USING THE JACKKNIFE TECHNIQUE TO APPROXIMATE SAMPLING ERROR FOR THE CRUISE -BASED LUMBER RECOVERY FACTOR By KAREN VERONICA JAHRAUS B.S.F., University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Faculty of Forestry) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1987 ©Karen Veronica Jahraus, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6f3/8-h ABSTRACT Timber cruises in the interior of British Columbia are designed to meet precision re-quirements for estimating total net merchantable volume. The effect of this single objective design on the precision of other cruise-based estimates is not calculated. One key secondary ob-jective, used in the stumpage appraisal of timber in the interior of the province, is estimation of the lumber recovery factor (LRF). The importance of the LRF in determining stumpage values and the fact that its precision is not presently calculated, prompted this study. Since the LRF is a complicated statistic obtained from a complex sampling design, standard methods of vari-ance calculation cannot be applied. Therefore, the jackknife procedure, a replication technique for approximating variance, was used to determine the sampling error for LRF. In the four cruises examined, the sampling error for LRF ranged from 1.27 fbm/m3 to 15.42 fbm/m3. The variability in the LRF was related to the number of sample trees used in its estimation. The impact of variations in the LRF on the appraised stumpage rate was influenced by the lumber selling price, the profit and risk ratio and the chip value used in the appraisal calculations. In the cruises investigated, the change in the stumpage rate per unit change in the LRF ranged between $0.17/m3 and $0.21/m3. As a result, sampling error in LRF can have a significant impact on assessed stumpage rates. Non-sampling error is also a major error source associated with LRF, but until procedural changes occur, control of sampling error is the only available means of increasing the precision of the LRF estimate. Consequently, it is recommended that the cruise design objectives be modified to include a maximum allowable level of sampling error for the LRF. ii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES . . . iv LIST OF FIGURES iv ACKNOWLEDGEMENTS v 1.0 INTRODUCTION 1 2.0 OVERVIEW 4 3.0 MATHEMATICAL DEVELOPMENT 10 3.1 SAMPLING ERROR 10 3.2 ANALYTICAL STATISTICS FROM COMPLEX SAMPLING DESIGNS 11 3.3 TAYLOR METHOD 12 3.4 BALANCED REPEATED REPLICATIONS 13 3.5 JACKKNIFE METHOD . 14 3.6 COMPARISON OF THE THREE METHODS 17 3.7 FORESTRY APPLICATIONS OF THE JACKKNIFE 19 4.0 METHODS 20 4.1 CHOOSING THE JACKKNIFE 20 4.2 DESCRIPTION OF STUDY DATA 20 4.3 PROGRAM DESCRIPTION 23 4.3.1 COMPUTATION OF LRF 23 4.3.2 COMPUTATION OF LRF VARIANCE . 25 4.4 IMPACT ANALYSIS OF LRF SAMPLING ERROR ON STUMPAGE 25 5.0 RESULTS AND DISCUSSION 27 5.1 LRF SAMPLING ERROR 27 5.2 EFFECT ON STUMPAGE 28 5.3 DETERMINATION OF LRF SAMPLING ERROR 33 5.4 OTHER SOURCES OF ERROR IN THE LRF 35 6.0 CONCLUSIONS AND RECOMMENDATIONS 37 LITERATURE CITED 39 APPENDICES A. Stumpage appraisal worksheet 43 B. Calculation of sampling error for total net volume 44 C. Loss factor tables 46 D. Height equations and tables 47 E. Listing of program LRFJACK 48 iii LIST OF TABLES Table Page 1 Metric soundwood lumber recovery factors (fbm/m3) 6 2 Chip yield factors (100 BDU/m3) for some common interior British Columbia species 8 3 Summary characteristics of the four timber cruises examined 21 4 Values used for each species in stumpage calculations 26 5 Sampling error for LRF estimates in four timber cruises 28 6 Effect of LRF sampling error on stumpage rates 30 7 Effect of LRF sampling error on spruce stumpage 33 LIST OF FIGURES Figure Page 1 Stumpage rate as a function of LRF for different lumber selling prices 31 iv ACKNOWLEDGEMENTS The preparation of a thesis often becomes the primary focus of a graduate student's life. As a result, many of the people that surround us in our academic and personal lives, whether consciously or not, have an influence on the final work that is produced. For me, these contributors have been numerous, and although I cannot begin to mention them all, there are those among them that deserve special thanks. I extend my sincere appreciation to Dr. Peter Marshall for his dauntless proofreading efforts in the latter stages of my thesis preparation and for the support he offered throughout. I am also thankful to Dr. Julien Demaerschalk for his suggestions and warm words of encourage-ment. To Dr. Tony Kozak, I wish to express my deep gratitude for the guidance he provided me throughout my graduate studies. In addition, I thank him for suggesting a thesis topic that proved to be challenging and rewarding. I gratefully acknowledge Mr. George Kondor and Industrial Forestry Service Ltd. in Prince George for supplying me with cruise data for my investigation. The information and cooperation provided by Dr. Gordon Bailey and other staff members of the Ministry of Forests and Lands is also appreciated. In addition, I acknowledge the Canadian Forestry Service for their financial support and the University of British Columbia for their computing facilities. Special thanks must also be given to all of those who made indirect contributions — to all of my friends who seemed never to tire of asking me how my thesis was progressing. My completion of the task was surely spurred by my great desire to tell them all that I had finally finished! My most heartfelt appreciation, however, goes to my husband Michael for his unbounded patience, understanding, advice and encouragement. His love and support have made and will forever make my life's challenges seem so much easier. v Dedication the memory of my dear, gentle grandmother who passed away on April 1, 1987. 1.0 INTRODUCTION The sampling designs of forest inventory systems are commonly based on a single pri-mary objective. If, however, one considers the many secondary objectives that exist in most . forest inventories, the choice of optimal sampling design is greatly complicated. Obtaining the desired precision for a single objective may result in oversampling or undersampling for other objectives. Although mathematical programming techniques can be used for determining op-timal sample allocation in multivariate forest inventories, the complexity of these techniques preclude their use in most practical applications (Marshall and Nautiyal 1980). The accept-ability of basing a sampling design on only one of several objectives is best determined by examining how the chosen design affects the ability of the inventory to satisfactorily meet its other objectives. Therefore, to judge the performance of such designs, the precision obtained for secondary objectives must be investigated. The most common type of forest inventory in British Columbia is an operational inven-tory or timber cruise. The primary objective of a timber cruise is to estimate the net volume of timber on a tract of land that is to be harvested (Ministry of Forests 1982). Therefore estima-tion of the total net volume to a desired level of precision is the criterion chosen to determine sample size. Timber cruises also provide essential data for determining the stumpage value of each species to be harvested, for establishing conditions of sale, and for operational planning by the company harvesting the timber. The optimal sampling procedure for volume estimation may not be the same as the optimal sampling procedure for these other objectives. However, the Ministry of Forests assumes that since the objective of timber volume estimation tends to result in more intensive sampling designs, the other objectives will usually be met as well (Ministry of Forests 1982). Total net volume is the only statistic for which sampling error is routinely computed in operational inventories. As the sampling error increases, the usefulness of the estimate of total net volume decreases until some point is reached beyond which the timber cruise is considered unacceptable. This upper level for allowable sampling error is stipulated as plus or minus 15 percent for total net volume estimation (Ministry of Forests 1982). Although knowledge of the 1 sampling error corresponding to a particular estimate is essential in evaluating its usefulness, the large number of estimates that result from a compilation of timber cruise data make it virtually impossible to calculate the sampling error for each. However, it is reasonable to suggest that sampling error be calculated for at least some of the more important estimates, particularly if these estimates correspond to major timber cruise objectives. As mentioned previously, one important objective of a timber cruise in British Columbia is to provide data which will be used in the valuation of harvested species for stumpage appraisal purposes. In the interior region of the province, the lumber recovery factor (LRF) is used in stumpage appraisal to determine the amount of lumber that can be recovered from the volume of timber present on a particular area. This volume of lumber represents a sales value upon which stumpage fees are based. As a result, the estimate of the LRF that is obtained from the timber cruise can have a significant impact on the stumpage fees that are charged for harvesting a tract of timber. For this reason, it would be advantageous to have knowledge of the precision of the cruise-based estimate of LRF by computing its sampling error. Since we do not presently compute the precision of the LRF the potential impact of sampling error on the stumpage fees assessed is not known. One of the reasons why the sampling error for the LRF is not routinely calculated is because there is no straightforward formula for its computation. With current statistical theory, it is possible to calculate sampling error of complicated relational or analytical statistics when the sampling design used is simple random sampling. It is also possible to calculate the sampling error of simple descriptive statistics for more complex sampling designs. However, no theory exists for computation of sampling error for complicated analytical statistics from complex sampling designs. (See section 3.0.) The LRF is a complex analytical statistic that measures the relationship between net volume, decay volume and gross volume and their effect on recoverable lumber. The design of an operational timber cruise is based on stratified systematic selection of clusters of trees. No statistical theory exists for exactly calculating the sampling error of such a statistic from this type of design. The objective of this thesis is to provide a solution to the problem of determining the sampling error associated with the estimate of LRF from a timber cruise and to analyze the impact of LRF sampling error on stumpage values. 2 A more detailed discussion of what the lumber recovery factor is, how it is calculated and where it fits in to both the timber cruising and appraisal systems in British Columbia is provided in the overview (section 2.0). The section on mathematical development begins by outlining the problem of determining the sampling error of an analytical statistic from a complex sample. A review of the literature finds three approximation methods that are applicable to this problem. These are described and compared (section 3.0). Based on these findings, the jackknife procedure is proposed as a means by which the sampling error for the cruise-based LRF estimate can be approximated. This technique is applied to data from four interior British Columbia timber cruises (section 4.0). The results of this study and an analysis of the effect of LRF sampling error on stumpage values are discussed (section 5.0). Relevant conclusions and recommendations for further investigation are then presented (section 6.0). 3 2.0 OVERVIEW In British Columbia, the stumpage value of timber to be harvested from Crown land by private forest companies is appraised by the Ministry of Forests (Juhasz 1976). The general procedure is to subtract all the necessary costs of production, including an allowance for the operator's profit and risk, from the price of the products which can be produced from the timber (Ministry of Forests 1980a). Stumpage is therefore a residual value. Minimum stumpage charges exist in the case where this residual value falls below a particular threshold level. In the coastal region of the province, where a competitive log market exists, the end product of forest harvesting operations for stumpage appraisal purposes is logs. The historical development of the forest industry in the interior resulted in logging and sawmilling being combined in a single enterprise and trading of logs between companies was never established. Hence adequate market value information for logs is unavailable for the interior. Since the Ministry's policy is to base appraisals on the first products of the forest for which values can be reasonably established from market information, lumber and chips have become the "end products" of forest harvesting for interior stumpage appraisal purposes. Stumpage assessment requires that the value for lumber and chips be converted back to a value for roundwood timber volume. The conversion ratio used to accomplish this is known as the LRF (Task Force 1974). Prior to a revision of the interior appraisal system in 1973, the potential lumber outturn of a stand of timber was estimated on the basis of zonal average LRF's for the various species (Ministry of Forests 1973). These zonal average LRF's reflected the lumber recovery possible in a stand of average quality. As a result, operators in stands of above average quality benefitted from the system, whereas operators in stands of below average quality were penalized. To resolve these inequities, the appraisal system was modified to provide for the calculation of LRF's specific to each stand being appraised. Timber size and quality, recognized as the two most variable and identifiable factors influencing the lumber recovery potential of a stand, were incorporated into the new system. Juhasz (1976) described how the current system for determining LRF's was developed. The original work on LRF's was done in imperial units. Theoretical LRF's were calculated for 4 16 foot sound logs in various two inch top diameter classes. The calculations assumed one inch per seven feet taper, one quarter inch saw kerf, no lumber recovery from the volume in the taper and cutting only 2 x 3, 2 x 4, 2 x 6, 2 x 8, 2 x 10 and 2 x 12 inch nominal dimensions. These theoretical LRF's were then reduced by 25 percent to allow for losses common to all species such as trim, irregular log shape, scale, mechanical defects, imperfect sawing, and kiln and planing losses. It was then necessary to develop a means by which these soundwood LRF's could be reduced for the effect of decay in a log. The effect of decay on lumber recovery is twofold. Firstly, the presence of decay effectively reduces the size of the log and secondly, there is ususally a shell of sound wood surrounding the decay that