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Simulation techniques for a stochastic model of the growth of douglas-fir Goulding, Christopher John 1972

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SIMULATION TECHNIQUES FOR A STOCHASTIC MODEL OF THE GROWTH OF DOUGLAS-FIR by C. J . GOULD IN G B.Sc. Hons. (For.), Aberdeen University 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT CF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Forestry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and s tudy . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copy ing o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of r"<TR£ S , Ry, The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date U -June 117Z 1 1 ABSTRACT Supervisors Dr. D.D. Munro This study is part of the overall problem of determining the most suitable management practises for second growth stands of Douglas-fir, Pseudotsuga menzesli (Mirb.) Franco, in B r i t i s h Columbia. A stochastic simulation model of the growth of a Douglas f i r stand was constructed capable of providing yield and stand tables in response to various con-ditions of site, I n i t i a l density and thinning regime. The model is suitable for incorporation as the growth module of a larger systems model of a forest estate. Execution time is in the order of seconds for 80 years of simulated growth on an IBM 3 6 O / 6 7 computer. The programme is written in Fortran IV. The basic component of the simulation model is a l i s t representing the dbh's of individual trees in a sample plot. The model does not consider lnter-tree distances, hence is restricted to simulating stands with no large, irregular gaps in tree cover. A black box approach was taken in formulating the components of the system of growth. The major components are two difference equations which predict mortality end dbh growth over a short increment period, six years or less. Growth over longer periods is obtained by using the output of one forecast as the input to the next. A stochastic addition to the mortality component was constructed. Random mortality had an interaction with the growth functions so that the complete model had smaller values of expected volume yield when compared to a model with the random component removed. The model Is capable of generating a r t i f i c i a l data for stands at age 20, of being thinned to various regimes, and of predicting both t.otal and merchantable volumes. The concept of validation was discussed. A three phase approach was takeni testing the individual components; direct comparison of the model's predictions with actual permanent sample plots; analysing and c r i t i c i s i n g the Inferences derived from the behaviour of the model as a whole. The average error in the prediction of total volume growth of l*J-plots measured over an average of 35 years was 10 cu. f t . per acre. The average error in numbers of trees was -? per acre. No trends in errors with respect to age, site index and density were found. The problems of multiple response and experimental size were important In designing experiments on the simulation model. Response surface techniques were used to obtain maxi-mum mean annual increment of total gross volume as a function of site index and i n i t i a l density for simulated unthinned stands and as a function of thinning intensity, site index and i n i t i a l density for a specific thinning regime. The response of merchantable volume was also examined. The model's behaviour was such that i n i t i a l densities affected volume yield more than did thinning intensity. Both factors interacted with si t e . More space was required per iv tree for the optimal development of the stand on poor sites than on good sites, where there was a strong tendency to lose production i f the density of the stand became low. The inferences derived from this partial analysis were compared to selected references of summaries of spacing and thinning t r i a l s . It was concluded that the confidence in the model's a b i l i t y to predict growth and yield was f a i r l y good. V TABLE OF CONTENTS Page ABSTRACT 11 TABLE OF CONTENTS v LIST OF TABLES v i i LIST CF FIGURES i x ACKNOWLEDGMENTS x i i Chapter 1. INTRODUCTION 1 1.1 Problem D e f i n i t i o n 1 1.2 Objectives . . . . . . . . . 3 2. A REVIEW OF SIMULATION 6 2.1 S i m u l a t i o n - I t s Advantages and Disadvantages 6 2.2 P r i n c i p l e s of S i m u l a t i o n 7 2.3 The Use of S i m u l a t i o n i n Forest Management . 