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Forest sampling on two occasions with partial replacement of sample units See, Thomas Elton 1974

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FOREST SAMPLING ON TWO OCCASIONS WITH PARTIAL REPLACEMENT OF SAMPLE UNITS by THOMAS ELTON SEE B.S.F., Northern Arizona University, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Faculty of Forestry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1974 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department: o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f F o r e s t r y The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date June 26, 1974 i i ABSTRACT Forest sampling i s conducted to determine current conditions and trends of change. When current conditions are estimated, most of the commonly used sampling designs specify the s p a t i a l d i s t r i b u t i o n of sample u n i t s . When estimation of change i s desired, several schemes may be employed. Some are combined with current condition inventories; some are inde-pendent. The former are r e l a t i v e l y imprecise; the l a t t e r r e l a t i v e l y expensive. A system of temporal d i s t r i b u t i o n of sample u n i t s , sampling on successive occasions with p a r t i a l replacement of sample u n i t s , has been developed for simultaneous estimation of current conditions and trends. As the emphasis i s on time, rather than area, t h i s system operates with conventional samp-l i n g designs to increase t h e i r e f f i c i e n c y . This study investigated the theory of sampling with p a r t i a l replacement to e s t a b l i s h the v a l i d i t y of the claims of increased e f f i c i e n c y i n comparison with conventional sys-tems. Three cases are examined, through example, f o r e s t i -mation of mean volume per acre and growth i n volume per acre. Sample sizes and costs are developed for the s i t u a t i o n of simple random sampling of both f i n i t e and i n f i n i t e populations. The comparisons are favorable to the proposed system. The p o s s i b i l i t i e s of using t h i s system with two recent develop-ments i n c r u i s i n g techniques are explored. F i n a l l y , the I l l a p p l i c a b i l i t y of t h i s system to B r i t i s h Columbia forest sur veys i s examined. A case i s made for t r a n s i t i o n of the ex-i s t i n g p r o v i n c i a l system to sampling with p a r t i a l replace-ment. i v TABLE OF CONTENTS Page LIST OF TABLES v i LIST OF FIGURES v i i SELECTED SYMBOLS v i i i Chapter INTRODUCTION 1 ONE THE DEVELOPMENT OF SPR 5 TWO THE DATA BASE AND METHODS OF ANALYSIS 8 THREE THEORY Estimation of a Mean 14 Estimation of Growth 18 Optimum Sample Size 23 FOUR COMPARISONS OF SPR WITH CONVENTIONAL INVENTORY Example One 29 Example Two 35 Example Three 40 FIVE BIAS IN SPR ESTIMATES 43 CONCLUSIONS 47 FUTURE CONSIDERATIONS 51 BIBLIOGRAPHY 55 APPENDIX I COMPUTING FORMULA FOR VOLUME OF HEIGHT-MEASURED TREES 58 APPENDIX II REGRESSION COEFFICIENTS FOR VOLUME COMPUTATION FROM /(D.B.H.O.B.) 59 V APPENDIX III COMPUTED STATISTICS OF FIFTY RANDOM SPR SAMPLES . . . . . . . . . . . 61 APPENDIX IV REQUIRED SAMPLE SIZES AT VARIOUS PLOT COST RATIOS 62 v i LIST OF TABLES Table Page 1. Parameters f o r example one. . . . . . . . . . 29 2. Comparison o f c o s t s between independent and SPR i n v e n t o r i e s . . . . . . . . . . . . 32 3. Parameters and sampling e r r o r s f o r example two. . . . . . . . . . . . . . . . 35 4. Comparison of c o s t s between independent and SPR i n v e n t o r i e s f o r example two. . . . 38 5. Optimal sample s i z e s a t v a r i o u s c o s t r a t i o s . . . . . . . . . . . . . . . . 62 v i i LIST OF FIGURES F i g u r e Page 1. The o p t i m a l numbers o f sample p l o t s t o remeasure and to r e p l a c e a t v a r i o u s c o s t r a t i o s . . . . . . . . . . . . 63 v i i i SELECTED SYMBOLS MEANING the t r u e p o p u l a t i o n v a r i a n c e o f the p l o t volume a t the i n i t i a l o c c a s i o n . the t r u e p o p u l a t i o n v a r i a n c e of the p l o t volume a t a subsequent o c c a s i o n . the t r u e p o p u l a t i o n c o r r e l a t i o n c o e f -f i c i e n t between p l o t volumes observed on the same p l o t on s u c c e s s i v e o c c a s i o n s = C o v a r i a n c e {XY)/o^Oy. the t r u e p o p u l a t i o n r e g r e s s i o n c o e f -f i c i e n t of the i t h v a r i a b l e ; = C o v a r i a n c e ( X Y ) / a 2 . the t r u e average volume per a c r e a t the ith o c c a s i o n . the change i n y between o c c a s i o n s . a n o r m a l l y d i s t r i b u t e d random v a r i a b l e w i t h mean zero and v a r i a n c e a2, i . e . £ .^N(0,a2) . v a s t a t i s t i c t h a t f o l l o w s the F - d i s t r i b u -t i o n (and can be used as a t e s t o f Ho°.&. = 0 i n l i n e a r r e g r e s s i o n ) . ^ an SPR e s t i m a t e of the mean volume per a c r e a t the second o c c a s i o n . the e r r o r v a r i a n c e o f y. an e s t i m a t e o f t h e growth i n volume per a c r e between subsequent p e r i o d s . the e r r o r v a r i a n c e o f g. the number o f i n i t i a l l y e s t a b l i s h e d sample p l o t s t h a t are not remeasured on the subsequent o c c a s i o n . the number of i n i t i a l l y e s t a b l i s h e d sample p l o t s t h a t a r e remeasured on the subsequent o c c a s i o n . i x PAGE FIRST SYMBOL USED MEANING n 14 the number of new plots established at a subsequent occasion to replace those not measured. u> 25 a Lagrangian function used i n a tech-nique to determine extremum of a func-ti o n / with constraint g. X 25 a Lagrangian m u l t i p l i e r ; a parameter (to be determined) which m u l t i p l i e s the constraint g. 3/ 25 the f i r s t p a r t i a l d e r i v a t i v e of the function / with respect to x{_; a l l other v a r i a b l e s remain constant. K 25 the s p e c i f i e d sampling e r r o r 2 of mean volume per acre. K 28 the s p e c i f i e d sampling e r r o r 2 of growth between periods. ^ 28 greater than or equal to. <_ 28 les s than or equal to. E( ) 15 the expected value. E 15 the summation of. X ACKNOWLEDGEMENT The acknowledgement of a l l those persons who have c o n t r i b u t e d i n some manner to the background f o r t h i s t h e s i s would be an i m p o s s i b l e task. But I wish to g i v e s p e c i a l thanks to c e r t a i n persons f o r t h e i r s p e c i a l h e l p and i n f l u -ence. F i r s t , I wish to thank those who reviewed t h i s t h e s i s f o r t h e i r comments and s u g g e s t i o n s . They are my committee members, Drs. A. Kozak, D.D. Munro, and J.H.G. Smith, and Dr. J u l i e n Demaerschalk o f the F a c u l t y of F o r e s t r y a t U.B.C. I e s p e c i a l l y wish to thank Drs. Kozak and Munro f o r t h e i r p e r s o n a l support d u r i n g my year of study a t U.B.C. I must take t h i s o p p o r t u n i t y to thank my f e l l o w graduate st u d e n t s and the s t a f f of the F a c u l t y of F o r e s t r y , e s p e c i a l l y Ms.'s Kerr and H e j j a s , f o r t h e i r acceptance of and a s s i s t a n c e to me. Without data, I c o u l d n o t proceed. F r a n c i s R. Herman o f the F o r e s t r y S c i e n c e s L a b o r a t o r y , U.S. F o r e s t Ser-v i c e , was k i n d enough to p r o v i d e t h i s n e c e s s a r y i n g r e d i e n t . But without f i n a n c i a l support, a l l these o t h e r i n g r e d i e n t s might have gone f o r naught. I wish t o thank the N a t i o n a l Research C o u n c i l of Canada f o r support i n d e f r a y i n g the c o s t s of computing s e r v i c e s and p a r t i a l p e r s o n a l support. The F a c u l t y o f F o r e s t r y was k i n d enough to p r o v i d e f u r t h e r support xi f o r those same needs. I wish to e s p e c i a l l y thank M a c M i l l a n B l o e d e l L i m i t e d f o r t h e i r F e l l o w s h i p i n F o r e s t Mensuration which p r o v i d e d the bul k o f my p e r s o n a l support and was i n s t r u -mental i n the p u r s u i t of my s t u d i e s a t U.B.C. F i n a l l y , I wish to thank Ms. Jean W i l l i a m s o n f o r the e x c e l l e n t t y p i n g of t h i s t h e s i s . INTRODUCTION In past decades, the forest industry was not con-cerned with determining growth and y i e l d of f o r e s t resources with much precision.^" In those times, there were vast acre-ages of old-growth timber s t i l l a v ailable for harvest. The l o c a t i o n of the timber was well known, i t was "out there". But now, much of t h i s type of timber resource has been cut over and some of the timberland converted to other uses. In t h i s same span of time, new uses f o r wood and wood f i b e r have been developed which have created new forest product markets. These and other factors have increased the pressures on a diminishing production base. Organizations must now prac-t i c e more intensive management and base t h e i r harvesting de-c i s i o n s on more precise information at reduced expenditure. The questions of where the trees are, how much of what kinds of products can be r e a l i z e d , and e s p e c i a l l y how much growth Decision-making requires a c c u r a t e information, e.g. i n ~ formation which i s representative of the true population mean y . However, due to some techniques employed, biases may be present i n our observations which are unavoidable or unknown. Therefore our information w i l l i n f a c t be p r e c i s e rather than accurate, i . e . successive measurements w i l l vary about a mean m, which d i f f e r s from y by the amount of bias present. Re-finement of our techniques may reduce or even eliminate these biases, but t h i s i s d i f f i c u l t to gauge. The technique under study i s t h e o r e t i c a l l y unbiased, therefore, we could use the term accuracy. However, the term p r e c i s i o n w i l l be the normal usage to avoid the complication of explaining at each step that bias from a n c i l l a r y considerations w i l l cause the e s t i -mates to be precise rather than accurate. For further d i s ~ cussion of t h i s t o p i c , the reader i s referred to Cochran and Cox (1957) or Cochran (1963). 2 i s occurring are v i t a l to management decisions. Davis ( 1 9 6 6 ) stated that . . . "accurate determination of growth becomes necessary, because c r u c i a l decisions rest d i r e c t l y on i t " . The same information requirements e x i s t for answering the other questions. The s o l u t i o n to the paradox of greater pre-c i s i o n at reduced expenditure l i e s i n use of good forest sur-vey^ techniques. The sol u t i o n of the mechanics of f o r e s t survey has been the aim of many in v e s t i g a t o r s . Many and varied schemes e x i s t , i n f a c t , they seem as numerous as the problems they attempt to solve. To an extent, t h i s i s c o r r e c t , because each inventory problem i s unique, but c e r t a i n methodology may be applied to many s i t u a t i o n s . One general s i t u a t i o n i s the estimation of the current volume of an area. One might use systematic sampling, s t r a t i f i e d random sampling, or pos-s i b l y multi-stage c l u s t e r sampling. In most cases, the sample has been a one-time event with l i t t l e or no consideration given to future surveys. This approach s i m p l i f i e d analyses, since independent samples give good information with few com-plex s t a t i s t i c a l manipulations needed. Growth estimation presented other problems. Inde-pendent samples on two occasions give larger variance of the estimate than i f a l l o r i g i n a l plots were remeasured [by a factor of 1 / ( 1 - P 2 ) ] . But permanent plo t s are expensive to The terms f o r e s t i n v e n t o r y and f o r e s t s u r v e y w i l l be used interchangeably and w i l l have the d e f i n i t i o n as given to forest survey by the Society of American Foresters ( 1 9 5 8 ) . 