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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 A r i z o n a U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the F a c u l t y of Forestry  We accept t h i s t h e s i s as conforming required  THE  t o the  standard  UNIVERSITY OF BRITISH COLUMBIA June, 1974  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  further  for  agree  scholarly  by  his  of  this  thesis at  it  may  representatives.  written  for  financial  is  June  26,  1974  of  Columbia,  British  by  for  gain  Columbia  shall  the  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  Forestry  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  for extensive  be g r a n t e d  It  fulfilment  available  permission.  Department o f  Date  freely  permission  purposes  thesis  partial  the U n i v e r s i t y  make that  in  of  this  be a l l o w e d  or  that  study. thesis  my Department:  copying  for  or  publication  without  my  ii  ABSTRACT F o r e s t sampling i s conducted t o determine c o n d i t i o n s and t r e n d s o f change.  current  When c u r r e n t c o n d i t i o n s are  e s t i m a t e d , most of the commonly used sampling d e s i g n s s p e c i f y the s p a t i a l d i s t r i b u t i o n o f sample u n i t s . change i s d e s i r e d , s e v e r a l schemes may  When e s t i m a t i o n o f  be employed.  Some are  combined w i t h c u r r e n t c o n d i t i o n i n v e n t o r i e s ; some are i n d e pendent.  The former are r e l a t i v e l y i m p r e c i s e ; the  latter  r e l a t i v e l y expensive. A system o f temporal d i s t r i b u t i o n o f sample u n i t s , sampling on s u c c e s s i v e o c c a s i o n s w i t h p a r t i a l replacement o f sample u n i t s , has been developed f o r simultaneous e s t i m a t i o n of c u r r e n t c o n d i t i o n s and t r e n d s .  As the emphasis i s on time,  r a t h e r than a r e a , t h i s system o p e r a t e s w i t h c o n v e n t i o n a l sampl i n g designs to increase t h e i r  efficiency.  T h i s study i n v e s t i g a t e d the t h e o r y o f sampling w i t h p a r t i a l replacement t o e s t a b l i s h the v a l i d i t y o f the c l a i m s o f i n c r e a s e d e f f i c i e n c y i n comparison w i t h c o n v e n t i o n a l s y s tems.  Three cases are examined, through example, f o r e s t i -  mation o f mean volume per a c r e and growth i n volume p e r a c r e . Sample s i z e s and c o s t s are developed f o r the s i t u a t i o n o f simple random sampling o f both f i n i t e and i n f i n i t e p o p u l a t i o n s . The comparisons  are f a v o r a b l e t o the proposed system.  The  p o s s i b i l i t i e s o f u s i n g t h i s system w i t h two r e c e n t d e v e l o p ments i n c r u i s i n g t e c h n i q u e s are e x p l o r e d .  Finally,  the  I l l  applicability  of this  system t o B r i t i s h Columbia f o r e s t  sur  veys i s examined.  A case i s made f o r t r a n s i t i o n  o f the ex-  isting provincial  system t o sampling w i t h p a r t i a l  replace-  ment.  iv  TABLE OF CONTENTS Page LIST OF TABLES  vi  LIST OF FIGURES  v i i  SELECTED SYMBOLS  viii  Chapter INTRODUCTION  1  ONE  THE DEVELOPMENT OF SPR  5  TWO  THE DATA BASE AND METHODS OF ANALYSIS  8  THREE  THEORY  FOUR  FIVE  E s t i m a t i o n o f a Mean  14  E s t i m a t i o n o f Growth  18  Optimum Sample S i z e  23  COMPARISONS OF SPR WITH CONVENTIONAL INVENTORY Example One  29  Example Two  35  Example Three  40  BIAS IN SPR ESTIMATES  43  CONCLUSIONS  47  FUTURE CONSIDERATIONS  51  BIBLIOGRAPHY  55  APPENDIX I APPENDIX I I  COMPUTING FORMULA FOR VOLUME OF HEIGHT-MEASURED TREES  58  REGRESSION COEFFICIENTS FOR VOLUME COMPUTATION FROM /(D.B.H.O.B.)  59  V  APPENDIX I I I COMPUTED STATISTICS OF FIFTY RANDOM SPR SAMPLES . . . . . . . . . . . APPENDIX IV  REQUIRED SAMPLE SIZES AT VARIOUS PLOT COST RATIOS  61 62  vi  L I S T OF  TABLES  Table  Page  1.  P a r a m e t e r s f o r example  2.  Comparison o f c o s t s a n d SPR  3.  inventories.  two.  29  independent  . . . . . . . . . . .  . . . . . . . . . . . . . . .  Comparison o f c o s t s and SPR  5.  between  . . . . . . . . .  32  P a r a m e t e r s and s a m p l i n g e r r o r s f o r example  4.  one.  between  inventories  O p t i m a l sample s i z e s a t cost  ratios.  35  independent  f o r example  two.  . . .  38  various  . . . . . . . . . . . . . . .  62  vii  L I S T OF  FIGURES  Figure 1.  Page The o p t i m a l numbers o f s a m p l e p l o t s t o r e m e a s u r e and t o r e p l a c e a t various cost ratios.  . . . . . . . . . . .  63  viii  SELECTED SYMBOLS  MEANING the true p o p u l a t i o n v a r i a n c e o f the volume a t the i n i t i a l o c c a s i o n .  plot  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 volume a t a subsequent o c c a s i o n .  plot  the true p o p u l a t i o n c o r r e l a t i o n coeff i c i e n t between p l o t volumes o b s e r v e d on t h e 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 = Covariance {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 f i c i e n t of the i t h v a r i a b l e ;  = Covariance  coef-  (XY)/a . 2  the ith  t r u e a v e r a g e volume per occasion.  the  change i n y between  acre at  the  occasions.  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 z e r o and v a r i a n c e a , i . e . 2  £ .^N(0,a ) . v 2  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 c a n b e u s e d a s a t e s t o f Ho°.&. = 0 in linear regression). ^  an SPR e s t i m a t e o f a c r e a t the second  t h e mean v o l u m e occasion.  the  of  error variance  per  y.  an e s t i m a t e o f t h e g r o w t h i n v o l u m e a c r e between s u b s e q u e n t p e r i o d s . the  error variance of  per  g.  t h e 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 a r e n o t remeasured the subsequent o c c a s i o n . t h e number o f i n i t i a l l y e s t a b l i s h e d s a m p l e p l o t s t h a t a r e r e m e a s u r e d on subsequent occasion.  on  the  ix  SYMBOL  PAGE FIRST USED  MEANING  n  14  the number o f new p l o t s e s t a b l i s h e d a t a subsequent o c c a s i o n t o r e p l a c e those n o t measured.  u>  25  a Lagrangian f u n c t i o n used i n a t e c h nique t o determine extremum o f a f u n c t i o n / w i t h c o n s t r a i n t g.  X  25  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 o f t h e f u n c t i o n / w i t h r e s p e c t to x{_; a l l o t h e r v a r i a b l e s remain c o n s t a n t .  K  25  t h e s p e c i f i e d sampling e r r o r volume p e r a c r e .  2  K  28  the s p e c i f i e d sampling e r r o r  2  a  L a g r a n g i a n 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 c o n s t r a i n t g.  growth between p e r i o d s . ^  28  g r e a t e r than o r e q u a l t o .  <_  28  l e s s than o r e q u a l t o .  E( )  15  t h e expected v a l u e .  E  15  t h e summation o f .  o f mean of  X  ACKNOWLEDGEMENT  The  acknowledgement o f  a l l t h o s e p e r s o n s who  contributed  i n some manner t o t h e b a c k g r o u n d  w o u l d be  impossible  an  thanks to c e r t a i n ence. for  First,  their  I  comments and  Julien  the  Kerr  special  Hejjas,  the  during  my  t o t h a n k my  the F a c u l t y for their  and  influ-  this  thesis  committee  J.H.G. S m i t h , of F o r e s t r y  Kozak and  year of  help  T h e y a r e my  Faculty  thesis  special  reviewed  Munro, and  thank Drs.  opportunity  s t a f f of  and  suggestions.  Kozak, D.D.  wish to  support  take t h i s and  A.  for this  I wish to give  for their  Demaerschalk of  especially  personal  persons  But  I w i s h t o t h a n k t h o s e who  members, D r s . Dr.  task.  have  study  a t U.B.C.  of F o r e s t r y ,  acceptance of  a t U.B.C.  Munro f o r  f e l l o w graduate  and  their I must  students  especially and  Ms.'s  assistance  to  me. Without data, Herman o f  the  vice,  kind  But  was  without  Forestry  I could not Sciences  support,  m i g h t h a v e gone f o r n a u g h t . Research C o u n c i l of  computing  Laboratory,  enough t o p r o v i d e  financial  of  proceed.  F a c u l t y o f F o r e s t r y was  a l l these other  I wish to  Ser-  ingredient. ingredients National  i n defraying support.  enough t o p r o v i d e  R.  Forest  thank the  p a r t i a l personal  kind  U.S.  this necessary  Canada f o r s u p p o r t  s e r v i c e s and  Francis  the  costs  The  further  support  xi  for  t h o s e same n e e d s .  Bloedel Limited which  I wish  for their  to especially  thank M a c M i l l a n  Fellowship i n Forest  Mensuration  p r o v i d e d t h e b u l k o f my p e r s o n a l s u p p o r t and was  mental  i n t h e p u r s u i t o f my  studies  a t U.B.C.  Finally,  w i s h t o t h a n k Ms. J e a n W i l l i a m s o n f o r t h e e x c e l l e n t of  this  thesis.  instruI  typing  INTRODUCTION In p a s t decades, the f o r e s t i n d u s t r y was cerned w i t h d e t e r m i n i n g growth and y i e l d o f f o r e s t w i t h much p r e c i s i o n . ^ " ages o f old-growth  resources  In those times, t h e r e were v a s t a c r e -  timber  still  l o c a t i o n o f the timber was But now,  not con-  available f o r harvest.  w e l l known, i t was  The  "out t h e r e " .  much of t h i s type o f timber r e s o u r c e has been c u t  over and some o f the t i m b e r l a n d c o n v e r t e d t o o t h e r u s e s . t h i s same span o f time, new been developed  In  uses f o r wood and wood f i b e r have  which have c r e a t e d new  f o r e s t product  markets.  These and o t h e r f a c t o r s have i n c r e a s e d the p r e s s u r e s on a d i m i n i s h i n g p r o d u c t i o n base.  O r g a n i z a t i o n s must now  prac-  t i c e more i n t e n s i v e management and base t h e i r h a r v e s t i n g dec i s i o n s on more p r e c i s e i n f o r m a t i o n a t reduced The q u e s t i o n s o f where the t r e e s a r e , how of  expenditure.  much o f what k i n d s  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 r e q u i r e s a c c u r a t e i n f o r m a t i o n , e.g. i n ~ f o r m a t i o n which i s r e p r e s e n t a t i v e of the t r u e p o p u l a t i o n mean y. However, due t o some techniques employed, b i a s e s may be p r e s e n t i n our o b s e r v a t i o n s which are unavoidable or unknown. T h e r e f o r e our i n f o r m a t i o n w i l l i n f a c t be p r e c i s e r a t h e r than a c c u r a t e , i . e . s u c c e s s i v e measurements w i l l v a r y about a mean m, which d i f f e r s from y by the amount o f b i a s p r e s e n t . Refinement o f our techniques may reduce o r even e l i m i n a t e these b i a s e s , but t h i s i s d i f f i c u l t t o gauge. The technique under study i s t h e o r e t i c a l l y unbiased, t h e r e f o r e , we c o u l d use the term a c c u r a c y . However, the term p r e c i s i o n w i l l be the normal usage t o a v o i d the c o m p l i c a t i o n o f e x p l a i n i n g a t each s t e p t h a t b i a s from a n c i l l a r y c o n s i d e r a t i o n s w i l l cause the e s t i mates t o be p r e c i s e r a t h e r than a c c u r a t e . F o r f u r t h e r d i s ~ c u s s i o n of t h i s t o p i c , the reader i s r e f e r r e d to Cochran and Cox (1957) or Cochran (1963).  2  is  o c c u r r i n g are v i t a l t o management d e c i s i o n s . Davis  (1966)  s t a t e d t h a t . . . "accurate d e t e r m i n a t i o n o f growth becomes n e c e s s a r y , because c r u c i a l d e c i s i o n s r e s t d i r e c t l y on i t " . The  same i n f o r m a t i o n requirements  other questions.  The  c i s i o n a t reduced  expenditure  vey^  e x i s t f o r answering the  s o l u t i o n t o the paradox of g r e a t e r p r e l i e s i n use o f good f o r e s t  sur-  techniques. The  s o l u t i o n o f the mechanics o f f o r e s t survey  been the aim o f many i n v e s t i g a t o r s .  has  Many and v a r i e d schemes  e x i s t , i n f a c t , they seem as numerous as the problems they attempt t o s o l v e .  To an e x t e n t , t h i s i s c o r r e c t , because  each i n v e n t o r y problem i s unique, may is  but c e r t a i n methodology  be a p p l i e d t o many s i t u a t i o n s .  One  general  situation  the e s t i m a t i o n o f the c u r r e n t volume of an a r e a .  use s y s t e m a t i c sampling,  stratified  random sampling,  s i b l y m u l t i - s t a g e c l u s t e r sampling.  might  o r pos-  In most c a s e s , the sample  has been a one-time event w i t h l i t t l e given to future surveys.  One  or no c o n s i d e r a t i o n  T h i s approach s i m p l i f i e d  analyses,  s i n c e independent samples g i v e good i n f o r m a t i o n w i t h few complex s t a t i s t i c a l  manipulations  needed.  