12 E c o l o g i c a l models 12 Forest e s t a t e models . . l4 S i n g l e stand models . . . . l6 2.4 Conclusions 2 i 3. THE FORMULATION OF THE MODEL 24 3.1 The Lev e l of R e s o l u t i o n of the Model . . . . 24 3.2 The Components of the System . 26 3.3 The J u s t i f i c a t i o n of a S t o c h a s t i c Component . 32 3.4 Several Approaches to the P r e d i c t i o n of Stand Growth 34 Markov chains . 35 Compatible growth and y i e l d models . . . 37 Competition d e n s i t y models . . . . . . . 38 Other growth models 40 Conclusions 4i 3.5 D e s c r i p t i o n of the Data . . 42 4. CONSTRUCTION OF THE MODEL . 48 4.1 I n t r o d u c t i o n 8^ 4.2 S i t e Index and Dominant Height 50 4.3 Volume C a l c u l a t i o n s 51 v l Page 4.4 T r i a l of K l r a ' s C - D Rule 55 4.5 Stand Generation 57 4.6 The Diameter Increment Function 66 4.7 M o r t a l i t y 72 4.8 The Si m u l a t i o n Programme 82 5. VALIDATION 90 5»1 General C o n s i d e r a t i o n s . 90 5.2 S p e c i f i c Problems i n V a l i d a t i n g TOPSY . . . . 93 5.3 V a l i d a t i o n of Long-Term P r e d i c t i o n s 96 5.4 V a l i d a t i o n of Short-Term (10 year) P r e d i c t i o n s 106 5.5 The "Turing" Test 108 5.6 Conclusions 109 6. THE DESIGN, ANALYSIS AND RESULTS OF SOME SIMULATION EXPERIMENTS I l l 6.1 I n t r o d u c t i o n I l l 6.2 The V a r i a b i l i t y of the Model and i t s Behaviour when Compared t o the Y i e l d Tables . ll2 6.3 The Design of Computer S i m u l a t i o n Experiments 130 6.4 Experiment l - Unthinned Stands 133 6.5 Experiment 2 - Thinned Stands l43 6.6 The V a l i d i t y of the Model when Thinned . . . 153 6.7 Summary of R e s u l t s 163 7. SUGGESTIONS FOR FUTURE WORK AND CONCLUSIONS . 166 7.1 Future Work 166 7.2 Summary and Conclusions . . . . . l68 BIBLIOGRAPHY 171 APPENDIX 1 A i t k i n ' s I n t e r p o l a t i o n 184 APPENDIX 2 Y i e l d and Stand Tables - Unthinned . . . . 186 APPENDIX 3 Y i e l d and Stand Tables - Thinned 199 APPENDIX 4 P l o t s of I n d i v i d u a l Runs f o r Normal Density Stands 208 APPENDIX 5 Example of C o n t r o l and Output a t a Remote Terminal 2 l 8 APPENDIX 6 Programme L i s t i n g f o r TOPSY 220 APPENDIX 7 M o d i f i c a t i o n s to TOPSY f o r M o r t a l i t y a f t e r Thinning 232 v i i LIST OP TABLES Table Page 3.1 Statistics of the Data Points 44 3.2 Correlation Matrix of the Data Points 4 7 4.1 The Klra C-D Function to Predict Average Dbh . 57 4.2 Plot Data for the Stand Generation Component . 59 4.3 X 2 Tests on the Dbh Distribution of 8 Plots at Ages 20-25 6 0 4.4 Regression Coefficients for the Inverse Cumul-ative Density Functions of Dbh Distributions . 6l 4.5 X 2 Tests on the Inverse C.D.F. ( a l l stands) for 8 Plots 6 4 4.6 Correlation Coefficients of Average Annual Dbh Increments with Independent Stand Variables . . 68 4.7 Coefficients of the Inverse C.D.F.'s of Residual Mortality 76 4.8 The Effect of the Random Residual Mortality on a Plot, Site 120, l600 Stems per acre at Age 20 79 4.9 Relative Probabilities of Mortality by Compet-itio n Class for Individual Trees 80 4.10 Definition of the Subroutines used in the Programme 84 5«1 Paired " t " Tests on Long-term Predictions Net Volume per acre cu. f t 98 5.2 Paired " t " Tests on Long-term Predictions Stems per acre 99 5.3 Paired " t " Tests on Long-term Predictions Basal Area per acre sq. f t . . 100 5.4 Paired " t " Tests on Long-term Predictions Average Dbh inches 101 5.5 Summary of Results of Paired " t " Tests on Short-term Predictions 107 v i i i Table Page 6.1 Maximum MAI of Gross Volume per acre (cu. f t . per acre per year) as a Function of Site Index and I n i t i a l Density for Unthinned Stands . . . 136 6.2 Close Utilization Volume per acre (cu. f t . per acre to 4" top) as a Function of Age, Site Index and I n i t i a l Density l 4 i 6.3 Maximum MAI of Gross Volume per acre (cu. f t . per acre per year) as a Function of Thinning Intensity, Site Index and I n i t i a l Density for a Specific Regime 152 6.4 Summary of the Effect of the Thinning Regime on Maximum MAI Gross Total Volume per acre . . . . 1 5 3 6.5 Summary of the Comparison between Unthinned, Low and Crown Thinning. I n i t i a l Density at Age 20, 800 stems per acre l60 i x L IST OF FIGURES F i g u r e Page 3.1 The I n f l u e n c e Diagram of Stand Growth . . . . 