3 e s t a b l i s h and r e l o c a t e . F u r t h e r , u s i n g permanent p l o t s f o r c u r r e n t volume e s t i m a t i o n has the drawback of not y i e l d i n g a d d i t i o n a l i n f o r m a t i o n about the s p e c i e s composition and t r e e s i z e d i s t r i b u t i o n i n a stand. D i f f e r e n t o r g a n i z a t i o n s s o l v e d these c o n f l i c t s i n s e v e r a l ways. Some accepted the a d d i t i o n a l problems and expense o f two c o m p l e t e l y d i f f e r e n t i n v e n t o r y systems and used independent surveys f o r c u r r e n t volume e s t i -mation on each o c c a s i o n and a permanent p l o t system f o r growth e s t i m a t i o n . Others r e s o r t e d t o i n c r e a s i n g the amount of i n -f o r m a t i o n gathered d u r i n g the independent surveys, and ob-t a i n e d increment c o r e s or o t h e r such e s t i m a t o r s of growth and a p p l i e d the r e s u l t s t o the stand a t l a r g e . A l l such methods i n c r e a s e the c o s t s i n v o l v e d i n i n v e n t o r y . In r e c e n t y e a r s , a system has been developed t h a t p u r p o r t e d l y can g i v e p r e c i s e answers to growth and y i e l d q u e s t i o n s a t a c o n s i d e r a b l e c o s t s a v i n g . T h i s scheme, termed sampling w i t h p a r t i a l replacement of sampling u n i t s on s u c c e s -s i v e o c c a s i o n s (SPR) i n v o l v e s the use of both remeasured p e r -manent p l o t s and temporary independent p l o t s . SPR i s not a sampling scheme t o ' d i s t r i b u t e sample p l o t s s p a t i a l l y , r a t h e r , i t i s concerned w i t h d i s t r i b u t i o n over time. For e s t i m a t i n g c u r r e n t volume on second and s u c c e s s i v e o c c a s i o n s , remeasure-ment of some of the i n i t i a l p l o t s , w i t h the replacement o f o t h e r s , w i l l g i v e i n c r e a s e d p r e c i s i o n over a s e r i e s o f i n d e -pendent surveys. The remeasured p l o t s a r e used t o e s t a b l i s h a l i n k i n g r e l a t i o n s h i p between p e r i o d s . T h i s r e l a t i o n i s 4 applied to the replaced plots and an estimate based on a l l plots measured on a l l occasions i s formed. The replacement plots can be back-linked again through the remeasured p l o t s . Thus, a new estimate of volume can be formed f o r any previous occasion, again based on a l l measured p l o t s . Since informa-t i o n gained by d i r e c t measurement i s considered more r e l i a b l e than linked information, the i n d i v i d u a l estimates are weighted by t h e i r r e l a t i v e importance. Several estimators of growth can be formed from the estimates of volume on successive occasions. An e f f i c i e n t estimator i s formed from the estimate of current volume and the revised estimate on the previous occasion. Through use of the expressions for the error v a r i -ances of the current volume and growth and a l i n e a r cost func-t i o n for t o t a l expenditure, the optimum numbers of pl o t s to remeasure and to replace can be determined. Two d i f f e r e n t situations can be handled. I f a s p e c i f i e d amount of money i s a v a i l a b l e , the optimum numbers for minimum sampling errors i n both growth and volume determination can be found. Con-versely, optimum numbers can be found to minimize costs at s p e c i f i e d sampling e r r o r s . This feature allows f l e x i b i l i t y i n designing surveys f o r changing s i t u a t i o n s and management goals. CHAPTER ONE THE DEVELOPMENT OF SPR SPR i s not new to inventory s i t u a t i o n s . In a g r i -culture, Jessen (1942) was the f i r s t i n v e s t i g a t o r to recog-nize i t s p o t e n t i a l . Many authorities i n the f i e l d of samp-l i n g (Patterson 1950, Cochran 1963, Hansen et a l . 1953, Kish 1965) have described the theory and use of SPR i n survey sampling. However, various r e s t r i c t i o n s and s p e c i a l cases were presented i n these des c r i p t i o n s , and no general s i t u a -t i o n or f o r e s t r y applications were discussed. Bickford (1956, 1959) f i r s t proposed using SPR i n extensive f o r e s t inventor-i e s , and l a t e r reported on the r e s u l t s obtained i n the north-eastern United States. The f i r s t presentation of a general u n i f i e d theory of SPR came from separate independent inves-tigations by two researchers, K. Ware and T. Cunia. Ware (1960) wrote h i s doctoral d i s s e r t a t i o n on the general theory, while Cunia's r e s u l t s came from his work as Research Biome-t r i c i a n f or Canadian International Paper Corporation. In 1962, these two authors collaborated on a j o i n t p u b l i c a t i o n on the theory of SPR and optimal replacement p o l i c y f o r the general case of simple random sampling on two occasions (Ware and Cunia 1962). Cunia has since continued his research i n t h i s f i e l d . He has extended SPR theory from the simple l i n e a r to the multiple l i n e a r regression case (Cunia 1965) . Mul t i p l e 6 regression SPR theory has many important ra m i f i c a t i o n s . For example, using multiple regression with dummy variables (Dra-per and Smith 1966, Cunia 1973), the separate s t r a t a from a s t r a t i f i e d random sample could be simultaneously analyzed. With simple l i n e a r regression SPR theory, each stratum had to be considered separately. More importantly, multiple regression SPR theory has allowed the use of many stand v a r i -ables to improve the c o r r e l a t i o n between periods i n the l i n k -ing expressions. Another important r e s u l t was the a p p l i c a -b i l i t y of multiple regression to multiple occasion SPR. Through the multiple regression and matrix manipulation, l i n k -ages between plots measured on various occasions can be formed (Cunia and Chevrou 1969). In t h i s s i t u a t i o n , say a t h i r d measurement period, plots measured on the f i r s t only, the f i r s t and t h i r d , second and t h i r d , etc. occasions can be combined to form an estimate of volume at any of the three occasions. As explained e a r l i e r , t h i s allows the estimate to be based on many more plots than were measured on a s p e c i f i c occasion. Newton (1971) examined the multiple regression case and hypothesized that m u l t i v a r i a t e sampling could be more e f f i c i e n t through use of SPR. He also proposed the exten-sion of SPR techniques to the f a m i l i a r case of double samp-l i n g using a e r i a l photographs for large-scale f o r e s t surveys. He f e l t that the addition of singly-measured ground p l o t s could increase the p r e c i s i o n of the airphoto p l o t estimates. Reports of t r i a l s of SPR i n simulated (Nyyssonen 1967) and 7 actual (Frayer et at. 1971) inventory s i t u a t i o n s have shown that SPR i s an e f f e c t i v e system i n saving both time and ex-pense. Recent l i t e r a t u r e regarding the theory and use of SPR has stated that considerable savings i n time and ex-pense with a high degree of p r e c i s i o n i n estimation can be r e a l i z e d . Since most budget analysts i n an organization's f i n a n c i a l control department look on inventory costs as an expense item with no v i s i b l e return, the v a l i d i t y of claims of reduced expenditure should be investigated. The objective of t h i s thesis i s to investigate SPR and attempt to e s t a b l i s h the v a l i d i t y i n c e r t a i n cases. Several such cases w i l l be looked at through example, with costs of SPR and equivalent normal inventory systems compared. Further, c e r t a i n c r u i s -ing techniques, as they might be employed within an SPR frame-work, w i l l be discussed. F i n a l l y , the a p p l i c a b i l i t y of SPR to f o r e s t r y practices i n B r i t i s h Columbia w i l l be given a cursory look. 1 CHAPTER TWO THE DATA BASE AND METHODS OF ANALYSIS The data for these analyses were furnished by the Forestry Sciences Laboratory, U.S. Forest Service, Olympia, Washington, U.S.A. These data consist of recorded measure-ments from permanent sample plots established i n the Cascade Head Experimental Forest near Otis, Oregon (Madison 1957). The species found i n t h i s stand include S i t k a spruce { P i c e a s i t c h e n s i s (Bong.) Carr.) and western hemlock (Tsuga h e t e r o -p h y l l a (Raf.) Sarg.) with a few scattered Douglas-fir ( P s e u d o -t s u g a m e n z i e s i i (Mirb.) Franco var. m e n z i e s i i ) . The f o r e s t of t h i s l o c a l e regenerated n a t u r a l l y following a large f i r e thought to have occurred i n the 1850's. In 1935, the U.S. Forest Service established eleven one-acre permanent sample plots i n the Experimental Forest. Complete enumeration of the trees growing on these plots was made by Forest Service personnel and the following measurements taken: d.b.h.o.b., the diameter outside bark at 4.5 feet above germination point; crown c l a s s , following the four class system of dominant, codominant, intermediate, and overtopped (Smith 1962); species of each tree; and t o t a l height, which was measured f o r 15-20 percent of the trees across the diameter range of the stand. The point of measure-ment of d.b.h.o.b. was marked with a metal tag which c a r r i e d a number fo r future i d e n t i f i c a t i o n . Remeasurement occurred 9 every f i v e years u n t i l 1950. A f i n a l remeasurement occurred i n 1968. These measurements, were coded onto e l e c t r o n i c data processing cards and v e r i f i e d . This then was the data base for t h i s analysis. The general procedure for the analysis involved f i r s t the computation of volume f o r each tree i n cubic feet for several measurement periods. The eleven plots were then divided i n t o sub-plots to create a population for the samp-l i n g experiments. Summary volumes for each p l o t were c a l c u -l a t e d which served as the data f o r computation of the popula-t i o n parameters o^2, o^z, p, p l f and u 2 . Given these para-meters, the various examples could be formulated and analyzed. As a l l trees were not measured for height, a method was required to r e l a t e volume to diameter. In previous For-est Service analyses, height-diameter curves were developed and heights were read from these curves for each tree. Vol-umes were then read from alinement charts. In t h i s analysis, a d i f f e r e n t and hopefully sounder method was used, the de-velopment of f (volume) - / (d.b.h.o.b.) equations. To