﻿60 
  

  

  50- 
  

  

  C 
  

  

  U 
  M- 
  

   B 
  

   I 
  

   C 
  

  

  F 
  

  

  30- 
  

  

  

  

  T 
  

  

  V 
  

   

  

  L 
  20- 
  

   U 
  

   M 
  

   E 
  

  

  10- 
  

  

  0- 
  

  

  A 
  t^' 
  A 
  

  

  A 
  

   A 
  

  

  A 
  

  

  A 
  ^--A-" 
  A 
  

  

  A 
  A 
  

  

  10 
  

  

  15 
  

  

  DRC 
  (INCHES) 
  

  

  20 
  

  

  ' 
  ' 
  ' 
  I 
  ' 
  ' 
  ' 
  

   25 
  

  

  30 
  

  

  Figure 
  1. 
  — 
  Volume 
  plotted 
  against 
  DRC 
  for 
  

   Utah 
  juniper 
  trees 
  from 
  the 
  Moab 
  BLM 
  

   District. 
  

  

  60- 
  

  

  50- 
  

  

  C 
  

  

  U 
  10- 
  

   B 
  

   I 
  

   C 
  

  

  F 
  

  

  30- 
  

   

  

  V 
  

   

  

  L 
  20- 
  

   U 
  

   M 
  

   E 
  

  

  10- 
  

  

  0- 
  

  

  A 
  

  

  A 
  

  

  a.Aa^^ 
  a 
  

  

  A 
  /A/^ 
  A^A 
  

  

  A 
  A 
  

   A 
  

  

  A 
  A 
  

  

  -r 
  

  

  -r 
  

  

  "T" 
  

  

  4000 
  8000 
  12000 
  16000 
  

  

  ORSQH 
  (DRC 
  SQUARED 
  TIMES 
  HEIGHT) 
  

  

  Figure 
  2. 
  — 
  Volume 
  plotted 
  against 
  DRSQH 
  

   for 
  Utah 
  juniper 
  trees 
  from 
  the 
  Moab 
  BLM 
  

   District. 
  

  

  20000 
  

  

  analyzed 
  in 
  exploratory 
  plots, 
  multiple 
  regressions, 
  and 
  

   stepwise 
  regressions. 
  Some 
  benefit 
  in 
  volume 
  predictions 
  

   resulted 
  from 
  adding 
  HT 
  and 
  STEMS 
  into 
  the 
  volume 
  

   prediction 
  model, 
  but 
  most 
  of 
  the 
  variability 
  in 
  the 
  

   DRC-volume 
  relationship 
  could 
  not 
  be 
  explained. 
  The 
  

   crown 
  variables 
  seemed 
  to 
  add 
  very 
  little 
  to 
  the 
  volume 
  

   prediction 
  model, 
  when 
  DRC 
  was 
  already 
  in 
  the 
  model. 
  

   The 
  DRC 
  and 
  HT 
  variables 
  were 
  combined 
  into 
  a 
  simple 
  

   variable, 
  DRSQH, 
  by 
  multiplying 
  DRC 
  squared 
  times 
  

   HT. 
  A 
  diameter 
  and 
  height 
  combination 
  variable 
  that 
  

   predicts 
  volume 
  well 
  for 
  commercial 
  timber 
  species 
  

   worked 
  as 
  well 
  for 
  P-J. 
  The 
  STEMS 
  variable 
  was 
  ren- 
  

   dered 
  almost 
  useless 
  because 
  of 
  an 
  apparent 
  interaction 
  

   between 
  stem 
  sizes 
  (not 
  measured) 
  and 
  number 
  of 
  stems 
  

   for 
  a 
  given 
  P-J 
  tree. 
  However, 
  it 
  helped 
  volume 
  predic- 
  

   tions 
  somewhat 
  to 
  use 
  a 
  dummy 
  variable 
  to 
  indicate 
  

   whether 
  a 
  tree 
  was 
  multiple-stem 
  or 
  single-stem. 
  