26 3.2 The V a r i a b i l i t y of P e r i o d i c A n n u a l M o r t a l i t y and Growth 28 3.3 Stand Component I n t e r a c t i o n M a t r i x 29 3.4 D i s t r i b u t i o n of Data by Increment P e r i o d s Age v s . S i t e 45 3.5 D i s t r i b u t i o n of Data by Increment P e r i o d s Stems per a c r e v s . S i t e 46 4.1 Flow Diagram of the C a l c u l a t i o n s of Stand Growth 49 4.2 P l o t of Average Stand T a r i f Number a g a i n s t Dominant H e i g h t 54 4.3 E m p i r i c a l C.D.F.'s f o r I n i t i a l Dbh D i s t r i b -u t i o n P l a n t a t i o n , N a t u r a l and Average Stands . 62 4.4 Dbh D i s t r i b u t i o n s a t Age 20 Generated by the Model 65 4.5 F o u r P l o t s of Dbh Increment a g a i n s t Tree Dbh w i t h i n a Stand 67 4.6 P l o t of M o r t a l i t y R e s i d u a l s and the Four " R i s k " C l a s s e s 75 4.7 D i s t r i b u t i o n of Random R e s i d u a l s of M o r t a l i t y P r e d i c t e d by the Model f o r Four C l a s s e s . . . 78 4.8 R a t i o s of P r o b a b i l i t i e s of M o r t a l i t y P l o t t e d a g a i n s t Age 8 l 4.9 A b b r e v i a t e d Flow Diagram of the S i m u l a t i o n Model 83 5.1 I l l u s t r a t i o n of the D i f f e r e n c e between Model B e h a v i o u r and P r e d i c t i o n 91 5.2 Comparison of R e a l and S i m u l a t e d P l o t s , B a s a l A r e a and Volume per a c r e 103 5.3 Comparison of R e a l and S i m u l a t e d P l o t s , Stems p e r a c r e and Average Dbh 104 X F i g u r e Page 5.4 R e s u l t s of Paired " t " Tests over 10, 20 and 30 Years of S i m u l a t i o n 105 6.1 The E f f e c t of the S t o c h a s t i c Component Volume per acre f o r S i t e 90 I l 6 6.2 The E f f e c t of the S t o c h a s t i c Component Volume per acre f o r S i t e 120 117 6.3 The E f f e c t of the S t o c h a s t i c Component Volume per acre f o r S i t e 150 118 6.4 The Growth Curves of the Unthlnned Model Ba s a l Area per acre f o r S i t e 90 120 6.5 The Growth Curves of the Unthinned Model Basal Area per acre f o r S i t e 120 121 6.6 The Growth Curves of the Unthinned Model B a s a l Area per acre f o r S i t e 150 122 6.7 The Growth Curves f o r the Unthinned Model Average Dbh . . . . . . . . . . . . . . . . . 123 6.8 The Growth Curves of the Unthinned Model Stems per acre 124 6.9 The Growth Curves of the Unthinned Model Standing Volume per acre 125 6.10 The Growth Curves of the Unthinned Model Gross Volume per acre 126 6.11 Diameter Frequency D i s t r i b u t i o n s f o r the Unthinned Model. Average Values f o r S i t e 90, I n i t i a l Density 800 stems per acre 127 6.12 Diameter Frequency D i s t r i b u t i o n s f o r the Unthinned Model. Average Values f o r S i t e 150, I n i t i a l Density 800 stems per acre 128 6.13 Response Surface of Gross Volume Maximum MAI to I n i t i a l Density and S i t e 135 6.14 Response Surface of Close U t i l i z a t i o n Volume per acre to Age and I n i t i a l Density, S i t e 90 . 138 6.15 Response Surface of Close U t i l i z a t i o n Volume per acre t o Age and I n i t i a l Density, S i t e 120 139 6.16 Response Surface of Close U t i l i z a t i o n Volume per acre t o Age and I n i t i a l Density, S i t e 150 l 4 0 x i F i g u r e Page 6.1? Response S u r f a c e o f Gross Volume Maximum MAI t o i n i t i a l D e n s i t y , S i t e and T h i n n i n g I n t e n s i t y l 5 l 6.18 Comparison of Net and Gross Volumes p e r a c r e f o r U n t h i n n e d and Low Thinned Model. S i t e 90 155 6 . 1 9 Comparison of Net and Gross Volumes p e r a c r e f o r U n t h i n n e d and Low Thinned Model. S i t e 120 156 6.20 Comparison of Net and Gross Volumes p e r a c r e f o r U n t hinned and Low Thinned Model. S i t e 150 157 6.21 Diameter Frequency D i s t r i b u t i o n s f o r t h e Low T h i n n i n g , S i t e 90, I n i t i a l D e n s i t y 800 stems p e r a c r e , 158 6.22 Diameter Frequency D i s t r i b u t i o n s f o r the Crown T h i n n i n g , S i t e 150, I n i t i a l D e n s i t y 800 stems pe r a c r e 159 x i i ACKNOWLEDGMENTS The author wishes to thank his supervisor, Dr. D.D. Munro, for the encouragement and help received during the work on this dissertation. The thesis was reviewed by Drs. D. Haley, A. Kozak, J.H.G. Smith, C.J. Walters and Mr. G.G. Young, who contrib-uted much useful advice and help. Drs. C. Boyd and J.V. Zidek also helped guide the early stages of development. The data used in the thesis was obtained from the Forestry Division of MacMlllan Bloedel Ltd., and from the Pacific Northwest Experiment Forest and Range Station, US Forest Service. The writer would like to gratefully acknowl-edge the permission to use this data. The author was awarded the Canadian Department of Forestry and Rural Development Forestry