de-velop these equations, f i r s t volumes had to be calculated f o r the trees for which heights had been measured. The For-est Survey group of the U.S.F.S. has extensively inventoried the same type of stands i n the same area. From these sur-veys, they have developed empirical equations for the c a l c u -l a t i o n of cubic volume from d.b.h.o.b., height, and Girard form c l a s s . But no form class measurements existed for t h i s 10 stand. Again the F o r e s t Survey p r o v i d e d the needed i n f o r m a -t i o n from t h e i r i n v e n t o r y data. These d a t a p r o v i d e d a v e r -age form c l a s s by v a r i a b l e diameter c l a s s e s f o r each of the s p e c i e s , again from l o c a l s urveys. A F o r e s t Survey volume program was then used to c a l c u l a t e the t r e e volumes. V a r i o u s l i n e a r and n o n - l i n e a r models were hypothe-s i z e d f o r the f (volume) - f (d.b.h.o.b.) r e l a t i o n s h i p . The r e g r e s s i o n a n a l y s e s of t h e s e models were c a r r i e d o u t on the U n i v e r s i t y of B r i t i s h Columbia IBM 370/168 computer. F o r the l i n e a r models, the a n a l y s e s u t i l i z e d a backward e l i m i n a -t i o n procedure developed by Dr. A. Kozak o f the U.B.C. F a c u l t y of F o r e s t r y . N o n - l i n e a r parameter e s t i m a t i o n came from use o f the U.B.C. Computing Centre's BMD:X85 N o n - l i n e a r L e a s t Squares programme. A n a l y s i s of the equations generated i n -v o l v e d comparisons o f standard e r r o r s of e s t i m a t e s (SE„), • hi m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s , p a r t i a l F and o v e r a l l F v a l u e s , and r e s i d u a l p l o t s t o determine the "best" u n b i a s e d r e l a t i o n s h i p s . The equations adopted were o f the form L o g 1 0 (volume) = b 0 + b j • d.b.h.o.b. + b 2 * (d.b.h.o.b.) 2* b 3 ' (d.b.h.o.b.) 3. For a s p e c i f i c measurement p e r i o d , two equations were used, one f o r western hemlock and one f o r S i t k a spruce and Douglas-fir."*" T h i s model n e a r l y always gave the lowest SE_ and the r e s i d u a l s were unbiased and showed homogeneity of v a r i a n c e . An a n a l y s i s of c o v a r i a n c e i n d i c a t e d There were on l y 12 D o u g l a s - f i r t r e e s on the e n t i r e 11 a c r e s sampled. Of these 12, o n l y a few were measured f o r h e i g h t on any o c c a s i o n . As t h i s d i d not g i v e a s u i t a b l e data base f o r the r e g r e s s i o n a n a l y s i s , i n t h i s r e s p e c t , the Douglas-f i r were combined with the S i t k a spruce. 11 the equations were s i g n i f i c a n t l y d i f f e r e n t between s p e c i e s . W i t h i n s p e c i e s and between time p e r i o d s , s l o p e s or i n t e r c e p t s were i n some cases n o n - s i g n i f i c a n t l y d i f f e r e n t , but these c o n d i t i o n s never e x i s t e d s i m u l t a n e o u s l y . T e s t s f o r goodness of f i t o f l o g t o the normal d i s t r i b u t i o n i n d i c a t e d no s i g -n i f i c a n t d i f f e r e n c e which s a t i s f i e d the requirement t h a t the e^'s s h o u l d f o l l o w the lognormal d i s t r i b u t i o n . For the SPR examples, s u b - p l o t s were needed f o r the sampling p o p u l a t i o n to have a u s e a b l e s i z e . A minor problem was encountered s i n c e no stem maps e x i s t e d t o a i d i n t h i s p l o t d i v i s i o n . I t was h y p o t h e s i z e d t h a t the s t a n d a r d method of permanent p l o t enumeration was f o l l o w e d so t h a t twenty-l i n k s t r i p s were c r u i s e d . As the t r e e data were s e q u e n t i a l l y numbered, s u c c e s s i v e l y numbered t r e e s s h o u l d be a d j a c e n t a l o n g these s t r i p s . A review of p e r t i n e n t l i t e r a t u r e gave c o n f l i c t -i n g views on the e f f e c t s of d i f f e r e n t - s h a p e d p l o t s . Some sources r e p o r t e d e f f e c t s , some d i d not. In any event, s i n c e o t h e r methods seemed unduly c o m p l i c a t e d , one-twentieth a c r e r e c t a n g u l a r p l o t s were h y p o t h e s i z e d w i t h a t w e n t y - l i n k s i d e . To determine a count of t r e e s t o be p l a c e d i n t o a p l o t , the t o t a l number of t r e e s on the a c r e was d i v i d e d by twenty. T h i s c o n s t i t u t e d the minimum number. I f the t o t a l number of t r e e s was n o t evenly d i v i s i b l e by twenty, the remainder was d i s t r i -buted among the twenty p l o t s . T h i s was done randomly and w i t h replacement of a s s i g n e d p l o t s . Once the number of t r e e s a l l o t t e d t o each p l o t was determined, the assignment of t r e e s 12 was made sequentially. I t i s hoped that t h i s method gave un-^ biased r e s u l t s . One major objection to t h i s method would involve the clumping of trees. Personal v i s i t s to these plots showed that the trees were reasonably evenly d i s t r i b u t e d , so only minor deviations from the s p e c i f i e d assignments might occur. f The actual subdivision and tree assignment was car-r i e d out by a computer programme which c o i n c i d e n t a l l y computed the parameters, mean volume per acre, variance within p l o t s , mean volume per tree, variance within trees, and correspond-ing c o e f f i c i e n t s of v a r i a t i o n . The U.B.C. Computing Centre's *SIMCORT programme was used to compute c o r r e l a t i o n c o e f f i c i e n t s between periods. The example analyses compare sample s i z e require-ments for the c l a s s i c a l and the corresponding SPR inventories. Sample s i z e determination f o r the c l a s s i c a l cases involved the population variances and s p e c i f i e d sampling errors. For the SPR cases, a l i n e a r cost function was optimized with samp-l i n g errors as constraints. The sampling errors were s p e c i -f i e d as percentage pr e c i s i o n requirements and hopefully r e -f l e c t management objective requirements. Estimates of p l o t survey costs were also required and f o r the d e t a i l e d analyses, $50.00 for remeasured plots and $25.00 for temporary plots were considered reasonable values. In the optimization rou-tines, p l o t costs are used as r a t i o s , i.e. 2:1. In the cost comparisons between the two types of inventory, these costs 13 were used d i r e c t l y . The costs given i n the analyses should therefore be viewed as order of magnitude only, as p l o t costs do vary. Because of t h i s v a r i a b i l i t y , sample sizes for cost r a t i o s of 1:1, 2:1, 3:1, 4:1, and 8:1 are tabulated (Appen-dix IV). The reader may thus extrapolate costs as may f i t his own needs. For current volume only, SPR sample sizes were computed on a desk c a l c u l a t o r . When simultaneous estimation of current volume and growth was required, the U.B.C. Comput-ing Centre programme COMPLX i n the public f i l e *NUMLIB was used. This programme i s a t y p i c a l non-linear optimization routine with provisions for constraining equations. Similar programmes should be a v a i l a b l e through most computer f a c i l i -t i e s . (•i CHAPTER THREE SAMPLING WITH PARITIAL REPLACEMENT OF SAMPLE UNITS THEORY The derivations presented herein are for the case of simple random sampling from a homogeneous population and for two successive occasions and closely follow those given by Ware and Cunia (1962). The inclusion of this section should be benef ic ia l to those readers who do not have a working know-ledge of this subject. In areas where the algebraic manipula-tions involved would become tedious and repet i t ious, these steps have been eliminated to improve the readabi l i ty . For the exact procedures involved, the interested reader is re -ferred to K. Ware's doctoral dissertat ion (Ware 1960). We wish to estimate for a population, e.g. a for -est stand, these parameters: y l f the true average volume per sampling unit at the f i r s t occasion; p 2 / the true average v o l -ume per sampling unit at the second occasion; A = p 2 ~ V l ' the true average growth in volume per sampling unit between these two occasions. For this estimation, we choose a simple random sample of units (plots) from the total population. We establish uX . and mX . plots on the f i r s t occasion. On ui mj the second occasion we remeasure ml . plots and establ ish nYn^ new plots which replace those not remeasured. From these observations, we obtain simple averages of the plot volumes 15 u m m n X = (ZX .)/u; X = (IX .)/m; Y = (ZY ,)/m; Y = (IY , ) / n . u u i ' m mo m mo n nh The X . and the X . have a t r u e p o p u l a t i o n v a r i a n c e of a 2 ui mo X and the Y . and the Y , have a t r u e p o p u l a t i o n v a r i a n c e of mo nn a . To e s t i m a t e v>2r we s h o u l d make use o f a l l the a v a i l a b l e i n f o r m a t i o n from both o c c a s i o n s , t h e r e f o r e we form the gen-e r a l e x p r e s s i o n y = a- (X ) + b- (Z ) + c- (Y ) + d' (Y ) (1) a u m m n which s h o u l d be unbiased, t h e r e f o r e E(y) = \*2- From simple random sampling E(X ) = E(X ) = M I and E(Y ) = E(Y ) = y 2 , u m 1 m n so to be unbiased, a + b = 0 and c + d = 1. E q u a t i o n (1) may be r e w r i t t e n as y = a- (X ) - a- (X ) + c (Y ) + (1 - c) • (Y ) (2) a u> y m m' v n where i . a and c are c o n s t a n t s to be determined; i i . X , X , I , and Y are sample e s t i m a t e s s u b j e c t u m m n c to sampling e r r o r ; i i i . due t o remeasurement and the independent random sampling used, Ym i s c o r r e l a t e d w i t h Xm, but s t a t i s t i c a l l y independent of X and Y , and Z i s s t a t i s t i c a l l y u n m •* independent of X and Y . c u n The v a r i a n c e of y can be found from expected v a l u e s as a-2 = a 2 a y 2 ( - + - ) + c 2 a v 2 ( - ) + ( l - c ) 2 a 2 ( - ) - 2 a c p a v a v ( - ) (3) y X u m Y m Y n X Y m 16 To obtain the most e f f ic ient estimate of y, we should determine a and c to minimize ° y 2 ' W e therefore use the technique of minimization of the par t ia l derivatives of a _ 2 with respect to a and c. The solutions for a and c from y this technique are m mu/[m+u) c = ; a = • p—. N2 - (unp2)/(m+u) N2 - (unp2)/(m+u) av where N2 i s the total sample size of the second occasion (m+n) and Covariance (XY) p = . oxoY where Covariance (XY) i s the true population covariance be-tween plot volumes observed on the same plot on successive periods. From linear regression theory, we know that the least squares estimate of the regression coeff ic ient of Y on X can be expressed as 3y^ = Covariance (XY)/a^2. From the expression for p , we see that °XaY °Y YX ~ P = P — ' ° x 2 ax From this we obtain a = c 3 ^u/ (m+u ) o We now define the quan-t i t i e s = u/ (m+u), the proportion of plots unmatched at Occasion 2; Pm = m/(m+u), the proportion of plots matched at occasion 2. 17 Substituting into equation (2), we have for the mean volume at occasion 2 y = cY + P B (X -X ) + (1-c) ( ? ) (4) m u IX u r n n We now form the o v e r a l l mean at the f i r s t occasion u m X = (IX . + EZ .)/(m+u) = P X + P X u% mo mm u u = X + P (X -X ) . m u u m' Rearranging, we obtain P (Z^-Z^) = (X-X^) . Substituting i n equation (4) y .