  

  Equation 
  Form 
  

  

  Modeling 
  the 
  DRSQH 
  to 
  volume 
  relationship 
  as 
  a 
  sim- 
  

   ple 
  linear 
  equation 
  would 
  be 
  desirable 
  for 
  field 
  use, 
  but 
  

   there 
  were 
  problems 
  with 
  this 
  choice 
  as 
  illustrated 
  in 
  fig- 
  

   ure 
  2. 
  Moab 
  juniper 
  data 
  show 
  the 
  variance 
  of 
  volume 
  

   increasing 
  with 
  tree 
  size. 
  This 
  created 
  a 
  problem 
  because 
  

   the 
  few 
  largest 
  trees 
  disproportionately 
  dominated 
  the 
  

   outcome 
  of 
  regression 
  coefficient 
  estimation. 
  

  

  The 
  log 
  transformation 
  is 
  commonly 
  used 
  to 
  deal 
  with 
  

   increasing 
  variance 
  problems 
  in 
  regression. 
  This 
  trans- 
  

   formation 
  rescales 
  data 
  so 
  that 
  small 
  and 
  large 
  trees 
  

   have 
  the 
  same 
  impact 
  upon 
  estimation 
  of 
  the 
  regression 
  

   coefficients. 
  Transforming 
  by 
  applying 
  fractional 
  powers 
  

   (such 
  as 
  X'% 
  X'\ 
  and 
  so 
  forth) 
  will 
  also 
  accomplish 
  

  

  the 
  same 
  purpose 
  as 
  the 
  log 
  transformation. 
  After 
  ex- 
  

   amining 
  several 
  transformations 
  on 
  a 
  subset 
  of 
  the 
  data, 
  

   the 
  log 
  and 
  cube 
  root 
  transformations 
  were 
  selected 
  for 
  

   comparison 
  on 
  all 
  data. 
  

  

  Figures 
  3 
  and 
  4 
  demonstrate 
  the 
  effect 
  of 
  the 
  log 
  and 
  

   cube 
  root 
  transformations 
  on 
  the 
  Moab 
  juniper 
  data. 
  

   The 
  log 
  transformation 
  appeared 
  to 
  compress 
  the 
  data 
  

   too 
  much 
  for 
  large 
  trees, 
  actually 
  decreasing 
  the 
  vari- 
  

   ance 
  with 
  increasing 
  tree 
  size. 
  The 
  cube 
  root 
  transforma- 
  

   tion 
  looked 
  more 
  reasonable. 
  

  

  All 
  data 
  for 
  the 
  other 
  species 
  from 
  other 
  areas 
  

   responded 
  to 
  the 
  transformations 
  the 
  same 
  way 
  the 
  

   Moab 
  data 
  did. 
  Additional 
  plots 
  of 
  DRSQH 
  against 
  vol- 
  

   ume 
  with 
  stem 
  counts 
  overlaid 
  showed 
  some 
  gain 
  from 
  

   inclusion 
  of 
  a 
  dummy 
  variable 
  to 
  distinquish 
  single- 
  from 
  

   multiple-stem 
  trees. 
  Therefore, 
  the 
  final 
  equation 
  form 
  

   selected 
  for 
  regression 
  estimation 
  of 
  the 
  coefficients 
  was: 
  

  

  Vj'''' 
  = 
  a 
  + 
  b(DRSQHi)'' 
  + 
  c(STEMi) 
  + 
  €; 
  (1) 
  

   where 
  

  

  Vj 
  = 
  visually 
  estimated 
  cubic 
  foot 
  volume 
  to 
  1.5-inch 
  

   minimum 
  branch 
  diameter 
  (includes 
  live 
  wood, 
  dead 
  

   wood, 
  and 
  bark) 
  of 
  the 
  ith 
  tree 
  

  

  DRSQH; 
  = 
  DRC 
  squared 
  times 
  total 
  height 
  of 
  the 
  ith 
  

   tree 
  

  

  STEMj 
  = 
  1 
  if 
  a 
  single-stem; 
  if 
  a 
  multiple-stem 
  of 
  the 
  

   ith 
  tree 
  

  

  a, 
  b, 
  c 
  = 
  coefficients 
  to 
  be 
  estimated 
  by 
  regression 
  

   ej 
  = 
  random 
  error 
  (assumed 
  to 
  be 
  zero 
  on 
  the 
  average) 
  

   of 
  the 
  ith 
  tree. 
  

  

  During 
  the 
  analysis, 
  I 
  uncovered 
  evidence 
  for 
  question- 
  

   ing 
  the 
  quality 
  of 
  some 
  of 
  the 
  visual 
  volume 
  data. 
  

  

  3 
  

  

  