5“ B3= 16.2“ 8,: 3.0 85: 0.0005” 4.1(vi): Y=Bo+BlA+BzH +8,B+8,s+e 716.7“ 21403 0.907 3.7 Bo=-661.3” 8,: -26.0“ 8,: 24.7“ B3= 13.4” B4= 0.2776” ** indicates significance at a=0.01 83 Table 16 (Cont'd.). Model F-value MSE adj-R2 (2 Parameter estimates 4.1(vii): Y=80+81A+82H +83B+8,I +858+e 573.6“ 21401 0.907 4.7 80=-641.8” 81= -28.4“ I32= 24.5” B3= 13.2“ 8,= 2.7 85= 0.2985“ ** indicates significance at a=0.01 84 models 4.1(iv), 4.1(v) and 4.1(vii) at a=0.05, even though all the three models had higher values of adjusted R2 (0.903, 0.907 and 0.907) than the remaining four models. Among these four models, 4.1(i) and 4.1(ii) had high values of Ch (39.9 and 15.1) and their MSE were on the higher side of the range. Four variable models 4.1(iii) and 4.1(vi) with variables A, H, B and $2, and A, H, B and S, had the highest value of adjusted R2 (0.907). Both of these models had all their parameter estimates significant at a = 0.05, and had the lowest CI, values (4.1 and 3.7). MSE values did not differ much for all seven models; it was, however, lowest in models 4.1(iii) and 4.1(vi). Models 4.1(iii) and 4.1(vi) were selected for additional analysis of multicollinearity, which might influence the least square estimates of the parameters. 5.1.3 MULTICOLLINEARITY One of the assumptions for multiple regression is that the independent variables (regressors) should be orthogonal, i.e. there should be no linear relationship among them. But as pointed out by Neter et al. (1985), in most sciences the independent variables tend to be correlated among themselves. Intercorrelation among regressors is termed multicollinearity. Neter et al. (1985), Belsley et al. (1980) and Montgomery and Peck (1982) have used the variance inflation factor to detect multicollinearity in the regressors. For the matrix, C = (X'X)4, the diagonal elements of C can be written as 85 Cji = (1-R2) , where R2 is the coefficient of determination when xiis regressed on the remaining p-1 regressors, and X is the matrix of independent variables. Marquardt (1970) has termed <% as the "variance inflation factor (VIF)." The VIF for each term in the:model.measures the combined effect of dependencies among the regressors on the variance of that term (Montgomery and Peck 1983). Ideally the VIF shouLd be equal to unity, indicating orthogonality in the regressors. VIF values greater than 10 can be regarded as an indication that multicollinearity may be influencing the least squares estimates (Neter et al. 1985; Chatterjee and Price 1977). Montogomery and Peck (1982) , however, suggested that VIF values exceeding 5 or 10 indicate the influence of multicollinearity on the estimates of regression coefficients. For this study, a VIF value of 10 or more will be used as an indicator of multicollinearity among the regressors. Other measures of collinearity are the values of characteristic roots, (pl, (192. (p3, . . .¢p, of the matrix X'X. This measure follows from the fact that every linear regression model with collinear regressors can be restated in terms of a set of orthogonal explanatory variables obtained as linear combinations of the original explanatory variables. If there are one or more near linear dependencies in the data, then one or’more characteristic roots (eigen values) will be small“ To avoid the descriptive sense of ‘small', Montogomery & Peck (1982) examined the condition number (CN) of X'X, defined as 86 CN = «pm/(pm. They indicated that if CN is less than 100, there is no serious problem of multicollinearity. Condition numbers between 100 and 1000 imply moderate to strong :multicollinearity, and. CN’ exceding 1000 indicates severe multicollinearity. VIF and CN values were calculated for models 4.1(iii) and 4.1(vi) along with the parameter estimates using the data without the 19 outliers (Table 17). Examination of Table 17 revealed that all VIF values for model 4.1(iii) were less than 10. For model 4.1(vi), the VIF was greater than 10 for A and S. CN values, however, were less than 100 for both.models. All of the parameter estimates were significant at a = 0.01 in both models. Although it is desireable to prefer the use of first-degree term of a variable