= cY + BVY(X-X ) + (1-c)(X ). (5) ° m YX * m n We note that i n t h i s equation, the expression Y + BVV(X-X ) s c m YX m i s a regression estimate of y from double sampling which we denote as Y , Equation (6) can now be written simply as y = cYr •+ d-c)rn... (6) The best estimate for y i s a weighted combination of Y from regression and Y from the replacement p l o t s . The weights c and (1-c) can be shown to be functions of the error variance of the mean volume based on the new plots and of the error variance of the mean volume based on double sampling with re-gression. These weights are °Y 2 °Y 2 C = -2 -; (1-C) = o-j 2 + oj 2 a- 2 + oj 2 n v n r 18 E q u a t i o n (6) becomes a- >CYR) + a - HYN) y = -2 r- (7) oy 2 +a- 2 n r w i t h e r r o r v a r i a n c e °v 2 = {AY 2°Y 2 > / ( a y 2 + °Y 2 ) ( 8 ) ^ v n r n To e s t i m a t e growth between o c c a s i o n s , we c o u l d form a number of e x p r e s s i o n s , b u t again we s h o u l d make use of a l l the i n f o r m a t i o n a t our d i s p o s a l . We t h e r e f o r e form the g e n e r a l e x p r e s s i o n g. = AJ + BX + CY + DX , (9) a b m m n u where g^ denotes the b e s t e s t i m a t o r of growth, For t h i s e x p r e s s i o n t o be unbiased, E(9fo) = ( y x - y 2) • From simple random sampling E(X ) = E{X ) = p,, and E(Y ) - E(Y ) = p,, Th e r e f o r e A + C = 1 and B + D = -1. S u b s t i t u t i n g these v a l -ues i n t o e q u a t i o n (9), we have H = ™m + ( 1 ~ A ) 1 „ + B'Xm " ( 1 " B ) * u <10> with e r r o r v a r i a n c e (from expected values) , °Y2 „AY2 .aX2 -ar2 ^ ' = A z H °- 2 = A 2 — + ( 1 - A ) 2 — + B 2-|- + ( 1 + B ) 2 ^ - + 2 A B p - ^ , (11) where a l l terms have been p r e v i o u s l y d e f i n e d . The b e s t un-b i a s e d e s t i m a t o r s h o u l d minimize the e r r o r v a r i a n c e , t h e r e f o r e 19 we take the p a r t i a l d i f f e r e n t i a l s of equation (11) with re-spect to A and B, set the r e s u l t s equal to zero, and solve for A and B. Using the previously defined notation, the re-s u l t of t h i s procedure provides the estimate for growth = { (- ) (_i) #1 (-1) (-Ibp2 u + m X )] m N1 N1 Ni [X 1 > n -{ (—) .u . n (I__) ( _ ) P 2 N2 N2 X )] m + ( £ - ) ( ! " £ " ) P 2 ( J . ) - ( _ ) ( _ _ ) P 2 N2 N2 N2 (12) I f we define H = N 2 - P^rcp 2, we can rewrite equation (12) as m n (1 - P p 2 ) gb = {[- ?p] + [ * ? ] > n H P u ( ^ 2 - n p 2 ) (13) 20 where X = X + Rvv(Y - Y ) is a regression estimate of the p Tn Al in mean volume at the f i r s t occasion from double sampling and the o v e r a l l mean at the second occasion Y i s given by 1 = <?—)Y + Nz Tn N2 n and i s the true population l e a s t squares regression co-e f f i c i e n t of X on I. The error variance of g^ i s 1 N2-npz o 2 = r _ ] { [ ] 0 2 + |i-p p 2 ] a 2 _ 2PooYoY} (14) 3b H N X Examining equation (13), we see that i t i s a combination of four estimates. The f i r s t two expressions (within the braces) form an equivalent equation to equation (7) f o r y. The se-cond two expressions are i d e n t i c a l with the information r e -versed, i . e . a revised estimate of volume at the f i r s t occa-sion using information not availa b l e u n t i l the second occa-sion. Thus i n finding the best estimate for growth, we have used a l l information i n forming new estimates f o r volume at both occasions and subtracted one from the other. One d i s -advantage of t h i s estimator i s the value obtained for growth, when added to the o r i g i n a l estimate of mean volume at the f i r s t occasion, w i l l not r e s u l t i n the previously derived estimator for volume at the second occasion. S t a t i s t i c a l l y , t h i s i s no problem, but may cause some concern to those not conversant with SPR theory. 21 Other estimators of growth may be formed. One was just i n d i r e c t l y mentioned, c a l l i t g , which i s formed by subtracting the mean volume estimated at occasion one from the SPR estimate of volume at occasion two. This can be expressed as a = (a-P, )X„ - (P +a)Z m + cY + (1-c)? (15) " c u u m m m n with a l l terms previously defined. This estimator has an error variance 0 a 2 1-P p 2 P a9*= — + * y 2 -2 2-po-ff. (16) Ni N2~P np2 1 Nz-P np2 This estimator i s consistent with the f i r s t estimation of mean volume and makes use of the information ava i l a b l e on both occasions f o r i t s estimation of volume on the second occasion. Another estimator i s the usual matched p l o t volume estimations. In our notation, t h i s estimator, g , i s ex-pressed as g = Y - X (17) am m m v ' with error variance aq 2 = ^x1 + °Y2 ~ 2pa a )/m (18) 777 Note that t h i s i s a s p e c i a l case of g^ where u = n = 0. 22 Another s p e c i a l case which r e s u l t s i n an estimator for growth occurs when no plots are matched (m = 0) which i s an independent sample case. The estimator g ., i s formed as g. = Y - X (19) with error variance o 2 = a 2/n + a 2/« (20) If the sample s i z e was the same on both occasions, t h i s e s t i -mator would have an error variance of is These l a s t several formulae f o r the error variances are f a -m i l i a r ones from basic s t a t i s t i c s , which i s an i n t e r e s t i n g check on the derivation of a . One f i n a l estimator that w i l l be shown i s a weighted estimator from the matched and independent samples. We use weights of and g which are proportional to t h e i r error variances to form g . A general formula for g,, i s 9W = w(^) + (i-»> (gm) (21) i n which w = (o 2 ) / ( a 2 + a 2) and (1-w) = (a 2 ) / ( a 2 + 9i 9m ffi 9i &m 23 The error variance, again i n general terms, i s O g w 2 = (w) 2c 2 + (l-w) 2o 2 (22) For the sp e c i a l case when sample s i z e and the population v a r i -ance are equivalent at both occasions, t h i s estimator i s equi-valent to g^, otherwise i t i s les s e f f i c i e n t . These then are the various estimators that can e a s i l y be formed to estimate growth between occasions. In sp e c i a l cases, such as equal variance on both occasions, some are equivalent, but i n general, the f i r s t derived estimator, g^, i s the most e f f i c i e n t as i t takes best advantage of informa-t i o n available only a f t e r the second measurement has occurred. A major feature of SPR i s the s p e c i f i c a t i o n of the optimum number of sample units to remeasure and to replace. This can be achieved for minimum sampling error with f i x e d t o t a l expenditure or f o r minimum cost at fix e d sampling er-ror. Several authors have derived these options under var-ious r e s t r i c t i o n s such as equal sample s i z e on both occasions, equal p l o t costs, e t a . (Patterson 1950, Cochran 1963, Kish 1965). Ware and Cunia (1962) removed a l l r e s t r i c t i o n s and presented general formulae. In the analyses i n t h i s report, a f i n i t e population of 220 plo t s was used. I t was thus nec-essary to derive formulae for that p a r t i c u l a r r e s t r i c t i o n . This derivation w i l l be presented as t y p i c a l of those of other authors. 24 A f i r s t d erivation was attempted using the t o t a l sample s i z e on occasion two as + n) . The solu t i o n was not found. A second attempt was made with the r e s t r i c t i o n that a constant N2, which w i l l ultimately be equal to {m + n) 3 was used i n the f i n i t e population correction (FPC) terra. This r e s t r i c t i o n necessarily makes the sample s i z e determina-t i o n f o r the number of new pl o t s an i t e r a t i v e procedure. To determine the optimum ,number of sample plots to remeasure and to replace at occasion two, we specify that the cost must be minimized subject to the r e s t r i c t i o n that the sampling error of the current mean be less than or equal to a s p e c i f i e d value, say The cost i s a l i n e a r combina-tion of the following components; 1. C y , the amount of money ava i l a b l e f o r estimating the current mean at the second occasion; 2. C , the cost per p l o t of the m remeasured p l o t s ; 3. C , the cost per p l o t of the n new p l o t s ; 4. Cy, the f i x e d costs i n the inventory. The cost function i s of the form CT - Cf = mCm + nCn (23) To minimize t h i s function with the s p e c i f i e d r e s t r i c t i o n , we may use the method of Lagrange m u l t i p l i e r s (Reif 1965, Ware and Cunia 1962). Given that u, m, and n are the unmatched, matched, and new numbers of plots r e s p e c t i v e l y i n the samples taken on the two occasions, we define the following q u a n t i t i e s s 25 1. N i , the t o t a l sample s i z e on the f i r s t occasion which i s given and consists of (m + u ) plots; 2. N-i, the t o t a l sample s i z e at the second occa-sion which i s equal to (m + n ) p l o t s ; 3. The estimates of the population parameters p and a y 2 are obtained independent of the samples. K, the square of the s p e c i f i e d sampling error, must be less than or equal to a_ 2, which i s expressed as N2 a 2 [ N i (1 - p 2 ) + mp2] [1 a - 2 = -1 * , (24) y mNi + nN i (1 - p 2 ) +mnp2 N2 which includes the f i n i t e population correc t i o n (1 - ^—) , where N i s the t o t a l number of units from which we can draw. We form the Lagrangian function a 2 [Ni (1-p 2) + mp2] [1 - ^ ] i> = [nC + mC + C . ] + A { — : - - K } . (25) n m J m N l + n N i d - p Z ) + m n p 2 We next take the p a r t i a l derivatives of ip with respect to m, n, and A , which are then, for minimization, set equal to zero. These p a r t i a l s are [mNi + n ^ i d - P 2 ) + mnp 2] [ p 2 ( l - ^ 2 - ) ] = C + Aa 2{ ^ } 3 m m 1 [mNi + n N i { 1 - p 2 ) + m n p 2 ] 2 tol(l-p2)+wp2 - ^ ( 1 - P 2 ) + * P 2 1 . ( 2 6 ) [mNi + n ^ ! ( l - p 2 ) + m n p 2 ] 2 2 6 i = V A f f y ( > [m^j + n ^ C L - p 2 ) + m n p 2 ] 2 3, a / [ ^ ( l - P 2 ) ( l - f i ) + m p 2 g . f i ] —_ _ ^ - Ki , (to) mN1 + n l j l l - p 2 ) + mnp2 The f i r s t two p a r t i a l derivatives [Eqn's (26) and (27)] are solved for X and set equal to each other. I f we define G -Ni (1 -p 2 ) + m p 2 , the e q u a l i t i e s are CG2 C G2 x = 2 ] r - 2 - ^ . (29) tfid-p2) ( l - ^ ) c 2 ( i - Si.) Substituting for G and rearranging, we have C mp2 = i y 1 [ _ 2 L ( i - p 2 ) ] * „ tfjd-p2) . m Solving t h i s expression for m and c o l l e c t i n g terms tfi/d-p2) /C m = 5 -ij^- - / ( 1 - p 2 ) } . (30) P' We now rearrange the p a r t i a l d e r i v a t i v e with respect to X a 7 2 [ t f 1 ( l - p 2 ) + m p 2 ] [ 1 _ t f 2 ] K = — - . (31) mNi + n#i(l-p2) + mnp2 Again we substitute G where possible f o r s i m p l i f i c a t i o n to arr i v e at „ mNlK + nKG = a y 2 G ( i _ £ 2 ) # 27 Solving for n mN\K ay2(1 - | i ) mNi n = KG KG K N1 (1-p 2) + mp2 #1 U - [ P 2 ^ ( 1 - P 2 ) (32) K C n Ware and Cunia (1962) presented an i d e n t i c a l expression for m, the optimum number of plots to remeasure. Their expres-sion for n, the optimum number to replace d i f f e r s from equa-t i o n (32) i n that the f i r s t part of the r i g h t hand side i s Oy2/K. Thus we see that the f i n i t e population c o r r e c t i o n factor a f f e c t s only the sample s i z e requirement for a given p r e c i s i o n and a given variance. The expression f o r reduction i n sample s i z e due to SPR i s as presented by Ware and Cunia when current volume and growth are simultaneously estimated, increases i n complexity due to the need to s a t i s f y two samp-l i n g error r e s t r i c t i o n s . These r e s t r i c t i o n s are non-linear i n that they involve cross-product terms i n m and n. For derivation of these formulae, the reader i s again r e f e r r e d to Ware (1960). The formulae presented by Ware and Cunia (1962) are given here without proof f o r the e d i f i c a t i o n of the reader. The l i n e a r cost function [Eqn (23)] i s optimized sub-je c t to the r e s t r i c t i o n s that the sampling error of the current (1962). The method of optimum sample siz e determination. 