Growth e s t i m a t i o n p r e s e n t e d o t h e r problems. pendent samples on two  o c c a s i o n s g i v e l a r g e r v a r i a n c e of the  e s t i m a t e than i f a l l o r i g i n a l p l o t s were remeasured factor of 1 / ( 1 - P ) ] . 2  Inde-  [by a  But permanent p l o 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 i n t e r c h a n g e a b l y and w i l l have the d e f i n i t i o n as g i v e n t o f o r e s t survey by the S o c i e t y o f American F o r e s t e r s ( 1 9 5 8 ) .  3  establish  and  relocate.  Further,  c u r r e n t volume e s t i m a t i o n additional  information  size distribution these  conflicts  p r o b l e m s and s y s t e m s and m a t i o n on  tained  i n a stand.  increase  during  results  the  costs  to  or  involved in  questions  can  sampling with  (SPR)  manent p l o t s and sampling it  replacement of  temporary  c u r r e n t v o l u m e on  others,  will  a linking  the  give  pendent surveys.  of  increased The  ob-  growth  been d e v e l o p e d  and  This  sampling  use  scheme, u n i t s on  termed succes-  of both remeasured  over  time.  successive plots,  that  yield  SPR  i s not  sample p l o t s s p a t i a l l y ,  distribution  initial  and  A l l such methods  independent p l o t s .  s e c o n d and  f o r growth  inventory.  i n v o l v e s the  i s concerned with  some o f  large.  cost saving.  scheme t o ' d i s t r i b u t e  ment o f  at  esti-  amount o f i n -  g i v e p r e c i s e a n s w e r s t o g r o w t h and  partial  sive occasions  system  such e s t i m a t o r s  a s y s t e m has  at a considerable  solved  inventory  independent surveys,  stand  tree  f o r c u r r e n t volume  i n c r e a s i n g the  other  In r e c e n t y e a r s ,  and  the a d d i t i o n a l  different  a permanent p l o t  the  the  composition  Some a c c e p t e d  completely  and  yielding  Different organizations  Others r e s o r t e d to  gathered  purportedly  species  used independent surveys  each o c c a s i o n  the  two  permanent p l o t s f o r  drawback o f n o t  i n s e v e r a l ways.  increment cores  applied  the  about the  expense o f  estimation. formation  has  using  with  occasions, the  p r e c i s i o n over a  remeasured p l o t s are  r e l a t i o n s h i p between p e r i o d s .  For  pera  rather,  estimating remeasure-  replacement s e r i e s of  of  inde-  used to e s t a b l i s h  This r e l a t i o n  is  4  a p p l i e d t o the r e p l a c e d p l o t s and an e s t i m a t e based p l o t s measured on a l l o c c a s i o n s i s formed.  The  on a l l  replacement  p l o t s can be b a c k - l i n k e d a g a i n through the remeasured Thus, a new  e s t i m a t e o f volume can be formed f o r any  o c c a s i o n , a g a i n based  on a l l measured p l o t s .  Since  plots. previous  informa-  t i o n gained by d i r e c t measurement i s c o n s i d e r e d more r e l i a b l e than l i n k e d i n f o r m a t i o n , the i n d i v i d u a l e s t i m a t e s are by t h e i r r e l a t i v e importance.  weighted  S e v e r a l e s t i m a t o r s o f growth  can be formed from the e s t i m a t e s o f volume on s u c c e s s i v e occasions. of  An e f f i c i e n t e s t i m a t o r i s formed from the  estimate  c u r r e n t volume and the r e v i s e d e s t i m a t e on the p r e v i o u s  occasion. Through use o f the e x p r e s s i o n s f o r the e r r o r  vari-  ances o f the c u r r e n t volume and growth and a l i n e a r c o s t f u n c t i o n f o r t o t a l e x p e n d i t u r e , the optimum numbers o f p l o t s t o remeasure and t o r e p l a c e can be determined. s i t u a t i o n s can be handled.  Two  different  I f a s p e c i f i e d amount o f money i s  a v a i l a b l e , the optimum numbers f o r minimum sampling  errors  i n both growth and volume d e t e r m i n a t i o n can be found. v e r s e l y , optimum numbers can be found t o minimize s p e c i f i e d sampling e r r o r s .  This feature allows  i n d e s i g n i n g surveys f o r changing goals.  Con-  costs at flexibility  s i t u a t i o n s and management  CHAPTER ONE THE  DEVELOPMENT OF SPR  SPR i s n o t new to i n v e n t o r y s i t u a t i o n s . c u l t u r e , Jessen  (1942) was the f i r s t i n v e s t i g a t o r t o r e c o g -  nize i t s potential. ling  In a g r i -  Many a u t h o r i t i e s i n the f i e l d  o f samp-  ( P a t t e r s o n 1950, Cochran 1963, Hansen et a l . 1953, K i s h  1965)  have d e s c r i b e d the t h e o r y and use o f SPR i n survey  sampling.  However, v a r i o u s r e s t r i c t i o n s and s p e c i a l  were p r e s e n t e d i n these d e s c r i p t i o n s , and no g e n e r a l t i o n o r f o r e s t r y a p p l i c a t i o n s were d i s c u s s e d . 1959) ies,  f i r s t proposed  cases situa-  Bickford  (1956,  u s i n g SPR i n e x t e n s i v e f o r e s t i n v e n t o r -  and l a t e r r e p o r t e d on the r e s u l t s o b t a i n e d i n the n o r t h -  eastern United States.  The f i r s t p r e s e n t a t i o n o f a g e n e r a l  u n i f i e d t h e o r y o f SPR came from s e p a r a t e independent  inves-  t i g a t i o n s by two r e s e a r c h e r s , K. Ware and T. Cunia. Ware (1960) wrote h i s d o c t o r a l d i s s e r t a t i o n on t h e g e n e r a l t h e o r y , while Cunia's r e s u l t s came from h i s work as Research t r i c i a n f o r Canadian I n t e r n a t i o n a l Paper C o r p o r a t i o n . 1962,  these two authors c o l l a b o r a t e d on a j o i n t  on the t h e o r y o f SPR and o p t i m a l replacement  BiomeIn  publication  p o l i c y f o r the  g e n e r a l case o f simple random sampling on two o c c a s i o n s  (Ware  and Cunia 1962). Cunia has s i n c e c o n t i n u e d h i s r e s e a r c h i n t h i s field.  He has extended  SPR theory from t h e simple l i n e a r t o  the m u l t i p l e l i n e a r r e g r e s s i o n case  (Cunia 1965) .  Multiple  6  r e g r e s s i o n SPR  theory has many important r a m i f i c a t i o n s .  For  example, u s i n g m u l t i p l e r e g r e s s i o n w i t h dummy v a r i a b l e s per and Smith 1966,  Cunia 1973), the s e p a r a t e s t r a t a from  s t r a t i f i e d random sample c o u l d be s i m u l t a n e o u s l y With simple l i n e a r r e g r e s s i o n SPR to  be c o n s i d e r e d s e p a r a t e l y .  r e g r e s s i o n SPR  (Dra-  analyzed.  t h e o r y , each stratum  had  More i m p o r t a n t l y , m u l t i p l e  theory has allowed the use o f many stand v a r i -  a b l e s t o improve the c o r r e l a t i o n between p e r i o d s i n the ing  expressions.  bility  a  Another important r e s u l t was  link-  the a p p l i c a -  of m u l t i p l e r e g r e s s i o n to m u l t i p l e o c c a s i o n  SPR.  Through the m u l t i p l e r e g r e s s i o n and m a t r i x m a n i p u l a t i o n ,  link-  ages between p l o t s measured on v a r i o u s o c c a s i o n s 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 p e r i o d , p l o t s measured on the f i r s t o n l y , the and t h i r d , second to  and t h i r d , etc.  first  o c c a s i o n s can be combined  form an e s t i m a t e o f volume a t any o f the t h r e e o c c a s i o n s .  As e x p l a i n e d e a r l i e r ,  t h i s a l l o w s the e s t i m a t e t o be  based  on many more p l o t s than were measured on a s p e c i f i c o c c a s i o n . Newton  (1971) examined the m u l t i p l e r e g r e s s i o n case  and h y p o t h e s i z e d t h a t m u l t i v a r i a t e sampling e f f i c i e n t through use o f SPR. s i o n o f SPR  c o u l d be more  He a l s o proposed  the  exten-  techniques t o the f a m i l i a r case o f double  samp-  l i n g u s i n g a e r i a l photographs f o r l a r g e - s c a l e f o r e s t s u r v e y s . He f e l t  t h a t the a d d i t i o n of singly-measured  ground p l o t s  c o u l d i n c r e a s e the p r e c i s i o n o f the a i r p h o t o p l o t e s t i m a t e s . Reports o f t r i a l s o f SPR  i n simulated  (Nyyssonen 1967)  and  7  actual  (Frayer et at.  t h a t SPR  1971)  i n v e n t o r y s i t u a t i o n s have shown  i s an e f f e c t i v e system i n s a v i n g both time and  ex-  pense. Recent l i t e r a t u r e r e g a r d i n g o f SPR  has  the theory  s t a t e d that considerable savings  and  use  i n time and  pense w i t h a h i g h degree of p r e c i s i o n i n e s t i m a t i o n can realized.  S i n c e most budget a n a l y s t s i n an  expense item w i t h no v i s i b l e r e t u r n , the v a l i d i t y of s h o u l d be  investigated.  of t h i s t h e s i s i s t o i n v e s t i g a t e SPR the v a l i d i t y  i n c e r t a i n cases.  and  The o b j e c t i v e  attempt t o e s t a b l i s h  looked a t through example, w i t h c o s t s o f SPR  ing techniques,  and  Finally,  be  equivalent  Further, c e r t a i n c r u i s -  as they might be employed w i t h i n an SPR  work, w i l l be d i s c u s s e d .  an  claims  S e v e r a l such cases w i l l  normal i n v e n t o r y systems compared.  be  organization's  f i n a n c i a l c o n t r o l department look on i n v e n t o r y c o s t s as  o f reduced e x p e n d i t u r e  ex-  the a p p l i c a b i l i t y o f  frameSPR  to f o r e s t r y p r a c t i c e s i n B r i t i s h Columbia w i l l be g i v e n a cursory  look.  1  CHAPTER THE  DATA BASE AND  TWO  METHODS OF ANALYSIS  The d a t a f o r these a n a l y s e s were f u r n i s h e d by 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. Washington, U.S.A.  the  F o r e s t S e r v i c e , Olympia,  These data c o n s i s t o f r e c o r d e d measure-  ments from permanent sample p l o t s e s t a b l i s h e d i n the Cascade Head E x p e r i m e n t a l F o r e s t near O t i s , Oregon  (Madison  1957).  The s p e c i e s found i n t h i s stand i n c l u d e S i t k a spruce  {Picea  sitchensis  hetero-  phylla tsuga of  (Bong.) Carr.) and western  hemlock  (Tsuga  (Raf.) Sarg.) w i t h a few s c a t t e r e d D o u g l a s - f i r menziesii  (Mirb.) Franco v a r . m e n z i e s i i ) .  The  forest  t h i s l o c a l e regenerated n a t u r a l l y f o l l o w i n g a large  thought t o have o c c u r r e d i n the 1850's.  In 1935,  (Pseudo-  the  fire U.S.  F o r e s t S e r v i c e e s t a b l i s h e d e l e v e n one-acre permanent sample p l o t s i n the E x p e r i m e n t a l F o r e s t . Complete enumeration  o f the t r e e s growing  on  these  p l o t s was made by F o r e s t S e r v i c e p e r s o n n e l and the f o l l o w i n g measurements taken: at  4.5  d.b.h.o.b., the diameter o u t s i d e bark  f e e t above g e r m i n a t i o n p o i n t ; crown c l a s s ,  following  the f o u r c l a s s system of dominant, codominant, i n t e r m e d i a t e , and overtopped  (Smith 1962); s p e c i e s of each t r e e ; and  h e i g h t , which was measured f o r 15-20  p e r c e n t of the t r e e s  a c r o s s the diameter range of the stand. ment o f d.b.h.o.b. was  total  The p o i n t of measure-  marked w i t h a metal tag which c a r r i e d  a number f o r f u t u r e i d e n t i f i c a t i o n .  Remeasurement o c c u r r e d  9  every f i v e y e a r s u n t i l i n 1968.  1950.  A f i n a l remeasurement o c c u r r e d  These measurements, were coded onto e l e c t r o n i c d a t a  p r o c e s s i n g cards and v e r i f i e d . for this  T h i s then was  the data base  analysis. The g e n e r a l procedure  f i r s t the computation  f o r the a n a l y s i s i n v o l v e d  o f volume f o r each t r e e i n c u b i c f e e t  f o r s e v e r a l measurement p e r i o d s .  The e l e v e n p l o t s were then  d i v i d e d i n t o s u b - p l o t s t o c r e a t e a p o p u l a t i o n f o r the sampl i n g experiments.  Summary volumes f o r each p l o t were c a l c u -  l a t e d which s e r v e d as the d a t a f o r computation t i o n parameters meters,  o^ , 2  o^ , z  p,  plf  and u 2 .  of the p o p u l a -  Given these p a r a -  the v a r i o u s examples c o u l d be f o r m u l a t e d and  analyzed.  As a l l t r e e s were not measured f o r h e i g h t , a method was  r e q u i r e d t o r e l a t e volume t o diameter.  In p r e v i o u s F o r -  e s t S e r v i c e a n a l y s e s , h e i g h t - d i a m e t e r curves were developed and h e i g h t s were read from these c u r v e s f o r each t r e e . umes were then r e a d from a l i n e m e n t c h a r t s .  In t h i s  a d i f f e r e n t and h o p e f u l l y sounder method was velopment of f  (volume) - /  used,  (d.b.h.o.b.) e q u a t i o n s .  v e l o p these e q u a t i o n s , f i r s t volumes had t o be  analysis, the deTo  de-  calculated  f o r the t r e e s f o r which h e i g h t s had been measured. e s t Survey  Vol-  The  For-  group of the U.S.F.S. has e x t e n s i v e l y i n v e n t o r i e d  the same type o f stands i n the same a r e a .  