OVERDUE FINES ARE 25¢ pER DAY PER ITEM Return to book drop to remove this checkout from your record. VOLUME PREDICTION FROM STUMP DIAMETER AND STUMP HEIGHT FOR SELECTED SPECIES IN NORTHERN MINNESOTA By Car] Victor Byiin A THESIS Submitted to Michigan State University in partial fulfiilment of the requirements for the degree of MASTER OF SCIENCE Department of Forestry 1979 ABSTRACT VOLUME PREDICTION FROM STUMP DIAMETER AND STUMP HEIGHT FOR SELECTED SPECIES IN NORTHERN MINNESOTA By Carl Victor Bylin Regression equations and volume tables are developed and pre- sented for predicting tree volumes from measurements of stump diam- eter and stump height. Volumes are presented in cubic feet units. Pulpwood volume tables are presented for aspen (Populus tremulodies), paper birch (Betula papyrifera), red pine (Pinus resinosa), jack pine (Pinus banksiana), black spruce (Picea mariana), and balsam fir (Abies balsamea). Sawlog volume tables are presented for aspen, red pine, and balsam fir. Sample sizes ranged from 42 for balsam fir to 147 for jack pine. Data were collected from 41 logging sites. Coeffic- ients of determination ranged from .733 for the aspen sawlog volume equation to .977 for the aspen pulpwood volume equation. Regression equations were evaluated by variable stump heights and by an independ- ent test data set. Site index was not found to be a significant pre- dictor variable. Volume tables and regression equations are applicable in northern Minnesota. ACKNOWLEDGMENTS I extend my thanks and my praise to the following persons for, without their help, this thesis would not have occurred. Mr. James Blyth. North Central Forest Experiment Station, St. Paul Minnesota. Dr. Wayne Myers. Professor of Forestry, Pennsylvania State University. Members of the forest survey crew (1976-l977), North Central Forest EXperiment Station, Grand Rapids, Minnesota. Nancy Bylin. My wife, my friend, and my proof reader. Members of my graduate committee, Michigan State University: Dr. James. B. Hart, Department of Forestry Dr. Victor J. Rudolph, Department of Forestry Dr. Robert J. Marty, Department of Forestry Dr. Carl w. Ramm, Department of Forestry 11° VITA CARL VICTOR BYLIN Candidate fur the degree of Master of Science FINAL EXAMINATION: February 8, 1979 GUIDANCE COMMITTEE: Dr. James 3. Hart, Department of Forestry Dr. Victor J. Rudolph, Department of Forestry Dr. Robert J. Marty, Department of Forestry Dr. Carl W. Ramn, Department of Forestry BIOGRAPHICAL ITEMS: Born - July 20, 1946, Sioux City, Iowa Married - September 5, 1970 EDUCATION: Morningside College, 1964-1967 Iowa State University, 1967-1968, 3.5.; Mathematics Colorado State University, 1972-1974, 3.5.; Forestry Michigan State University, 1977-1979 PROFESSIONAL EXPERIENCE: Graduate Assistant, Michigan State University, Department of Forestry, 1977-1979 Forester, U.S. Forest Service, North Central Experiment Station, Grand Rapids, Minnesota, May, 1976-September, 1977 iii \ rw——“ Forester, U.S. Forest Service, Chippewa National Forest, Deer River, Minnesota, June, 1975-December, 1975 Agriculture Statistician, S.E.S.A.-Bureau of the Census, Agriculture Division, Jeffersonville, Indiana, July, 1974- May, 1975 ORGANIZATIONS: Kappa Mu Epsilon Xi Sigma Pi Gamma Sigma Delta Society of American Foresters iv TABLE OF CONTENTS LIST OF TABLES ........................ LIST OF FIGURES ....................... INTRODUCTION ......................... LITERATURE REVIEW ...................... Studies Utilizing Charts, Graphs, or Curves ...... Studies Utilizing Tables and "Rule of Thumb" Equations ...................... Studies Utilizing Regression Equations ......... Equations Using Fixed Stump Height ......... Equations Using Variable Stump Heights ....... Summary ........................ METHODS AND MATERIALS .................... Selection of Sites and Trees .............. Data Collection .................... Tree Volume Calculations ................ Statistical and Computer Methods ............ STATISTICAL ANALYSIS AND DISCUSSION ............. Equation Development and Selection ........... Equations with Transformation of Volume ........ Summary of Equation Selection and Justification . Anomalies in the Statistical Analysis ......... Commonalities of Selected Equations .......... Equation Verification ................. Equation Verification Using Varying Stump Heights Equation Verification Using Data in Different Areas ...................... TABULAR RESULTS AND DISCUSSION ................ Pulpwood Volume Tables and Their Use .......... Sawlog Volume Tables and Their Use ........... Page vii ix 37 38 47 Page CONCLUSIONS ......................... 54 APPENDICES ......................... 56 A EQUATIONS DEVELOPED FOR PULPwOOD VOLUME ........ 56 B EQUATIONS DEVELOPED FOR SAwLOC VOLUME ......... 59 LITERATURE CITED ...................... 61 vi Table 10. ll. 12. 13. LIST OF TABLES Species, Number of Sites, and Range of Site Indices Sampled ........................ Number of Trees Sampled by Species and Stump Diameter Class ......................... Mean and Range of Selected Tree Parameters for Species Studied ........................ Pulpwood Volume (Cubic Feet) Equations With Coeffic- ients of Determination (R2), Standard Errors (SE), and Sample Sizes ................... Sawlog Volume (Cubic Feet) Equations with Coefficients with Determination (R2 ), Standard Errors (SE), and Sample Sizes ..................... Results of Prediction Sum Of Squares Test, Sample Size, and Stump Diameter Range of Independent Test Data . . Number of Volume Residuals Within + One and + Two Standard Errors and Sample Size of Indepéhdent TeSt Data . . . . Aspen Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Paper Birch Pulpwood Volume (Cubic Feet) Predictions from Stamp Diameter and Stump Height ......... Hardwood Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ......... Red Pine Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ......... Jack Pine Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ......... Balsam Fir Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ......... vii Page 12 33 35 39 Table 14. 15. 16. 17. 18. 19. 20. 21. Black Spruce Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height .......... Conifer Pulpwood Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height .......... Aspen Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Hardwood Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Red Pine Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Jack Pine Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Balsam Fir Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ Conifer Sawlog Volume (Cubic Feet) Predictions from Stump Diameter and Stump Height ............ viii Page 45 46 LIST OF FIGURES Figure Page 1. General Location of Logging Sites Sampled in Minnesota ....................... 13 2. Schematic Representation of Stump Variations and Methods of Field Measurement ............. l7 ix INTRODUCTION During my experience in a national forest district, there were occasions when loggers cut over a boundary on a logging contract, an action called "trespass cutting." Since the trees are usually re- moved from the site in such cases, managers can only approximate the volume of the tree. This volume is then used to estimate the value of the trees according to current timber prices so that restitution may be made. The objective of this study was to provide acceptable prediction equations for tree volumes using measurements Of stump parameters and other site information. Other possible uses of the results are: 1. Conducting growth and yield studies 2. Evaluating growth of previous stands 3. Check-cruising on marked timber sales (used fOr trees that were cut but not marked) 4. Estimating volumes from stump tallies The criteria for selecting the best equations for such studies commonly includes small standard errors of estimate (SE), large coef- ficient of determination (R2), examination of residuals of each equa- tion, and use of the partial F-test on coefficients in the equation. Other selection criteria considered important in this study are a minimum number of variables in the equations, similar variables be- tween species, and easy measurement of variables in the field. 