28 mean, ^K, i s less than or equal to o_, and the sampling error of growth, say SK , i s less than or equal to a „ The i n -9 9b equality, a_ 2 <_ K, i s reduced to m[KN 1 - p 2 o J 2 ] + n [KN'i { 1 - p 2 ) ] + m n [ K p 2 ] - a . y 2 N 1 ( l - p 2 ) >_ 0 (33) and the inequ a l i t y , o : <_ K , to m i K ^ N i - i a ^ p a ^ ) 2 ] + n [ ( 1 - p 2 ) (K J^-a.^2) ] + m n [ p 2 K ] - (i_.p2)- >_ o (34) These expressions are set equal to zero. Ware and Cunia (1962) presented a graphical method to solve these i n e q u a l i t i e s f o r m and n. The method used i n the analyses for t h i s t h e s i s has been outlined e a r l i e r . Many of the present-day methods of mathematical, dynamic, and l i n e a r programming were not highly developed when Ware and Cunia developed t h e i r report. As was v e r i f i e d i n t h i s study, these systems can e a s i l y solve problems of t h i s kind. 2? CHAPTER FOUR COMPARISONS OF SPR WITH'CONVENTIONAL INVENTORY Example One In t h i s example, we w i l l compare conventional i n -ventory with an SPR system for a small area s i t u a t i o n . F i r s t we w i l l determine sample sizes for the conventional case. We w i l l then follow the establishment of an SPR system with s i m i l a r goals, following which, the costs of the two systems w i l l be compared. F i n a l l y , r e l a t i v e merits of the system w i l l be discussed. TABLE 1. Parameters for example one. PERIOD SYMBOL a2 P 2 cu.ft./ac ( c u . f t . / a c ) 2 1 X 20403.4 40947200 2 I 21558.8 47215200 0.9620 Al-2 9 1155.4 1911533 We must specify our p r e c i s i o n requirements f o r current volume and growth. We decide to take a one-in-twenty chance and choose a p r o b a b i l i t y l e v e l of 0.95. We f i n d that, for our management requirements, mean volume per acre should l i e within ± 6 percent of the true mean value. Growth must be ± 20 percent of the true value. 30 We now determine required sample sizes for a con-ventional inventory. For the i n i t i a l survey, the confidence i n t e r v a l half-width £ = (0. 06) (20403.4) = 1224,20 cu.ft./ac. From the formula for sample s i z e of a f i n i t e population t 2 a 2 n0 "o = ; n = —- (35) E2 1 + £ L we need 71 p l o t s . These can be low cost temporary p l o t s . For estimating current volume f i v e years l a t e r , we w i l l r e -quire a half-width of the confidence i n t e r v a l of E^ = (0.06) (21558.8) = 1293.53 cu.ft./ac. The sample s i z e w i l l be (Eqn. 35) 73 plots, again of the temporary type. We want to e s t i -mate growth over t h i s f i v e year period, so we decide to i n s t a l l a permanent growth p l o t system. The confidence i n t e r v a l h a l f -width i s E = (0.20)(1155.37) = 231.07 c u . f t . / a c / f i v e years. The sample siz e required i s 85 p l o t s , but we decide to estab-l i s h 88 as a hedge against loss of some pl o t s . Summarizing, at the f i r s t occasion we e s t a b l i s h 88 permanent p l o t s . We also need 71 plots for volume determination which may or may not be from the 88 permanent p l o t s . At the second occasion, we need to measure 73 temporary plots for volume, and to r e -measure at l e a s t 85 permanent p l o t s . For the SPR inventory, we assume the establishment of 88 pl o t s , some of which w i l l be permanent and some tempor-ary. For the second occasion, we determine the optimum num-ber to remeasure and how many we must replace considering 31 only the measurement of current volume. The cost r a t i o of permanent to temporary plots i s established as 2:1. From equation (32), we c a l c u l a t e that nine plots should be remea-sured. The solution for the replacement sample s i z e i s i t e r -a t i v e , and the f i n a l s o l u t i o n i s 45. I f we are only i n t e r -ested i n measuring the current volume afte r f i v e years, these are the numbers of samples required. I f we want to simul-taneously estimate growth, we check whether these sample sizes w i l l s a t i s f y inequation (34) . In t h i s case, they do not. We must now optimize a cost function subject to the r e s t r a i n t s a _ 2 < K and a 2 < K . As these r e s t r a i n t s are y - g - 9 non-linear, we resort to a non-linear optimization programme. This programme computes the optimum number to remeasure as 74 and to replace as 25. So of the 88 plots we e s t a b l i s h at occasion one, only the 74 need be permanent. Using the following table (TABLE 2)i l e t us com-pare and see i f SPR can save money for us. I f we conducted completely independent current volume and growth inventories on both occasions, we would spend $12,250.00 (a+b+c+d). We could reduce t h i s cost by using the permanent plots for cur-rent volume information at the f i r s t occasion only. This would increase the p r e c i s i o n of the f i r s t volume inventory and re-duce the cost to $10,475.00 (b+c+d). Using the permanent plots at the second occasion would not add any information about the stand structure and would r e i n f o r c e any f a i l u r e of the f i r s t sample to be representative of the stand. The TABLE 2. C o m p a r i s o n of c o s t s between i n d e p e n d e n t and SPR i n v e n t o r i e s . Inventory type (1) occasion (2) p l o t nos. (3) p l o t cost (4) (2) x (3) d o l l a r s d o l l a r s Independent (volume only) one 71 25. 00 1775.00 (a) two 73 25.00 1825.00 (b) (growth only) one 88 50.00 4400.00 (c) two 85 50.00 4250.00 (d) SPR one 77 50.00 3850.00 (e) 11 25.00 275.00 (f) (growth and volume) two 74 50.00 3700. 00 (g) 25 25. 00 625.00 (h) (volume only) two 9 50. 00 450.00 (i) 28 25.00 700.00 (j) 33 SPR inventory for the same information would cost $8,450.00 (e+f+g+h). This would save a minimum of $2,025.00 and allow us the option of measuring current volume only at a cost of $1,150.00 (i+j). Several i n t e r e s t i n g by-products of the SPR system can be seen i n t h i s example. One, the option of measurement of current volume only at a greatly reduced cost, w i l l be discussed i n a l a t e r example. Another i s increased p r e c i s i o n i n the current volume estimate when simultaneous estimation i s used. I f we had measured only current volume at the se-cond occasion, we would have needed only 37 p l o t s to meet our p r e c i s i o n requirements. As i t turned out, we used 99 plots because of the growth p r e c i s i o n needed. We can calcu-l a t e the increased p r e c i s i o n by solving equation (33) f o r K, the sampling e r r o r 2 . From t h i s value, we f i n d that the con-fidence i n t e r v a l half-width has been reduced to ±4.37 per-cent of the population mean. The p r e c i s i o n of the estimate of current volume on the f i r s t occasion i s changed to ±5.07 percent of the mean whether we use the SPR system or a l l of the permanent growth p l o t s . Perhaps th i s p r e c i s i o n i s not required, but i t i s there and we should take advantage of i t i n our decision-making processes. Another advantage of SPR occurs when, as i n t h i s example, the minimum variance estimator of growth i s used. A revised estimate of volume at the f i r s t occasion can be ob-tained from the computations. This new estimate takes i n f o r -mation available only at the second occasion and through 34 regression combines i t with the mean of the non-remeasured plots from the f i r s t occasion. In t h i s example, the degrees of freedom (for variance estimation) for the o r i g i n a l estimate would be 88 - 1 = 87. For the SPR estimate, a t o t a l of 99 + 14 observations are av a i l a b l e . There are three means (assum-ing the regression c o e f f i c i e n t i s independently known) e s t i -mated from these observations. Thus we have 113 - 3 = 110 degrees of freedom f o r the variance of the revised estimate. This estimate should be more e f f i c i e n t . One f i n a l point can be made from t h i s example. I f we are always only interested i n current volume, we have a new s i t u a t i o n i n which only 71 pl o t s would be established at the f i r s t occasion. A t o t a l of 73 plots would s t i l l be needed at the second occasion under the conventional system. But only 8 remeasured and 19 new plots are needed f o r an SPR e s t i -mate with the same second occasion p r e c i s i o n . The t o t a l SPR cost (exclusive of fix e d costs) would be $2,850.00 versus $3,6 00.00 for the conventional survey. There would undoubt-edly be a time savings as well since, even though remeasured plots require more time spent on a l o c a t i o n , the very small number of plo t s needed at the second occasion should require less time than for the 73 p l o t s of the conventional system. 35 Example Two For t h i s example we w i l l assume a large acreage, reasonably homogeneous stand i s i n need of inventory. Stand age i s somewhere past the middle of r o t a t i o n . The area i s high s i t e II and therefore w i l l be i n our management scheme for some years. For decision-making purposes, we desire a continuing current volume estimate within ± 3 per cent of true mean volume per acre and a growth estimate within ± 10 per cent of the true value for a ten year period. Estimates of the population parameters and sampling error requirement are given i n TABLE 3. TABLE 3. Parameters and sampling e r r o r s 2 for example two. PERIOD SYMBOL 3 A l - 3 mt Y cu.ft./ac 20403.4 21558.8 22705.2 2301.8 (cu. f t./ac) 2 40947200 47215200 58756000 5272314 1-2 0.9620 1-3 0.9266 SAMPLING ERROR2 ( c u . f t . / a c ) 2 K = 120776.1 K = 13790.8 5/ For a conventional inventory, the c a l c u l a t i o n of sample size from equation (34) (without the FPC) indicates we need 420 temporary plots at the f i r s t occasion to e s t i -mate the current mean volume. At the second occasion, ten years l a t e r , 487 p l o t s are needed for the same estimate. For growth over the ten year period, 383 permanent sample plo t s are required. For an SPR inventory, we assume the same 420 p l o t s on the f i r s t occasion, some of which are permanent and some temporary. The numbers of each type come from our c a l c u l a -tions. For current volume only on the second occasion, solutions of equations (30) and (32) i n d i c a t e we w i l l re-quire 54 permanent and 207 temporary p l o t s . These numbers of p l o t s w i l l not s a t i s f y our growth p r e c i s i o n requirements, so we must optimize a cost function - = $50.00*m + $25.00«n with the constraints that the maximum m i s 420, minimum m and n are zero, and the following non-linear i n -q u a l i t i e s (Eqn's 33 and 34)% -3717344.66m + 3723285. 