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An assessment of variable radius plot sampling techniques for measuring change over time : a simulation… Carter, David Hugh Harrison 2007

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A N ASSESSMENT OF V A R I A B L E RADIUS PLOT SAMPLING TECHNIQUES FOR MEASURING C H A N G E OVER TIME: A SIMULATION STUDY by DAVID H U G H HARRISON CARTER B.Sc. (Hons.), University of Waterloo, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Forestry) The University of British Columbia October 2007 © David Hugh Harrison Carter, 2007 ABSTRACT The most commonly used approach for measuring change over time involves using Fixed Radius Plots (FRPs). A major disadvantage of using FRPs for detecting change is the cost of implementation, both statistically and monetarily. Variable Radius Plots (VRP) can also used for measuring change over time; however, VRPs have not been widely used due to perceived problems including: 1. sudden additions to the value of interest which produce high variability; 2. complex mathematical formulae used for computation; 3. use of an angle for measuring trees; and 4. lack of studies on the use of VRPs for measuring change over time. In this thesis four VRP change estimators were evaluated: (1) the traditional subtraction estimator; (2) the Grosenbaugh estimator; (3) the Distance Variable (DV) estimator; arid (4) the Flewelling estimator. Tree and stand data were generated using a stand simulator (StandSim), and stands were sampled using a series of SAS programs. Volume change, basal area change, and stems per hectare change were calculated for a series of 54 stand conditions in which density, dbh distribution, mortality, and spatial distribution were varied. Relative efficiencies were calculated comparing each VRP estimator to both the FRP estimator, and the subtraction estimator. The D V estimator had a relative efficiency greater than 1.0 (i.e., it was more precise) in 59.% of the scenarios for volume change, 15% of the scenarios for basal area change, and 0% of scenarios for stems per hectare change when compared to FRP. Mortality is a high source of variability for all estimators and in another comparison where high mortality scenarios were excluded, the D V estimator had a relative efficiency greater than 1.0 in 74% of scenarios for volume change, 30% of scenarios for basal area change, and 0% of scenarios for stems per hectare change. The D V estimator had a relative efficiency greater than 1.0 in at least 70% of the scenarios for each attribute when compared to the traditional subtraction method, excluding the high mortality scenarios. This thesis demonstrates that VRP estimators for measuring change over time (in particular the D V estimator) can reduce the variability introduced by sudden additions of value. VRP estimators are not so complex that they should be excluded as an option for use, and require no new field techniques. Further studies should be completed to support the use of VRPs for change detection. ii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS .. .iii LIST OF TABLES vi LIST OF FIGURES viii ACKNOWLEDGEMENTS .x 1.0 INTRODUCTION 1 2.0 THEORETICAL BACKGROUND 4 2.1 PROBABILITIES 4 2.2 T R E E - C E N T R E D V I E W 5 2.3 F I X E D R A D I U S P L O T S 6 2.4 V A R I A B L E RADIUS P L O T S 7 2.5 C H A N G E O V E R T I M E 10 2.5.1 Components of Growth / / 2.5.2 Compatibility 12 3.0 CHANGE ESTIMATORS 13 3.1 F I X E D R A D I U S P L O T E S T I M A T O R 13 3.1.1 Components of Growth and Change 14 3.1.2 Mechanics 15 3.2 V A R I A B L E R A D I U S P L O T - S U B T R A C T I O N 16 3.2.1 Components of Growth and Change 17 3.2.2 Mechanics 18 3.3 V A R I A B L E R A D I U S P L O T - G R O S E N B A U G H (1958) 19 3.3.1 Components of Growth and Change 20 3.3.2 Mechanics 21 3.4 V A R I A B L E R A D I U S P L O T - D I S T A N C E V A R I A B L E A P P R O A C H 22 3.4.1 Critical Height Sampling 22 3.4.1.1 Change Over Time 23 3.4.2 DV Sampling 24 3.4.3 Components of Growth and Change 26 3.4.4 Mechanics 2 7 iii 3.5 V A R I A B L E R A D I U S P L O T - F L E W E L L I N G ' S (1981) A P P R O A C H 28 3.5.1 Components of Growth and Change 30 3.5.2 Mechanics 31 4.0 SIMULATION 33 4.1 S T A N D S I M U L A T I O N 33 4.1.1 Stand Density 34 4.1.2 Diameter Distribution 34 4.1.3 Spatial Distribution 35 4.1.3.1 Random Distribution 35 4.1.3.2 Uniform Distribution 35 4.1.3.3 Clumped Distribution 36 4.1.4 Growth •. 37 4.1.5 Mortality 39 4.2 S A M P L E S I M U L A T I O N 39 5.0 MATERIAL AND METHODS 41 5.1 S T A N D S , S T A N D G E N E R A T I O N , G R O W T H , A N D IMPORT 42 5.7.7 Density 43 5.1.2 Diameter Distributions 43 5.1.3 Mortality .-. 44 5.1.4 Spatial Distribution 44 5.1.5 Stand Variability 46 5.1.6 Growth 46 5.1.7 Stand Data Import 46 5.2 S T A N D S A M P L I N G A N D V A L U E D E T E R M I N A T I O N 47 5.2.7 Volume 48 5.2.2 Basal Area 48 5.3 V A L U E STABILIZATION 48 5.4 P L O T SIZES 52 5.5 O U T P U T S 53 5.6 COMPARISONS 55 6.0 RESULTS 57 6.1 R E L A T I V E EFFICIENCIES F O R E R R O R A N D S A M P L E 59 6.2 D E N S I T Y 61 iv 6.3 D B H DISTRIBUTIONS 62 6.4 M O R T A L I T Y 63 6.5 S P A T I A L DISTRIBUTIONS 64 6.6 R E M O V A L OF H I G H M O R T A L I T Y 65 6.7 V R P E S T I M A T O R S R E L A T I V E TO O N E A N O T H E R 66 7.0 D I S C U S S I O N 68 7.1 C O M P A R I N G E S T I M A T O R M E C H A N I C S 68 7.2 BIAS A N D PRECISION G A I N S 69 7.3 I M P A C T O F S T A N D CONDITIONS 70 7.4 LIMITATIONS 72 8.0 F U T U R E D I R E C T I O N S A N D C O N C L U S I O N S 74 8.1 S T A N D S I M D E V E L O P M E N T 74 8.2 M E A S U R E M E N T E R R O R 75 8.3 S T A N D CONDITIONS 75 8.4 G R O W T H 76 8.5 C O S T 76 8.6 C O N C L U S I O N S 76 9.0 R E F E R E N C E S 78 A P P E N D I X I 82 A P P E N D I X II 9 0 A P P E N D I X III 98 v LIST OF TABLES Table 1. An example of change over time using the FRP estimator 15 Table 2. An example of change over time using the VRP - Subtraction estimator 18 Table 3. An example of change over time using the VRP - Grosenbaugh estimator 21 Table 4. An example of change over time using the VRP - D V method 27 Table 5. An example of change over time using the VRP - Fie welling method 31 Table 6. Stand Scenarios 42 Table 7. Dbh distributions used for stand simulation 44 Table 8. Values used for the spatial distribution of simulated stands 45 Table 9. FRP sizes and BAFs used for comparisons 53 Table 10. Statistical outputs for generated attributes 54 Table 11. Plot sizes and trees per plot used for comparisons 57 Table 12. Performance results for the VRP estimators relative to FRPs for three densities 61 Table 13. Performance results the alternate VRP estimators relative to the Subtraction estimator for three densities 62 Table 14. Performance results for the VRP estimators relative to FRPs for three dbh distributions 62 Table 15. Performance results for the three alternate VRP estimators relative to the Subtraction estimator for three dbh distributions 63 Table 16. Performance results for the VRP estimators relative to FRPs under different mortality levels 63 Table 17. Performance results for the three alternate VRP estimators relative to the Subtraction estimator under different mortality levels 63 Table 18. Performance results for the VRP estimators relative to FRPs for different spatial distributions 64 Table 19. Performance results for the three alternate VRP estimators relative to the Subtraction estimator for the different spatial distributions 65 Table 20. REs of the VRP estimators relative to each other 66 vi Table 21. Relative Efficiencies of VRP estimators relative to FRP 83 Table 22. REs of VRPs relative to the FRP Estimator for varying densities 84 Table 23. REs of VRPs relative to the FRP Estimator for varying diameter distributions 85 Table 24. REs of VRPs relative to the FRP estimator for varying levels of mortality 87 Table 25. REs of VRPs relative to the FRP estimator for varying spatial distributions. 88 Table 26. VRP REs of all scenarios relative to the Subtraction Estimator 91 Table 27. REs of VRPs relative to the Subtraction Estimator for varying densities 92 Table 28. REs for VRPs relative to the Subtraction Estimator for varying diameter distributions. 93 Table 29. REs for VRPs relative to the Subtraction Estimator for normal and high mortality. ... 95 Table 30. REs of VRPs relative to the Subtraction Estimator for varying spatial distributions... 96 Table 31. Variance of error for volume and basal area for each estimator 99 Table 32. Error and Sample REs for volume and basal area compared to FRP 100 vii LIST OF FIGURES Figure 1. Plot-centered view (a) versus tree-centered view (b) 5 Figure 2. Inclusion zones for FRPs.... 6 Figure 3. Inclusion zone for VRPs 8 Figure 4. Geometric selection of trees for VRP 8 Figure 5. Example of "IN" and "OUT" trees using VRP 9 Figure 6. Illustration of V B A R 10 Figure 7. Geometrical representation of FRP for change over time 14 Figure 8. Geometrical representation of change over time using Subtraction 17 Figure 9. Geometrical representation of change over time using Grosenbaugh 20 Figure 10. Geometrical representation of critical height sampling 23 Figure 11. Geometrical representation of change over time using D V 25 Figure 12. Geometrical representation of change over time using Flewelling 29 Figure 13. Population frame in StandSim 34 Figure 14. Stem maps of generated stands with random distribution 35 Figure 15. Stem maps of generated stands with uniform distribution 36 Figure 16. Stem maps of generated stands with equal sized clumps ; 37 Figure 17. Stem maps of generated stands with random sized clumps 37 Figure 18. Growth curves used for simulation 38 Figure 19. Determining distance between a tree and sample point 40 Figure 20. The steps required for comparing FRP and VRP estimators for measuring change over time 41 Figure 21. Stem maps of generated stands at three chosen densities 43 Figure 22. Stem maps of three generated stands with different spatial distributions 45 Figure 23. Graphical representation of the plot layout with scenarios and replicates 47 Figure 24. Cumulative mean of volume change for three different densities 49 Figure 25. Cumulative mean of Basal Area change for three different densities 50 viii Figure 26. Cumulative mean SPH change for three different densities 51 Figure 27. VRPs compared to the FRP estimator 58 Figure 28. Other VRP estimators compared to the Subtraction estimator 59 Figure 29. Comparison of the error REs versus the sample REs for each VRP estimator for volume and basal area 60 Figure 30. VRP performance compared to FRP without high mortality 65 Figure 31. VRP relative to Subtraction without high mortality 66 Figure 32. Volume REs for VRP estimators relative to FRPs 89 Figure 33. Basal area REs for VRP estimators relative to FRPs. 89 ix ACKNOWLEDGEMENTS I am grateful to have had the opportunity to work with my supervisor, Dr. Peter Marshall. Peter has always provided me a listening ear and excellent guidance while I have worked with him over the past few years. Peter has been a great mentor, and his dedication, excitement, and commitment to the world of forestry and forest biometrics has help guide me in my own endeavors. I have no doubt that I will continue to have a positive relationship with Peter throughout my career. I am extremely thankful to have had the support and guidance of an exceptional group of colleagues from J.S. Thrower & Associates. The enthusiasm for their work and their innovation has inspired me to pursue a career in forest biometrics. Thank you to Guillaume Therien, Gyula Gulyas, Eleanor McWilliams, and Jim Wilson for all your encouragement and help. My sincere appreciation also goes to the members of my supervisory committee, Dr. Kim lies, Dr. Kevin Lyons and Mr. Norm Shaw for their mentoring, support, and guidance throughout my graduate studies. x To my wife Lindsay, you have inspired me more than you will know. Thank you for your love, support, motivation, and friendship. To my parents Andrea Russell and David Carter, and my grandmother Orla Angle. You have given me a great gift, the thirst for knowledge. And to my sisters, Erin Carter and Leslie Cunsolo, for their unconditional love and encouragement. Thank you! xi 1.0 INTRODUCTION Sampling in forestry is used frequently as it is unreasonable to do a census of all trees to determine values of interest. Plot sampling is one way of obtaining a sample of trees in a forest. Plots are essentially clusters of the object of interest, with probabilities assigned to each object as a result of the sampling technique being used. Two commonly used plot-sampling techniques in forestry are fixed radius plots (FRPs) and variable radius plots (VRPs) (lies 2003, p. 360). The most commonly used approach for measuring change over time involves using FRPs. FRPs are easily understood and implemented, resulting in data that are easily compiled for a variety of stand and tree attributes (Husch et al. 1972, p.203 and Avery and Burkhart 1983, p. 176). A major disadvantage to the use of FRPs for change detection is the cost of implementation, both statistically and monetarily. FRPs, by design, require the measurement of many trees which may lead to statistical inefficiency, as the sampler may collect more information than is required to get a precise answer for the value of interest. If a dollar cost is associated with each tree measured, then measuring many trees is clearly more costly then measuring fewer trees. It is also possible to use VRPs for measuring change over time. The use of VRPs in change estimation has been a topic of discussion for over four decades (Grosenbaugh, 1958, 1959, Beers and Miller, 1964, lies 1979, 1981, 1987, lies and Beers, 1983, Flewellirig, 1981, Bitterlich, 1984, Van Deusen, 1986, Scott, 1990, Gregoire, 1993, lies and Carter, 2007). VRPs are commonly used as temporary sample plots in operational settings. The VRP approach has been proven to be more efficient and cost effective as an estimator of some standing tree attributes for forest, landscape, and stand level inventories (Grosenbaugh and Stover, 1957, Arvantis and O'Reagan, 1967, Oderwald, 1981, Martin, 1983, Ranneby, 1980). However, VRPs have not been widely used for measuring change due to some perceived problems including: 1. sudden additions to the value of interest from ingrowth, ongrowth, and nongrowth1, which produce high variability; 2. complex mathematical formulae used to compute the values of interest; 3. use of an angle for selecting trees; and 4. lack of studies on the use of VRPs for measuring change over time. The final product for both FRP and VRP sampling methods in change detection should be an unbiased estimate of growth. It is possible that the same precision might be achieved using VRPs 1 Ingrowth occurs when a tree is in the sample at Time 1, but not of merchantable size, and is of merchantable size at Time 2. Ongrowth occurs when a tree is not of merchantable size and not in a sample at Time 1 but is of merchantable size and in the sample at Time 2. Nongrowth occurs when a tree is merchantable and out of the sample at Time 1, but grows sufficiently that it is in the sample at Time 2. 1 with greater efficiency (lies 2003, p. 676). However, the perceived issues listed above have limited the use of VRPs in change detection; solutions to these issues that are practically implemented, relatively easy to understand, and which include data to support the methodology have not been presented. The purpose of this study is to provide an objective assessment of the performance of VRPs for measuring change over time. Through comparing different VRP estimation methods with FRPs, and by testing performance under a range of conditions, the intent is to provide forest managers with support when deciding on a sampling technique for their change measurement needs. The intent of this study is not to discredit or replace the use of FRPs; nor should VRPs be viewed as a panacea to solve all sampling issues regarding change over time. Rather the purpose of this study is to provide evidence that the use of VRPs is a viable choice for measuring change over time. This study is comprised of three objectives: 1. identify and explain the differences between FRPs and VRPs for measuring change over time; 2. identify precision gains (if any) over FRPs by using VRPs for measuring change over time; and 3. compare the statistical efficiency of various VRP methods for measuring change over time over a range of different conditions using simulation. The remainder of this document is separated into seven chapters. The following is a list of subject headings with explanation of their purpose: 1. Theoretical Background - information on theoretical concepts related to probability sampling, FRPs, VRPs and change over time, which provides the basic information required to understand the estimators being tested. 2. Change Estimators - explanations (with a unified form) of the change estimators with geometrical representations and examples of calculations. 3. Simulation - explanation of the simulator used for generating stands and the sample simulation process employed. 4. Material and Methods - outline of the data and methods used for testing the performance of the change estimators. 5. Results - data outputs used to determine the performance of the estimators under a variety of conditions explained in the material and methods chapter. 2 6. Discussion - analysis and interpretation of the results and explanations of encountered issues. 7. Future Directions and Conclusions- discussion of aspects of the study that could be addressed in further studies and presentation of the overall conclusions drawn from this study. 3 2.0 THEORETICAL BACKGROUND The following section provides basic information on the theory of probability sampling with a brief history and explanation of FRPs, VRPs, and change over time. 2.1 P R O B A B I L I T I E S Horvitz and Thompson (1952) explained equal selection probabilities in terms of numbers of objects as follows: p i = (i = l , 2 , . . . ,N ) where P is the selection probability of object i, and n is the frequency of the object in a population of N. These probabilities are based on the assumption that the sampler knows the total number of elements in the population. It is uncommon in forest sampling for a sampler to know the total number of elements (trees) within a population prior to sampling; however, the sampler will more commonly know the area of the population of interest. Areas can be substituted for number of objects if each element in the population occupies an area. Through this substitution the probability of selecting an object based on the area it occupies in a population of known size can be described: Pi = — (i= 1,2, . . . ,N) A where &\ is the area occupied by object i in a population of area A. Based on the typical Horwitz-Thompson (HT) estimator, the value of interest ( V H ) for an individual object is obtained as: p, where V H is the value of interest, Vj is the measured value of object i, and R is the probability of selecting the object i. i i I i = l V H T o t a , = ^ x A [1] 4 where v H Tola] is the total value for the area of interest based for sample size n, V is the measured value of object i, a] is the area occupied by object i, and A is the total area of the population. where V H A r e a is the value per unit area based on a sample size of n, Vj is the measured value for object i, and a; is the area occupied by object i. 2.2 TREE-CENTRED VIEW For consistency in visualizing how objects may be sampled using probability sampling, it is best to visualize the inclusion areas of individual objects. A tree-centered approach (Grosenbaugh, 1958, Beers, 1964) deals with each tree in a population as an individual. Each object in a sampled population has an inclusion zone that extends beyond its actual occupancy of the area. If a sample point lands within an object's inclusion zone then the object is included in the sample. This approach works for both FRPs and VRPs. The differences between a plot-centered and tree-centered view are illustrated in Figure 1. Traditionally, FRPs have been viewed as an area (plot) around a sample point with the sampler selecting any trees falling inside the plot. However, by visualizing trees from a tree-centered view the probabilities of each tree in a sample can be decomposed, and the value of individual trees can be modified. This approach is of particular use when trees have varying individual plot sizes or inclusion probabilities. V, [2] H Area n Sample Point Inclusion Zone a) Plot-centered view b) Tree-centered view Figure 1. Plot-centered view (a) versus tree-centered view (b). 5 A tree-centered view w i l l be used for visualizing inclusion probabilities and tree plots throughout this document. 2.3 FIXED RADIUS PLOTS FRPs are an example of an equal-probability sampling method. This means that each object within a population has the same probability of being sampled. From a tree-centered view, each object has the same size inclusion zone surrounding it. Given a stand of trees, it is useful to visualize each tree with a basal area expanded to a constant user-defined radius. In a stand, these expanded basal areas wi l l overlap and i f a randomly placed sample point is established it wi l l l ikely land in the inclusion zone of more than one tree (Figure 2). Figure 2. Inclusion zones for FRPs. Each tree represents the same relative basal area, which is based on the size of the F R P . Thus, a constant multiplier can be applied to obtain the value of the i * tree (Vj) per unit area. The multiplier for each tree wi l l be based on the relationship of the area of the F R P and the area units; because both are constant, a constant called the per ha factor (phf) can be derived: , . 10,000m2/ha phf = a where phf is the per ha factor and a is the area of the user defined F R P . In H T terms, the value obtained from a sample is as follows (using [1]): 6 Sfaxphf) V«™=- x A [3] n Or using [2] Z ( V , x p h f ) ^ H A r e a t^] n Commonly used FRP shapes include squares, rectangles, and circles. The sampler determines the size and shape of FRPs (Husch et al. 1972, p. 204). Choosing a plot size can be dictated by the tree count desired and will most likely be influenced by the variability of the forest. The use of FRPs within a probability sampling application will provide an unbiased estimate of stand parameters since the probabilities of selecting the objects are known. In other words, if sampling is repeated many times in a specific population, the average of the sample answers will approach the true answer for the population. 