cannot be converted into lumber. Unfortunately, no quantitative studies of losses in recovery due to decay were available. Therefore it was arbitrarily decided to reduce the LRF of a log by one half percent for every one percent of decay volume present in the log. Hence the equation for LRF reduced for decay was determined to be: LRF d e c a y = LRF s o u n d x ( l - 0,5(^cayvolumex\ \ v gross volume / / Several mill studies were carried out under the auspices of the Joint Interior Lumber Outturn Committee (including representatives from industry, the Ministry of Forests and the Canadian Forest Service) to evaluate the system (eg. Dobie 1978, Dobie and Kasper 1975). Based on these findings some further adjustments were made to the soundwood LRF values. It was found that LRF's increased with log size up to a limit beyond which they decreased. This was partly because certain physical defects, such as checks and shakes for example, were more prominent in larger size logs (Task Force 1974). It was also discovered that the system predicted higher lumber recovery for cedar logs than was experienced by industry. Consequently, a new set of soundwood LRF's and a new decay-adjusted LRF equation . were established for this species. Additional modifications were made to the system when the Ministry of Forests changed to metric measure in 1979. In imperial units, LRF's represented the number of board feet of lumber recoverable per cunit of log volume for logs in two inch top diameter classes. New tables 5 of metric LRF's were produced representing board feet per cubic metre (fbm/m3) of log volume for logs in five cm top diameter classes. The values for soundwood LRF's that are in current use are shown in table 1. Table 1. Metric soundwood lumber recovery factors (fbm/m3). ameter Class (cm) Mid-Point (cm) H,CY,PW,and F f All Others Cedar * 10.0 - 14.9 12.5 151 158 132 15.0 - 19.9 17.5 172 185 156 20.0 - 24.9 22.5 187 202 177 25.0 - 29.9 27.5 199 217 197 30.0 - 34.9 32.5 210 226 215 35.0 - 39.9 37.5 217 234 227 40.0 - 44.9 42.5 222 240 235 45.0 - 49.9 47.5 228 246 238 50.0 - 54.9 52.5 233 252 239 55.0 - 59.9 57.5 237 258 239 60.0 - 64.9 62.5 240 263 237 65.0 - 69.9 67.5 241 266 233 70.0 - 74.9 72.5 240 267 228 75.0 - 79.9 77.5 237 265 223 80.0 - 84.9 82.5 234 263 217 85.0 - 89.9 87.5 231 261 210 90.0 - 94.9 92.5 227 259 201 95.0 - 99.9 97.5 224 257 193 100.0+ 221 254 184 t H=hemlock (Tsuga heterophylla (Raf.) Sarg.), CY=yellow cedar (Chamaecyparis nootkatensis (D.Don) Spach.), PW=white pine (Pinus monticola Dougl. ex D.Don in Lamb.), F=Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) % Thuja plicata Donn ex D.Don in Lamb. In order to apply these soundwood LRF's to individual stands of timber, a timber cruise must initially be carried out to determine the volume of timber by species, size and quality. Then the cruise information is converted into estimates of recoverable lumber and chips (Task Force 1974). In a timber cruise, several measurements and observations are made on all of the sample trees in each plot. Diameter and height information is used to estimate the gross merchantable volume of each tree. The limits of merchantability depend on the utilization standards that are specified by the Ministry. The occurence of certain visual decay 6 indicators are among the observations that are recorded for each tree. These decay indicators are used to classify the trees into one of three risk groups. Based on volume and decay studies carried out by the Ministry of Forests Inventory Branch, tables of loss factors for each of the three risk groups have been compiled for all commercial species in various regions of the province (Ministry of Forests 1976). These loss factors represent the percentage loss in gross merchantable volume due to decay, waste (the volume of soundwood in logs or trees that are more than fifty percent defective), and breakage (the volume of soundwood expected to be broken in falling and yarding). When these losses are subtracted from the gross merchantable volume of the tree or log, the resulting volume is referred to as net merchantable volume. For the LRF, all calculations are based on five metre log lengths. Thus all of the sample trees must be mathematically divided into five metre logs. Top logs that are less than five metres in length are added onto the log beneath if they are less than 2.5 metres long or considered a separate "short" log if they are greater than or equal to 2.5 metres long. Using taper equations, gross merchantable volume is then calculated for each log in a tree. Net merchantable volume is determined by subtracting volume losses due to decay, waste and breakage. All logs from one tree share the same risk group and therefore the same loss factors. Thus the top log and the butt log would have the same percentage decay assigned to them. The logs that result from the sample trees are grouped according to species, top diameter, and risk group. Logs below 30 cm top diameter are referred to as "small logs". Those above 30 cm top diameter are called "large logs". LRF's for each species are calculated for both the large and small log categories of each risk group. LRF's for large and small logs across all risk groups are also computed. The final LRF for each species is obtained by prorating these LRF's over the proportion of large and small logs of that species in the stand. The end products considered for stumpage appraisal in the interior include chips as well as lumber. Therefore the average chip yield for each species is also calculated in the cruise compilation. Every cubic metre of solid wood potentially contains 423.72 board feet1. The Ministry of Forests (1980b) assumes that in the process of milling small logs, 15 percent of the 1 This value is obtained by converting fbm/ft3 to fbm/m3. (12 fbm/ft3x 35.31 ft3/m3 = 423.72 fbm/m3) 7 total possible 423.72 fbm/m3 will be lost as fibre and sawdust. For large logs this fibre and sawdust loss is assumed to be 20 percent. After subtracting the fibre and sawdust component, the remaining recoverable product volume from a cubic metre of solid wood is 360 fbm/m3 and 339 fbm/m3 for small and large logs respectively. This product volume is converted into lumber and chips. The volume of lumber that is recoverable for a particular species is determined by the cruise calculated LRF. The remaining volume becomes chips. Chips are sold in bone dry units (BDU), thus it is convenient to state chip yield in this measure. Yield factors (in units of 100 BDU/fbm) have been computed (Ministry of Forests 1980b) for each species based on the unit weight the wood (table 2). Table 2. Chip yield factors (100 BDU/m3) for Species Chip Yield Factor some common interior British Columbia species. Species Chip Yield Factor Douglas-fir 0.098 Cedar 0.067 Hemlock 0.089 Balsam 0.072 Spruce 0.076 Lodgepole Pine 0.087 The chip yield (BDU/m3) for small logs is computed as: Chip Yield s m a l J l o g = (360 - L R F s m a l l l o g ) x chip yield factor/100 . (2) Similarly, the chip yield (BDU/m3) for large logs is computed as: Chip Yield l a r g e l o g = (339 - L R F l a r g e l o g ) x chip yield factor/l00 . (3) The chip yield for large and small logs is then prorated over the proportion of large and small logs in the stand (by net volume) to obtain an average chip yield. The LRF and average chip yield calculated for each species in the cruise is entered directly onto the stumpage appraisal worksheet (appendix A). The LRF is multiplied by the average selling price per boardfoot of lumber for a particular species to obtain the contribution of lumber to the total selling price per cubic metre of timber. The value of the chips produced is calculated by multiplying the average chip yield by the price per BDU of chips. These two 8 product values, chips and lumber, are combined to form the total selling price of the timber. The stumpage value of each species is then computed as SP stumpage rate = ——— — OC (4) P 6 1 + P+R v ; where SP=selling price ($/m3), P+R=profit and risk allowance (expressed as a decimal fraction) and OC=operating cost ($/m3). 9 3.0 MATHEMATICAL DEVELOPMENT 3.1 SAMPLING ERROR Sampling error is defined by Cunia (1981) as the error associated with the random selection of sample elements. If, for example, the same sampling procedure were applied to a given population on repeated occasions, a different set of sample elements would likely result each time. Thus a different set of sample-based estimates would be obtained. The estimate of the mean or the total of a parameter obtained from a sample is generally different from the corresponding true value over the whole population (Fontaine 1973). The inherent value of a sample increases as the estimate becomes more accurate (i.e. as the estimate approaches the true population value). Although the precision of an estimate cannot be expressed in an absolute manner, it can be expressed in terms of a probability statement (Fontaine 1973). For example, the precision of an estimate may be stated as plus or minus ten percent at a 0.95 probability level. This would correspond to stating that the true population parameter would fall within the plus or minus ten percent interval of the estimated value given by every sample for ninety-five percent of samples of a similar size selected using the same sampling procedure and drawn from the same population. In making inferences regarding population parameters, sampling error is computed as k/2.»-i x Se[/(y)] (5) where Se [f(y)] is the standard error of the estimate f(y) and i a/2,n- i i s 'he value from Student's t distribution corresponding to a particular confidence coefficient, (a), and the degrees of freedom (n — 1) associated with the standard error. The standard error of the estimate f(y) is simply the square root of the variance of the estimate. Statistical inference requires that three basic assumptions are met. Firstly, the sample estimate of the population parameter must be approximately unbiased. Secondly, an approx-imately unbiased estimate of the variance must be computable from the sample. Thirdly, the 10 distribution of the sample estimate minus its expected value divided by its estimated standard error must approximate a Student's t distribution. If the sample design is simple random sam-pling, the sample size is "large", and the estimate is a mean, a proportion or a total, then it is reasonable to believe that these assumptions will hold (Frankel 1971). 3.2 ANALYTICAL STATISTICS FROM COMPLEX SAMPLING DESIGNS Shah (1978) indicated that computation of the variance of an estimate is the first step toward making any inference regarding a population parameter. The variance of a sample-based estimate depends on the design of the sample as well as the form of the estimator. In discussions regarding classification of estimators, a distinction is often made between analytical and descriptive statistics (Kish 1965, Cunia 1981). Descriptive statistics generally include the estimation of means and totals whereas analytical statistics measure relationships (eg. differences between means, indices, etc.). For many sample designs, estimators of the variance of descriptive statistics are readily available. For simple random sampling, estimators of the variance of more complex analytical statistics are also available. However, when sample designs other than simple random sampling are considered and the sample estimates are for analytical rather than descriptive parameters, the validity of the assumptions for statistical inference mentioned in section 3.1 become questionable (Frankel 1971). In fact, very little theory is available for dealing with analytical statistics from complex sampling designs (Kish 1965, Kish and Frankel 1974). For practical purposes, however, it would be extremely useful to have at least some approximate techniques to provide quantitative measures for the sampling errors in such estimates (Kish 1965, Frankel 1971, Shah 1978, and others). Considerable research has been dedicated to developing methods of approximating the standard error of analytical statistics from complex samples. Due to the lack of statistical theory in this area, much of the work that has been done has been empirical (Frankel 1971). Three main approximate methods have been applied. All are non-parametric methods (i.e. they do not depend on the underlying distribution of the statistics) (Efron 1981). These three methods are: 1) Taylor series expansion; 2) balanced repeated replications; and 3) the jackknife method (Cochran 1977). 11 3.3 TAYLOR METHOD The Taylor series expansion or Taylor method is known by several other names includ-ing propagation of errors, linearization, method of statistical differentials and delta method (Kish 1968, Frankel 1971, Parr 1983). The use of the Taylor method for obtaining an estimate of the variance of sample-based statistics has long been recognized (Frankel 1971). The Taylor method is the basis for determining the variance of a ratio estimate (Deming 1950, Cochran 1977). Deming (1960), Kish (1962), and Parratt (1961) also described its use in the propagation of variance for other functions of the sample data. The first detailed published extension of this method to more complex statistics is attributed by Frankel (1971) to Tepping (1968). Tepping (1968) described the procedure of the Taylor method as follows. Let y — (j/i, V2, • • •, yk) be a vector of sample statistics whose expected value, E(y) = Y = (Y\, Yi, • • • , Y^), is a corresponding vector of population values. Also let the function /(Y) be the population parameter of interest that is to be estimated by f(y). The first assumption of the Taylor method is that the sampling variance of f(y) is approximately equal to the sampling variance of the first degree terms of the Taylor series approximation of f(y) near Y (Frankel 1971). Thus: k Var [^»)] =V™[f(Y) + J2(yi-Yd3-§p-]- (6) This reduces to: Var[ / ( y ) ]=Var [ X > ^ ] (7) where the partial derivatives are to be evaluated at y = Y (Tepping 1968). Frankel (1971) expresses (7) in a perhaps more familiar form as: »=i 1 t j=i ' ; ¥i In either case, the values of df(Y)/dY{, which are unknown population parameters, must be estimated from the sample (Tepping 1968). The precise manner in which the variance of the linear combination ,=i 0 y> 12 is estimated depends upon the sample design (Tepping 1968). Woodruff (1971) and Woodruff and Causey (1976) gave a generalized short-cut approach to computing the variance by the Taylor method when stratified designs are used, based on reversing the order between selection units and component variables in the linear expression above. 