as compared to its higher- degree terms in linear regression analysis, model 4.1(iii) was taken as more suitable model than model 4.1(vi), because the latter might influence the parameter estimates through collinearity. No model produced a significant parameter estimate for irrigation. This suggests that more reliable data for irrigation is needed for regression analysis. Irrigation water being one of the environmental factors for tree growth, its response on stand development is associated with site quality. The variable of site index used in this study' may be considered to have included the irrigation effect on stand development. Table 17. Multicollinearity 87 diagnosis Model Variable in the model Intercept A H B 52 4.1(vi) Intercept mtnuz> 1.00 3.12 9.65 16.13 17.03 1.00 5.19 11.19 16.85 37.11 5.2 COMPARISON WITH CONCEPTUAL MODELS One of the objectives of this study was to test the commonly used models available in the literature and compare them to the developed empirical model. models were tested: 1) Schumacher (1939) model: The following yield In (V) = b0 + bl S + b2 In (B) + b3 (A)'1 2) Clutter (1963) model: In (V) = 160+ b, s + b2 ln(B) + b3A+ b, s ln(B) +-kx S (A) +-k% A ln(B) 88 3) Buckman (1962) model: V=bo+blB+b2(B)2+b3A+b,(A)2+bss where V = cubic-foot volume per acre, 8 = site index, A = age, and B = stand basal area. 4) Chapman-Richards model: Y = P1[1 " exp (P2 X) ]p3 where Y is the response and X is the indicator variable, P1 is the upper asymptote, and P2 and P3 are the coefficients to be estimated. The regression statistics and multicollinearity diagonoses obtained for the first three equations using data without 19 outliers are given in Table 18, along with the empirical model. Table 18. Regression statistics and multicollinearity diagonosis for conceptual models Equation F value adj -R2 Cl, (VIF) m ( CN) m Schumacher 361.3” 0.787 4.0 3.4 50.6 Buckman 164.3" 0.736 6.0 66.9 72.0 Clutter 187.4“ 0.792 7.0 57865 2138.9 Regression model 4.1(iii) 715.5” 0.907 4.1 3.9 17.0 89 Out of these three equations, Schumacher's model gave the lowest value of Cp (4.0), which was very close to that of regression model 4.1(iii). There seemed to be not. much difference in the adjusted R2 among the three conceptual models. The VIF of Schumacher model was the minimum (3.4) and comparable to that of the regression model (3.9). Model 4.1(iii) had regression statistics which seemed better as far as adjusted R2 and F statistics are concerned. The Buckman and Clutter models had higher Cp, VIF and CN values than the Schumacher as well as the regression models. In addition to the regression statistics, the above mentioned three models evaluated for for their residuals by plotting standardized residuals against the independent variables as ‘well as jpredicted values of the. dependent variable. For the Schumacher model, the residual plots did not show any pattern except for 1/age, where a somewhat funnel shaped pattern indicated non-homogeneity of errors for the model. Among the three parameters of the Schumacher model, parameter for the site was nonsignificant at a=0.05. The Buckman model behaved better, there were no pattern in the residual plots. Four out of five parameters (square of basal area, site, age and square of age) in the Buckman model were nonsignificant at a=0.05 level. This indicated that the Buckman model did not fit the data. For the Clutter model, the residual plots showed patterns for site*log(basal area) and site*age, where residuals concentrated on the lower range of lumn..: - __ 90 the values. The parameter estimates were nonsignificant for intercept and site*age at a=0.05 level. On the basis of these analyses, parameter estimates and regression statistics, it was concluded that Schumacher model could be regarded as the one closest to the regression model developed above. The Chapman-Richards model was evaluated with the data using nonlinear regression. Only one variable was used for prediction of the response variable of volume per ha because of the complication for the interpretations of multiple parameters in. nonlinear regression. As stand age is an important variable for yield prediction, and