2 There would be a need for volume predictions of pulpwood and sawtimber for many commercial species. The estimates should be suf- ficiently accurate to give reasonable predictions of volume. Some variables that might be used to predict tree volume would include information on species, stump, and site quality. A clear definition of that part of the tree which is considered merchantable would also be important. The trend in the U.S. Forest Service today is towards utiliza- tion of the whole tree. Cubic feet volumes are better representations r3";— Of the solid wood of a tree than either cords or board feet. A prob- lem with cords is that the volumes are different in different areas of the country. A problem with board feet is that there are differ- ent log rules which produce different volumes for the same tree. To alleviate these problems and to conform with the current trend, all vOlumes calculations in this study are expressed in cubic feet units. Information collected as part of a wood utilization study in 1976 and 1977 served as the basic data for this study. The age and height parameters needed to determine site indices were obtained in 1978. The utilization study used U.S. Forest Service survey stand- ards to classify whether a tree was sawtimber or pulpwood. A tree is classified as pulpwood if it is between 4.0" and 9.0" diameter at breast height (dbh) for softwood and between 4.0" and 11.0" dbh 1 for hardwoods. A tree is classified as sawtimber if it is greater than 9.0" dbh for softwoods and greater than 11.0" dbh for hardwoods.2 1Forest Survey Handbook guidelines. 21bid. 3 Foresters would be the principal users of the volume tables and regression equations presented in this study. Extension agents and owners Of timbered lands would also be possible users. LITERATURE REVIEW Available literature on prediction Of tree volume from stump diameter was relatively sparse. The majority of the articles pre- dicted diameter at breast height, rather than volume, from stump "'1 diameter. Generally, two phases of estimation were suggested. . n- zr After dbh was predicted from stump diameter, estimated volumes were obtained by the use of local volume tables. The results from vari- ous studies have been presented in many different fOrms--charts, curves, graphs, tables, regression equations, "rule of thumb" equa- tions, or combinations Of the above. The earlier studies used charts whereas the latter studies used tables and regression equa- tions. Studies Utilizing Charts, Graphs,,or Curves Alignment charts were presented by Hough (1930) for beech in northwest Pennsylvania and by Ostrom and Taylor (1938) for beech, black cherry, sugar maple, and yellow poplar in Pennsylvania. Rapraeger (1941) presented Charts for western white pine, ponderosa pine, western larch, Douglas fir, and Engelmann spruce in Idaho using Stump height and stump diameter to predict dbh. Endicott (1959) presented a family of harmonized taper curves for eucalyptus species to provide estimation of dbh. 5 Studies Utilizing Tables and "Rfile of Thumb" Equations Some Of the dbh predictions appear in tabular form. Cunning- ham gt_al, (1947) presented two sets of tables for 15 different trees species in Pennsylvania: one showing stump diameter when dbh is known; and the other showing dbh when stump diameter is known. McCormack (1953) presented tables of predicted dbh's for yellow pine and hardwoods in Georgia and North Carolina. He used stump diameters measured at stump heights of 6, 12, 18, and 30 inches as independent variables. Eie (1959) presented taper tables for five diameter classes of silver fir, spruce, Scotch pine, Austrian pine, beech, and oak. Stump measurements were taken at a height Of one-third of the stump diameter at ground level. Decourt (1973) presented tables and graphs to predict tree volumes for eight softwoods in France. He claimed that it is impossible to obtain an unbiased and reasonably accurate estimate of volume (error less than 10%) removed in a thin- ning by subsequent measurement of the girth (diameter) Of the stump unless thinning has taken place within the last five years. Almedag and Honer (1973, 1977) presented dbh--stump diameter relationships for eleven species in eastern and central Canada, in both English and metric units. Quigley (1954) presented a table for the average number of 16 foot logs per tree by dbh and a table of gross volume by dbh and numbers of 16 foot logs. He utilized a fixed stump height of one foot in his measurement Of Central States hardwoods. Horn and Keller (1957) used a fixed stump height of 1.0 foot for sawtimber and a stump height of 0.5 foot for poletimber in his tables. They pre- sented a “rule of thumb" equation of the form: 6 dbh = (diameter of stump) - ((diameter of stump/10) +1) He also developed dbhzdiameter of stump ratios fOr softwoods, hard- woods, and aspen in Minnesota. Studies UtilizinggRegression Equations Studies using linear and multiple regression equations consti- tute the majority Of the articles. These were divided into two cat- egories: those that used stump height as an independent variable, and those that did not. Equations Using Fixed Stump_Height Ostrom and Taylor (1938), in addition to alignment charts, presented regression equations using a fixed stump height of one foot for four species. Schaeffer (1953) presented equations: dbh = 01 + 02 for oak, beech, hornbeam, and maple and dbh = 1.2(01 + 02) for elm, poplar, Scotch pine, cherry, alder, robinia, and birch where 01 and D2 are the least and the greatest diameters of the stump respectively. Church (1953), working with Virginia pine in Maryland, presented a graph based on equations regressing stump diameter and stump diameter squared on dbh for fixed stump heights of 0.5 and 1.0 foot. Vimmerstedt (1957) used a stump height of 0.5 foot with a stump diameter measurements outside bark and inside bark when he de- veloped seven regression equations and tables for yellow poplar, red maple, chestnut oak, black locust, yellow pine, and white pak. Bones (1960, 1961) presented dbhzstump diameter ratios for ponderosa pine, Douglas fir, white fir, western larch, lodgepole pine, subalpine fir, 7 and Engelmann spruce in Washington and Oregon and fOr Sitka spruce and western hemlock in Alaska. Meyers (1963), working with ponderosa pine in the Southwest, gave two equations fOr predicting dbh; one fOr immature ponderosa pine (4-11 inches stump diameter) and one for Old growth ponderosa pine (12-40 inches stump diameter) at a stump height of one foot. He gave a five step procedure to Obtain volume from estimated dbh. Valiquette (1964) presented the relationship between dbh and stump diameter at different heights fOr Abigs_baj- samea, Picea mariana, Picea glauca, Pinus banksiana, Populus tremu- 1oides, and Betula papyrifera in Canada. Decourt (1964) gave equa- tions and tables fOr the relationship between girth at breast height and butt girth for Pinus sylvestris, Pinus nigra var Austrica and corsicanna, Picea abies, Pseudotsuga taxifolia, Picea sitchensis, and Abies alba in France. Beck gt_al, (1966) presented regression equations fOr predicting dbh from stump diameter under (inside) bark and found that accurate results were obtained for Pinus ponderosa of greater than 33 inches dbh. Over- and under-estimation occurred