From these s u r -  veys, they have developed e m p i r i c a l e q u a t i o n s f o r t h e c a l c u l a t i o n of c u b i c volume from d.b.h.o.b., h e i g h t , and form c l a s s .  Girard  But no form c l a s s measurements e x i s t e d f o r t h i s  10  stand. tion age  Again  from  the F o r e s t Survey  their  form c l a s s  species,  inventory data.  then used  Various sized  local  f o r the  f  classes  surveys.  to c a l c u l a t e  linear  and  f o r each  the  tree  of B r i t i s h  the l i n e a r models, tion  procedure  of F o r e s t r y .  Dr.  A.  N o n - l i n e a r parameter  o f t h e U.B.C. C o m p u t i n g Squares  IBM  programme.  o u t on  a backward  values,  and  residual  relationships. Log b  3  1 0  '  The  (volume) = b  elimina-  of estimates  (SE„),  hi  coefficients, plots  partial  to determine  equations adopted  F and  the  "best" unbiased  were o f t h e  For  a s p e c i f i c measurement p e r i o d ,  one  f o r western  Douglas-fir."*"  g a v e t h e l o w e s t SE_  and  homogeneity of v a r i a n c e .  2  *  form  (d.b.h.o.b.) .  s p r u c e and  • d.b.h.o.b. + b  overall F  + bj  3  use  of the equations generated i n -  0  e q u a t i o n s were u s e d , Sitka  For  e s t i m a t i o n came f r o m  of standard errors  correlation  the  K o z a k o f t h e U.B.C. F a c u l t y  • multiple  The  C e n t r e ' s BMD:X85 N o n - l i n e a r L e a s t  Analysis  v o l v e d comparisons  hypothe-  370/168 c o m p u t e r .  the analyses u t i l i z e d by  volume  (d.b.h.o.b.) r e l a t i o n s h i p .  Columbia  developed  the  volumes.  r e g r e s s i o n a n a l y s e s o f t h e s e models were c a r r i e d University  averof  A F o r e s t Survey  n o n - l i n e a r models were  f  (volume) -  informa-  These d a t a p r o v i d e d  by v a r i a b l e d i a m e t e r  again from  p r o g r a m was  p r o v i d e d the needed  (d.b.h.o.b.) *  h e m l o c k and  T h i s model n e a r l y  2  one  analysis  for  always  t h e r e s i d u a l s were u n b i a s e d An  two  of covariance  and  showed  indicated  T h e r e were o n l y 12 D o u g l a s - f i r t r e e s on t h e e n t i r e 11 a c r e s sampled. Of t h e s e 12, o n l y a few were m e a s u r e d 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 n o t g i v e a s u i t a b l e d a t a 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 Douglasf i r were c o m b i n e d w i t h t h e S i t k a s p r u c e .  11  the equations Within  were s i g n i f i c a n t l y  species  and  were i n some c a s e s  between t i m e  different  f i t of  nificant e^'s  log  non-significantly  different,  f o l l o w the  For sampling was  the  division.  examples,  s i n c e no I t was  of permanent p l o t link  SPR  lognormal  but  these  f o r goodness  i n d i c a t e d no  the requirement  that  hypothesized  e n u m e r a t i o n was As  size.  A minor  the  this  that the standard  f o l l o w e d so  that  method  twenty-  t r e e d a t a were s e q u e n t i a l l y adjacent  these  gave  i n g v i e w s on sources  A review  of p e r t i n e n t l i t e r a t u r e  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 .  reported effects,  some d i d n o t .  o t h e r m e t h o d s seemed u n d u l y rectangular  t o t a l number o f constituted was  not  buted  among t h e  allotted  t r e e s t o be  t r e e s on  divisible  by  t o each p l o t  plots.  was  T h i s was  event,  since acre  a twenty-link  side.  total  the  twenty.  number o f  r e m a i n d e r was  This trees  distri-  done r a n d o m l y and  Once t h e number o f  determined,  Some  one-twentieth  d i v i d e d by  I f the  along  conflict-  placed into a plot,  twenty, the  twenty p l o t s .  of assigned  with  t h e a c r e was  t h e minimum number.  evenly  replacement  of  In any  complicated,  p l o t s were h y p o t h e s i z e d  To d e t e r m i n e a c o u n t  the  problem  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 strips.  the  s u b - p l o t s were n e e d e d f o r  s t e m maps e x i s t e d t o a i d i n  s t r i p s were c r u i s e d .  sig-  distribution.  p o p u l a t i o n to have a u s e a b l e  encountered  plot  Tests  t o the normal d i s t r i b u t i o n  d i f f e r e n c e which s a t i s f i e d  should  species.  p e r i o d s , slopes or i n t e r c e p t s  c o n d i t i o n s never e x i s t e d simultaneously. of  between  the  with  trees  assignment of  trees  12  was  made s e q u e n t i a l l y .  biased r e s u l t s .  One  I t i s hoped t h a t t h i s method gave  un-^  major o b j e c t i o n to t h i s method would  i n v o l v e the clumping of t r e e s .  Personal  visits  to these  p l o t s showed t h a t the t r e e s were r e a s o n a b l y evenly d i s t r i b u t e d , so o n l y minor d e v i a t i o n s from the s p e c i f i e d assignments might occur. f  The  a c t u a l s u b d i v i s i o n and  t r e e 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 a c r e , v a r i a n c e w i t h i n p l o t s , mean volume per  tree, variance  within  i n g c o e f f i c i e n t s of v a r i a t i o n . *SIMCORT programme was between  The  t r e e s , and  correspond-  U.B.C. Computing C e n t r e ' s  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  periods. The  example analyses  ments f o r the c l a s s i c a l and Sample s i z e d e t e r m i n a t i o n  compare sample s i z e r e q u i r e -  the c o r r e s p o n d i n g SPR  f o r t h e c l a s s i c a l cases  the p o p u l a t i o n  variances  the SPR  a l i n e a r c o s t f u n c t i o n was  cases,  and  l i n g e r r o r s as c o n s t r a i n t s .  The  optimized  survey c o s t s were a l s o r e q u i r e d and $50.00 f o r remeasured p l o t s and  For  w i t h samp-  sampling e r r o r s were s p e c i -  f l e c t management o b j e c t i v e r e q u i r e m e n t s .  hopefully  re-  Estimates of p l o t  f o r the d e t a i l e d  analyses,  $25.00 f o r temporary p l o t s  reasonable values.  In the o p t i m i z a t i o n  t i n e s , p l o t c o s t s are used as r a t i o s , comparisons between the two  involved  s p e c i f i e d sampling e r r o r s .  f i e d as percentage p r e c i s i o n requirements and  were c o n s i d e r e d  inventories.  i.e.  2:1.  types o f i n v e n t o r y ,  In the these  roucost  costs  13  were used d i r e c t l y .  The c o s t s g i v e n i n the analyses  should  t h e r e f o r e be viewed as order of magnitude o n l y , as p l o t c o s t s do v a r y .  Because of t h i s v a r i a b i l i t y ,  sample s i z e s f o r c o s t  r a t i o s of 1:1, 2:1, 3:1, 4:1, and 8:1 a r e t a b u l a t e d dix  IV).  The reader may  his  own needs.  (Appen-  thus e x t r a p o l a t e c o s t s as may f i t  For c u r r e n t volume o n l y , SPR sample s i z e s were  computed on a desk c a l c u l a t o r .  When simultaneous e s t i m a t i o n  of c u r r e n t volume and growth was r e q u i r e d , the U.B.C. Computing  Centre programme COMPLX i n the p u b l i c f i l e  used.  T h i s programme i s a t y p i c a l n o n - l i n e a r  *NUMLIB was optimization  r o u t i n e with p r o v i s i o n s f o r c o n s t r a i n i n g equations. programmes should be a v a i l a b l e through most computer ties.  Similar facili-  (•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 c l o s e l y follow those given by Ware and Cunia (1962).  The i n c l u s i o n of this section should  be b e n e f i c i a l to those readers who do not have a working knowledge of this subject.  In areas where the algebraic manipula-  tions involved would become tedious and r e p e t i t i o u s ,  these  steps have been eliminated to improve the r e a d a b i l i t y .  For  the exact procedures involved, the interested reader i s r e ferred to K. Ware's doctoral d i s s e r t a t i o n (Ware 1960). We wish to estimate for a population, e.g. a f o r est stand, these parameters:  y  l f  the true average volume per  sampling u n i t at the f i r s t occasion; p / the true average v o l 2  ume per sampling unit at the second occasion; A = p  2  ~ Vl '  the true average growth in volume per sampling u n i t between these two occasions.  For t h i s estimation, we choose a simple  random sample of units (plots) from the t o t a l population. We e s t a b l i s h uX . and mX . plots on the f i r s t occasion. ui mj  On  the second occasion we remeasure ml . p l o t s and e s t a b l i s h nY ^ n  new p l o t s which replace those not remeasured.  From  these observations, we obtain simple averages of the p l o t volumes  15  u  X  =  m  (ZX  u  The  X  and  the Y  a .  =  . and mo  . and  mo  =  (ZY  m  Y  ,)/m;  =  ,)/n.  (IY  n  mo  nh  . have a t r u e p o p u l a t i o n v a r i a n c e o f a  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  nn  X  2  of  v>2r we s h o u l d make u s e o f a l l t h e a v a i l a b l e  To e s t i m a t e  information  Y  .)/m; mo  the X  n  m  (IX  m  '  u i  ui  X  .)/u;  f r o m b o t h o c c a s i o n s , t h e r e f o r e we  form  the gen-  eral expression  ) + b- (Z ) + c- (Y  y = a- (X u  a  m  ) + d' (Y n  m  w h i c h s h o u l d be u n b i a s e d , t h e r e f o r e E(y) random s a m p l i n g E(X  )  = E(X  u  s o t o be u n b i a s e d , a + b = may  be r e w r i t t e n  where  -  a- (X  >  a and X  ii.  u  ,  to i i i .  X  1  0 and  c + d = 1.  ) + c  , I  m  X,  with  m  , and  Y  u  The v a r i a n c e o f y  2  = a a 2  2  X  y  (-+-) u  m  =  y , 2  (1)  Equation  and  Y  n  c a n be + c a 2  2  Y v  (2)  )  determined;  a r e sample e s t i m a t e s  n  c  the  Y  subject  , and u  Z  m  and  is Y  independent  i s correlated  m  but s t a t i s t i c a l l y  m  c  y  ) n  error;  independent of X  a-  simple  n  v  m'  t o r e m e a s u r e m e n t and  X  = E(Y  (1 - c) • (Y  ) +  (Y  random s a m p l i n g u s e d ,  of  From  m  c a r e c o n s t a n t s t o be  sampling  due  \*2)  = M I and E(Y  m  m  y  u  i .  =  as  y = a- (X ) a  )  (1)  )  independent  statistically •*  n  .  found from e x p e c t e d v a l u e s as  (-) m  + (l-c)2a  2  Y  (-) - 2acpa a (-) n  v  X Y v  m  (3)  16  To obtain the most e f f i c i e n t estimate of y, we should determine a and c to minimize ° y ' 2  W  e  therefore use  the technique of minimization of the p a r t i a l derivatives of a_  2  with respect to a and c.  The solutions f o r a and c from  y  t h i s technique are m  c = N  mu/[m+u)  ; a =  - (unp )/(m+u)  N  2  2  where N  2  • p—.  - (unp )/(m+u)  a  2  2  v  i s the t o t a l sample s i z e of the second occasion  (m+n) and p =  Covariance (XY)  .  oo x  Y  where Covariance (XY) i s the true population covariance between p l o t volumes observed on the same p l o t on successive periods.  From l i n e a r regression theory, we know that the  least squares estimate of the regression c o e f f i c i e n t of Y on X can be expressed as 3y^ = Covariance (XY)/a^ . 2  From  the expression for p , we see that °X Y a  YX ~  P  =  °x  2  P  °Y — '  x  a  From t h i s we obtain a = c 3 ^ u / ( m + u ) o  We now define the quan-  tities = u/ (m+u), the proportion of plots unmatched at Occasion 2; P  m  = m/(m+u), the proportion of plots matched at occasion 2.  17  S u b s t i t u t i n g i n t o e q u a t i o n (2), we have f o r the mean volume at occasion 2  y = cY + P B (X -X ) + (1-c) ( ? ) m u IX u r n n We now form t h e o v e r a l l mean a t the f i r s t  u m X = (IX . + EZ .)/(m+u) u% mo  = P X mm  (4)  occasion  + PX u u  = X + P (X -X ) . m u u m' Rearranging, we o b t a i n P (Z^-Z^) = (X-X^)  .  Substituting i n  equation (4) y  (X-X ° .= cY m + B YX * VY  m) + ( 1 - c ) ( X n ).  (5)  We note t h a t i n t h i s e q u a t i o n , the e x p r e s s i o n Y s  m  c  + B (X-X  )  VV  YX  m  i s a r e g r e s s i o n e s t i m a t e o f y from double sampling which we denote as Y ,  E q u a t i o n (6) can now be w r i t t e n simply as  y = cY •+ d-c)r ... r  (6)  n  The b e s t e s t i m a t e f o r y i s a weighted combination o f Y from r e g r e s s i o n and Y from the replacement p l o t s .  The weights  c and (1-c) can be shown t o be f u n c t i o n s o f the e r r o r v a r i a n c e of t h e mean volume based on the new p l o t s and o f the e r r o r v a r i a n c e o f t h e mean volume based on double sampling w i t h r e gression.  These weights a r e  C  °Y  =  -2  o-j  2  n  °Y  2  +  -; ( 1 - C )  oj  2  v  2  =  a-  2  n  +  oj  2  r  18  Equation  (6) becomes a-  +  >CY ) R  y = -2  ar  o  +a-  2  y  error  -  (7)  r  variance  °v  2  Y °Y > / y  {A  =  ^  To  2  2  v  estimate  ( a  °Y  2 )  +  r  ( 8 )  n  g r o w t h b e t w e e n o c c a s i o n s , we c o u l d  a l l the information  the general  2  n  f o r m a number o f e x p r e s s i o n s , of  N  2  n with  HY )  b u t a g a i n we s h o u l d make u s e  a t our d i s p o s a l .  