consequently differentiating the yield model with respect to age to obtain stand growth, it was considered appropriate to use age as the independent variable for the Chapman-Richards model. In solving the nonlinear regression, it is essential to guess a priori values of the parameters to be estimated. Since 15 is the asymptote, the upper boundary of the volume per ha in the data was assigned to it. For the other two parameters, the literature was consulted and the following values were assigned as the grid search for the computer iterations: P,== 1000, 3000, 4000, 5000,- P2 = -0.01, -0.1, -0.5, 0.0, 0.5, 1.0; and P3 = 1, 2, 3, 4, 5, 10. After 19 iterations, the convergence criterion was met. The results obtained for the data without outliers are given in Table 19. The results of the Chapman-Richards model could not be compared with the 91 other conceptual models because of the nonlinear solution of the model. Table 19. Regression statistics for the Chapman-Richards model Source DF Sum of Squares Mean Square Regression 3 420147717 140049239 Residual 291 38068203 130819 Total 294 458215920 (uncorrected) Total 293 67546339 (corrected) Parameters Estimate Asymptotic Standard Error Pl 9212.1537 97317.1242 P2 -0.0017 0.0376 P3 0.5646 0.1854 The asymptotic correlation matrix for the parameters indicated that a high correlation, ranging from 0.716 to 0.966, existed between all the three parameters. In order to see whether an estimate is well determined in the model, Ratkowsky (1983) indicates that the criterion of the Student's t-value associated with the parameter estimate may be useful. Ratkowsky defines the t-value as the ratio of the parameter estimate to its standard error, the. latter being estimated by the square root of the asymptotic variance of the estimate, i.e., 92 A high t-value associated with a parameter estimate tends to indicate that the estimate is well determined in the model. The t—values in the present case are 0.0947, 0.0452 and 3.0453 for the parameters P1, P2 and P3. This indicates that only parameter P3, significant at a = 0.05, is well determined in the model and the remaining two parameter estimates are poorly determined. Although the apparent result is that the Chapman-Richards model could not function well for the prediction purposes of shisham.yield, Ratkowsky (1983) points out that sometimes a't- value in a multiparameter model may be low because of high asymptotic correlation of the parameter with other parameters in the model. This seems to be the case in the present situation. The studentized residuals could not be accomplished due to overflow of the calculations, so categorizing model fitness for different portions of data set was not possible. Keeping in view the nonsignificance of two out of three parameters, it was concluded that the Chapman-Richards model did not fit the data well. 5.3 MODEL VALIDATION Model validation is an important step to see if the model will function successfully for its intended use. Montgomery 93 and Peck (1982) indicated the following three types of procedures were useful for validating a regression model: 1. Analysis of the model coefficients and predicted values including comparisons with.prior experience. 2. Collection of fresh data with which to investigate the model's predictive performance. 3. Data splitting; setting aside some of the original data to use at later stage to investigate model's predictive performance. Of the three methods, analysis of fresh data is the most effective if the resources and time permit, which is usually not the case. Data splitting can be done in cases which involve very large number of observations. In the present study, 313 observations may be considered a large data set but taking into consideration the large extent of geographical area being covered by the yield model, it was decided not to split data. Moreover, Table 6 reveals that the number of measurements were as low as 5 for one plantation, which suggests that all measurements are needed for model building. Analysis of model's predicted values and coefficients is mostly used where the other methods are not practically possible. In this study, a comparison