3.4 BALANCED REPEATED REPLICATIONS Balanced repeated replications require exactly two primary sampling units per stratum in all strata. Variance estimation is based on a number of half-samples formed by deleting one primary sampling unit from each stratum in the sample (Kish and Frankel 1970). This method was first proposed by McCarthy (1966, 1969). The variance estimates produced by the balanced repeated replication method are described by Frankel (1971) as follows. Assume a stratified sample design with two primary sampling units per stratum. As in the previous description for the Taylor method, let the function f(Y) be the population parameter of interest that is to be estimated by f(y). Now let f(S) denote the estimate of f(Y) made from the whole sample. Let /(#,) denote the estimate of f(Y) made from the Ith half-sample formed by including only one of the two primary units in each of the strata. Let /(Ci) denote the estimate of f(Y) made from the i01 complement half-sample. If A; half-samples H\,...,Hk and corresponding complement half-samples C\,...,Ck are formed, then four estimates of variance for f(y) can be produced. The first is based on the half-sample minus the total Vari[/(</)] =l/*£[/(tf.-)-/(S)]2. (10) i=i The second is based on the half-sample complement minus the total k Yav2{f(y)}=l/kJ2[f(Ct)-f(S)}2. (11) i=i The third is an average of (10) and (11) (12) 13 and finally the fourth is based on the difference between the half-sample and the complement Var4[/(j,)] = 1/(4*) £ [/(#,) - f(Ct)}2 . (13) i=i The method most commonly used to choose the pattern of primary units that constitute the half-samples is known as "full orthogonal balance" (Frankel 1971). 3.5 JACKKNIFE METHOD The balanced repeated replications method just discussed falls into a general class of methods often simply called replication methods. In this general class, the sample variance of a statistic is estimated by computing it for several subsamples and examining its variability over the subsamples (Lemeshow and Levy 1979). The jackknife is also included in this same general class of methods. The jackknife method was originally proposed by Quenouille (1956) as a technique for reduction of bias in serial correlation estimators. There has been much attention given to the jackknife technique since its introduction. Miller (1974) and Parr and Schucany (1980) have surveyed some of the important literature in this area. In the jackknife method, "pseudo-values" of the estimated parameter are computed by repeatedly omitting a single observation. An estimate of the variance of the estimated parameter is then obtained by examining the variance of the pseudo-values (Lemeshow and Levy 1979). Miller (1974) gave a more formal description of this technique. Let 6 (equivalent to f(y) ) be an estimator of the population parameter 6 (equivalent to f(Y)) based on a sample of size n. Let the sample be split into g groups each of size h such that n = gh. Now let 0_, be an estimator corresponding to 9 but based on a sample of size (g — l)h, where the Ith group of size h has been deleted. Define h=gO-(g-l)e-i i = l . . . g . (14) The jackknife estimator of 9 is then 14 According to McCarthy (1966) it was John W. Tukey who coined the term pseudo-values for the values in (14) and created the name "jackknife" estimator in the hope that this method would become a rough-and-ready statistical tool. A more important contribution by Tukey however was the result that the pseudo-values could be treated as approximately independent and identically distributed random variables in many situations (Miller 1974). The statistic 6-9 (16) 5 ^ H a fr"')2 should then have an approximate t distribution with g — 1 degrees of freedom (Miller 1974). The jackknife estimator of the variance of both 9 and 9 is given by Var(0) = Var® = ^ (17) (Wolter 1985). This is equivalent to Var(0) = ^ il^(0_,-0) 2 (18) 3 i=i where 6 = S f = 1 9-</g. For small samples (i.e. when only a small number of pseudo-values can be created), the Student t approximation for jackknife confidence limits can be improved using an Edgeworth expansion (Hinkley and Wei 1984). The case where g = n and h = 1, or one-at-a-time omission of observations, is most common because it eliminates any arbitrariness that may be involved in forming groups and is also the most efficient (Miller 1974). However, a study by Hinkley (1977) suggested that the Student t approximation for jackknife confidence intervals with g — 1 degrees of freedom may be better for h > 1. Based on limited simulation results, Hinkley (1977) found that grouping the data omissions usually improves the probability accuracy of confidence limits with only a slight loss in efficiency. However, in non-homogeneous situations (eg. stratified sampling) A-at-a-time omissions become more complex (Hinkley 1977). Farewell (1978) demonstrated that the jackknife technique can be improved if the pattern or special structure in the data is reflected in the jackknife calculations. Several au-thors have discussed jackknife variance estimation in the context of stratified random sampling 15 (Frankel 1971, Kish and Frankel 1974, Jones 1974, Lemeshow and Levy 1979, Krewski and Rao 1981, Rao and Wu 1985). Since the extension of the jackknife to stratified samples is not imme-diate, several different versions have been proposed. Krewski and Rao (1981) and Rao and Wu (1985) have provided excellent summaries of the more common jackknife variance estimators for this sample design. The Jones (1974) jackknife estimator of Var(0) is given by Var ( « ) = E ^ F i ii;(«-«-« L*) 2 h=i n i=l where h = 1... L strata, i = 1... nw observations in stratum h, 9-M is an estimator corresponding to 9 but based on a sample with the Ith observation in stratum h omitted and 9-h = Yl"=\ 9-hilnh-Efron and Stein (1981) also used this form of the jackknife variance estimator. In their study they found that (19) was biased upward. A modified version of (19) replaces 9-h by 9 to give V a r ( 0 ) = X ; ^ ^ £ ( L f t l - ^ (20) nh (Krewski and Rao 1981). Kish and Frankel (1974) and Frankel (1971) considered (20) in the special case of % = 2 for all strata. Their value of 9-w, was computed by omitting the hi01 observation and including the hj 01 (j^i= 1;2) observation twice. Four estimators of Var(^ ) were then produced in a manner analogous to (10), (11), (12) and (13) of the balanced repeated replications method (Frankel 1971). Two additional jackknife variance estimators are h=l i=l X h=l i=l 7 and h=l i=l h=l (Krewski and Rao 1981). If we define the pseudo-values as 9\i = n^ti - (n^ - 1) 9-hi, both L nh «' = E E (23) h=l i=l 16 and e" = v% E h i 1 (24) h=l i=l are natural extensions of (15) (Krewski and Rao 1981). The corresponding jackknife variance estimators are given by Var® = Var® = £ 1 )2 (25) and Va t(«)=VarW=g- I_ T yy:(i - -S» f . (26) Folsom et al. (1971) used a jackknife variance estimator of the form Var® = Var® = £ £ ( * « - £ W " * ) ' • (27) A=l ^ h ' t=l i=l Krewski and Rao (1981) pointed out that (27) is equivalent to (19). All of these estimators assume either sampling with replacement or sampling without replacement when the sampling fractions are negligible (Wolter 1985). 3.6 COMPARISON OF THE THREE METHODS The majority of the studies comparing the Taylor method, balanced repeated replica-tions and the jackknife have been empirical. Kish and Frankel (1970) found that the jackknife and the balanced repeated replications methods are closely related and thus they suggested that there would be no large differences in the results using either of the two techniques. Frankel (1971) and Kish and Frankel (1974) compared all three methods in a large empirical study and found that for the case of stratified random sampling for = 2 all three methods give good results. Based on the criterion of the closeness of the actual tail values to the tabular Student t distribution tail values, the balanced repeated replications method performs consistently better than both the jackknife and the Taylor method. Using the same criterion, the jackknife in turn performs better than the Taylor method. When judged by several criteria, none of the three methods shows up strongly and consistently better or worse (Kish and Frankel 1974). 17 Lemeshow and Levy (1979) compared the jackknife and balanced repeated replications for the same case as above. Their findings indicated that either technique may be used effectively for variance estimation. They pointed out however that because balanced repeated replications is limited to the case where rih ~ 2 for all strata, it is not as widely applicable as the jackknife (Lemeshow and Levy 1979, Krewski and Rao 1981). Bean (1975) compared the Taylor and balanced repeated replications method for approximating the variance of a ratio-type estimator in a more complex sample design situation. He found that both methods perform satisfactorily. Shah (1978) used four evaluation criteria to compare the three methods: 1) the validity or number of assumptions required; 2) restrictions on sample design; 3) computational problems for large data sets; and 4) flexibility of applications. He suggested that the method of balanced repeated replications is least favorable due to restrictions on sample design and computational problems. Furthermore, he concluded that the Taylor method is likely better than the jackknife because of computational considerations. However, a large disadvantage of the Taylor method is that the function for which a variance estimate is required must be differentiable (Shah 1978, Krewski and Rao 1981). Krewski and Rao (1981) provided one of the first analytical comparisons of the three methods. Their study found that all three estimators are asymptotically consistent to the first order. Parr (1983) used simulation results and asymptotic expansions to show that the jackknife and the Taylor method coincide to at least first order terms. More recently, Rao and Wu (1985) obtained second-order asymptotic expansions of the three approximate variance estimators for non-linear statistics. Several interesting conclusions arose from the study of Rao and Wu (1985). The bal-anced repeated replications and jackknife variance estimators corresponding to (13) and the Taylor method estimator are all identical for quadratic statistics in the special case of = 2 in all strata. Furthermore, the six jackknife estimators (19), (20), (21), (22), (25), and (26) are asymptotically equal, to higher order terms. In addition, these are asymptotically equal, to higher order terms, to the Taylor method variance estimator in the case nw = 2. In contrast, the balanced repeated replications variance estimators are not asymptotically equivalent to the 18 Taylor method variance estimators, to higher order terms (Rao and Wu 1985). Rao and Wu also showed that replacing 9 in (20) by any jackknifed estimator (eg. (23), (24) ), leads to in-consistent variance estimates. In comparing (19), (20), (21) and (22) Wolter (1985) determined that (21) and (22) were conservative estimators of variance relative to (19) and that (20) was the most conservative. 3.7 FORESTRY APPLICATIONS OF THE JACKKNIFE The wide applicability and general simplicity of the jackknife method has prompted its use in many situations where some approximation of the variance of an analytical statistic from a complex sample is required. The jackknife procedure has recently been recognized for its potential value in forestry sampling applications (Schreuder and Brink 1983). Yang and Kung (1983) used the jackknife to reduce bias in estimating bole volume. The confidence interval for their estimate was based on the jackknife variance estimate. Schreuder et al. (1984a) recommended use of the jackknife procedure to produce a variance estimate for point-Poisson (point 3P) sampling. They found that the jackknife-based estimate was equally as good as the variance estimate suggested by Grosenbaugh (1971) and was much simpler to compute. The jackknife variance estimate for point-Poisson sampling was also used by Schreuder et al. (1984b) in their comparison of model-based sampling and point-Poisson sampling. The jackknife was applied by Schreuder and Thomas (1985) to obtain a variance estimate for total stand volume estimated using a purposive sampling design (pscx or purposive sampling from the cumulated z's). Gregoire (1984) applied the jackknife to determine the variance of quadratic mean stem diameter from prism data. Both Gregoire (1984) and Schreuder and Brink (1983) emphasized the great opportunity for applying the jackknife procedure to many of the complex sampling situations encountered in forestry. 19 4.0 METHODS 4.1 CHOOSING THE JACKKNIFE The jackknife was selected to approximate the sampling error for the LRF computed for each species in each timber cruise in this investigation. Although all three methods described in section 3.0 have been found to produce asymptotically equivalent results, the jackknife pro-cedure was chosen over the Taylor and balanced repeated replications methods for a number of reasons. The method of balanced repeated replications requires two primary sampling units per stratum, therefore it was not applicable in this study where strata sizes differed. The Taylor method requires evaluation of partial derivatives for the LRF and calculation of the variance of all the functional components of the LRF (ie. net volume, decay volume and gross volume of five metre logs). The independence of net volume, decay volume and gross volume cannot be assumed therefore the covariance between these variables must be determined to use the Taylor method. Estimation of the variance and covariance of these variables is complicated by the fact that they are not normally distributed. There is a very high proportion of low volume logs (every tree has a small top log) but a much lower proportion of high volume logs (only very large trees produce large logs). This results in highly skewed distribution functions, the variances of which are difficult to estimate. In contrast, the jackknife procedure is conceptu-ally straightforward and readily applied given sufficient computing facilities. Therefore it was chosen as being most applicable for this study. 4.2 DESCRIPTION OF THE STUDY DATA Data from four operational timber cruises were provided by Industrial Forestry Service (IFS) Ltd. The data consisted of cruise plot information, the final cruise compilations and additional information regarding the height equations and loss factor tables that were used in each compilation. All of the areas for which the cruise information was gathered are located in the Prince George region of British Columbia. Summary characteristics of the four areas are shown in table 3. The primary coniferous species found in these areas are balsam (Abies lasiocarpa (Hook.) Nutt.), interior spruce (Picea spp.), and lodgepole pine (Pinus contorta Dougl. ex Loud.). 