with smaller Pinus ponderosa, Pinus lambertina, Abies concolor, Pseu- dotsuga taxifolia, and Lebocedrus decurrens in California. Kim and Yoo (1966) using linear relationships between dbh and stump diameter, found that dbh was approximately 86% of stump diameter for Pinus koraiensis, Pinus densiflora, Pinus rigjda, Larix leptolepis, Abies holophylla and various hardwood species in Taiwan. Sukwong (1971) found a Significant relationshp between dbh and stump diameter for teak (Tectona grandis) in Thailand. Lange (1973) presented tables and equations for Pinus ponderosa, Pinus contorta, Pseudotsuga taxi- folia, and Larix occidentalis in Montana. Van Deusen (1975) using 8 stump diameter measurements both inside bark and outside bark and fixed stump heights Of 0.5 and 1.0 foot, presented equations and tables for ponderosa pine in South Dakota's Black Hills. Hann (1976) using stump heights of 1.0 and 1.2 feet presented tables and equations for ponderosa pine, Douglas fir, aspen, white fir, south- western white pine, Engelmann spruce, and corkbark fir in Arizona and New Mexico. Equations Using Variable Stump Heights Stump height and stump diameter were used as variables in mul- tiple regression equations for predicting dbh by several authors. Hampf presented regression equations and graphs for white pine (1954), sugar maple (l955a), American beech (1955b), yellow birch (l955c), northern red oak (l955d), yellow poplar (1955e), pitch pine (1957a), and white oak (1957b) in the northeast. Miller (1957) developed equations for dbh using stump diameter (inside and outside bark) and stump height on lowland and hill Sites for slash pine in Georgia. McClure (1968) developed the fOllowing regression equation to pre- dict dbh of 53 species in North Carolina, Virginia, and South Caro- lina: dbh = 0 (D0 + b1 (log(H + 1.0) - log(5.5)) + b2 (log(H + 1.0) - 109(5.s))2 + b3(o 22.5 2 l l Tota1 42 147 62 53 117 120 9 15 .xeen mevmuzo agape: «memes we empmsepo .ugmpm; mo_3em mpneueegueme we xeen murmuso emHmEePo N F o.o~-o.~m m.__-o.e a.e~-a.a o.~-a.o mm. o.~m-o.o_ om.ep were mace: o.ea-o.a_ o.e-o.e a.m_-e.m _._-_.o mm. o.m_-m.a ap.mp samflwm o._m-o.m. P.__-o.~ a.mp-a.a F._-o.o Fm. m.PN-F.m am.pp aawa xaaa o.ao-o.- N.__-o.a m.--m.m m._-_.o am. m.m~-m.o am.m_ acea nae o.mm-o.o~ m.e-o.e F.o.-o.m m.o-N.o we. e.__-a.m ma.m auseam xaapm . - . . - . . - . . - . . . - . . eaten semi: mm: :2; Zoo Na 3:: SS tag o._m-o.m_ m.._-o.m m.ep-m.m N.P-_.o am. m.--~.m ma.~. :aamq magma magma mmeem magma cam: magma new: Asmace Pm uupaaz Amwuuuww Ammsucmv Apmmev Ammgucwv lucegoemz N mmwomnm Pfiaaveaa Semeaz aszam cauaEapo asspm mama xuwpmamucmeuemz mopzem voweaum mmwuwam so; memumseema mosh umpumpmm mo macem use new: .m open» 16 segment. Merchantable sawlog heights were measured to a 7.0" min- 1 imum dob for softwoods and 9.0" minimum dob for hardwoods. Mer- chantable pulpwood heights were measured to 4.0" minimum dob for 2 Merchantable sawlogs were a minimum length of eight all species. feet and merchantable pulpwood bolts were a minimum length Of four feet. Diameters were measured with either diameter tape (D-tape) or tree calipers. Bark thickness was measured with a Swedish bark guage. Site index (SI) parameters were collected during the summer of 1978. Site index parameters (age at dbh and total tree height) were measured on dominant trees within the logging sites. The site index was interpreted from regional site index curves.3 All diameters were measured and rounded down to the nearest tenth of an inch. In measuring stumps, the D-tape was located at the edge of the cut surface closest to the ground (Figures 2a, 2b, 2c, and 2d). All diameters were measured on a plane perpendicular to the centerline of the bole. Stump height was measured to the nearest tenth Of a foot, from ground level tO the point of measure- ment Of stump diameter (Figures 2a, 2b, and 2C). On a hill, measurement was on the uphill side of the tree (Figures 2d and 2e). 0n leaning trees, stump diameter was measured at the shortest length parallel to the stump (Figure 2f). When two stumps occurred on the same tree with a fork within one foot of ground level, the situation was treated as two separate trees. The 1Forest Survey Handbook guidelines. 21bid. b) Stump with sloping cut \ sdob a) Normal stump sdob sh sh \L c) Stump with uneven cut d) Hillside stump sdob - sdob sh sh e) Hillside stump with uneven cut f) Stump of leaning tree sdob sh WW sh m. 9) Double stump sh sdob = Stump diameter sdob sdob outside bark "‘ sh = Stump height Figure 2. Schematic representation of stump variations and methods of field measurement 18 stump height of trees which forked between dbh and one foot was ' measured to the lowest point in the fork which produced the two separate trees (Figure 29). Bark thickness was recorded to the nearest tenth of an inch. The length of each bole segment was measured to the nearest tenth of a foot. Total tree height was recorded to the nearest fOot and age at dbh was recorded to the nearest year. Many problems were encountered during data collection. Stump diameter was measured with a D-tape whenever possible, and by tree caliper otherwise. It was assumed that all stumps were perfectly round while, in fact, most stumps were either fluted, oblong, or irregular in shape. Measurements were taken on all stumps that were not split, regardless of their shape. Stumps cut flush with the ground, beside being irregular in shape, were difficult to measure with either the D-tape or tree cal- iper;. These trees resulted in a volume:stump diameter ratio larger than that which actually occurred. A later reference to this prob- lem is made in this report. When all or part of the bark was missing from a stump (as was comnonly caused by a feller buncher), a "best estimation" of stump diameter was made using the bark thickness and measured stump diameter. Tree Volume Calculations The volume Of each tree was calculated by using Smalian's fOr- mula for each segment and adding all segment volumes to Obtain the volume for pulpwood and for sawtimber. The formula is: 19 v = 2( L'" ((9%9192 + (9%9212)) W where dib1 = Diameter inside bark at lower or larger end (inches) dib2 = Diameter inside bark at upper or smaller end (inches) L = Length of tree segment (feet) V = Volume (feet3) Z = Summation symbol n = Pi Statistical and Computer Methods Prediction equations were developed by regression analysis. Equations are Of the form: f(V) = D0 + b1 - f(sdob) + b2 - sh + b3 - SI where f(V) = Volume, volume'], or volume2 (cubic feet) f(sdob) = stump diameter outside bark or (stump diameter out- side bark)2 (inches) sh = Stump height (feet) SI = Site index (based on age 50) b. = Regression coefficients; i = 0, 1, 2, 3 1 Regression equations were developed for each Of the six species and for several species combinations. The methodology and theory of regression equations and analysis of variance are explained by Draper and Smith (1966), Cochran and Cox (1957) and Snedecor and Cochran (1971). Statistical Packagg_for the Social Science (Nie, gt_al,, 1975) was used fOr the analysis of the data using a CDC 6500 computer. FORTRAN programs were used to construct volume tables based on the 20 regression equations. Several variations of the above equations were examined and compared for each species. The specific equations that were de- veloped for each species and combinations Of species were as fol- lows: a) V = D0 + b1 . sdob + b2 - sh + b3 - SI - 2 . b) V - b0 + b1 - sdob + b2 - Sh + b3 SI - , 2 . , c) V - b1 sdob + b2 sh + b3 51 The fOllowing equations were developed for selected species to eval- uate the potential for using them as alternative equation forms: d) V = b1 - sdob + b2 - sh + b3 - SI 2 _ 2 . e) V - b0 + b1 - sdob + b2 - sh + b3 SI 2 f) 1/V = 60 + b1 - sdob + b2 - sh + b3 - SI Volume equations were developed for both pulpwood volumes (Vp) and fOr sawtimber volumes (Vs). The criteria used in developing and selecting the best equations were: 1. The significance of coefficients in the equations based upon the partial F-test with significance level of .05 Lack of trends in scatter plots of residuals Minimum number of independent variables Small standard error of estimate (SE) Large coefficient of determination (R2) Similar equations variables between species \I 01 01 h (A) N o o o o o 0 Ease Of field measurement of variable in the equation Draper and Smith (1966, p. 163) stated "there is no unique procedure for selecting the best regression equation and personal judgment will be a necessary part of any statistical method . . . 