of the predicted yield values was done with the values obtained through a previous yield study done on a provisional basis i.e., provisional yield tables of shisham for the irrigated plantations of the Punjab as reported by Hussain and Glennie (1978). !T.Til ' 94 The predicted values of yield were calculated with the regression model using average, minimum and maximum observed values of the independent variables for each age class. The predicted values were plotted against stand age superimposing the values of the provisional yield tables for all the three site indices, SI I, SI II and SI III, (Figures 7, 8 and 9). The difference of yield between SI I and maximum predicted yield (using' maximum lobserved ‘values of the independent variables) ranged from 4.65 to 48.83 m3/ha. This difference ranged from 3.13 to 18.22 nfifha in case of SI II and average predicted yield, and from 22.28 to 39.60 nfifha in case of SI III and minimum predicted values. This indicated that model prediction was more precise for the SI II yield than the remaining two site indices. An approximation of squared errors was calculated by taking the squares of the difference of observed yield, Yw" and predicted yield, me, and the difference of observed yield and site index yield, Y“. These approximations are given in Table 20. The difference of these yields indicated that the model prediction was more precise in case of average site index. Montgomery and Peck (1982) indicated that the VIF's also are an important guide to the validity of the model. If any VIF exceeds 5 or 10, that particular coefficient is poorly estimated or unstable. In case of the regression model developed in this study, the maximum VIF is 3.9, which 95 H noose open you moaoww omuowomun can oo>u0mno .b ousmwm ..o> 5m 0 ._o> BE 1. 50> wmo 0 38.0 uo< ozfim mm mm ea ea or a. n e 4 a m a m a! 7 q 7 w d . _ _ a U L .r ..L a U D + 1 D n. D o c D o + + 1 a o o o 1 n. o + 1 a a a 4 l + 0 0 + + + + 4. + J 0 + + o + + 1 o + + a ... + I. 0 D D + + III. + Ill + II. I? ll .... L 0.. ON on O¢ on On on on om 00F ow p our On P 0.: Om — Om F on; one om? CON 0 a N (CW/urn?) awmoxx '0~v1s HH xmosw open now undue» omuOHomHQ can om>uomno .w muzuflm 392.8 6 30> BE 1. 30> mmo n. A8620 “.3 ozfiw mN VN ON or N— m 96 1 on m o 104 a + 1 on 1 on 1 on 1 oo 1 o? o 1 o: L 02 + D + L one 1 oi m o 1 one 1 of 1 on. oar (OH/WM) 314101071 .ONV1S 9.... '1' .....UF‘t HHH xmoca open you moaofla omuowooum can om>uomno .m whamflh 40> ammo + 4,2690 No.4 ozfiw 40>=5m 0 40>mm0 D 97 Nn ON VN ON 2. fi _ u OT. 09 ON On O? On On Oh Om Om Oo— OF w ON p On — 0.. P On _. Om « ON a On P Om? (Om/w'n 3) 39101071 0111115 98 Table 20. Comparison of observed and predicted yield by site index class Site Index 2(Yobs YW)2 2(Yobs Y”)2 I 12405.81 6337.07 II 46.88 1394.67 III 15482.26 3681.80 indicates the stability of parameter estimates. From both the procedures mentioned above, it is concluded that the regression model developed for shisham yield has acceptable prediction accuracy, particularly in case of average sites. -..— 5‘32? —r. . «.- W.‘FI§V\Q ., a .o CHAPTER 6 SUMMARY AND CONCLUSION Multiple regresssion was used to model shisham yield for irrigated plantations of the Punjab. The sample plantations were first divided into four groups on the basis of their extent of area and the climatic zonation. An analysis of covariance (ANCOVA) , taking age as covariate, was done to test the significance of plantations grouping. Although the ANCOVA test indicated significant difference in case of two plantations, it did not however indicate any significant results for the predetermined contrasts of plantation groups. Thus, the null hypothesis that these groups of plantations came from the same population was not rejected. Volume per ha was predicted as a function of stand age, mean stand dbh, stand height, number of trees per ha, and basal area per ha. Two more variables were constructed to represent site quality and irrigation quantity. All variables were analyzed for their normality using the Kolmogorov test of normality and measures of skewness and kurtosis. No obvious departures from normality were observed in any of the variables. 