20 Table 3. Summary characteristics of the four timber cruises examined. Cruise Number 1 2 3 4 Licence/Cutting Permit A08686 CP127 CP273 CP809 Forest Inventory Zone H I I I Public Sustained Yield Unit Nechako Crooked Carp Parsnip Number of Plots 129 32 47 31 Basal Area Factor 6 6 6 6 Strata or Types 15 4 7 4 Total Merch. Area (ha) Average Net Volume (m3/ha) 245.0 127.7 170.7 99.0 277.89 443.18 400.86 226.07 Total Net Volume (m3) 68082 56594 68428 22381 Standard Deviation (m3/ha) 107.9 146.8 100.9 94.3 Coeff. of Variation (%) 38.8 33.1 25.2 41.7 Sampling Error (%) (a = .05) 6.8 12.0 7.4 15.3 The timber cruises used in this study were designed and carried out in accordance with the Ministry of Forests cruising procedures (Ministry of Forests 1982). The intensity of each cruise, or the number of prism points (n) that were required, was determined such that a sampling error of plus or minus 15% for total net merchantable volume at close utilization with a confidence level of 95% could be achieved. The formula used to compute n was: i 2 x C V 2 , . n = - p W ~ ( 2 8 ) where PE is percentage error, CV is coefficient of variation and t is the appropriate value from Student's t distribution. This sample size formula assumes simple random sampling. However, when the sampling is carried out in the field the area is stratified into homogenous forest types. Thus although the cruise intensity is based on a simple random sampling design, all of the calculations in the compilation of the cruise data are based on a stratified sampling design. After the completion of each cruise, the Ministry of Forests requires the calculation of the actual sampling error for total net merchantable volume that was attained. This calculation is based on the generalized equation for estimating the overall variance in a stratified sample: s f - E ^ f i ) ' <29) h=i v ' where h = 1... L is the number of types or strata, is the estimated variance of each strata and Nh/N is the proportion of the population in type h. The specific procedure outlined by the 21 Ministry of Forests (1982) for sampling error calculation and its derivation from (29) is detailed in appendix B. If the sampling error is greater than 15% the cruise is generally not accepted by the Ministry of Forests. The sampling errors that were achieved in the four study cruises are shown in table 3. Although cruise four resulted in 15.3% sampling error, it was still approved by the Ministry of Forests. Due to the difficulty in ensuring randomness, all plot centres must be located using systematic sampling techniques. At each plot, sample trees are selected with probability pro-portional to basal area using Bitterlich's point method. The basal area factor of the angle gauge device used to select sample trees is chosen to provide an average of seven trees per plot. All commercial species are tallied as sample trees. This procedure was followed for all of the timber cruises used in this study. The information that was gathered in the timber cruises included plot terrain charac-teristics, and several measurements and quality observations on the sample trees. Since this study focused on the LRF, not all of this information was utilized. Stumpage payment calcula-tions are generally limited to coniferous species therefore all data regarding deciduous species were ignored. The information that was extracted from the cruise plot data consisted of species, diameter at breast height (dbh) and pathological indicators for all coniferous sample trees that were either alive or dead but with potential recoverable volume (ie. those denoted as tree class 1,2 or 3 according to Ministry of Forests (1982) definitions). Based on the pathological indica-tors, each sample tree was placed into one of three risk groups. Using the applicable loss factor tables (Ministry of Forests 1976) percentage losses for decay, waste and breakage were assessed for each sample tree. The loss factor tables used for each cruise are listed in appendix C. In addition to this information, total tree height for each sample tree was also required for the computation of LRF. When a timber cruise is carried out, total tree height is only measured on a portion of all sample trees. In the course of compiling the cruise data, a height-diameter regression equation that best "fits" the data is used to estimate the total tree height for those sample trees whose height was not directly measured. If the number of measured tree heights is not sufficient to confidently apply regression techniques, hand-drawn curves converted 22 to tabular form are utilized. Unfortunately the cruise plot information that was provided for this study did not include the measured tree heights. However, a listing of the regression equation or table values that were applicable to each species in every cruise was supplied (appendix D). Thus total heights of all sample trees were estimated using either regression equations or tables in this study. 4.3 PROGRAM DESCRIPTION The use of the jackknife procedure to approximate the LRF variance involved repeated calculation of the LRF. A computer program called LRF JACK was developed by the author to carry out a partial cruise compilation. The algorithm used was validated by comparing the results of this program to the completed cruise compilation produced by IFS. LRF JACK computed the LRF from the cruise plot information and produced the jackknife estimate of the variance for the LRF. A listing of LRF JACK can be found in appendix E. 4.3.1 COMPUTATION OF LRF Two data files were created to provide the information specifically required to compute the LRF and its sampling error. The first file contained individual sample tree records which included information on the stratum or type number, plot number, tree number, species, dbh, tree class, risk group, and decay, waste and breakage factors. The second file prepared for each cruise included information such as number of plots, number of strata or forest types, and number of species. For each type, the number of plots and the area in the type was listed. The utilization standards (minimum top diameter inside bark and minimum dbh) and a code denoting the height equation type or table were included for each species. Parameters for the height equations and/or table values were also included in this file. These data files were then read by LRFJACK. One run of the program was performed for each cruise. Once the data files had been read the actual compilation calculations were repeated for each species in the cruise. Based on the height table or height equation that was specified for each species, an estimated height was obtained for each sample tree. This information, along with dbh and utilization standards, was 23 forwarded into a subroutine called LOG2. This subroutine was based on the whole-bole taper equation system of Demaerschalk and Kozak (1977). In LOG, the merchantable portion of each sample tree was mathematically divided into five metre logs. LOG provided the top diameter inside bark and the gross volume of each log as well as other supplementary information. These results were then returned to the main program, LRFJACK. In LRFJACK, top logs that were less than 2.5 metres in length were added onto the log beneath. Then the tree factor for each sample tree was computed by dividing the basal area factor by the basal area of the tree. The volume of the log was then multiplied by its particular tree factor to yield the volume per hectare represented by that log. This value was divided by the number of plots to obtain the contribution of this log to the average volume per hectare which was then multiplied by the total area to obtain the contribution to total volume. This gross merchantable volume was adjusted for decay, waste and breakage to yield net merchantable volume. The logs from the sample trees were then grouped according to risk group and five centimetre top diameter inside bark class. For each class the total gross merchantable volume, decay percentage, net merchantable volume and volume of recoverable lumber were computed. The volume of recoverable lumber was calculated by where FBM is recoverable lumber (board feet), NMV is net merchantable volume (m3),LRFsoun<j is the soundwood LRF value (from table 1), and DV and GV are the decay volume and gross volume (m3) of the log respectively. These values were summarized for the small and large log categories in each risk group. Based on this summary a decay-adjusted LRF was computed for each risk group for both small and large logs. The decay-adjusted LRF was computed as 2 The subroutine LOG was provided by Dr. A. Kozak, Faculty of Forestry, University of British Columbia. (30) decay — FBM NMV (31) 24 The decay-adjusted LRF was also computed for large and small logs averaged over all risk groups. The final LRF for each species was then obtained by prorating the small log and large log LRF's by the proportion of total net volume represented by each log size grouping. 4.3.2 COMPUTATION OF LRF VARIANCE The variance of the prorated LRF for each species was computed using the jackknife procedure. LRFJACK was designed to repeat the entire compilation process, deleting one cruise plot per repetition. The total number of repetitions required was therefore equal to the total number of plots in a cruise. The jackknife variance estimate used in this study was based on equation (20): VariLRF] = £ £ fLBF_M _ L R F \ 2 ( 3 2 ) h= 1 1= 1 where LRF is the LRF estimate based on all observed plots, L is the total number of types in the cruise, is the number of plots in type h and LRF_/j, is the LRF estimate calculated with the Xth plot in type h omitted. This form of the jackknife variance estimator was chosen because it provides a conservative estimate of variance and it reflects the stratification in the cruise design. The standard error of the LRF estimate was found by taking the squareroot of the variance. Sampling error was computed as the standard error multiplied by the t value with (n — l) degrees of freedom (n is the total number of cruise plots) and a confidence coefficient (a) of 0.05. Percentage error was calculated by expressing the sampling error as a percentage of the LRF estimate. 4.4 IMPACT ANALYSIS OF LRF SAMPLING ERROR ON STUMPAGE To determine the effect of LRF sampling error on the stumpage value of a species, an appraisal analysis was carried out for each species in each cruise. The calculations were based on a July 1986 appraisal worksheet from the Prince George area (appendix A). The operating costs, profit and risk ratio, lumber selling price and chip selling price used for all four cruises were assumed to be the same as those given in this sample appraisal worksheet. Table 4 lists the values of these variables for each species. 25 Table 4. Values used for each species in stumpage calculations. Species Operating Profit and Risk Lumber Selling Chip Selling Cost ($/m3) Ratio Price ($/Mfbm) Price ($/BDU) Balsam 42.77 .20 251+ 10.50 Spruce 39.43 .18 251+ 10.50 Lodgepole Pine 42.41 .17 207* 10.50 f selling price based on random dimension lumber 1 selling price based on studs Three LRF values were used in each appraisal calculation. These corresponded to the LRF point estimate and the upper and lower confidence limits of this estimate as calculated for each species in each of the four study cruises. Thus three stumpage values were computed for each species from each cruise. The range of these values was then examined as an indication of the potential impact of LRF sampling error on appraised stumpage rates. 26 5.0 RESULTS AND DISCUSSION 5.1 LRF SAMPLING ERROR Using the jackknife, the variance of the LRF estimates was determined for each species in all four cruises. The highest variability was exhibited by the LRF estimate for balsam in cruise three with a standard error of 7.66 fbm/m3. This estimate was based on five trees observed in 47 plots. In contrast, the least variable LRF estimate, that for lodgepole pine in cruise one, was based on 628 trees observed in 129 plots. The standard error for this estimate was 0.65 fbm/m3. In general, as the number of sample trees decreased, the variability associated with the LRF estimates tended to increase. The pattern in the variability of the LRF estimates noted above was also reflected in the sampling error. For example, balsam in cruise three had a sampling error of plus or minus 15.42 fbm/m3 (the highest) whereas lodgepole pine in cruise one had a sampling error of plus or minus 1.27 fbm/m3 (the lowest). The trend of increased precision with increased sample size was also observed for cases between the two extremes just mentioned. Further investigation into the relationship between sampling error and number of sample trees found that these two variables showed a significant (a = 0.05) negative correlation (correlation coefficient r = -0.64). The arithmetic average of all the sampling error values obtained was 6.07 fbm/m3. A summary of the results for all cruises is presented in table 5. The precision of an estimate can be discussed either in absolute terms (ie. in the same units as the estimate) or in relative terms (ie. as a proportion of the estimate). Both methods are included in table 5 in the sampling error and percentage error columns respectively. If the magnitudes of the estimates differ then percentages are most convenient for comparisons of variability3. If, however, estimates are of the same magnitude, the use of relative measures of variability can be misleading. In such instances, the variability of an estimate is confounded with the size of the estimate making any comparisons using percentages obscure. On this basis, 3 As an example, relative measures of error such as percentages are often used in mor-phological comparisons. If one were to compare the variability in the size of mouse ears to the variability in the size of elephant ears, the differences in the magnitude of the estimates would necessitate expressing the variability or error as a percentage (Zar 1984). 27 Table 5. Sampling error for LRF estimates in four timber cruises. Cruise Plots (n) Species Trees LRF (fbm/m Variance 3) (fbm/m3)2 Standard Error (fbm/m3) *.05/2,n-l Sampling Error (fbm/m3) Percent Error (%) 1 129 Balsam 2 161.5 13.1815 3.63 1.96 7.11 4.41 Spruce 35 183.8 11.7891 3.43 1.96 6.72 3.66 Pine 628 177.4 0.4228 0.65 1.96 1.27 0.72 2 32 Balsam 8 170.7 17.6926 4.21 2.0395 8.59 5.03 Spruce 108 201.4 4.4264 2.10 2.0395 4.28 2.13 Pine 123 188.8 2.0125 1.42 2.0395 2.90 1.53 3 47 Balsam 5 180.7 58.7119 7.66 2.0129 15.42 8.53 Spruce 97 196.6 6.2990 2.51 2.0129 5.05 2.57 Pine 222 186.9 1.1845 1.09 2.0129 2.19 1.17 4 31 Balsam 32 184.3 17.0206 4.13 2.0423 8.43 4.58 Spruce 117 210.3 5.5480 2.36 2.0423 4.82 2.29 discussion of sampling error in the units of the estimate is likely more appropriate than using percentage error with regard to LRF precision. 5.2' EFFECT ON STUMPAGE Once the precision of an estimate has been determined it is possible to make some judgement as to the usefulness or value of the estimate. To be able to make such a judgement requires some notion of the consequences of various error levels. That is, the indirect costs associated with "poor" or imprecise estimates must be considered. When the sampling error results in table 5 are examined, similar consideration must be given to determining what con-stitutes a "good" or a "poor" estimate of LRF. For example, it must be decided whether or not 15.42 fbm/m3 is an acceptable level of precision. To enable such a decision, the consequences of various levels of LRF precision must be investigated. As was discussed in section 2.0 the value of the LRF has a direct effect on the stumpage rate assessed for each species in each cruise. Therefore the consequence of LRF sampling error is best illustrated by a comparison of appraised stumpage rates. The results of this comparison are outlined in table 6. 