2.4 VARIABLE RADIUS PLOTS VRPs are an example of unequal probability sampling, more specifically, probability proportional to size (PPS) sampling. Bitterlich (1948) first developed Angle Count Sampling (ACS), which later came to be known as VRP, as a way to determine the basal area of a stand of trees by counting trees rather than measuring them. In ACS, each tree is given a relative area (disc) that is the basal area of the tree expanded by a constant (the blow up factor). If a sample point were established in a stand with these discs, the sample point will likely land on more than one overlapping disc. Depending on the blow up factor and the number of tree discs in which the sample point lands, the basal area of the stand can be derived as (lies 2003, p. 504): B A = — xl0,000m 2 /ha [5] B F where B A s is the basal area of the stand, tc is the average tree count at the sample point, and tc B F is the blow up factor ( is the percent of basal area of the stand, regardless of units, which B F can be converted using any unit system). If [5] is rearranged: 7 BA„ = tc x 10,000m2/ha BF [6] Further, since BF is a constant along with the units of area, these can be isolated and called the basal area factor (BAF in m2/ha) leading to [7]. BA, = tcxBAF [7] Keeping the blow up factor constant means the discs for each tree will be of varying size and trees of larger basal area will have a higher probability of being selected in a sample (Figure 3). T r e e ' Figure 3. Inclusion zone for VRPs. Sample Point Inc lus ion Zone Bitterlich needed to determine a way of selecting trees that did not require distance measurements to decide when a point was in or out of a tree plot. By associating the blow up factor to an angle Bitterlich was able to determine the maximum distance that a tree could be away from a sample point before it was considered out of the plot (Figure 4). Zone Figure 4. Geometric selection of trees for VRP. 8 Using two concentric circles with a similar area relationship (i.e., constant blow up factor) the angle will not change. If one angle is chosen for a survey, then all trees can be compared to it and the trees within the plot can be determined. If a tree subtends the angle, the tree is "IN" the plot; if not, the tree is "OUT" (Figure 5). Figure 5. Example of "IN" and "OUT" trees using V R P Expanding on the idea presented by Bitterlich, Grosenbaugh (1958) extended the use of ACS to estimate values other than basal area on forest stands. Grosenbaugh determined that the B A F and the ratio of the value of interest to basal area (VBAR) could be estimated separately then multiplied together to yield the value of interest for a unit of area. Value/ha = V B A R x B A F [8] Essentially Grosenbaugh took a two-dimensional concept and extended it to the third dimension. Visually, Grosenbaugh's concept can be viewed as a disc with a height and area. The volume of the disc would represent the value of interest stacked on top of the basal area of the tree (Figure 6). 9 Sample Point f VBAR Figure 6. Illustration of V B A R . In HT terms, the value obtained from a sample using Grosenbaugh's estimator is: n V H T M T = ^ ( V B A R x B AF) x A [9] i = l For both the Bitterlich and Grosenbaugh estimators, the basal area/unit area is the same for all trees and equals the BAF. Consequently, the B A F is the average basal area/unit area and thus each "in" tree represents the average basal area for the area of interest. This means that when a tree is selected in a sample it will represent the full value of the basal area, regardless of the size of tree. 2.5 CHANGE OVER TIME Foresters estimate forest growth over time to forecast timber supply and availability of other forest values, as well as to assess the response to various stand-level treatments. The dominant approach for measuring change over time for almost fifty years has been the use of permanent sample plots (PSP), where successive measurements over time are completed at the same sample point. Conversely, a forester could use successive forest inventories with temporary sample plots on the same sampling unit. Grosenbaugh (1959) recognized that recurring measurement of continuously identifiable trees by some form of probability sampling is the most efficient way to assess volume change (or growth); however, he warned that the use of PSPs should not replace the use of temporary sample plots for other operational needs. Hall (1959) explained the efficiencies of using PSPs for measuring change over time. He stated that sampling errors are reduced to one fourth that attained by successive inventories, and as a result forest managers can obtain a much more precise answer with less effort using PSPs. 10 2.5.1 Components of Growth Change measurement requires that each tree condition at successive times in a PSP be evaluated for its contribution to change on a sample. Beers (1962) first introduced the concept of components of growth, in an effort to formalize tree conditions in changing stands over time. Beers focused on the components of growth for FRPs and presented terms with this in mind. The use of VRP for change measurement introduces situations that do not fall into Beers' (1962) definitions. Situations encountered with VRP were addressed by Myers and Beers (1968), and expanded upon in Van Deusen et al. (1986). Many authors use varying terms for the same tree conditions. In an attempt to reduce confusion, the following terms will be used throughout the rest of this thesis: • Survivor (S) - A tree that is in the sample at Time 1 as well as the subsequent measurement. • Cut (C) - A tree that is in the sample at Time 1 and is removed by cutting prior to the subsequent measurement. • Mortality (M) - A living tree that is in the sample at Time 1, which dies prior to the subsequent measurement. • Ingrowth (I) - A tree that is in the sample at Time 1 but not tallied as a result of being below the merchantable size criteria, and which grows to merchantable size prior to the subsequent measurement. • Ongrowth (O) (VRP only) - A tree that is out of the sample and below the merchantable size criteria at Time 1, and grows to merchantable size and is in at the subsequent measurement. Nongrowth (N) (VRP only) - A tree that is not in the sample at Time 1 but is of merchantable size, and grows to a size such that the sample point falls within its inclusion zone prior to the subsequent measurement. The calculation of change using components of growth can generally be described as follows (Gregoire 1993): A V = A s - C - M + I forFRPs [10] A V = A s - C - M + I + 6 + N for VRPs [11] 11 2.5.2 Compatibility Flewelling (1981) formalized compatibility for growth estimators. Compatible estimators are those that estimate a value (V) through an unbiased method, and growth is estimated as the difference in successive value estimates. A V = V 2 - V , Noncompatible estimators are those that estimate a value and its change (growth) independently, meaning that growth cannot be calculated by the simple difference between an estimate and its successive measurement. A V * V 2 - V , Van Deusen (1986) stated that "non-additivity" (i.e., incompatibility) leads to a lack of confidence in estimates and the components of growth, and becomes a larger concern when the forest survey specialist is unable to explain this effect to users of the survey results. Gregoire (1993) challenged the concern of having compatible estimates, as compatible estimates may be futile unless the target population remains static over time. With this in mind the sampling techniques used in this study will only be identified as compatible or noncompatible. Further investigation related to compatibility and its effects will not be completed. 12 3.0 CHANGE ESTIMATORS There are a variety of methods to choose from when measuring change over time using plot-based systems. FRPs can vary in the size and shape, and the intensity of samples will determine the precision of the answer produced. In the case of VRPs, the B A F used will determine plot size, and together with the estimator employed, will determine the precision. A variety of VRP estimators have been derived in an attempt to address the perceived issues stated in Chapter 1. In this study four VRP estimators and the FRP estimator will be compared. These estimators represent work on the use of VRPs for measuring change over time, covering a span of almost 50 years. The VRP estimators tested in this study include: 1. Traditional Subtraction Method (Grosenbaugh, 1958 and FRP principles); 2. Grosenbaugh's Method (Grosenbaugh, 1958); 3. Flewelling's Method (Flewelling, 1981); and 4. lies Method/Distance Variable Method (lies 1979 and lies and Carter 2007). The following paragraphs explain the differences among the FRP and VRP estimators, along with the mechanics of each, using an example involving one plot and one change interval. All formula terminology followed the terms used in Gregoire 1993. 3.1 FIXED RADIUS PLOT ESTIMATOR FRPs have been used for change measurement because of their ease of implementation and compilation. However, because all trees carry the same weight, the number of trees cannot be controlled and many trees will be measured within a plot. This makes FRPs potentially inefficient. The values for each tree selected in a sample are multiplied by the per hectare factor (phf) to determine the value that the trees represent at the stand level. When calculating stems/ha each tree represents a value equal to the phf. For each of three attributes of interest (e.g., volume, basal area, stems per hectare), the values are summed across the plot and the result is an estimate for the stand. When establishing many plots, these plot-based values are averaged over all plots in the sample. Change over time is calculated by subtracting the estimate at Time 1 from the estimate at Time 2. An illustration of a tree-centered view for an individual tree using FRPs is provided in Figure 7, where V, is the value of interest at Time 1, V 2 is the value of interest at Time 2, and R is the radius of the plot. 13 F i x e d Figure 7. Geometrical representation of FRP for change over time. 3.1.1 Components of Growth and Change The following equations can be used to determine the change at the stand level. In each case the change value could be multiplied by the area of the stand to get total change for the stand. In each of the following equations, m is the number of plots measured, Vj is the value at Time 1, V 2 is the value at Time 2, 5 I j k represents the Bernoulli indicator of inclusion (0 or 1) at Time 1, 5 2 i k represents the Bernoulli indicator of inclusion at Time 2, and phf is the per hectare factor as explained in Section 2.2. Survivor tree estimate (As) contribution: m £ s = _k=Lj6S x p h f m 14 Mortality estimate ( M ) contribution: M = J s L t e M x p h f m Cut estimate (C ) contribution: m £=*=Uec x p n f m Ingrowth (I) contribution: m l = ±=LM xphf m Substituting the values of each component contribution into the total value calculation yields: E(V 2 l5 2 l k - V ^ J - X V A * - £ v H 5 l i k + V>2 i52 i k . ieS ieM ieC iel m xphf [12] The FRP method is a compatible estimator since the calculation of change is the simple difference between the value at Time 1 and the value at Time 2. The calculation of value and change are not completed independently. 3.1.2 Mechanics An example of the mechanics of FRP for measuring change over time using a 3.99m radius plot is presented in Table 1. Table 1. An example of change over time using the FRP estimator. Tree Volume Tree Tree Tree D B H Distance Sample Status phf vol/ha BA/ha S P H (m3/ha) (m2/ha) 1 9.5 0.0320 0.6 IN 200 6.4 1.4 200 2 4.5 0.0000 0.8 < Min. D B H 200 0.0 0.0 0 E 3 15.0 0.0957 1.6 IN 200 19.1 3.5 200 H 4 18.3 0.1905 3.99 IN 200 38.1 5.3 200 5 10.0 0.0370 0.9 IN 200 7.4 1.8 200 15 Tree D B H Tree Volume Distance Sample Status phf Tree vol/ha (nrVha) Tree BA/ha (m2/ha) S P H 1 9.8 0.0378 0.6 CUT 200 0.0 0.0 0 <^  2 5.0 0.0070 0.8 IG 200 1.4 0.4 200 E 3 15.5 0.1133 1.6 SUR. 200 22.7 3.8 200 H 4 19.0 0.2290 3.99 SUR. 200 45.8 5.7 200 5 10.5 0.0455 0.9 MORT. 200 0.0 0.0 0 Tree Tree vol/ha BA/ha S P H (m3/ha) (nrVha) Survivor Change + 3.2 0 -20 Cut Change -24.8 -5.5 -775 Mortality Change -25.9 -5.5 -700 Ingrowth Change + 19.5 + 5.5 + 2801 Total Change + 16.4 0.0 + 1500 3.2 VARIABLE RADIUS PLOT - SUBTRACTION Based on the value estimator developed by Grosenbaugh (1958) for determining the value of a tree using VRP, the same principles used for FRP can be applied, subtracting the value at Time 1 from the value at Time 2. This method is reasonable in that it provides an unbiased estimate of change over time; however, because the value of the tree is based on a stand average, the amount of value that is added through ingrowth and ongrowth is the full amount of the average for the stand. This results in the introduction of a large amount of variability over time. This increased variability is one of the main reasons why VRPs have not been used for measuring change over time. For this method, basal area/hectare and volume/hectare are calculated as explained in Section 2.4. Stems per hectare for each tree are represented by the B A F divided by the basal area of the individual tree or the stand table factor (STF). This represents the number of trees of a certain basal area that will fit in the square units/unit area represented by the BAF. Keeping the square units/unit area constant with a tree growing in basal area, it can readily be seen that the number of stems/unit area represented by a given tree will decrease over time. Thus, stems per hectare will decrease over time if ingrowth, ongrowth, and nongrowth are not present. An illustration of a tree centered view for an individual tree using the Subtraction Method is provided in Figure 8, where V i is the value of interest at Time 1, and V 2 is the value of interest at Time 2. 16 Figure 8. Geometrical representation of change over time using Subtraction. 3.2.1 Components of Growth and Change In each of the following equations m is the number of plots measured, V! is the value of interest as calculated from the tree measurement at Time 1, V 2 is the value of interest at Time 2, gi and g 2 is the basal area of the individual tree at Time 1 and Time 2 respectively, 8 l i k represents the Bernoulli indicator of inclusion (0 or 1) at Time 1, 5 2 i k represents the Bernoulli indicator of inclusion at Time 2, and BAF is the Basal Area Factor as explained in Section 2.4. In each case, the change value could be multiplied by the total area of the stand to get total change for the stand. Survivor tree estimate (As) contribution: A ^ J s z L i s s — i i i — x B A F m Mortality estimate ( M ) contribution: M = k = l f e M g l i x B A p m 17 Cut estimate (C ) contribution: C = k=l ieC g l i m -xBAF Ingrowth estimate (I) contribution: V 2 i ^ 2 i k l = ^ U B l § 2 ^ x B A p m Nongrowth estimate (N) contribution: ^ _ ^ 2 i ° 2 i k N k=l ieN g -—xBAF m Substituting the values of each component contribution into the total value calculation yields: .ieS ' v 2 i 5 2 i k V H 5 l i k A ^ v n ° i i k V § 2 i gli J i e M Sli feC gli " g2i feN § 2 i , V 2 i 5 2 i k ^ V 2 i 5 2 i k - x B A F [13] This method is compatible because value change and the growth estimates were not derived independently. 3.2.2 Mechanics An example of the mechanics of the Subtraction Method for measuring volume change using a BAF of 5.5 over time is presented in Table 2. Table 2. An example of change over time using the V R P - Subtraction estimator. Tree D B H Tree Vol. Dist. Sample Status Max. Dist. V B A R Tree vol/ha (m3/ha) Tree BA/ha (nrVha) SPH 1 9.5 0.0320 0.6 IN 2.03 4.512 24.8 5.5 775 Timel 2 4.5 0.0000 0.8 < Min. D B H 0.96 0.000 0.0 0.0 0 Timel 3 15.0 0.0957 1.6 IN 3.20 5.413 29.8 5.5 311 4 18.3 0.1905 3.99 OUT 3.90 0.000 0.0 0.0 0 5 10.0 0.0370 0.9 IN 2.13 4.714 25.9 5.5 700 18 Tree D B H Tree Vol. Dist. Sample Status Max. Dist. V B A R Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.8 0.0378 0.6 CUT 2.09 0.000 0.0 0.0 0 2 5.0 0.0070 0.8 IG 1.07 3.541 19.5 5.5 2801 3 15.5 0.1133 1.6 SUR. 3.30 6.004 33.0 5.5 291 4 19.0 0.2290 3.99 N G 4.05 8.076 44.4 5.5 194 5 10.5 0.0455 0.9 MORT. 2.24 0.000 0.0 0.0 0 Tree Tree vol/ha BA/ha S P H (m3/ha) (mVha) Survivor Change + 3.2 0.0 -20 Cut Change -24.8 -5.5 -775 Mortality Change -25.9 -5.5 -700 Ingrowth Change + 19.5 + 5.5 + 2801 Nongrowth Change + 44.4 + 5.5 + 194 Total Change + 16.4 0.0 + 1500 3.3 VARIABLE RADIUS PLOT - GROSENBAUGH (1958) Grosenbaugh (1958) recognized the variability increases that would occur from ongrowth and nongrowth when measuring change over time. His solution was to eliminate the effects of these components of growth by fixing the plot size for each tree that was in the sample at Time 1, which does not allow other trees to grow over the sample point. This means that all change would be evaluated by examining only those trees that were in the sample at Time 1. This set of trees would also include ingrowth trees that would be eligible for evaluation when they became of merchantable size. Grosenbaugh dealt with ingrowth trees by fixing the plot size of the ingrowth tree when it satisfied the diameter limit. Although this approach to dealing with ingrowth is unbiased, it is noncompatible, as the estimate of change is derived independently of the tree growth. Beers and Miller (1964), Martin (1982), Van Duesen (1986), and Roesch et al. (1989), modified either the ingrowth estimate or the survivor tree estimate based on the complete set of trees included at Time 1 to make Grosenbaugh's estimate compatible. In this study Grosenbaugh's original estimate of ingrowth will be used. However, it is noted that this estimate is not compatible. An illustration of a tree centered view for an individual tree using the Grosenbaugh estimator is provided in Figure 9, where Vi is the value of interest at Time 1, V 2 is the value of interest at Time 2, and R is the fixed radius of the tree plot at Time 1. 19 Fixed Figure 9. Geometrical representation of change over time using Grosenbaugh. 3.3.1 Components of Growth and Change In each of the following equations m is the number of plots measured, V, is the value of interest as calculated from the tree measurement at Time 1, V 2 is the value of interest at Time 2, g[ and g 2 is the basal area of the individual tree at Time 1 and Time 2 respectively, 8lik represents the Bernoulli indicator of inclusion (0 or 1) at Time 1, 52ik represents the Bernoulli indicator of inclusion at Time 2, and BAF is the Basal Area Factor as explained in Section 2.4. In each case the change value could be multiplied by the total area of the stand to get total change. Survivor tree estimate (As ) contribution: ^ ^ A V , 5 , i k A s = ^ = ^ — — — x B A F m 20 Mortality estimate (M) contribution: Z Z ^ -M = k = 1 i e M g l i xBAF m Cut estimate (C) contribution: VH5lik C = k = 1 i £ C g l i xBAF m Ingrowth estimate (I) contribution: 1 = ^TJ^ V 2 i ° 2 i k k=l iel § 2 -xBAF m Substituting the values of each component contribution into the total value calculation yields: AV = Z k=l 5 V 5 A v 2 i u 2 i k v l i " l i k ieS V 61i ' l i J V l i ° l i k V l i ° l i k , V 2 i ° 2 i k ieM E l i ieC § l i iel § 2 i -xBAF [14] m This method is unbiased in most cases, and provides a good answer for estimating growth; however, because the initial value added to the estimate of growth did not give the final value with new trees added, it is not compatible. Although fixing plot sizes provides consistent estimates of growth and essentially eliminates the compatibility issue, the fact that the new trees are being ignored has limited the use of this estimator. 3.3.2 Mechanics An example of the mechanics of the Grosenbaugh estimator for measuring volume change over time is presented in Table 3. These calculations assume a BAF of 5.5. Table 3. An example of change over time using the VRP - Grosenbaugh estimator. Tree D B H Tree Vol. Dist. (m) Sample Status Max. Dist. V B A R Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.5 0.0320 0.6 IN 2.03 4.512 24.8 5.5 775 rime 1 2 4.5 0.0000 0.8 < Min. D B H 0.96 0.000 0.0 0.0 0.0 rime 1 3 15.0 0.0957 1.6 IN 3.20 5.413 29.8 5.5 310 4 18.3 0.1905 3.99 OUT 3.90 0.000 0.0 0.0 0.0 5 10.0 0.0370 0.9 IN 2.13 4.714 25.9 5.5 705 21 Tree D B H Tree Vol. Dist. (m) Sample Status Max. Dist. Tree Tree V B A R v o l / h a B A / h a S P H (m3/ha) (m2/ha) 41 s 1 2 3 4 5 9.8 5.0 15.5 19.0 10.5 0.0378 0.0070 0.1133 0.2290 0.0455 0.6 0.8 1.6 3.99 0.9 CUT IG SUR. N G MORT. 2.09 1.07 3.30 4.05 2.24 0.000 3.541 6.401 0.000 0.000 0.0 19.5 35.2 0.0 0.0 5.8 5.5 5.9 0.0 0.0 0 2750 310 0 0 Tree Tree vol/ha BA/ha (m3/ha) (m2/ha) S P H Survivor Change + 5.4 + 0.3 0 Cut Change - 24.8 -5.5 - 775 Mortality Change -25.9 -5.5 -705 Ingrowth Change + 19.5 + 5.5 + 2750 Nongrowth Change 0 0 0 Total Change +25.8 -5.2 +1270 3.4 VARIABLE RADIUS PLOT - DISTANCE VARIABLE APPROACH Distance Variable (DV) sampling (lies and Carter 2007) is an extension of critical height (CH) sampling that applies to any plot-based sampling system for any measured variable. Critical height sampling is actually a specific application of the more generalized D V estimate and it is important to understand the work done on C H to understand the gains that are obtained using D V sampling. 3.4.1 Critical Height Sampling Critical height sampling was first developed by Kitamura (1962), and independently by lies (1979), as a way of directly measuring tree volume for standing inventories. The concept is that trees are sampled in three dimensions so the actual shape of the tree falling within the plot is represented in the sample. Essentially Kitamura and lies took Bitterlich's two-dimensional concept and expanded it to the third dimension in the form of a tree. As with the other VRP sampling schemes, the process of C H sampling involves the use of an angle gauge with a chosen BAF. If a tree subtends the angle at zero height, the angle will be "levitated" above the sample point to the point on the tree where the sample point is on the edge of the inclusion zone at that point on the bole (borderline). This point on the tree bole is called the critical height (Figure 10). The B A F multiplied by the sum of the critical heights will give the volume for that sample point. C H sampling is an excellent estimator of volume as it provides the sampler with a direct measurement of volume without having to use complicated taper equations. In papers by McTague and Bailey (1985) and Van Duesen and Meerschaert (1986), C H sampling was again proven to be an unbiased estimator of volume. However, some practical complications arise 22 when implementing it, and overall the method showed no significant gains in the estimate of standing volumes over the well understood VRP methods explained by Grosenbaugh (lies, 1979). 3.4.1.1 Change Over Time The natural progression for the use of C H sampling was to extend it to the fourth dimension of time. lies (1979) speculated that the variance of volume growth would be reduced relative to the growth estimates suggested by Grosenbaugh (1958), particularly in terms of ongrowth and nongrowth. Bitterlich (1984, p. 123-125) referred to the use of C H sampling for measuring volume growth as the "lies Method". Graphically, the variance reduction is evident as the leading edge of the tree shape allows for a gradual addition of volume rather than the full value of the average as suggested by Grosenbaugh (1958) (Figure 9). Many shapes could be considered for variance control when using C H ; however, the most success has been achieved through the use of a cone-shaped estimator, as the sloping angle of a cone allows for a "built-in" gradient of values depending on the distance the tree is from the sample point. Using a cone is not only easy to visualize, it allows the use of easily understood and implemented calculations of volume. Although volume change can be measured directly using C H for change over time based on the taper of the tree, other shapes were tested to provide a "pseudo-tree" (lies, 1979) approach as a variance control measure (Van Duesen 1987). 