83n + 111911.14/rm - 1811329968. > 0; 4832580.63m - 2580382.lln + 12778.54mn - 1811329968. > 0„ The optimization program calculates the sample s i z e as m = 347 and n = 73. I n t e r e s t i n g l y , the t o t a l sample s i z e on the second occasion i s i d e n t i c a l to the number of plots established on the f i r s t occasion. We would measure 347 permanent plots on both occasions, 73 temporary pl o t s on the f i r s t occasion, and 73 d i f f e r e n t temporary plots on the second occasion. 37 The t o t a l cost of completely independent inventor-i e s for volume and growth on two occasions (not including fixed costs) i s $60,975.00 (a+b+c+d+e, TABLE 4 ) . Assuming we used the 383 permanent plots i n the f i r s t occasion volume survey (thus reducing the necessary temporary sample s i z e to 37), t h i s cost could be reduced to $51,400.00 (b+d+e+37•$25.00) plus f i x e d costs. We see that the SPR inventory costs are only $50.00 more than the cost of the growth p l o t s . One of the benefits of SPR, mentioned i n example one, can be r e a d i l y demonstrated i n t h i s s i t u a t i o n . Say we decided to conduct a new survey f o r current volume only at f i v e years a f t e r p l o t establishment. At the same p r e c i s i o n , 434 new temporary plots would be required. But, we can take advantage of our permanent plots and the linkage a v a i l a b l e through them and we need measure only 36 permanent pl o t s and e s t a b l i s h 173 temporary p l o t s . The t o t a l cost of t h i s s i n g l e inventory would be $6,125.00, a saving of $4,725.00 over the cost of a s i n g l e conventional volume inventory, Summing the costs f o r both methods: Conventional (3 current volume measurements + 1 growth mea-surement) : $10,500.00 + 10,850.00 + 12,175.00 + 38,300.00 = $71,825.00; SPR (3 current volume measurements + 1 growth measurement): $38,350.00 + 6,125.00 = $44,175.00. TABLE 4. C o m p a r i s o n of c o s t s between i n d e p e n d e n t and SPR i n v e n t o r i e s f o r example two. Inventory type (1) occasion (2) p l o t nos. (3) p l o t cost (4) (2) x (3) d o l l a r s d o l l a r s Independent (volume only) one 420 25.00 10500.00 (a) two 434 25.00 10850.00 (b) three 487 25.00 12175.00 (c) (growth only) one 383 50.00 19150.00 (d) three 383 50.00 19150.00 (e) SPR (growth and volume) one 347 50. 00 17350.00 (f) 73 25.00 1825.00 (g) (volume only) two 36 50.00 1800.00 (h) 173 1 25.00 4325.00 (i) (growth and volume) three same as one 19175.00 (j) 39 The demonstrated saving of $27,650.00 i s considerable. At any of these remeasurement periods, we might want to spend some of these savings to e s t a b l i s h permanent plots i n l i e u of the indicated temporary p l o t s . This would allow us to use these pl o t s on t h i r d , fourth, or subsequent remeasure-ments. By adding, deleting, and/or bringing plots back i n t o our inventory system, we can change our p r e c i s i o n require-ments at almost any stage of management. The system i s very f l e x i b l e . 40 Example Three In some f o r e s t management s i t u a t i o n s , we wish to combine or fragment our holdings. For inventory, t h i s can be very disconcerting. To be t r u l y universal i n a p p l i c a t i o n , SPR should help overcome some of the problems we encounter i n t h i s s i t u a t i o n . For t h i s example, we assume that a stand with an area of 11 square miles has been i n our holdings for some time. Inventory on t h i s stand has been with the require-ments of the half-width of the confidence i n t e r v a l f o r current volume be ± 6 percent of the mean volume per acre. We have purchased a forested area, surrounding our holding, of 49 square miles f o r a t o t a l area of 60 square miles. This new area i s forested with species, stocking, and s i t e s i m i l a r to our present stand. A new inventory, that w i l l encompass the t o t a l of our holding i s necessary for our management of t h i s area. From our previous inventories i n the 11 square mile stand, we have parameters as given i n TABLE 1, which have been compiled from 220 permanent plots located i n the o r i g i n a l stand. An area compensating formula w i l l be used to de-termine the required sample s i z e at the f i r s t occasion (Ware and Cunia 19 62)% _ (confidence i n t e r v a l half-width i n p e r c e n t ) ( / o r i g i n a l area) (/new area of estimate) (36) 41 For t h i s stand, e = 2.569 percent. Therefore the allowable error variance i s K = 71518.97 c u . f t . 2 / a c 2 . From t h i s we compute the sample s i z e on the f i r s t occasion as 573 p l o t s . Some of these w i l l be temporary and some permanent. The sam-ple s i z e f o r estimating sample s i z e on the second occasion i n an SPR sample i s 60 remeasured plots and 229 temporary plots for a t o t a l of 289. This sample s i z e w i l l not f u l f i l l our p r e c i s i o n requirements f o r growth which we have estab-l i s h e d as ± 16 percent of the mean growth per acre per f i v e years. We therefore use our non-linear optimization routine with the following constraints: i . the minimum m and n equal zero; i i . the maximum m i s equal to 573; i i i . o_ 2 < K and a 2 < K . y - g - 9 The optimum number of pl o t s to remeasure i s 534 and to add i s 89. We may now proceed towards our two goals, the e s t i -mation of the current volume of the 60 square miles and the provision f o r estimation of volume and growth i n the future. As we have a 220 p l o t e x i s t i n g base, we remeasure these p l o t s , and e s t a b l i s h 314 new permanent and 89 temporary p l o t s . Most, i f not a l l , of these new plots should be established i n our new holdings as we have a s u f f i c i e n t number of plots i n the o r i g i n a l 11 square miles. In addition, we should consider e s t a b l i s h i n g the 89 replacement plots as permanent to pro-vide f o r p l o t losses and f o r the p o s s i b i l i t y that our new 42 holdings w i l l introduce added v a r i a b i l i t y . Regardless, we have extended our sampling scheme to the area surrounding our o r i g i n a l stand with a minimum of e f f o r t . CHAPTER FIVE BIAS IN SPR ESTIMATES Bias i n sampling can cause estimates to vary con-siderably from t h e i r expected values. This v a r i a b i l i t y may occur such that not only are the estimates biased, but they are also inconsistent. These conditions can be minimized by c a r e f u l l y examining the estimation methods. I f a l t e r n a t i v e methods are not ava i l a b l e or applicable, steps can usually be taken to estimate the amount of bias present and to make the sample estimates consistent. In SPR, the basic assumptions that o^,2, ° j 2 ^ P> and therefore 6 are either known without sampling error or obtained independent of the sample allow the current volume and growth estimates to be unbiased. When these conditions are not met, or i f the sample population i s f i n i t e , c e r t a i n steps can be taken to a l l e v i a t e the problem. One p o t e n t i a l source of bias occurs i n the e s t i -mation of y from equation 7 involving the error variance a- 2. Ware and Cunia (1962) present t h i s variance as r o y 2 ( l - p 2 ) p 2 a / r m (m+u) Cochran (1963) and others have derived t h i s same expression. When regression estimates are used with double sampling, the variance of the estimate i s dependent on the x^ chosen f o r that p a r t i c u l a r sample. Cochran averaged the variance over 44 a l l possible draws to obtain an expression that w i l l give the average mean square error which i s equivalent to our 2 . This expression i s r a 2 ( l - p 2 ) p 2 a 2 o 2 ( l - p 2 ) (w) (1) a- 2 = - 1 + — + — (37) r m (m+u) m (m+u) (m-3) The f e e l i n g i s that i f the term l/(w-3) i s n e g l i g i b l e , the much simpler expression i s acceptable. With the population parameters used i n these examples, which are not t y p i c a l , o-j, 2 was increased by approximately 54 parts i n 363860, or r about 0.015 percent. This i s indeed n e g l i g i b l e and the cor-r e c t i n g formula was not used i n the computations involving the population parameters. When p and 3 are estimated from the sample, the reduced form of the expression f o r oy 2 i s no longer unbiased, v but i s only consistent (Ware and Cunia 1962) . A sampling experiment was conducted to estimate the amount of t h i s bias fo r the population used f o r these analyses. F i f t y r e p e t i -tions of the sampling experiment with a random s t a r t f o r each run were made. For each run, °YX,Z w a s estimated from popu-l a t i o n parameters, the sample i t s e l f using equation (36) , and from the sample using equation (37) . Further, these e s t i -mates were used to estimate the mean volume at the second occasion. The object was to determine differences i n s-p 2 r by the d i f f e r e n t methods and to further determine the e f f e c t s upon mean volume estimation. The r e s u l t s are tabulated i n 45 A P P E N D I X I I I . Analyzing the r e s u l t s , the l a r g e s t deviation of s-y 2 as computed from the reduced form from that computed r from the f u l l equation was 1 part i n 300,000. The reduced form gave a smaller average deviation (mean volume subtracted from the population mean volume)(on the order of -0.0164) than did the f u l l expression (order -0.0705). For t h i s sam-ple then, the reduced form gave s u f f i c i e n t l y precise, i f not better, estimates of both a-^ 2 and y. r Cochran (1963) suggested a possible answer for t h i s very small amount of bias. The basic derivations f o r double sampling with regression assume a simple random sample on both occasions. From l e a s t squares regression theory, the stan-dard error of the regression c o e f f i c i e n t i s minimized i f the sampled values are at the extremes of the range. This i s the technique used i n these analyses. The 88 o r i g i n a l p l o t s selected i n each case were sorted i n order of increasing p l o t volume. The 74 remeasured pl o t s were selected by choosing the lowest 30 and the highest 30 plots by volume. The remaining 14 were systematically selected as every other p l o t from the 32nd to the 58th, i n c l u s i v e . Thus the p r e c i s i o n of the r e -gression c o e f f i c i e n t should be high. Cochran suggested that t h i s form of sampling may reduce the term i n l/(m-3), . . . "perhaps considerably". In small populations, i t becomes necessary to use a f i n i t e population correction i n SPR sampling. Indeed, i n the examples presented where the t o t a l population was 220 p l o t s , i t was nearly impossible to obtain any solu t i o n to 46 the doubly-constrained cost function u n t i l these corrections had been applied. A derivation of optimum numbers to remea-sure and to replace was given (Eqn's 30 and 32). As can be seen, the number to remeasure i s not affected, but, as would be expected, the number to replace i s reduced i n the term i n o2/K. This formula should be used i f there i s any question of small sample s i z e or sampling without replacement. Other sources of error, though not a l l s t r i c t l y forms of bias, include some f a m i l i a r problems from