28 Table 6. Effect of LRF sampling error on stumpage rates. Cruise Species LRF1 used in Lumber Chips Selling Appraised2 Minimum3 Final3 Appraisal Value Value Price Stumpage Stumpage Stumpage (fbm/m3) (S/m3) ($/m3) ($/m3) ($/m3) ($/m3) ($/m3) Balsam lower 154.4 38.75 1.55 point 161.5 40.54 1.50 upper 168.6 42.32 1.45 Spruce lower 177.1 44.45 1.44 point 183.8 46.13 1.38 upper 190.5 47.82 1.33 Pine lower 176.1 36.45 1.67 point 177.4 36.72 1.66 upper 178.7 36.99 1.65 Balsam lower 162.1 40.69 1.50 point 170.7 42.85 1.43 upper 179.3 45.00 1.37 Spruce lower 197.1 49.47 1.25 point 201.4 50.55 1.21 upper 205.7 51.63 1.18 Pine lower 185.9 38.48 1.56 point 188.8 39.08 1.54 upper 191.7 39.68 1.51 Balsam lower 165.3 41.49 1.46 point 180.7 45.36 1.34 upper 196.1 49.22 1.23 Spruce lower 191.5 48.07 1.30 point 196.6 49.35 1.26 upper 201.7 50.63 1.22 Pine lower 184.7 38.23 1.58 point 186.9 38.69 1.56 upper 189.1 39.14 1.54 Balsam lower 175.9 44.15 1.37 point 184.3 46.26 1.30 upper 192.7 48.37 1.24 Spruce lower 205.5 51.58 1.14 point 210.3 52.79 1.10 upper 215.1 53.99 1.07 40.30 -9.19 1.21 1.21 42.04 -7.74 1.26 1.26 43.77 -6.30 1.31 1.31 45.89 -0.54 1.38 1.38 47.51 0.83 1.43 1.43 49.15 2.22 1.47 2.22 38.12 -9.83 1.14 1.14 38.38 -9.61 1.15 1.15 38.64 -9.38 1.16 1.16 42.19 -7.61 1.27 1.27 44.28 -5.87 1.33 1.33 46.37 -4.13 1.39 1.39 50.72 3.55 1.52 3.55 51.76 4.43 1.55 4.43 52.81 5.32 1.58 5.32 40.04 -8.19 1.20 1.20 40.62 -7.69 1.22 1.22 41.19 -7.20 1.24 1.24 42.95 -6.98 1.29 1.29 46.70 -3.85 1.40 1.40 50.45 -0.73 1.51 1.51 49.37 2.41 1.48 2.41 50.61 3.46 1.52 3.46 51.85 4.51 1.56 4.51 39.81 -8.38 1.19 1.19 40.25 -8.01 1.21 1.21 40.68 -7.64 1.22 1.22 45.52 -4.84 1.37 1.37 47.56 -3.14 1.43 1.43 49.61 -1.43 1.49 1.49 52.72 5.25 1.58 5.25 53.89 6.24 1.62 6.24 55.06 7.23 1.65 7.23 The LRF's used in the appraisal calculations corresponded to the LRF point estimate and the upper and lower confidence limits (LRF + sampling error, LRF — sampling error). 2 The operating costs, profit and risk ratio, and lumber and chip selling prices used for each species in each appraisal are listed in table 4. 3 Current minimum stumpage rates in the interior of British Columbia are 3% of the selling price. The final stumpage rate payable by the licencee is the greater of minimum stumpage or the stumpage rate indicated by the appraisal calculations. 29 Many of the species in table 6 exhibit appraised stumpage values that are negative. In such cases, minimum stumpage rates would apply. For these species, the variability associated with the LRF has no impact on the stumpage rate payable unless the stumpage rate based on the upper confidence limit of the LRF extends above the level of minimum stumpage. Thus given the current market conditions and operating costs associated with many low value, poor recovery species in the interior, the LRF sampling error is inconsequential with respect to the resulting stumpage rates. This, however, is not always the case. For some high value species that yield good lumber recovery, appraised stumpage rates are substantially above the minimum level. For such species, LRF sampling error can have a significant impact on the stumpage rate payable. The effect of the value of the LRF on the stumpage rate varies with the lumber sales value. Higher lumber sales values at any given level of LRF will result in a higher selling price for logs and thus higher stumpage rates. This relationship is shown graphically in figure 1 for lumber sales values ranging between $180/Mfbm and $260/Mfbm. The points in this figure were based on a profit and risk ratio of 0.2 and operating costs of $40/m3. To simplify the graph, chip values were ignored in the computation of the stumpage rate. Chip values have a compensatory effect with regard to lumber values. If lumber recovery is low (and hence the lumber value represented by a cubic metre of timber is low), the chip recovery will be higher (and the chip value will be higher). However, given the current low valuation of chips in appraisals, the contribution of chips to the total selling price is small. Chips presently contribute between two and five percent of the total selling price per cubic metre of timber. Therefore, chip values do not have a major effect on the stumpage rates. To a limited extent, the value of the profit and risk ratio (which functions as a discount factor on the selling price) also affects the appraised stumpage rate. Increases in the profit and risk ratio result in decreases in the appraised stumpage rate. The rate of change of the stumpage rate per unit increase in LRF (as represented by the slope of the lines in figure 1) also differs with lumber sales value. Higher lumber sales values will result in slightly more pronounced changes in stumpage rates per unit change in LRF. Based on figure 1, a lumber sales value of $180/Mfbm would result in a $0.15/m3 increase in 30 Stumpage rate over LRF for various levels of lumber sales. Legend A lumbar v»lu« tieO/Mlbm x lumbar v«lu« tiOO/Mlbm • lumbar valua ta2Q/Mfbm a lumbar valua 1240/Mlbm B lumbar valu«l260/Mlbm 230 LRF board feet/cubic metre stumpage rate for every unit increase in LRF. At $260/Mfbm, one would expect a $0.22/m3 increase in the stumpage rate per unit change in LRF. The profit and risk ratio and the chip value would influence these rates of change to a lesser degree. For the cruises examined in this investigation, balsam showed a $0.20/m3 change in stumpage rate per unit change in LRF. Spruce, which had the same lumber sales value as balsam ($251/Mfbm) but a lower profit and risk ratio (0.18 for spruce compared to 0.20 for balsam) showed a $0.21/m3 change in stumpage rate per unit change in LRF. The lower valued lodgepole pine (lumber sales value of $207/Mfbm, profit and risk ratio of 0.17) exhibited a $0.17/m3 change in stumpage rate per unit change in LRF. The impact of LRF sampling error on the actual amount of stumpage that is paid will vary with the volume of timber in the cutting permit represented by that species. For example, a $0.50/m3 difference in stumpage rates over a volume of 20,000 m3 would result in a $10,000 difference in the stumpage payable. The volume of timber used for stumpage payment calculations is determined either from the scale of the harvested timber or the cruise. For cruise-based stumpage assessment the precision requirement for total net volume estimation is higher than that stipulated for a standard cruise such as those used in this study. For illustrative purposes, however, the net merchantable volume estimates from the four cruises examined were used to calculate the range of possible stumpage payments associated with LRF sampling error for spruce. These are given in table 7. Substantial overpayment to the Crown by the licencee or conversely lost Crown stum-page revenue may be occurring as a result of LRF sampling error. For example for spruce in cruise 2, the stumpage payable is $108,796 based on the LRF point estimate from the cruise. Discussions in terms of point estimates, however, are not particularly useful. If the distribution of all possible estimates is continuous, no statistical probability can be assessed for the occurence of the point estimate. However, the probability of the true population value being found within a specified range or interval of estimated values can be calculated. Thus for spruce in cruise 2 one can be 95% confident that the true population value for LRF will fall in the interval between 197.1 fbm/m3 and 205.7 fbm/m3. As a result, it is far more meaningful to discuss the 32 Table 7. Effect of LRF sampling error on spruce stumpage. Lower Bound Point Upper Bound LRF LRF LRF Cruise 1 $1.38/m3 $1.43/m3 $2.22/m3 Stumpage Rate (Net Volume = 3,119m3) $4,304 $4,460 $6,924 Stumpage Payable Range of stumpage payable is between $4,304 and $6,924. This interval is $2,620. Cruise 2 S3.55/m3 $4.43/m3 $5.32/m3 Stumpage Rate (Net Volume = 24,559m3) $87,184 $108,796 $130,654 Stumpage Payable Range of stumpage payable is between $87,184 and $130,654. This interval is $43,470. Cruise 3 $2.41/m3 $3.46/m3 $4.5l/m3 Stumpage Rate (Net Volume = 20,976m3) $50,552 $72,577 $94,602 Stumpage Payable Range of stumpage payable is between $50,552 and $94,602. This interval is $44,050. Cruise 4 $5.25/m3 $6.24/m3 $7.23/m3 Stumpage Rate (Net Volume = 18,602m3) $97,661 $116,076 $134,492 Stumpage Payable Range of stumpage payable is between $97,661 and $134,492. This interval is $36,831. interval estimate of LRF rather than the point estimate. If one considers the interval estimate for LRF, a range of stumpage payable values can be calculated that relate to the limits of the interval. For spruce in cruise 2, this range due to imprecision in the LRF is $43,470. That is, the "true" stumpage payable resulting from the "true" LRF value might be anywhere between $87,184 and $130,654 (with a probability of 95%). In financial terms, such imprecision is clearly unacceptable. 5.3 DETERMINATION OF LRF SAMPLING ERROR The successful application of the jackknife method to approximate the LRF sampling error depends largely on the computing resources available. The cost in computing time of using the jackknife is determined by the design efficiency of the compilation program and the number of plots in the cruise. Since the number of plots in a cruise dictates the number of times that the LRF calculations must be replicated, using the jackknife to determine the LRF sampling error in a large cruise can be costly. In addition, the LRF must be calculated for each species. This causes the computing cost for cruises with many species to increase greatly. 33 The computing time required to process cruise 2 and cruise 3 in the program LRFJACK was investigated. Computation of the jackknifed LRF variance estimate for cruise 2, with 32 plots, three species and a total of 239 sample trees, took 35.445 CPU (central processing unit) seconds. The same run through LRFJACK for cruise 3 took 76.329 CPU seconds. Although cruise 3 took twice as long to process as cruise 2, cruise 3 had 47 plots, three species and a total of 324 trees. If the jackknife procedure was disregarded and only the time taken to compute the LRF estimate itself was considered, the computing time was 2.669 CPU seconds and 2.889 CPU seconds for cruise 2 and cruise 3 respectively. Thus for cruise 2, incorporation of the jackknife procedure increased the computing time related to the LRF calculation by 13.3 times. The computing time for LRF in cruise 3 was increased 26 fold by including the jackknife. For the individual species in each cruise, the computing time required also depended on the number of sample trees. For example, 1.172 CPU seconds were required to compute the LRF estimate for lodgepole pine in cruise 3 (222 sample trees). Computation of jackknifed LRF variance for this species increased the computing time requirement to 49.846 CPU seconds. In contrast, 0.049 CPU seconds were required to compute the LRF estimate for balsam in cruise 3 (five sample trees). With the jackknife procedure for LRF variance approximation included, the required computing time increased to 2.174 CPU seconds for this species. It must be emphasized that the previous computing time comparisons are specific to the program LRFJACK and cannot be extrapolated to all compilation programs in general. More research on program design could reduce the additional computing time required to compute the jackknife variance estimates. In addition, one must consider that the LRF computation represents only a portion of a total cruise compilation program. As a result, it is possible that the overall compilation time would be so large that the extra time involved in jackknifing would not be noticeable. Due to computing time constraints, some indirect means of approximation might be more practical and less costly than using the jackknife for routine computation of LRF sampling error in operational cruises. An alternative approach could take the form of a regression to predict LRF sampling error from more readily available variables. However, the development of such a technique would be complex and is beyond the scope of this study. A convenient 34 approach would be to relate LRF sampling error to the cruise computed sampling error for total net volume. Unfortunately, there is little reason to suspect a good correlation between these two errors. The sampling error for total net volume is based on an aggregation of all species. In contrast, LRF is computed on a per species basis. The variation of the total net volume of a species is determined largely by the spatial distribution of the species. A homogenous distribution would result in lower variability whereas a clumped distribution would result in some high plot volumes and some low plot volumes and thus higher variability. LRF however is essentially a ratio of the decay volume to gross volume in a log. This ratio dampens much of the variability associated with high and low plot volumes. LRF variability is influenced primarily by the variability of the decay percentage between trees in a particular species. Based on the four cruises in this study, a simple correlation test was used to confirm that no significant (a = 0.05) relationship existed between LRF sampling error and total net volume sampling error (correlation coefficient r = —0.05). As a result, until an alternative means of assessing the LRF sampling error can be found, the jackknife is likely the best option available. 