21 The partial F-test on the regression coefficients was the first criteria used to eliminate nonsignificant variables. Differences (residuals) between the predicted volumes and the actual volumes were examined and related to the standard error of the equation. Residuals were also examined in a test procedure developed by Cady and Allen (1972). Personal judgment and experience, as well as the previously stated criteria, were used in the selection of the best equations. Resultant pulpwood vOlume equations were verified by two methods. First, calculated individual tree volumes were compared to predicted volumes based on a range of stump heights and diameters. Second, volumes based on species specific equations were compared with tree volumes of an independent data set Of the same species. These inde- pendent data sets were obtained from different logging sites. It was desirable to have similar predictor variables between species. The user of these equations and volume tables would only need to measure the same variables for all species. When species have equations of the same general form, it is desirable to compare equations to determine if they could be combined into one. Bartlett's chi-squared was used to test for homogeniety of variances. F-tests were then used to determine if two regression equations were the same. Comparison could not be made between re- gression equations having different variables or forms. STATISTICAL ANALYSIS AND DISCUSSION Equation Development and Selection Regression equations were first developed using pulpwood vol- ume as the dependent variable and different combinations Of the three independent variables. These equations were developed with non-zero intercepts. Regression equations were then developed by fOrcing the equations through the origin using stump diameter squared and stump height as the independent variables. Analysis indicated that the variable stump diameter squared was better. The partial F-test of the regression coefficients showed that, in most cases, the site index variable wasn't significant and it was not in- cluded in any equation. The equation selection, using the stated criteria, is dis- cussed in detail for the aspen pulpwood volume equation. Selection for all other equations followed the same process. All equations developed for pulpwood volume are found in Appendix A, and all equa- tions developed fOr sawlog volume are found in Appendix B. The notations used for variables in the equations are: sdob = Stump diameter outside bark (inches) sh = Stump height (feet) . SI = Site index Vp = Pulpwood volume (cubic feet) Vs = Sawlog volume (cubic feet) 22 23 2 R Coefficient of determination SE Standard error of estimate For aspen, based on a sample size of 117 trees, the equations developed were: R2 SE (1) Vp = -38.8206 + 5.5200 sdob + 17.7354 sh .875 7.38 (2) Vp - -35.2275 + 5.1032 sdob .839 8.33 (3) Vp = -3.7023 + 0.2110 sdob2 + 18.3884 sh .874 7.41 (4) Vp = 3.1512 + 0.2294 sdob2 .837 8.39 (5) Vp = 0.2031 sdob2 + 15.0507 sh .977 7.48 Other equations were disregarded because the R2 were less than .30, usually with a standard error of greater than 17 cubic feet. The regression coefficients of the variables in all equations were tested for significance using partial F-test with an alpha significance level of 0.05. Site index coefficients were not signif- icant and equations with site index variables were eliminated. Both stump height and stump diameter coefficients were significant. Equa- tions 2 and 4 are simpler forms of the equations that might be se- lected for use if only stump diameter information were available. 2 and These two equations were not considered here because higher R lower SE are Obtained by using equations 1, 3, or 5. Criteria 2, 4, and 5 were used simultaneously to determine the best equation of equations 1, 3, and 5. Studying the scatter plots of the residuals, there was a slight indication that equations 3 and 5 had a better pattern (less scattering from the equation estimates and less numbers of residuals of :_two standard error) than equation 1. There was little difference in the scatter plots between equa- tions 3 and 5 and little difference in standard error (SE) among 24 the three equations. The coefficient of determination (R2) was best for equation 5. Equation 5 was the equation selected for aspen pulp- wood volume. The same procedures were used to determine the best equations for all other species or combination of Species. Table 4 presents the equations selected for pulpwood volume (cubic feet) and Table 5 presents the equations selected for sawlog volume (cubic feet). The site index variable was not selected for jack pine because of the desire for minimum number of independent varialbes and similar equations variables between species. Equations with Transformation Of Volume Several transformations on the volume were developed. A regres- sion was fitted to pulpwood volume squared for paper birch using a sample of 120 trees: 2 Vp = -1079.1521 + 8.7424 sdob2 + 268.7942 sh + 8.710 SI; R2 = .705, SE 322.17 = -544.5193 + 8.8550 sdob 2 2 2 Vp + 291.9912 sh; R = .693, SE 327.63 sz = -440.5150 + 9.0435 sdob 2 2. R 341.63 = .663, SE Another transformation involved reciprocal of the volume as the de- pendent variable in the regression equation. This was done for black spruce pulpwood volume using a sample of 53 trees: l/Vp = .1080 - .0013 sdob2 - .0702 sh + .0025 SI; 2 R = .646, SE = .03 l/Vp = .2440 - .0013 sdob2 - .1253 sh; 25 Table 4. Pulpwood Volume (Cubic Feet) Equations1 with Coefficients of Determination (R2), Standard Errors (SE), and Sample Sizes Specjes b0 b1 b2 R2 SE Sgigle Aspen .2031 15.0507 .977 7.48 117 Paper birch .1689 5.9720 .955 5.02 120 Hardwoodz .2045 9.7315 .943 8.05 237 Red pine -14.5711 .2751 21.1505 .958 7.23 52 Jack pine .1857 5.8589 .974 5.38 147 Black spruce .1353 9.0759 .941 3.78 53 Balsam fir .1352 10.1287 .973 5.25 42 Conifer3 -13.4839 .2552 19.8723 .945 7.57 271 1Vp = b0 + b1 - (stump diameter)2 + 02 - (stump height) 2Applicable for aspen and paper birch. 3Applicable for red pine, jack pine, white pine, and black Spruce. 26 Table 5. Sawlog Volume (Cubic Feet) Equations1 with Coefficients of Determination (R2), Standard Errors (SE), and Sample Sizes . 2 Sample Spe01es b0 b1 02 R SE Size Aspen -l3.2597 .2075 20.7462 .733 9.74 61 Hardwoodz -16.8581 .2098 22.9846 .700 10.22 72 Red pine -20.4576 .2833 20.6706 .954 8.40 45 Jack pine .1651 8.3650 .967 6.99 82 Balsam fir .1206 6.8955 .961 5.92 35 3 Conifer ~22.5873 .2669 21.5344 .931 9.69 152 1VS = b0 + b] - (stump diameter)2 + b2 - (stump height) 2Applicable for aspen and paper birch. 