23 3.4.2 DV Sampling The original intent of C H sampling was to measure volume change over time and intrinsically basal area change. Little effort was devoted to extending this method to other values such as stems per hectare, quality, etc., although they were briefly eluded to in lies (1981) and Bitterlich (1984, p. 123). Scott and Alegria (1990) stated that the VRP method they used did not efficiently estimate stems per hectare compared to FRPs for some of the components of growth. This prompted the development of an estimator by lies and Carter (2007) that extended beyond volume change to any variable for any plot-based system. This estimator was formally termed the "distance variable estimator". Traditional estimators work under the assumption that the value is the same (the average of that value for the stand) no matter where a sample point falls within the inclusion zone of a tree. In terms of change over time, this introduces high variability for newly qualifying trees due to ongrowth and nongrowth. Graphically, the traditional method can be viewed as the height of the geometric representation of the value stacked on the basal area of the object (Figure 9). The geometric height remains the same wherever the sample point lands. This is unbiased in that the expectation of the value is the average. With the D V estimate, the requirement for the values to be identical is relaxed, as long as the average of all the possile values equals the expected value of the tree. This can be achieved by using a cone shape with a volume equal to that of the traditional disc. The height of the geometric representation of the value when using a cone will vary depending on the distance of the sample point relative to the tree; however, the average of all potential values (geometric heights) will be the same as the traditional value. The expectation that the average of all possible values is equal to the expected value makes the DV estimate unbiased. Comparing the traditional estimate and the D V estimate in HT terms yields: fv 1 1,S f v , l V a i J l a i J where V i s i s the estimated value using DV. Typically the traditional estimate would be the same as [1]. Substituting V i s into [1] yields: ' V l . S V B M = — X A [15] 24 Using a cone shaped estimator with the same volume as the traditional estimator for the value of interest, the height of the cone will be three times higher at the tree centre than that of the traditonal disc-shaped estimator. Conversely the height of the cone will be zero at the edge of the inclusion zone, thereby reducing the variabilty introduced by newly qualifying trees due to ongrowth and nongrowth (Figure 11). Figure 11. Geometrical representation of change over time using D V . To determine the height of the conical estimator for DV, the relative distance of the sample point to the tree is determined and multiplied by three times the value of the object [17]. ( V De-Ds De x 3 x V ; [17] 25 where Dei is the distance from the tree centre to the edge of the inclusion at Time 1, De 2 is the distance from the tree centre to the edge of the inclusion at Time 2, Ds is the distance from the tree centre to the sample point, Vj is the value of the object at Time 1, V 2 is the value of the object at Time 2. When measuring for change over time using the D V estimator, the same methods are used as Equation [9], where change is the simple difference between values at Time 1 and Time 2. 3.4.3 Components of Growth and Change In each of the following equations m is the number of plots measured, V t is the value of interest as calculated from the tree measurement at Time 1, V 2 is the value of interest at Time 2, gi and g 2 is the basal area of the individual tree at Time 1 and Time 2 respectively, 8 l i k represents the Bernoulli indicator of inclusion (0 or 1) at Time 1, 8 2 i k represents the Bernoulli indicator of inclusion at Time 2, and B A E is the Basal Area Factor as explained in Section 2.4. In each case, the change value could be multiplied by the total area of the stand to get total change for the stand. De, - Ds \ D; = Survivor estimate ( A s D V ) contribution: As xBAF DV m Mortality estimate ( M D V ) contribution: M xBAF DV m Cut estimate ( C D V ) contribution: xBAF DV m Ingrowth estimate (IDV) contribution: 26 f V 5 ^ ^ D V k=l iel V g 2i y xBAF m Nongrowth ( N D V ) contribution: m ' V 5 ^ k=l ieN N D V = If DVi = Dj V g Ii J xBAF m f V N 3 x - ^ v Sny Then: Ay = z k=l J > V 2 5 2 i k - D V , 5 l i k ) - ^ D V , S l i k - ^ D Y , ^ + 2>V 2 5 2 i k +Z D V 2 5 2ik ieS ieM ieC IE I ieN m *BAF [18] The DV estimator is unbiased and compatible. The expectation is that the estimator will reduce the variability introduced from ingrowth and nongrowth when compared to other VRP methods. The DV estimator is also applicable for all variables in any plot based system including FRP. There are situations when the DV estimator could introduce higher variability particularly for the mortality component of growth. This could occur if a tree is in close proximity to the sample point and dies, eliminating the higher weighted value, and potentially introducing greater variability in the estimate. 3.4.4 Mechanics An example of the mechanics of the DV Method for measuring volume change over time is presented in Table 4. These calculations assume a BAF of 5.5. Table 4. An example of change over time using the VRP - DV method. Tree D B H Tree Vol. Dist. (m) Sample Status Max. Dist. V B A R Rel. Dist. Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.5 0.0320 0.6 IN 2.03 4.512 0.704 9.529 3.872 1639 s 2 4.5 0.0000 0.8 < Min. DBH 0.96 0.000 0.167 0.000 0.000 0 3 15.0 0.0957 1.6 IN 3.20 5.413 0.500 8.120 8.250 467 4 18.3 0.1905 3.99 OUT 3.90 0.000 0.000 0.000 0.000 0 5 10.0 0.0370 0.9 IN 2.13 4.714 0.577 8.160 9.521 1212 27 Tree D B H Tree Vol Dist. Sample Status Max. Dist. V B A R Rel. Dist. Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.8 0.0378 0.6 CUT 2.09 0.000 0.713 0.000 0.000 0 E 2 5.0 0.0070 0.8 IG 1.07 3.541 0.252 2.677 4.158 2118 3 15.5 0.1133 1.6 SUR. 3.30 6.401 0.515 9.890 8.498 450 H 4 19.0 0.2290 3.99 N G 4.05 8.076 0.015 0.363 0.248 9 5 10.5 0.0455 0.9 MORT. 2.24 0.000 0.598 0.000 0.000 0 Tree Tree vol/ha BA/ha S P H (nrVha) (m2/ha) Survivor Change + 1.8 + 0.2 17 Cut Change -9.5 - 3.9 - 1639 Mortality -8.2 -9.5 - 1212 Change Ingrowth Change + 2.7 + 4.2 + 2118 Nongrowth + 0.4 + 0.2 + 9 Change Total Change - 12.8 - 8.8 +707 3.5 VARIABLE RADIUS PLOT - FLEWELLING'S (1981) APPROACH Flewelling (1981) developed a distance dependent estimator that had a similar concept to CH sampling, with an emphasis on modifying the BAP used for trees rather than the actual value being estimated. This estimator accommodated compatibility, as well as reduced the sudden addition of volume. Flewelling's method used the premise that the average BAF that would be applied to each tree could vary, but the average of all possible BAFs would equal the expected BAF for the tree. This involved determining the maximum value of the inclusion zone for a tree and using an arbitrary ratio to determine a smaller value to create another inclusion zone. As with the DV estimate, there would be a gradient between the outer and inner inclusion zones; depending on the relative distance of the sample point to the tree, the variable BAF would be assigned to the tree at the time of measurement. A visual representation of this method is provided in Figure 12. VBAF 1 is the variable BAF at Time 1, VBAF 2 is the variable BAF at Time 2, and x is the distance from the tree center to the sample point. Visually, one would expect this method to have a higher variability for ingrowth and nongrowth than the DV estimate, as there is a distinct leading edge that would introduce a minimum value greater than that of zero (which occurs with the cone shaped estimator used in the DV estimate). Also there is a maximum value that will be achieved if the sample point were to fall in the inner circle. 28 The calculation of this estimator is more complicated than the other estimates. It also has the disadvantage that the sampler must set the ratio of the outer to inner plot, which will impact the value of the VBAP. i i i i 1 w 1 L -*. X - J Figure 12. Geometrical representation of change over time using Flewelling. The process required to determine the VBAFs of an individual tree at one time follows. Again, this estimator uses the difference between the value calculated at Time 1 and Time 2 for determining the change over time. 1. Find the angle of the B A F 2. Determine the gauge constant (K) 3. Determine the K L / K Ratio, where K L is a numerical constant less than K (defines inner plot). 4. Determine the Variable B A F (VBAF) by: V B A F : L 81n(K/KL) + 4 x ( Z i ) 2 [19] where z = \ c KL i t U±<± K x, KL if - L S A KL x, [20] 29 3.5.1 Components of Growth and Change In each of the following equations m is the number of plots measured, is the value of interest as calculated from the tree measurement at Time 1, V 2 is the value of interest at Time 2, g] and g 2 is the basal area of the individual tree at Time 1 and Time 2 respectively, 8 l i k represents the Bernoulli indicator of inclusion (0 or 1) at Time 1, 8 2 i k represents the Bernoulli indicator of inclusion at Time 2, and V B A F is the variable Basal Area Factor as explained in Section 3.5. In each case, the change value could be multiplied by the total area of the stand to get total change for the stand. Survivor estimate (As) contribution: J ^ A V i o i i k _ x V B A F As = k = 1 i £ S g l i m Mortality estimate ( M ) contribution: m V ft £ ^ ^ l i O i i ! L x V B A F _ k=I ieM g l i m Cut estimate ( C ) contribution: £ £ ^ w x V B A F £ _ k=l ieC g l i m Ingrowth estimate (I) contribution: m V ft £ £ ^ a x V B A F j _ k=l i£ l g 2 j m Nongrowth estimate ( N ) contribution: m V ft £ £ _ ^ ! i ! L X V B A F J^T _ k=l ieN g l i m 30 Substituting the values of each component contribution into the total value calculation yields: A V = s k=l Z iES V 2 i 5 2 i k V 1 j 5 l i k x V B A F - V V ' ' 5 ' i k x V B A F - V V l ' 5 l i k x VBAF+ V V 2 ' S 2 i k xVBAF+V V 2 ' 5 2 i k x V B A F fri Sii gn g 2 i ^ g 2 i i sM ieC ie! leO 1 v 2 i 8 2 i k [21] The Flewelling estimator is unbiased and compatible. The expectation is that the estimator will reduce the variability introduced from ongrowth and nongrowth when compared to traditional VRP methods. 3.5.2 Mechanics An example of the mechanics of the Flewelling Method for measuring volume change over time is presented in Table 5. These calculations assume a B A F of 5.5, with a K of 42.66 and a K L of 17.06. Table 5. An example of change over time using the V R P - Flewelling method Tree D B H Tree Vol. Dist. Sample Status Max. Dist. V B A R V B A F Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.5 0.0320 0.6 IN 2.03 4.512 12.12 54.7 12.1 1707 ^ 2 0) •§ 3 4.5 0.0000 0.8 < Min. D B H 0.96 0.000 0.00 0.0 0.0 0 15.0 0.0957 1.6 IN 3.20 5.413 7.73 41.8 7.7 436 H 4 18.3 0.1905 3.99 OUT 3.90 0.000 0.00 0.0 0.0 0 5 10.0 0.0370 0.9 IN 2.13 4.714 10.89 51.3 10.9 1388 Tree D B H Tree Vol. Dist. Sample Status Max. Dist. V B A R V B A F Tree vol/ha (m3/ha) Tree BA/ha (m2/ha) S P H 1 9.8 0.0378 0.6 CUT 2.09 0.000 0.00 0.0 0.0 0 <^  2 5.0 0.0070 0.8 IG 1.07 3.541 3.45 12.2 3.4 1756 E 3 15.5 0.1133 1.6 SURV. 3.30 6.401 8.29 53.0 8.3 440 H 4 19.0 0.2290 3.99 N G 4.05 8.078 2.00 16.2 2.0 71 5 10.5 0.0455 0.9 MORT. 2.24 0.000 0.00 0.0 0.0 0 Tree Tree vol/ha BA/ha S P H (m3/ha) (m2/ha) Survivor Change + 11.2 +0.6 +4 Cut Change -54.7 -12.1 -1707 Mortality Change -51.3 -10.9 -1388 Ingrowth Change + 12.2 +3.4 + 1756 Nongrowth Change + 16.2 +2.0 +71 Total Change -66.4 -17 -1264 31 Although this method is compatible and less variable than the subtraction method, it is expected that it will be more variable than the D V method because the leading edge of the plot still allows for a sudden increase in volume by trees growing over the sample point. 32 4.0 SIMULATION Simulation can be a valuable tool when trying to generate data for analysis. Due to the high cost of actual measurement it is impractical to acquire data for all potential scenarios. To deal with this information gap, researchers often resort to artificially generated data that mimic the situations in which they are interested. To make comparisons between FRPs and VRPs for measuring change over time it is difficult to get data that have both FRPs and VRPs for the same sample points in any stand. It is also difficult to obtain data that cover a wide range of conditions. Available data tend to only cover the range of conditions based on the objectives of the data collection exercise. For the purposes of this study, a simulator (StandSim) was created in Microsoft Excel to produce stem maps of trees with attributes for making reasonable conclusions about the use of VRP for measuring change over time. The three tree characteristics simulated for the purposes of this study were height, diameter, and location. StandSim was used to generate a variety of stand conditions. The resulting stands were imported into SAS where a series of programs were developed to generate samples, and calculate estimates and appropriate statistics for the values of interest (volume, basal area, and number of stems) using each of the estimators explained in Section 3.0. This chapter will explain the way in which the trees were generated and how the resultant stem maps were used to simulate sampling. 4.1 S T A N D S I M U L A T I O N The focus of this study is to assess the performance of VRP estimators relative to FRP for measuring change over time under varying stand conditions. The intent of StandSim is to control stand conditions from a spatial distribution perspective, as well as to control other factors that potentially impact the way in which a change measurement sampling technique performs. Five main factors were considered when creating StandSim: 1. stand density - refers to the number of stems per hectare; 2. diameter distribution - refers to the percentage of the stems per hectare that fall into specified diameter classes; 3. spatial distribution - refers to the spatial arrangement of trees within the defined population; 4. growth - refers to an increase in tree attributes over time; and 5. mortality - refers to trees that initially had a value and over time contribute no value. 33 Although there are many other aspects of stands that could impact on the performance of sampling techniques, the above aspects were the only ones simulated for the purposes of this study. 4.1.1 Stand Density The population frame used for simulation was 140 m x 140 m, with a total area of 1.96 ha, and was designed to allow for a 20 m buffer around a 1 ha area (Figure 13). Stand density is defined on a per ha basis and the number is expanded to the size of the sample frame. All trees within the frame are assigned attributes depending on the stand characteristics chosen by the user, including spatial arrangement of the trees. „ Sample Area E o 140m Figure 13. Population frame in StandSim. - Buf fer 4.1.2 Diameter Distribution Diameter distribution is defined by the percentage of stems falling within certain dbh classes. At the time of simulation, dbhs are drawn from a uniform random distribution, meaning that the number of stems defined in any given dbh class will be generated randomly (within the range of the class), resulting in a subset that has a uniform distribution within the class. This allows each tree within a class to have an equal chance of being assigned any diameter within the class. 34 The heights are generated using a simple height to diameter (HD) ratio. The minimum HD ratio is 0.5 and the user defines the maximum ratio. The HD ratio for each tree is randomly chosen within the range, and the resulting HD ratios follow a uniform random distribution (within the range of HD ratios defined). 4.1.3 Spatial Distribution Each stem is given a set of random x and y coordinates. The user is able to modify the way in which stems are arranged within the frame. 4.1.3.1 Random Distribution In the random distribution, the x and y coordinates are produced using a uniform distribution; this is the default for the simulator. Trees can land anywhere within the population frame. A random distribution is unlikely; however, it is a good benchmark for testing other spatial distributions, such as uniform and aggregated. Indices of spatial distributions typically use the random situation as the basis of comparison for all other spatial arrangements (Pielou 1960, Hurlbert 1990, Stoyan and Petinen 1998), as uniformity and aggregation are forms of departure from random. A graphical representation of three random stands is presented in Figure 14. a) 500 stems/ha b) 1500 stems/ha c) 3000 stems/ha Figure 14. Stem maps of generated stands with random distribution. 4.1.3.2 Uniform Distribution In a uniform arrangement all the distances between the trees are equal. Within StandSim this arrangement is based on the number of stems per hectare regardless of tree size. Uniform stands are useful for simulating planted or managed stands. A graphical representation of three uniform stands is presented in Figure 15. 35 a) 500 stems/ha b) 1500 stems/ha c) 3000 stems/ha Figure 15. Stem maps of generated stands with uniform distribution. 4.1.3.3 Clumped Distribution Clumped stands are the most likely condition in which trees are arranged in a stand. Typically tree aggregation will depend oh ground conditions (soil, moisture, and light availability) and species (Oliver and Larson 1996, p.227). StandSim can assign a species to each tree based on a defined species distribution; however, this assignment is random and is not based on neighbouring species. Through the elimination of species designations, the intent is to look at only clumping and not stand dynamics. To simulate clumping, four main aspects of clumps are controlled: a) number of clumps - the number of clumps falling within the population frame; b) clumped area - the maximum area of the population frame that could be covered by clumps; c) trees in clumps - the minimum percentage of the density that will fall within clumps; and d) equal vs. random clump size - the size of each clump relative to each other. The logic used for creating clumps assumes that there will be a number of clumps covering a certain amount of area in the population frame. The first step was to randomly establish clump centers within the population frame. Based oh circular clumps of known size around a center, a radius can be determined and a range of coordinates can be used for generating the points within the clumps for the number of trees assigned to fall within clumps. For equal clump sizes, each clump will have the same number of trees. A graphical representation of three clumped stands with equal size clumps is presented in Figure 16. 36 a) 500 stems/ha b) 1500 stems/ha ''"4 c) 3000 stems/ha Figure 16. Stem maps of generated stands with equal sized clumps. The determination of unequal clump sizes is an iterative process whereby the user defines the minimum and maximum clump sizes, and the simulator optimizes to ensure the resulting clump sizes for the set of clumps fall within the defined sizes. The number of trees assigned to each clump is relative to the clump size area compared to the total area the clumps will cover and this percentage is applied to the number of trees that will land in the clumps. A graphical representation of three clumped stands with random sized clumps is presented in Figure 17. a) 500 stems/ha b) 1500 stems/ha c) 3000 stems/ha Figure 17. Stem maps of generated stands with random sized clumps. 4.1.4 Growth The focus of StandSim is to produce a stand composed of specific attributes by defining their limits. Growth is arbitrary in this study since the purpose is to measure the performance of various estimators of change. The accuracy of the change is not relevant as the only requirement is that the stands change over time. 37 Growth in StandSim is based on the dbh of a tree at establishment. Each tree is placed in one of five growth categories based on the dbh of an individual tree relative to the mean dbh of the all trees. Each growth category is associated with a curve in which a multiplier is assigned to the tree for every year of growth based on its dbh position on the curve. The five growth categories are as follows: 1. poor growth - trees less than or equal to 40% of the average dbh are placed into the poor growth category and follow the curve associated with poor growth; 2. fair growth -trees within 41% to 80% of the average dbh are placed into the fair growth category and follow the curve associated with fair growth; 3. average growth - trees within 81% to 120% of the average dbh are placed into the average growth category and follow the curve associated with average growth; 4. good growth - trees within 121% to 160% of the average dbh are placed into the good growth category and follow the curve associated with good growth; and 5. excellent growth - trees greater than or equal to 161% of the average dbh are placed into the excellent growth category and follow the curve associated with excellent growth. Once a tree has been assigned to a growth category, it remains there until the end of the growth interval that is defined by the user. A graphical representation of the growth curves used is presented in Figure 18. Growth Curves for the Growth Categories 100 150 Diameter (cm) 200 250 -Excellent Growth -Good Growth -Average Growth - Fair Growth -Poor Growth Figure 18. Growth curves used for simulation. 38 4.1.5 Mortality Mortality is a source of high variability in both FRP and VRP sampling (lies 2003, p. 669). This prompted the development of a mortality component to StandSim. Mortality in StandSim is addressed by eliminating the value of a tree at a future time in the growth interval. There are two ways in which mortality can occur within a stand. First, constant mortality could occur over time, where a consistent proportion of the trees are eliminated from year to year. Second, episodes of mortality could occur at points within the growth interval of the stand. Episodic mortality could mimic a major event (e.g., disease, wind throw, cut) occurring within the life of the trees. In a natural stand it is understood that trees with less vigor are more likely to die over time due to a variety of factors (Oliver and Larson 1996, p.78, 227). To account for this varying likelihood of mortality in StandSim, the likelihood of mortality depends on the growth category to which each tree is assigned. The percentages of trees eliminated for each growth category were determined arbitrarily and could be modified: 1. poor growth mortality - 60% of the mortality on a yearly basis; 2. fair growth mortality - 20% of the mortality on a yearly basis; 3. average growth mortality - 10% of the mortality on a yearly basis; 4. good growth mortality - 7% of the mortality on a yearly basis; and 5. excellent growth mortality - 3% of the mortality on a yearly basis. 4.2 SAMPLE SIMULATION Every stand generated by StandSim needed to be sampled using a FRP and VRP at the same point. A series of programs were developed in SAS to import, compile, and sample the generated tree data. StandSim was set up to generate samples for every stand created; however, all stand sampling was compiled in SAS. All samples were based on the relationship between the tree coordinates and sample point coordinates. The sample points for all stands are located at the same coordinates. The distance from each tree to the sample point was determined using the absolute difference between the tree x coordinate and the sample point x coordinate and the absolute difference between the tree y coordinate and the sample point y coordinate. These differences were used to determine the distance along the hypotenuse of the triangle created between the sample point and the tree (Figure 19). 39 Difference between Point Y and Tree Y »X,Y of Sample Point Distance from tree \ to sample point • X,Y of Tree Difference between Point X and Tree X Figure 19. Determining distance between a tree and sample point. Trees were selected in the FRP if the calculated distance between the tree and sample point was less than or equal to the distance of the plot radius. For the VRPs, the critical distance for each tree was determined based on the selected B A F and the dbh. Using the plot radius factor (prf) to the center of tree, the maximum distance the tree could be away from the sample point was calculated. Those trees that had a distance less than the critical distance for the chosen B A F were considered in the VRP plot. Volume per ha, basal area per ha, and stems per ha were determined using the estimators presented in Section 3.0. 