5.4 OTHER SOURCES OF ERROR IN THE LRF Error in the determination of the LRF can be divided into sampling and non-sampling errors. By definition, non-sampling errors include all sources of error other than sampling error. Non-sampling errors are, in general, difficult to quantify and even more difficult to reduce. As a result, control joi the sampling error is often the only means available to directly increase the precision of an estimate. Although the estimation of non-sampling error for the LRF is beyond the scope of this study, it is useful to discuss them to put the effect of the sampling error on the stumpage rates into perspective. Much of the non-sampling error associated with the LRF is built into the cruising and appraisal systems. Since the procedures in these systems are essentially Ministry of Forests policy, any associated inaccuracies cannot truly be considered error. However as procedures and policy are subject to reevaluation and revision it is worthwhile to discuss some of these inaccuracies here. 35 One source of inaccuracy in LRF estimation is the loss factor for decay that is applied to each tree. Decay losses are applied as an equal percentage along the entire length of the stem. The result is that the butt log of a tree is assigned the same percentage of decay as the top of the tree. Therefore the true distribution of decay amongst the logs in a tree is not reflected. In addition, this procedure also makes the LRF estimates that result appear less variable than they really are since the variability of decay within a tree is eliminated. Another possibility for inaccuracy in LRF estimation is the manner in which the sound-wood LRF's are adjusted for the decay present in a log. The current one-half percent reduction for every one percent of decay is an arbitrary rule. Very little investigation has been directed toward testing and validating this assumption. Perhaps the most serious contribution to the error in LRF is the soundwood LRF values. These values (table 1) have remained virtually unchanged since the late 1970's. In the interim, advances in sawmilling technology have spread rapidly throughout the interior of British Columbia. Features such as computerized bucking optimization, improved log quality control, more rigorous sorting, laser scanning for headrig sawing, computerized edging optimiza-tion and thinner saw kerfs have all contributed to increases in lumber recovery. As a result, the level of average mill recovery in the interior has been significantly elevated. Consequently it is questionable whether the present soundwood LRF values used in appraisals actually reflect the current lumber recovery of an operator of average efficiency as they are intended to do. All three of these factors warrant further investigation with regard to their effect on the accuracy of the LRF estimate. It is likely that the magnitude of these non-sampling errors far outweighs that of sampling error. However, until procedural and policy changes occur, sampling error remains as the only means by which the error associated with LRF can be controlled. 36 6.0 CONCLUSIONS AND RECOMMENDATIONS The jackknife can be used successfully as a method of approximating the sampling error for the cruise-based LRF. Although the jackknife procedure is conceptually straightforward, its use results in a substantial increase in computing time which may limit its practical application. In the four cruises investigated, the LRF sampling error for individual species ranged from plus or minus 15.4 fbm/m3 to 1.3 fbm/m3. All of these cruises had been designed to meet the Ministry of Forests' precision requirement of plus or minus 15 percent for total net volume. The LRF sampling error was significantly correlated to the number of sample trees of a particular species. There was no significant correlation between the LRF sampling error and the sampling error attained for total net volume. LRF sampling errors of the magnitude found in this study had a significant impact on the appraised stumpage rates. Depending on the market value of the species (and to a lesser extent the profit and risk ratio) a $0.17 - 0.21/m3 change in stumpage rate was observed for every fbm/m3 change in' the LRF. For those species with good recovery and high market value that are currently above minimum stumpage levels, the LRF sampling error can have a dramatic effect on the stumpage payable. In operational timber cruises in the interior of British Columbia the estimation of LRF is considered a secondary objective. All specifications for cruise design are based on estimation of total net merchantable volume. There are currently no standards for acceptable levels of variability for the LRF estimate. This study has shown that the existing levels of precision associated with LRF estimation can have a significant effect on the appraised stumpage rates. Therefore, it is reasonable to suggest that minimum requirements be established to ensure that an acceptable level of LRF sampling error is maintained. The design objectives of the cruise should be adjusted to place more emphasis on the precise estimation of LRF. Based on this limited study, no specific recommendations for revised cruise design criteria can be made except that some minimum number of sample trees per species be established. Sampling error is only a small component of the total error associated with LRF esti-mation and it is suggested that more effort be devoted to examining some of the other error 37 sources. In particular, the soundwood LRF values should be reevaluated on a regular basis to ensure that they represent the current levels of average lumber recovery. The arbitrary one-half percent reduction in the soundwood LRF for every one percent of decay found in a log should be tested and validated. In addition, an alternative to decay factors that are applied as an equal percentage over the length of the bole of a tree should be investigated. 38 LITERATURE CITED Bean, J.A. 1975. Distribution and properties of variance estimators for complex multistage probability samples. Vital and Health Statistics. National Centre for Health Statistics. Wash. D.C. Series 2 No.65. Cochran, W. 1977. Sampling Techniques. John Wiley and Sons, Inc., New York. 428p. Cunia, T. 1981. Needs and basis of sampling, pp.315-325 IN In-place Resource Inventories Principles and Practices: Proceedings of a National Workshop. University of Maine, Orono. HOlp. Demaerschalk, J.P. and A. Kozak. 1977. The whole-bole system: a conditioned dual equation system for precise prediction of tree profiles. Can. J. For. Res. 7:488-497. Deming, W.E. 1950. Some Theory of Sampling. John Wiley and Sons, Inc., New York. 602p. Deming, W.E. 1960. Sample Design in Business Research. John Wiley and Sons, Inc., New York. 517p. Dobie, J. and J.B. Kasper. 1975. Log values for hemlock and cedar from northwestern B.C. Western For. Prod. Lab. CFS Information Report. VP-X-144. Dobie, J. 1978. Small-log sawmill yields in the B.C. interior. Western For. Prod. Lab. CFS Information Report. VP-X-167. Efron, B. 1981. Nonparametric estimates of standard error: the jackknife, the bootstrap and other methods. Biometrika 68:589-599. Efron, B. and C. Stein. 1981. The jackknife estimate of variance. Ann. Stat. 9:586-596. Farewell, V.T. 1978. Jackknife estimation with structured data. Biometrika 65:444-447. Folsom, R.E., D.L. Bayless and B.V. Shah. 1971. Jackknifing for variance components in com-plex sample survey designs. Proceedings of the Social Stat. Sec, Amer. Stat. Assoc. pp.36-39. Fontaine, R.G.(ed.). 1973. Manual of Forest Inventory With Special Reference to Mixed Tropical Forest. FAO, Rome. 199p. Frankel, M.R. 1971. Inference from Survey Samples. Inst, for Soc. Res., Ann Arbor, Mich. 173p. Gregoire, T.G. 1984. The jackknife: an introduction with applications in forestry data analysis. Can. J. For. Res. 14:493-497. 39 Grosenbaugh, L.R. 1971. STX1-11-71 for dendrometry or multistage 3P samples. USDA Forest Service Publ. FS-277. 63p. Hinkley, D.V. 1977. Jackknife confidence limits using Student t-approximations. Biometrika 64:21-28. Hinkley,D.V. and B.-C. Wei. 1984. Improvements of jackknife confidence limit methods. Biometrika 71:331-339. Jones, H.L. 1974. Jackknife estimation of functions of stratum means. Biometrika 61:343-348. Juhasz, J.J. 1976. Methods of crown timber appraisal in B.C. pp.56-88 IN Timber Policy Issues in B.C. Edited by W. McKillop and W.J. Mead. University of B.C. Press, Vancouver. 277p. Kish, L. 1962. Variances for indexes from complex samples. Proceedings of the Social Stat. Sec, Amer. Stat. Assoc. p.190-199. Kish, L. 1965. Survey Sampling. John Wiley and Sons, Inc., New York. 643p. Kish, L. 1968. Standard errors for indexes from complex samples. J. Am. Stat. Assoc. 63:512-529. Kish, L. and M.R. Frankel. 1970. Balanced repeated replication for standard errors. J. Am. Stat. Assoc. 65:1071-1094. Kish, L. and M.R. Frankel. 1974. Inference from complex samples. Jour. Roy. Stat. Soc. B 36:1-37. Krewski, D. and J.N.K. Rao. 1981. Inference from stratified samples: properties of the lineariza-tion, jackknife and balanced repeated replications methods. Ann. of Stat. 9:1010-1019. Lemeshow, S. and P.S. Levy. 1979. Estimating the variances of ratio estimates in complex sur-veys with two primary sampling units per stratum: a comparison of balanced repeated replication and jackknife techniques. J. of Stat. Comp. and Sim. 8:191-205. Marshall, P.L. and J.C. Nautiyal. 1980. Optimal sampling size in multivariate forest inventories: a programming procedure. Can. J. For. Res. 10:579-585. McCarthy, P.J. 1966. Replication: an approach to the analysis of data from complex surveys. Vital and Health Statistics. National Centre for Health Statistics. Wash., D.C. Series 2, no.14. McCarthy, P.J. 1969. Pseudoreplication: half-samples. Internat. Stat. Rev. 37:239-264. Miller, R.G. 1974. The jackknife—a review. Biometrika 61:1-15. 40 Ministry of Forests. 1973. Information report on the revised appraisal system in the B.C. inte-rior. Province of B.C. Victoria. Ministry of Forests. 1976. Metric diameter class decay, waste and breakage factors for all forest inventory zones. Forest Inventory Division. Province of B.C. Victoria. Ministry of Forests. 1980a. White paper on crown timber disposal. Province of B.C. Victoria. Ministry of Forests. 1980b. Documentation and cruise compilation. Valuation Branch. Province of B.C. Victoria. Ministry of Forests. 1982. Manual of cruising procedures and cruise compilation. Province of B.C. Victoria. Parr, W.C. 1983. A note on the jackknife, the bootstrap and the delta method estimators of bias and variance. Biometrika 43:353-360. Parr, W.C. and W.R. Schucany. 1980. The jackknife: a bibliography. Internat. Stat. Rev. 48:73-78. Parratt, L.G. 1961. Probability and Experimental Errors in Science. John Wiley and Sons, Inc., New York. 255p. Quenouille, M.H. 1956. Notes on bias in estimation. Biometrika 43:353-360. Rao, J.N.K. and C.F.J. Wu. 1985. Inference from stratified samples: second-order analysis of three methods for non-linear statistics. J. Am. Stat. Assoc. 80:620-630. Schreuder, H.T. and G.E. Brink. 1983. The jackknife—a useful statistical tool. pp.531-535 IN Renewable Resources Inventories Workshop Proceedings. College of Forestry, O.S.U. Corvallis, Oregon. 737p. Schreuder, H.T., G.E. Brink and R. L. Wilson. 1984a. Alternative estimators for point-Poisson sampling. For. Sci. 30:803-812. Schreuder, H.T., G.E. Brink, D.I. Schroeder and R. Dieckman. 1984b. Model-based sampling versus point-Poisson sampling on a timber sale in the Roosevelt National Forest in Colorado. For. Sci. 30:652-656. Schreuder, H.T. and CE. Thomas. 1985. Efficient sampling techniques for timber sale surveys and inventory updates. For. Sci. 31:857-866. Shah, B.V. 1978. Variance estimates for complex statistics from multistage sample surveys. pp.25-34 IN Survey Sampling and Measurement. Edited by N.K. Namboordiri. Aca-demic Press Inc., New York. 364p. 41 Task Force on Crown Timber Disposal. 1974. Timber appraisal: policies and procedures for eval-uating crown timber in British Columbia. Second Report. Ministry of Forests. Province of B.C. Victoria. Tepping, B. 1968. The estimation of variance in complex surveys. Proceedings of the Social Stat. Sec, Amer. Stat. Assoc. pp.11-18. Wolter, K. 1985. An Introduction to Variance Estimation. Springer-Verlag, New York. 427p. Woodruff, R.S. 1971. A simple method for approximating the variance of a complicated estimate. J. Am. Stat. Assoc. 66:411-414. Woodruff, R.S. and B.D. Causey. 1976. Computerized method for approximating the variance of a complicated estimator. J. Am. Stat. Assoc. 71:315-321. Yang, Y.C. and F.H. Kung. 1983. Method for estimating bole volume. J. For. 81:224-227. Zar, J.H. 1984. Biostatistical Analysis. Prentice-Hall, Englewood Cliffs, N.J. 718p. 42 APPENDIX A Rothery Calculation END PRODUCT APPRAISAL WORKSHEET APPRAISAL DATA Species BA&O/S FI LO SP Small Op. Indicator 0 0 0 0 PRICES AMV-Base Selling Price 251 233 207 251 Chips 10.50 10.50 10.50 10.50 Pulp Logs 0.00 0.00 0.00 0.00 LOG PERCENTAGES Small Wood % 90 41 76 58 Large Wood % 10 59 24 42 Pulp logs % 0 0 0 0 RECOVERIES SW LRF 173 195 186 189 LW LRF 211 226 219 226 Graph Factor 0.072 0.098 0.087 0.076 COSTS Logging(S/m3) 19.64 19.64 19.64 19.64 Milling(S/m3) 23.13 22.66 22.77 19.79 Forestry ($/m 3) 0.00 0.00 0.00 0.00 OTHER DATA Profit Ratio 20 17 17 18 Bonus Bid 0.00 0.00 0.00 0.00 Effective Date 860701 860701 860701 860701 CALCULATED VALUES Prorated LRF 177 213 192 205 Chip Recovery 0.1304 0.1316 0.1420 0.1114 Selling Price - Lumber 44.38 49.70 39.69 51.34 Selling Price - Chips 1.37 1.38 1.49 1.17 S.P. - Pulp Logs 0.00 0.00 0.00 0.00 Total Selling Price 45.75 51.08 41.18 52.51 Total Operating Costs 42.77 42.30 42.41 39.43 Discount Value 38.12 43.66 35.20 44.50 Profit and Risk 7.63 7.42 5.98 8.01 Indicated Stumpage -4.65 1.36 -7.21 5.07 Minimum Stumpage 1.37 1.53 1.24 1.58 Recommended Stumpage 1.37 1.53 1.24 5.07 Bonus Bid 0.00 0.00 0.00 0.00 Final Stumpage 1.37 1.53 1.24 5.07 43 APPENDIX B The procedure for calculating the sampling error for prism plots when the area has been strat-ified into types is described in section 5.31 of the Ministry of Forests cruising manual (Ministry of Forests 1982). A summary of this procedure is as follows: Step 1. For each type, compute the average volume per hectare,yh, and the standard deviation, Syh. Step 2. Compute the volume per type, VOLh, by multiplying the average volume per hectare in the type, yh, by the area (in hectares), Ah, represented by the type. Step 3. Compute the proportional volume, PVh, for each type by dividing the volume per type, VOLh, by the sum of all the type volumes, VOL. Step 4. For each type, compute what is referred to as sampling error % or SE% (Ministry of Forests 1982) by dividing the coefficient of variation (multiplied by 100%) for each type by the square root of the number of plots in the type. Step 5. For each type, compute (PVh)2 x (SE%h)2 • Sum this quantity over all types. Step 6. The percentage error (or the sampling error expressed as a percentage of the mean) for total volume is then calculated as the square root of the sum computed in step 5 multiplied by a t value with degrees of freedom corresponding to the total number of plots minus the number of types. The manner in which the preceding procedure relates to equation (29) is given in the following proof. The standard formula for calculation of the variance in stratified sampling is: If Nh/N is substituted by Ah/A (the area represented by type h over the total area) then, Multiplying the RHS by y\j yh results in and then multiplying both sides by 1002 yields: 44 Dividing both sides by y2 (overall average volume squared) gives 1002 y = E100 Vh 2 (W (VhM A y which is equivalent to step 5 in the Ministry of Forests (1982) procedures. Taking the square root of both sides results in: x 100 £ ( ^ x W o ) W Multiplying both sides by the appropriate t value (see step 6 above) yields the percentage error, PE: S-PE = t x -4 x 100 y E = - x 100 y where £=precision or sampling error = r x Sy, thus completing the proof. 