3 spruce. Applicable for red pine, jack pine, white pine, and black 27 2 R = .615, SE = .03 1/Vp = .1788 - .0012 sdobz; R2 = .418, SE = .04 Based upon R2, the above transformations generally did not improve the accuracy of the selected equations (Table 4) for paper birch and for black spruce. Further transformations of the dependent variable were not tried. Summary of Equation Selection and Justification Stump diameter squared, instead of stump diameter, was used as an independent variable in the equations selected because it gave a better volume predictiability. Diameter squared and volume are often highly correlated. This fact was used in the judgment decision whether stump diameter squared or stump diameter was better in the selected equations. Height of the stump was used as an independent variable be- cause it contributed significantly to all equations. This agreed with the results of Nyland (1975). Reviewing all equations that had site index as an independent variable, the partial F-test on the coefficients (criterion 1) in- dicated that the site index variable was not significant enough to justify its presence in the selected equations. This trend was evident in all species equations for both pulpwood and sawlog vol- umes. The range of the site indices was limited due to the fact that the logging sites were usually on higher site index lands. Sawlog volume prediction equations for paper birch, white pine, and black spruce were not developed because of the sample sizes 28 (11, 9, and 6) were too small to make statistical inference with any degree of accuracy. Pulpwood volume prediction equation for white pine was not developed because of insufficient sample size (n = 9). The scatter plots of the combined conifer residuals indicated that balsam fir volumes were always overestimated. Therefore, com- bined conifer pulpwood and sawlog equations were developed with the exclusion of balsam fir data. The conifer equations were developed from data that included the nine white pine trees. Prediction of white pine volume can therefore be made from the conifer equations in Tables 4 and 5. Anomalies in the Statistical Analysis During the analysis of the data, certain trends emerged. The coefficients of determination (R2) were usually larger for all equa- tions that were forced through the origin. The difference in R2 ranged from .173 for paper birch to .007 for red pine in the pulpwood equations. In almost every case, the standard error was also larger for equations with non-zero intercept. For those equations with non-zero intercept, the residuals (observed volume minus estimated volume) increased in a linear trend as the stump diameter increased. The equations generally over- estimated the volumes of smaller trees (cira 4.0" - 7.0" stump diam- eters). The residuals of the equations in predicting volume of larger trees (cira 20.0" stump diameter and larger) indicated that an underestimation occurred. The red pine diameter distribution differed from that of the other species. Table 2 Shows there was little variation in the 29 1 number of trees in each diameter class. The fact that the standard errors differed between equations (see Appendix A) with stump diam- eter and those with stump diameter squared may be attributed to this distribution. 2 was best for equations with zero in- In most cases where R tercept and simultaneously SE was better for equations with non-zero intercept, the scatter plots of residuals were better for the equa- tion with non-zero intercept. it Residuals tended to increase linearly as the stump diameter I increases because larger trees have more variability in volume than smaller trees. Hann (1976) reported that the squared residuals in- crease as a linear function of stump diameter and he used the recip- rocal of stump diameter in weighted least squares regression equa- tions. Another technique that could be used to correct this trend would be to divide the data into two parts: one with smaller trees and the other with larger trees. This would create two separate equations for each Species and fOr both pulpwood and sawlog volumes. The user would then have four equations (two for pulpwood and two for sawlog) for each species and their use would be more complex. These techniques were not utilized in this study but warrant future investigation. After the volume tables were prepared using the selected re- gression equations, negative volumes resulted in the smaller diam- eter classes. These entries were eliminated because they were extra- polations outside the distribution of the species stump diameter and stump height ranges. 3O Commonalities of Selected Equations Tests were made to determine if regression Equations for any of the species were the same. Only those selected equations that had the same form and the same variables were tested. Bartlett's chi-squared (Snedecor and Cochran, 1971, p. 296) and the two tail F-test of homogeneity of variances were used to compare residual variances. The two tests produced the same results for comparison of two equations. is The results of conifer species comparison with each other and with the group conifer data (without balsam fir) showed that vari- f ances were heterogeneOus for all except the jack pine - balsam fir, and red pine - conifer combinations. Since the variances of the two combinations noted above were homogeneous, the next step was to com- pare the slopes of the two individual equations. The slopes were different using the F-test with a significance level of 0.05. The variances between aspen and paper birch were heterogeneous. The variances between aspen and hardwood were homogeneous but the Slopes were significant at the 0.05 level. Therefore, no two equations or combination of equations can be combined into one equation. Equation Verification The equations were verified using two different techniques. The first compared the volume obtained from the regression equations, using stump heights from 0.5 feet to 2.5 feet and their respective diameters at these heights, with the actual measured tree volume. The second technique used independent data from trees in areas other 31 than those used for equation development. Each Species specific equation was compared with the same Species in the other areas. Equation VErification Using Varying_Stump Heights The selected equations (Tables 4 and 5) were verified using the volume data and the diameter at stump heights of 0.5, 1.0, 1.5, 2.0, and 2.5 feet. Residuals scatter graphs were used to determine how well the calculated volumes fit the data for each test height. All pulpwood volume equations using fixed heights of greater than 1.5 feet had residuals that were large indicating that the predic- tion equations used with stump heights of greater than 1.5 feet were very inaccurate. Stump heights greater than 1.5 feet were not within the range of sampled heights as shown in Table 3. Therefore, these equations are valid only for stump heights of less than 1.5 feet and the best volume prediction occurs at the mean stump height of each species. The estimation of volume was good for fixed stump heights of 0.5, 1.0 and 1.5 feet. Good estimations of pulpwood volume for aspen were obtained with stump heights at 0.5 and 1.0 feet. Paper birch volume estimation was best at stump height of 1.0 foot and good at stump heights of 0.5 and 1.5 feet. Balsam fir volume estima- tion was equally good at stump heights of 0.5 and 1.0 feet. Black spruce, red pine, and jack pine volume estimations were best at a stump height of 0.5 foot. N Aspen and balsam fir sawlog volume prediction equations were good at stump heights of 0.5, 1.0, and 1.5 feet. Jack pine sawlog volume equation was good at stump heights of 0.5 and 1.0 feet. The —r 32 best sawlog volume prediction equation for red pine was at a stump height of 0.5 feet where sawlog volumes were overestimated by approx- imately 5 to 15 cubic feet. Equation Verification Using Data in Different Areas Regression equations for pulpwood and sawlog volumes (Tables 4 and 5) were tested with data of their respective species that were from other areas. Residuals were examined and the quantity