40 5.0 MATERIAL AND METHODS This section explains the approach used to compare the FRP and VRP estimators for measuring change over time. The steps required for the comparison are explained in the flow chart in Figure 2 0 . Generate Stands Using StandSim Import Stands into SAS Sample stands in SAS Calculate values of interest for* each estimator . Determine value ^.stabilization Determine FRP and a^'BAF-size -fpfj*w;» comparison J\ Generate Statistics' for Vol/ha, BA/ha, and SPH change Generate Statistics ; for Q M I ^ Height, .' - i and trees per plot 7 Compare estimates using relative, efficiency I Make conclusions regarding V R P for change over time Figure 20. The steps required for comparing FRP and V R P estimators for measuring change over time. 41 5.1 STANDS, STAND GENERATION, GROWTH, AND IMPORT All stands were simulated using StandSim, as explained in Chapter 4. Fifty-four scenarios were developed to ensure a broad range of stand characteristics. Although any number of scenarios could be simulated, it was decided that this number was reasonable as a primer for further work, while still identifying potential areas of concern for the estimators. Table 6 describes the stand scenarios generated for this study. The main factors that varied for each stand were: density, diameter distribution, mortality, and clumping. Table 6. Stand Scenarios Sen S P H Dia. Dist. Annual Mort. Spatial Dist. Sen S P H Dia. Dist. Annual Mort. Spatial Dist. 1 500 Understory 0.5% - 1.0% Random 28 1500 Mixed 3.0% - 5.0% Random 2 500 Understory 0.5% - 1.0% Clumped 29 1500 Mixed 3.0% - 5.0% Clumped 3 500 Understory 0.5% - 1.0% Very 30 1500 Mixed 3.0% - 5.0% Very Clumped Clumped 4 500 Understory 3.0% - 5.0% Random 31 1500 No Understory 0.5% - 1.0% Random 5 500 Understory 3.0%- 5.0% Clumped 32 1500 No Understory 0.5% - 1.0% Clumped 6 500 Understory 3.0%- 5.0% Very 33 1500 No Understory 0.5% - 1.0% Very Clumped Clumped 7 500 Mixed 0.5% - 1.0% Random 34 1500 No Understory 3.0% - 5.0% Random 8 500 Mixed 0.5% - 1.0% Clumped 35 1500 No Understory 3.0% - 5.0% Clumped 9 500 Mixed 0.5% - 1.0% Very 36 1500 No Understory 3.0% - 5.0% Very Clumped Clumped 10 500 Mixed 3.0% - 5.0% Random 37 3000 Understory 0.5% - 1.0% Random 11 500 Mixed 3.0% - 5.0% Clumped 38 3000 Understory 0.5% - 1.0% Clumped 12 500 Mixed 3.0% - 5.0% Very 39 3000 Understory 0.5%- 1.0% Very Clumped Clumped 13 500 No Understory 0.5% - 1.0% Random 40 3000 Understory 3.0% - 5.0% Random 14 500 No Understory 0.5% - 1.0% Clumped 41 3000 Understory 3.0% - 5.0% Clumped 15 500 No Understory 0.5% - 1.0% Very 42 3000 Understory 3.0% - 5.0% Very Clumped Clumped 16 500 No Understory 3.0% - 5.0% Random 43 3000 Mixed 0.5% - 1.0% Random 17 500 No Understory 3.0% - 5.0% Clumped 44 3000 Mixed 0.5% - 1.0% Clumped 18 500 No Understory 3.0% - 5.0% Very 45 3000 Mixed 0.5% - 1.0% Very Clumped Clumped 19 1500 Understory 0.5%- 1.0% Random 46 3000 Mixed 3.0%- 5.0% Random 20 1500 Understory 0.5% - 1.0% Clumped 47 3000 Mixed 3.0%- 5.0% Clumped 21 1500 Understory 0.5% - 1.0% Very 48 3000 Mixed 3.0% - 5.0% Very Clumped Clumped 22 1500 Understory 3.0%- 5.0% Random 49 3000 No Understory 0.5% - 1.0% Random 23 1500 Understory 3.0%- 5.0% Clumped 50 3000 No Understory 0.5% - 1.0% Clumped 24 1500 Understory 3.0%- 5.0% Very 51 3000 No Understory 0.5% - 1.0% Very Clumped Clumped 25 1500 Mixed 0.5% - 1.0% Random 52 3000 No Understory 3.0% - 5.0% Random 26 1500 Mixed 0.5% - 1.0% Clumped 53 3000 No Understory 3.0%- 5.0% Clumped 27 1500 Mixed 0.5% - 1.0% Very 54 3000 No Understory 3.0%- 5.0% Very Clumped Clumped 42 5.1.1 Density Three tree densities (500 sph, 1500 sph, and 3000 sph) were chosen to investigate the impacts of tree spacing and density on the estimators for change over time. The 500 stems per ha density is quite open and intertree spacing is the highest of the three. The 1500 stems per ha density represents a moderate density with smaller inter-tree distances. Finally, the 3000 stems per ha will be used to investigate a high-density stand. The three density situations are illustrated in Figure 21 using a random distribution. a) 500 stems/ha b) 1500 stems/ha c) 3000 stems/ha .!«'/Ii^..''0V.rV:\*^.-,-' . v. ..... **',•'4 " , ' . \ , ' <•" •* .'• * • ,'v\'.'' : Figure 21. Stem maps of generated stands at three chosen densities. 5.1.2 Diameter Distributions The dbh distributions chosen were used to explore the impacts of varying dbh on the performance of the estimators. There were four dbh classes used: Class 1: 0 c m - 1 0 cm; Class 2: 10 c m - 3 0 cm; Class 3: 30 cm - 50 cm; and Class 4: 50 cm - 70 cm. The dbhs assigned to each tree in a class were generated randomly from a uniform distribution. The maximum diameter used was 70 cm to represent a typical coastal second growth stand. The three dbh distributions used in the simulation are described in Table 7. 43 Table 7. Dbh distributions used for stand simulation. Diameter Distribution Label Diameter Class 1 % Diameter Class 2 % Diameter Class 3 % Diameter Class 4 % Total % Understory Mixed No Understory 50 25 0 20 25 0 20 25 50 10 25 50 100 100 100 The intent of using the "Understory" distribution was to simulate ingrowth for the samples. The dbh cut-off for each tree to be in the sample was 10 cm. Trees lower than a 10 cm dbh at Time 1 were not used for the calculation of the plot value. If the tree grew enough to be greater than the dbh limit at Time 2, the value would contribute to the plot value for that time, and the change would be the value at Time 2 minus a value of zero at Time 1. The "Mixed" distribution was used to assess the performance of the estimators in a complex stand where there is the possibility of all components of growth. Finally the intent of the "No Understory" distribution was to assess the performance of the estimators with only larger trees present (i.e., no ingrowth). 5.1.3 Mortality Mortality is a source of large variability when measuring for change over time, regardless of the estimator used. With this in mind, it was decided to test two mortality levels to explore the impacts of varying amounts of mortality on estimator performance. The first mortality level (0.5% - 1.0% annual mortality) was used to mimic the typical mortality that would occur in a forest stand under natural processes. This level means that for every year of simulated growth a random percentage of the trees (between 0.5% and 1.0%) would be eliminated (i.e., produce a value of zero for the height and dbh). The second level of mortality (3.0% - 5.0% annual mortality) was used to assess the performance of the estimators for stands where a large mortality event occurred between measurements. This would mimic a disturbance event such as wind throw. 5.1.4 Spatial Distribution Three spatial distributions were used to assess the impact of tree position within the population on the estimators for change over time. Each pattern had three variables to provide varying spatial distributions: number of clumps, % area occupied by clumps, and % of trees falling within clumps. Table 8 shows the values for each of the spatial distribution variables used in this study. 44 Table 8. Values used for the spatial distribution of simulated stands. Spatial Distribution Number of % Area Occupied by % of trees Falling Label Clumps Clumps within the clumps Random 0 0 0 Clumped 10 25 25 Very Clumped 10 25 75 The spatial distributions given in Table 8 represent only a small fraction of all possible spatial distributions; however, this set of conditions provides a reasonable cross-section of distributions, and will provide a basis to determine impacts of spatial distributions on the performance of the estimators. Figure 22 provides visual representations of the spatial distributions used on a generic stand with 1500 stems per ha. a) Random b) Clumped i° I "• • ' . 'v' • ' . s < . * ;'; v-' . ' • ' V.. • • .V . 1 • 1 ' *•' • •' ", • •" i-*C> ",".V . ' ' • • * * ••' •* ." ' • . « \ V.7 • ' »•*••' ;- ,*;" " • * . * • * • ' ' • : ' • . • / * " V - ' ' V "^'X'ffevv.i:;;^: 1 : . ' •;"'.*':'»-;•.••'. '•'j\5>.v.*-V-.' - . • • v'.' c) Very Clumped Figure 22. Stem maps of three generated stands with different spatial distributions. A random spatial distribution is not a likely scenario in a real stand, but it represents a control situation in terms of spatial distributions. Clumping or tree aggregation is a departure from the random situation; some degree of clumping is often found in real stands. The two clumped situations use equal clump sizes meaning that each clump simulated will contain the same basal area. Although natural clumping patterns will likely have unequal clump sizes, this condition was difficult to control for comparison purposes. To provide a valid comparison and eliminate the variability introduced by having unequal clump sizes, all clumped distributions were created with equal sizes. The number of clumps and the amount of area occupied by each clump were held constant for this study. This reduced the number of comparisons to be evaluated. The degree of clumping was modified by varying the number of trees included in each of the clumps. 45 5.1.5 Stand Variability Although each stand generated was a random representation of the attributes defined by the specified parameters, it was recognized that a single representation of each stand could produce specific results that may be misleading when making conclusions about the estimators' performance. Consequently, for each set of stand attributes defined by the scenarios, a series of thirty iterations (or replicates) were generated. Although this number of iterations was arbitrary, it did address the variability that could occur within each set of stand attributes. Essentially this meant that the number of stands generated was expanded thirty fold, resulting in 1620 (54 x 30) generated stands created using StandSim. 5.1.6 Growth Each stand generated was grown for twenty-five years. Although comparisons could be done for a variety of measurement intervals, this study focused on one interval of five years of growth. The amount of growth simulated was based on the tree dbh at the first measurement. Based on its dbh, a tree was placed into a growth category for the remainder of the growth interval. Its position on the growth curve determined a multiplier applied to the dbh simulating tree growth. The height to diameter ratio established for each tree remained constant for the full growth interval, so the new tree height was easily generated from the new dbh. Although this is a very crude way of simulating growth, for this study the only concern is that the tree dimensions increase in size with time. 5.1.7 Stand Data Import The output of StandSim is a series of .CSV files generated for each stand iteration (or replicate) containing both tree data and tree coordinates. These files were imported into SAS using the PROC IMPORT procedure and combined to produce a file that contained the tree data and tree coordinates. These SAS files were subsequently used for the sample simulation and all other analysis. 46 5.2 STAND SAMPLING AND VALUE DETERMINATION To provide a proper comparison of the estimators, the sample points in each stand had to be positioned at the same location. The sample population frame for each stand was a 1 ha area. There is a possibility of achieving variable results depending on the location of the plot and the representation of the stand attributes defined. This variability (measured as the standard error) should be reduced by using the 30 replicates as explained in Section 5.1.5; however, to further reduce the variability within each replicate, five plot locations were established in each replicate. This resulted in a total of 8,100 sample points for the entire study (54 scenarios x 30 replicates x 5 plots/replicate = 8,100 sample points). Each plot centre had a set of coordinates and all trees in the sample population were compared to these coordinates to determine whether the trees were "in" or "out" of the plot. One plot was established at the center of the population frame and the four other plots were established 25 m from the center in the cardinal directions. A graphical representation of the plot locations is presented in Figure 23. Replicate 1 Replicate 2 Replicate 30 | 100 m 4 100 m -< - - J Figure 23. Graphical representation of the plot layout with scenarios and replicates Each of the five sample points had three FRP and four VRP sizes associated with them. This resulted in a total of 56,700 plots (8,100 sample points x 7 plot sizes per point = 56,700 plots) of varying size that would then be used for comparison. For each of the different plot types and 47 sizes, all trees that were considered "in" at some time in the measurement interval were tallied (refer to Section 4.2 for details). Ingrowth and nongrowth trees were given a value of zero at Time 1, and at Time 2 they were given a value appropriate for the estimator being used. Trees that died (mortality) were given a value of zero at the second measurement. Al l values of interest (change in volume/ha, basal area/ha, and stems per ha) were summed for each of the trees in each plot and averaged over all the plots of that type. 5.2.1 Volume The volume calculation used for each tree was the basic volume of a cone with a base equal to dbh and a height equal to the height of the tree. While volume or taper equations are normally used for volume calculation, it was not essential in this study to have an accurate calculation of tree volume. This is particularly true considering the trees in this study do not have a species associated with them, and volume and taper equations are usually species dependent. The following is the basic volume equation used for each tree sampled: •tbaxtht 3 where T V is the tree volume, tba is the tree basal area (in m2), and tht is the tree height (in m). This basic volume was converted to a per ha value depending on the estimator used, as explained in Section 3.0. The change in volume per ha was separated into the components of growth for each plot and the total volume change was the combination of the components of growth as explained in Section 2.0. 5.2.2 Basal Area Basal area was calculated for each tree as: tba = (d/200)2X7t where tba is the tree basal area and d is the dbh (in cm). Each tree basal area was converted to a per ha value according to the estimator used. The components of growth were determined for each plot and the change in basal area was calculated depending on the estimator used. Values were summed within plots and averaged amongst plots. 5.3 VALUE STABILIZATION To ensure reliability in the comparison of each value, the number of plots required to achieve stabilization needed to be determined (i.e., the number of plots that produced an average value 48 that did not change appreciably with the addition of extra plots). To do this 150 plots were generated and the number of plots (rounded to the nearest 5) to achieve stable averages for each value of interest was determined. Three "generic" stands (Scenarios 7, 25, and 43) representing each density class with a random spatial distribution were assessed by graphing the results, with average change on the y-axis and number of plots on the x-axis. The graphs are shown in Figures 24-26. 0 ) CT c ( 0 .c O Q) E 3 O > 0 ) CT C ( 0 .c O 0 ) E 3 O > V CT C ( 0 .c O 0 ) E o > 500 SPH 50 40 30 20 10 -FRP -DV -FLEW -GROS -SUB -Actual 20 40 60 80 100 # of Plots 120 140 160 1500 SPH -FRP -DV -FLEW -GROS -SUB -Actual 20 40 60 80 100 120 140 160 # of Plots 3000 SPH 50 -FRP -DV -FLEW -GROS -SUB -Actual 20 40 60 80 100 # of Plots 120 140 160 Figure 24. Cumulative mean of volume change for three different densities. 49 a> D) c ( 0 .E O ( 0 Q) ( 0 V) 03 oo. 500 SPH 2.5 2 1.5 0.5 I 1 - FRP -DV -FLEW -GROS -SUB -Actuall 0 20 40 60 80 100 120 140 160 # of Plots D) c J= O re Q) ( 0 w is m 1500 SPH 10 -FRP -DV -FLEW -GROS -SUB -Actuall 20 40 60 80 100 # of Plots 120 140 160 0 ) o> c re .c O re o re v> re 00 3000 SPH 20 iHHIHlHIHlHir*-* -FRP -DV -FLEW -GROS -SUB -Actual I 20 40 60 80 1C # of Plots 120 140 160 Figure 25. Cumulative mean of Basal Area change for three different densities. 50 3000 SPH For most stands the value of interest stabilized before 150 plots. Based on this assessment it decided to use all 150 plots to provide a reliable and consistent comparison. 51 5.4 PLOT SIZES For appropriate comparisons of the values of interest, it was decided to compare each estimator with relatively equal numbers of trees per plot. For VRPs the number of trees per plot from one estimator to the next would be equal at Time 1 as the same B A F is used. However, when comparing the VRP methods to the FRP method, FRP sizes had to be adjusted to ensure a comparable number of trees were being sampled. Comparing equal tree numbers helped ensure relatively equal measurement costs among the different sampling approaches. By design, FRPs will pick up more small trees than VRPs, and these smaller trees could potentially have lower measurement costs than larger ones; however, without introducing complicated cost equations this is the best approach to achieve a valid comparison. This comparison is possible as each tree represents the same value in terms of the number of objects measured. To determine the B A F or radius of the plots for reasonable comparisons, the 54 scenarios were broken into a series of nine groups that were expected to have comparable variability based on density and spatial distributions. One stand was arbitrarily selected from each of the nine groups and the average numbers of trees per plot were calculated over 150 plots. The three FRP and four VRP plots were assessed for the number of trees per plot and their sizes adjusted to produce between 10 and 20 trees per plot. The final plot sizes were determined based on the combination of FRP and B A F sizes that gave a comparable number of trees per plot within the defined range. By using three FRP sizes and four BAFs the resulting range of plot sizes allowed for a reasonable range of trees per plot to be chosen for comparison. The plot sizes expected to give comparable numbers of trees per plot are presented in Table 9. 52 Table 9. FRP sizes and BAFs used for comparisons Scenario 1-6 F R A D 1 11.28 m F R A D 2 12.62 m F R A D 3 13.82 m B A F 1 2.00 m2/ha B A F 2 3.00 m2/ha B A F 3 4.00 m2/ha B A F 4 5.00 m2/ha Scenario 19-24 F R A D 1 6.91 m F R A D 2 7.98 m F R A D 3 9.77 m B A F 1 4.00 m2/ha B A F 2 6.00 m2/ha B A F 3 8.00 m2/ha B A F 4 10.00 m2/ha Scenario 37-42 F R A D 1 5.64 m F R A D 2 6.31 m F R A D 3 6.91 m B A F 1 8.00 m2/ha B A F 2 10.00 m2/ha B A F 3 12.00 m2/ha B A F 4 14.00 m2/ha Scenario 7-12 F R A D 1 9.77 m F R A D 2 11.28 m F R A D 3 12.62 m B A F 1 2.50 m2/ha B A F 2 3.00 m2/ha B A F 3 3.50 m2/ha B A F 4 4.00 m2/ha Scenario 25-30 F R A D 1 4.89 m F R A D 2 5.64 m F R A D 3 6.91 m B A F 1 10.00 m2/ha B A F 2 12.00 m2/ha B A F 3 14.00 m2/ha B A F 4 16.00 m2/ha Scenario 42-48 F R A D 1 3.99 m F R A D 2 4.89 m F R A D 3 5.64 m B A F 1 15.00 m2/ha B A F 2 18.00 m2/ha B A F 3 21.00 m2/ha B A F 4 24.00 m2/ha Scenario 13-18 F R A D 1 7.98 m F R A D 2 9.77 m F R A D 3 11.28 m B A F 1 5.00 m2/ha B A F 2 7.00 m2/ha B A F 3 9.00 m2/ha B A F 4 11.00 m2/ha Scenario 31-36 F R A D 1 4.89 m F R A D 2 5.64 m F R A D 3 6.31 m B A F 1 16.00 m2/ha B A F 2 18.00 m2/ha B A F 3 20.00 m2/ha B A F 4 22.00 m2/ha Scenario 49-54 F R A D 1 2.82 m F R A D 2 3.99 m F R A D 3 4.89 m B A F 1 29.00 m2/ha B A F 2 32.00 m2/ha B A F 3 35.00 m2/ha B A F 4 38.00 m2/ha 5.5 OUTPUTS The output types from the sample simulation for each solution are presented in Table 10. This is a comprehensive list and all of the statistics will not be used in this study. If further work were to be done on various aspects of the simulated samples, producing these values will preclude having to rerun the data. 53 Table 10. Statistical outputs for generated attributes. Statistic Mean Standard Standard Variance Quadratic Deviation Error Mean Volume/ha Change •/ S V V Basal Area/ha Change Stems/ha Change V Trees/plot D B H 1 S Height 1 S Tree Vol 1 •/ 3 Tree Vol 2 Attri Tree B A 1 S S Attri Tree B A 2 S Vol/ha 1 S Vol/ha 2 S BA/ha 1 V BA/ha 2 S Stems/ha 1 S •/ Stems/ha 2 •/ The statistics produced for each scenario were based on all plots for the 30 replicates. This means that the values of interest were calculated for each plot (150 plots total for a given scenario) by summing the components of growth based on each estimator, then calculating the statistics over all plots for each estimator and plot size. The result is one statistic for each scenario, estimator and plot size. The values for change in volume/ha, basal area/ha, and stems/ha are the values that will be used for comparisons and conclusions. The remaining attributes will be used as auxiliary or descriptive values to help explain some of the results. Quadratic mean diameter (QMD) is used as a descriptor because it is useful for subsequent volume and basal area calculations if required. The formula used for the calculation of Q M D is: Q M D = where n is the number of trees sampled and d is the dbh of the trees sampled. The average dbh and height of the trees in the plots were produced to illustrate differences between the FRP and VRP methods in terms of what is being sampled. 54 5.6 COMPARISONS Relative efficiency (RE) is a method for comparing the precision of one estimator to another. In terms of exploring the precision of one estimator versus another for a value of interest, the variance of the value over a sample can be determined and the ratio of variances for the two methods of interest can be compared. The variances used for this study were obtained from the 150 plots sampled for each scenario. The variance provides a measure of variability within a scenario around the sample mean. A comparatively lower variance indicates lower variability from one plot to the next for a scenario. Two variances were used for comparisons in this study. First, the variance calculated from the sample means (plot estimate of change - sample mean) was used to determine relative efficiencies for the comparisons used to make conclusions, and assumes no bias (Husch et al. 1972, p. 200). Second, the variance of the error of the estimate (plot estimate of change - actual change) was calculated and includes bias. If the relative efficiencies obtained using the error are approximately equal to the sample mean relative efficiencies then there is no bias. The formula for determining the relative efficiency of the two estimators (FRP versus VRP) for a value of interest was: RE = O" VRP where RE is the relative efficiency of the VRP method, 0"2FRP is the variance of the estimate of the value of interest for the FRP method, and 0"2VRP is the variance of the estimate of the value of interest for the VRP method. It is possible to compare the variances generated for all values of interest for each estimator and scenario to determine their relative efficiency compared to the FRP method. This would allow ranking the methods in terms of precision gains for each value of interest for each stand scenario relative to FRPs. Using this methodology, the method with the lower variance of the two methods will be more efficient (i.e., more precise), given that the samples are equal in terms of numbers of trees per plot sampled. A relative efficiency greater than one means that the denominator (i.e., test method) has a lower variance, and thus is more precise given the trees per plot equalization. A value less than one indicates that the numerator has a smaller variance than the test method and is therefore more precise given the trees per plot equalization. 