We t h e r e f o r e f o r m  expression  (9)  g. = A J + BX + CY + DX , b m m n u a  where g^ d e n o t e s t h e b e s t expression  t o be u n b i a s e d ,  random s a m p l i n g Therefore ues  2  ,  x  °Y  —  2  For this From  simple  = ™m  2  S u b s t i t u t i n g these  +  ( 1  ~  A ) 1  „  +  'm "  B  X  (from e x p e c t e d „Y A  2  + (1-A) — 2  .X a  2  ( 1  "  B )  *u  < > 10  values) -r a  ^  2  + B -|- + (1+B) ^- + 2 A B p - ^ , 2  2  where a l l t e r m s h a v e b e e n p r e v i o u s l y d e f i n e d . biased estimator  val-  (9), we h a v e  error variance  z 2  (y - y ) •  A + C = 1 a n d B + D = -1.  H  ° - ' == AA H  E(9fo) =  o f growth,  E(X ) = E{X ) = p,, a n d E(Y ) - E(Y ) = p,,  into equation  with  estimator  should minimize  (11)  The b e s t un-  the error variance,  therefore  19  we  take the p a r t i a l  differentials  spect t o A and B, s e t the r e s u l t s f o r A and B.  of e q u a t i o n  (11) w i t h r e -  equal t o z e r o , and  solve  Using the p r e v i o u s l y d e f i n e d n o t a t i o n , the r e -  s u l t of t h i s procedure  p r o v i d e s the e s t i m a t e f o r growth  (- )  u  = {  (-1) (-Ibp  2  (_i) #1  +  m  X )] m  [X 1 > n N  N  1  1  Ni  (—)  -{  X )]  .u . ( _n) (I__) N  P  m  2  N  2  2  (£-)(!  "  £")P  2  +  (12) (J.)  -  (_)  N  I f we  g  b  =  define m {[- ? ] p  H  +  =  N  2  -  n (1 [  P  N  2  we  P^rcp , 2  P  (__) 2  N  2  2  can r e w r i t e e q u a t i o n  (12)  as  p ) 2  *  ? n  ]>  P (^ u  2  -  np ) 2  (13) H  20  where X = X + R (Y p Tn Al  -  vv  Y ) is a r e g r e s s i o n e s t i m a t e o f t h e in  mean volume a t t h e f i r s t  o c c a s i o n from double  the o v e r a l l mean a t the second 1  =  <?—)Y z  and  o c c a s i o n Y i s g i v e n by +  Tn  N  N  n  2  i s t h e t r u e p o p u l a t i o n l e a s t squares  e f f i c i e n t o f X on I .  o  3b  2  Examining  r_  =  regression co-  The e r r o r v a r i a n c e o f g^ i s  N -np  1  sampling and  z  2  ]  {  [  ]  H  N  equation  four estimates.  0  2  |i-p p 2  +  ] a  2 _ 2Poo o } Y  (14)  Y  X  (13), we see t h a t i t i s a combination o f  The f i r s t  two e x p r e s s i o n s  form an e q u i v a l e n t e q u a t i o n t o e q u a t i o n  ( w i t h i n t h e braces)  (7) f o r y.  The s e -  cond two e x p r e s s i o n s a r e i d e n t i c a l w i t h t h e i n f o r m a t i o n r e versed, i . e .  a r e v i s e d e s t i m a t e o f volume a t t h e f i r s t  s i o n u s i n g i n f o r m a t i o n n o t a v a i l a b l e u n t i l t h e second sion.  occaocca-  Thus i n f i n d i n g the b e s t e s t i m a t e f o r growth, we have  used a l l i n f o r m a t i o n i n forming new e s t i m a t e s f o r volume a t both o c c a s i o n s and s u b t r a c t e d one from t h e o t h e r .  One d i s -  advantage o f t h i s e s t i m a t o r i s t h e v a l u e o b t a i n e d f o r growth, when added t o the o r i g i n a l e s t i m a t e o f mean volume a t t h e f i r s t o c c a s i o n , w i l l n o t r e s u l t i n the p r e v i o u s l y d e r i v e d e s t i m a t o r f o r volume a t the second t h i s i s no problem, but may cause conversant w i t h SPR t h e o r y .  occasion.  Statistically,  some concern t o those n o t  21  Other e s t i m a t o r s o f growth may be formed.  One was  j u s t i n d i r e c t l y mentioned, c a l l i t g , which i s formed by s u b t r a c t i n g the mean volume e s t i m a t e d the SPR estimate  a t o c c a s i o n one from  o f volume a t o c c a s i o n two.  T h i s can be  expressed as a = (a-P, )X„ - (P +a)Z "c u u m m  + cY  m  + (1-c)?  m  w i t h a l l terms p r e v i o u s l y d e f i n e d .  (15) n  T h i s e s t i m a t o r has an  error variance  0  9*=  a  a —  1-P p  2  Ni  +  N ~P  P  2  np  2  * 2  2 y  -2  1  N -P z  2-po-ff.  np  (16)  2  This estimator i s c o n s i s t e n t with the f i r s t estimation o f mean volume and makes use o f t h e i n f o r m a t i o n a v a i l a b l e on b o t h o c c a s i o n s f o r i t s e s t i m a t i o n o f volume on t h e second occasion. Another e s t i m a t o r i s t h e u s u a l matched p l o t volume estimations.  In our n o t a t i o n , t h i s e s t i m a t o r , g , i s ex-  p r e s s e d as g m  = Y - X m m  a  (17)  v  '  with e r r o r variance  a  q  2  = ^x  1  +°  2 Y  ~ 2pa a )/m  777 Note t h a t t h i s i s a s p e c i a l case o f g^ where u = n = 0.  (18)  22  Another s p e c i a l case which r e s u l t s i n an e s t i m a t o r for  growth o c c u r s when no p l o t s are matched  i s an independent sample  case.  (m = 0) which  The e s t i m a t o r g ., i s formed  as  g. = Y  - X  (19)  with error variance  o  = a /n + a /«  2  2  (20)  2  I f the sample s i z e was the same on both o c c a s i o n s , t h i s  esti-  mator would have an e r r o r v a r i a n c e o f  is  These l a s t  s e v e r a l formulae f o r t h e e r r o r v a r i a n c e s a r e f a -  m i l i a r ones from b a s i c s t a t i s t i c s , check on the d e r i v a t i o n o f a  .  which i s an i n t e r e s t i n g One f i n a l e s t i m a t o r  that  w i l l be shown i s a weighted e s t i m a t o r from t h e matched and independent samples. proportional  We use weights o f  and g  which a r e  t o t h e i r e r r o r v a r i a n c e s t o form g .  A general  formula f o r g,, i s  9  W  i n which w = (o  2  = w ( ^ ) + (i-»> (g )  )/(a  2  9i &m  (21)  m  + a  2  ) and (1-w) = (a  9m  ffi  2  )/(a  2  +  9i  23  The  e r r o r v a r i a n c e , again i n g e n e r a l terms, i s  O g  = (w) c 2  w  2  2  + (l-w) o 2  (22)  2  For t h e s p e c i a l case when sample s i z e and the p o p u l a t i o n  vari-  ance a r e e q u i v a l e n t a t both o c c a s i o n s , t h i s e s t i m a t o r i s e q u i v a l e n t t o g^, otherwise  i t i s less  efficient.  These then a r e the v a r i o u s e s t i m a t o r s t h a t can e a s i l y be formed t o e s t i m a t e growth between o c c a s i o n s .  In s p e c i a l  c a s e s , such as equal v a r i a n c e on both o c c a s i o n s , some a r e e q u i v a l e n t , b u t i n g e n e r a l , t h e f i r s t d e r i v e d e s t i m a t o r , g^, i s t h e most e f f i c i e n t as i t takes b e s t advantage o f informat i o n a v a i l a b l e o n l y a f t e r t h e second measurement has o c c u r r e d . A major f e a t u r e o f SPR i s t h e s p e c i f i c a t i o n o f the optimum number o f sample u n i t s t o remeasure and t o r e p l a c e . T h i s can be achieved t o t a l expenditure ror.  f o r minimum sampling  error with  fixed  o r f o r minimum c o s t a t f i x e d sampling e r -  S e v e r a l authors have d e r i v e d these o p t i o n s under v a r -  i o u s 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 c o s t s , e t a . ( P a t t e r s o n 1950, Cochran 1963, K i s h 1965).  Ware and Cunia  presented  (1962) removed a l l r e s t r i c t i o n s and  g e n e r a l formulae.  In t h e a n a l y s e s  a f i n i t e p o p u l a t i o n o f 220 p l o t s was used.  i n this report, I t was thus  e s s a r y t o d e r i v e formulae f o r t h a t p a r t i c u l a r T h i s d e r i v a t i o n w i l l be p r e s e n t e d authors.  nec-  restriction.  as t y p i c a l o f those o f o t h e r  24 A f i r s t d e r i v a t i o n was sample s i z e on o c c a s i o n not  found.  + n) .  the t o t a l  The s o l u t i o n  A second attempt was made w i t h the  that a constant N ,  used i n the f i n i t e p o p u l a t i o n  correction  restriction  t i o n f o r the number of new  s i z e determina-  p l o t s an i t e r a t i v e procedure.  To determine the optimum ,number of sample and t o r e p l a c e  3  (FPC) terra.  T h i s r e s t r i c t i o n n e c e s s a r i l y makes the sample  to remeasure  was  which w i l l u l t i m a t e l y be e q u a l t o {m + n)  2  was  two as  attempted u s i n g  at occasion  two, we  plots  specify  the c o s t must be m i n i m i z e d s u b j e c t t o t h e r e s t r i c t i o n  that  that  the sampling e r r o r of the c u r r e n t mean be l e s s than or e q u a l to a s p e c i f i e d v a l u e ,  say  t i o n of the f o l l o w i n g  components;  1.  The c o s t i s a l i n e a r  combina-  C y , the amount of money a v a i l a b l e f o r e s t i m a t i n g the c u r r e n t mean a t the second  occasion;  2.  C , the c o s t per p l o t of the m remeasured  3.  C , the c o s t per p l o t o f the n new  4.  Cy,  the f i x e d c o s t s i n the  plots;  plots;  inventory.  The c o s t f u n c t i o n i s of the form  C  T  - C  f  = mC  m  + nC  (23)  n  To minimize t h i s f u n c t i o n w i t h the s p e c i f i e d r e s t r i c t i o n , may  we  use the method of Lagrange m u l t i p l i e r s ( R e i f 1965, Ware  and Cunia 1962). matched, and new  Given t h a t u, m, and n a r e t h e unmatched, numbers of p l o t s r e s p e c t i v e l y i n the  taken on the two o c c a s i o n s ,  we d e f i n e  samples  the f o l l o w i n g 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 which i s g i v e n and c o n s i s t s of  2.  occasion  (m + u ) p l o t s ;  N-i, the t o t a l sample s i z e a t the second s i o n which i s e q u a l t o  3.  occa-  (m + n ) p l o t s ;  The  e s t i m a t e s of the p o p u l a t i o n parameters p  and  ay  are o b t a i n e d independent  2  of the  samples. K, the square  of the s p e c i f i e d sampling  than or e q u a l t o a _ ,  e r r o r , must be  which i s expressed  2  as N  a a-  =  2  2  [ N i (1 - p ) + mp ]  -  2  mNi  2  [1  2  * +mnp  1  y  + nN i (1 - p ) 2  ,  (24)  2  N which i n c l u d e s the f i n i t e p o p u l a t i o n c o r r e c t i o n  can draw.  form the Lagrangian f u n c t i o n a  i>  2  (1 - ^—) ,  where N i s the t o t a l number of u n i t s from which we We  less  [nC  =  + mC n  + C  .]  J  m  +  [Ni ( 1 - p ) + mp ]  2  2  [1 -  2  A{ —  ^] -  :  m  N  + nNid-pZ)  l  +  m  n  p  K}.  -  (25)  2  We  next take the p a r t i a l d e r i v a t i v e s of ip w i t h r e s p e c t to m,  n,  and A , which are then, f o r m i n i m i z a t i o n , s e t e q u a l t o z e r o .  These p a r t i a l s  are  [mNi  = 3 m  + n^id-P )  + mnp ] [p (l-^ -)] 2  2  2  2  ^  + Aa {  C  2  m  [mNi + n N i { 1 - p )  1  2  - ^ ( 1 - P  to l-p2) 2 l(  +wp  [mNi  2  +mnp ] 2  )  + n^!(l-p ) 2  }  2  +  +  mnp ] 2  2  *P 1 2  .  (  2  6  )  26  i  =  V  A  f  y  f  >  (  + n^CL-p )  [m^j  3, —_  a / [ ^ ( l - P ) ( l  first  2  2  m p 2 g . f i ] -  + nljll-p )  + mnp  2  1  two p a r t i a l d e r i v a t i v e s  solved for Ni ( 1 - p )  +  mnp ] 2  ^  mN The  - f i )  2  _  +  2  X and s e t  Ki  ,  (to)  2  [Eqn's  (26) and (27)] a r e  equal t o each o t h e r .  I f we d e f i n e G -  + m p , the e q u a l i t i e s are 2  CG  C G  2  2  2  x =  ]  tfid-p ) Substituting  2 - ^  .  (29)  c ( i - Si.)  - ^)  (l  2  -  r  2  f o r G and r e a r r a n g i n g , we have  mp  C = iy [_2L(i-p2) * „  2  1  tfjd-p )  .  2  ]  m S o l v i n g t h i s e x p r e s s i o n f o r m and c o l l e c t i n g tfi/d-p )  /C -ij^-  2  m =  5  P'  -  /(1-p ) } 2  We now r e a r r a n g e t h e p a r t i a l d e r i v a t i v e a  2 7  [tf (l- 2) 1  p  +  m  p  2  ]  [  K = — mNi Again we s u b s t i t u t e arrive  + n#i(l-p2) +  1  terms  .  (30)  with respect t o X  _ t f 2  mnp  ]  . 2  G where p o s s i b l e f o r s i m p l i f i c a t i o n t o  at mN K l  + nKG = a  y  2  G  „ (i_£ ) 2  #  (31)  27  Solving f o r n mN\K n =  KG  a (1  - |i)  2  y  KG #1  K  P  Ware and Cunia  (1962)  K  mNi N  ( 1 - p ) + mp 2  1  2  ^(1-P ) 2  U-[  C  2  (32)  n  presented  an i d e n t i c a l e x p r e s s i o n f o r  m, t h e optimum number o f p l o t s t o remeasure.  T h e i r expres-  s i o n f o r n, the optimum number t o r e p l a c e d i f f e r s from equation  (32)  Oy /K.  i n t h a t t h e f i r s t p a r t o f the r i g h t hand s i d e i s  Thus we see t h a t the f i n i t e p o p u l a t i o n c o r r e c t i o n  2  f a c t o r a f f e c t s o n l y the sample s i z e requirement p r e c i s i o n and a g i v e n v a r i a n c e .  f o r a given  The e x p r e s s i o n f o r r e d u c t i o n  i n sample s i z e due t o SPR i s as p r e s e n t e d by Ware and Cunia (1962). The method o f optimum sample s i z e  determination.  when c u r r e n t volume and growth a r e s i m u l t a n e o u s l y i n c r e a s e s i n complexity ling error restrictions.  estimated,  due t o t h e need t o s a t i s f y two sampThese r e s t r i c t i o n s a r e n o n - l i n e a r  i n t h a t they i n v o l v e c r o s s - p r o d u c t terms i n m and n. F o r d e r i v a t i o n o f these formulae,  the reader i s again  t o Ware (1960).  