within :_ one standard error and withing :_two standard errors were noted. Cady and Allen's (1972) predictions sum of squares test was also used. The test used the following equation: c 2 T2 = 2(Vi "' Vi) ‘ 2v. - (2V1)4 1 n where Vi = Volume from the independent data set Vi = Volume from the prediction equation n = Numbers of Vi's The regression equation being tested has a better fit as T2 approches zero. Table 6 presents the results of this prediction sum of squares test of the selected equations with their respective in- 2 2 dependent test sets. Balsam fir volume equations had the best T values. Black spruce pulpwood volume equation had the largest T value of all pulpwood volume equations tested. This value is not conclusive as to whether or not the equation is a “good fit" be- cause of the small sample. Although balsam sawlog volume equation 2 has the best T value, it also was not conclusive due to small test sample size. Paper birch, jack pine, and aspen pulpwood 33 Table 6. Results of Prediction Sum of Squares Test, Sample Size, and Stump Diameter Ranges of Independent Test Data Stump Species 12 5:?gle Dagmgger (inches) Pulpwood equations1 Aspen .233 41 5.0-18.7 Paper birch .183 18 6.0-15.7 L Black spruce .305 7 8.2-13.7 "5 Balsam fir .158 19 5.5-15.3 L Jack pine .224 23 9.0-23.4 Sawlogequations1 Aspen .479 25 10.8-18.7 Jack pine .428 21 10.5-23.4 Balsam fir .067 4 10.8-16.3 1Equations from Tables 4 and 5. 34 volume equations, in respect to their T2 values, were concluded to have good fits. The T2 values for jack pine and aspen sawlog volume equations, .43 and .48 respectively, indicate that the residuals were large for the above equations and that these equations did not fit their respective independent data sets as well as the other equations. Another similar test involved the examination of the distri- bution of the residuals. A good prediction equation would have the 1s residual values near zero. As the residual's values deviate from zero, the accuracy of the predictability of the equation decreased. (1 A good prediction equation would have all residuals within : one standard error. All of the fOllowing species were tested with their respective pulpwood volume equation. Balsam fir, with 19 test trees, had all 19 residuals (100%) within :_one standard error. Paper birch, with 18 test trees, bad 9 residuals (50%) within :_one standard error and 18 residuals (100%) within :_two standard errors. Jack pine, with 23 test trees, had 9 residuals (39%)within ione standard error and 17 residuals (74%) within :_two standard errors. Black spruce, with 7 test trees, had 4 residuals (57%) within :_0ne stand- ard error and 5 residuals (71%) within : two standard errors. As- pen, with 286 test trees, had 185 residuals (65%) within :_one standard error and 271 residuals (95%) within 3 two standard errors. Table 7 presents the number of residuals and percentages within :_one standard error and within :_two standard error for test data of pulpwood and sawlog of selected species. \ Testing sawlog volume equations using their respective data 35 Table 7. Number of Volume Residuals Within :_0ne and :_Two Stand- ard Errors and Sample Size of Independent Test Data Number (Percent) of Species nggle Residuals Within :lSE 3;st Pulpwood Balsam fir 19 19(100%) 19(100%) Paper birch 18 9(50%) 18(100%) Jack pine 23 9(39%) 17(74%) Black spruce 7 4(57%) 5(71%) Aspen 286 185(65%) 271(95%) Same Ba1sam fir 4 4(100%) 4(100%) Jack pine 21 10(48%) l7(81%) Aspen 80 56(70%) 71(89%) 1Standard Error 36 sets, the following residuals distributions were obtained. Balsam fir, with 4 test trees, had all 4 residuals (100%) within :_one standard error. Jack pine, with 21 test trees, had 10 residuals (48%) within :_one standard error and 17 residuals (81%) within 1 two standard errors. Aspen, with 80 test trees, had 56 residuals (70%) within 1 one standard error and 71 residuals (89%) within 1 two standard errors. Balsam fir had the best residuals distribution. Aspen and 1. paper birch had good residuals distributions. Jack pine and black spruce residuals distributions were not as good as the others. The residuals distributions were an indication of the "good- ness of fit" or accuracy of the prediction equations. The conclu- sion was that balsam fir, aspen, and paper birch prediction equa- tions had good predictability of volume. Jack pine and black spruce predictability of volume equations were not as accurate as other species prediction equations. It is noted that this comparison of residuals distribution results were analogous to the T2 test results. Based upon the T2 test and the residual distributions compar- ison test, several equations were considered to be accurate fOr the prediction of volume. Pulpwood volume equations were acceptable for aspen, paper birch, balsam fir, and black spruce. Sawlog volume equations were acceptable for aspen, jack pine, and balsam fir. Red pine did not have an independent test sample. Its equa- tions were accepted on the comparison of various stump heights pro- cedures. TABULAR RESULTS AND DISCUSSION The volume tables in this section were calculated directly from the selected regression equations found in Tables 4 and 5. The range of stump heights (Table 3) of each species were usually between 0.0 and 1.5 feet with the majority were between 0.0 and 1.0 feet. The mean stump height was approximately 0.4 foot. The pre- dictions using stump heights of greater than 1.5 feet were very inaccurate. Therefore, the stump heights in the volume tables ranged from 0.0 to 1.2 feet due to the above reasons. Trees with flaring characteristics and those that were measured at stump heights less than 0.2 feet often have estimated volumes larger than the actual volume. If a stump is measured within the flaring part of the stump, than the volume tables should be used with caution. The volume tables and regression equations can be used with confidence in northern Minnesota. The validity of the equations should be tested in any other area by using a sample Of trees of the appropriate species within that area. These tables are valid only for the prediction of volumes of aspen, paper birch, red pine, jack pine, balsam fir, and black spruce. Volume predictions for white pine can be obtained from the conifer volume tables. The hardwood volume tables are only valid for the prediction of aspen and paper birch. The conifer tables 37 38 are valid for the prediction of volume of red pine, jack pine, white pine, and black spruce, but not balsam fir. To use the volume tables, stump diameter outside bark and stump height are required. Stump diameter is measured to the nearest inch and stump height is measured to the nearest tenth of a foot. The intersection of the stump height and stump diameter is the tree's predicted volume (cubic feet). If either the stump height or stump diameter are not presented in the volume table, than interpolation may be used. The regression equation can be used to calculate the volume as an alternative method. Caution is necessary if one extrapolates equations beyond the ranges given in the tables. Pulpwood Volume Tables and Their Use Pulpwood volume tables are presented for aspen (Table 8), paper birch (Table 9), hardwood (Table 10), red pine (Table 11), jack pine (Table 12), balsam fir (Table 13), black spruce (Table 14), and conifer (Table 15). This cubic feet volume could be converted into cords. A cord is defined as a stack of logs 4 feet by 4 feet by 8 feet of wood, bark, and air. The range of solid wood in a cord is 60 to 95 cubic feet. Some typical values used fOr a cord are 79 cubic feet in the Lake States, 72 cubic feet for southern pines, and 79 cubic feet for pulping hardwoods (Avery, 1967). 