55 The relative efficiency of volume/ha, basal area/ha, and stems/ha change will be calculated for each scenario using each of the four VRP estimators as the test methods. By calculating these relative efficiencies, the precision gains of each estimator compared to the FRP can be obtained, as well as a ranking of performance between VRP estimators. Many comparisons can be explored using the relative efficiencies produced. For this study general comparisons for all scenarios (using the RE of VRP compared to FRP) will be completed to obtain an idea of which estimator might be best for a given attribute. Comparisons will also be completed for each of the stand attributes that varied when generating the 54 scenarios. All scenarios related to a varied attribute will be compared regardless of their other attributes for the different densities, dbh distributions, mortality levels, and spatial distributions. By doing this, the impact of the stand attribute grouping on the performance of the estimator for each of the change attributes of interest can be evaluated. Comparisons of the DV, Flewelling, and Grosenbaugh estimators with the subtraction estimator will also be completed, as the subtraction method is the estimator that would have been used historically for VRPs. 56 6.0 RESULTS This chapter provides a summary of the simulation results, highlighting and briefly explaining the main points. More details can be found in Appendix I. The plot sizes used for comparisons, along with the average number of trees per plot, are presented in Table 11. The average FRP size for all 54 scenarios was 7.73m (-188 m2) in radius and the average B A F for all scenarios was 13.08m2/ha. The average number of trees per plot for FRP and VRP across all scenarios was 15.3 and 15.5 respectively. Table 11. Plot sizes and trees per plot used for comparisons. F R P F R P V R P V R P F R P F R P V R P V R P Scenario Size Trees Size Trees Scenario Size Trees Size Trees (m) /plot (m2/ha) /plot (m) /plot (m2/ha) /plot 1 13.82 14.9 2.00 15.3 28 6.91 16.6 10.00 16.7 2 13.82 14.7 2.00 14.4 29 6.91 17.4 10.00 17.4 3 13.82 16.3 2.00 16.7 30 5.64 13.6 14.00 14.1 4 13.82 14.9 2.00 15.5 31 5.64 15.0 20.00 15.6 5 13.82 15.6 2.00 15.3 32 5.64 14.7 20.00 15.2 6 13.82 16.5 2.00 16.7 33 5.64 14.6 20.00 14.9 7 11.28 14.8 3.50 15.6 34 5.64 15.1 20.00 15.5 8 11.28 14.5 3.50 15.4 35 5.64 15.3 20.00 15.7 9 11.28 15.4 4.00 14.3 36 5.64 13.2 20.00 13.9 10 11.28 14.6 4.00 13.8 37 5.64 14.9 12.00 14.9 11 11.28 14.5 3.50 15.2 38 5.64 14.7 12.00 15.3 12 11.28 15.1 4.00 14.2 39 5.64 18.0 12.00 17.7 13 9.77 15.0 7.00 14.7 40 5.64 14.6 12.00 15.4 14 9.77 14.3 7.00 14.2 41 5.64 15.6 12.00 15.5 15 9.77 13.6 7.00 13.1 42 5.64 13.5 12.00 13.8 16 9.77 14.7 7.00 14.5 43 4.89 17.0 21.00 15.9 17 9.77 14.6 7.00 14.3 44 4.89 16.9 18.00 18.1 18 9.77 14.3 7.00 13.9 45 4.89 16.4 21.00 15.4 19 7.98 14.9 6.00 14.8 46 4.89 17.3 18.00 18.5 20 7.98 14.7 6.00 15.3 47 4.89 16.9 21.00 16.0 21 7.98 13.5 6.00 14.0 48 4.89 15.2 21.00 14.6 22 7.98 15.2 6.00 15.2 49 3.99 15.5 38.00 16.6 23 7.98 14.9 6.00 14.9 50 3.99 14.2 38.00 15.4 24 7.98 13.1 6.00 13.4 51 4.89 26.3 29.00 25.1 25 5.64 11.3 14.00 12.0 52 4.89 22.4 29.00 21.4 26 5.64 11.1 14.00 11.8 53 3.99 14.2 38.00 15.3 27 6.91 14.6 10.00 14.7 54 3.99 16.5 38.00 17.7 M E A N 7.73 15.3 13.08 15.5 M I N . 3.99 11.1 2.00 11.8 M A X . 13.82 26.3 38.00 25.1 R A N G E 9.83 15.3 36.00 13.3 57 The results of the performance for each estimator for change in volume, basal area, and stems per hectare are presented in Figure 27. Recall that a RE value of greater than 1.0 represents a situation where the test estimator is more efficient (more precise) than the estimator it is tested against. VRP change estimator performance when tested against FRP. DISTANCE VARIABLE FLEWELLING GROSENBAUGH Change Estimator SUBTRACTION • Volume Change • Basal Area Change • SPH Change Figure 27. VRPs compared to the FRP estimator. Figure 27 shows that the Grosenbaugh estimator performs best for estimating volume change with 83% of the stand scenarios yielding a RE of greater than 1.0. The DV estimator also performed well with 59% of the stand scenarios yielding a RE of greater than 1.0. The poorest performance came from the subtraction estimator, with only 2% of the scenarios yielding an RE of greater than 1.0. In terms of basal area change, none of the VRP estimators exhibited more than 50% of the scenarios with a R E greater than 1.0. For change in stems per ha, none of the estimators performed well compared to FRP. The Subtraction change estimator is the approach that would have most often been used with VRPs for change measurement. The comparison of the DV, Flewelling, and Grosenbaugh estimators to the traditional Subtraction method are presented in Figure 28. All of these estimators performed better than the Subtraction method for volume change, with the Grosenbaugh and DV estimators having a RE of greater than 1.0 for all 54 stand scenarios. The Subtraction estimator performed better than the DV and Flewelling method 50% of the time for basal area change. The Grosenbaugh and D V estimators out-performed the Subtraction method for estimating change in stems per ha, with the Grosenbaugh estimator always the best. 58 VRP change estimator performance when tested against Subtraction 120% 100% q T— A 80% -I Ul rr § 60% -I *c <0 § 40% o w O 20% + 0% D I S T A N C E V A R I A B L E F L E W E L L I N G Change Estimator 100% 100% G R O S E N B A U G H • Volume Change • Basal Area Change • S P H Change Figure 28. Other V R P estimators compared to the Subtraction estimator. 6.1 RELATIVE EFFICIENCIES FOR ERROR AND SAMPLE The REs for volume and basal area obtained from the error were compared against the REs from the sample (estimate) and plot in the graphs presented in Figure 29. Since the error REs account for bias and the sample mean REs do not, it can be concluded that if the values are approximately equal, then the sample REs are not biased and thus the estimators used to obtain these values are not biased. The REs used were the VRP estimators compared to FRP. All values for the error are presented in Appendix III DV Volume Error RE versus Estimate RE 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 l . i Error RE% Grosenbaugh Volume Error RE versus Estimate RE 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 E r r o r R E % 59 F l e w e l l i n g V o l u m e E r r o r RE v e r s u s E s t i m a t e RE 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Error RE°/o S u b t r a c t i o n V o l u m e E r r o r RE v e r s u s E s t i m a t e RE 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Error RE% Figure 29. Comparison of the error REs versus the sample REs for each V R P estimator for volume and basal area. 60 All REs for the error are approximately equal to the sample REs with the exception of the Subtraction basal area. The differences are exclusively in the "High Mortality" scenarios and in all cases the sample estimate was overestimating the RE. 6.2 D E N S I T Y The performances of the different VRP estimators versus the FRP estimator for the various densities included in the simulations are presented in Table 12. The scenarios that fell into each density class (18 scenarios per class) were compared separately to explore the performance of each estimator. Table 12. Performance results for the VRP estimators relative to FRPs for three densities. % of Scenarios with R E % of Scenarios with R E > % of Scenarios with R E > 1.0 for Vol. Change 1.0 for of B A Change > 1.0 for S P H Change Density D V F L G R SU D V F L G R SU D V F L G R SU 500 sph 50 33 78 0 17 0 33 28 0 0 0 0 1500sph 67 44 89 6 17 17 33 28 0 0 0 0 3000 sph 61 33 83 0 11 0 50 22 0 0 0 0 None of the estimators performed particularly well for any of the attributes of interest for 500 sph, with the exception of the Grosenbaugh and D V estimators for volume change. For scenarios with 1500 sph, an increase in the performance of the estimators for volume change was observed. The D V estimator had a 17% increase in performance over the scenarios with 500 sph. Scenarios with 3000 sph showed a slight decrease in volume change performance compared to 1500 sph. The results also show a decrease in the performance of basal area change for all of the estimators with 3000sph, except for the Grosenbaugh estimator, which increased by 17% relative to 500 sph and 1500 sph. The DV, Flewelling and Grosenbaugh estimators were generally more precise than the Subtraction estimator (Table 13). The poorest performance by the D V estimator was with basal area change particularly in the 1500 sph scenarios. The Flewelling estimator generally performed well compared to the Subtraction estimator except for change in stems per ha, but was the poorest performer of the three alternative estimators tested. 61 Table 13. Performance results the alternate V R P estimators relative to the Subtraction estimator for three densities. % of Scenarios with RE % of Scenarios with RE % of Scenarios with RE > 1.0 for Vol. Change > 1.0 for of B A Change > 1.0 for S P H Change Density D V F L GR D V F L GR D V F L GR 500 sph 100 94 100 50 50 67 61 17 100 1500 sph 100 100 100 44 44 61 50 11 100 3000 sph 100 94 100 56 56 61 44 17 100 6.3 D B H DISTRIBUTIONS Three dbh distributions were used to investigate the sensitivity of the estimators to dbh distribution. Table 14 provides RE results for the scenario groupings that fell within each of the dbh distributions (18 scenarios per dbh distribution). Table 14. Performance results for the V R P estimators relative to FRPs for three dbh distributions. % Scenarios with RE > 1.0 for Vol. Change % Scenarios with RE > 1.0 for B A Change % Scenarios with RE > 1.0 for S P H Change D B H Distribution D V F L GR SU D V F L GR SU D V F L GR SU Understory Mixed No Understory 94 61 100 6 67 50 100 0 17 0 50 0 17 28 .0 6 50 22 11 56 50 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 The DV and Grosenbaugh estimators almost always out performed the FRP for volume change in a stand with a high level of understory. Conversely, the subtraction method performed poorly, with only 6% of the scenarios performing better than the FRP for volume change. Again, none of the VRP estimators performed particularly well for change in basal area or stems per ha. From Table 14, it can be seen for the "Mixed" dbh distribution there was a slight decrease in the volume change performance compared to the "Understory" results; however, there was increased performance for estimates of basal area change. There was a decrease in performance for the attributes of interest for the "No Understory" distribution. Table 15 shows that all three estimators were better than the subtraction estimator for volume change. Half of the time the subtraction estimator was more efficient than the DV and Flewelling estimators for basal area change. The Flewelling method performed poorly when estimating change in stems per ha. 62 Table 15. Performance results for the three alternate V R P estimators relative to the Subtraction estimator for three dbh distributions. % Scenarios with % Scenarios with % Scenarios with R E > 1.0 for Vol. R E > 1.0 for B A R E > 1.0 for S P H Change Change Change D B H Distribution D V F L G R D V F L G R D V F L G R Understory 100 100 100 50 50 78 56 44 100 Mixed 100 94 100 50 50 50 44 0 100 No Understory 100 94 100 50 50 61 56 0 100 6.4 M O R T A L I T Y Mortality introduces high levels of variability when sampling for change over time. Tables 16 and 17 give the performance results for the VRP estimators for two mortality levels. From Table 16, it can be seen that the Grosenbaugh and DV estimators performed best for volume change. The Grosenbaugh estimator performed best in terms of basal area change. None of the estimators performed well compared to the FRP for change in stems per ha. There were large decreases in estimator performance for high levels of mortality. The D V estimator showed the largest decrease in performance with an increased level of mortality (30% decrease for volume change). As is evident in Table 16, all estimators performed poorly when estimating change in basal area and change in stems per ha for high levels of mortality. For the normal level of mortality the three estimators were more efficient than the subtraction method in all cases with the exception of the Flewelling method for estimating change in the stems per hectare (Table 17). Table 16. Performance results for the V R P estimators relative to FRPs under different mortality levels. % Scenarios with R E > 1.0 for Vol. Change % Scenarios with R E > 1.0 for B A Change % of Scenarios with R E > 1.0 for S P H Change Mortality Level D V F L G R SU D V F L G R SU D V F L G R SU Normal High 74 44 89 0 44 30 78 4 30 11 52 0 0 0 26 52 0 0 0 0 0 0 0 0 Table 17. Performance results for the three alternate V R P estimators relative to the Subtraction estimator under different mortality levels. % Scenarios with R E > % Scenarios with R E > % Scenarios with R E > 1.0 for Vol. Change 1.0 for B A Change 1.0 for S P H Change Mortality Level D V F L G R D V F L G R D V F L G R Normal High 100 100 100 93 100 100 96 4 96 4 100 26 70 33 7 22 100 100 63 6.5 SPATIAL DISTRIBUTIONS Three spatial distributions were used to cover the range of possible stand arrangements. Table 18 shows the performance results of the scenarios falling into each of these distributions. Table 18. Performance results for the V R P estimators relative to FRPs for different spatial distributions. % Scenarios with R E > 1.0 for Vol. Change % Scenarios with R E > 1.0 for BA Change % Scenarios with R E > 1.0 for S P H Change Spatial Distribution D V F L G R SU D V F L G R SU D V F L G R SU Random Clumped Very Clumped 44 22 83 0 67 50 83 6 67 39 83 0 6 17 22 6 28 33 6 50 28 6 39 17 0 0 0 0 0 0 0 0 0 0 0 0 The Grosenbaugh estimator performed the best for volume change. The Subtraction estimator performed best for basal area change, although the number of scenarios with a RE > 1.0 was below 50%. As with the other varied stand attributes, none of the estimators performed well for change in stems per ha compared to the FRP method. The clumped distribution is the spatial distribution that is most realistic. There was an increase in performance for the DV, Flewelling, and Subtraction estimators in volume change estimation for the clumped distribution compared to the random distribution. There were also performance gains in estimation of basal area change for the DV and Grosenbaugh estimators. Performance for predicting volume change remained the same for D V and Grosenbaugh estimators for the "Very Clumped" spatial distribution. The performance of the D V estimator for basal area change increased compared to the clumped spatial distribution. All three alternate VRP estimators performed better than the Subtraction estimator (Table 19). All estimators performed more poorly for basal area change than for volume change, and the D V and Flewelling estimators were generally poorer than the subtraction method for estimating stems per ha. 64 Table 19. Performance results for the three alternate V R P estimators relative to the Subtraction estimator for the different spatial distributions. % Scenarios with % Scenarios with % Scenarios with RE > 1.0 for Vol. RE > 1.0 for BA RE > 1.0 for SPH Change Change Change Spatial Distribution DV FL GR DV FL GR DV FL GR Random 100 94 100 50 50 56 44 17 100 Clumped 100 100 100 44 44 61 50 28 100 Very Clumped 100 94 100 56 56 72 61 0 100 6.6 REMOVAL OF HIGH MORTALITY By removing the "High" level of mortality a significant portion of variability was eliminated. As seen in Table 15 and 17, the estimators performed poorly when the scenarios had the "High" level of mortality. The results for the 27 scenarios that had only "Normal" mortality are presented in Figures 30 and 31. By eliminating the scenarios that had high mortality all estimators performed better, with the exception of the Subtraction estimator, which seemed to perform better when there were high levels of mortality. The DV, Flewelling and Grosenbaugh estimators were all more precise than the Subtraction estimator for all stand change attributes, with the exception of the Flewelling method for stems per ha change (Figure 31). VRP change estimator performance when tested against FRP without "High" mortality. 100% 90% O 80% A 70% f [J! 60% in 50% o •J5 40% CD % 30% W 20% -| O 10% * 0% 89% 74% DISTANCE VARIABLE FLEWELLING GROSENBAUGH Change Estimator 0% 0% 0% • Volume Change • Basal Area Change • SPH Change SUBTRACTION Figure 30. V R P performance compared to FRP without high mortality. 65 120% y 100% -q A 80% -UJ EC (fl 60% -,o 'n ca 40% -c 0 o 00 20% -9^ 0% --V R P change estimator perfomance when tested against the Subtraction estimator without "High" mortality DISTANCE VARIABLE FLEWELLING GROSENBAUGH Change Estimator • Volume Change • Basal Area Change • SPH Change Figure 31. VRP relative to Subtraction without high mortality. 6.7 V R P ESTIMATORS RELATIVE TO ONE ANOTHER T o gauge the performance of the V R P estimators relative to one another, the REs of each estimator compared to the other estimator were analyzed (Table 20). A l l REs of the V R P estimators are presented in Table 1 of Appendix I. A l l REs were given equal weighting when producing the mean; consequently, the means might be misleading, as some scenarios had extremely high REs compared to the other estimators. Table 20. REs of the V R P estimators relative to each other. VRP Estimators VRP Estimators VRP Estimators VRP Estimators Relative to Relative to Relative to Relative to Subtraction Distance Variable Flewelling Grosenbaugh DV FL GR FL GR SU DV GR SU DV FL SU Mean 2.6 2.1 3.2 0.9 1.3 0.5 1.2 1.5 0.6 0.8 0.7 0.4 S DO = 5 Min 1.0 1.0 1.2 0.5 0.8 0.1 0.9 0.9 0.2 0.4 0.3 0.1 —• «* o -S > U Max 10.7 5.2 9.3 1.2 2.4 1.0 2.1 3.0 1.0 1.2 1.2 0.8 STD 1.9 1.0 1.9 0.1 0.4 0.2 0.2 0.4 0.2 0.2 0.2 0.2 a Mean 1.6 1.4 2.1 0.9 1.5 1.2 1.1 1.6 1.2 0.7 0.7 0.8 Basal Area Tiane Min 0.3 0.2 0.4 0.6 0.8 0.2 0.9 0.7 0.3 0.4 0.3 0.2 Basal Area Tiane Max 4.8 3.3 6.3 1.1 2.6 3.9 1.6 2.9 4.2 1.2 1.3 2.3 STD 1.2 0.9 1.5 0.1 0.4 0.9 0.2 0.4 0.9 0.2 0.2 0.5 SPH Change Mean 1.3 1.3 14.7 1.1 15.1 1.1 1.0 14.3 1.1 0.4 0.4 0.3 SPH Change Min 0.3 0.3 1.1 0.7 0.6 0.2 0.6 0.6 0.2 0.0 0.0 0.0 SPH Change Max 5.5 4.0 147.8 1.8 139.0 3.9 1.4 128.4 2.9 1.7 1.8 0.9 SPH Change STD 0.9 0.8 28.5 0.2 26.7 0.8 0.2 25.3 0.6 0.4 0.4 0.3 66 When the VRP estimators were compared to the Grosenbaugh estimator all estimators were less precise for all attributes of interest. When the estimators were compared to the Subtraction estimator, all performed better. 67 7.0 DISCUSSION The purpose of this study was to provide an objective assessment of the performance of VRP estimators for measuring change over time, with the intent of providing forest managers with decision support when deciding on a sampling technique. In the past, forest managers have been reluctant to use VRPs for measuring change over time, as they perceived the following problems: 1. sudden additions to the value of interest from ingrowth and nongrowth, which produce high variability; 2. complex mathematical formulae required to compile the values of interest; 3. use of different, sometimes more abstract field techniques for measuring and selecting trees; and 4. lack of studies on the use of VRPs for measuring change over time. To satisfy this purpose and address the perceived problems three objectives were developed that guided this study: 1. identify and explain the differences and methodologies between FRPs and VRPs for measuring change over time; 2. identify precision gains (if any) by using VRP for measuring change over time; and 3. compare the statistical efficiency of VRP estimators for measuring change over time over a range of different conditions using simulation. 7.1 COMPARING ESTIMATOR MECHANICS The change estimators were explained using a unified form to ensure consistency. Illustrations were used to visually explain the estimators; these illustrations were augmented with an example that explained the mechanics of each estimator. By doing this, it was evident that the FRP method was the simplest. However, the VRP estimators were only slightly more complex and only minor modifications to the already well-understood methods of FRP compilation would have to occur. The exception to this was the Flewelling estimator, which required considerable computation to get the final result. VRP sampling is widely used for operational inventories throughout North America. For all VRP estimators, the only extra measurement that may be required, beyond what would normally be required in a VRP sample taken for inventory purposes, would be the distance from the sample point to each tree (particularly for the D V and Flewelling estimators). However, in some PSP installations a stem map is required to ensure the trees are easily relocated for future measurements. If a stem map is required, then the distances to trees will be available, and thus no extra measurements would be required. 68 For estimating change in stems per ha, FRPs outperformed all VRP estimators for all scenarios. This is most likely due to the way in which stems per ha is calculated for FRPs. In the absence of ingrowth and mortality, FRPs will always be less variable for estimates of stems per ha, as the number of stems will never change over time. Conversely, because the calculation of stems per hectare for VRPs is a function of the B A F used and the basal area of the trees, if a stand is growing in the absence of ingrowth and mortality, the number of stems over time will almost always decrease. Normally, the expectation is that ingrowth and nongrowth will offset this decrease; however, the increases and decreases introduce variability that FRPs do not encounter. In many cases, change in stems per ha is not an attribute of great concern. Overall, VRP estimators are more complex than the FRP estimator. However they are not so complex that they should not be considered if there are existing VRPs that a forester manager would want to remeasure, or if a forest manager had a reason for using VRPs for a long term growth and yield or change monitoring program. 