Fellowship and two MacMlllan Bloedel Forest Mensuration Fellowships. Additional financial support was given by the Faculty of Forestry, University of Brit i s h Columbia in the form of research asslstantships. The writer was employed by MacMlllan Bloedel Ltd. for two summers and would like to thank a l l the members of the Forestry Division at Nanalmo for a stimulating experience. Thanks are due to Mr. M. Bonita, F. Bunnell and S. Smith for valuable discussion, Mrs. M. Home for help with the drafting and to Mrs. A. Daly for typing this manuscript. Finally, the author would like to sincerely thank his x i i l wife, Isobel and daughter, Nicola f o r t h e i r constant encour-agement . 1 CHAPTER 1 INTRODUCTION 1.1 Problem Definition The timber of Douglas-fir, Pseudotsuga menzlesll (Mlrb.) Franco, Is one of the most Important and prized timbers in the world. In 1970 i t was the most important species In terms of stumpage revenue to the B r i t i s h Columbia Forest Service ( l 9 7 l ) , but only third in terms of volume cut in the province. Faced with increasing consumption and dwindling supplies of old growth timber - Palmer (19&5) estimated Its complete exhaustion by 1983 - more use w i l l be made of second growth stands, usually younger than the previous crop and with a considerable difference in quality and size of tree. Paille (1968) and Smith (1971) argued strongly for the i n t e n s i f i -cation of management, both to Improve quality and increase yield. However there is l i t t l e information and less experi-ence on the consequences of such practises in the Pacific North West. What w i l l be the rotation age, the yield, the product mix, the costs, the pr o f i t a b i l i t y ? How stable are these regimes to changes in the price/size class structure? With the increased use of operations research, detailed information w i l l be required. Forest management suffers from the i n a b i l i t y to see the long term consequences of policy and stand treatments, with many different regimes on many different stands interacting 2 with future costs and revenues. Regimes that are optimal for a single stand may not be optimal when considered over many stands. Moreover, there Is an increasing pressure for hunt-ing, camping and wilderness uses of the forest which w i l l c o nflict with intensive timber management and possibly modify some practises. The problem of how to determine the most suitable management practises can be broken down into six-phases . 1. Prediction of the growth of natural second growth stands. 2. Prediction of the results of s i l v l c u l t u r a l tech-niques such as planting, spacing and thinning. 3. Allocation of costs and revenues. 4. Optimization and enumeration of viable alternatives. 5 . Construction and analysis of a systems model of a forest estate capable of predicting the effects of future changes in policy, timber demand, costs and prices. 6 . Optimization of treatments and policy with respect to the whole forest estate. The traditional methods of solving the whole problem have been through yield tables and working plans. The older yield tables are not easily adaptable to a computer and are often too inflexible, providing data for one type of regime only. The working plan has often been formulated after con-sideration of only one or two alternatives. The annual cut 3 has been determined from a simplistic formula or from a desire to convert the forest to an idealized state. Recently large scale optimization models have been formulated, for example, linear programming with decomposition, or simulation models of the forest estate. These either require growth and yield data to be provided or supply their own estimates by overly simple functions. Cn the other hand several simul-ation models of stand growth based on individual trees have been created. These often take too long to run to be incorp-orated into a large forest estate model with many different stands. The important stages of testing and analysing the behaviour of the model were often neglected. 