99 -ffila if 2‘ TV \Vm ‘1: VET—A-x 4|", 100 Biplots were used to examine the possible relationship of each indicator variable with mean stand volume per ha. The variable of site quality showed a somewhat curvilinear trend so a second-degree term was included in the model. Number of trees per ha was log-transformed because of its J-shaped relationship with volume. The transformed irrigation variable had an improved linear relationship with the volume. A number of indices were constructed to represent various site potentials to stand yield. The selection of a site quality index (S = B x A) was done on the basis of high correlation with volume and low values of skewness and kurtosis. After transformation, all variables were put to variable selection methods for inclusion in the final regression model. Application. of all-possible :regressions and. backward elimination methods were used to select a subset of models for the final selection. Different selection criteria such as adjusted R2, MSE and Cl, were applied. Graphical techniques were used to select the suitable values of these criteria. The general yield model developed for the Punjab zone was of the form: Y = b0+ b1A+ b2H+ b3B+ b4 52 Standardized residuals were jplotted against predicted yield, and against the independent variables to test the assumptions of regression and to detect outliers. The regression 101 assumptions were found to be satisfactory. Cook's D statistics was calculated to determine influential observations. Ninteen observations were found as outliers and were deleted from a total number of 313 observations. Residual analysis was redone and the regression coefficients were recalculated for the above mentioned model after deleting outliers. Various conceptual models were compared to the empirically developed model. The criteria of comparison used were adjusted R2, MSE, Cp, (VIF)mand (CN)“. Results showed that the regression model with the variables A, H, B, and $2 behaved comparably with Schumacher model in case of linear models. Among nonlinear models, the Chapman-Richards function was selected for comparison purposes because of its extensive use in the literature. The t-values for the parameters of the Chapman-Richard model indicated that only one out of three parameters was significant at a = 0.05 level. It was thus concluded that the linear function was more suitable for shisham yield prediction as compared to a nonlinear function. However, the Chapman-Richards function needs to be tested further by changing the number of parameters. FUTURE RECOMMENDATIONS 1. The yield model recommended in this study must be validated by using the latest collected data set from each irrigated plantation. 2. Use of the model for younger plantations needs special 102 attention, because of the one negative prediction for the 5 year old stand. It is recommended that yield data from younger plantations may be collected for cross-validation of the model. 3. In order to get growth of a stand at a particular age, the yield model can be differentiated with respect to age. 4. To have a deeper insight to the nonlinearity of Shisham yield data, if any, the Chapman-Richards model may be examined further by changing the number of parameters in the model and testing for the significance of the estimates. 5. Keeping in view the importance of irrigation water for the development of plantations, it is recommended that irrigation should be better documented. 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z. z. ouzzox» no~.n.. ooo.o onn.. omn.. ooo.o vvu.c ooo.o n «o 2.x» son..n oun.~. ouo.oc oo..~o on...n ocn..~ no..oo n «o 2.4: n.c.«v onn.v. ooo.om ooo.oo ooo.on oo..m« ooo.o» a noun» o 2.4: ooo.o. o...o ooo..u ooo.oo ooo.om ooo.o. ooo.o. a broom: o...» ooo.o ooo.u ooo.o. ooo.o. «o... ons.c. n :oo -------------- --- oouuo< :uuu ---uuu--u---u-------u---u-u- omu.on m.o..o« oo...~o. oov.«oo« ous.ooo vo...co oon..~.« o 4o) z~ ouzz~zp unm.v~. on..n oo..o« oo..o~ ooo.o «no.» n...o o co 2.:. ooo.o“ ..n.o ooo.oo o.o..o. ooo.o. ooo.o" ooo.oo o «- z~ woo<> zo..«~>uo .