7.2 BIAS AND PRECISION GAINS The comparisons of the error based REs with the sample based REs show little difference and it can be concluded that the assumption of no bias is valid except in the case of the Subtraction estimates of basal area for the "High Mortality" scenarios. The "all or nothing" inclusion or subtraction aspect of the subtraction method for basal area could have lead to these results. This occurrence needs to be further explored to rule out the possibility of an error and explain why this phenomenon is happening, as it could impact the performance estimates of the subtraction method, particularly with high levels of mortality. The Subtraction estimator would likely have been the approach that most forest managers would have used if they were going to implement VRPs for measuring change over time. A major concern with this estimator is the introduction of high variability from trees growing onto plots, as the full value of the tree will be added to the sample once the new tree is included. To reduce this variability, other change estimators including D V and Flewelling estimators have been developed. By reducing the variability, the estimates of the value of interest become more precise; consequently, a sampler can reduce the amount of work required to achieve the level of precision required. Variance was used in this study as a measure of variability and precision gains were measured in terms of RE. The DV, Flewelling and Grosenbaugh estimators all exhibited precision gains compared to the Subtraction estimator, particularly when the high mortality scenarios were eliminated. For volume growth, each of the other three VRP estimators 6 9 was at least twice as efficient as the Subtraction method. Some scenarios showed REs as high as 11. The most precise estimator of the four VRP estimators was the Grosenbaugh estimator. This was expected as it intrinsically eliminates sources of variability such as nongrowth. This estimator only deals with the change of the trees originally sampled. Although this estimator performed best, it is incompatible (as explained in Section 2.5.2), meaning that the growth estimate is derived independent of the cumulative values that are calculated at Time 1 and 2. Depending on one's point of view regarding compatibility, this may or may not be an important issue. However, this estimator has been in existence since 1952 and has not been widely used, which lends credence to the fact that compatibility is important to most potential users of VRPs for change detection. The D V estimator was the most precise of the compatible change estimators, particularly for volume change. The D V estimator is an extension of C H sampling. D V estimates are easily implemented, compatible, require no new field measurements, and can be used for any value of interest for any plot-based system (including FRP). In the scenarios explored, the D V estimator was almost always more efficient than the Subtraction estimator for the three attributes of interest. The Flewelling estimator is the most complex of the VRP methods tested in this study. Using a K L / K ratio of 0.4, the Flewelling estimator had precision gains over the Subtraction estimator in most cases; however, overall there were no precision gains over the Grosenbaugh and D V estimators. It is possible that varying the K L / K ratio may lead to further precision gains. However, given the complexity of this estimator and the fact that the method has been in existence since 1981 with little use, it is unlikely that this estimator will be implemented in an operational setting. The D V estimator is the most likely of the four VRP estimators examined to be used for change measurement, even though it performed poorly in estimating change in stems per ha. This estimator is flexible and compatible and based on its performance in this study, there is little evidence to suggest that this method (along with the others) could not be used for measuring change over time in an operational setting. This being said, it is important that this estimator be tested using real data rather than simulated data to ensure that these conclusions are valid. 7.3 IMPACT OF STAND CONDITIONS Due to the low likelihood of using the Subtraction, Grosenbaugh, or Flewelling approaches this section will focus mostly on the D V estimator. History has excluded these three methods for use 70 in an operational setting, even though these methods performed better than the D V estimator in specific scenarios. The D V estimator performed poorly in most scenarios where there was a high level of mortality. As explained by lies and Carter (2007), there is the possibility that mortality could actually introduce higher than normal variability. This could occur when a tree in close proximity to the sample point (i.e. highly weighted tree) dies, and the associated weighted value is eliminated from the sample, consequently adding variability to the estimate. However, it is known that mortality would be a high source of variability regardless of the estimator used. Due to the fact that the comparisons were completed using the combined components of growth (including mortality), it is important for future studies to focus on the precision and efficiency gains that could result for each of the individual components of growth. When the high mortality scenarios were eliminated, the D V estimator was more efficient than the Subtraction estimator in most cases. Thus it is difficult to see the stand conditions where the D V estimator performed poorly using this comparison. To overcome this, the comparison of D V relative to the FRP estimator will be explored for only the change in volume and basal area. For the three densities tested in this study, the D V estimator performed poorest for scenarios with 500 stems per ha, where the D V estimator had REs greater than 1.0 for only 56% of the scenarios for volume change. However, only two of the nine scenarios for this density class were below 90% (RE = 0.9) efficiency and these scenarios contained other stand attributes values where the D V estimator also performed poorly. For the three dbh distributions tested, the D V estimator performed poorest for scenarios with the "No Understory" distribution, where the D V estimator had REs greater than 1.0 for only 33% of the scenarios for volume change. Again, the scenarios that had REs less than 90% (RE = 0.9) contained other stand attributes where the D V estimator also performed poorly. For the three spatial distributions tested, the D V estimator performed poorest for scenarios with "random" distributions, where the D V estimators had REs greater than 1.0 for only 56% of the scenarios for volume change. As expected the scenarios that contained at least two stand attribute values where the D V estimator performed poorly were the scenarios that had the lowest REs. Generally, basal area change in VRPs is a less stable attribute than volume change, which is stabilized through VBAR. This could explain the overall poor performance of basal area change using the D V estimator. As with stems per ha estimates, the FRP method will only have additions of small amounts of basal area due to ingrowth. Overall, this will contribute less variability to the sample than what is evident from the combination of nongrowth and ingrowth 71 on VRPs. However, the REs show that very few of the REs for the D V estimator are below 0.8, which suggest that these estimates are almost as efficient as the FRP method. Thus, this may not be an area of concern when deciding on using D V estimates for measuring change in basal area. The D V estimator was more precise than the Subtraction estimator in almost all of the scenarios. Overall, the D V estimator performed well in all scenarios except for those that had some combination of 500 stems per ha, no understory, and a random spatial distribution. These stands comprised approximately 15% of all the stands with the high mortality excluded. Given the range of stand conditions where the D V estimator performed well, it is reasonable to conclude that the D V estimate is as good as, or better, than the FRP estimator for volume change. Further work needs to be completed to determine the causes for the relatively poor performance of basal area change using the D V estimator. 7.4 LIMITATIONS Because this study was based on simulated data there are some limitations that may influence the results and the conclusions drawn. Although the simulation components were tested, further testing could be completed to help determine the accuracy of all assumptions. Testing is an integral part of any software development and a more structured testing regime for StandSim could be implemented to identify any inconsistencies that may be present in the data. This study did not incorporate stand dynamics directly, which might have affected the outcomes, particularly regarding density. Each tree in each stand had equal growth regardless of density. In a real stand, density would impact on how fast the trees would grow, as well as the way they grow. Considering the number and robustness of existing growth models it is unreasonable to think that much effort should be put into developing a robust growth model in StandSim. However, StandSim can generate a variety of forest spatial distributions, and this is where its strength lies. It would be interesting to further develop this aspect of StandSim and import the outputs produced into other well-established growth models. By doing this, the impacts of more realistic stand dynamics on the estimators could be investigated. While the scenarios used cover a broad range of stand conditions, they do not address all possible situations; this has resulted in broad conclusions. Readers should be aware of this and should not think that these conclusions apply to all stand types. More intensive studies on important or commonly encountered stand types could help draw further conclusions about the use of VRPs for change measurement. 72 The results produced only cover a single five-year time interval. The amount of time between measurements is an aspect that may influence the results. This could also be viewed as the amount of change from one measurement to the next, regardless of time. Essentially by varying the amount of change between two measurements, different growth period lengths could be explored. In this study the measurements were assumed to have no error. This would be an unreasonable assumption with real data. By investigating the amount of measurement error that results from using VRPs versus FRPs one might gain more confidence in the answer obtained. This would be a particularly important study as there is some thought that there is a higher potential for missed trees and potential measurement errors when using VRPs. However, one would expect that the potential for error when measuring more trees as in a FRP is higher. It was originally thought that a dollar cost would be associated with each of the methods used to determine the actual cost difference of using one estimator over another. However, this would require the development of a cost function that ultimately was considered to be beyond the scope of this project. While the results presented provide information on the statistical cost of using one estimator over another, it is important to associate a dollar cost for practical implementation. 73 8.0 FUTURE DIRECTIONS AND CONCLUSIONS Most future work should concentrate on the DV estimator as it is the most likely VRP to be used for change measurement in the future. The most important future work would be testing the estimators using real forest data to prove their use as a viable choice for forest managers. Currently, operational studies are being completed in British Columbia. 8.1 STANDSIM DEVELOPMENT Further testing and development of StandSim should be completed to increase its robustness and ability to more accurately mimic stands of varying spatial distributions. StandSim could prove to be a valuable tool for research on sampling stands or other forms of stand simulation. If StandSim is to be used at length for any future work one major improvement might be to reprogram it in a programming language other than Visual Basic (Microsoft Excel). The following improvements could be made to StandSim to increase its abilities for stand simulation. 1. Clump Shape - Currently the generated clump shapes are circular. It is unlikely that clumps will all be circular in a forest. By allowing the simulator to create irregular polygons for clumps the simulator would be more accurate in its generation of stands. One way of approaching this might be to have a clump axis of a random length and at random points on this axis, a number of random length polygon radii could be created. As long as the total area of the polygons equals the one defined by the user, then the clumps could be optimized the same way they are with the random clump size logic. 2. Stand Dynamics - Currently the trees generated in StandSim are assigned species randomly and the clumping of trees is independent of individual species. An improvement would be to determine what species would normally clump together. If tests were to be completed where species is a concern it would be important to have this aspect of stand dynamics. One approach for this might be to determine the probability of certain species of a certain size to clump with other species of certain size. Once these probabilities were determined, each tree could be assigned a clumping likelihood based on the previous tree's placement. Doing this would require that each tree species and size be evaluated before placement in the population frame. This would require more computing time and may require programming StandSim outside of the Microsoft Excel environment. 3. Clumping indices - There are many existing indices of clumping (e.g. Pielou 1960, Clark Evans 1954, and Ripley 1976). StandSim currently only uses the attributes of clumps to produce clumping. An improvement would be to allow the user to choose a level of 74 clumping based on an index. It would also be advantageous to produce an index value that represents the created stand configuration. This would ensure that clumping levels cover an appropriate range for research, beyond just a visual inspection. 4. Stand Vigor - Currently there is no site productivity or stand vigor values used StandSim. If the outputs of StandSim were to be exported and used in other growth models, a value for site productivity would need to be generated. This may be as simple as just arbitrarily assigning a value to any given stand; however, exploration of options for this type of stand attribute need to be completed. 8.2 MEASUREMENT ERROR All measurements in this study were assumed to have no error. Realistically, there would be some error in the measurements, whether through a missed tree or through an error in the dbh or height measurement. A future study could look at the efficiency of VRP estimators relative to the FRP estimator where some measurement error is introduced. By excluding different trees of varying size and value from plots, an assessment of using VRP estimators over the FRP in terms of the impact of measurement error could be obtained. Studies might need to be done to provide the expected amount of measurement error from one method to the next; this may be a function of a series of factors (i.e., weather, slope, down material, tree size, topography, number of trees measured, etc.). Completing this type of study could provide forest managers with a level of confidence in the change measurement estimator they decide to use. 8.3 STAND CONDITIONS The stand conditions used in this study represent a broad range. In some cases the scenarios used represent extremes of stand conditions that would not normally occur in a real forest. Future studies could concentrate on a specific set of stand conditions with modifications to other factors within that set of conditions. Simulation of more likely occurring stand conditions would be important to explore to help make conclusions on the possible limitations of the estimator for change measurement in those stand types. Mortality is a source of high variability for estimating change over time. Future studies could investigate the threshold of mortality before an estimator starts to perform poorly. Future work also needs to be completed on how to directly sample for mortality in a way that would reduce the variability of losing a tree in a sample. Although much work has been completed on how to model mortality in a stand, little work has been completed on the development of a mortality estimator. The D V estimator shows promise for estimating mortality in nongrowth trees, 75 particularly if a nongrowth tree dies soon after it has grown onto the sample. However this will only constitute a minor component of the contributing parts of a sample in many cases. Other components of growth using any of the estimators still introduce large amounts of variability from mortality. 8.4 GROWTH The growth function used for this study was basic in that it did not account for stand dynamics, particularly how the stands would grow under different densities. This might be addressed by exporting the stem maps of the stands generated in StandSim and growing them using a growth model which accounts for a variety of aspects of stand dynamics. This would require some way of assigning stand vigor or productivity to the simulations, which could be addressed in further development of StandSim. This study explored the performance of the combined components of growth. Future studies will need to investigate the performance of the individual components, as it may provide insight into the components that are best estimated using VRP methods. 8.5 COST This study only investigated the statistical differences between the VRP and FRPs estimators. While theoretically these comparisons could be related to dollar cost, more work needs to be done on investigating the actual dollar cost differences of using one change estimator over the other. A cost function could be developed that incorporates aspects of field measurements. One aspect of the actual dollar cost of sampling using one method over another is the tree size being sampled. FRPs would typically contain more small trees compared to VRPs of a similar size. The cost of plot establishment in terms of measuring trees in a plot versus with an angle gauge and the time required to measure distances and determining borderline trees could also be incorporated into this cost function. Scott (1990) investigated cost comparisons for FRPs and VRPs; however, the D V estimator had not been developed at that time and was not used in the test. More studies of this nature would provide further insight into the benefits of one estimator over another. 8.6 CONCLUSIONS There is no evidence to suggest that VRPs could not be used for measuring change over time based on the results of this study. VRP formulae and compilation procedures are slightly more complex than the FRP estimator when measuring change over time; however, not so much as to exclude their use. The use of the traditional Subtraction estimator introduces high variability for 76 volume change, but this has been eliminated through the development of the DV estimator. The VRP field techniques for PSPs do not differ from operational VRPs, except for the need for stem mapping. This study should be viewed as a primer for future work, particularly on aspects of the DV estimator performance. 77 9.0 REFERENCES Arvantis, L . G . , and W.G. O'Reagan. 1967. Computer simulation and economic efficiency in forest sampling. Hilgardia 38:133-164. Avery, T.E. , and H.E. Burkhart. 1983. Forest Measurements. 3 r d ed. McGraw-Hill Publishing, New York. 331 p. Beers, T.W. 1962. Components of Forest Growth. J For 60:245-248. Beers, T.W., and C.I. Miller. 1964. Point sampling: research results, theory and application. Purdue Univ, Agric Exp Stn., Layfayette Ind, Res Bull 786, 56 p. Bitterlich, W. 1948. Die Winkelzahl-probe Alg. Forst - U. Holzw. Ztg 59:4-5. (Original not consulted, Abstracted 10:2314, 1949). Bitterlich, W. 1984. The Relaskop Idea. Page Bros (Norwich) Ltd, London, UK. 242p. Clark, P.J. and F.C. Evans. 1954. Distance to the nearest neighbour as a measure of spatial relationships in populations. Ecology 35:445-453. Flewelling, J.W. 1981. Compatible estimates and basal area growth from remeasured point samples. For Sci 27(1): 191-203. Flewelling, J.W. and C.E. Thomas. 1984. An improved estimator for merchantable basal area growth on point samples. For Sci. 30(3):831-821. Gregoire, T.G. 1993. Estimation of forest growth from successive surveys. For Ecol Manage. 56:267-278. Grosenbaugh, L.R. 1958. Point-sampling and line sampling: probability theory, geometric implications, synthesis. USDA Forest Serv, South Forest Exp Stn Occas Pap 160, 34 p. Grosenbaugh, L.R. 1959. Should continuity dominate forest inventories? Paper presented at short course on continuous inventory control in forest management. May, 1959. Athens Georgia. Grosenbaugh, L.R., and W.F. Stover. 1957. Point sampling compared with plot sampling in southeast Texas. For Sci 3(1):2-14. Hall, O.F. 1959. The contribution of re-measured sample plots to the precision of growth estimates. J For 57:807-811. Horvitz, D.G, and D.J. Thompson. 1952. A generalization of sampling without replacement from a finite universe. Amer Stat Assoc J. 47:663-685. 78 Hurlbert, S.H., 1990. Spatial distribution of the montane unicorn. Oikos 58:257-271. Husch, B., C L Miller, and T.W. Beers. 1972. Forest Mensuration. 2 n d ed. Ronald Press, New York. 410 p. lies, K., 1974. Penetration sampling - an extension of the Bitterlich system to the third dimension. Oregon State Univ, Corvallis, Oregon. Unpublished manuscript. lies, K., 1979a. Some techniques to generalize the use of variable plot and line intersect sampling. Forest resource inventories workshop proceedings. Colo State Univ, Fort Collins. P. 270-278. lies, K., 1981. Permanent "variable" plots for forest growth. To have and to hold...? Paper presented at Western Mensurationist Meeting, Ketchum Idaho. 13 p. lies, K., 1987. Growth on permanent angle-count plots, historical procedures up through critical height sampling. IUFRO Forest Growth Modelling and Prediction Conference. Minneapolis, Minnesota, p. 1105-1113. lies, K., 1989. Critical height sampling: A workshop on the current state of the technique. Invited paper in the proceedings: State of the art methodology of forest inventory. Syracuse, New York. p. 74-83. lies, K., 2003. A Sampler of Inventory Topics. Kim lies & Associates Ltd., Nanaimo, B.C., 869 P-lies, K., and T.W. Beers. 1983. Growth information from variable plot sampling. Int'l conf. Renewable Resource Inventories For Monitoring Changes and Trends. Corvallis, Oregon, p. 693-695. lies, K , and H.H. Carter. 2007. Distance variable estimators for sampling and change measurement. Can J For Res. In Press (Tentatively Sept. 2007). Kitamura, M., 1962. On an estimate of the volume of trees in a stand by the sum of critical heights (In Japanese). Kai Nichi Rin Ko 73:64-67. Martin, G.L., 1982. A Method for estimating ingrowth on permanent horizontal sample points. For Sci 28(1): 110-114. Martin, G.L., 1983. The relative efficiency of some forest growth estimators. Biometrics. 39(3):639-650. McTague, J.P. and Bailey, R.L., 1985. Critical height sampling for stand volume estimation. For Sci 28(1):899-911. 79 Myers, C.C. and Beers, T.W., 1968. Point sampling for forest growth estimating. J For 66: 927-929. Oderwald, R.G. 1981. Comparison of point and plot sampling basal area estimators. For Sci 27(l):42-48. Oliver, D.O. and B.C. Larson., 1996. Forest Stand Dynamics. Update ed. John Wiley & Sons, New York. 520 p. Pielou, E.C, 1960. A single mechanism to account for regular, random and aggregated populations. The Journal of Ecology 48:575-584. Ranneby, B., 1980. The precision of the estimates of the growth in basal area. Swedish Univ of Agric Sci, Section of Forest Biometry, Report no. 19, 25p. Ripley, B.D. 1976. The second-order analysis of stationary processes. J. Appl. Prob. 13:255-266. Roesch, F.A., E.J Green, and C T . Scott. 1989. New compatible estimators for survivor growth and ingrowth from re-measured horizontal point samples. For Sci. 35(2):281-293. Roesch, F.A., E.J Green, and CT. Scott. 1991. Compatible basal area and number of trees estimators from re-measured horizontal point samples. For Sci. 37(1): 136-145. Scott, C.T., 1990. 'An Overview of Fixed Versus Variable Radius Plots for Successive Inventories'. Proc. Symposium, State-of-the-art Methodology of Forest Inventory. USDA Forest Service, Pacific Northwest Res Stn General Technical Report 263 p.97-104. Scott, C.T., and J. Alegria., 1990. Fixed-versus variable-radius plots for change estimation. USDA Forest Serv, Northeastern Forest Exp Stn General Tech Report 163. p. 126-132. Stage, A.R., 1960. Computing growth from increment cores with point sampling. J For. 58:531-533. Stoyan, D. and A. Penttinen. 2000. Recent applications of point process methods in forestry statistics. Statistical Science 15:61-78. Thomas, C E . and, F.A. Roesch. 1990. Basal area estimators for survivor component: A quality control application. S J App For 14(1): 12-18. Van Deusen, P.C and T.B. Lynch., 1987. Efficient unbiased tree-volume estimation. For Sci. 33(2):583-590. Van Deusen, P.C. and, W.J. Meerschaert. 