45 APPENDIX C The loss factor tables (Ministry of Forests 1976) that were applicable to each species in each study cruise are listed below. CRUISE SPECIES TABLE NUMBER 1 Balsam 413 Spruce 511 Lodgepole pine 812 2 Balsam 491 Spruce 597 Lodgepole pine 812 3 Balsam 491 Spruce 597 Lodgepole pine 812 4 Balsam 491 Spruce 597 46 APPENDIX D The height equation or table used for each species in each cruise is listed below. CRUISE SPECIES HEIGHT Balsam Spruce Lodgepole pine Balsam Spruce Lodgepole pine Balsam Spruce Lodgepole pine Balsam Spruce see height table A below see height table B below Height Height Height Height Height Height Height Height Height 1.3 + 14.6967 0.30484 (DBH) 1.3 + 18.448 ^ S K ) + 0.29609 (DBH) 1.3 + 0.97673 (DBH) - 0.00637 (DBH) 2 1.3 + 18.448 {DDB$X) + 0.29609 (DBH) 1.34137 (DBH) - 0.01452 (DBH) 2 0.54773 (£>£#) 1.34137 (DBH) - 0.01452 2 = 1.3 = 1.3 - 8.4735 = 1.3 1.3+ 0.68894 (£>£#) 1.3 + 0.75339 (DBH) 0.0024 (DBH)2 0.00415 (DBH)2 The heights corresponding to height tables A and B referred to above are listed below. DBH CLASS (cm) HEIGHT TABLE A (metres) HEIGHT TABLE B (metres) < 7.5 3.3 6.2 7.5-12.4 7.1 10.2 12.5-17.4 11.3 13.6 17.5-22.4 15.1 16.6 22.5-27.4 17.9 19.3 27.5-32.4 19.9 21.6 32.5-37.4 21.1 23.7 37.5-42.4 21.8 25.5 42.5-47.4 22.2 27.2 47.5-52.4 22.4 28.6 52.5-57.4 22.4 29.9 57.5-62.4 22.5 31.1 62.5-67.4 22.5 32.0 > 67.5 22.5 32.0 47 APPENDIX E The following is a listing of the program LRFJACK. 1 c 2 c 3 c 4 C ********** PROGRAM LRFJACK ********** 5 C 6 C 7 C T h i s program performs a p a r t i a l c r u i s e c o m p i l a t i o n i n which the 8 C LRF f o r each s p e c i e s found i n the c r u i s e i s computed. F u r t h e r 9 C d e t a i l s can be found i n s e c t i o n 4.3 on program d e s c r i p t i o n . 10 C 11 C 12 C 13 C 14 C 15 COMMON /TAPER/ HHP,DDIN,PPER,ZZZZ,CCC1,CCC2,CCC3,CCC4,RRH 16 REAL DBH(700),GOL,HT,DVOL,NVOL,DECAY(700),WASTE(700),BREAK(700) 17 REAL TF,PI.SMALL,LARGE,PROLRF,AVELRF,dAKLRF,VARLRF 18 REAL DWO(150),STUMP,T0P(4),PLOTN,HA(15),DBHLIM(4),BAF 19 INTEGER TREE,DIB,LNUM,AGE,ZONE,PTYPE(15),NUMTYP,NPLOT 20 INTEGER PNPLOT,NUMSPP,SPNAME(4),HTEQN(4) 21 INTEGER TC(700),TYPE(700),PLOT(700),SPP(700),RG(700) 22 DIMENSION V0LL(40) , T0PL(40), V0LDL(40), V0LNL(40) 23 DIMENSION LRF(19),DC(19),LRFH(19).LRFC(19),DIBM(19) 24 REAL NMV(40), SUMNMV(19.3), SUMGMV(19,3), 25 1SDECAY(19,3) , SUMFBM(19,3), STGMV(3), 26 2STDEC(3 ) , STNMV(3), STFBM(3 ) , SLRF(3), 27 3LTGMVO), LTDEC ( 3 ) , LTNMV(3), LTFBM( 3 ) 28 REAL LLRF(3) , GMVS(19), NMVS(19), DECS(19), 29 1FBMS(19), SGMV, SDEC, SNMV. SFBM. LRFSM, LGMV, 30 2LDEC. LNMV, LFBM, LRFBIG. SUMSOR, HC(20) 31 C 32 C 33 C 34 C Soundwood LRF v a l u e s : 35 C 36 DATA LRF/ 158, 185, 202, 217, 226. 234, 240. 246. 252, 37 1 258. 263, 266, 267. 265, 263. 261. 259, 257, 38 2 254/ 39 C 40 DATA LRFC/ 132, 156, 177, 197, 215, 227, 235, 238, 239, 41 1 239. 237, 233, 228, 223, 217, 210, 201, 193. 42 2 184/ 43 C 44 DATA LRFH/ 151. 172. 187, 199, 210, 217, 222, 228, 233, 45 1 237, 240, 241, 240, 237, 234. 231, 227, 224, 46 2 221/ 47 C 48 C 49 C Top d i a m e t e r i n s i d e bark c l a s s e s : 50 C 51 DATA DC/ 14.9. 19.9, 24.9, 29.9, 34.9, 39.9. 44.9. 49.9, 52 1 54.9, 59.9, 64.9, 69.9, 74.9, 79.9. 84.9, 89.9. 53 2 94.9. 99.9 / 54 C 55 C 56 C M i d p o i n t s f o r top di a m e t e r i n s i d e bark c l a s s e s : 57 C 58 DATA DIBM/ 12.5, 17.5, 22.5, 27.5, 32.5, 37.5. 42.5, 48 59 1 47.5, 52.5, 57.5, 62.5, 67.5, 72.5. 77.5, 60 2 82.5, 87.5, 92.5, 97.5, 97.5/ 61 C 62 C 63 C 64 C NUM=counter f o r number of sample t r e e s ; 65 C AGE=age g r o u p i n g (1=immature 2=mature); 66 C Z0NE=1nventory zone (1=A,B,C 2=D-J 3=K,L); 67 C G0L=1og l e n g t h (m); 68 C BAF=basal a r e a f a c t o r ; 69 C STUMP=stump h e i g h t (m). 70 C 71 NUM=0 72 AGE = 2 73 Z0NE=2 74 STUMP=0.3 75 G0L=5. 76 BAF=6.0 77 PI=3.141593 78 AVELRF=0.0 79 C 80 C 81 C 82 C C r u i s e d a t a i s r e a d from input f i l e : 83 C 84 C 85 CALL TIME(O) 86 READ (4,40) NUMTYP.NPLOT.NUMSPP 87 40 F0RMAT(12,13,12) 88 DO 46 J=1,NUMTYP 89 READ (4.41) HA(J ) ,PTYPE(J) 90 41 FORMAT (F5.1,13) 91 46 CONTINUE 92 DO 45 K=1.NUMSPP 93 READ (4,42) SPNAME(K) ,TOP(K) ,DBHLIM(K),HTEQN(K) 94 42 FORMAT (I2.2F5.1,12) 95 45 CONTINUE 96 C 97 DO 66 J=1,700 98 READ (4, 1,END = 200) TYPE(J ). PLOT(J ),SPP(u),DBH(J),TC(J),RG(J) 99 1DECAY(J),WASTE(J),BREAK(J) 100 1 FORMAT (12.213.F6 . 1 ,212,3F5 . 1 ) 101 NUM=NUM+1 102 66 CONTINUE 103 200 CONTINUE 104 C 105 C 106 C A l l c a l c u l a t i o n s a r e performed f o r each s p e c i e s : 107 C 108 DO 100 N=1.NUMSPP 109 CALL TIME(2, 1 ) 1 10 WRITE (6,57) SPNAME(N) 1 1 1 57 FORMAT ('-', 'SPECIES= '.13) 1 12 C 1 13 c 1 14 c H e i g h t d a t a a p p r o p r i a t e f o r s p e c i e s N i s read: 1 15 c 1 16 IF (HTEQN(N).EQ.1.OR.HTEQN(N).EQ.2) READ (5,43,END=47) 49 1 17 1X1 ,X2 1 18 43 FORMAT (F8.5.F9.5) 1 19 IF (HTEON(N).EO.3) READ ( 5.44,END = 47) (HC( I ) ,I = 1.20) 120 44 FORMAT (20F4.1) 121 47 CONTINUE 122 C 123 C 124 C T h i s s e c t i o n of code keeps t r a c k of p l o t d e l e t i o n s f o r the 125 C j a c k k n i f e p r o c e d u r e . 126 C 127 NP=NPL0T+1 128 DO 16 M=1,NP 129 CALL TIME(2,1) 130 PLOTN=NPLOT 131 C 132 C 133 C V a r i a b l e I n i t i a l i z a t i o n : 134 C 135 TREE =0 136 DO 22 1=1,3 137 STGMV(I)=0.0 138 STDEC(I)=0.0 139 STNMV(I)=0.0 140 STFBM(I)=0.0 141 LTGMV(I)=0.0 142 LTNMV(I)=0.0 143 LTFBM(I)=0.0 144 LTDEC(I)=0.0 145 DO 23 d= 1 , 19 146 SUMNMV(J,I)=0.0 147 SUMGMV(d,I)=0.0 148 SDECAY(J,I)=0.0 149 SUMFBM(J,I)=0.0 150 23 CONTINUE 151 22 CONTINUE 152 DO 24 1=1,19 153 GMVS(I)=0.0 154 NMVS(I)=0.0 155 DECS(I)=0.0 156 24 CONTINUE 157 SGMV=0.0 158 SDEC=0.0 159 SNMV=0.0 160 SFBM=0.0 161 LGMV=0.0 162 LDEC=0.0 163 LNMV=0.0 164 LFBM=0.0 165 C 166 C 167 C T h i s s e c t i o n of code a l s o keeps t r a c k of p l o t d e l e t i o n s . 168 C 169 L=M-1 170 MINUS=0 171 NTYPE=0 172 DO 55 J=1,NUM 173 IF (MINUS. EQ.O GO TO 777 174 IF (PLOT(d) .NE.L) GO TO 777 50 175 PTYPE(TYPE(d))=PTYPE(TYPE(d))-1 176 NTYPE=TYPE(J) 177 MINUS=1 178 777 IF (PLOT(d).EO.L) GO TO 55 179 IF (SPP(d).NE.SPNAME(N)) GO TO 55 180 C 181 C 182 C Only t r e e s above the minimum dbh l i m i t a r e i n c l u d e d i n the 183 C volume c o m p i l a t i o n . 184 C 185 DLIM=DBHLIM(N)-. 1 186 IF (DBH(d).LE.DLIM) GO TO 55 187 C 188 C 189 C H e i g h t f o r sample t r e e i s computed: 190 C 191 IF (HTEON(N).EQ.1) HT=1.3+X1*DBH(J)+X2*DBH(J)**2 192 IF (HTEQN(N).EQ.2) HT=1.3-X1*DBH(d)/(DBH(d)+1)+X2*DBH(d) 193 IF (HTEQN(N).EQ.3) CALL FINDHT(DBH(d),HC,HT) 194 C 195 C 196 C S u b r o u t i n e LOG i s c a l l e d to m a t h e m a t i c a l l y d i v i d e the t r e e 197 C i n t o 5m l o g s and compute volume f o r each l o g : 198 C 199 CALL LOG(SPP(d).ZONE,AGE,DBH(d),HT.STUMP,TOP(N),GOL, 200 1LNUM,VOLL,TOPL,TLH,BUTD,VOLM,VOLG,BBH) 201 TREE=TREE+1 202 C 203 C 204 C T r e e f a c t o r o r e x p a n s i o n f a c t o r i s c a l c u l a t e d f o r sample t r e e 205 C 206 TF = (BAF *4. )/(PI*(DBH(d)/100.0)**2) 207 c 208 c 209 c Top l o g s l e s s than 2.5m a r e added onto l o g below: 210 c 211 IF (TLH.GE.2.50) GO TO 800 212 VOLL(LNUM-1)=VOLL(LNUM-1)+VOLL(LNUM) 213 TOPL(LNUM-1)=TOPL(LNUM) 214 LNUM=LNUM-1 215 800 CONTINUE 216 C 217 C 218 C A l l the f o l l o w i n g c a l c u l a t i o n s a r e done on a per l o g b a s i s : 2 19 c 220 DO 801 1 = 1 ,LNUM 221 VOLL(I)=VOLL(I)*HA(TYPE(d))*TF/(PTYPE(TYPE(d))* 1.0) 222 NMV(I)=VOLL(I)*(100.0-OECAY(d)-WASTE(d)-BREAK(d))/100.0 223 c 224 c 225 c B i n a r y s e a r c h to f i n d a p p r o p r i a t e top d i b c l a s s f o r . l o g : 226 c 227 IF (TOPL(I) .GT.DC(18)) GO TO 554 228 IF (TOPL(I).GT.DC(8)) GO TO 500 229 IF (T0PL(I).GT.DC(4)) GO TO 501 230 IF (TOPL(I).GT.DC(2)) GO TO 502 231 IF (TOPL(I).GT.DC(1)) GO TO 503 232 DIB=1 51 233 GO TO 555 234 503 DIE = 2 235 GO TO 555 236 502 IF (TOPL(I ) GT DC(3)) DIB=4 237 IF (TOPL(I ) . LE DC(3)) DIB=3 238 GO TO 555 239 501 IF (TOPL(I ) .GT. DC(6)) GO TO 511 240 IF (TOPL (I ) GT DC(5)) DIB=6 241 IF (TOPL(I ) .LE. DC(5)) DIB=5 242 GO TO 555 243 511 IF (TOPL(I ) .GT. DC(7)) DIB=8 244 IF (TOPL(I ) . LE . DC(7)) DIB=7 245 GO TO 555 246 500 IF (TOPL(I ) .GT. DC(14)) GO TO 510 247 IF (TOPL(I ) .GT. DC( 12) ) GO TO 520 248 IF (TOPL(I ) .GT. DC(10)) GO TO 530 249 IF (TOPL(I ) .GT. DC(9)) DIB=10 250 IF (TOPL( I ) . LE . DC(9)) DIB=9 251 GO TO 555 252 530 IF (TOPL(I ) .GT. DC( 1 1 )) DIB=12 253 IF (TOPL(I ) . LE . DC(1 1 ) ) DIB=11 254 GO TO 555 255 520 IF (TOPL(I ).GT. DC(13)) DIB=14 256 IF (TOPL(I ) . LE . DC( 13) ) DIB=13 257 GO TO 555 258 510 IF (TOPL( I ) .GT. DC(16)) GO TO 540 259 IF (TOPL(I ) .GT. DC(15)) DIB=16 260 IF (TOPL(I ) . LE . DC(15)) DIB=15 261 GO TO 555 262 540 IF (TOPL(I ) .GT. DC( 17) ) DIB=18 263 IF (TOPL(I ) .LE. DC(17)) DIB=17 264 GO TO 555 265 554 DIB = 19 266 555 CONTINUE 267 . C 268 C 269 C For each DIB c l a s s and r i s k group, the net merch volume (SUMNMv) 270 C g r o s s volume (SUMGMV), decay p r o p o r t i o n (SDECAY) and 271 C boar d f e e t of lumber (SUMFBM) r e p r e s e n t e d by a l l l o g s of a 272 C s p e c i e s a r e computed: 273 C 274 SUMNMV(DIB.RG(d)) = SUMNMV(DIB,RG(d) ) + NMV(I) 275 SUMGMV (DIB, RG( J ) ) =SUMGMV (DIB, RG( <J ) ) + VOLL(I) 276 SDECAY(DIB,RG(J) )=SDECAY(DIB,RG(d)) + (DECAY(d)*VOLL(I)) 277 IF (SPP(d) . EQ.3) GO TO 420 278 IF (SPP(d) .EQ.2) GO TO 421 279 SUMFBM(DIB,RG(J)) = SUMFBM(DIB,RG(d)) + (NMV(I) * LRF(DIB ) * 280 1(1- -DECAY(d)/200.0)) 281 GO TO 801 282 420 SUMFBM(DIB,RG(d))=SUMFBM(DIB,RG(d)) + (NMV(I) * LRFH(DIB) * 283 1(1. -DECAY(d)/200.0) ) 284 GO TO 801 285 421 SUMFBM(DIB ,RG(d))=SUMFBM(DIB,RG(d)) + NMV(I) * (LRFC(DIB) -286 134 . 256 * (0.147 - 0.00122 * DIBM(DIB)) * DECAY(d)/2.0) 287 801 CONTINUE 288 55 CONTINUE 289 IF (NTYPE. NE.O) PTYPE(NTYPE)=PTYPE(NTYPE)+1 290 DO 850 1=1 ,3 52 291 DO 860 d=1,19 292 IF (SUMGMV(J,I).EO.0.0) GO TO 99 293 SDECAY(d.I)=SDECAY(d, I )/SUMGMV(J,I) 294 GO TO 860 295 99 SDECAY(J,I)=0.0 296 860 CONTINUE 297 850 CONTINUE 298 IF (L.EO.O) WRITE (6,3) 299 3 FORMAT (' ' , 3 ( ' GMV DECAY NMV FBM ')) 300 DO 11 K=1, 19 301 IF (L.EQ.O) WRITE (6,2) (SUMGMV(K,J),SDECAY(K,d). 302 1SUMNMV(K,J),SUMFBM(K,d ) ,d=1.3) 303 2 FORMAT (' ', 3(F9.1,F7.2,F9.1,F13.1)) 304 1 1 CONTINUE 305 C 306 C 307 C These f i g u r e s a r e summarized f o r the small l o g g r o u p i n g 308 C ( l e . l o g s < 30cm top DIB): 309 C 310 DO 600 K=1,3 311 DO 700 d=1,4 312 STGMV(K)=STGMV(K) + SUMGMV(d.K) 313 STDEC(K)=STDEC(K)+ (SDECAY(J,K)*SUMGMV(d,K)) 314 STNMV(K)=STNMV(K)+SUMNMV(d,K) 315 STFBM(K)=STFBM(K)+SUMFBM(d,K) 316 700 CONTINUE 317 IF (STNMV(K).EO.0.0) GO TO 98 318 SLRF(K)=STFBM(K)/STNMV(K) 319 GO TO 97 320 98 SLRF(K)=0.0 321 97 IF (STGMV(K).EQ.O.O) GO TO 96 322 STDEC(K)=STDEC(K)/STGMV(K) 323 GO TO 95 324 96 STOEC(K)=0.0 325 95 IF (L.EO.O) WRITE(6.4) K 326 4 FORMAT('0','Smal1 l o g t o t a l s f o r r i s k group ',12) 327 IF (L.EO.O) WRITE(6,5) STGMV(K),STDEC(K),STNMV(K),STFBM(K) 328 1SLRF(K ) 329 5 FORMAT(' ',F9.3,F7.2,F9.3.F12.3,F8.2) 330 600 CONTINUE 331 C 332 C 333 C S i m i l a r summary f o r l a r g e l o g g r o u p i n g ( i e . l o g s > 30cm 334 C top DIB): 335 C 336 DO 660 K=1,3 337 DO 550 d = 5 , 19 338 LTGMV(K)=LTGMV(K)+SUMGMV(d,K) 339 LTDEC(K)=LTDEC(K)+(SDECAY(d,K)*SUMGMV(d,K)) 340 LTNMV(K)=LTNMV(K)+SUMNMV(d,K) 341 LTFBM(K)=LTFBM(K)+SUMFBM(d,K) 342 550 CONTINUE 343 IF (LTNMV(K).EQ.O.O) GO TO 94 344 LLRF(K)=LTFBM(K)/LTNMV(K) 345 GO TO 93 346 94 LLRF(K)=0.0 347 93 IF (LTGMV(K).EQ.O.O) GO TO 92 348 LTDEC(K)=LTDEC(K)/LTGMV(K) 53 349 1 GO TO 91 350 92 LTDEC(K)=0.0 351 91 IF (L.EO.O) WRITE(6,6) K 352 6 F0RMAT('0','Large l o g t o t a l s f o r r i s k group ',12) 353 IF (L.EO.O) WRITE(6.7) LTGMV(K).LTDEC(K),LTNMV(K) , 354 1LTFBM(K).LLRF(K) 355 7 FORMAT (' ',F9.3,F7.2,F9.3,F12.3,F8.2) 356 660 CONTINUE 357 IF (L.EO.O) WRITE(6,8) 358 8 FORMAT ( ' 0 ' , ' A l l r i s k groups - by d i b c l a s s ' ) 359 C 360 C 361 C Summary a c r o s s a l l r t s k groups: 362 C 363 DO 300 d=1 , 19 364 DO 350 1=1,3 365 GMVS(d)=GMVS(d)+SUMGMV(d,I ) 366 NMVS(J ) =NMVS(d) + SUMNMV(d,I) 367 DECS(d)=DECS(d)+(SDECAY(d,I)*SUMGMV(J,I)) 368 350 CONTINUE 369 IF (GMVS(d).EQ.O.O) GO TO 90 370 DECS(d)=D£CS(d)/GMVS(d) 371 GO TO 89 372 90 DECS(d)=0.0 373 89 FBMS(d)=LRF(d)*NMVS(d)*(1.-DECS(d)/200.) 374 IF (L.EQ.O) WRITE(6,9) GMVS(d).DECS(d),NMVS(d),FBMS(d) 375 9 FORMAT (' ',F9.3,F7.2,F9.3,F12.3) 376 300 CONTINUE 377 DO 999 K=1 ,4 378 SGMV=SGMV+GMVS(K) 379 SDEC=SDEC+(DECS(K)*GMVS(K)) 380 SNMV=SNMV+NMVS(K) 381 SFBM=SFBM+FBMS(K) 382 999 CONTINUE 383 IF (SGMV.EQ.O.O) GO TO 88 384 SDEC=SDEC/SGMV 385 GO TO 87 386 88 SDEC=0.0 387 87 IF (SNMV.EQ.O.O) GO TO 86 388 C 389 C 390 C Small l o g LRF 1s computed: 391 C 392 LRFSM=5FBM/SNMV 393 GO TO 85 394 86 LRFSM=0.0 395 85 IF (L.EQ.O) WRITE(6,10) 396 10 FORMAT('0','A11 r i s k groups - small l o g t o t a l s ' ) 397 IF (L.EQ.O) WRITE(6,12) SGMV,SDEC,SNMV,SFBM,LRFSM 398 12 FORMAT(' ',F9.3,F7.2,F9.3,F12.3,F8.2) 399 DO 998 K=5,19 400 LGMV=LGMV+GMVS(K) 401 LDEC=LDEC+(DECS(K)*GMVS(K)) 402 LNMV=LNMV+NMVS(K) 403 LFBM=LFBM+FBMS(K) 404 998 CONTINUE 405 IF (LGMV.EQ.O.O) GO TO 82 406 LDEC=LDEC/LGMV 54 407 GO TO 81 408 82 LDEC=0.0 409 81 IF (LNMV.EO.O.O) GO TO 80 4 10 C 41 1 C 412 C Larg e l o g LRF 1s computed: 413 C 414 LRFBIG=LFBM/LNMV 415 GO TO 79 416 80 LRFBIG=0.0 417 79 IF (L.EQ.O) WRITE(6,13) 418 13 FORMAT('0','A11 r i s k groups - l a r g e l o g t o t a l s ' ) 419 IF (L.EQ.O) WRITE(6,14) LGMV,LDEC,LNMV,LFBM,LRFBIG 420 14 FORMAT(' ',F9.3,F7.2,F9.3,F12.3,F8.2) 421 IF (L.EQ.O) WRITE(6,15) TREE 422 15 FORMAT('0','The number of t r e e s 1s ',14) 423 SMALL=SNMV/(SNMV+LNMV)* 100.0 424 LARGE = LNMV/(SNMV+LNMV)* 100.0 425 IF (L.EQ.O) WRITE(6,17) SMALL. LARGE 426 17 FORMATCO' , 'SMALL LOG %= ',F6.2,' LARGE LOG %= '.F6.2) 427 C 428 C 429 C P r o r a t e d LRF (by net volume) over l a r g e and small l o g s : 430 C 431 PROLRF = SMALL/100.0*LRFSM+LARGE/100.0*LRFB IG 432 IF (L.EQ.O) WRITE(6,18) PROLRF 433 18 FORMAT( '0' , 'PRORATED LRF = ',F7.2) 434 IF (L.NE.O) GO TO 30 435 AVELRF=PROLRF 436 GO TO 16 437 30 DWO(L)=PROLRF 438 16 CONTINUE 439 CALL TIME(2,1) 440 C 441 C 442 C C a l c u l a t i o n of j a c k k n i f e v a r i a n c e e s t i m a t e : 443 C 444 L=0 445 VARLRF=0.0 446 DO 67 M=1.NUMTYP 447 SUMSQR=0.0 448 DO 68 1=1,100 449 L = L+1 450 SUMSQR=SUMSQR+(DWO(L)-AVELRF)**2 451 IF (I.EQ.PTYPE(M)) GO TO 63 452 68 CONTINUE 453 63 CONTINUE 454 SUM=((PTYPE(M)-1)/(PTYPE(M)*1.0))*SUMSQR 455 VARLRF=VARLRF+SUM 456 67 CONTINUE 457 WRITE(6,21) VARLRF 458 21 FORMATC-'.' VARIANCE OF LRF:',F9.4) 459 100 CONTINUE 460 CALL TIME(2,1) 461 RETURN 462 END 463 C 464 C 55 465 C Subrout me to d e t ermine sample t r e e h e i g h t i f h e i g h t t a b l e used: 466 C 467 SUBROUTINE FINDHT(D,HC,HT) 468 REAL 0,HT,HC(20) 469 IF (0 GE . 97 5) GO TO 60 470 IF (D GE . 53 • 5) GO TO 50 471 IF (D GE . 27 .5) GO TO 40 472 IF (D GE. 12 • 5) GO TO 30 473 IF (D GE . 7. 5) GO TO 20 474 HT=HC( 1 ) 475 20 HT=HC(2) 476 GO TO 66 477 30 IF (D GE . 17 • 5) GO TO 31 478 IF (D LT. 17 .5) HT=HC(3) 479 GO TO 66 480 31 IF (D GE. 22 5) HT=HC(5) 481 IF (D LT. 22 5) HT=HC(4) 482 GO TO 66 483 40 IF (D GE . 37 • 5) GO TO 41 484 IF (D GE . 32 5) HT=HC(7) 485 IF (D LT. 32 .5) HT=HC(6) 486 GO TO 66 487 41 IF (D GE. 42 .5) GO TO 42 488 IF (0 LT. 42 5) HT=HC(8) 489 GO TO 66 490 42 IF (D GE . 47 5) HT=HC(10) 491 IF (D LT . 