p r e s e n t e d by Ware and Cunia  (1962) the  The formulae  a r e g i v e n here w i t h o u t  referred  p r o o f f o r the e d i f i c a t i o n o f  reader. The  l i n e a r c o s t f u n c t i o n [Eqn (23)]  j e c t t o the r e s t r i c t i o n s t h a t the sampling  i s optimized  sub-  e r r o r o f the c u r r e n t  28  mean, ^K, i s l e s s than o r e q u a l t o o_, and t h e sampling of  growth, say SK , i s l e s s than o r equal t o a  9  b  2  m[KN  + n [KN'i { 1 - p ) ] + m n [ K p ]  and  -p o 2  2 J  ]  The i n -  9  equality, a_  1  „  error  <_ K, i s reduced t o  2  the i n e q u a l i t y ,  o  2  2  2 y  N (l-p ) 2  1  >_ 0  (33)  : <_ K , t o  miK^Ni-ia^pa^) ]  + mn[p K  - a.  2  + n [ ( 1 - p ) (K J^-a.^ ) ]  ]-  2  2  (i_.2)- >_ o  (34)  p  These e x p r e s s i o n s a r e s e t e q u a l t o zero.  Ware and Cunia  (1962)  presented a g r a p h i c a l method t o s o l v e these i n e q u a l i t i e s f o r m and n.  The method used i n t h e a n a l y s e s f o r t h i s t h e s i s has  been o u t l i n e d e a r l i e r . mathematical, developed  Many of the present-day methods o f  dynamic, and l i n e a r programming were n o t h i g h l y  when Ware and Cunia developed  their 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 s o l v e problems o f t h i s k i n d .  2?  CHAPTER FOUR COMPARISONS OF SPR WITH'CONVENTIONAL INVENTORY Example One In t h i s example, we w i l l  compare c o n v e n t i o n a l i n -  v e n t o r y w i t h an SPR system f o r a s m a l l a r e a s i t u a t i o n .  First  we w i l l determine sample s i z e s f o r t h e c o n v e n t i o n a l case. will  We  then f o l l o w the e s t a b l i s h m e n t o f an SPR system w i t h  s i m i l a r g o a l s , f o l l o w i n g which, t h e c o s t s o f t h e two systems w i l l be compared.  F i n a l l y , r e l a t i v e m e r i t s o f the system  w i l l be d i s c u s s e d .  TABLE PERIOD  A  1. Parameters  for example  one.  a  SYMBOL  2  cu.ft./ac  (cu.ft./ac)  1  X  20403.4  40947200  2  I  21558.8  47215200  l-2  9  1155.4  1911533  P  2  2  0.9620  We must s p e c i f y our p r e c i s i o n requirements f o r c u r r e n t volume and growth.  We d e c i d e t o 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 o f 0.95. for  our management requirements,  We f i n d t h a t ,  mean volume per a c r e should  l i e w i t h i n ± 6 p e r c e n t o f t h e t r u e mean v a l u e . be ± 20 p e r c e n t o f the t r u e v a l u e .  Growth must  30  We now determine r e q u i r e d sample s i z e s f o r a conventional inventory.  F o r the i n i t i a l survey,  interval half-width £ From the formula  =  (0. 06) (20403.4) = 1224,20  f o r sample s i z e o f a f i n i t e  t a 2  "o =  cu.ft./ac.  population  n  2  0  ;  n =  —-  E  1  2  we need 71 p l o t s .  the c o n f i d e n c e  +  (35)  £L  These can be low c o s t temporary p l o t s .  For e s t i m a t i n g c u r r e n t volume f i v e y e a r s l a t e r , we w i l l r e q u i r e a h a l f - w i d t h of the c o n f i d e n c e i n t e r v a l of E^ = (21558.8) = 1293.53 c u . f t . / a c . 35) 73 p l o t s ,  (0.06)  The sample s i z e w i l l be (Eqn.  again of the temporary type.  We want t o e s t i -  mate growth over t h i s f i v e year p e r i o d , so we d e c i d e t o i n s t a l l a permanent growth p l o t system. width i s E  =  The c o n f i d e n c e i n t e r v a l h a l f -  (0.20)(1155.37) = 231.07 c u . f t . / a c / f i v e  years.  The sample s i z e r e q u i r e d i s 85 p l o t s , but we d e c i d e t o e s t a b lish  88 as a hedge a g a i n s t l o s s o f some p l o t s .  a t the f i r s t o c c a s i o n we e s t a b l i s h a l s o need 71 p l o t s  Summarizing,  88 permanent p l o t s .  We  f o r volume d e t e r m i n a t i o n which may o r may  not be from the 88 permanent p l o t s . we need t o measure 73 temporary p l o t s  A t the second o c c a s i o n , f o r volume, and t o r e -  measure a t l e a s t 85 permanent p l o t s . For the SPR i n v e n t o r y , we assume the e s t a b l i s h m e n t of 88 p l o t s , some of which w i l l be permanent and some temporary.  F o r the second o c c a s i o n , we determine the optimum num-  ber t o remeasure and how many we must r e p l a c e c o n s i d e r i n g  31  o n l y the measurement of c u r r e n t volume.  The  cost r a t i o of  permanent t o temporary p l o t s i s e s t a b l i s h e d as 2:1. equation sured.  (32), we The  a t i v e , and  From  c a l c u l a t e t h a t n i n e p l o t s should be  remea-  s o l u t i o n f o r the replacement sample s i z e i s i t e r the f i n a l s o l u t i o n i s 45.  I f we  are o n l y  inter-  e s t e d i n measuring the c u r r e n t volume a f t e r f i v e y e a r s , are the numbers o f samples r e q u i r e d . taneously  estimate growth, we  sizes w i l l satisfy not.  We must now  restraints  a_  optimize  < K and  2  y  a  -  n o n - l i n e a r , we  2  g  I f we want to s i m u l -  check whether these  inequation  (34) .  these  sample  In t h i s case,  they  do  a c o s t f u n c t i o n s u b j e c t to  the  < K .  are  -  As these r e s t r a i n t s  9  r e s o r t to a n o n - l i n e a r o p t i m i z a t i o n programme.  T h i s programme computes the optimum number to remeasure as and t o r e p l a c e as 25. o c c a s i o n one,  the f o l l o w i n g t a b l e  see i f SPR  completely  establish at  o n l y the 74 need be permanent.  Using pare and  So of the 88 p l o t s we  74  can  (TABLE  save money f o r us.  2)i  l e t us com-  I f we  conducted  independent c u r r e n t volume and growth i n v e n t o r i e s  on both o c c a s i o n s , we  would spend $12,250.00 (a+b+c+d).  We  c o u l d reduce t h i s c o s t by u s i n g the permanent p l o t s f o r c u r r e n t volume i n f o r m a t i o n a t the f i r s t o c c a s i o n o n l y .  T h i s would  i n c r e a s e the p r e c i s i o n of the f i r s t volume i n v e n t o r y and duce the c o s t to $10,475.00 (b+c+d).  Using  p l o t s a t the second o c c a s i o n would not add  the permanent any  information  about the stand s t r u c t u r e and would r e i n f o r c e any of the f i r s t  re-  failure  sample to be r e p r e s e n t a t i v e of the stand.  The  TABLE  2.  Comparison  Inventory  type  of c o s t s  between  independent  and SPR i n v e n t o r i e s .  (1)  (2)  (3)  (4)  occasion  p l o t nos.  plot cost  (2) x (3)  dollars  dollars  Independent (volume only)  (growth only)  one  71  25. 00  1775.00  (a)  two  73  25.00  1825.00  (b)  one  88  50.00  4400.00  (c)  two  85  50.00  4250.00  (d)  one  77  50.00  3850.00  (e)  11  25.00  275.00  (f)  74  50.00  3700. 00 (g)  25  25. 00  625.00 (h)  9  50. 00  450.00  (i)  28  25.00  700.00  (j)  SPR  (growth and volume)  (volume o n l y )  two  two  33  SPR i n v e n t o r y f o r the same i n f o r m a t i o n would c o s t $8,450.00 (e+f+g+h).  T h i s would save a minimum o f $2,025.00 and a l l o w  us the o p t i o n of measuring c u r r e n t volume o n l y a t a c o s t o f $1,150.00  (i+j). S e v e r a l i n t e r e s t i n g by-products of t h e SPR system  can be seen i n t h i s example.  One, the o p t i o n o f measurement  of c u r r e n t volume o n l y a t a g r e a t l y reduced c o s t , w i l l be d i s c u s s e d i n a l a t e r example. in  Another i s i n c r e a s e d p r e c i s i o n  the c u r r e n t volume e s t i m a t e when simultaneous  i s used.  estimation  I f we had measured o n l y c u r r e n t volume a t the s e -  cond o c c a s i o n , we would have needed o n l y 37 p l o t s t o meet our p r e c i s i o n requirements.  As i t turned o u t , we used 99  p l o t s because o f the growth p r e c i s i o n needed.  We can c a l c u -  l a t e t h e i n c r e a s e d p r e c i s i o n by s o l v i n g e q u a t i o n the sampling e r r o r . 2  (33) f o r K,  From t h i s v a l u e , we f i n d t h a t the con-  f i d e n c e i n t e r v a l h a l f - w i d t h has been reduced t o ±4.37 p e r cent o f the p o p u l a t i o n mean.  The p r e c i s i o n o f the e s t i m a t e  of c u r r e n t volume on t h e f i r s t o c c a s i o n  i s changed to ±5.07  p e r c e n t o f the mean whether we use the SPR system o r a l l o f the permanent growth p l o t s .  Perhaps t h i s p r e c i s i o n i s n o t  r e q u i r e d , b u t i t i s there and we s h o u l d take advantage o f i t in  our decision-making  processes.  Another advantage o f SPR o c c u r s when, as i n t h i s example, the minimum v a r i a n c e e s t i m a t o r o f growth i s used. A r e v i s e d estimate  o f volume a t the f i r s t o c c a s i o n can be ob-  t a i n e d from t h e computations.  T h i s new e s t i m a t e  takes  infor-  mation a v a i l a b l e o n l y a t the second o c c a s i o n and through  34  r e g r e s s i o n combines i t with the mean o f the non-remeasured p l o t s from the f i r s t o c c a s i o n . of  freedom  (for variance estimation)  would be 88 - 1 = 87.  f o r the o r i g i n a l  For the SPR e s t i m a t e ,  14 o b s e r v a t i o n s are a v a i l a b l e . ing  In t h i s example, the degrees  a t o t a l o f 99 +  There a r e t h r e e means  t h e r e g r e s s i o n c o e f f i c i e n t i s independently  mated from these o b s e r v a t i o n s .  estimate  (assum-  known) e s t i -  Thus we have 113 - 3 = 110  degrees o f freedom f o r t h e v a r i a n c e o f the r e v i s e d e s t i m a t e . T h i s estimate  should be more e f f i c i e n t .  One f i n a l p o i n t can be made from t h i s example.  If  we a r e always o n l y i n t e r e s t e d i n c u r r e n t volume, we have a new s i t u a t i o n i n which o n l y 71 p l o t s would be e s t a b l i s h e d a t the f i r s t o c c a s i o n . at  A t o t a l o f 73 p l o t s would s t i l l be needed  the second o c c a s i o n under t h e c o n v e n t i o n a l system.  But  o n l y 8 remeasured and 19 new p l o t s a r e needed f o r an SPR mate with the same second o c c a s i o n p r e c i s i o n . cost  esti-  The t o t a l SPR  ( e x c l u s i v e o f f i x e d c o s t s ) would be $2,850.00 v e r s u s  $3,6 00.00 f o r t h e c o n v e n t i o n a l survey.  There would undoubt-  e d l y be a time savings as w e l l s i n c e , even though remeasured p l o t s r e q u i r e more time spent on a l o c a t i o n , the v e r y  small  number o f p l o t s needed a t the second o c c a s i o n s h o u l d r e q u i r e l e s s time than f o r the 73 p l o t s o f the c o n v e n t i o n a l system.  35  Example Two F o r t h i s example we w i l l assume a l a r g e acreage, r e a s o n a b l y homogeneous stand i s i n need o f i n v e n t o r y . age i s somewhere p a s t the middle o f r o t a t i o n .  The area i s  h i g h s i t e I I and t h e r e f o r e w i l l be i n our management for  some y e a r s .  F o r decision-making  Stand  purposes,  scheme  we d e s i r e a  c o n t i n u i n g c u r r e n t volume e s t i m a t e w i t h i n ± 3 per c e n t o f t r u e mean volume p e r a c r e and a growth e s t i m a t e w i t h i n ± 10 per c e n t o f the t r u e v a l u e f o r a ten year p e r i o d . of  the p o p u l a t i o n parameters and sampling  a r e g i v e n i n TABLE  TABLE  3.  Parameters  and sampling example  mt  A  errors  2  two.  SYMBOL cu.ft./ac  3  requirement  3.  for  PERIOD  error  Estimates  Y  (cu. f t./ac)  2  20403.4  40947200  21558.8  47215200  1-2 0.9620  22705.2  58756000  1-3 0.9266  2301.8  5272314  l-3  SAMPLING ERROR (cu.ft./ac) K = 120776.1  2  2  K  = 13790.8 5/  For a c o n v e n t i o n a l sample s i z e from e q u a t i o n we  need 420  i n v e n t o r y , the c a l c u l a t i o n (34)  (without  temporary p l o t s a t the f i r s t  mate the c u r r e n t mean volume. years  the FPC)  later,  487  For growth over  of  indicates  occasion  to  esti-  A t the second o c c a s i o n ,  ten  p l o t s are needed f o r the same e s t i m a t e . the ten year p e r i o d , 383  permanent sample  p l o t s are r e q u i r e d . F o r an SPR  i n v e n t o r y , we  assume the same 420  plots  on the f i r s t o c c a s i o n , some of which are permanent and temporary. tions.  The  numbers o f each type come from our  F o r c u r r e n t volume o n l y on the second  s o l u t i o n s of equations  (30) and  q u i r e 54 permanent and  207  (32)  must o p t i m i z e  a cost function  -  will  requirements,  = $50.00*m +  $25.00«n w i t h  the c o n s t r a i n t s t h a t the maximum m i s  minimum m and  n are  qualities  zero, and  (Eqn's 33 and  the f o l l o w i n g n o n - l i n e a r i n -  + 111911.14/rm - 1811329968. > 0;  4832580.63m - 2580382.lln +  12778.54mn - 1811329968. > 0„  o p t i m i z a t i o n program c a l c u l a t e s the sample s i z e  m = 347  and  n = 73.  420,  34)%  -3717344.66m + 3723285. 83n  The  re-  These numbers  of p l o t s w i l l n o t s a t i s f y our growth p r e c i s i o n so we  calcula-  occasion,  i n d i c a t e we  temporary p l o t s .  