>.u coaam ohm moz<¢ aoaux Ao.pcouv mn< xHozmmm4 APPENDIX B: ALL No. of Variables in Model 1 (A(.000wwwwwwwwwwNNNNNNNNNNNNNNNNNNNNNNNNNNNNHF‘Hr-‘h-‘u-‘H R-Square 0.227396 0.437271 0.443435 0.495564 0.628660 0.697511 0.703065 0.730367 0.438277 0.458083 0.489461 0.526265 0.540300 0.611423 0.629811 0.642645 0.654872 0.657068 0.698252 0.710200 0.715500 0.717127 0.720187 0.731191 0.731776 0.732459 0.733583 0.738139 0.749482 0.750753 0.765534 0.791238 0.794953 0.815100 0.827894 0.895628 0.489968 0.542694 0.615546 0.621803 0.643153 0.657354 0.659830 0.672986 0.710701 0.715246 0.717489 0.720350 0.722452 111 POSSIBLE REGRESSIONS Adjusted R-Square 0.224750 0.435344 0.441529 0.493836 0.627388 0.696475 0.702048 0.729444 0.434416 0.454358 0.485952 0.523009 0.537141 0.608753 0.627267 0.640189 0.652500 0.654711 0.696178 0.708208 0.713545 0.715183 0.718264 0.729343 0.729933 0.730620 0.731752 0.736340 0.747760 0.749040 0.763922 0.789803 0.793544 0.813829 0.826712 0.894911 0.484692 0.537963 0.611568 0.617891 0.639462 0.653810 0.656311 0.669603 0.707708 0.712301 0.714566 0.717458 0.719581 MSE 178721.] 130172.0 128746.3 116687.6 85899.47 69972.68 68687.87 62372.13 130385.9 125788.6 118505.1 109962.4 106704.5 90195.53 85927.33 82948.51 80110.29 79600.58 70041.03 67267.79 66037.34 65659.86 64949.45 62395.4 62259.44 62100.98 61840.02 60782.44 58149.69 57854.64 54423.75 48457.32 47594.91 42918.58 39948.69 24226.4 118795.7 106514.9 89546.4 88088.87 83116.06 79808.34 79231.75 76167.58 67382.98 66324.26 65801.92 65135.41 64645.88 Hallows CD 2137.577 1478.133 1458.767 1294.974 876.776 660.442 642.99 557.204 1476.973 1414.741 1316.148 1200.509 1156.409 932.934 875.158 834.835 796.415 789.516 660.112 622.572 605.916 600.807 591.19 556.617 554.777 552.632 549.099 534.783 499.145 495.151 448.709 367.944 356.27 292.969 252.767 39.94137 1316.556 1150.888 921.982 902.32 835.237 790.615 782.837 741.502 622.997 608.715 601.669 592.677 586.074 Variables X25 X2 X23 X26 S3 X3 X6 X4 X2 X25 X23 X25' X2 X23 X2 X26 X25 X26 X23 X26 X25 S3 S3 X26 X2 S3 X23 83 X3 X2 X6 X3 X3 X2 X25 X3 X26 X26 S3 X25 X4 83 X25 X6 X23 X4 X23 X26 S3 X23 X6 X6 X23 X25 X25 X26 X23 X25 X26 X2 X23 X26 X25 S3 X26 X23 X25 S3 X23 S3 X26 X2 X2 X2 X3 X3 X3 X25 83 X3 X25 S3 X26 X25 X26 S3 X26 X25 S3 aoaoAAbabbwwcowwwwwwwwwwwwwonwww-wwwwwwwwwwwwwwmwwwwuwww 0.734477 0.737204 0.738175 0.738184 0.739787 0.743189 0.743649 0.749544 0.754064 0.756331 0.756399 0.761402 0.762796 0.765706 0.765755 0.765997 0.775996 0.791255 0.793760 0.796093 0.797010 0.804358 0.806474 0.815170 0.815322 0.815730 0.815960 0.818623 0.824267 0.827910 0.827924 0.829880 0.830145 0.839267 0.842531 0.844590 0.883516 0.895648 0.897089 0.897543 0.898742 0.898810 0.904164 0.641148 0.661440 0.720944 0.722495 0.738187 0.743552 0.743905 0.743954 0.757092 0.765109 APPENDIX B (Cont'd) 0.731730 0.734486 0.735467 0.735476 0.737095 0.740532 0.740997 0.746953 0.751520 0.753810 0.753879 0.758933 0.760343 0.763282 0.763332 0.763576 0.773679 0.789095 0.791626 0.793984 0.794910 0.802334 0.804472 0.813258 0.813412 0.813824 0.814056 0.816747 0.822449 0.826129 0.826144 0.828120 0.828388 0.837604 0.840902 0.842982 0.882310 0.894568 0.896025 0.896483 0:897695 0.897763 0.903173 0.636181 0.656754 0.717082 0.718654 0.734563 0.740002 0.740361 0.740410 0.753730 0.761858 112 61845.11 61209.81 60983.73 60981.55 60608.22 59815.93 59708.68 58335.69 57282.83 56754.95 56738.97 55573.82 55248.98 54571.31 54559.8 54503.49 52174.52 48620.53 48036.98 47493.47 47280 45568.47 45075.6 43050.23 43014.71 42919.8 42866.19 42245.87 40931.45 40082.91 40079.65 39624.02 39562.21 37437.61 36677.39 36197.8 27131.27 24305.49 23969.72 23864.05 23584.69 23568.86 22321.78 83872.41 79129.61 65222.03 64859.66 61192.07 59938.1 59855.47 59844.02 56773.39 54899.57 548.291 539.721 536.671 536.642 531.606 520.918 519.471 500.949 486.746 479.625 479.409 463.692 459.309 450.168 450.012 449.253 417.835 369.891 362.019 354.687 351.808 328.719 322.07 294.748 294.269 292.989 292.265 283.897 266.166 254.719 254.675 248.529 247.695 219.034 208.778 202.309 80.00112 41.88135 37.35179 35.92634 32.15777 31.94420 15.12111 843.538 779.778 592.812 587.94 538.635 521.778 520.667 520.513 479.233 454.042 X2 X6 x2 X2 X6 X6 X2 X4 X3 X2 X2 X2 X2 X3 X3 X3 X2 X3 X4 X4 X3 X4 X2 X6 X6 X4 X2 X6 X2 X3 X3 X4 X3 X3 X3 X2 X2 X4 X3 x4 X4 X4 x2 X2 X4 X25 S3 X26 X6 93 X6 X26 X25 X26 X25 s3 X6 X25 X23 x25 X4 X25 X3 X26 X3 X4 X23 83 X4 X23 . X23 X25 X23 X26 x23 93 x3 x23 X4 X26 . X25 X26 F ...