1.2 Objectives Because of the size of the problem, this thesis was restricted to phases l and 2, with the constraint that i t be capable of extension to the construction and analysis of a systems model of the timber and value production of a forest estate composed of pure, even-aged, second growth Douglas* f i r , phases 3 to 6. With this constraint, and in view of the complexity of the systems model, a simulation study of growth using a d i g i t a l computer was the only feasible method. The objective of the thesis can be stated a s i "to construct, to attempt to validate and to at least partially analyse a growth and yield simulation model of a pure, even-aged Douglas-fir stand, capable of simulating the 4 tree growth on a wide range of natural conditions, of being thinned and of predicting both yield and stand tables." Depending on their objectives, many modelers do not claim to predict results with any accuracy; rather they seek to explain the nature of the system and the relative effects of one strategy versus another. However, some policy decis-ions are based on the attainment of a particular level of some variable. If such a policy is adopted, with a large commitment of capital and effort, i t is necessary to be able to state with some confidence that this level w i l l be reached, i f not exceeded. Also the results from a growth predicter may be used in further analyses where levels are important, for example, optimization of harvesting equipment and log allocation. In order to explore the growth characteristics of the model, interventions by and interactions with the researcher are desirable. This would allow a secondary use of the model as a game for teaching purposes. Because of this, and because of the need to use the model as the growth module of a much larger system, results must be obtained quickly, in the order of a few seconds computer time for 100 years growth. Thus four secondary objectives of the thesis can be stated. 1. It must be predictive. 2. It should be structurally suitable for incorpor-ation as the growth module of a larger model. 3« It should be capable of being used as an instant-aneous interactive model. 4. Execution time must be minimal. This thesis draws heavily on the body of literature from the socio-economic f i e l d of simulation models for design and techniques. The formulation of the model through a black box approach emphasises the Importance of the inter-actions between components on the behaviour of the model. The concept of validation is introduced and explained. This involves not only testing the individual components, but also testing the results of the model as a whole. A stochastic component is used with i t s consequent problems and the thesis illustrates some of the methods used to design and analyse the behaviour of a stochastic model. The model has one important restriction. In order to minimize execution time, no attempt was made to take into account the effects of the spatial dis-tribution of individual trees. Thus the model is assumed to apply only to stands with no large, irregular gaps in tree cover. 6 CHAPTER 2 A REVIEW OF SIMULATION 2 . 1 Simulation - Its Advantages and Disadvantages In Wagner's ( 1 9 6 * 9 ) "book on operations research, the f i r s t section in the chapter on simulation is entitled "When a l l else f a i l s . . . " Simulation in i t s e l f is not an optim-izing technique such as those of mathematical programming, nor does i t have any unifying set theory or algorithm. Thus any solution obtained by simulation is usually very much more expensive than i f i t had been obtained by mathematical pro-gramming. Simulation is therefore used when the system i s too complicated to be formulated In a standard way; when an analytic solution is either too d i f f i c u l t or impossible; or when an optimal solution is not required at a l l , but rather an understanding of the underlying mechanism or principles of the system. With the use of the computer, a simulation experiment can perform many calculations and test many alternatives, too dangerous, too expensive or too time con-suming to perform in the real world. It can do this in a very short period of time, a feature termed "time compres-sion." The 'raison d'etre* of simulation is best described by T.S. E l l i o t . Do I dare Disturb the Universe? In a minute there is time For decisions and revisions that a minute w i l l reverse. 7 However, computer simulation is a d i f f i c u l t and expensive technique, and although the model may run quickly, the whole project may take a long time. Moreover, in order to model a system adequately the simulation programme may become so complicated that the results are incomprehensible. Finally i t is so easy to produce r e a l i s t i c looking results and a r t i s -t i c graphs that the unwary may overlook basic deficiencies in the model. 