some  as  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 o c c a s i o n i s i d e n t i c a l t o the number of p l o t s e s t a b l i s h e d on  the f i r s t o c c a s i o n .  permanent p l o t s on both o c c a s i o n s , the f i r s t o c c a s i o n , second  occasion.  and  We  would measure  347  73 temporary p l o t s on  73 d i f f e r e n t temporary p l o t s on  the  37  The  t o t a l c o s t o f completely  i e s f o r volume and  growth on  two  independent i n v e n t o r -  occasions  f i x e d c o s t s ) i s $60,975.00 (a+b+c+d+e, TABLE we  used the 383  survey  (not i n c l u d i n g 4).  Assuming  permanent p l o t s i n the f i r s t o c c a s i o n volume  (thus r e d u c i n g  the n e c e s s a r y  temporary sample s i z e t o  37), t h i s c o s t c o u l d be reduced to $51,400.00 (b+d+e+37•$25.00) plus f i x e d c o s t s .  We  see t h a t the SPR  inventory costs  are  o n l y $50.00 more than the c o s t of the growth p l o t s . One one,  o f the b e n e f i t s of SPR,  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 .  decided  to conduct a new  survey  new  Say  we  f o r c u r r e n t volume o n l y a t  f i v e years a f t e r p l o t e s t a b l i s h m e n t . 434  mentioned i n example  A t the same p r e c i s i o n ,  temporary p l o t s would be r e q u i r e d .  advantage o f our permanent p l o t s and  But, we  can  take  the l i n k a g e a v a i l a b l e  through them and we need measure o n l y 36 permanent p l 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  this  s i n g l e i n v e n t o r y would be $6,125.00, a s a v i n g of $4,725.00 over the c o s t of a s i n g l e c o n v e n t i o n a l volume i n v e n t o r y , Summing the c o s t s f o r both methods: Conventional  (3 c u r r e n t volume measurements + 1 growth mea-  surement) : $10,500.00 + 10,850.00 + 12,175.00 + 38,300.00 = $71,825.00; SPR  (3 c u r r e n t volume measurements + 1 growth measurement):  $38,350.00 + 6,125.00  = $44,175.00.  TABLE  4.  Comparison  of c o s t s  Inventory  type  between  independent  and SPR i n v e n t o r i e s  (1)  (2)  occasion  plot nos.  (3) plot  cost  dollars  f o r example  (4) (2) x  (3)  dollars  Independent (volume only)  (growth only)  one  420  25.00  10500.00  (a)  two  434  25.00  10850.00  (b)  three  487  25.00  12175.00  (c)  one  383  50.00  19150.00  (d)  three  383  50.00  19150.00  (e)  one  347  50. 00  17350.00  (f)  73  25.00  1825.00  (g)  36  50.00  1800.00  (h)  25.00  4325.00  (i)  19175.00  (j)  SPR (growth and volume)  (volume only)  two  173 (growth and volume)  three  1  same as one  two.  39  The  demonstrated s a v i n g of $27,650.00 i s c o n s i d e r a b l e .  any o f these remeasurement  p e r i o d s , we might want t o spend  some o f these s a v i n g s t o e s t a b l i s h permanent of  At  the i n d i c a t e d temporary p l o t s .  plots i n l i e u  T h i s would a l l o w us t o  use these p l o t s on t h i r d , f o u r t h , or subsequent remeasurements. our  By adding, d e l e t i n g , and/or b r i n g i n g p l o t s back  i n v e n t o r y system, we can change our p r e c i s i o n  ments a t almost any s t a g e of management. flexible.  into  require-  The system i s v e r y  40  Example Three In some f o r e s t management s i t u a t i o n s , we wish t o combine or fragment our h o l d i n g s . be v e r y d i s c o n c e r t i n g . SPR  For i n v e n t o r y , t h i s  To be t r u l y u n i v e r s a l i n a p p l i c a t i o n ,  s h o u l d h e l p overcome some o f the problems we  in this situation.  can  For t h i s example, we  encounter  assume t h a t a s t a n d  w i t h an a r e a o f 11 square m i l e s has been i n our h o l d i n g s f o r some time.  Inventory on t h i s s t a n d has been w i t h the r e q u i r e -  ments o f the h a l f - w i d t h o f the c o n f i d e n c e i n t e r v a l f o r c u r r e n t volume be ± 6 p e r c e n t of the mean volume per a c r e . purchased  a f o r e s t e d area, surrounding  We  our h o l d i n g , of  square m i l e s f o r a t o t a l area of 60 square m i l e s .  This  have 49 new  a r e a i s f o r e s t e d w i t h s p e c i e s , s t o c k i n g , and s i t e  similar  t o our p r e s e n t s t a n d .  encompass  A new  inventory, that w i l l  the t o t a l of our h o l d i n g i s n e c e s s a r y t h i s area.  From our p r e v i o u s i n v e n t o r i e s i n the 11  m i l e stand, we  have parameters as g i v e n i n TABLE  have been compiled original  f o r our management of  from 220  square  1, which  permanent p l o t s l o c a t e d i n the  stand. An a r e a compensating formula w i l l be used t o de-  termine  the r e q u i r e d sample s i z e a t the f i r s t o c c a s i o n  and Cunia  _  (Ware  19 62)%  (confidence i n t e r v a l h a l f - w i d t h 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 p e r c e n t .  T h e r e f o r e the a l l o w a b l e  e r r o r v a r i a n c e i s K = 71518.97 c u . f t . / a c . 2  From t h i s  2  we  compute the sample s i z e on the f i r s t o c c a s i o n as 573  plots.  Some o f these w i l l be temporary and some permanent.  The  sam-  p l e s i z e f o r e s t i m a t i n g sample s i z e on the second o c c a s i o n i n an SPR  sample i s 60 remeasured p l o t s and  229  temporary  p l o t s f o r a t o t a l of 289.  T h i s sample s i z e w i l l n o t  our p r e c i s i o n requirements  f o r growth which we  fulfill  have e s t a b -  l i s h e d as ± 16 p e r c e n t of the mean growth per acre per years.  We  five  t h e r e f o r e use our n o n - l i n e a r o p t i m i z a t i o n r o u t i n e  w i t h the f o l l o w i n g c o n s t r a i n t s : i.  the minimum m  ii.  and n equal  the maximum m i s e q u a l t o  iii.  o_  y  2  < K and  -  a  2  g  zero; 573;  < K .  -  9  The optimum number of p l o t s t o remeasure i s 534 i s 89.  We may  now  proceed  towards our two  and  t o add  g o a l s , the  esti-  mation of the c u r r e n t volume of the 60 square m i l e s and  the  p r o v i s i o n f o r e s t i m a t i o n of volume and growth i n the f u t u r e . As we  have a 220  p l o t e x i s t i n g base, we  and e s t a b l i s h 314 new i f not a l l , new  permanent and  of these new  remeasure these  plots,  89 temporary p l o t s .  Most,  p l o t s s h o u l d be e s t a b l i s h e d i n our  h o l d i n g s as we have a s u f f i c i e n t number of p l o t s i n the  o r i g i n a l 11 square m i l e s .  In a d d i t i o n , we  e s t a b l i s h i n g the 89 replacement v i d e f o r p l o t l o s s e s and  should consider  p l o t s as permanent to p r o -  f o r the p o s s i b i l i t y  t h a t our  new  42  h o l d i n g s w i l l i n t r o d u c e added v a r i a b i l i t y . have extended our sampling our o r i g i n a l  Regardless,  scheme t o the a r e a  stand w i t h a minimum of e f f o r t .  we  surrounding  CHAPTER FIVE BIAS IN SPR ESTIMATES B i a s i n sampling  can cause e s t i m a t e s t o v a r y  s i d e r a b l y from t h e i r expected v a l u e s .  con-  T h i s v a r i a b i l i t y may  occur such t h a t n o t o n l y a r e t h e e s t i m a t e s b i a s e d , b u t they are a l s o i n c o n s i s t e n t . c a r e f u l l y examining  These c o n d i t i o n s can be minimized by  the e s t i m a t i o n methods.  I f alternative  methods a r e n o t a v a i l a b l e o r a p p l i c a b l e , steps can u s u a l l y be taken t o e s t i m a t e the amount o f b i a s p r e s e n t and t o make the sample e s t i m a t e s c o n s i s t e n t . In SPR, the b a s i c assumptions t h a t o^, , ° j ^ 2  and t h e r e f o r e 6 a r e e i t h e r known without sampling o b t a i n e d independent  2  P>  error or  o f the sample a l l o w the c u r r e n t volume  and growth e s t i m a t e s t o be unbiased.  When these c o n d i t i o n s  a r e n o t met, or i f t h e sample p o p u l a t i o n i s f i n i t e ,  certain  steps can be taken t o a l l e v i a t e the problem. One p o t e n t i a l source o f b i a s o c c u r s i n the e s t i mation o f y from e q u a t i o n 7 i n v o l v i n g the e r r o r v a r i a n c e a- . r 2  Ware and Cunia  (1962) p r e s e n t t h i s v a r i a n c e as o  r Cochran  2 y  (l-p ) 2  m  p a/ 2  (m+u)  (1963) and o t h e r s have d e r i v e d t h i s same e x p r e s s i o n .  When r e g r e s s i o n e s t i m a t e s a r e used w i t h double  sampling, t h e  v a r i a n c e o f the e s t i m a t e i s dependent on the x^ chosen f o r t h a t p a r t i c u l a r sample.  Cochran averaged  the v a r i a n c e over  44  a l l p o s s i b l e draws t o o b t a i n an e x p r e s s i o n  that w i l l  give  the average mean square e r r o r which i s e q u i v a l e n t t o our 2  .  This expression i s  r  a (l-p ) 2  a-  2  = -  r  p  2  +  1  m  2  a  o (l-p )  2  —  2  (w)  2  (1)  + —  (37)  (m+u)  (m+u) (m-3)  m  The f e e l i n g i s t h a t i f the term l/(w-3) i s n e g l i g i b l e , t h e much s i m p l e r e x p r e s s i o n i s a c c e p t a b l e . parameters used i n these examples, o-j,  2  r  With t h e p o p u l a t i o n  which a r e n o t t y p i c a l ,  was i n c r e a s e d by a p p r o x i m a t e l y 54 p a r t s i n 363860, o r  about 0.015 p e r c e n t .  T h i s i s indeed n e g l i g i b l e and t h e c o r -  r e c t i n g f o r m u l a was n o t used i n t h e computations  involving  the p o p u l a t i o n parameters. When p and 3 a r e e s t i m a t e d from t h e sample, t h e reduced form o f t h e e x p r e s s i o n f o r o  2 y  i s no l o n g e r u n b i a s e d ,  v but i s o n l y c o n s i s t e n t  (Ware and Cunia 1962) .  A sampling  experiment was conducted t o e s t i m a t e t h e amount o f t h i s f o r t h e p o p u l a t i o n used f o r t h e s e a n a l y s e s .  Fifty  bias  repeti-  t i o n s o f t h e sampling experiment w i t h a random s t a r t f o r each run were made.  F o r each run, °Y ,  Z  w  a  X  s  e s t i m a t e d from popu-  l a t i o n parameters, t h e sample i t s e l f u s i n g e q u a t i o n (36) , and from t h e sample u s i n g e q u a t i o n (37) .  F u r t h e r , these e s t i -  mates were used t o e s t i m a t e t h e mean volume a t the second occasion.  The o b j e c t was t o determine d i f f e r e n c e s i n s-p r  2  by t h e d i f f e r e n t methods and t o f u r t h e r determine t h e e f f e c t s upon mean volume e s t i m a t i o n .  The r e s u l t s a r e t a b u l a t e d i n  45  A P P E N D I X  of  s-y  2  r  A n a l y z i n g the r e s u l t s , the l a r g e s t d e v i a t i o n  III.  as computed from the reduced form from t h a t computed  from the f u l l  e q u a t i o n was  1 p a r t i n 300,000.  form gave a s m a l l e r average d e v i a t i o n from the p o p u l a t i o n mean volume)(on than d i d the f u l l ple  The  reduced  (mean volume s u b t r a c t e d  the o r d e r of -0.0164)  e x p r e s s i o n (order -0.0705).  F o r t h i s sam-  then, the reduced form gave s u f f i c i e n t l y p r e c i s e , i f n o t  b e t t e r , e s t i m a t e s of both a-^  2  and  y.  r Cochran  (1963) suggested a p o s s i b l e answer f o r t h i s  v e r y s m a l l amount of b i a s .  The b a s i c d e r i v a t i o n s f o r double  sampling w i t h r e g r e s s i o n assume a simple random sample on b o t h occasions.  From l e a s t squares r e g r e s s i o n t h e o r y , the s t a n -  dard e r r o r of the r e g r e s s i o n c o e f f i c i e n t i s minimized i f the sampled v a l u e s are a t the extremes o f the range. the t e c h n i q u e used i n these a n a l y s e s .  The  This i s  88 o r i g i n a l  plots  s e l e c t e d i n each case were s o r t e d i n o r d e r of i n c r e a s i n g volume.  The  74 remeasured  plot  p l o t s were s e l e c t e d by choosing the  lowest 30 and the h i g h e s t 30 p l o t s by volume.  The  remaining  14 were s y s t e m a t i c a l l y s e l e c t e d as every o t h e r p l o t from the 32nd t o 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 -  g r e s s i o n c o e f f i c i e n t s h o u l d be h i g h . t h i s form of sampling may  Cochran  suggested  reduce the term i n l/(m-3),  that  . . .  "perhaps c o n s i d e r a b l y " . In  s m a l l p o p u l a t i o n s , i t becomes n e c e s s a r y t o use  a f i n i t e p o p u l a t i o n c o r r e c t i o n i n SPR  sampling.  Indeed, i n  the examples p r e s e n t e d where the t o t a l p o p u l a t i o n was p l o t s , i t was  220  n e a r l y i m p o s s i b l e t o o b t a i n any s o l u t i o n t o  46  the d o u b l y - c o n s t r a i n e d had  been a p p l i e d .  sure and  c o s t f u n c t i o n u n t i l these c o r r e c t i o n s  A d e r i v a t i o n of optimum numbers to remea-  to r e p l a c e was  given  (Eqn's 30 and  32).  