47 • 5) HT=HC(9) 492 GO TO 66 493 50 IF (D GE . 72 • 5) GO TO 51 494 IF (D GE . 62 5) GO TO 52 495 IF (D GE . 57 .5) HT=HC(12) 496 IF (D LT. 57 5) HT=HC(11) 497 GO TO 66 498 52 IF (D GE . 67 • 5) HT=HC(14) 499 IF (D LT. 67 5) HT=HC(13) 500 GO TO 66 501 51 IF (D GE . 82 .5) GO TO 53 502 IF (0 GE . 77 5) HT=HC(16) 503 IF (D LT. 77 • 5) HT=HC(15) 504 GO TO 66 505 53 IF (D GE . 92 .5) GO TO 54 506 IF (D. GE . 87 .5) HT=HC(18) 507 IF (D LT . 87 .5) HT=HC(17) 508 GO TO 66 509 54 HT=HC( 19) 510 GO TO 66 511 60 HT=HC(20) 512 66 CONTINUE 513 RETURN 514 END 515 SUBROUTINE LOG (10,IZ,IM,DBH,HT.SH.TD.GOL.NL,VLOG,TDL,HLL,DBT,TVOL 516 1,GROS, BAR) 517 C 518 CC ***** M E T R I C V E R S I O N ***** 519 CC 520 CC IS=SPECIES CODE (1-1 6 ) . IZ=ZONE CODE (1 - 3 ) , IM IS 1=IMMATURE, 2=MATU 521 CC 522 CC SPECIES CODE 1 = F 2=C 3=H 4=B 5=S 6=CY 7=PW 8=PL 9=PY 10=L 56 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC C ZONE CODE 1 1=CT 1=A.B,C 12=D 13=MB 14=BI 15=A 2=D,E,F.G.H.I,d 3=K,L 16=WP DBH=DIAMETER OUTSIDE BARK IN CM. HT=TOTAL HEIGHT IN M. SH=STUMP HEIGH IN M. TD=TOP DIAMETER FOR UTILIZATION IN CM. GOL=LOG LENGTH IN M. NL=NO. OF LOGS. VOLUMES AND TOP DIAMETERS CALCULATED IN THE PROGRAM WILL BE IN METRIC UNITS. VLOG=VOLUMES FOR EACH LOG ( 4 0 ) . TDL=TOP DIAMETERS FOR EACH LOG (40) HLL=LENGTH OF TOP LOG, DBT=BUTT DIAMETER OF FIRST LOG TVOL=VOLUME BETWEEN STUMP HT AND TOP DIAMETER, GROS=TOTAL VOLUME BAR=BARK THICKNESS AT BH DIMENSION VL0G(40),TDL(40),PP(16,3),A1(16,3),A2(16,3),A3(16,3) DIMENSION B1(16,3),B2(16,3),B3(16,3),C1(16,3),C2(16,3),AI1(4) DIMENSION A I 2 ( 4 ) , A I 3 ( 4 ) , B I 1 ( 4 ) , B I 2 ( 4 ) , B I 3 ( 4 ) , C i l ( 4 ) . C I 2 ( 4 ) , P P I ( 4 ) INTEGER IO.IZ.IM.NL COMMON /TAPER/ HP,DIN,PER,ZZZ.CC1,CC2,CC3.CC4,RH PP=INFLECTION POINT AS A % OF HEIGHT BY SPECIES, A 1,A2,A3 = C0EFFICI EN TO CALCULATE DIB AT BH BY SPECIES AND ZONE, B1 ,B2,B3=C0EFFICIENTS TO CALCULATE DIB AT INFLECTION POINT BY SPECIES AND ZONE, C1,C2=C0EFFI-CIENTS TO CALCULATE THE TAPER BETWEEN INFLECTION POINT AND THE TOP OF THE TREE BY SPECIES AND ZONE, AI 1,A12,A13,B11,B12,813,CI1,C12 = SAME A A1.A2.A3.B1,B2.B3,C1,C2 FOR THE IMMATURE SPECIES, PPI=AS PP FOR IMMA 546 DATA A l / 0. .749864, -o. ,430824, -0. ,348494, -0. 319971 , 547 1 -1 . 049790, 0. ,110498, -0. ,486151, -0. 180926, 548 2 -0. . 153580, -0, ,882733, 0, .747572, -0. 228798, 549 3 -0. 360590. -0. ,405545. 0. .783003, -0. , 180926, 550 4 -0. 065296, 0. 081798, 0. 309797, 0. 005222, 551 5 -0. .533352, 0. ,110498. 0, , 149363, -0. 180926, 552 6 -0. .153580, -0. .882733, 0. .747572, -0. ,228798. 553 7 -0. 360590, -0. 405545, 0. 783003, -0. 180926, 554 C 555 8 -O. .065296, 0. ,081798, 0. . 309797, -0. ,454276, 556 9 -0, .748257, 0, .110498. 0. .149363. -0, ,610711, 557 1 -0. .153580, -0. ,397575, -0, ,196503, -0. 228798, 558 2 -0. .360590, -0. .136714, -0. .240408, -0. ,610711/ 559 C 560 DATA A2/ 0. .769619, O. .947383. 0. .911785, 0. 948196, 561 1 0. 976369, 0. .950495. 0. ,952628, 0. 941988, 562 2 0. 804892, 0. .902009, 0. ,863199, 0. ,968877, 563 3 0. ,970778, 0. .964192. 0. ,861413. 0. 941988, 564 C 565 4 0. 859519, 0. .921453, 0. .882731, 0. ,932597, 566 5 0. .962901, 0. .950495, 0, ,937943, 0. .941988, 567 6 0. ,804892, 0 .902009, 0, .863199. 0, ,968877, 568 7 0. ,970778, 0. ,964192, 0, ,861413, 0. .94 1988, 569 C 570 8 0. .859519, 0, ,921453, 0. .882731, 0, .970500, 571 9 0. .981445, 0. .950495, 0 .937943, 0, .995397, 572 1 0. .804892, 0, ,934639, 0, .885095, 0, .968877, 573 2 0, ,970778. 0, .939769, 0, .950504, 0, .995397/ 574 C 575 DATA A3/ o. .00053 1. 0 .000215, o .000457, 0 .000203, 576 1 0. .000062, 0, .000227, 0 .000179, 0, .000492, 577 2 0. .001041, -0 .000871, 0 .000254, -0, .000282, 578 3 -0. .000419, -0 .000152. 0 .000783. 0 .000492, 579 C 580 4 -0. ,000305, 0, .000260, 0 .000458, 0, .000200, 57 581 5 582 6 583 7 584 C 585 8 586 9 587 1 588 2 589 C 590 DATA B1/ 591 1 592 2 593 3 594 C 595 4 596 5 597 6 598 7 599 C 600 8 601 9 602 1 603 2 604 C 605 DATA B2/ 606 1 607 2 608 3 609 C 610 4 611 5 612 6 613 7 614 C 615 8 616 9 617 1 618 2 619 C 620 DATA B3/ 621 1 622 2 623 3 624 C 625 4 626 5 627 6 628 7 629 C 630 8 631 9 632 1 633 2 634 C 635 DATA C1/ 636 1 637 2 638 3 -0. . 000035, 0. 000227. 0. ,001041, -0. .000871, -0. ,000419, -o. .000152, -0. ,000305, 0. ,000260, -0. ,000172, 0. 000227. 0. 001041, 0. 000756, -0. 000419, 0. 000323, 2 . ,838860, 3. ,647350, 1. 056450, 2. 451860, 0. ,214856, 1 . 974160, 0. ,518366, -1 . ,019400. 1. ,763810, 2. . 180160, o. .715862, 2 . ,451860. 0. ,214856, 1 . ,974160, 0. .518366. -1 . ,019400, 1. .763810, 2. , 180160, 0. ,907790. 2. ,451860, 0. .214856, 0. .943628, 0. .518366, 0. .378956, 0, .787982. 0. .688175, 0 .801271. 0. .750033, 0. ,895951 , 0, ,807766, 0. .801880, 1 . .017740. 0. .779846, 0. .747715, 0. .861977 , 0. .750033, 0. ,895951 , 0. .807766. 0. ,801880, 1 . .017740. 0. .779846, 0 .747715, 0. .831680, 0, .750033. 0. ,895951 , 0. ,865595, 0. .801880, 0. .831479, -0, ,000637, -o, .000781, -0. .000965, -o .000905, -o. .000518, -o, ,001590. 0. .000080, -o. .003981, 0. .000160, -0. .000939. -0. .0016 1 1 , -0 .000905. -0. .000518, -0, .001590. 0. .000080, -0. .003981, 0 .000160, -0 .000939. -0 .001142, -o .000905. -o. .000518, -o .003139. 0. .000080, -0 .000256, 0. ,973202, 1, .003030, 0. .991697, 1 .258210, 1, .036000. 0 .990570, 1, .044820, 1 .274900, 0. 000161, 0. 000492. 0. 000254. -0. 000282, 0. 000783, 0. 000492, 0. 000458, -o. 000467, O. 000161. -o. 000444, 0. 000007, -o. 000282, -0. .000715. -o. 000444/ 1 . 675580, 1 . 447370, 0. 395600. 0. 301781. -0. .890373, 0. 050854, -0. .980107, 0. .301781, 0. ,521024, • 1 . ,017510, 1 . 508640, 0. 301781, -0. ,890373. 0. ,050854, -0. .980107, 0. ,301781, 0. .521024, 1. .280360, 1. .508640. 0. ,274859, -0. .312463, 0. .050854. -0 . 162158, 0. , 274859/ 0. .882285, 0. ,911030, 0 .916258, 0. ,951416, 0. ,964179, 0. ,916894, 1. .048720, 0. ,951416, o. .941583, 0. ,853919. 0 .840049, 0. ,951416, 0. .964179, 0. 916894, 1. .048720, 0. ,951416, 0. .941583, 0. ,838295, 0. .840049, 0, .935927, 0, .909975, 0. ,916894, 0 .952760, o .935927/ -0. .001335, -0. 001311, -0 .001495, -0. .002899. -0 .002565. -0, .002170, -0 .004548, -0, .002899, -0 .001381, -o. .0O0850, -0. .000744, -0, .002899, -0. .002565, -o. .002170. -0 .004548, -0 .002899, -0 .001381. -0 .000820, -0 .000744, -o .001720, -0 .001763, -0 .002170, -0 .002111. -o .001720/ 1 . 180280. 0. .899432, 0 .927138, 0 .814998, 1 . 267300, 1 .186810, 1 . 174460, 0 .814998, 58 639 C 640 4 1 . .069940, 641 5 0 .888013. 642 6 1 . .036000. 643 7 1 . .044820. 644 C 645 8 1 , ,069940. 646 9 0. .806778. 647 1 1 . ,036000. 648 2 1 . ,044820. 649 C 650 DATA C2/ 0. .641689, 651 1 0, .467846, 652 2 o. .653812, 653 3 0. , 37131 1 , 654 C 655 4 0. .670164, 656 5 0. .311883, 657 6 0. .653812, 658 7 0. .371311. 659 C 660 8 0, .670164, 661 9 0, .169605, 662 1 0. .653812. 663 2 0. .371311, 664 C 665 C 666 DATA PP/ 0. 25, 667 1 0. 25, 668 2 0. 25, 669 3 0. . 25, 670 C 671 4 0. 25, 672 5 0. 25. 673 6 0. 25, 674 7 0. .25, 675 C 676 8 0. 25, 677 9 0. 25, 678 1 0. 25. 679 2 0. 25, 680 c 68 1 DATA AI1/ -0. .364587. 682 1 AI2/ 0. ,901858, 683 2 AI3/ -0. .000500, 684 c 685 3 BI 1/ 0. ,338355, 686 4 BI2/ 0. 912187. 687 5 BI3/ -0. 002251. 688 c 689 6 CI1/ 0. 949986, 690 7 CI2/ 0. 561960. 69 1 c 692 8 PPI/ 0. 25, 693 c 694 c C FIND THE PROPER COEF 695 c 696 IS = I0 1 . .065620, 1 . ,140220, 0. 916023, 1 , ,258210, 0, .915427, O. .814998, 0. .990570, 1 . ,267300, 1 . 186810, 1 . ,274900, 1 . ,174460, 0. 814998, 1 . .065620, 1 . ,140220, 0. 880668, 1 . ,258210, 0. ,915427, 0. 859318, 0. .938843, 1 . ,234620, 1 . 186810, 1 . ,257200, 1 . ,238800, 0. .859318/ 0. .403223, 0. ,762552, 0. .436096, 0, .924172, 0, .510404, 0. .409437, 0. ,664786, 0. ,760254, 0. 781076, 0. .755804, 0. ,850223, 0. .409437, 0. ,581790, 0, ,677481 , 0. .306823, 0. .924172, 0. .427401, 0. ,409437, o. .664786, 0. .760254, 0. .781076, 0. ,755804, 0. ,850223, 0. .409437, 0. .581790, 0. .677481 , 0. .241800, 0. .924172, 0. .427401, 0. .534771, 0. ,320945, 0. .748465, 0. .781076, 0. ,741735, 0. ,916040, 0. .534771/ 0. ,25, 0. .20, 0. 20, 0. • 25, 0. 20, 0. 20, 0. 25, 0. 25, 0. 25, 0. 20, 0 20, 0. 20. 0. .25, 0, .20, 0. 25, 0. 25, 0. .20, 0. .20, 0. 25. 0. 25, 0. 25, 0. , 20. 0. .20, 0. 20, 0. . 25, 0, . 20. 0. .25, 0. 25, 0. .20, 0. 20, 0. 25, 0. 25, 0. ,25, 0. . 25, 0. . 20, 0. . 20/ 0. .055910, -o. .421692, -0. ,442280/, 0. .937633. 0 .946279, 0, .969271/, 0. .000412, 0 .000188, 0. .000053/, 1. ,851750. 1. ,026030. -0 .248163/, 0. ,810763, 0. ,902576, 0. .920830/, -0. .002617, -0. .001138. -0. ,002530/, 1. .136790, 1. .186970. 0. .960912/, 0. ,709180, 0. . 762531 , 0. ,426897/, 0. 25, 0. .20, 0. 25/ CINTSS 59 697 IF (I0.GT.10) IS=I0-1 698 IF (10.EQ.10) IS=16 699 ZZZ=PP(IS.IZ) 700 AA1=A1(IS.IZ) 701 AA2=A2(IS,IZ) 702 AA3=A3(IS,IZ) 703 BB1=B1(IS,IZ) 704 BB2=B2(IS,IZ) 705 BB3=B3(IS,IZ) 706 CC1=C1(IS,IZ) 707 CC2 = C2(IS, IZ) 708 IF (IZ.GT.1) GO TO 10 709 IF (IM.GT.1) GO TO 10 710 IF (IS.GT.5.OR.IS.EQ.4) GO TO 10 711 IIS=IS 712 IF (IIS.EQ.5) IIS=4 713 AA1=AI1(IIS) 714 AA2=AI2(IIS) 715 AA3=AI3(IIS) 716 BB1=BI1(IIS) 717 BB2=BI2(IIS) 718 BB3=BI3(IIS) 719 CC1=CI1(IIS) 720 CC2=CI2(IIS) 721 ZZZ=PPI(IIS) 722 10 DIB=AA1+AA2*DBH+AA3*DBH*DBH 723 IF (DIB.GT.DBH) DIB=DBH-0.10 724 IF (HT.LT.7.50) HT=7.5 725 HP=HT 726 DIN=BB1+BB2*DIB+BB3*DIB*DIB 727 PER=1.0-ZZZ 728 IF (DIB.LT.DIN) DIN=DIB-0.25 729 C 730 C C DIB=DIAMETER INSIDE BARK AT BH, DIN=DIAMETE INSIDE BARK AT 731 C INFLECTIO 732 C C POINT 733 C 734 CC8=CC2 735 CC2=EXP(CC2) 736 C 737 C C CALCULATE THE COEFFICIENT FOR THE BUTT EQUATION CC3 AND CC4 738 C 739 A=DIB/DIN 740 C=(-1.0/ZZZ) 741 B=((1.0-(HT-1.3)/HT)/ZZZ) 742 RK=(CC1-CC8)/PER 743 CC3=0.1 744 NN=0 745 20 E=B**CC3 746 F=RK-(1.0-A)*C*CC3/(1.0-E) 747 NN=NN+1 748 IF (ABS(F).LT.0.01) GO TO 30 749 IF (NN.GT.6) GO TO 30 750 DF=(A-1.0)*C*((1.0-E)+CC3*E*ALOG(B))/(1.0-E)**2 751 CC3=CC3-F/DF 752 GO TO 20 753 30 IF (CC3.GT.0.9) CC3=0.9 754 IF (CC3.LT.0.1) CC3=0.1 60 755 DUM=((1.0-(HT-1.3)/HT)/ZZZ)**CC3 756 CC4=(DIB/DIN-0UM)/(1.0-DUM) 757 C 758 C C CALCULATE DBT 759 C 760 DO 40 1=1,40 761 TDL(I)=0. 762 VLOG(I)=0. 763 40 CONTINUE 764 NL=0 765 HLL=0. 766 TV0L=O. 767 GR0S=O. 768 DBT=0. 769 BAR=0. 770 SHA=1.O-SH/HT 771 HHA=SHA/PER 772 IF (SHA.GT.PER) GO TO 50 773 DBT=DIN*(HHA**CC1*CC2**(1.O-HHA)) 774 GO TO 60 775 50 DBT=DIN*(CC4-(CC4-1.0)*((1.O-SHA)/ZZZ)**CC3) 776 60 IF (DBT.LT.TD) RETURN 777 BAR=DBH-DIB 778 C 779 C C CALCULAT MERCH HEIGHT HME 780 C 781 70 A=1.O/HT/PER 782 IF (DIN.LE.TD) GO TO 100 783 HME=3.5 784 NN=0 785 80 B=HME/HT/PER 786 C=CC2**(1.0-B) 787 F=TD-(B**CC1*C)*DIN 788 NN=NN+1 789 IF (ABS(F).LE.O.01) GO TO 90 .790 IF (NN.GT.6) GO TO 90 791 DF = DIN*(A**CC1*CC1*HME**(CC1- 1 .0)*C + B**CC1*C*CC8*(-A)) 792 HME=HME+F/DF 793 IF (HME.LT.0) HME =0. 1 794 GO TO 80 795 90 CONTINUE 796 GO TO 1 10 797 100 HME=HT*(1.0-ZZZ*((TD/DIN-CC4)/(1.0-CC4))**(1.0/CC3)) 798 1 10 HME = HT-HME 799 C 800 C C CALCULATE NO. OF LOGS NL AND CLEAR STORAGE FOR TDL AND VLOG 801 C 802 NL=(HME-SH)/G0L+1 .0 803 C 804 C C CALCULATE HLL 805 C 806 HLL=HME-(NL-1.0)*GOL-SH 807 RH=ZZZ*HT 808 C 809 C C CALCULATE VOLUMES. TVOL, GROS AND VLOG, AND TOP DIAMETERS TDL 810 C 81 1 STMV=VLM(0.0,SH,DBT,DBT) 812 T0PV=VLM(HME,HT,TD,O.O) 61 813 D1=DBT 814 X1=SH 815 X2=SH 816 DO 130 I=1,NL 817 IF (I.GT.1) D1=TDL(I-1) 818 X2=X2+G0L 819 IF (I.EQ.NL) X2=HME 820 SHA=1.0-X2/HT 821 HHA=SHA/PER 822 IF (SHA.GT.PER) GO TO 120 823 TDL(I)=DIN*(HHA**CC1*CC2**(1.O-HHA)) 824 GO TO 13 825 120 TDL(I)=DIN*(CC4-(CC4-1.0)*((1.O-SHA)/ZZZ)**CC3) 826 13 D2=TDL(I) 827 VLOG(I)=VLM(X1,X2,D1,D2) 828 TVOL=TVOL+VLOG(I) 829 X1=X2 830 130 CONTINUE 831 GROS=TVOL+STMV+TOPV 832 TDL(NL)=TD 833 RETURN 834 END 835 FUNCTION VLM(HL,HU,D1,D2) 836 DIMENSION D(51) 837 COMMON /TAPER/ HT,DIN,P.ZZZ,C1,C2,C3.C4,RH 838 SS=0. 839 DD=D1 840 N=18 841 IND=0 842 SV=0. 843 HZ=HL 844 HY =HU 845 FF=ABS(HU-HL) 846 IF (FF.GT.5.0) N=37 847 IF (FF.LT.0.01) GO TO 60 848 D(1)=D1 849 D(N+2)=D2 850 XA=HL/HT 851 IF (XA.GT.ZZZ) GO TO 1 852 XB=HU/HT 853 IF (XB.LE.ZZZ) GO TO 31 854 XB=ZZZ 855 HY=RH 856 IND=1 857 GO TO 31 858 1 XA=(HU-HZ)/(N+1.0) 859 XB=HZ 860 DO 2 1 = 1 ,N 861 XB=XB+XA 862 SHA=1.O-XB/HT 863 BB=SHA/P 864 0(1+1)=DIN*(BB**C1*C2**(1.O-BB)) 865 2 SS = SS+D(1 + 1 )*D(1 + 1 ) 866 SS=0.00007854*XA*(SS+0.5*D(1)*D(1)+0.5*0(N+2)*D(N+2)) 867 GO TO 60 868 31 AK=0.00007854*DIN*DIN 869 AP=XA/ZZZ 870 AO=XB/ZZZ 62 871 SV=(HY-HL)*C4*C4*AK 872 BB=C3*2.+1. 873 CC=C3+1. 874 EE=(C4-1.0)*AK 875 SV=(AP**BB-A0**BB)*(C4-1.0)*EE*(P-1.)*HT/BB+SV 876 SV=(AP**CC-AQ**CC)*2.0*C4*EE*ZZZ*HT/CC+SV 877 IF (IND.LT.1) GO TO 60 878 D(1)=DIN 879 HZ=RH 880 GO TO 1 881 60 VLM=SS+SV 882 RETURN 883 END 63
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Using the jackknife technique to approximate sampling error for the cruise-based lumber recovery factor Jahraus, Karen Veronica 1987
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Title | Using the jackknife technique to approximate sampling error for the cruise-based lumber recovery factor |
Creator |
Jahraus, Karen Veronica |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | Timber cruises in the interior of British Columbia are designed to meet precision requirements for estimating total net merchantable volume. The effect of this single objective design on the precision of other cruise-based estimates is not calculated. One key secondary objective, used in the stumpage appraisal of timber in the interior of the province, is estimation of the lumber recovery factor (LRF). The importance of the LRF in determining stumpage values and the fact that its precision is not presently calculated, prompted this study. Since the LRF is a complicated statistic obtained from a complex sampling design, standard methods of variance calculation cannot be applied. Therefore, the jackknife procedure, a replication technique for approximating variance, was used to determine the sampling error for LRF. In the four cruises examined, the sampling error for LRF ranged from 1.27 fbm/m³ to 15.42 fbm/m³. The variability in the LRF was related to the number of sample trees used in its estimation. The impact of variations in the LRF on the appraised stumpage rate was influenced by the lumber selling price, the profit and risk ratio and the chip value used in the appraisal calculations. In the cruises investigated, the change in the stumpage rate per unit change in the LRF ranged between $0.17/m³ and $0.21/m³. As a result, sampling error in LRF can have a significant impact on assessed stumpage rates. Non-sampling error is also a major error source associated with LRF, but until procedural changes occur, control of sampling error is the only available means of increasing the precision of the LRF estimate. Consequently, it is recommended that the cruise design objectives be modified to include a maximum allowable level of sampling error for the LRF. |
Subject |
Forests and forestry -- Measurement -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0075406 |
URI | http://hdl.handle.net/2429/26419 |
Degree |
Master of Science - MSc |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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