__‘,.,~-’ 1' " ‘1‘.1.V1'.‘Jm . I . ‘3 S3 X26 X4 83 X25 83 X3 83 X23 X26 X23 X25 X23 X26 X4 X26 X23 83 X6 X23 X6 X25 X6 X23 X23 83 X6 X26 X6 83 X4 X23 X3 X6 X4 83 X6 X26 X4 X6 X6 X25 X6 53 X6 X23 X4 X6 X23 X25 X26 X23 X25 S3 X26 X2 X3 X2 X6 X2 X2 X2 X2 X25 S3 X26 X25 S3 X26 X6 S3 X26 X25 S3 X26 X6 X25 83 X6 X25 X26 X3 X4 X25 X3 X25 X26 hhhbkhuh-glbfihfi-bb‘bbbbbkb-finufi-ufi-bbhkb-finhéfifinbbbkhfibbbfibfifibhfibfibéb 0.765913 0.766044 0.766300 0.767256 0.777083 0.779664 0.783046 0.793934 0.797562 0.801585 0.804376 0.806201 0.808440 0.815400 0.815963 0.816498 0.817534 0.819661 0.823006 0.823698 0.825371 0.826859 0.827131 0.827536 0.827938 0.828482 0.829998 0.830174 0.830399 0.835776 0.839494 0.839705 0.844632 0.844653 0.844776 0.844981 0.845472 0.846714 0.847079 0.847790 0.848513 0.883994 0.884007 0.885429 0.897253 0.897543 0.898451 0.898774 0.898919 0.899077 0.899460 0.899880 0.899886 0.900040 APPENDIX B (Cont'd) 0.762673 0.762806 0.763066 0.764035 0.773998 0.776614 0.780043 0.791082 0.794760 0.798839 0.801668 0.803519 0.805788 0.812845 0.813416 0.813958 0.815008 0.817165 0.820557 0.821258 0.822954 0.824463 0.824739 0.825149 0.825557 0.826108 0.827645 0.827823 0.828052 0.833503 0.837272 0.837487 0.842482 0.842503 0.842628 0.842836 0.843333 0.844592 0.844963 0.845683 0.846417 0.882388 0.882401 0.883844 0.895831 .896125 .897046 .897373 .897520 .897681 .898069 .898494 .898500 0.898657 OOOOOCOO 113 54711.61 54681.03 54621.16 54397.73 52100.96 51497.87 50707.27 48162.58 47314.54 46374.28 45722.08 45295.46 44772.19 43145.37. 43013.86 42888.88 42646.69 42149.56 41367.58 41205.95 40815.03 40467.17 40403.5 40308.97 40214.89 40087.85 39733.48 39692.42 39639.76 38383.02 37514.15 37464.69 36313.13 36308.18 36279.4 36231.54 36116.95 35826.63 35741.16 35575.12 35405.99 27113.39 27110.38 26777.85 24014.39 23946.46 23734.25 23658.75 23624.88 23587.94 23498.44 23400.46 23399.06 23362.94 451.515 451.104 450.299 447.296 416.419 408.311 397.683 363.473 352.073 339.432 330.664 324.929 317.895 296.025 294.257 292.576 289.321 282.637 272.125 269.952 264.697 260.02 259.164 257.893 256.629 254.921 250.157 249.605 248.897 232.002 220.321 219.656 204.175 204.109 203.722 203.079 201.538 197.635 196.486 194.254 191.98 80.49869 80.45829 75.98781 38.83737 37.92416 35.07127 34.05627 33.60091 33.10430 31.90118 30.58398 30.56518 30.07951 X23 X25 X26 X23 S3 X26 X23 X25 S3 X4 X3 X23 X23 X25 X23 X25 X25 83 X3 X23 X26 X4 X4 X25 X26 83 X26 X23 83 X26 X25 33 X26 X4 X3 X25 S3 X23 S3 X23 X25 X26 X3 X25 S3 X23 X25 X26 X4 X25 X26 X23 X25 S3 X4 X3 X3 X6 X23 83 X6 X6 X6 X23 83 X6 X6 X6 X23 X26 X4 X26 S3 X26 X23 S3 X26 X26 X25 X25 X26 X26 X26 X23 X23 X23 X25 X23 X23 125 S3 X23 S3 X25 S3 X6 S3 X6 X23 X23 X25 X6 X26 X4 X23 S3 X26 X23 X26 X6 X25 X23 S3 X4 83 X23 S3 X25 S3 X6 X26 X25 X26 X6 X25 X6 83 X23 X26 X6 X23 X25 S3 X23 S3 S3 X26 X23 X25 APPENDIX 8 (Cont'd) mmmmmwmmmmmmmmmumwmmwmmmwmmmmmmmmmmmmmonoummalmmmmmabhbaa 0.904164 0.904186 0.904704 0.906662 0.908282 0.908427 0.743958 0.766518 0.787463 0.806441 0.809192 0.816638 0.823523 0.825434 0.826044 0.827672 0.829886 0.830427 0.833743 0.834112 0.835162 0.837604 0.839954 0.844750 0.845200 0.846128 0.846721 0.846785 0.847890 0.848349 0.848523 0.849232 0.849336 0.851262 0.852814 0.858275 0.885614 0.886098 0.887541 0.898613 0.899098 0.899493 0.899889 0.900114 0.900174 0.900182 0.900462 0.900591 0.900976 0.904187 0.904715 0.905826 0.906663 0.906856 0.902838 0.902860 0.903385 0.905370 0.907013 0.907160 0.739513 0.762465 0.783773 0.803080 0.805879 0.813455 0.820459 0.822403 0.823024 0.824680 0.826933 0.827483 0.830857 0.831232 0.832300 0.834784 0.837175 0.842055 0.842513 0.843456 0.844060 0.844125 0.845249 0.845716 0.845893 0.846614 0.846720 0.848679 0.850259 0.855814 0.883628 0.884121 0.885589 0.896853 0.897346 0.897748 0.898151 0.898380 0.898441 0.898449 0.898734 0.898865 .0.899256 0.902523 0.903061 0.904191 0.905042 0.905239 114 22399.02 22393.9 22272.82 21815.19 21436.52 21402.72 60050.97 54759.7 49847.38 45396.5 44751.3 