As  seen, the number t o remeasure i s not a f f e c t e d , but,  can  be  as would  be expected, the number to r e p l a c e i s reduced i n the term i n o /K. 2  T h i s formula should  be used i f t h e r e i s any  question  of s m a l l sample s i z e o r sampling without replacement. Other sources o f e r r o r , though not a l l s t r i c t l y forms of b i a s , i n c l u d e some f a m i l i a r problems from f o r e s t inventory. devices,  Some examples are, i n a c c u r a t e  i n c o r r e c t tree species  bias i n on-location  identification,  sample p l o t s e l e c t i o n , and  conventional  measuring observer  i n the  SPR  case, because r e g r e s s i o n e s t i m a t e s are used, the f a i l u r e the x^  t o be known w i t h o u t e r r o r .  should  be aware of these b i a s e s  and  Users or p o t e n t i a l u s e r s s h o u l d make e f f o r t s t o  see t h a t t h e i r e s t i m a t e s are e i t h e r not s u b j e c t t o or r e c t e d f o r these problems.  of  cor-  f7  CONCLUSIONS In t h i s t h e s i s we examined some a p p l i c a t i o n s o f sampling  w i t h p a r t i a l replacement o f sample u n i t s on s u c c e s -  sive occasions.  An e x p r e s s i o n f o r e s t i m a t i n g t h e mean on a  second o c c a s i o n was o b t a i n e d .  V a r i o u s e s t i m a t o r s of change  between measurement p e r i o d s were formed and d i s c u s s e d . examples o f f o r e s t i n v e n t o r y p l a n n i n g were p r e s e n t e d  Two  that  compared sample s i z e s and c o s t s o f independent mean volume and growth e s t i m a t i o n and SPR e s t i m a t i o n of t h e same parameters. SPR  Changing s i z e o f i n v e n t o r y a r e a was examined from t h e  viewpoint.  The main q u e s t i o n i n v e s t i g a t e d i n these ex-  amples i s whether SPR can r e a l i z e the h y p o t h e t i c a l savings i n time and expense w i t h equal  precision.  In t h e h y p o t h e t i c a l case  t h a t o n l y mean volume  was r e q u i r e d on an o c c a s i o n o t h e r than t h e f i r s t ,  SPR would  equal the c o n v e n t i o n a l method o f independent samples i n p r e c i s i o n a t reduced  costs.  As was shown, a s a v i n g s  occurs  whether the g o a l o f the survey i s o n l y a two o c c a s i o n volume survey o r an o c c a s i o n sometime between t h e p e r i o d o f growth and volume d e t e r m i n a t i o n .  I conclude  that f o r estimation  of a mean on more than one o c c a s i o n , SPR i s s u p e r i o r t o conv e n t i o n a l independent surveys  i n c o s t and presumably i n time  expenditure. In t h e s i t u a t i o n where both c u r r e n t mean and changes are r e q u i r e d on s u c c e s s i v e o c c a s i o n s , SPR i s a g a i n a s u p e r i o r system i n terms of c o s t .  In these cases, an i n c r e a s e i n the  48  p r e c i s i o n of the e s t i m a t e of the mean can as l o n g as the numbers of p l o t s r e q u i r e d of t h i s parameter on the f i r s t for  estimation  of change.  occasion  a l s o be f o r the  exceed those  than t h a t of  increment p r e c i s i o n , and  required  these g r e a t e r numbers g i v e  more p r e c i s e e s t i m a t e of the mean than was Time savings may  required  increment.  T h i s t r a n s l a t e s t o , g r e a t e r numbers of p l o t s are  fied.  estimation  T h i s , because the magnitude o f  the mean i s n o r m a l l y much g r e a t e r  for  expected,  or may  initially  a  speci-  not be i n f a v o r o f SPR  in this  s i t u a t i o n as the l a r g e number o f permanent p l o t s t h a t must be e s t a b l i s h e d and  subsequently remeasured w i l l r e q u i r e a  d e a l o f time e x p e n d i t u r e . r e p o r t of an a c t u a l sis)  SPR  trial  There has  been o n l y one  ( r a t h e r than s i m u l a t e d  (Prayer  e f f e c t i v e i n reducing  et a l . 1971). b o t h c o s t and  great  recent  as i n t h i s  analy-  In t h a t study SPR  was  time e x p e n d i t u r e s .  t i l more a c t u a l t r i a l s can be analyzed, t h i s q u e s t i o n  Unmust  remain unanswered. An  i n t e r e s t i n g question  i s how  or p l o t l e s s c r u i s i n g be used w i t h SPR? t i o n i s p e r i p h e r a l t o SPR,  inevitable.  used i n two  One,  of prism  variable-radius  Although t h i s ques-  the obvious c o s t r e d u c t i o n s  make c o n s i d e r a t i o n ways.  can  possible  T h i s type of p l o t c o u l d  the more obvious, i n v o l v e s the  use  p l o t s as the temporary non-remeasured p l o t s .  c o u l d b r i n g about a dramatic d i f f e r e n c e i n p l o t c o s t s reduce the number o f permanent p l o t s needed. i n c o m p a t i b i l i t i e s w i t h SPR  theory.  be  This and  There are  no  Again i t i s a matter of  49  actual t r i a l s t i o n i s can measured?  to determine the g a i n s .  prism  The  p l o t s be used as permanent p l o t s and  I f e e l that t h i s i s e n t i r e l y possible.  ways, i t c o u l d have d i s t i n c t advantages. vertex  more complex ques-  The  re-  In some  p o i n t of  the  of the sweep would have to be p r e c i s e l y l o c a t e d f o r  f u t u r e measurements.  The  " i n " t r e e s should  p o i n t of d.b.h.o.b. measurement.  But  n e c e s s a r y than w i t h c o n v e n t i o n a l  be  tagged a t  the  l e s s marking would  permanent p l o t s .  be  T h i s would  cause a more n a t u r a l treatment of the area which would r e move some of the b i a s e s  t h a t do occur i n t h i s s i t u a t i o n .  T h i s would i n t u r n g i v e a b e t t e r p i c t u r e of f o r e s t d e v e l o p ment and  use. Another i n n o v a t i o n  proposed f o r use  w i t h SPR,  i n sampling t e c h n i q u e has  that i s , tree s e l e c t i o n with pro-  b a b i l i t y p r o p o r t i o n a l to p r e d i c t i o n (3-P). use of 3-P porary  I envision  as a stage i n a two-stage system.  the  In both tem-  and permanent p l o t s i t u a t i o n s , the p l o t b o u n d a r i e s  c o u l d be l o c a t e d and and  been  the t r e e s w i t h i n the p l o t would be  s e l e c t e d f o r measurement by a 3-P  system.  This  visited  could  work q u i t e e a s i l y f o r the temporary p l o t s , but more d e l i b e r a t i o n i s n e c e s s a r y f o r the other  type.  are s e l e c t e d f o r the f i r s t o c c a s i o n  be marked and  remeasured on a l l subsequent o c c a s i o n s ? c o n s t i t u t e a new  3-P  selection?  the l a t t e r , as the t r e e s would be to d e s c r i b e  Should the t r e e s  Or should  that  therefore each  visit  I p e r s o n a l l y tend towards s e l e c t e d on each  occasion  the p l o t which i s then used f o r l i n k i n g between  50  periods.  These two  areas should  be  subjects for further  investigations. Another obvious q u e s t i o n does SPR  have i n B r i t i s h Columbia?  i s what a p p l i c a b i l i t y T h i s q u e s t i o n must be  d i s c u s s e d w i t h i n the framework of the B.C.  Forest  as most f o r e s t l a n d i s owned by the p r o v i n c e . i e s o f the P u b l i c S u s t a i n e d  One  Inventory  s e c t i o n bears r e s p o n s i b i l i t y f o r estimates  r e n t volumes, the o t h e r  i s i n v o l v e d w i t h growth.  t i o n e s t a b l i s h e s i t s own i s made o f i n f o r m a t i o n  sampling network and  from the o t h e r group.  The  Each s e c little  use  Politically,  The  survey  Both s e c t i o n s e s t a b l i s h s i m i l a r type  growth s e c t i o n p l o t s are more permanently marked  and more t r e e s are measured f o r c e r t a i n v a r i a b l e s , but are few  cur-  s e c t i o n s would not i n -  v o l v e a g r e a t change from the p r a c t i c a l s i d e .  The  of  happen.  amalgamation of the two  schemes are s i m i l a r .  Divi-  framework t o  the survey requirements a t reduced c o s t s .  I do not b e l i e v e t h i s can  plots.  very  two  Probabilistically,  these s e c t i o n s c o u l d be combined w i t h i n an SPR achieve  inventor-  Y i e l d U n i t s are conducted by  s e c t i o n s of the B.C.F.S. F o r e s t Surveys and sion.  The  Service,  striking differences.  Given the immensity o f  there the  f o r e s t survey r e q u i r e d throughout B r i t i s h Columbia, the i n c o r p o r a t i o n o f SPR  i n t o the sampling design  could r e s u l t i n  r e d u c t i o n of manpower requirements, time, and  costs.  FUTURE CONSIDERATIONS In t h i s study, SPR  was  considered,  i.e.  mean volume per acre and to p r e s e n t SPR  o n l y an elementary a p p l i c a t i o n of simple  random sampling f o r e s t i m a t i n g  i t s change o v e r time.  the b a s i s of SPR  This  served  and t o e s t a b l i s h i t s e f f i c i e n c i e s *  i s an e f f i c i e n t temporal sampling t e c h n i q u e  which must be  a p p l i e d i n c o n j u n c t i o n w i t h s p a t i a l sampling methods* we must c o n s i d e r where SPR  Now  can be a p p l i e d t o e x p l o i t i t s de-  sirable features. A s i t u a t i o n s i m i l a r i n nature outdoor r e c r e a t i o n .  T h i s f i e l d has  l e c t i o n and a n a l y s i s .  The  i s in  had problems i n data  very s p e c i f i c area o f  campsite use c o u l d be a t a r g e t of SPR a highly-developed  to t h i s study  col-  estimating  application.  Within  park, a simple random sample of v i s i t s  campsites c o u l d be made d u r i n g a high-use season.  to  From t h i s  sample, a mean and v a r i a n c e o f number o f v i s i t s c o u l d be termined.  The  f o l l o w i n g season some campsites c o u l d be  measured and new estimates  ones measured as replacement u n i t s •  In subsequent y e a r s , more  samples c o u l d be made and a n a l y z e d theory  re-  SPR  c o u l d then be made o f means f o r both o c c a s i o n s  f o r growth i n v i s i t o r use.  SPR  de-  (Cunia and  and  SPR  using multiple-occasion  Chevrou 1969).  F u r t h e r , w i t h i n a park,  t h e r e might be campsites w i t h v a r y i n g l e v e l s of f a c i l i t i e s fered.  These c o u l d be s t r a t i f i e d  and u s e - l e v e l s and  e s t a b l i s h e d f o r each type o f f a c i l i t y .  An a t t r a c t i v e  of-  trends 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 u n i t s (campsites)  and  the 1:1  correspondence  o f c o s t s o f remeasured  and temporary p l o t s . Moving from a simple a p p l i c a t i o n t o the o t h e r  ex-  treme, the knowledge of f o r e s t s t a n d c h a r a c t e r i s t i c s and namics may  dy-  be the most important c o n t r i b u t i o n o f f o r e s t samp-  l i n g t o management's decision-making  processes.  At p r e s e n t ,  the d e s c r i p t i v e parameters a r e i n t e r r e l a t e d , but  usually  s e p a r a t e l y i n v e n t o r i e d , f e a t u r e s of the f o r e s t .  True, many  v a r i a b l e s are measured d u r i n g an i n v e n t o r y , but these are normally a p p l i e d 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 v a r i a b l e ) o r used i n separate a n a l y s e s to d e s c r i b e s e v e r a l such f a c t o r s . t i s t i c a l procedures  Through the use o f m u l t i v a r i a t e s t a -  many f a c t o r s c o u l d be j o i n t l y  analyzed  w i t h an i n c r e a s e i n the d i s c e r n i n g power of the a n a l y s i s . C o n s i d e r the c o n s t r u c t i o n and use o f growth and y i e l d tables. temporal  These t a b l e s n o r m a l l y i n v o l v e  simultaneous  p r e s e n t a t i o n o f number of stems, b a s a l a r e a , mean  diameter, mean h e i g h t , stand form f a c t o r , and  stand volume.  Many measured v a r i a b l e s are used t o determine  the a p p r o p r i a t e  e x p r e s s i o n of these f a c t o r s .  These i d e a l l y i n c l u d e age,  de-  gree o f s t o c k i n g , s i t e index, s i l v i c u l t u r a l programs, e t c . The f o r m u l a t i o n o f these t a b l e s i s an expensive procedure.  F u r t h e r , the a p p l i c a b i l i t y  to stands w i t h l i k e  characteristics.  time-consuming  i s limited  ( i n theory)  53  We  s h a l l now  put the c a r t b e f o r e the horse and  s i d e r the use o f y i e l d t a b l e s .  I f the a p p r o p r i a t e v a r i a b l e s  were m u l t i v a r i a t e l y measured and analyzed, the s h i p s would become more apparent fidence regions. proach  from  interrelation-  the use of j o i n t  con-  We might see, f o r i n s t a n c e , whether an  to n o r m a l i t y of a stand was  i n f l u e n c e d by  i n stocking.  