conventional f o r e s t inventory. Some examples are, inaccurate measuring devices, i n c o r r e c t tree species i d e n t i f i c a t i o n , observer bias i n on-location sample p l o t s e l e c t i o n , and i n the SPR case, because regression estimates are used, the f a i l u r e of the x^ to be known without error. Users or p o t e n t i a l users should be aware of these biases and should make e f f o r t s to see that t h e i r estimates are either not subject to or cor-rected for these problems. f7 CONCLUSIONS In t h i s thesis we examined some applications of sampling with p a r t i a l replacement of sample units on succes-sive occasions. An expression f o r estimating the mean on a second occasion was obtained. Various estimators of change between measurement periods were formed and discussed. Two examples of f o r e s t inventory planning were presented that compared sample sizes and costs of independent mean volume and growth estimation and SPR estimation of the same parame-ter s . Changing s i z e of inventory area was examined from the SPR viewpoint. The main question investigated i n these ex-amples i s whether SPR can r e a l i z e the hypothetical savings i n time and expense with equal p r e c i s i o n . In the hypothetical case that only mean volume was required on an occasion other than the f i r s t , SPR would equal the conventional method of independent samples i n pre-c i s i o n at reduced costs. As was shown, a savings occurs whether the goal of the survey i s only a two occasion volume survey or an occasion sometime between the period of growth and volume determination. I conclude that f o r estimation of a mean on more than one occasion, SPR i s superior to con-ventional independent surveys i n cost and presumably i n time expenditure. In the s i t u a t i o n where both current mean and changes are required on successive occasions, SPR i s again a superior system i n terms of cost. In these cases, an increase i n the 48 p r e c i s i o n of the estimate of the mean can also be expected, as long as the numbers of plots required for the estimation of t h i s parameter on the f i r s t occasion exceed those required for estimation of change. This, because the magnitude of the mean i s normally much greater than that of increment. This translates to, greater numbers of plots are required for increment p r e c i s i o n , and these greater numbers give a more precise estimate of the mean than was i n i t i a l l y s p e c i -f i e d . Time savings may or may not be i n favor of SPR i n t h i s s i t u a t i o n as the large number of permanent p l o t s that must be established and subsequently remeasured w i l l require a great deal of time expenditure. There has been only one recent report of an actual (rather than simulated as i n t h i s analy-sis) SPR t r i a l (Prayer et a l . 1971). In that study SPR was e f f e c t i v e i n reducing both cost and time expenditures. Un-t i l more actual t r i a l s can be analyzed, t h i s question must remain unanswered. An i n t e r e s t i n g question i s how can variable-radius or p l o t l e s s c r u i s i n g be used with SPR? Although t h i s ques-ti o n i s peripheral to SPR, the obvious cost reductions possible make consideration i n e v i t a b l e . This type of p l o t could be used i n two ways. One, the more obvious, involves the use of prism plots as the temporary non-remeasured p l o t s . This could bring about a dramatic d i f f e r e n c e i n p l o t costs and reduce the number of permanent plots needed. There are no i n c o m p a t i b i l i t i e s with SPR theory. Again i t i s a matter of 49 actual t r i a l s to determine the gains. The more complex ques-t i o n i s can prism plots be used as permanent p l o t s and re-measured? I f e e l that t h i s i s e n t i r e l y possible. In some ways, i t could have d i s t i n c t advantages. The point of the vertex of the sweep would have to be p r e c i s e l y located for future measurements. The " i n " trees should be tagged at the point of d.b.h.o.b. measurement. But less marking would be necessary than with conventional permanent p l o t s . This would cause a more natural treatment of the area which would r e -move some of the biases that do occur i n t h i s s i t u a t i o n . This would i n turn give a better picture of f o r e s t develop-ment and use. Another innovation i n sampling technique has been proposed f o r use with SPR, that i s , tree s e l e c t i o n with pro-b a b i l i t y proportional to p r e d i c t i o n (3-P). I envision the use of 3-P as a stage i n a two-stage system. In both tem-porary and permanent p l o t s i t u a t i o n s , the p l o t boundaries could be located and the trees within the p l o t would be v i s i t e d and selected f o r measurement by a 3-P system. This could work quite e a s i l y for the temporary p l o t s , but more d e l i b e r a -t i o n i s necessary for the other type. Should the trees that are selected f o r the f i r s t occasion be marked and therefore remeasured on a l l subsequent occasions? Or should each v i s i t c onstitute a new 3-P selection? I personally tend towards the l a t t e r , as the trees would be selected on each occasion to describe the p l o t which i s then used for l i n k i n g between 50 periods. These two areas should be subjects for further in v e s t i g a t i o n s . Another obvious question i s what a p p l i c a b i l i t y does SPR have i n B r i t i s h Columbia? This question must be discussed within the framework of the B.C. Forest Service, as most f o r e s t land i s owned by the province. The inventor-i e s of the Public Sustained Y i e l d Units are conducted by two sections of the B.C.F.S. Forest Surveys and Inventory D i v i -sion. One section bears r e s p o n s i b i l i t y f o r estimates of cur-rent volumes, the other i s involved with growth. Each sec-tion establishes i t s own sampling network and very l i t t l e use i s made of information from the other group. P r o b a b i l i s t i c a l l y , these sections could be combined within an SPR framework to achieve the survey requirements at reduced costs. P o l i t i c a l l y , I do not believe t h i s can happen. The amalgamation of the two sections would not i n -volve a great change from the p r a c t i c a l side. The survey schemes are s i m i l a r . Both sections e s t a b l i s h s i m i l a r type plo t s . The growth section plots are more permanently marked and more trees are measured for c e r t a i n v a r i a b l e s , but there are few s t r i k i n g differences. Given the immensity of the f o r e s t survey required throughout B r i t i s h Columbia, the i n -corporation of SPR into the sampling design could r e s u l t i n reduction of manpower requirements, time, and costs. FUTURE CONSIDERATIONS In t h i s study, only an elementary a p p l i c a t i o n of SPR was considered, i.e. simple random sampling f o r estimating mean volume per acre and i t s change over time. This served to present the basis of SPR and to e s t a b l i s h i t s e f f i c i e n c i e s * SPR i s an e f f i c i e n t temporal sampling technique which must be applied i n conjunction with s p a t i a l sampling methods* Now we must consider where SPR can be applied to e x p l o i t i t s de-s i r a b l e features. A s i t u a t i o n s i m i l a r i n nature to t h i s study i s i n outdoor recreation. This f i e l d has had problems i n data c o l -l e c t i o n and analysis. The very s p e c i f i c area of estimating campsite use could be a target of SPR a p p l i c a t i o n . Within a highly-developed park, a simple random sample of v i s i t s to campsites could be made during a high-use season. From t h i s sample, a mean and variance of number of v i s i t s could be de-termined. The following season some campsites could be r e -measured and new ones measured as replacement units• SPR estimates could then be made of means for both occasions and for growth i n v i s i t o r use. In subsequent years, more SPR samples could be made and analyzed using multiple-occasion SPR theory (Cunia and Chevrou 1969). Further, within a park, there might be campsites with varying l e v e l s of f a c i l i t i e s of-fered. These could be s t r a t i f i e d and use-levels and trends established for each type of f a c i l i t y . An a t t r a c t i v e feature 52 of t h i s a p p l i c a t i o n i s the permanency of the sample units (campsites) and the 1:1 correspondence of costs of remeasured and temporary p l o t s . Moving from a simple application to the other ex-treme, the knowledge of f o r e s t stand c h a r a c t e r i s t i c s and dy-namics may be the most important contribution of f o r e s t samp-l i n g to management's decision-making processes. At present, the d e s c r i p t i v e parameters are i n t e r r e l a t e d , but usually separately inventoried, features of the f o r e s t . True, many variables are measured during an inventory, but these are normally applied to describe a s i n g l e f a c t o r - o f - i n t e r e s t (de-pendent variable) or used i n separate analyses to describe several such f a c t o r s . Through the use of m u l t i v a r i a t e s t a -t i s t i c a l procedures many factors could be j o i n t l y analyzed with an increase i n the discerning power of the analysis. Consider the construction and use of growth and y i e l d tables. These tables normally involve simultaneous temporal presentation of number of stems, basal area, mean diameter, mean height, stand form factor, and stand volume. Many measured variables are used to determine the appropriate expression of these factors. These i d e a l l y include age, de-gree of stocking, s i t e index, s i l v i c u l t u r a l programs, e t c . The formulation of these tables i s an expensive time-consuming procedure. Further, the a p p l i c a b i l i t y i s l i m i t e d (in theory) to stands with l i k e c h a r a c t e r i s t i c s . 53 We s h a l l now put the cart before the horse and con-sider the use of y i e l d tables. I f the appropriate variables were m u l t i v a r i a t e l y measured and analyzed, the i n t e r r e l a t i o n -ships would become more apparent from the use of j o i n t con-fidence regions. We might see, for instance, whether an ap-proach to normality of a stand was influenced by c e r t a i n v ariables and not others, or indeed i f our stand was t r u l y changing i n stocking. Once we e s t a b l i s h the l e v e l of stocking i n our stand, along with other multivariate f a c t o r s - o f - i n t e r e s t , we should be able to use the multivariate growth and y i e l d table to a s s i s t i n management decisions. But what has t h i s to do with SPR, and how would we formulate these multivariate growth and y i e l d tables? A usual method of formulation has been to u t i l i z e permanent sample plo t s i n homogeneous even-aged portions of stands. But f o r e s t con-d i t i o n s are not necessarily i n t h i s state of homogeneity. So the permanent sample plots are not c h a r a c t e r i s t i c of the en-t i r e stand. "This disadvantage can be s u b s t a n t i a l l y reduced by using as the source of y i e l d table data a combination of permanent full y - s t o c k e d sample pl o t s and temporary p l o t s , o b j e c t i v e l y selected i n the course of a f o r e s t inventory" (Loetsch et al. Vol. I I , 1973). This s i t u a t i o n i s one of the predominant e f f i c i e n c i e s of the SPR system. The growth and y i e l d tables could be formulated i n an unbiased manner with SPR techniques and likewise SPR techniques might prove most e f f i c i e n t to use with the subsequent sampling for the necessary 54 variables to u t i l i z e these tables. Conjecturally, the new view of the interrelationships might shed l ight on some of the plaguing problems in forestry, such as mortality. This is the direct ion recommended for further study of SPR; a multivariate SPR procedure for use with the most e f f ic ient spat ia l sampling methods. These must be procedurally simpli f ied for use by personnel not conversant with this type of sampling theory (among whom the author may be counted). Then a long-range plan of implementation should be formulated and begun in cooperation with a l l agencies interested in eval -uating their holdings with a view toward f u l l real izat ion of the production potent ia l . 