43004.87 41390.06 40941.88 40798.88 40417.1 39897.68 39770.9 38993.11 38906.55 38660.35 38087.65 37536.51 36411.6 36305.95 36088.46 35949.28 35934.37 35675.1 35567.5 35526.73 35360.49 35336.06 34884.4 34520.23 33239.51 26827.47 26713.94 26375.5 23778.76 23665.07 23572.31 23479.39 23426.63 23412.7 23410.8 23345.04 23314.81 23224.65 22471.58 22347.65 22087.12 21890.8 21845.41 17.12105 17.05220 15.42449 9.272343 4.181735 3.7274 522.501 451.614 385.804 326.176 317.532 294.135 272.502 266.497 264.582 259.467 252.508 250.81 240.39 239.23 235.932 228.259 220.876 205.805 204.39 201.476 199.612 199.412 195.939 194.497 193.951 191.724 191.396 185.345 180.467 163.309 77.40705 75.88608 71.35199 36.56356 35.04041 33.79772 32.55287 31.84606 31.65940 31.63393 30.75297 30.34794 29.14014 19.05120 17.39096 13.90069 11.27051 10.66242 X3 X4 X3 X3 X6 X3 X6 X6 X6 X25 X6 X23 X4 X6 83 X26 X6 X26 X6 83 X25 S3 X26 X23 X25 S3 X26 X3 X23 X25 X26 X4 X25 S3 X26 X23 X25 S3 X26 X23 X25 S3 X23 X25 X26 X23 83 X26 X4 X25 X26 X23 S3 X26 X25 S3 X26 X23 X25 X23 X25 X23 X25 83 X6 X23 X25 X26 ' X23 X25 83 X6 X3 X3 X3 X3 X6 X3 X3 X4 X4 X3 X6 X4 X3 X4 X3 X3 X4 X4 X4 X3 X4 X4 X6 X4 X23 X25 X26 S3 X26 X26 83 X6 X23 83 X6 X23 X26 X4 X23 X25 X6 83 X26 X23 S3 X26 X6 X23 X25 X6 X25 S3 X23 83 X26 X23 X25 X26 X6 X25 X26 X25 S3 X26 X23 X25 83 X4 X23 X26 X23 X25 83 X4 X25 83 X4 X23 83 X6 X25 X26 X6 X23 X26 X6 X25 83 X4 S3 X26 X23 S3 X26 X25 S3 X26 X23 X25 X26 X6 X23 X25 X23 X25 83 X6 X23 S3 X6 X23 X25 X4 X6 X25 X4 X6 X23 X23 S3 X26 X25 S3 X26 4 : '11. x) ~31»: 5‘. no Wit-fli- “I _ n . 7w .' 4'" m: «zmmmmmmmmmmmmmmmmmmmmmwmmmmmmmmmwmamw ‘3 ‘3 ‘1 0.907019 0.908296 0.908430 0.908469 0.908615 0.908617 0.908671 0.908751 0.829888 0.835407 0.846785 0.848753 0.849945 0.851308 0.851501 0.853398 0.858446 0.888715 0.900191 0.900319 0.900481 0.900595 0.901674 0.905826 0.906860 0.907299 0.907566 0.908475 0.908707 0.908732 0.908776 0.908825 0.908833 0.908876 0.908996 0.909098 0.851623 0.900774 0.907754 0.908875 0.908930 0.909124 0.909160 0.909256 0.909295 APPENDIX B (Cont'd) 0.905404 0.906704 0.906840 0.906880 0.907028 0.907030 0.907086 0.907167 0.826332 0.831966 0.843582 0.845591 0.846808 0.848199 0.848397 0.850333 0.855487 0.886388 0.898105 0.898235 0.898401 0.898517 0.899618 0.903858 0.904913 0.905361 0.905634 0.906562 0.906798 0.906824 0.906869 0.906919 0.906927 0.906971 0.907093 0.907197 0.847991 0.898346 0.905496 0.906645 0.906701 0.906900 0.906937 0.907035 0.906749 115 21807.33 21507.68 21476.43 21467.22 21432.95 21432.49 21419.77 21401 40036.19 38737.37 36059.43 35596.33 35315.69 34995.05 34949.53 34503.19 33314.98 26191.25 23490.21 23460.13 23421.88 23395.19 23141.31 22163.95 21920.63 21817.46 21754.48 21540.58 21485.98 21480.15 21469.75 21458.12 21456.36 21446.24 21418 21394 35042.95 23434.57 21786.14 21521.46 21508.54 21462.56 21454.13 21431.48 21497.39 10.15236 6.137897 5.719195 5.595806 5.136704 5.130609 4.96022 4.708765 254.502 237.162 201.41 195.227 191.481 187.2 186.592 180.633 164.77 69.66526 33.60504 33.20348 32.69283 32.33657 28.94713 15.89892 12.65043 11.27316 10.43226 7.576655 6.84771 6.769844 6.631015 6.475766 6.452258 6.317193 5.940209 5.619765 188.209 33.77219 11.84151 8.320279 8.148401 7.536627 7.424459 7.123116 9 X3 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X3 X3 X2 X2 X2 X2 X3 X2 X3 X4 X3 X3 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X26 X2 X3 X26 X3 X4 X26 X2 X4 X26 X2 X3 X26 X2 X3 S3 X2 X3 X26 X2 X3 X26 X2 X3 83 X26 S3 X26 X23 X26 X23 S3 S3 X26 X6 X26 X6 83 X25 X26 X25 S3 X23 X25 S3 X26 X23 X25 S3 X26 X6 X23 S3 X26 X6 X23 X25 83 X6 X23 X25 X26 X6 X25 S3 X26 X23 X25 S3 X26 X23 X25 S3 X26 X4 X23 X25 X26 X4 X23 X25 S3 X4 X25 S3 X26 X23 X25 S3 X26 X6 X23 X25 X26 X4 X23 S3 X26 X6 X23 X25 83 X4 X6 X23 X25 X23 X25 S3 X26 X6 X25 S3 X26 X6 X23 S3 X26 X6 X23 83 X26 X4 X6 83 X26 X6 X23 X25 X26 X6 X23 X25 S3 X4 X6 X23 83 X6 X25 S3 X26 X4 X6 X23 X26 X4 X6 X25 83 X4 X6 X25 X26 X6 X23 X25 83 X4 X23 X25 33 X6 X23 X25 83 X6 X23 X25 83 X4 X6 X23 S3 X4 X6 X23 X25 X4 X6 X25 83 X4 X6 X23 X25 X4 X6 X23 X25 APPENDIX B (Cont'd) 116 VARIABLES LEGEND X2 X3 X4 X6 S3 X23 X25 X26 STAND AGE AV. 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