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LOHOEO ..O .NeeNO NNN.ez NsNNN NNNNN NOO.NO OENOO ONO smumeN.O OEONO NOON Nco.uu.umge .uomm o.n=OV we:.o> eoozn.ae N.O ENN.NO .O. m.ON. 45 NN.N n secsN NNNNNNNN .eO. n .va :O.e~:.2gmumO No uco.u.mmmou .NNN.5= NNNNNO NNNN.N + N.NNNNNN.N NNNNNO NNN.. n NNN.N> OO eO OO NO .O OO OO OO OO NO OO OO eO ON OO OO OO NO NO OO eO OO NO NO .O OO Oe O. OO eO OO NO .O OO Oe Oe Ne Ne Oe Oe ee O. OO Oe Oe Ne Oe Oe Oe ee Oe Ne .e oe OO N. Oe Oe ee Oe Ne .e oe OO OO NO OO OO OO O. .e oe oe OO OO NO OO OO eO OO NO .O OO O. NO OO OO OO eO OO NO .O OO ON ON NN NN e. eO OO NO .O OO ON ON NN ON ON ON eN ON O. OO ON ON ON NN ON ON eN ON NN .N ON O. N. NN ON ON ON eN ON NN .N ON O. O. N. O. .. eN eN ON NN .N ON O. O. N. O. O. e. e. O. NN .N ON O. O. N. O. O. O. e. O. N. .. O ON O. O. N. O. O. e. O. N. .. O. O. O O O. N. O. O. e. O. N. .. O. O O O N N O. O. e. O. N. .. O. O O O N O O O e. O. N. N. .. O. O O N O O e O O O. N. .. O. O O O N O O e i O N e N.. ... O.. 0.0 0.0 N.O 0.0 0.0 e.O 0.0 N.O ..O 0.0 .NNONNNO emumENNO .Neeev NNN.NN NNNNN NNNNN NOO.m: OENNO OON LmumeN.O OEOOO seem Neo.uu.umse .ummm N.ONOO we:.o> Ooo3N.=O moseNO xuN.O .e. N.ON. um.%En gage—m GLOUCOHW OO.—um. u Ammv :Owumcwfihmug h..O HcmewLJNmoo .NOO.m= New. ONN0.0. + N.NmumeNNO ENNOO NONN. + OOOe.O.1 n we:.o> moasam on.O OON .mc.N NN.N: .mc.q NNNO .o=.N Owe LON N.ONo..NN<. N.N O.N O.N ..N OON NON OON OON .ON ON NON OON OO. OO. OO. OO. .O. OO. NO. ON OO. OO. eO. NO. OO. ON. NN. ON. ON. .NN ON. ON. .N. OO. NO. OO. OO. .O. OO. ON NO. OO. OO. OO. eO. NO. OO. Oe. Oe. ON Oe. Ne. Oe. Oe. .e. OO. NO. OO. OO. eN NO. OO. OO. .O. ON. NN. ON. ON. .N. ON ON. eN. NN. ON. O.. . O.. e.. N.. O.. NN O.. O.. ... OO. NO. OO. OO. .O. OO .N eO. NO. .O. OO NO OO OO .O OO ON ON OO .O OO NO OO OO .O ON O. OO OO .O ON NN ON ON .N OO O. ON eN NN ON OO OO eO NO ON N. OO OO eO NO OO OO OO eO NO O. OO OO OO eO NO ON Oe Oe ee O. NO OO Oe Oe ee Ne .e OO NO e. Oe ee Ne Oe OO OO eO NO OO O. OO NO OO OO .O ON NN ON ON N. OO .O ON NN ON ON .N O. N. .. ON ON eN. NN ON O. O. e. N. O. ON .N O. N. O. O. .. O N O O. N. O. O. .. O N O O O O. O. .. O N O O . O N N. O. O O e N O O O O O N O O . O O O O O O O O . O O O O O e N.N N.N N.N N.N N.N N.N N.N ..N N.N .Neeee.v ewumEN.O .Neuev NNN.e= NENNN NENNN NOO.m: OEOOO OON empoeN.O OENNO SOLO Neo.uo.umee .umom N.OOOV we:.o> Ooo3N.:N LNN.OOO .O. N.ON. . 47 Sawlog_V01ume Tables and Their Use Sawlog volume tables are presented for aspen (Table 16), hardwood (Table 17), red pine (Table 18), jack pine (Table 19), balsam fir (Table 20), and conifer (Table 21). The volume of paper birch sawlogs can be obtained from the hardwood sawlog volume table. The volume of white pine and black spruce sawlogs can be obtained from the conifer sawlog volume table. The volume for black spruce sawlogs should be used with caution in the conifer sawlog volume table because black spruce sawlog diameters were near the low diam- eter classes. There were a wide range of sawlog heights for each stump diam- eter Class. The sawlog volume tables gives the volumes for the average sawlog height. The range of the sawlog heights are pre- sented in Table 3. The unit of measurement for sawlog volume is cubic feet. To convert volume into board feet, multiply it by twelve and compen- sate for saw kerf. This conversion would not produce the same re- sults for all log rules (i.e. International 1/4 or Scribner rules). If one would want the sawlog volume of a removed aspen tree which has a stump diameter of 20.0" and a stump height of 0.6 feet, Table 16 gives a volume of 82 cubic feet. 48 em.m u NOLLm ULOUCOHW NNN. H .mm. 55.555.5e555N .5 555.5...55N .NNN.5= NNNNNO NN5N.NN + N.N5N5NNNN 555N. NNNN. + NNNN.N.- n 555.5. NN. 5N. NN. NN. NN. NN. 5N. NN. NN. NN. NN. NN. .N. NN 5N. NN. NN. NN. NN. 5N. NN. NN. NN. NN. 5N. NN. N5. NN NN. .N. NN. NN. NN. NN. NN. N5. N5. 55. N5. N5. NN. NN NN. NN. N5. N5. 55. N5. NN. NN. NN. NN. .N. NN. NN. NN .5. 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N.ON. 49 NNN. n .555.5= 55=NNO N5NN.NN + N.N55555.N O NN.N. n c5ceN 5555555N NO 55.555.55555N .5 555.5...555 555NO NNNN. + .NNN.N.- u 555.5. .zoc.n Luann O55 :mONN sow w.amo..na<. NN. NN. NN. NN. NN. NN. NN. .N. NN. NN. 5N. NN. NN. NN NN. NN. .N. NN. NN. 5N. .N. NN. NN. NN. NN. NN. N5. NN 5N. .N. NN. NN. 5N. NN. NN. N5. N5. N5. .5. NN. NN. NN NN. NN. N5. N5. N5. .5. NN. NN. 5N. NN. NN. NN. .NN. NN N5. N5. NN. NN. NN. NN. NN. NN. NN. .N. N.. N.. 5.. NN NN. NN. NN. NN. NN. NN. N.. N.. N.. ... NN. NN. 5N. 5N NN. N.. N.. N.. N.. N.. NN. NN. NN. .N. NN NN 5N NN N.. N.. NN. NN. NN. .N. NN NN 5N NN NN NN NN NN NN. .N. NN NN 5N NN NN NN NN NN NN NN NN .N NN NN NN NN NN NN .N NN NN 5N NN NN NN NN NN 5N NN NN NN NN NN NN NN NN NN .N NN N. NN NN 5N NN NN NN NN NN NN NN NN NN .N N. .N NN NN 5N NN NN NN NN NN .N N5 N5 55 N. 5N NN NN NN NN NN .N N5 N5 55 .5 NN NN N. NN NN NN .N N5 N5 55 N5 N5 NN NN NN NN N. NN NN N5 N5 N5 N5 NN NN NN .N NN NN 5N 5. N5 N5 N5 NN NN NN NN NN NN NN NN .N N. N. .5 NN NN 5N NN NN NN NN NN NN N. N. N. N. NN 5N NN NN NN NN NN NN N. N. N. .. N .. N.. ... N.. N.N N.N N.N N.N N.N 5.N N.N N.N ..N N.N .55555.. swam—OO....— .555NO 555.5: 5555N 5555N ”EOE: 9.5.3 .O.—O ..wumEOE O..—sum EON... 23:039.; 30mm 0333 0...: .o> OO....OO @003..ng .N. v.22. . 50 oe.O n LOLLO OLNOONNO E eOO. n .mmv co.uec.eemumo No 555.5.NNNOO .N:O.52 Nanamv OON0.0N + N.NNNNENNO seam. OOON. + ONOe.ON1 n me=.o> OeN .eN OON OON eON NON OON ONN ONN eNN NNN ONN O.N ON ONN eNN NNN ONN O.N O.N e.N N.N O.N OON OON eON NON ON ..N OON NON OON OON .ON OO. OO. eO. NO. OO. OO. OO. NN OO. eO. NO. OO. OO. OO. OO. .O. ON. NN. ON. ON. .N. ON .O. ON. NN. ON. ON. .N. OO. NO. OO. OO. .O. OO. NO. ON OO. OO. OO. .O. OO. NO. OO. OO. .O. Oe. Ne. Oe. Oe. eN eO. NO. OO. Oe. Oe. ee. Ne. oe. OO. OO. eO. .O. ON. ON .e. OO. NO. OO. OO. .O. ON. NN. ON. ON. .N. O.. N.. NN ON. NN. ON. ON. .N. O.. N.. O.. O.. ... OO. NO. eO. .N O.. O.. e.. ... OO. NO. OO. OO. .O. OO NO OO OO ON NO. OO. NO. OO. OO OO eO NO OO OO OO eO NO O. OO eO NO OO OO OO eO NO OO ON ON ON .N O. OO eO NO OO ON ON eN NN ON OO OO OO .O N. NN ON ON .N NO NO eO NO ON OO OO eO NO O. OO OO eO NO OO OO OO eO NO Oe Ne Oe Oe O. OO OO OO eO NO ON Ne Oe Oe .e OO NO OO e. NO OO Oe Oe ee Ne oe OO OO eO NO ON NN O. Oe Oe .e NO NO OO OO .O ON NN eN NN ON N. NO NO eO NO OO ON ON eN NN ON O. O. e. .. OO .O ON ON eN NN ON O. O. e. N. O. O O. NN ON ON .N O. N. O. O. .. O N O N O N.. ... O.. 0.0 0.0 N.O 0.0 0.0 e.O 0.0 N.O ..O 0.0 .NNONNNO emumeNNO .uomNV NOO.5: OENNO OEONO NOO.5= OENNO OON ewumsN.O OENNO soem NOONNNNOmeN .NNNN N.O:Ov ms=.o> OO.3NO NN.N ONN .O. «.ON. 51 OO.N n eoeem OLNOONOO NNN. n .NNV 55.555.55555N .5 555.5.5555N .NNN.5= N5NNNO N5NN.N + N.N5N555.N 5555NO .NN.. n 555.55 Oe. Oe. Ne. Oe. Oe. Oe. ee. Oe. Ne. .e. oe. oe. OO. ON OO. OO. OO. NO. OO. OO. eO. eO. OO. NO. .O. OO. ON. ON OO. OO. ON. ON. NN. ON. ON. ON. eN. ON. NN. .N. ON. NN NN. .N. ON. O.. O.. N.. N.. O.. O.. e.. O.. N.. N.. ON O.. N.. N.. ... O.. OO. OO. NO. NO. OO. OO. eO. OO. ON OO. eO. OO. OO. NO. .O. OO. OO OO OO NO NO OO eN NO NO NO NO eO OO NO NO .O OO OO OO NO ON ON OO OO NO NO OO OO eO NO NO NO .O OO NN OO NO .O OO ON ON ON NN ON ON eN eN ON .N ON ON eN eN ON NN .N ON OO OO OO NO NO ON ON ON OO NO ON ON OO eO OO NO .O ON OO O. eO NO NO .O OO OO OO OO NO NO OO eO OO O. OO NO NO OO eO eO NO NO .O OO Oe Oe Oe N. NO .O .O OO Oe Oe Ne Oe Oe Oe ee Oe Ne O. Ne Oe Oe Oe ee Oe Ne .e O5 oe OO OO NO O. Ne Ne .e oe OO OO NO NO NO OO eO OO NO e. OO NO ON OO OO eO NO NO .O OO OO ON ON O. eO OO NO .O OO ON ON ON NN ON ON ON eN N. OO ON ON ON NN ON ON eN ON NN NN .N ON .. NN ON ON eN ON NN NN .N ON O. O. N. N. O. ON ON NN .N ON O. O. O. N. O. O. e. O. O N.. ... O.. 0.0 0.0 N.O 0.0 0.0 e.O 0.0 N.O ..O 0.0 .Nocuc.v emamse.O .NNNNO p:O.N: OENNO OENNO NOONN= OENNO e55 LN»¢EN.O asaum seem Nco.uu.emee .NNNN N.NNOO mE:.o> Oo.3NO NN.N some .O. N.ON. 52 NO.O n eoeem OLNONNNO NOO. u ANN. OO.NOONELOHOO NO OOONONNNOOO .NNN.5N 5555NV NNNN.N + N .emmeN.O.Ne:uOv OON.. n 55:.o> N.. ... N.. N.N N.N N.N N.N N.N 5.N N.N N.N ..N N.N .5555. 