1986. On critical height sampling. Can J For Res. 16:1310-1313. 80 Van Deusen, P.C, T.R. Dell, and C E . Thomas. 1986. Volume growth estimation from permanent horizontal points. For Sci. 32(2):415-422. 81 APPENDIX I RESULTS FOR ALL SCENARIOS RELATIVE TO FRP 82 Table 21. Relative Efficiencies of VRP estimators relative to FRP. A L L SCENARIOS RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 1 1.7 0.8 1.4 0.2 0.5 0.3 0.5 0.1 0.0 0.0 0.1 0.0 2 1.4 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 3 1.5 1.1 1.4 0.5 1.2 0.8 1.2 0.3 0.1 0.1 0.7 0.0 4 1.7 1.5 2.4 0.8 0.8 0.7 1.3 1.2 0.0 0.0 0.1 0.0 5 1.2 1.4 1.7 0.8 0.7 0.8 1.0 1.3 0.0 0.0 0.3 0.0 6 1.1 1.0 1.6 0.6 0.4 0.4 1.0 0.6 0.0 0.0 0.2 0.0 7 0.9 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 8 1.8 1.4 1.7 0.4 1.1 1.0 1.1 0.3 0.4 0.4 0.3 0.1 9 1.3 1.2 1.3 0.7 1.0 0.8 1.0 0.4 0.2 0.2 0.1 0.1 10 0.6 0.6 1.1 0.5 0.5 0.4 0.8 1.2 0.0 0.0 0.1 0.0 11 1.1 1.1 1.9 0.8 0.7 0.7 1.4 1.7 0.0 0.0 0.3 0.0 12 1.0 0.8 1.7 0.9 0.5 0.5 0.9 2.0 0.0 0.0 0.3 0.0 13 0.5 0.4 1.0 0.2 0.4 0.3 0.8 0.2 0.3 0.3 0.5 0.2 14 0.7 0.6 1.0 0.2 0.5 0.4 0.7 0.2 0.2 0.2 0.4 0.2 15 0.9 0.8 1.1 0.3 0.7 0.6 0.9 0.2 0.5 0.5 0.6 0.3 16 0.5 0.5 0.9 0.4 0.4 0.5 0.8 1.0 0.3 0.3 0.5 0.4 17 0.7 0.6 0.9 0.3 0.6 0.5 0.7 0.6 0.4 0.4 0.5 0.4 18 0.8 0.7 0.9 0.4 0.5 0.5 0.6 0.8 0.5 0.5 0.6 0.5 19 1.8 1.1 1.7 0.3 0.8 0.7 1.1 0.3 0.0 0.0 0.1 0.0 20 1.2 0.9 1.4 0.4 0.3 0.2 0.6 0.3 0.0 0.0 0.1 0.0 21 1.3 1.3 1.2 1.0 1.1 1.2 0.9 0.5 0.2 0.3 0.2 0.1 22' 1.1 1.1 1.7 0.8 0.5 0.5 0.9 1.1 0.0 0.0 0.2 0.0 23 1.5 1.5 2.1 1.4 0.8 0.8 1.3 1.9 0.0 0.0 0.3 0.0 24 1.4 1.1 1.7 0.8 0.7 0.6 1.1 0.7 0.0 0.0 0.4 0.0 25 1.9 1.7 3.1 0.9 1.6 1.5 2.2 0.8 0.3 0.2 0.2 0.1 26 1.7 1.6 1.4 0.5 1.2 1.4 1.0 0.5 0.1 0.2 0.3 0.1 27 1.1 1.1 1.3 0.8 1.0 1.0 1.2 0.7 0.0 0.0 0.2 0.0 28 0.9 0.9 1.5 0.6 0.7 0.6 1.0 1.4 0.0 0.0 0.2 0.0 29 0.9 0.9 1.2 0.5 0.5 0.5 0.8 1.2 0.0 0.0 0.2 0.0 30 1.0 0.9 1.6 0.7 0.5 0.5 0.9 1.1 0.1 0.1 0.6 0.1 31 0.4 0.4 1.1 0.2 0.3 0.3 0.9 0.2 0.2 0.2 0.5 0.2 32 1.1 0.9 1.1 0.3 0.8 0.7 0.9 0.4 0.4 0.4 0.6 0.2 33 1.0 0.9 1.0 0.5 0.8 0.7 0.9 0.4 0.3 0.2 0.9 0.2 34 0.6 0.5 0.8 0.4 0.5 0.5 0.8 0.9 0.4 0.4 0.7 0.5 35 0.5 0.5 1.0 0.4 0.4 0.4 0.8 0.9 0.3 0.4 0.6 0.5 36 0.7 0.7 1.1 0.3 0.5 0.5 0.8 0.6 0.5 0.6 0.7 0.5 37 1.4 0.9 1.2 0.2 0.7 0.5 0.8 0.2 0.1 0.1 0.1 0.1 38 1.2 1.1 1.6 0.5 0.8 0.7 1.2 0.4 0.0 0.0 0.1 0.0 39 1.3 1.1 - 1.2 0.8 1.2 1.0 1.0 0.5 0.0 0.0 0.2 0.0 40 0.8 0.9 1.1 0.5 0.5 0.5 0.7 0.8 0.0 0.0 0.3 0.0 41 1.6 1.4 2.1 0.8 0.8 0.8 1.4 0.9 0.0 0.0 0.2 0.0 42 1.2 1.0 1.2 0.6 0.6 0.6 0.8 0.4 0.0 0.0 0.6 0.0 43 1.3 0.7 1.2 0.2 0.8 0.5 1.1 0.2 0.0 0.0 0.3 0.0 44 1.5 1.1 2.1 0.5 1.2 0.9 1.8 0.4 0.2 0.2 0.6 0.2 45 1.2 1.3 1.2 0.9 0.9 0.9 1.0 0.5 0.0 0.0 0.1 0.0 46 1.1 0.9 1.2 0.7 0.8 0.7 0.9 1.8 0.0 0.0 0.3 0.0 47 1.2 1.1 1.6 0.7 0.7 0.7 1.0 1.5 0.0 0.0 0.2 0.1 48 0.8 0.8 1.2 0.8 0.6 0.7 0.9 1.1 0.0 0.0 0.7 0.1 49 0.7 0.5 1.5 0.2 0.7 0.5 1.2 0.2 0.4 0.4 0.7 0.3 50 0.9 0.6 1.0 0.2 0.8 0.5 0.8 0.3 0.5 0.4 0.7 0.3 51 1.0 0.9 1.1 0.6 0.9 0.8 1.0 0.4 0.4 0.4 0.5 0.3 52 0.5 0.5 0.9 0.5 0.4 0.4 0.7 1.4 0.3 0.3 0.4 0.3 53 0.7 0.7 1.2 0.5 0.5 0.5 0.9 1.0 0.3 0.3 0.5 0.4 54 0.6 0.5 0.7 0.3 0.5 0.5 0.6 0.7 0.7 0.6 0.6 0.6 % GT 1.0 59% 37% 83% 2% 15% 6% 39% 26% 0% 0% 0% 0% % GT 0.9 65% 50% 93% 4% 20% 13% 54% 31% 0% 0% 0% 0% % GT 0.8 74% 67% 98% 17% 28% 24% 72% 35% 0% 0% 2% 0% % GT 0.7 83% 76% 100% 31% 46% 33% 93% .44% 2% 0% 4% 0% 83 Table 22. REs of VRPs relative to the FRP Estimator for varying densities. DENSITY 500 Stems Per Hectare RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 1 1.7 0.8 1.4 0.2 0.5 0.3 0.5 0.1 0.0 0.0 0.1 0.0 2 1.4 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 3 1.5 1.1 1.4 0.5 1.2 0.8 1.2 0.3 0.1 0.1 0.7 0.0 4 1.7 1.5 2.4 0.8 0.8 0.7 1.3 1.2 0.0 0.0 0.1 0.0 5 1.2 1.4 1.7 0.8 0.7 0.8 1.0 1.3 0.0 0.0 0.3 0.0 6 1.1 1.0 1.6 0.6 0.4 0.4 1.0 0.6 0.0 0.0 0.2 0.0 7 0.9 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 8 1.8 1.4 1.7 0.4 1.1 1.0 1.1 0.3 0.4 0.4 0.3 0.1 9 1.3 1.2 1.3 0.7 1.0 0.8 1.0 0.4 0.2 0.2 0.1 0.1 10 0.6 0.6 1.1 0.5 0.5 0.4 0.8 1.2 0.0 0.0 0.1 0.0 11 1.1 1.1 1.9 0.8 0.7 0.7 1.4 1.7 0.0 0.0 0.3 0.0 12 1.0 0.8 1.7 0.9 0.5 0.5 0.9 2.0 0.0 0.0 0.3 0.0 13 0.5 0.4 1.0 0.2 0.4 0.3 0.8 0.2 0.3 0.3 0.5 0.2 14 0.7 0.6 1.0 0.2 0.5 0.4 0.7 0.2 0.2 0.2 0.4 0.2 15 0.9 0.8 1.1 0.3 0.7 0.6 0.9 0.2 0.5 0.5 0.6 0.3 16 0.5 0.5 0.9 0.4 0.4 0.5 0.8 1.0 0.3 0.3 0.5 0.4 17 0.7 0.6 0.9 0.3 0.6 0.5 0.7 0.6 0.4 0.4 0.5 0.4 18 0.8 0.7 0.9 0.4 0.5 0.5 0.6 0.8 0.5 0.5 0.6 0.5 %GT1.0 50% 33% 78% 0% M E A N RE 1.1 0.9 1.4 0.5 M A X RE 1.8 1.5 2.4 0.9 MIN RE 0.5 0.4 0.9 0.2 RANGE 1.3 1.1 1.5 0.7 17% 0% 33% 28% 0.6 0.6 0.9 0.7 1.2 1.0 1.4 0.4 0.3 0.5 0.8 0.6 0.9 2.0 0.1 1.9 0% 0.2 0.5 0.0 0.5 0% 0.2 0.5 0.0 0.5 0% 0.3 0.7 0.1 0.6 0% 0.1 0.5 0.0 0.5 1500 Stems Per Hectare RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 19 1.8 1.1 1.7 0.3 0.8 0.7 1.1 0.3 0.0 0.0 0.1 0.0 20 1.2 0.9 1.4 0.4 0.3 0.2 0.6 0.3 0.0 0.0 0.1 0.0 21 1.3 1.3 1.2 1.0 1.1 1.2 0.9 0.5 0.2 0.3 0.2 0.1 22 1.1 1.1 1.7 0.8 0.5 0.5 0.9 1.1 0.0 0.0 0.2 0.0 23 1.5 1.5 2.1 1.4 0.8 0.8 1.3 1.9 0.0 0.0 0.3 0.0 24 1.4 1.1 1.7 0.8 0.7 0.6 1.1 0.7 0.0 0.0 0.4 0.0 25 1.9 1.7 3.1 0.9 1.6 1.5 2.2 0.8 0.3 0.2 0.2 0.1 26 1.7 1.6 1.4 0.5 1.2 1.4 1.0 0.5 0.1 0.2 0.3 0.1 27 1.1 1.1 1.3 0.8 1.0 1.0 1.2 0.7 0.0 0.0 0.2 0.0 28 0.9 0.9 1.5 0.6 0.7 0.6 1.0 1.4 0.0 0.0 0.2 0.0 29 0.9 0.9 1.2 0.5 0.5 0.5 0.8 1.2 0.0 0.0 0.2 0.0 30 1.0 0.9 1.6 0.7 0.5 0.5 0.9 1.1 0.1 0.1 0.6 0.1 31 0.4 0.4 1.1 0.2 0.3 0.3 0.9 0.2 0.2 0.2 0.5 0.2 32 1.1 0.9 1.1 0.3 0.8 0.7 0.9 0.4 0.4 0.4 0.6 0.2 33 1.0 0.9 1.0 0.5 0.8 0.7 0.9 0.4 0.3 0.2 0.9 0.2 34 0.6 0.5 0.8 0.4 0.5 0.5 0.8 0.9 0.4 0.4 0.7 0.5 35 0.5 0.5 1.0 0.4 0.4 0.4 0.8 0.9 0.3 0.4 0.6 0.5 36 0.7 0.7 1.1 0.3 0.5 0.5 0.8 0.6 0.5 0.6 0.7 0.5 % GT 1.0 67% 44% 89% 6% 17% 17% 33% 28% 0% 0% 0% 0% M E A N RE 1.1 1.0 1.4 0.6 0.7 0.7 1.0 0.8 0.2 0.2 0.4 0.1 M A X RE 1.9 1.7 3.1 1.4 1.6 1.5 2.2 1.9 0.5 0.6 0.9 0.5 MIN RE 0.4 0.4 0.8 0.2 0.3 84 3000 Stems Per Hectare RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 37 1.4 0.9 1.2 0.2 0.7 0.5 0.8 0.2 0.1 0.1 0.1 0.1 38 1.2 1.1 1.6 0.5 0.8 0.7 1.2 0.4 0.0 0.0 0.1 0.0 39 1.3 1.1 1.2 0.8 1.2 1.0 1.0 0.5 0.0 0.0 0.2 0.0 40 0.8 0.9 1.1 0.5 0.5 0.5 0.7 0.8 0.0 0.0 0.3 0.0 41 1.6 1.4 2.1 0.8 0.8 0.8 1.4 0.9 0.0 0.0 0.2 0.0 42 1.2 1.0 1.2 0.6 0.6 0.6 0.8 0.4 0.0 0.0 0.6 0.0 43 1.3 0.7 1.2 0.2 0.8 0.5 1.1 0.2 0.0 0.0 0.3 0.0 44 1.5 1.1 2.1 0.5 1.2 0.9 1.8 0.4 0.2 0.2 0.6 0.2 45 1.2 1.3 1.2 0.9 0.9 0.9 1.0 0.5 0.0 0.0 0.1 0.0 46 1.1 0.9 1.2 0.7 0.8 0.7 0.9 1.8 0.0 0.0 0.3 0.0 47 1.2 1.1 1.6 0.7 0.7 0.7 1.0 1.5 0.0 0.0 0.2 0.1 48 0.8 0.8 1.2 0.8 0.6 0.7 0.9 1.1 0.0 0.0 0.7 0.1 49 0.7 0.5 1.5 0.2 0.7 0.5 1.2 0.2 0.4 0.4 0.7 0.3 50 0.9 0.6 1.0 0.2 0.8 0.5 0.8 0.3 0.5 0.4 0.7 0.3 51 1.0 0.9 1.1 0.6 0.9 0.8 1.0 0.4 0.4 0.4 0.5 0.3 52 0.5 0.5 0.9 0.5 0.4 0.4 0.7 1.4 0.3 0.3 0.4 0.3 53 0.7 0.7 1.2 0.5 0.5 0.5 0.9 1.0 0.3 0.3 0.5 0.4 54 0.6 0.5 0.7 0.3 0.5 0.5 0.6 0.7 0.7 0.6 0.6 0.6 % GT 1.0 61% 33%) 83% 0% 11% 0% 50% 22% 0% 0%) 0% 0% M E A N RE 1.0 0.9 1.3 0.5 0.7 0.7 1.0 0.7 0.2 0.2 0.4 0.1 M A X RE 1.6 1.4 2.1 0.9 1.2 1.0 1.8 1.8 0.7 0.6 0.7 0.6 MIN RE 0.5 0.5 0.7 0.2 0.4 0.4 0.6 0.2 0.0 0.0 0.1 0.0 RANGE 1.1 1.0 1.4 0.7 0.8 0.5 1.2 1.7 0.7 0.6 0.6 0.6 Table 23. REs of VRPs relative to the FRP Estimator for varying diameter distributions. DIAMETER DISTRIBUTIONS Understory RE of Vo . Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 1 1.7 0.8 1.4 0.2 0.5 0.3 0.5 0.1 0.0 0.0 0.1 0.0 2 1.4 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 3 1.5 1.1 1.4 0.5 1.2 0.8 1.2 0.3 0.1 0.1 0.7 0.0 4 1.7 1.5 2.4 0.8 0.8 0.7 1.3 1.2 0.0 0.0 0.1 0.0 5 1.2 1.4 1.7 0.8 0.7 0.8 1.0 1.3 0.0 0.0 0.3 0.0 6 1.1 1.0 1.6 0.6 0.4 0.4 1.0 0.6 0.0 0.0 0.2 0.0 19 1.8 1.1 1.7 0.3 0.8 0.7 1.1 0.3 0.0 0.0 0.1 0.0 20 1.2 0.9 1.4 0.4 0.3 0.2 0.6 0.3 0.0 0.0 0.1 0.0 21 1.3 1.3 1.2 1.0 1.1 1.2 0.9 0.5 0.2 0.3 0.2 0.1 22 1.1 1.1 1.7 0.8 0.5 0.5 0.9 1.1 0.0 0.0 0.2 0.0 23 1.5 1.5 2.1 1.4 0.8 0.8 1.3 1.9 0.0 0.0 0.3 0.0 24 1.4 1.1 1.7 0.8 0.7 0.6 1.1 0.7 0.0 0.0 0.4 0.0 37 1.4 0.9 1.2 0.2 0.7 0.5 0.8 0.2 0.1 0.1 0.1 0.1 38 1.2 1.1 1.6 0.5 0.8 0.7 1.2 0.4 0.0 0.0 0.1 0.0 39 1.3 1.1 1.2 0.8 1.2 1.0 1.0 0.5 0.0 0.0 0.2 0.0 40 0.8 0.9 1.1 0.5 0.5 0.5 0.7 0.8 0.0 0.0 0.3 0.0 41 1.6 1.4 2.1 0.8 0.8 0.8 1.4 0.9 0.0 0.0 0.2 0.0 42 1.2 1.0 1.2 0.6 0.6 0.6 0.8 0.4 0.0 0.0 0.6 0.0 % GT 1.0 94% 61% 100%) 6% 17% 6% 50% 22% 0% 0% 0% 0%) M E A N RE 1.3 1.1 1.6 0.6 0.7 0.6 1.0 0.6 0.0 0.0 0.2 0.0 M A X RE 1.8 1.5 2.4 1.4 1.2 1.2 1.4 1.9 0.2 0.3 0.7 0.1 MIN RE 0.8 0.8 1.1 0.2 0.3 0.2 0.5 0.1 0.0 0.0 0.1 0.0 RANGE 0.9 0.7 1.3 1.2 0.9 1.0 0.8 1.8 0.2 0.3 0.6 0.1 85 RE of Vol. Change Mixed RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 7 0.9 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 8 1.8 1.4 1.7 0.4 1.1 1.0 1.1 0.3 0.4 0.4 0.3 0.1 9 1.3 1.2 1.3 0.7 1.0 0.8 1.0 0.4 0.2 0.2 0.1 0.1 10 0.6 0.6 1.1 0.5 0.5 0.4 0.8 1.2 0.0 0.0 0.1 0.0 11 1.1 1.1 1.9 0.8 0.7 0.7 1.4 1.7 0.0 0.0 0.3 0.0 12 1.0 0.8 1.7 0.9 0.5 0.5 0.9 2.0 0.0 0.0 0.3 0.0 25 1.9 1.7 3.1 0.9 1.6 1.5 2.2 0.8 0.3 0.2 0.2 0.1 26 1.7 1.6 1.4 0.5 1.2 1.4 1.0 0.5 0.1 0.2 0.3 0.1 27 1.1 1.1 1.3 0.8 1.0 1.0 1.2 0.7 0.0 0.0 0.2 0.0 28 0.9 0.9 1.5 0.6 0.7 0.6 1.0 1.4 0.0 0.0 0.2 0.0 29 0.9 0.9 1.2 0.5 0.5 0.5 0.8 1.2 0.0 0.0 0.2 0.0 30 1.0 0.9 1.6 0.7 0.5 0.5 0.9 1.1 0.1 0.1 0.6 0.1 43 1.3 0.7 1.2 0.2 0.8 0.5 1.1 0.2 0.0 0.0 0.3 0.0 44 1.5 1.1 2.1 0.5 1.2 0.9 1.8 0.4 0.2 0.2 0.6 0.2 45 1.2 1.3 1.2 0.9 0.9 0.9 1.0 0.5 0.0 0.0 0.1 0.0 46 1.1 0.9 1.2 0.7 0.8 0.7 0.9 1.8 0.0 0.0 0.3 0.0 47 1.2 1.1 1.6 0.7 0.7 0.7 1.0 1.5 0.0 0.0 0.2 0.1 48 0.8 0.8 1.2 0.8 0.6 0.7 0.9 1.1 0.0 0.0 0.7 0.1 % GT 1.0 67% 50% 100% 0% 28% 11% 56% 50% 0% 0% 0% 0% M E A N RE 1.2 1.1 1.5 0.6 0.8 0.8 1.1 1.0 0.1 0.1 0.3 0.1 M A X RE 1.9 1.7 3.1 0.9 1.6 1.5 2.2 2.0 0.4 0.4 0.7 0.2 MIN RE 0.6 0.6 1.1 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 RANGE 1.2 1.1 2.0 0.7 1.2 1.1 1.5 1.8 0.4 0.4 0.6 0.2 RE of Vol. Change No Understory RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 13 0.5 0.4 1.0 0.2 0.4 0.3 0.8 0.2 0.3 0.3 0.5 0.2 14 0.7 0.6 1.0 0.2 0.5 0.4 0.7 0.2 0.2 0.2 0.4 0.2 15 0.9 0.8 1.1 0.3 0.7 0.6 0.9 0.2 0.5 0.5 0.6 0.3 16 0.5 0.5 0.9 0.4 0.4 0.5 0.8 1.0 0.3 0.3 0.5 0.4 17 0.7 0.6 0.9 0.3 0.6 0.5 0.7 0.6 0.4 0.4 0.5 0.4 18 0.8 0.7 0.9 0.4 0.5 0.5 0.6 0.8 0.5 0.5 0.6 0.5 31 0.4 0.4 1.1 0.2 0.3 0.3 0.9 0.2 0.2 0.2 0.5 0.2 32 1.1 0.9 1.1 0.3 0.8 0.7 0.9 0.4 0.4 0.4 0.6 0.2 33 1.0 0.9 1.0 0.5 0.8 0.7 0.9 0.4 0.3 0.2 0.9 0.2 34 0.6 0.5 0.8 0.4 0.5 0.5 0.8 0.9 0.4 0.4 0.7 0.5 35 0.5 0.5 1.0 0.4 0.4 0.4 0.8 0.9 0.3 0.4 0.6 0.5 36 0.7 0.7 1.1 0.3 0.5 0.5 0.8 0.6 0.5 0.6 0.7 0.5 49 0.7 0.5 1.5 0.2 0.7 0.5 1.2 0.2 0.4 0.4 0.7 0.3 50 0.9 0.6 1.0 0.2 0.8 0.5 0.8 0.3 0.5 0.4 0.7 0.3 51 1.0 0.9 1.1 0.6 0.9 0.8 1.0 0.4 0.4 0.4 0.5 0.3 52 0.5 0.5 0.9 0.5 0.4 0.4 0.7 1.4 0.3 0.3 0.4 0.3 53 0.7 0.7 1.2 0.5 0.5 0.5 0.9 1.0 0.3 0.3 0.5 0.4 54 0.6 0.5 0.7 0.3 0.5 0.5 0.6 0.7 0.7 0.6 0.6 0.6 % GT 1.0 17% 0% 50% 0% 0% 0% 11% 6% 0% 0% 0% 0% M E A N RE 0.7 0.6 1.0 0.3 0.6 0.5 0.8 0.6 0.4 0.4 0.6 0.3 M A X RE 1.1 0.9 1.5 0.6 0.9 0.8 1.2 1.4 0.7 0.6 0.9 0.6 MIN RE 0.4 0.4 0.7 0.2 0.3 0.3 0.6 0.2 0.2 0.2 0.4 0.2 RANGE 0.6 0.6 0.7 0.5 0.6 0.5 0.6 1.1 0.6 0.4 0.5 0.4 86 Table 24. REs of VRPs relative to the FRP estimator for varying levels of mortality. Normal Mortality RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 1 1.7 0.8 1.4 0.2 0.5 0.3 0.5 0.1 0.0 0.0 0.1 0.0 2 1.4 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 3 1.5 1.1 1.4 0.5 1.2 0.8 1.2 0.3 0.1 0.1 0.7 0.0 7 0.9 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 8 1.8 1.4 1.7 0.4 1.1 1.0 1.1 0.3 0.4 0.4 0.3 0.1 9 1.3 1.2 1.3 0.7 1.0 0.8 1.0 0.4 0.2 0.2 0.1 0.1 13 0.5 0.4 1.0 0.2 0.4 0.3 0.8 0.2 0.3 0.3 0.5 0.2 14 0.7 0.6 1.0 0.2 0.5 0.4 0.7 0.2 0.2 0.2 0.4 0.2 15 0.9 0.8 1.1 0.3 0.7 0.6 0.9 0.2 0.5 0.5 0.6 0.3 19 1.8 1.1 1.7 0.3 0.8 0.7 1.1 0.3 0.0 0.0 0.1 0.0 20 1.2 0.9 1.4 0.4 0.3 0.2 0.6 0.3 0.0 0.0 0.1 0.0 21 1.3 1.3 1.2 1.0 1.1 1.2 0.9 0.5 0.2 0.3 0.2 0.1 25 1.9 1.7 3.1 0.9 1.6 1.5 2.2 0.8 0.3 0.2 0.2 0.1 26 1.7 1.6 1.4 0.5 1.2 1.4 1.0 0.5 0.1 0.2 0.3 0.1 27 1.1 1.1 1.3 0.8 1.0 1.0 1.2 0.7 0.0 0.0 0.2 0.0 31 0.4 0.4 1.1 0.2 0.3 0.3 0.9 0.2 0.2 0.2 0.5 0.2 32 1.1 0.9 1.1 0.3 0.8 0.7 0.9 0.4 0.4 0.4 0.6 0.2 33 1.0 0.9 1.0 0.5 0.8 0.7 0.9 0.4 0.3 0.2 0.9 0.2 37 1.4 0.9 1.2 0.2 0.7 0.5 0.8 0.2 0.1 0.1 0.1 0.1 38 1.2 1.1 1.6 0.5 0.8 0.7 1.2 0.4 0.0 0.0 0.1 0.0 39 1.3 1.1 1.2 0.8 1.2 1.0 1.0 0.5 0.0 0.0 0.2 0.0 43 1.3 0.7 1.2 0.2 0.8 0.5 1.1 0.2 0.0 0.0 0.3 0.0 44 1.5 1.1 2.1 0.5 1.2 0.9 1.8 0.4 0.2 0.2 0.6 0.2 45 1.2 1.3 1.2 0.9 0.9 0.9 1.0 0.5 0.0 0.0 0.1 0.0 49 0.7 0.5 1.5 0.2 0.7 0.5 1.2 0.2 0.4 0.4 0.7 0.3 50 0.9 0.6 1.0 0.2 0.8 0.5 0.8 0.3 0.5 0.4 0.7 0.3 51 1.0 0.9 1.1 0.6 0.9 0.8 1.0 0.4 0.4 0.4 0.5 0.3 % G T 1.0 74% 44% 89% 0% 30% 11% 52% 0% 0% 0% 0% 0% M E A N R E 1.2 1.0 1.4 0.4 0.8 0.7 1.0 0.3 0.2 0.2 0.3 0.1 M A X RE 1.9 1.7 3.1 1.0 1.6 1.5 2.2 0.8 0.5 0.5 0.9 0.3 MIN R E 0.4 0.4 1.0 0.2 0.3 0.2 0.5 0.1 0.0 0.0 0.1 0.0 RANGE 1.4 1.4 2.2 0.8 1.3 1.3 1.7 0.7 0.5 0.5 0.8 0.3 High Mortality RE of Vol. Change R E of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 4 1.7 1.5 2.4 0.8 0.8 0.7 1.3 1.2 0.0 0.0 0.1 0.0 5 1.2 1.4 1.7 0.8 0.7 0.8 1.0 1.3 0.0 0.0 0.3 0.0 6 1.1 1.0 1.6 0.6 0.4 0.4 1.0 0.6 0.0 0.0 0.2 0.0 10 0.6 0.6 1.1 0.5 0.5 0.4 0.8 1.2 0.0 0.0 0.1 0.0 11 1.1 1.1 1.9 0.8 0.7 0.7 1.4 1.7 0.0 0.0 0.3 0.0 12 1.0 0.8 1.7 0.9 0.5 0.5 0.9 2.0 0.0 0.0 0.3 0.0 16 0.5 0.5 0.9 0.4 0.4 0.5 0.8 1.0 0.3 0.3 0.5 0.4 17 0.7 0.6 0.9 0.3 0.6 0.5 0.7 0.6 0.4 0.4 0.5 0.4 18 0.8 0.7 0.9 0.4 0.5 0.5 0.6 0.8 0.5 0.5 0.6 0.5 22 1.1 1.1 1.7 0.8 0.5 0.5 0.9 1.1 0.0 0.0 0.2 0.0 23 1.5 1.5 2.1 1.4 0.8 0.8 1.3 1.9 0.0 0.0 0.3 0.0 24 1.4 1.1 1.7 0.8 0.7 0.6 1.1 0.7 0.0 0.0 0.4 0.0 28 0.9 0.9 1.5 0.6 0.7 0.6 1.0 1.4 0.0 0.0 0.2 0.0 29 0.9 0.9 1.2 0.5 0.5 0.5 0.8 1.2 0.0 0.0 0.2 0.0 30 1.0 0.9 1.6 0.7 0.5 0.5 0.9 1.1 0.1 0.1 0.6 0.1 34 0.6 0.5 0.8 0.4 0.5 0.5 0.8 0.9 0.4 0.4 0.7 0.5 35 0.5 0.5 1.0 0.4 0.4 0.4 0.8 0.9 0.3 0.4 0.6 0.5 36 0.7 0.7 1.1 0.3 0.5 0.5 0.8 0.6 0.5 0.6 0.7 0.5 40 0.8 0.9 1.1 0.5 0.5 0.5 0.7 0.8 0.0 0.0 0.3 0.0 41 1.6 1.4 2.1 0.8 0.8 0.8 1.4 0.9 0.0 0.0 0.2 0.0 42 1.2 1.0 1.2 0.6 0.6 0.6 0.8 0.4 0.0 0.0 0.6 0.0 46 1.1 0.9 1.2 0.7 0.8 0.7 0.9 1.8 0.0 0.0 0.3 0.0 47 1.2 1.1 1.6 0.7 0.7 0.7 1.0 1.5 0.0 0.0 0.2 0.1 48 0.8 0.8 1.2 0.8 0.6 0.7 0.9 1.1 0.0 0.0 0.7 0.1 52 0.5 0.5 0.9 0.5 0.4 0.4 0.7 1.4 0.3 0.3 0.4 0.3 53 0.7 0.7 1.2 0.5 0.5 0.5 0.9 1.0 0.3 0.3 0.5 0.4 54 0.6 0.5 0.7 0.3 0.5 0.5 0.6 0.7 0.7 0.6 0.6 0.6 % G T 1.0 44% 30% 78% 4% 0% 0% 26% 52% 0% 0% 0% 0% M E A N R E 1.0 0.9 1.4 0.6 0.6 0.6 0.9 1.1 0.2 0.2 0.4 0.2 M A X RE 1.7 1.5 2.4 1.4 0.8 0.8 1.4 2.0 0.7 0.6 0.7 0.6 MIN R E 0.5 0.5 0.7 0.3 0.4 0.4 0.6 0.4 0.0 0.0 0.1 0.0 RANGE 1.3 1.1 1.7 1.1 0.4 0.4 0.8 1.6 0.7 0.6 0.6 0.6 87 Table 25. REs of VRPs relative to the FRP estimator for varying spatial distributions. SPATIAL DISTRIBUTIONS Random RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 1 1.7 0.8 1.4 0.2 0.5 0.3 0.5 0.1 0.0 0.0 0.1 0.0 4 1.7 1.5 2.4 0.8 0.8 0.7 1.3 1.2 0.0 0.0 0.1 0.0 7 0.9 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 10 0.6 0.6 1.1 0.5 0.5 0.4 0.8 1.2 0.0 0.0 0.1 0.0 13 0.5 0.4 1.0 0.2 0.4 0.3 0.8 0.2 0.3 0.3 0.5 0.2 16 0.5 0.5 0.9 0.4 0.4 0.5 0.8 1.0 0.3 0.3 0.5 0.4 19 1.8 1.1 1.7 0.3 0.8 0.7 1.1 0.3 0.0 0.0 0.1 0.0 22 1.1 1.1 1.7 0.8 0.5 0.5 0.9 1.1 0.0 0.0 0.2 0.0 25 1.9 1.7 3.1 0.9 1.6 1.5 2.2 0.8 0.3 0.2 0.2 0.1 28 0.9 0.9 1.5 0.6 0.7 0.6 1.0 1.4 0.0 0.0 0.2 0.0 31 0.4 0.4 1.1 0.2 0.3 0.3 0.9 0.2 0.2 0.2 0.5 0.2 34 0.6 0.5 0.8 0.4 0.5 0.5 0.8 0.9 0.4 0.4 0.7 0.5 37 1.4 0.9 1.2 0.2 0.7 , 0.5 0.8 0.2 0.1 0.1 0.1 0.1 40 0.8 0.9 1.1 0.5 0.5 0.5 0.7 0.8 0.0 0.0 0.3 0.0 43 1.3 0.7 1.2 0.2 0.8 0.5 1.1 0.2 0.0 0.0 0.3 0.0 46 1.1 0.9 1.2 0.7 0.8 0.7 0.9 1.8 0.0 0.0 0.3 0.0 49 0.7 0.5 1.5 0.2 0.7 0.5 1.2 0.2 0.4 0.4 0.7 0.3 52 0.5 0.5 0.9 0.5 0.4 0.4 0.7 1.4 0.3 0.3 0.4 0.3 % GT 1.0 44% 22% 83% 0% 6% 6% 28% 33 % 0% 0% 0% 0% M E A N RE 1.0 0.8 1.4 0.4 0.6 0.6 1.0 0.7 0.1 0.1 0.3 0.1 M A X RE 1.9 1.7 3.1 0.9 1.6 1.5 2.2 1.8 0.4 0.4 0.7 0.5 MIN RE 0.4 0.4 0.8 0.2 0.3 0.3 0.5 0.1 0.0 0.0 0.1 0.0 RANGE 1.4 1.4 2.3 0.7 1.3 1.2 1.7 1.7 0.4 0.4 0.6 0.5 Clumped RE of Vol . Change RE of BA Change RE of SPH Change Scenario DV F L GR SU DV F L GR SU DV F L GR SU 2 1.4 0.8 1.4 0.2 0.4 0.4 0.7 0.2 0.0 0.0 0.1 0.0 5 1.2 1.4 1.7 0.8 0.7 0.8 1.0 1.3 0.0 0.0 0.3 0.0 8 1.8 1.4 1.7 0.4 1.1 1.0 1.1 0.3 0.4 0.4 0.3 0.1 11 1.1 1.1 1.9 0.8 0.7 0.7 1.4 1.7 0.0 0.0 0.3 0.0 14 0.7 0.6 1.0 0.2 0.5 0.4 0.7 0.2 0.2 0.2 0.4 0.2 17 0.7 0.6 0.9 0.3 0.6 0.5 0.7 0.6 0.4 0.4 0.5 0.4 20 1.2 0.9 1.4 0.4 0.3 0.2 0.6 0.3 0.0 0.0 0.1 0.0 23 1.5 1.5 2.1 1.4 0.8 0.8 1.3 1.9 0.0 0.0 0.3 0.0 26 1.7 1.6 1.4 0.5 1.2 1.4 1.0 0.5 0.1 0.2 0.3 0.1 29 0.9 0.9 1.2 0.5 0.5 0.5 0.8 1.2 0.0 0.0 0.2 0.0 32 1.1 0.9 1.1 0.3 0.8 0.7 0.9 0.4 0.4 0.4 0.6 0.2 35 0.5 0.5 1.0 0.4 0.4 0.4 0.8 0.9 0.3 0.4 0.6 0.5 38 1.2 1.1 1.6 0.5 0.8 0.7 1.2 0.4 0.0 0.0 0.1 0.0 41 1.6 1.4 2.1 0.8 0.8 0.8 1.4 0.9 0.0 0.0 0.2 0.0 44 1.5 1.1 2.1 0.5 1.2 0.9 1.8 0.4 0.2 0.2 0.6 0.2 47 1.2 1.1 1.6 0.7 0.7 0.7 1.0 1.5 0.0 0.0 0.2 0.1 50 0.9 0.6 1.0 0.2 0.8 0.5 0.8 0.3 0.5 0.4 0.7 0.3 53 0.7 0.7 1.2 0.5 0.5 0.5 0.9 1.0 0.3 0.3 0.5 0.4 % GT 1.0 67 % 50% 83% 6% 17% 6% 50% 28%; 0% 0% 0% 0% M E A N RE 1.2 1.0 1.5 0.5 0.7 0.7 1.0 0.8 0.2 0.2 0.3 0.1 M A X RE 1.8 1.6 2.1 1.4 1.2 1.4 1.8 1.9 0.5 0.4 0.7 0.5 MIN RE 0.5 0.5 0.9 0.2 0.3 0.2 0.6 0.2 0.0 0.0 0.1 0.0 RANGE 1.3 1.1 1.2 1.2 0.9 1.2 1.2 1.8 0.5 0.4 0.6 0.5 88 Very Clumped RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV FL GR SU DV F L GR SU DV F L GR SU 3 1.5 1.1 1.4 0.5 1.2 0.8 1.2 0.3 0.1 0.1 0.7 0.0 6 1.1 1.0 1.6 0.6 0.4 0.4 1.0 0.6 0.0 0.0 0.2 0.0 9 1.3 1.2 1.3 0.7 1.0 0.8 1.0 0.4 0.2 0.2 0.1 0.1 12 1.0 0.8 1.7 0.9 0.5 0.5 0.9 2.0 0.0 0.0 0.3 0.0 15 0.9 0.8 1.1 0.3 0.7 0.6 0.9 0.2 0.5 0.5 0.6 0.3 18 0.8 0.7 0.9 0.4 0.5 0.5 0.6 0.8 0.5 0.5 0.6 0.5 21 1.3 1.3 1.2 1.0 1.1 1.2 0.9 0.5 0.2 0.3 0.2 0.1 24 1.4 1.1 1.7 0.8 0.7 0.6 1.1 0.7 0.0 0.0 0.4 0.0 27 1.1 1.1 1.3 0.8 1.0 1.0 1.2 0.7 0.0 0.0 0.2 0.0 30 1.0 0.9 1.6 0.7 0.5 0.5 0.9 1.1 0.1 0.1 0.6 0.1 33 1.0 0.9 1.0 0.5 0.8 0.7 0.9 0.4 0.3 0.2 0.9 0.2 36 0.7 0.7 1.1 0.3 0.5 0.5 0.8 0.6 0.5 0.6 0.7 0.5 39 1.3 1.1 1.2 0.8 1.2 1.0 1.0 0.5 0.0 0.0 0.2 0.0 42 1.2 1.0 1.2 0.6 0.6 0.6 0.8 0.4 0.0 0.0 0.6 0.0 45 1.2 1.3 1.2 0.9 0.9 0.9 1.0 0.5 0.0 0.0 0.1 0.0 48 0.8 0.8 1.2 0.8 0.6 0.7 0.9 1.1 0.0 0.0 0.7 0.1 51 1.0 0.9 1.1 0.6 0.9 0.8 1.0 0.4 0.4 0.4 0.5 0.3 54 0.6 0.5 0.7 0.3 0.5 0.5 0.6 0.7 0.7 0.6 0.6 0.6 % GT 1.0 67% 39% 83% 0% 22% 6% 39 % 17% 0% 0% 0% 0% M E A N RE 1.1 1.0 1.2 0.6 0.8 0.7 0.9 0.7 0.2 0.2 0.5 0.2 M A X RE 1.5 1.3 1.7 1.0 1.2 1.2 1.2 2.0 0.7 0.6 0.9 0.6 MIN RE 0.6 0.5 0.7 0.3 0.4 0.4 0.6 0.2 0.0 0.0 0.1 0.0 RANGE 0.9 0.8 1.0 0.7 0.7 0.8 0.6 1.8 0.7 0.6 0.8 0.5 Per fo rmance o f Es t imato rs f o r V o l u m e C h a n g e • GTFE1.0 • GTFE0.9 • GTFEQ8 • GTRE0.7 •STANCE VARIABLE FLEWELUN3 GFC6ENBALGH SLBTRACTlCN Figure 32. Volume REs for V R P estimators relative to FRPs. Per fo rmance o f Es t imato rs f o r Basa l Area C n a n g e 80% 7D% 40% 30% 20% 1ff% 0% s s i 1 • GTRE 1.0 • GTPE Q9 a GTRE Q8 • GTFEQ7 •STANCE VAHABLE FLEWBJJN3 GROSEr*AU3H C h a n g e E s U m a t o r SUBTRACTION Figure 33. Basal area REs for V R P estimators relative to FRPs. 89 APPENDIX II RESULTS FOR A L L SCENARIOS RELATIVE TO SUBTRACTION 90 Table 26. V R P REs of all scenarios relative to the Subtraction Estimator. A L L SCENARIOS RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 1 10.7 5.2 9.3 4.6 3.0 4.9 1.7 0.8 11.1 2 7.2 4.1 7.2 2.8 2.3 4.7 1.1 1.2 25.6 3 2.7 2.1 2.6 3.3 2.3 3.5 1.5 0.0 17.0 4 2.1 1.9 3.0 0.6 0.6 1.0 0.9 3.6 21.6 5 1.5 1.8 2.1 0.6 0.6 0.8 0.7 2.3 26.5 6 1.8 1.7 2.7 0.7 0.7 1.7 0.5 0.2 11.0 7 4.8 3.9 7.0 1.7 1.7 2.9 2.2 0.3 10.6 8 4.9 3.7 4.6 3.4 3.0 3.5 3.6 0.0 2.8 9 1.9 1.7 1.9 2.5 2.0 2.5 2.1 0.0 1.2 10 1.4 1.3 2.4 0!4 0.4 0.6 1.1 0.7 5.4 11 1.5 1.4 2.5 0.4 0.4 0.8 1.3 0.5 25.1 12 1.1 1.0 1.9 0.3 0.2 0.4 0.7 0.1 8.4 13 2.3 1.9 4.9 1.6 1.4 3.2 1.8 0.0 3.1 14 3.5 2.7 4.7 2.1 1.8 3.1 0.8 0.0 1.7 15 2.7 2.4 3.3 3.3 2.9 4.4 1.8 0.0 2.3 16 1.3 1.4 2.4 0.4 0.5 0.8 1.0 0.0 1.5 17 2.4 2.0 2.9 0.9 0.8 1.1 1.0 0.0 1.3 18 2.0 1.8 2.3 0.7 0.6 0.8 1.1 0.0 1.3 19 5.9 3.7 5.8 3.2 2.6 4.3 0.3 0.4 4.1 20 2.7 2.1 3.4 1.0 0.8 1.9 0.6 0.2 7.5 21 1.3 1.4 1.2 2.0 2.3 1.8 1.4 0.0 1.4 22 1.4 1.4 2.1 0.5 0.5 0.9 0.4 5.8 47.9 23 1.1 1.1 1.5 0.4 0.4 0.7 5.5 4.9 135.6 24 1.6 1.4 2.0 0.9 0.8 1.4 1.2 0.2 16.1 25 2.2 2.0 3.6 2.0 1.9 2.8 3.7 0.1 3.3 26 3.6 3.5 3.0 2.4 2.7 2.0 2.3 0.0 4.7 27 1.4 1.4 1.6 1.5 1.5 1.8 0.7 0.0 6.2 28 1.5 1.4 2.3 0.5 0.5 0.7 0.7 1.0 11.5 29 1.7 1.6 2.2 0.4 0.4 0.7 1.1 0.2 5.0 30 1.4 1.3 2.2 0.5 0.5 0.9 0.8 0.0 6.8 31 2.9 2.3 7.0 1.5 1.3 4.0 1.0 0.0 2.6 32 3.5 2.7 3.4 2.2 1.8 2.6 2.1 0.0 2.9 33 2.0 1.8 1.9 2.3 1.8 2.5 1.5 0.0 4.4 34 1.7 1.5 2.4 0.6 0.5 0.9 0.8 0.0 1.4 35 1.2 1.3 2.7 0.4 0.4 0.9 0.7 0.0 1.3 36 2.0 2.1 3.1 0.8 0.9 1.4 1.1 0.0 1.6 37 7.0 4.9 6.4 4.7 3.3 5.1 1.8 0.2 2.3 38 2.7 2.4 3.5 2.0 1.7 2.9 1.9 2.1 77.8 39 1.6 1.4 1.5 2.4 2.0 2.1 1.4 0.0 7.0 40 1.6 1.7 2.1 0.6 0.6 0.9 1.1 9.6 147.8 41 2.1 1.9 2.8 0.9 0.9 1.6 0.4 1.5 25.8 42 2.0 1.7 2.0 1.4 1.3 2.0 1.0 0.2 20.2 43 7.3 4.1 6.7 4.8 2.9 6.3 0.7 0.3 29.8 44 2.8 2.1 3.9 2.9 2.2 4.4 1.0 0.0 2.7 45 1.4 1.5 1.4 1.8 1.8 2.0 0.3 0.0 2.6 46 1.6 1.4 1.9 0.4 0.4 0.5 0.9 0.5 8.7 47 1.6 1.5 2.2 0.5 0.5 0.7 0.6 0.1 3.3 48 1.0 1.1 1.6, 0.5 0.6 0.8 0.3 0.1 11.6 49 3.9 2.6 7.8 2.9 2.1 5.1 1.5 0.0 2.5 50 4.0 2.8 4.4 2.8 2.1 3.2 1.8 0.0 2.4 51 1.7 1.5 1.7 2.6 2.2 2.8 1.5 0.0 1.9 52 1.0 1.0 1.9 0.3 0.3 0.5 0.8 0.0 1.3 53 1.4 1.3 2.4 0.5 0.5 0.9 0.8 0.0 1.3 54 1.8 1.7 2.3 0.7 0.7 0.8 1.3 0.0 1.1 % GT 1.0 100% 96% 100% 50% 50% 63% 52% 15% 100% % GT 0.9 100% 100% 100% 56% 52% 67% 63% 17% 100% % GT 0.8 100% 100% 100% 59% 56% 81% 70% 19% 100% % GT 0.7 100% 100% 100% 63% 61% 87% 78% 20% 100% 91 Table 27. REs of VRPs relative to the Subtraction Estimator for varying densities. DENSITY 500 Stems Per Hectare RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 1 10.7 5.2 9.3 4.6 3.0 4.9 1.7 0.8 11.1 2 7.2 4.1 7.2 2.8 2.3 4.7 1.1 1.2 25.6 3 2.7 2.1 2.6 3.3 2.3 3.5 1.5 0.0 17.0 4 2.1 1.9 3.0 0.6 0.6 1.0 0.9 3.6 21.6 5 1.5 1.8 2.1 0.6 0.6 0.8 0.7 2.3 26.5 6 1.8 1.7 2.7 0.7 0.7 1.7 0.5 0.2 11.0 7 4.8 3.9 7.0 1.7 1.7 2.9 2.2 0.3 10.6 8 4.9 3.7 4.6 3.4 3.0 3.5 3.6 0.0 2.8 9 1.9 1.7 1.9 2.5 2.0 2.5 2.1 0.0 1.2 10 1.4 1.3 2.4 0.4 0.4 0.6 1.1 0.7 5.4 11 1.5 1.4 2.5 0.4 0.4 0.8 1.3 0.5 25.1 12 1.1 1.0 1.9 0.3 0.2 0.4 0.7 0.1 8.4 . 