55 BIBLIOGRAPHY Avery, T. Eugene. 1967. Forest measurements. McGraw-Hill Book Company, Inc., New York. 290 p. Bickford, C A . 1956. Proposed design for continuous inven-tory: a system of perpetual Forest Survey for the North-east. U.S. Forest Service Eastern Techniques Meeting, Forest Survey, Cumberland F a l l s , Ky. Oct. 8-13, 1956. 37 p. . 1959. A te s t of continuous inventory f o r National Forest management based on a e r i a l photographs, double sampling, and remeasured p l o t s . Soc. Amer. For-esters Proc. 1959:143-148. Cochran, W.G. 1963. Sampling techniques, 2nd ed. John Wiley & Sons, Inc., New York. 413 p. Cochran, W.G., and G.M. Cox. 1957. Experimental designs, 2nd ed. John Wiley & Sons, Inc., New York. 611 p. + tables. Cunia, T. 1964. What i s sampling with p a r t i a l replacement and why use i t i n continuous f o r e s t inventory? Soc. Amer. Foresters Proc. 1964:207-211. . 1965. Continuous f o r e s t inventory, p a r t i a l replace-ment of samples and multiple regression. For. S c i . 11: 480-502. . 1973. Dummy variables and some of t h e i r uses i n regression analysis. Proc. of the meeting of IUFRO Sub-j e c t Group S4.02, Nancy, France. June 25-29, 1973. Vol. 1. 146 p. , and R.B. Chevrou. 1969. Sampling with p a r t i a l r e -placement on three or more occasions. For. S c i . 15:204-224. Davis, Kenneth P. 1966. Forest management: regulation and valuation, 2nd ed. McGraw-Hill Book Company, Inc., New York. 519 p. Draper, N.R., and H. Smith. 1966. Applied regression analysis. John Wiley & Sons, Inc., New York. 407 p. Frayer, W.E., R.C. VanAken, and R.D. S u l l i v a n . 1971. Applica-tion of sampling with p a r t i a l replacement to timber i n -ventories, Central Rocky Mountains. For. S c i . 17:160-162. 56 Hansen, M.H., W.N. Hurwitz, and W.G. Madow. 1953. Sample sur-vey methods and theory. V o l . I, Methods and applic a t i o n s , 638 p. Vol. I I , Theory, 332p. John Wiley & Sons, Inc., New York. Husch, B., C.I. M i l l e r , and T.W. Beers. 1972. Forest mensura-t i o n , 2nd ed. The Ronald Press Company, New York. 410 p. Jessen, R.J. 1942. S t a t i s t i c a l investigations of a sample survey f o r obtaining farm f a c t s . Iowa Agr. Expt. Sta. B u l l . 304. 10 p. Kramer, C Y . 1972. A f i r s t course i n m u l t i v a r i a t e analysis. P r i v a t e l y published. 331 p. Kish, L e s l i e . 1965. Survey sampling. John Wiley & Sons, Inc., New York. 643 p. L i , J.C.R. 1964. S t a t i s t i c a l inference. Vol I, 658 p. V o l . I I , 575 p. Edwards Brothers, Inc., Ann Arbor, Michigan. Loetsch, F., F. Zohrer, and K.E. H a l l e r . 1964, Vol, I, 436 p.; 1973, V o l . I I , 469 p. (English by K.F. Panzer). BLV Verlagsgesellschaft mbH, Munchen, Germany. Madison, R.W. 19 57. A guide to the Cascade Head Experimental Forest. U.S. Forest Service P a c i f i c Northwest Forest and Range Experiment Station, Portland, Oregon, 14 p. + map. Newton, CM. 1971. Sampling with p a r t i a l replacement as ap-p l i e d to two successive mul t i v a r i a t e f o r e s t measurements, Ph.D. Thesis. Syracuse Univ. 83 p. + computer program l i s t i n g . Univ. Microfilms, Ann Arbor, Michigan. Nyyssonen, Aarne. 1967. Remeasured sample p l o t s i n f o r e s t inventory. Medd. Det. Norske Skogfors^ksves, 84(22): 195-220. Patterson, H.D. 1950. Sampling on successive occasions with p a r t i a l replacement of u n i t s . J. Royal S t a t i s , Soc., Ser. B, 12:241-255. Reif, F. 1965. Fundamentals of s t a t i s t i c a l and thermal physics, McGraw-Hill Book Company, Inc., New York. 651 p. Society of American Foresters. 1958. Forest terminology, 3rd ed. Washington, D.C. Smith, D.M. 1962. The p r a c t i c e of s i l v i c u l t u r e , 7th ed. (pub-l i s h e d as a f i r s t e d i t i o n i n March, 1921, by R.C. Hawley), - <^ -v,c i n c . , New York. 578 p. 57 Snedecor, George W., and William G. Cochran. 1971. S t a t i s t i -c a l methods, 6th ed. The Iowa State Univ. Press, Ames, Iowa. 593 p. Ware, K.D. 1960. Optimum regression sampling design f o r f o r -est populations on successive occasions. Ph.D. Thesis. Yale Univ. 154 p. + appendices. Ware, K.D., and Tiberius Cunia. 1962. Continuous f o r e s t i n -ventory with p a r t i a l replacement of samples. For. S c i , Monog. 3. 4 0 p. 58 APPENDIX 1 COMPUTING FORMULA FOR VOLUME OF HEIGHT-MEASURED TREES The empirical volume formula developed by the For est Survey, U.S. Forest Service i s VOLUME = 8.436 - 2.608*DIA + 0.070242*DIA*DIA + 3.1278*CH - 12.18*FC*FC + (0.002474*CH*CH - 0.00792)*(HT -17.0), where DIA i s diameter outside bark at breast height (4.5 f t above mean germination point; FC i s Girard form c l a s s ; CH i s DIA*FC; HT i s measured tree height. 59 APPENDIX II REGRESSION COEFFICIENTS FOR VOLUME COMPUTATION FROM /(D.B.H.O.B.) Tsuga heterophyI la Period b 2 b 3 Obns. 1 -0.313092 0.225714 -0.00717811 0.0000884926 236 2 -0.577882 0.275449 -0.01007860 0.0001425050 266 3 -0.471625 0.250275 -0.00834343 0.0001065710 243 4 -0.256965 0.211510 -0.00623550 0.0000717721 259 Picea s i t c h e n s i s and Pseudotsuga menziesii Period b 0 bi bz b 3 Obns. 1 0.420500 0.122713 -0.00255568 0.0000214305 155 2 0.345056 0.129963 -0.00273977 0.0000228607 199 3 0.456704 0.118534 -0.00227102 0.0000168226 144 4 0.472309 0.113621 -0.00204826 0.0000145269 .165 These c o e f f i c i e n t s describe the r e l a t i o n s h i p s de-veloped for tree volume as a function > of diameter. The model i s L o g i o ^ : = So + 6 i ^ i + B 2 * l 2 + 3 3 * i 3 + ^ where I. = volume i n cubic feet; Xi = diameter outside bark at breast height; e. = random error, ^N(0,a2); = true population regression c o e f f i c i e n t s of which the b^  are l e a s t squares estimates. 60 The diameter ranges used i n these analyses were: Tsuga heterophylla 6.9 inches <_ d.b.h.o.b. <_ 31.2 inches; Picea sitchensis -i n -> • , , , , , , „ , , . . . 10.2 inches < d.b.h.o.b. < 51.4 inches. Pseudotsuga menz%es%% — — 61 APPENDIX I I I COMPUTED STATISTICS OF FIFTY RANDOM SPR SAMPLES p population p reduced p f u l l o- 2 reduced a- 2 f u l l mean volume per acre computed from population parameters mean volume .per acre computed from sample sta-t i s t i c s and the reduced form of mean volume per acre computed from sample sta-t i s t i c s and the f u l l form of error variance of regression estimate computed from sample s t a t i s t i c s (reduced equation) error variance of regression estimate computed from sample s t a t i s t i c s ( f u l l equation) Obsn y population y reduced Sy 2 reduced y f u l l Sy 2 f u l l 1 21421.39 21424.85 363826 21424.85 I* 363827 2 21065.33 21065.13 319095 21065.14 319096 3 21884.93 21884.63 430616 21884.63 430617 4 21601.29 21601.45 376216 21601.46 376217 5 21508.96 21507.72 372901 21507.72 372901 6 22197.78 22194.10 368583 22194.10 368583 7 21131.95 21132.27 435015 21132.27 435015 8 21969.85 21969.39 340193 21969.39 340194 9 21949.69 21948.03 437433 21948.02 437433 10 21116.18 21117.87 415434 21117.86 415434 11 22280.21 22276.36 321704 22276.36 321704 12 21033.21 21033.48 425515 21033.49 425515 13 21791.20 21789.77 321563 21789.77 321564 14 21264.52 21262.50 423523 21262.50 423523 15 20569.65 20569.06 345330 20569.05 345331 16 21754.28 21753.25 295008 21753.26 295008 17 21718.28 21717.08 391610 21717.08 391611 18 22280.85 22280.29 392562 22280.28 392562 19 21773.03 21773.60 404994 21773.61 404995 20 21755.93 21756.81 427902 21756.81 427903 21 22262.70 22263.09 306629 22263.08 306629 22 21462.05 21457.68 342772 21457.67 342773 23 22159.20 22157.54 383534 22157.54 383534 24 22393.57 22393.55 381479 22393.55 381479 25 20488.13 20491.84 355906 20491.84 355907 26 21115.20 21112.83 302759 21112.82 302760 27 21837.08 21833.08 326493 21833.08 326493 28 21457.13 21459.39 387506 21459.39 387506 29 21556.88 21556.29 325142 21556.29 325143 30 21549.00 21549.97 398870 21549.93 398871 31 21542.62 21542.50 366954 21542.51 366955 32 21142.55 21143.76 324399 21143.75 324400 33 22030.74 22031.05 366341 22031.05 366341 34 21940.78 21941.01 338800 21941.01 338801 35 21732.68 21733.38 375625 21733.37 375626 36 21127.29 21134.52 372133 21134.52 372134 37 21298.20 21297.79 385756 21297.79 385757 38 20843.33 20848.98 314216 20648.99 314216 39 22320.38 22318.95 439907 22318.94 439907 40 21277.65 21275.57 455254 21275.57 455255 41 21285.03 21284.64 378870 21284.63 37e870 42 21785.73 21787.33 320947 21787.33 320947 43 22363.77 22364.07 415439 22364.07 415440 44 22416.03 22416.39 386259 22416.39 386259 45 21706.61 21707.04 431039 21707.04 431040 46 22221.19 22227.27 379272 22227.77 379272 47 21421.71 21419. 14 374690 21419.14 374691 48 22104.85 22104.38 349350 22104.37 349350 49 21943.27 21943.34 454156 21943.34 454156 50 21941.61 21940.70 384738 21940.70 384738 mean 21655.9 21655.8 374684 21655.8 374685 std. dev. 477.389 476.780 42673.5 475.779 42673.5 V = 21558.8 oy2 = 325782 62 APPENDIX IV REQUIRED SAMPLE SIZES AT VARIOUS COST RATIOS For a given inventory s i t u a t i o n , the optimal num-bers of sample units to remeasure m and to replace n w i l l vary with the r a t i o of the costs of these p l o t s , c m / c n ° The constraints i n the cost function optimization, when simultan-eously estimating volume and growth involve the s p e c i f i e d sampling errors squared, K and K . This makes a general solu-t i o n impractical. A s p e c i f i c case, Example One of t h i s t h e s i s , was analyzed to convey an impression of how the sample sizes w i l l vary. The parameters and confidence i n t e r v a l half-widths are as s p e c i f i e d f o r that example. The r e s u l t s of the analy-s i s are presented both numerically (Table 5) and gr a p h i c a l l y (Figure 1). Table 5. Optimal sample sizes at various cost r a t i o s . c /c m n m n 1 : 1 86 6 2 : 1 74 25 3 : 1 65 46 4 : 1 61 60 8 : 1 49 122 63 FIGURE 1. The optimal numbers of sample plots to remeasure and to replace at various cost r a t i o s , ea co 01 a CQ CO H— -1°. (0 U J z: UJ aim LU a: tH a CD. a o 8:1 4:1 3:1 2:1 1:1 i 1 1 — 15.0 31.0 47.0 63.0 REMEASURED PLOTS 79.0 9SJ 

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