555.5: 5555N .Nmnoc.v ewumsN.O OEOOO HOONNI OEONO O55 empweN.O OENNO seem Nco.pu.umse .NNNO 5.N:OV we:.o> Oo.3NO 5.N ENN.NO .ON m.OON 53 .NNN.5N 5555NO 55NN .N + N.N55555.N .OO. u ..N. OO.O u eoegm OeNOONuO NO 55.555.55555N .5 555.5..N55N SOHO. OOON. + ONOO.NN1 u we=.o> .moasnm xue.n OON .o:.N xuNO .55.N NO.O: .me.N Owe so» N.ONN..NN<. ‘ NNN NNN NNN .NN N.N N.N N.N N.N N.N NNN NNN 5NN NNN NN N.N N.N NNN NNN 5NN NNN NNN NN. NN. NN. .N. NN. NN. NN NN. NN. NN. .N. NN. NN. NN. NN. .N. NN. NN. 5N. NN. NN 5N. .N. NN. NN. NN. NN. .N. NN. NN. 5N. NN.. NN. NN. NN NN. NN. NN. 5N. .N. NN. NN. NN. NN. .N. N5. N5. -.55. NN NN. NN. NN. NN. N5. N5. 55. N5. N5. NN. NN. NN. .N. 5N 55. N5. N5. NN. NN. 5N. .N. NN. NN. NN. NN. .N. N.. NN NN. NN. NN. NN. 5N. NN. N.. N.. N.. N.. ... NN. NN. NN .N. N.. N.. 5.. N.. N.. NN. NN. 5N. NN. NN NN NN .N N.. NN. NN. 5N. .N. NN NN NN NN .N NN NN 5N NN NN. NN NN NN .N NN NN NN NN NN NN NN 5N N. NN NN NN NN .N NN NN NN NN NN NN NN 5N N. NN NN NN 5N NN NN NN NN NN .N NN NN NN N. NN NN NN NN NN .N NN NN 5N NN NN N5 N5 N. NN .N NN NN NN NN NN N5 N5 55 N5 N5 NN N. NN NN .N N5 N5 N5 N5 N5 NN NN 5N NN NN 5. N5 N5 55 N5 N5 NN NN NN .N NN NN NN NN N. N5 N5 NN NN NN .N NN NN 5N NN NN N. N. N. NN NN .N NN NN NN NN NN N. N. 5. N. N. .. NN NN NN NN .N N. N. N. N. .. N N 5 N. NN NN .N N. N. 5. N. N. N N N . N N N.. ... N.. N.N N.N N.N N.N N.N 5.N N.N N.N ..N N.N .55555.. $3.855 .N55NO NNN.5= 55=5N 5555N NO0.00 OEOOO OON emumEN.O OEONO seem Nco.uu.OmeO .umou u.O=OV mE:.o> OO.35O LNNNOOO ..N m.ON. . CONCLUSIONS 'The regression equations and volume tables developed and pre- sented can be used to predict volumes of missing trees. Only stump diameter outside bark and stump height are needed to use these tables. The regression equations can also be used to predict vol- umes. To use the tables with confidence in areas other than north- ern Minnesota, sample trees must be used to verify the accuracy of the eqUations. The equations and volume tables tend to underestimate the volume of smaller trees (less than 8.0" sdob) and overestimated the volume of larger trees (greater than 20.0" sdob). These errors are not serious if the stump height of the smaller trees are at least 0.4 feet. Estimation of the volume of a tree between 8.0" and 20.0" sdob one can be confident that 68% of the time the actual volume will be within :_one standard deviation and that 95% of the time the actual volume will be within :th0 standard deviations. The sawlog volume tables and regression equations predicted volumes with less accuracy than the pulpwood equations due to the wide range of sawlog heights and the range of the diameters at the top height (Table 3). Volume equations fOr white pine were not developed due to an insufficient sample size. White pine sample trees were incorporated in the development Of the conifer volume equations and tables. Paper 54 55 birch sawlog volumes can be predicted from the hardwood sawlog vol- ume equation and table. Black spruce sawlogs were incorporated in the development of the conifer sawlog volume equation. This equa- tion could be used to predict black spruce sawlog volumes although the predicted volume would not be very accurate because the range of black spruce was near the low end of the stump diameter class. Other areas of investigation are suggested by this study. Cur- f tis and Arney (1977) used weights in the prediction of dbh from stump diameter. The trends of linearly increasing residuals with stump diameters indicates that better equations may be developed by using weighted regression equations. Another approach to improve the re- gression equations would be to separate the sample into two diameter ranges and develop separate equations for each range. These areas warrant future studies. APPENDICES APPENDIX A EQUATIONS DEVELOPED FOR PULPWOOD VOLUME APPENDIX A EQUATIONS DEVELOPED FOR PULPWOOD VOLUME Species: Aspen Sample size: 117 R2 SE Vp = -38.8206 + 5.5200 sdob + 17.7354 sh .875 7.38 Vp - -35.2275 + 5.1032 sdob .839 8.33 Vp = -3.7023 + .2110 sdob2 + 18.3884 sh .875 7.41 Vp = 3.1512 + .2294 sdob2 .837 8.39 Vp = .2031 sdob2 + 15.0507 sh .977 7.48 Species: Paper birch Sample size: 120 Vp = -20.0575 + 3.7777 sdob + 5.0434 sh .792 4.93 Vp = -18.3278 + 3.8583 sdob .754 5.33 Vp = -.3450 + .1709 55552 + 5.1380 sh .782 5.04 Vp = -l.8488 + .1745 sdob2 .744 5.44 Vp - .1589 sdob2 + 5 9720 sh .955 5.02 Species: Aspen and paper birch combined Sample size: 237 (Hardwood) Vp = -5.4581 + .2258 sdob2 + 11.2335 sh .851 7.52 Vp = -1.7741 + .2377 sdob2 .834 8.19 Vp = .2045 sdob2 + 9.7315 sh .943 8.05 56 ———- 57 Species: Red pine Sample size: 62 R2 SE Vp = -54.5973 + 7.7598 sdob + 23.5302 sh .939 10.15 Vp = -53.3709 + 8.5187 sdob .918 11.59 Vp = -14.5711 + .2751 sdob2 + 21.1505 sh .968 7.23 Vp = -9.1861 + .3000 sdob2 .952 8.93 Vp = .2531 sdob2 + 7.0579 sh .975 10.09 Species: Jack pine Sample size: 147 ' Vp = -45.4575 + 4.5504 sdob + 7.5991 sh + .2991 SI .921 5.05 Vp = -31 5157 + 4.8553 sdob + 12.9914 sh .911 5.34 Vp = -28.7475 + 4.9592 sdob .893 5.85 Vp = -l9.9108 + .1884 sdob2 + 5.2481 sh + .3052 SI .927 4.87 Vp - -3.5049 + .1955 sdob2 + 11.5501 sh .917 5.18 Vp = - 5739 + .2005 sdob2 .902 5.50 Vp = .1857 sdob2 + 6.8689 sh .974 5.38 Species: Black spruce Sample size: 53 Vp = -23.0927 + 3.3282 sdob + 18.0544 sh .759 3.15 Vp = -13.2445 + 3.2032 sdob .542 4.39 Vp = -9.4545 + .1953 sdob2 + 18.3115 sh .781 3.07 Vp = .0778 + .1859 sdob2 .548 4.35 Vp = .1353 sdob2 + 9.0959 sh .941 3.78 Species: Balsam fir Sample size: 42 Vp = -34.8265 + 4.4332 sdob + 14.2794 sh .856 4.92 58 R2 SE Vp = -29 5703 + 4.332 sdob .783 5.97 Vp = -8.5492 + .1590 sdob2 + 15.0590 sh .870 4.57 Vp = .8002 + .1548 sdob2 .777 5.05 Vp = .1332 sdob2 + 10.1287 sh .973 5.25 Species: All conifers combined Sample size: 313 Vp = -13.1434 + .2455 sdob2 + 15.8927 sh .914 8.99 Vp = -8.1981 + .2535 sdob2 .895 9.88 Species: Conifer without balsam fir Sample Size: 271 Vp = -13.4839 + .2552 sdob2 + 19.8723 sh .945 7.57 APPENDIX B EQUATIONS DEVELOPED FOR SAWLOG VOLUME APPENDIX B EQUATIONS DEVELOPED FOR SAWLOG VOLUME Species: Aspen Sample size: 61 R2 SE Vs = -66.6867 + 5.7185 sdob + 21.1427 sh .748 9.47 Vs = -13.2597 + .2075 55552 + 20.7452 sh .733 9.74 Vs = 2.8395 sdob + 10.5340 sh .937 13.20 Vs = .1752 sdob2 + 12.5348 sh .953 10 15 Species: Aspen and paper birch combined Sample size: 72 (Hardwood) Vs = -70 7515 + 5.7022 sdob + 25.1518 sh .710 10.10 Vs = -16.8581 + .2098 sdob2 + 22.9845 sh .570 10.22 Vs = 2.5372 sdob + 12.3715 sh .925 13.55 Species: Red pine Sample Size: 45 Vs = -9l.2989 + 9.1989 sdob + 21.0971 sh .944 9.34 Vs = -20 4575 + .2833 sdob2 + 20.5705 sh .954 8.40 Vs = 3.5844 sdob + 20.2372 sh .896 23.55 Vs = .2391 sdob2 + 9.2084 sh ' .974 11.80 Species: Jack pine Sample size: 82 Vs = -48.7921 + 5.5979 sdob + 14.0957 sh .854 5.41 Vs = -8.4512 + .1929 sdob2 + 14.6583 sh .847 5.49 59 Vs Vs Species: Vs VS Vs Vs Species: Vs Vs Vs Vs Species: VS 60 = 2.484 sdob + 4.8065 sh = .1651 sdob Balsam fir 2 + 8.3650 sh Sample size: 35 -47.0082 + 4.6526 sdob + 16.5611 Sh -14.1411 + .1519 stb 2 1.8755 sdob + 3.0220 sh .1206 sdob All conifers combined 2 + 6.8955 sh Sample size: + 16.2356 sh 177 -91.5122 + 8.5750 sdob + 22.9465 sh -24.2758 + .2686 sdob 2 2.5297 sdob + 20.8128 sh .2079 sdob2 + 3.5418 sh Conifer without balsam fir + 18.0678 sh Sample Size: = -22.5873 + .2559 sdob2 + 21.5344 sh .920 .967 .789 .801 .926 .961 .829 .893 .789 .921 142 .931 SE 10.87 6.99 5.26 5.10 8.12 5.92 14.17 11.11 24.71 15.11 9.69 r—- LITERATURE CITED LITERATURE CITED Almedag, I. 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