13 2.3 1.9 4.9 1.6 1.4 3.2 1.8 0.0 3.1 14 3.5 2.7 4.7 2.1 1.8 3.1 0.8 0.0 1.7 15 2.7 2.4 3.3 3.3 2.9 4.4 1.8 0.0 2.3 16 1.3 1.4 2.4 0.4 0.5 0.8 1.0 0.0 1.5 17 2.4 2.0 2.9 0.9 0.8 1.1 1.0 0.0 1.3 18 2.0 1.8 2.3 0.7 0.6 0.8 1.1 0.0 1.3 % GT 1.0 100% 94% 100% 50% 50% 67% 61% 17% 100% M E A N RE 3.1 2.3 3.8 1.7 1.4 2.3 1.4 0.5 9.9 M A X RE 10.7 5.2 9.3 4.6 3.0 4.9 3.6 3.6 26.5 MIN RE 1.1 1.0 1.9 0.3 0.2 0.4 0.5 0.0 1.2 RANGE 9.6 4.2 7.4 4.3 2.8 4.4 3.1 3.6 25.3 1500 Stems Per Hectare RE of Vol. Change RE of B A Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 19 5.9 3.7 5.8 3.2 2.6 4.3 0.3 0.4 4.1 20 2.7 2.1 3.4 1.0 0.8 1.9 0.6 0.2 7.5 21 1.3 1.4 1.2 2.0 2.3 1.8 1.4 0.0 1.4 22 1.4 1.4 2.1 0.5 0.5 0.9 0.4 5.8 47.9 23 1.1 1.1 1.5 0.4 0.4 0.7 5.5 4.9 135.6 24 1.6 1.4 2.0 0.9 0.8 1.4 1.2 0.2 16.1 25 2.2 2.0 3.6 2.0 1.9 2.8 3.7 0.1 3.3 26 3.6 3.5 3.0 2.4 2.7 2.0 2.3 0.0 4.7 27 1.4 1.4 1.6 1.5 1.5 1.8 0.7 0.0 6.2 28 1.5 1.4 2.3 0.5 0.5 0.7 0.7 1.0 11.5 29 1.7 1.6 2.2 0.4 0.4 0.7 1.1 0.2 5.0 30 1.4 1.3 2.2 0.5 0.5 0.9 0.8 0.0 6.8 31 2.9 2.3 7.0 1.5 1.3 4.0 1.0 0.0 2.6 32 3.5 2.7 3.4 2.2 1.8 2.6 2.1 0.0 2.9 33 2.0 1.8 1.9 2.3 1.8 2.5 1.5 0.0 4.4 34 1.7 1.5 2.4 0.6 0.5 0.9 0.8 0.0 1.4 35 1.2 1.3 2.7 0.4 0.4 0.9 0.7 0.0 1.3 36 2.0 2.1 3.1 0.8 0.9 1.4 1.1 0.0 1.6 % GT 1.0 100% 100% 100% 44% 44% 61% 50% 11% 100% M E A N RE 2.2 1.9 2.9 1.3 1.2 1.8 1.4 0.7 14.7 M A X RE 5.9 3.7 7.0 3.2 2.7 4.3 5.5 5.8 135.6 MIN RE 1.1 1.1 1.2 0.4 0.4 0.7 0.3 0.0 1.3 RANGE 4.8 2.6 5.8 2.8 2.3 3.6 5.2 5.8 134.3 92 3000 Stems Per Hectare RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 37 7.0 4.9 6.4 4.7 3.3 5.1 1.8 0.2 2.3 38 2.7 2.4 3.5 2.0 1.7 2.9 1.9 2.1 77.8 39 1.6 1.4 1.5 2.4 2.0 2.1 1.4 0.0 7.0 40 1.6 1.7 2.1 0.6 0.6 0.9 1.1 9.6 147.8 41 2.1 1.9 2.8 0.9 0.9 1.6 0.4 • 1.5 25.8 42 2.0 1.7 2.0 1.4 1.3 2.0 1.0 0.2 20.2 43 7.3 4.1 6.7 4.8 2.9 6.3 0.7 0.3 29.8 44 2.8 2.1 3.9 2.9 2.2 4.4 1.0 0.0 2.7 45 1.4 1.5 1.4 1.8 1.8 2.0 0.3 0.0 2.6 46 1.6 1.4 1.9 0.4 0.4 0.5 0.9 0.5 8.7 47 1.6 1.5 2.2 0.5 0.5 0.7 0.6 0.1 3.3 48 1.0 1.1 1.6 0.5 0.6 0.8 0.3 0.1 11.6 49 3.9 2.6 7.8 2.9 2.1 5.1 1.5 0.0 2.5 50 4.0 2.8 4.4 2.8 2.1 3.2 1.8 0.0 2.4 51 1.7 1.5 1.7 2.6 2.2 2.8 1.5 0.0 1.9 52 1.0 1.0 1.9 0.3 0.3 0.5 0.8 0.0 1.3 53 1.4 1.3 2.4 0.5 0.5 0.9 0.8 0.0 1.3 54 1.8 1.7 2.3 0.7 0.7 0.8 1.3 0.0 1.1 % GT 1.0 100% 94%) 100% 56% 56%) 61% 44%) 17% 100% M E A N RE 2.6 2.0 3.1 1.8 1.5 2.4 1.0 0.8 19.5 M A X RE 7.3 4.9 7.8 4.8 3.3 6.3 1.9 9.6 147.8 MIN RE 1.0 1.0 1.4 0.3 0.3 0.5 0.3 0.0 1.1 RANGE 6.3 3.9 6.4 4.5 3.0 5.8 1.6 9.6 146.7 Table 28. REs for VRPs relative to the Subtraction Estimator for varying diameter distributions. DIAMETER DISTRIBUTIONS Understory RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 1 10.7 5.2 9.3 4.6 3.0 4.9 1.7 0.8 11.1 2 7.2 4.1 7.2 2.8 2.3 4.7 1.1 1.2 25.6 3 2.7 2.1 2.6 3.3 2.3 3.5 1.5 0.0 17.0 4 2.1 1.9 3.0 0.6 0.6 1.0 0.9 3.6 21.6 5 1.5 1.8 2.1 0.6 0.6 0.8 0.7 2.3 26.5 6 1.8 1.7 2.7 0.7 0.7 1.7 0.5 0.2 11.0 19 5.9 3.7 5.8 3.2 2.6 4.3 0.3 0.4 4.1 20 2.7 2.1 3.4 1.0 0.8 1.9 0.6 0.2 7.5 21 1.3 1.4 1.2 2.0 2.3 1.8 1.4 0.0 1.4 22 1.4 1.4 2.1 0.5 0.5 0.9 0.4 5.8 47.9 23 1.1 1.1 1.5 0.4 0.4 0.7 5.5 4.9 135.6 24 1.6 1.4 2.0 0.9 0.8 1.4 1.2 0.2 16.1 37 7.0 4.9 6.4 4.7 3.3 5.1 1.8 0.2 2.3 38 2.7 2.4 3.5 2.0 1.7 2.9 1.9 2.1 77.8 39 1.6 1.4 1.5 2.4 2.0 2.1 1.4 0.0 7.0 40 1.6 1.7 2.1 0.6 0.6 0.9 1.1 9.6 147.8 41 2.1 1.9 2.8 0.9 0.9 1.6 0.4 1.5 25.8 42 2.0 1.7 2.0 1.4 1.3 2.0 1.0 0.2 20.2 % GT 1.0 100% 100% 100% 50% 50%) 78%) 56% 44% 100%) M E A N RE 3.2 2.3 3.4 1.8 1.5 2.3 1.3 1.8 33.7 M A X RE 10.7 5.2 9.3 4.7 3.3 5.1 5.5 9.6 147.8 MIN RE 1.1 1.1 1.2 0.4 0.4 0.7 0.3 0.0 1.4 RANGE 9.6 4.1 8.1 4.3 2.9 4.5 5.2 9.6 146.4 93 Mixed RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 7 4.8 3.9 7.0 1.7 1.7 2.9 2.2 0.3 10.6 8 4.9 3.7 4.6 3.4 3.0 3.5 3.6 0.0 2.8 9 1.9 1.7 1.9 2.5 2.0 2.5 2.1 0.0 1.2 10 1.4 1.3 2.4 0.4 0.4 0.6 1.1 0.7 5.4 11 1.5 1.4 2.5 0.4 0.4 0.8 1.3 0.5 25.1 12 1.1 1.0 1.9 0.3 0.2 0.4 0.7 0.1 8.4 25 2.2 2.0 3.6 2.0 1.9 2.8 3.7 0.1 3.3 26 3.6 3.5 3.0 2.4 2.7 2.0 2.3 0.0 4.7 27 1.4 1.4 1.6 1.5 1.5 1.8 0.7 0.0 6.2 28 1.5 1.4 2.3 0.5 0.5 0.7 0.7 1.0 11.5 29 1.7 1.6 2.2 0.4 0.4 0.7 1.1 0.2 5.0 30 1.4 1.3 2.2 0.5 0.5 0.9 0.8 0.0 6.8 43 7.3 4.1 6.7 4.8 2.9 6.3 0.7 0.3 29.8 44 2.8 2.1 3.9 2.9 2.2 4.4 1.0 0.0 2.7 45 1.4 1.5 1.4 1.8 1.8 2.0 0.3 0.0 2.6 46 1.6 1.4 1.9 0.4 0.4 0.5 0.9 0.5 8.7 47 1.6 1.5 2.2 0.5 0.5 0.7 0.6 0.1 3.3 48 1.0 1.1 1.6 0.5 0.6 0.8 0.3 0.1 11.6 % GT 1.0 100% 94% 100% 50% 50% 50% 44% 0% 100% M E A N RE 2.4 2.0 2.9 1.5 1.3 1.9 1.3 0.2 8.3 M A X RE 7.3 4.1 7.0 4.8 3.0 6.3 3.7 1.0 29.8 MIN RE 1.0 1.0 1.4 0.3 0.2 0.4 0.3 0.0 1.2 RANGE 6.2 3.1 5.6 4.5 2.7 5.9 3.4 1.0 28.6 RE of Vol. Change No Understory Scenario DV F L GR DV F L GR DV F L GR 13 2.3 1.9 4.9 1.6 1.4 3.2 1.8 0.0 3.1 14 3.5 2.7 4.7 2.1 1.8 3.1 0.8 0.0 1.7 15 2.7 2.4 3.3 3.3 2.9 4.4 1.8 0.0 2.3 16 1.3 1.4 2.4 0.4 0.5 0.8 1.0 0.0 1.5 17 2.4 2.0 2.9 0.9 0.8 1.1 1.0 0.0 1.3 18 2.0 1.8 2.3 0.7 0.6 0.8 1.1 0.0 1.3 31 2.9 2.3 7.0 1.5 1.3 4.0 1.0 0.0 2.6 32 3.5 2.7 3.4 2.2 1.8 2.6 2.1 0.0 2.9 33 2.0 1.8 1.9 2.3 1.8 2.5 . 1.5 0.0 4.4 34 1.7 1.5 2.4 0.6 0.5 0.9 0.8 0.0 1.4 35 1.2 1.3 2.7 0.4 0.4 0.9 0.7 0.0 1.3 36 2.0 2.1 3.1 0.8 0.9 1.4 1.1 0.0 1.6 49 3.9 2.6 7.8 2.9 2.1 5.1 1.5 0.0 2.5 50 4.0 2.8 4.4 2.8 2.1 3.2 1.8 0.0 2.4 51 1.7 1.5 1.7 2.6 2.2 2.8 1.5 0.0 1.9 52 1.0 1.0 1.9 0.3 0.3 0.5 0.8 0.0 1.3 53 1.4 1.3 2.4 0.5 0.5 0.9 0.8 0.0 1.3 54 1.8 1.7 2.3 0.7 0.7 0.8 1.3 0.0 1.1 % GT 1.0 100% 94% 100% 50% 50% 61% 56% 0% 100% M E A N RE 2.3 1.9 3.4 1.5 1.3 2.2 1.2 0.0 2.0 M A X RE 4.0 2.8 7.8 3.3 2.9 5.1 2.1 0.0 4.4 MIN RE 1.0 1.0 1.7 0.3 0.3 0.5 0.7 0.0 1.1 RANGE 3.0 1.8 6.1 3.0 2.5 4.6 1.4 0.0 3.2 94 Table 29. REs for VRPs relative to the Subtraction Estimator for normal and high mortality. Normal Mortality RE of Vol. Change R E of BA Change Scenario DV F L GR DV F L GR DV F L GR 1 10.7 5.2 9.3 4.6 3.0 4.9 1.7 0.8 11.1 2 7.2 4.1 7.2 2.8 2.3 4.7 1.1 1.2 25.6 3 2.7 2.1 2.6 3.3 2.3 3.5 1.5 0.0 17.0 7 4.8 3.9 7.0 1.7 1.7 2.9 2.2 0.3 10.6 8 4.9 3.7 4.6 3.4 3.0 3.5 3.6 0.0 2.8 9 1.9 1.7 1.9 2.5 2.0 2.5 2.1 0.0 1.2 13 2.3 1.9 4.9 1.6 1.4 3.2 1.8 0.0 3.1 14 3.5 2.7 4.7 2.1 1.8 3.1 0.8 0.0 1.7 15 2.7 2.4 3.3 3.3 2.9 4.4 1.8 0.0 2.3 19 5.9 3.7 5.8 3.2 2.6 4.3 0.3 0.4 4.1 20 2.7 2.1 3.4 1.0 0.8 1.9 0.6 0.2 7.5 21 1.3 1.4 1.2 2.0 2.3 1.8 1.4 0.0 1.4 25 2.2 2.0 3.6 2.0 1.9 2.8 3.7 0.1 3.3 26 3.6 3.5 3.0 2.4 2.7 2.0 2.3 0.0 4.7 27 1.4 1.4 1.6 1.5 1.5 1.8 0.7 0.0 6.2 31 2.9 2.3 7.0 1.5 1.3 4.0 1.0 0.0 2.6 32 3.5 2.7 3.4 2.2 1.8 2.6 2.1 0.0 2.9 33 2.0 1.8 1.9 2.3 1.8 2.5 1.5 0.0 4.4 37 7.0 4.9 6.4 4.7 3.3 5.1 1.8 0.2 2.3 38 2.7 2.4 3.5 2.0 1.7 2.9 1.9 2.1 77.8 39 1.6 1.4 1.5 2.4 2.0 2.1 1.4 0.0 7.0 43 7.3 4.1 6.7 4.8 2.9 6.3 0.7 0.3 29.8 44 2.8 2.1 3.9 2.9 2.2 4.4 1.0 0.0 2.7 45 1.4 1.5 1.4 1.8 1.8 2.0 0.3 0.0 2.6 49 3.9 2.6 7.8 2.9 2.1 5.1 1.5 0.0 2.5 50 4.0 2.8 4.4 2.8 2.1 3.2 1.8 0.0 2.4 51 1.7 1.5 1.7 2.6 2.2 2.8 1.5 0.0 1.9 % G T 1.0 100% 100% 100% 96% 96% 100% 70% 7% 100% M E A N R E 3.6 2.7 4.2 2.6 2.1 3.3 1.5 0.2 8.9 M A X RE 10.7 5.2 9.3 4.8 3.3 6.3 3.7 2.1 77.8 MIN RE 1.3 1.4 1.2 1.0 0.8 1.8 0.3 0.0 1.2 RANGE 9.4 3.8 8.1 3.8 2.5 4.6 3.4 2.1 76.5 R E of Vol. Change High Mortality RE of BA Change R E of SPH Change Scenario DV F L GR DV F L GR DV F L GR 4 2.1 1.9 3.0 0.6 0.6 1.0 0.9 3.6 21.6 5 1.5 1.8 2.1 0.6 0.6 0.8 0.7 2.3 26.5 6 1.8 1.7 2.7 0.7 0.7 1.7 0.5 0.2 11.0 10 1.4 1.3 2.4 0.4 0.4 0.6 1.1 0.7 5.4 11 1.5 1.4 2.5 0.4 0.4 0.8 1.3 0.5 25.1 12 1.1 1.0 1.9 0.3 0.2 0.4 0.7 0.1 8.4 16 1.3 1.4 2.4 0.4 0.5 0.8 1.0 0.0 1.5 17 2.4 2.0 2.9 0.9 0.8 1.1 1.0 0.0 1.3 18 2.0 1.8 2.3 0.7 0.6 0.8 1.1 0.0 1.3 22 1.4 1.4 2.1 0.5 0.5 0.9 0.4 5.8 47.9 23 1.1 1.1 1.5 0.4 0.4 0.7 5.5 4.9 135.6 24 1.6 1.4 2.0 0.9 0.8 1.4 1.2 0.2 16.1 28 1.5 1.4 2.3 0.5 0.5 0.7 0.7 1.0 11.5 29 1.7 1.6 2.2 0.4 0.4 0.7 1.1 0.2 5.0 30 1.4 1.3 2.2 0.5 0.5 0.9 0.8 0.0 6.8 34 1.7 1.5 2.4 0.6 0.5 0.9 0.8 0.0 1.4 35 1.2 1.3 2.7 0.4 0.4 0.9 0.7 0.0 1.3 36 2.0 2.1 3.1 0.8 0.9 1.4 1.1 0.0 1.6 40 1.6 1.7 2.1 0.6 0.6 0.9 1.1 9.6 147.8 41 2.1 1.9 2.8 0.9 0.9 1.6 0.4 1.5 25.8 42 2.0 1.7 2.0 1.4 1.3 2.0 1.0 0.2 20.2 46 1.6 1.4 1.9 0.4 0.4 0.5 0.9 0.5 8.7 47 1.6 1.5 2.2 0.5 0.5 0.7 0.6 0.1 3.3 48 1.0 1.1 1.6 0.5 0.6 0.8 0.3 0.1 11.6 52 1.0 1.0 1.9 0.3 0.3 0.5 0.8 0.0 1.3 53 1.4 1.3 2.4 0.5 0.5 0.9 0.8 0.0 1.3 54 1.8 1.7 2.3 0.7 0.7 0.8 1.3 0.0 1.1 % G T 1.0 100% 93% 100% 4% 4% 26% 33% 22% 100% M E A N RE 1.6 1.5 2.3 0.6 0.6 0.9 1.0 1.2 20.4 M A X RE 2.4 2.1 3.1 1.4 1.3 2.0 5.5 9.6 147.8 MIN RE 1.0 1.0 1.5 0.3 0.2 0.4 0.3 0.0 1.1 RANGE 1.4 1.2 1.6 1.2 1.1 1.5 5.2 9.6 146.7 95 Table 30. REs of VRPs relative to the Subtraction Estimator for varying spatial distributions. SPATIAL DISTRIBUTIONS Random RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 1 10.7 5.2 9.3 4.6 3.0 4.9 1.7 0.8 11.1 4 2.1 1.9 3.0 0.6 0.6 1.0 0.9 3.6 21.6 7 4.8 3.9 7.0 1.7 1.7 2.9 2.2 0.3 10.6 10 1.4 1.3 2.4 0.4 0.4 0.6 1.1 0.7 5.4 13 2.3 1.9 4.9 1.6 1.4 3.2 1.8 0.0 3.1 16 1.3 1.4 2.4 0.4 0.5 0.8 1.0 0.0 1.5 19 5.9 3.7 5.8 3.2 2.6 4.3 0.3 0.4 4.1 22 1.4 1.4 2.1 0.5 0.5 0.9 0.4 5.8 47.9 25 2.2 2.0 3.6 2.0 1.9 2.8 3.7 0.1 3.3 28 1.5 1.4 2.3 0.5 0.5 .0.7 0.7 1.0 11.5 31 2.9 2.3 7.0 1.5 1.3 4.0 1.0 0.0 2.6 34 1.7 1.5 2.4 0.6 0.5 0.9 0.8 0.0 1.4 37 7.0 4.9 6.4 4.7 3.3 5.1 1.8 0.2 2.3 40 1.6 1.7 2.1 0.6 0.6 0.9 1.1 9.6 147.8 43 7.3 4.1 6.7 4.8 2.9 6.3 0.7 0.3 29.8 46 1.6 1.4 1.9 0.4 0.4 0.5 0.9 0.5 8.7 49 3.9 2.6 7.8 2.9 2.1 5.1 1.5 0.0 2.5 52 1.0 1.0 1.9 0.3 0.3 0.5 0.8 0.0 1.3 % GT 1.0 100% 94% 100% 50%) 50% 56% 44% 17% 100% M E A N RE 3.4 2.4 4.4 1.7 1.4 2.5 1.2 1.3 17.6 M A X RE 10.7 5.2 9.3 4.8 3.3 6.3 3.7 9.6 147.8 MIN RE 1.0 1.0 1.9 0.3 0.3 0.5 0.3 0.0 1.3 RANGE 9.7 4.2 7.4 4.5 3.0 5.8 3.4 9.6 146.5 Clumped RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 2 7.2 4.1 7.2 2.8 2.3 4.7 1.1 1.2 25.6 5 1.5 1.8 2.1 0.6 0.6 0.8 0.7 2.3 26.5 8 4.9 3.7 4.6 3.4 3.0 3.5 3.6 0.0 2.8 11 1.5 1.4 2.5 0.4 0.4 0.8 1.3 0.5 25.1 14 3.5 2.7 4.7 2.1 1.8 3.1 0.8 0.0 1.7 17 2.4 2.0 2.9 0.9 0.8 1.1 1.0 0.0 1.3 20 2.7 2.1 3.4 1.0 0.8 1.9 0.6 0.2 7.5 23 1.1 1.1 1.5 0.4 0.4 0.7 5.5 4.9 135.6 26 3.6 3.5 3.0 2.4 2.7 2.0 2.3 0.0 4.7 29 1.7 1.6 2.2 0.4 0.4 0.7 1.1 0.2 5.0 32 3.5 2.7 3.4 2.2 1.8 2.6 2.1 0.0 2.9 35 1.2 1.3 2.7 0.4 0.4 0.9 0.7 0.0 1.3 38 2.7 2.4 3.5 2.0 1.7 2.9 1.9 2.1 77.8 41 2.1 1.9 2.8 0.9 0.9 1.6 0.4 1.5 25.8 44 2.8 2.1 3.9 2.9 2.2 4.4 1.0 0.0 2.7 47 1.6 1.5 2.2 0.5 0.5 0.7 0.6 0.1 3.3 50 4.0 2.8 4.4 2.8 2.1 3.2 1.8 0.0 2.4 53 1.4 1.3 2.4 0.5 0.5 0.9 0.8 0.0 1.3 % GT 1.0 100%; 100% 100% 44% 44% 61% 50% 28% 100% M E A N RE 2.7 2.2 3.3 1.5 1.3 2.0 1.5 0.7 19.6 M A X RE 7.2 4.1 7.2 3.4 3.0 4.7 5.5 4.9 135.6 MIN RE 1.1 1.1 1.5 0.4 0.4 0.7 0.4 0.0 1.3 RANGE 6.1 3.0 5.7 3.1 2.5 4.0 5.1 4.9 134.3 9 6 Very Clumped RE of Vol. Change RE of BA Change RE of SPH Change Scenario DV F L GR DV F L GR DV F L GR 3 2.7 2.1 2.6 3.3 2.3 3.5 1.5 0.0 17.0 6 1.8 1.7 2.7 0.7 0.7 1.7 0.5 0.2 11.0 9 1.9 1.7 1.9 2.5 2.0 2.5 2.1 0.0 1.2 12 1.1 1.0 1.9 0.3 0.2 0.4 0.7 0.1 8.4 15 2.7 2.4 3.3 3.3 2.9 4.4 1.8 0.0 2.3 18 2.0 1.8 2.3 0.7 0.6 0.8 1.1 0.0 1.3 21 1.3 1.4 1.2 2.0 2.3 1.8 1.4 0.0 1.4 24 1.6 1.4 2.0 0.9 0.8 1.4 1.2 0.2 16.1 27 1.4 1.4 1.6 1.5 1.5 1.8 0.7 0.0 6.2 30 1.4 1.3 2.2 0.5 0.5 0.9 0.8 0.0 6.8 33 2.0 1.8 1.9 2.3 1.8 2.5 1.5 0.0 4.4 36 2.0 2.1 3.1 0.8 0.9 1.4 1.1 0.0 1.6 39 1.6 1.4 1.5 2.4 2.0 2.1 1.4 0.0 7.0 42 2.0 1.7 2.0 1.4 1.3 2.0 1.0 0.2 20.2 45 1.4 1.5 1.4 1.8 1.8 2.0 0.3 0.0 2.6 48 1.0 1.1 1.6 0.5 0.6 0.8 0.3 0.1 11.6 51 1.7 1.5 1.7 2.6 2.2 2.8 1.5 0.0 1.9 54 1.8 1.7 2.3 0.7 0.7 0.8 1.3 0.0 1.1 % GT 1.0 100% 94% 100 % 56% 56%. 72% 61%) 0%) 100%) M E A N RE 1.8 1.6 2.1 1.6 1.4 1.9 1.1 0.0 6.8 M A X RE 2.7 2.4 3.3 3.3 2.9 4.4 2.1 0.2 20.2 MIN RE 1.0 1.0 1.2 0.3 0.2 0.4 0.3 0.0 1.1 RANGE 1.7 1.4 2.1 3.1 2.6 4.0 1.9 0.2 19.1 97 APPENDIX III ERROR VERSUS SAMPLE COMPARISONS 98 FRP BAF Used Used FRP Vol. Error Variance FRP Vol. Sample Variance FRP BA Error Variance FRP BA Sample Variance DV Vol. Error Variance DV Vol. Sample Variance DV BA Error Variance DV BA Sample Variance Flew Vol. Error Variance Flew Vol. Sample Variance Flew Ba Error Variance Flew BA Sample Variance Gros Vol. Error Variance Gros Vol. Gros BA Sample Error Variance Variance Gros BA Sub Vol. Sub Vol. Sub BA Sub BA Sample Error Sample Error Sample Variance Variance Variance Variance Variance 1 3 1 2 3 1 3 3 1 4 3 1 5 3 1 6 3 1 7 2 3 8 2 3 9 2 4 10 2 4 11 2 3 12 2 4 13 2 2 14 2 2 15 2 2 16 2 2 17 2 2 18 2 2 19 2 2 20 2 2 21 2 2 22 2 2 23 2 2 24 2 2 25 2 3 26 2 3 27 3 1 28 3 1 29 3 1 30 2 3 31 2 3 32 2 3 33 2 3 34 2 3 35 2 3 36 2 3 37 1 3 38 1 3 39 1 3 40 1 3 41 1 3 42 1 3 43 2 3 44 2 2 45 2 3 46 2 2 47 2 3 48 2 3 49 2 4 50 2 4 51 3 1 52 3 1 53 2 4 54 2 4 H 5' 3 o CD ct o < o c* 3 ft> 3 a. to 3 CD CD Co o 3 65 46.68 77.01 351.27 482.22 384.62 423.54 228.30 381.11 1361.80 1264.37 1719.26 2538.30 892.65 756.36 2030.06 2284.05 2350.72 2244.54 955.83 1263.12 4045.74 3678.27 7264.04 4098.77 11242.22 8231.35 13610.80 18172.74 12673.89 25983.12 5391.17 10615.78 29600.82 18559.70 22501.64 15968.60 2675.06 6552.41 32268.54 10568.15 17582.99 16572.53 8987.97 26669.01 65983.53 46348.69 63517.04 60431.35 25588.04 25638.32 133048.93 80338.61 93546.26 78412.00 47.68 0.23 0.24 27.44 28.72 0.46 0.47 56.93 59.04 0.70 0.71 31.59 33.17 0.43 0.44 303.49 77.86 0.35 0.35 54.46 54.67 0.82 0.81 96.44 96.22 1.00 0.99 53.07 53.98 0.49 0.49 391.50 355.02 1.23 1.24 240.51 244.84 1.05 1.07 305.88 309.09 1.53 1.55 251.00 254.93 1.00 1.02 658.84 495.27 2.65 2.70 279.95 286.49 3.47 3.47 319.85 323.11 3.71 3.71 198.57 203.71 2.07 2.10 606.65 376.00 2.72 2.72 310.12 308.53 3.80 3.83 265.19 262.84 3.41 3.45 227.54 224.10 2.59 2.62 459.06 426.58 2.17 2.20 401.49 404.43 5.22 5.20 431.42 435.80 5.01 5.01 276.20 273.95 2.24 2.23 744.34 233.25 1.28 1.29 243.73 250.58 3.19 3.22 301.36 309.98 3.13 3.15 163.60 171.03 1.79 1.83 1180.23 386.55 1.71 1.75 214.06 215.26 1.54 1.56 278.73 280.62 1.80 1.82 225.53 228.20 1.52 1.55 1044.70 1372.42 3.60 3.66 1018.59 1029.85 3.45 3.53 1145.78 1154.96 4.26 4.33 1039.75 1049.82 3.49 3.55 1934.51 1293.63 8.07 8.33 1980.44 1998.47 17.64 17.78 2082.85 2098.77 18.55 18.73 1130.63 1149.62 10.61 10.79 2731.11 1746.95 8.41 8.57 1525.93 1541.76 11.54 11.67 1602.92 1613.73 11.68 11.76 899.63 912.46 6.04 6.15 2291.89 2566.69 13.06 13.23 2617.40 2633.90 25.96 25.75 3015.19 3022.66 28.20 27.87 1515.39 1528.82 15.37 15.49 2979.66 899.53 4.92 4.98 1910.83 1912.32 12.88 12.91 2244.72 2268.74 14.84 15.02 882.90 888.25 6.26 6.33 4321.03 751.92 3.34 3.33 1012.33 1017.35 6.90 6.95 1290.78 1282.37 7.84 7.80 756.93 756.94 4.56 4.58 3536.61 1953.08 5.86 5.64 2245.68 2208.95 8.51 8.43 2542.40 2547.15 9.59 9.68 1850.76 1808.15 6.40 6.29 5990.50 2225.11 13.84 13.64 4277.73 4266.56 30.89 30.98 4167.32 4162.31 29.94 30.09 2347.46 2358.99 17.90 18.11 5713.58 2367.15 15.96 16.04 3152.58 3168.21 28.85 28.99 3832.77 3789.76 31.74 31.58 2624.02 2632.51 22.65 22.73 7979.69 2247.32 13.96 14.06 2732.56 2779.56 25.53 25.82 3153.62 3196.58 27.93 28.19 2444.30 2479.90 22.35 22.66 5571.90 958.04 5.06 5.12 545.49 538.28 6.32 6.33 840.91 851.46 7.64 7.73 555.10 550.83 4.73 4.71 3135.54 1261.42 5.56 5.53 1101.15 1096.59 17.84 17.86 1438.12 1433.14 22.50 22.33 894.53 883.64 9.20 9.18 3020.47 4091.64 12.82 12.94 3089.44 3116.72 11.92 11.98 3025.67 3032.78 10.57 10.54 3457.94 3488.08 13.65 13.71 4180.28 3704.34 19.70 19.75 3315.87 3351.15 37.78 37.44 3234.51 3245.19 39.49 38.92 2167.14 2178.96 21.60 21.60 4623.08 7193.22 34.65 34.15 4874.14 4847.37 45.54 45.30 4937.00 4886.85 42.74 42.36 3507.08 3479.33 26.60 26.44 5348.43 4100.21 17.72 17.92 2990.51 2983.67 26.55 26.84 3598.11 3568.24 30.03 30.16 2469.80 2481.71 16.73 17.01 4896.88 11318.09 45.45 45.76 5963.07 6062.80 28.19 28.51 6540.58 6622.69 30.11 30.46 3523.66 3603.72 20.10 20.35 13019.64 8288.34 38.17 38.29 4846.55 4898.86 30.49 30.69 5019.85 5092.73 26.95 27.20 5759.93 5837.61 36.19 36.55 17544.38 13531.96 40.34 40.29 11861.03 11807.17 41.99 41.90 12421.48 12317.57 42.55 42.24 10468.59 10455.03 34.38 34.38 17141.84 18208.67 93.28 93.47 19245.38 19359.67 139.52 141.08 20705.43 20846.95 143.55 145.51 12014.55 12109.57 94.77 95.43 28432.87 12825.21 75.28 75.90 14441.73 14707.13 148.80 150.49 14708.81 14968.41 147.31 148.96 10926.69 11110.66 90.47 90.96 24323.10 26240.01 145.09 146.46 25415.41 25821.21 279.02 282.12 27664.58 28022.74 281.73 284.49 16729.22 16790.49 153.72 154.64 36797.65 5444.44 31.21 31.67 12207.97 12309.04 97.51 98.24 15328.12 15423.53 108.26 109.08 5014.26 5067.10 36.61 37.08 35130.04 10717.76 46.18 46.69 9841.45 9872.80 59.23 59.14 12407.56 12520.31 71.21 71.62 9958.37 10031.66 49.54 49.77 34290.38 29664.73 75.65 75.67 28656.59 28798.58 91.67 92.14 32268.22 32459.95 115.09 115.77 29802.81 29932.64 85.03 85.34 56682.11 18557.82 131.43 131.86 31112.74 30866.07 259.26 258.07 36026.30 35887.77 282.99 282.56 22340.65 22403.35 170.34 170.74 52670.67 22268.93 155.71 154.55 47722.90 47447.15 410.66 408.80 46659.14 46590.91 384.39 384.25 21878.13 21998.62 186.66 188.03 57948.90 16129.47 95.93 97.47 23107.11 23326.51 209.01 210.24 21521.33 21908.97 192.03 193.98 15036.81 15242.01 121.67 122.79 46062.67 2661.51 14.68 14.70 1951.57 1967.77 19.53 19.66 2808.77 2817.81 28.27 28.32 2147.06 2141.61 17.97 18.05 13690.00 6543.96 27.02 27.09 5380.63 5386.44 32.89 32.97 6027.17 6017.08 38.89 38.91 4181.90 4183.57 23.21 23.01 14426.87 31687.82 105.98 103.81 25657.30 25270.05 90.37 89.44 28275.58 27835.67 110.00 108.47 27727.47 27262.92 103.99 101.60 40427.99 10350.69 62.98 61.31 12541.10 12438.79 130.93 131.40 11599.09 11559.55 119.88 120.14 9402.65 9437.96 84.71 84.78 19661.80 17517.97 99.99 100.05 10913.49 10925.08 126.06 126.28 12026.96 12092.49 121.41 122.02 8332.34 8351.15 72.48 72.36 22817.62 16674.98 73.57 73.80 14013.09 14102.23 118.81 118.81 16757.51 16989.00 130.02 131.11 13968.64 14215.00 85.87 87.16 28177.70 8904.42 48.67 48.04 6955.29 6695.76 58.93 57.01 12432.15 11921.74 95.92 92.46 7487.93 7299.38 45.81 43.08 49622.16 26717.55 96.36 96.71 17991.50 18178.13 81.15 82.36 23483.12 23779.58 107.38 108.20 12812.00 12898.93 54.37 54.78 50114.17 66621.49 167.64 169.29 53123.76 53670.91 186.01 187.47 50960.78 51409.21 179.49 180.13 54965.18 55500.12 165.90 167.66 76050.52 46567.42 282.83 281.82 44183.10 44295.31 373.05 372.14 49554.32 49823.51 397.72 397.09 37839.92 38330.43 319.40 320.54 70535.84 64709.93 333.81 342.23 53255.68 54305.51 471.63 479.52 57626.07 58595.78 469.84 476.34 38814.19 39971.75 328.94 337.56 86512.84 60789.89 361.84 369.33 76155.09 77175.73 584.45 598.32 72373.54 73454.59 546.02 560.03 50297.73 50537.83 393.38 401.64 78600.42 25620.77 135.51 135.98 34569.95 34670.63 200.36 201.02 51545.88 51801.28 273.76 274.48 17503.96 17361.41 116.01 115.36 135300.94 25542.04 129.37 130.27 29626.01 29798.60 170.21 171.65 41766.21 41983.58 236.17 236.98 26666.72 26823.41 152.66 154.16 117564.55 131467.19 338.60 334.32 127172.93 126102.11 358.22 355.35 143169.82 141266.49 416.21 410.88 123717.15 122283.13 332.17 327.88 211768.99 79205.54 507.25 498.62 169776.80 167394.99 1216.24 1205.01 173312.58 170625.00 1228.71 1217.01 90876.45 90620.77 707.57 705.90 169287.53 93703.49 550.48 548.00 129973.68 130924.63 996.35 1001.80 140042.12 140939.06 1030.55 1037.04 79184.86 79377.26 606.44 606.70 188368.76 77370.27 698.47 686.74 140751.59 138905.74 1356.07 1333.53 147132.76 145022.51 1377.34 1346.92 108175.18 106295.78 1145.03 1117.61 247628.68 307.50 391.32 659.33 604.65 460.89 744.83 1201.24 1044.80 1944.35 2741.49 2299.21 2965.29 4368.67 3520.63 6039.63 5739.57 7709.89 5639.23 3171.77 3013.02 4154.58 4602.87 5286.65 4849.99 13113.84 17686.37 17077.61 28441.74 24662.78 36624.62 35450.60 34428.93 57140.25 52728.98 58583.60 46853.13 13702.84 14454.15 40044.05 19854.34 22969.87 28677.40 48756.74 50239.73 76465.87 71452.26 88121.98 79712.25 135708.88 118036.93 209511.10 168874.66 188532.53 245898.45 2.16 2.15 2.28 2.30 3.64 3.57 3.52 2.17 3.45 2.16 4.94 3.72 5.32 5.37 5.41 5.38 8.86 8.75 13.93 6.81 9.61 4.95 11.13 6.62 21.42 20.51 14.16 14.25 28.36 27.62 23.02 13.70 43.34 24.82 26.30 17.97 20.26 20.07 17.18 17.25 24.13 24.04 31.64 18.65 31.26 17.63 33.31 24.50 57.66 57.43 75.32 74.40 62.05 61.34 148.40 66.15 139.74 65.33 220.35 133.44 148.80 147.20 131.55 131.53 210.70 210.13 244.95 149.19 255.67 168.68 241.85 173.53 95.34 92.58 67.62 66.85 225.06 218.31 123.21 76.67 177.46 114.58 229.15 172.21 291.50 272.30 264.59 238.75 328.70 329.04 402.39 153.03 443.70 234.04 584.67 324.09 605.77 586.71 505.99 486.44 949.83 913.21 692.08 368.76 833.54 548.91 1274.42 940.96 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 51 52 53 54 FRP Used 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 2 2 2 2 2 2 2 BAF Used DV Vol. Error RE i~70 1.41 1.46 1.72 1.24 1.05 0.94 1.78 1.34 0.64 1.13 0.97 0.47 0.75 0.90 0.53 0.75 0.82 1.75 1.15 1.31 1.11 1.49 1.37 1.89 1.70 1.15 0.94 0.88 1.02 0.44 1.08 1.03 0.60 0.47 0.69 1.37 1.22 1.26 0.84 1.61 1.18 1.29 1.48 1.24 1.05 1.19 0.79 0.74 0.87 1.05 0.47 072 0.56 • V Vol. Sample RE T~66 1.42 1.45 1.73 1.22 1.05 0.93 1.80 1.33 0.65 1.13 0.97 0.47 0.74 0.88 0.52 0.75 0.81 1.78 1.15 1.31 1.11 1.48 1.37 1.87 1.69 1.15 0.94 0.87 1.02 0.44 1.09 1.03 0.60 0.47 0.69 1.35 1.21 1.25 0.83 1.60 1.18 1.33 1.47 1.24 1.05 1.19 0.79 0.74 0.86 1.04 0.47 0.72 0.56 Flew Vol. Error RE 5~82 0.80 1.15 1.51 1.45 0.98 0.76 1.37 1.19 0.61 1.07 0.84 0.40 0.59 0.80 0.55 0.61 0.71 1.14 0.88 1.34 1.14 1.47 1.14 1.72 1.64 1.10 0.88 0.86 0.94 0.35 0.86 0.92 0.52 0.48 0.74 0.95 1.09 1.14 0.91 1.46 0.99 0.72 1.14 1.29 0.94 1.10 0.83 0.50 0.61 0.93 0.46 0.67 0.53 Flew Vol. Sample RE a l i 0.81 1.15 1.53 1.43 0.98 0.75 1.38 1.19 0.62 1.08 0.85 0.40 0.59 0.77 0.53 0.62 0.70 1.13 0.88 1.35 1.14 1.47 1.15 1.71 1.63 1.10 0.87 0.86 0.94 0.35 0.86 0.91 0.52 0.48 0.74 0.94 1.09 1.14 0.90 1.45 0.98 0.75 1.12 1.30 0.93 1.10 0.83 0.49 0.61 0.93 0.46 0.66 0.53 Gros Vol. ErTor RE T~48 1.45 1.40 2.43 1.69 1.53 1.40 1.69 1.31 1.12 1.91 1.68 1.01 1.00 1.10 0.97 0.90 0.92 1.72 1.41 1.17 1.70 2.07 1.66 3.19 1.43 1.30 1.51 1.16 1.55 1.08 1.07 0.99 0.83 1.03 1.06 1.25 1.57 1.16 1.12 2.11 1.19 1.20 2.08 1.20 1.22 1.64 1.20 1.46 0.96 1.08 0.88 1.18 0.72 Gros Vol. Sample RE i"44 1.44 1.39 2.43 1.68 1.56 1.36 1.69 1.31 1.13 1.91 1.68 1.01 0.99 1.08 0.94 0.90 0.91 1.74 1.43 1.17 1.70 2.07 1.65 3.14 1.42 1.29 1.50 1.15 1.56 1.07 1.07 0.99 0.83 1.01 1.06 1.24 1.56 1.16 1.10 2.10 1.17 1.22 2.07 1.20 1.21 1.62 1.20 1.48 0.95 1.08 0.87 1.18 0.73 Sub Vol. Error RE 015 0.20 0.53 0.79 0.84 0.57 0.19 0.36 0.70 0.46 0.75 0.85 0.21 0.21 0.34 0.40 0.29 0.40 0.30 0.42 0.97 0.80 1.36 0.84 0.86 0.47 0.79 0.64 0.52 0.71 0.15 0.31 0.52 0.35 0.39 0.35 0.20 0.45 0.80 0.54 0.77 0.59 0.18 0.53 0.87 0.66 0.73 0.77 0.19 0.22 0.63 0.47 0.50 0.32 Sub. Vol. DV BA Error DV BA Flew BA Flew BA Gros BA Gros BA Sub BA Sub BA Sample RE RE Sample RE Error RE Sample RE Error RE Sample RE Error RE Sample RE 0.16 0.51 0.51 0.33 0.33 0.54 0.54 0.11 0.11 0.20 0.42 0.43 0.34 0.35 0.71 0.72 0.15 0.15 0.54 1.17 1.16 0.81 0.80 1.23 1.22 0.34 0.35 0.82 0.76 0.78 0.71 0.73 1.28 1.29 0.75 1.24 0.82 0.71 0.71 0.80 0.79 1.05 1.04 0.79 1.26 0.57 0.42 0.42 0.43 0.44 0.97 0.99 0.44 0.59 0.19 0.40 0.40 0.41 0.41 0.72 0.71 0.24 0.24 0.37 1.11 1.12 0.95 0.96 1.12 1.13 0.32 0.32 071 1.04 1.04 0.84 0.84 1.03 1.03 0.41 0.42 0.47 0.46 0.47 0.43 0.44 0.76 0.77 0.58 1.22 0.76 0.73 0.73 0.72 0.73 1.39 1.39 0.88 1.73 0.87 0.50 0.51 0.46 0.47 0.85 0.85 1.17 2.00 0.21 0.38 0.39 0.33 0.33 0.79 0.79 0.23 0.24 0.21 0.48 0.48 0.43 0.43 0.73 0.73 0.24 0.23 0.32 0.69 0.67 0.61 0.58 0.92 0.90 0.21 0.20 0.39 0.45 0.44 0.46 0.45 0.77 0.75 0.60 1.00 0.31 0.55 0.55 0.50 0.51 0.70 0.71 0.37 0.65 0.40 0.55 0.54 0.50 0.50 0.62 0.62 0.53 0.78 0.30 0.80 0.81 0.66 0.66 1.07 1.09 0.25 0.25 0.42 0.31 0.31 0.25 0.25 0.60 0.60 0.32 0.32 0.98 1.08 1.08 1.21 1.23 0.94 0.94 0.53 0.54 0.80 0.52 0.53 0.50 0.51 0.91 0.91 0.62 . 1.06 1.36 0.76 0.75 0.81 0.81 1.30 1.29 1.11 1.94 0.85 0.67 0.67 0.59 0.59 1.06 1.05 0.53 0.73 0.86 1.61 1.61 1.51 1.50 2.26 2.25 0.79 0.80 0.47 1.25 1.25 . 1.42 1.41 1.05 1.05 0.51 0.51 0.79 0.96 0.96 0.95 0.95 1.17 1.17 0.65 0.66 0.64 0.67 0.66 0.65 0.64 0.98 0.98 0.63 1.41 0.52 0.51 0.50 0.51 0.51 0.83 0.83 0.54 1.16 0.72 0.52 0.52 0.52 0.51 0.94 0.95 0.66 1.10 0.15 0.32 0.32 0.29 0.29 0.85 0.85 0.21 0.22 0.31 0.78 0.79 0.65 0.65 0.93 0.94 0.35 0.36 0.52 0.83 0.82 0.66 0.65 0.89 0.89 0.36 0.36 0.35 0.51 0.51 0.46 0.47 0.77 0.77 0.54 0.88 0.38 0.38 0.38 0.41 0.40 0.83 0.82 0.61 0.92 0.34 0.46 0.46 0.50 0.50 0.79 0.79 0.40 0.56 0.19 0.75 0.75 0.52 0.52 0.82 0.81 0.15 0.16 0.45 0.82 0.82 0.69 0.70 1.16 1.18 0.40 0.41 0.79 1.17 1.16 0.96 0.96 1.02 1.02 0.47 0.48 0.52 0.48 0.47 0.53 0.51 0.74 0.72 0.51 0.80 0.76 0.79 0.79 0.82 0.82 1.38 1.38 0.56 0.87 0.58 0.62 0.62 0.57 0.56 0.86 0.85 0.32 0.43 0.18 0.83 0.84 0.51 0.52 1.06 1.12 0.17 0.18 0.53 1.19 1.17 0.90 0.89 1.77 1.77 0.36 0.41 0.87 0.90 0.90 0.93 0.94 1.01 1.01 0.51 0.51 0.65 0.76 0.76 0.71 0.71 0.89 0.88 0.70 1.84 0.73 0.71 0.71 0.71 0.72 1.01 1.01 0.75 1.46 0.76 0.62 0.62 0.66 0.66 0.92 0.92 0.62 1.14 0.19 0.68 0.68 0.50 0.50 1.17 1.18 0.22 0.23 0.22 0.76 0.76 0.55 0.55 0.85 0.84 0.26 0.27 0.63 0.95 0.94 0.81 0.81 1.02 1.02 0.36 0.37 0.47 0.42 0.41 0.41 0.41 0.72 0.71 0.73 1.35 0.50 0.55 0.55 0.53 0.53 0.91 0.90 0.66 1.00 0.31 0.52 0.51 0.51 0.51 0.61 0.61 0.55 0.73 tn o a. c/3 SO m EL 3 o 3 T3 a. o Tl so 13 

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