ap-  certain  v a r i a b l e s and n o t o t h e r s , or indeed i f our stand was changing  con-  truly  Once we e s t a b l i s h the l e v e l of s t o c k i n g  i n our stand, a l o n g w i t h o t h e r m u l t i v a r i a t e f a c t o r s - o f - i n t e r e s t , we  s h o u l d be a b l e to use the m u l t i v a r i a t e growth and  yield  t a b l e t o a s s i s t i n management d e c i s i o n s . But what has  t h i s t o do w i t h SPR,  and how  formulate these m u l t i v a r i a t e growth and y i e l d  would  tables?  we  A usual  method o f f o r m u l a t i o n has been to u t i l i z e permanent sample p l o t s i n homogeneous even-aged p o r t i o n s of stands.  But f o r e s t  con-  d i t i o n s are not n e c e s s a r i l y i n t h i s s t a t e o f homogeneity. the permanent sample p l o t s are not c h a r a c t e r i s t i c of the t i r e stand.  "This disadvantage  by u s i n g as the source o f y i e l d  can be s u b s t a n t i a l l y  So en-  reduced  t a b l e d a t a a combination  of  permanent f u l l y - s t o c k e d sample p l o t s and temporary p l o t s , o b j e c t i v e l y s e l e c t e d i n the course of a f o r e s t i n v e n t o r y " (Loetsch et al. V o l . I I , 1973).  T h i s 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 t a b l e s c o u l d be f o r m u l a t e d i n an unbiased manner w i t h SPR  t e c h n i q u e s and l i k e w i s e SPR  techniques might prove most  e f f i c i e n t to use w i t h the subsequent sampling  f o r the n e c e s s a r y  54  variables to u t i l i z e  these tables.  Conjecturally, the new  view of the i n t e r r e l a t i o n s h i p s might shed l i g h t on some of the plaguing problems i n f o r e s t r y , such as m o r t a l i t y . This i s the d i r e c t i o n recommended f o r further of SPR; a multivariate efficient  study  SPR procedure f o r use with the most  s p a t i a l sampling methods.  These must be procedurally  s i m p l i f i e d f o r use by personnel not conversant with t h i s type of sampling theory  (among whom the author may be counted).  Then a long-range plan of implementation should be formulated and begun i n cooperation with a l l agencies interested i n e v a l uating t h e i r holdings with a view toward f u l l r e a l i z a t i o n of the production p o t e n t i a l .  55  BIBLIOGRAPHY Avery, T. Eugene. 1967. F o r e s t measurements. Book Company, Inc., New York. 290 p.  McGraw-Hill  B i c k f o r d , C A . 1956. Proposed d e s i g n f o r c o n t i n u o u s i n v e n t o r y : a system of p e r p e t u a l F o r e s t Survey f o r the N o r t h e a s t . U.S. F o r e s t S e r v i c e E a s t e r n Techniques Meeting, F o r e s t Survey, Cumberland F a l l s , Ky. Oct. 8-13, 1956. 37 p. . 1959. A t e s t of continuous i n v e n t o r y f o r N a t i o n a l F o r e s t management based on a e r i a l photographs, double sampling, and remeasured p l o t s . Soc. Amer. F o r e s t e r s Proc. 1959:143-148. Cochran, W.G. 1963. Sampling t e c h n i q u e s , 2nd ed. John W i l e y & Sons, I n c . , New York. 413 p. Cochran, W.G., and G.M. Cox. 1957. Experimental designs, 2nd ed. John Wiley & Sons, I n c . , New York. 611 p. + tables. Cunia, T. 1964. What i s sampling w i t h p a r t i a l replacement and why use i t i n c o n t i n u o u s f o r e s t i n v e n t o r y ? Soc. Amer. F o r e s t e r s Proc. 1964:207-211. . 1965. Continuous f o r e s t i n v e n t o r y , p a r t i a l r e p l a c e ment o f samples and m u l t i p l e r e g r e s s i o n . F o r . S c i . 11: 480-502. . 1973. Dummy v a r i a b l e s and some o f t h e i r uses i n regression analysis. Proc. o f the meeting o f IUFRO Subj e c t Group S4.02, Nancy, France. June 25-29, 1973. V o l . 1. 146 p. , and R.B. Chevrou. 1969. Sampling w i t h p a r t i a l r e placement on t h r e e o r more o c c a s i o n s . F o r . S c i . 15:204224. Davis, Kenneth P. 1966. F o r e s t management: r e g u l a t i o n and v a l u a t i o n , 2nd ed. McGraw-Hill Book Company, Inc., New York. 519 p. Draper, N.R., and H. Smith. 1966. John W i l e y & Sons, Inc., New  Applied regression York. 407 p.  analysis.  F r a y e r , W.E., R.C. VanAken, and R.D. S u l l i v a n . 1971. Applicat i o n of sampling w i t h p a r t i a l replacement t o timber i n v e n t o r i e s , C e n t r a l Rocky Mountains. F o r . S c i . 17:160-162.  56  Hansen, M.H., W.N. Hurwitz, and W.G. Madow. 1953. Sample s u r vey methods and t h e o r y . V o l . I , Methods and a p p l i c a t i o n s , 638 p. V o l . I I , Theory, 332p. John W i l e y & Sons, I n c . , New York. Husch, B., C . I . M i l l e r , and T.W.  Beers. 1972.  F o r e s t mensura-  t i o n , 2nd ed. The Ronald P r e s s Company, New J e s s e n , R.J. 1942.  York. 410  p.  S t a t i s t i c a l i n v e s t i g a t i o n s o f a sample  s u r v e y f o r o b t a i n i n g farm f a c t s . Iowa Agr. E x p t . S t a . Bull.  304.  Kramer, C Y .  10 p.  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 p u b l i s h e d . 331 p. K i s h , L e s l i e . 1965. New  Survey sampling.  John W i l e y & Sons, I n c . ,  York. 643 p.  L i , J.C.R. 1964. S t a t i s t i c a l i n f e r e n c e . V o l I , 658 p. V o l . I I , 575 p. Edwards B r o t h e r s , Inc., Ann A r b o r , M i c h i g a n . L o e t s c h , F., F. Zohrer, and K.E. H a l l e r . 1964, V o l , I , 436 p.; 1973, V o l . I I , 469 p. ( E n g l i s h by K.F. P a n z e r ) . BLV V e r l a g s g e s e l l s c h a f t mbH, Munchen, Germany. Madison, R.W. 19 57. A g u i d e t o t h e Cascade Head E x p e r i m e n t a l F o r e s t . U.S. F o r e s t S e r v i c e P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon, 14 p. + map. Newton, C M . 1971. Sampling w i t h p a r t i a l replacement as app l i e d t o two s u c c e s s i v e m u l t i v a r i a t e f o r e s t measurements, Ph.D. T h e s i s . Syracuse U n i v . 83 p. + computer program l i s t i n g . Univ. M i c r o f i l m s , Ann A r b o r , M i c h i g a n . Nyyssonen, Aarne. 1967. Remeasured sample p l o t s i n f o r e s t i n v e n t o r y . Medd. Det. Norske S k o g f o r s ^ k s v e s , 84(22): 195-220. P a t t e r s o n , H.D. 1950. Sampling on s u c c e s s i v e o c c a s i o n s w i t h p a r t i a l replacement o f u n i t s . J . Royal S t a t i s , Soc., Ser. B, 12:241-255. R e i f , F. 1965. Fundamentals o f s t a t i s t i c a l and thermal p h y s i c s , McGraw-Hill Book Company, I n c . , New  York. 651  p.  S o c i e t y o f American F o r e s t e r s . 1958. F o r e s t t e r m i n o l o g y , 3rd ed. Washington, D.C. Smith, D.M. 1962. The p r a c t i c e o f s i l v i c u l t u r e , 7th ed. (publ 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, i n c . , New York. 578 p. c  57  Snedecor, George W., and W i l l i a m G. Cochran. 1971. Statistic a l methods, 6th ed. The Iowa S t a t e Univ. P r e s s , Ames, Iowa. 593 p. Ware, K.D. 1960. Optimum r e g r e s s i o n sampling d e s i g n f o r f o r e s t p o p u l a t i o n s on s u c c e s s i v e o c c a s i o n s . Ph.D. T h e s i s . Y a l e Univ. 154 p. + appendices. Ware, K.D., and T i b e r i u s Cunia. 1962. Continuous f o r e s t i n v e n t o r y w i t h p a r t i a l replacement o f samples. F o r . S c i , Monog. 3. 4 0 p.  58  APPENDIX 1 COMPUTING FORMULA FOR VOLUME OF HEIGHT-MEASURED TREES The e m p i r i c a l volume formula developed by t h e F o r e s t Survey, U.S. F o r e s t S e r v i c e i s VOLUME = 8.436 - 2.608*DIA + 0.070242*DIA*DIA - 12.18*FC*FC + (0.002474*CH*CH  + 3.1278*CH  - 0.00792)*(HT -  17.0), where DIA i s diameter o u t s i d e bark a t b r e a s t h e i g h t (4.5 f t above mean germination FC  i s G i r a r d form c l a s s ;  CH  i s DIA*FC;  HT  i s measured t r e e h e i g h t .  point;  59  APPENDIX I I REGRESSION COEFFICIENTS FOR VOLUME COMPUTATION FROM /(D.B.H.O.B.) Tsuga  heterophyI  la  Period  b  b  2  Obns.  3  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  sitchensis  Period  b  and Pseudotsuga bi  0  menziesii bz  b  Obns.  3  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 veloped  for  d e s c r i b e the r e l a t i o n s h i p s de-  t r e e volume as a f u n c t i o n >of diameter.  The model  is  Logio^  :  = So + 6 i ^ i + B * l  2  2  + 3 *i 3  3  +  ^  where I.  = volume i n c u b i c f e e t ;  Xi = diameter o u t s i d e bark a t b r e a s t e. = random e r r o r ,  height;  ^N(0,a ); 2  = true population regression c o e f f i c i e n t s of which t h e b^ a r e l e a s t squares e s t i m a t e s .  60  The diameter Tsuga  ranges  used i n these a n a l y s e s were:  heterophylla  6.9 inches <_ d.b.h.o.b. <_ 31.2 i n c h e s ;  Picea sitchensis „ , , ... Pseudotsuga menz%es%%  -i n -> • , , , , , , 10.2 i n c h e s < d.b.h.o.b. < 51.4 i n c h e s . — —  61 APPENDIX I I I COMPUTED STATISTICS OF F I F T Y RANDOM SPR SAMPLES  p population  mean volume per acre computed from population parameters  p reduced  mean volume .per acre computed from sample s t a t i s t i c s and the reduced form of  p  mean volume per acre computed from sample s t a t i s t i c s and the f u l l form of  full  o-  2  a-  2  Obsn  reduced  e r r o r variance of regression estimate computed from sample s t a t i s t i c s (reduced equation)  full  error variance of regression estimate computed from sample s t a t i s t i c s ( f u l l equation)  y population  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  21421.39 21065.33 21884.93 21601.29 21508.96 22197.78 21131.95 21969.85 21949.69 21116.18 22280.21 21033.21 21791.20 21264.52 20569.65 21754.28 21718.28 22280.85 21773.03 21755.93 22262.70 21462.05 22159.20 22393.57 20488.13 21115.20 21837.08 21457.13 21556.88 21549.00 21542.62 21142.55 22030.74 21940.78 21732.68 21127.29 21298.20 20843.33 22320.38 21277.65 21285.03 21785.73 22363.77 22416.03 21706.61 22221.19 21421.71 22104.85 21943.27 21941.61  mean  21655.9  std. dev.  477.389  y reduced  Sy  21424.85 21065.13 21884.63 21601.45 21507.72 22194.10 21132.27 21969.39 21948.03 21117.87 22276.36 21033.48 21789.77 21262.50 20569.06 21753.25 21717.08 22280.29 21773.60 21756.81 22263.09 21457.68 22157.54 22393.55 20491.84 21112.83 21833.08 21459.39 21556.29 21549.97 21542.50 21143.76 22031.05 21941.01 21733.38 21134.52 21297.79 20848.98 22318.95 21275.57 21284.64 21787.33 22364.07 22416.39 21707.04 22227.27 21419. 14 22104.38 21943.34 21940.70  2  y full  reduced  Sy  2  full  I*  363826 319095 430616 376216 372901 368583 435015 340193 437433 415434 321704 425515 321563 423523 345330 295008 391610 392562 404994 427902 306629 342772 383534 381479 355906 302759 326493 387506 325142 398870 366954 324399 366341 338800 375625 372133 385756 314216 439907 455254 378870 320947 415439 386259 431039 379272 374690 349350 454156 384738  21424.85 21065.14 21884.63 21601.46 21507.72 22194.10 21132.27 21969.39 21948.02 21117.86 22276.36 21033.49 21789.77 21262.50 20569.05 21753.26 21717.08 22280.28 21773.61 21756.81 22263.08 21457.67 22157.54 22393.55 20491.84 21112.82 21833.08 21459.39 21556.29 21549.93 21542.51 21143.75 22031.05 21941.01 21733.37 21134.52 21297.79 20648.99 22318.94 21275.57 21284.63 21787.33 22364.07 22416.39 21707.04 22227.77 21419.14 22104.37 21943.34 21940.70  363827 319096 430617 376217 372901 368583 435015 340194 437433 415434 321704 425515 321564 423523 345331 295008 391611 392562 404995 427903 306629 342773 383534 381479 355907 302760 326493 387506 325143 398871 366955 324400 366341 338801 375626 372134 385757 314216 439907 455255 37e870 320947 415440 386259 431040 379272 374691 349350 454156 384738  21655.8  374684  21655.8  374685  476.780  42673.5  475.779  42673.5  V = 21558.8  oy  2  = 325782  62  APPENDIX IV  REQUIRED SAMPLE SIZES AT VARIOUS COST RATIOS For a g i v e n i n v e n t o r y s i t u a t i o n , t h e o p t i m a l numbers o f sample u n i t s t o remeasure m and t o r e p l a c e n w i l l v a r y w i t h the r a t i o o f t h e c o s t s o f these p l o t s ,  c m  /  c n  °  The  c o n s t r a i n t s i n the c o s t f u n c t i o n o p t i m i z a t i o n , when s i m u l t a n e o u s l y e s t i m a t i n g volume and growth i n v o l v e the s p e c i f i e d sampling  e r r o r s squared, K and K .  tion impractical.  T h i s makes a g e n e r a l  A s p e c i f i c case, Example One o f t h i s  soluthesis,  was a n a l y z e d t o convey an i m p r e s s i o n o f how t h e sample s i z e s w i l l vary.  The parameters and c o n f i d e n c e  are as s p e c i f i e d f o r t h a t example. s i s a r e p r e s e n t e d both n u m e r i c a l l y  (Figure Table  i n t e r v a l half-widths  The r e s u l t s o f t h e a n a l y (Table  5) and g r a p h i c a l l y  1). 5.  Optimal  sample  c  /c  m  n  1 : 1  sizes  at various  m  cost  n  86  6  2  :  1  74  25  3  :  1  65  46  4  :  1  61  60  8  :  1  49  122  ratios.  63  FIGURE  1.  remeasure  The and  optimal to  replace  numbers at  of  various  sample cost  plots  to  ratios,  ea  8:1  co 01  a  CQ  CO H—  -1°. (0  4:1  UJ  z:  UJ  aim  3:1  LU  a:  tH 2:1 a  CD.  1:1 a o  i 15.0  1  1—  31.0 47.0 63.0 REMEASURED PLOTS  79.0  9SJ  

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