2.2 Principles of Simulation In a l l the subsequent discussion, simulation is referred to only in the context of d i g i t a l computer simulation as opposed to analog or physical simulation. A convenient div-ision is Monte Carlo methods, socio-economic simulation, biological simulation and simulation games, although basic principles and philosophy may be common to a l l . Monte Carlo methods deal with the use of random variables to explore complex probablistic and deterministic models (Hammersley and Handscomb, 1 9 6 4 ) . There have been a host of recent elementary books on simulation in the socio-economic f i e l d , e.g. Emshoff (1970), Evans et a l . ( 1 9 6 8 ) , and Schmidt and Taylor ( 1 9 7 0 ) . A thorough review is beyond the scope of this thesis. Meier et a l . (1969) dealt primarily with economic material. MacMlllan and Gonzales (1965) is similar and has excellent discussions on techniques. Forrester (i960) wrote a definitive text. Industrial 8 Dynamics, on the philosophy which underlies a l l simulation models. (There is a student's primer, Forrester ( 1 9 6 8 ) . ) He was concerned with "Systems Engineering," defined as "a formal awareness of the interactions between parts of the system." His Industrial dynamics approach to the design of an enterprise was* 1 . A definition of the goals (the problem must be worthy and significant). 2. A description of the situation. 3. The formulation of a mathematical model. 4 . The simulation. 5 . An interpretation of the system. 6 . The revision of the model. 7 . Repeated experimentation. He suggested that the absolute worth of a model can be no greater than i t s objectives. The basic methods of model construction were described in de t a i l , along with examples and a description of the simulation language, DYNAMO. Less philosophical than Forrester, Naylor et a l . (19^8) described simulation, i t s methods and techniques with examples from the business world, queuing theory and simulation lang-uages. Four classes of models were recognised; Deterministic or Stochastic, Static (time Invariant) or Dynamic. They gave nine stages in the design of a simulation model. 1 . Problem formulation. 2. Collecting and processing the data. 3. Formulating a mathematical model. 9 4. Estimating parameters. 5. Evaluation of the model and the parameters. If unsuccessful return to stage 2. 6. Construction and verification of the computer model. 7 . Validation. 8. Design of computer simulation experiments. 9. Analysis and interpretation. The design sequency of Naylor et a l . (1966) differed from that of Forrester (i960) only in d e t a i l . Moreover the order in which the stages were implemented remained f a i r l y arbitrary and the cyclic process of testing and revision was important. Data collection can precede, be simultaneous with and follow stages three and four. Formulating a mathematical model is both an art and a science. It requires decisions on whether the model is to be deterministic or stochastic, how many variables to include, how these are related to one another and what transformations or functions are necessary. Evaluation of the model and the parameter estimates precedes the construction of the simulation model. This evaluation is concerned with testing the assumptions and the significance of the components of the model. Once the model has success-f u l l y passed this evaluation, the computer programme can be written and verified to ensure that compilation and execution is correct. Stage seven, validation, on the other hand is concerned with ensuring that the output and the behaviour of the model is a "correct" representation of r e a l i t y . What constitutes validity and how the valid i t y of the model is 10 decided is not a t r i v i a l problem and is discussed further in Chapter five. Using the simulation model involves consider-ations of experimental design and analysis. This is also discussed in more detail In Chapter six. A second book edited by Naylor (1969) contains articles concerned with stages 7 and 8. The previous five books contain much that is relevant to biological simulation. There are few books dealing with the philosophy of biological simulation. Most contain selected articles on particular systems, e.g. Patten (1970). Biological simulation is characterized by great attention to the underlying mechanism of the system. More importance is placed on exposing the underlying principles and observing the behaviour of the system than in predicting absolute values. This is in accordance with Forrester, but contrasts with Naylor et a l . (1968) who suggested that at some stage of the evolution of simulation i t w i l l become necessary to attach confidence limits to the result of the model. Watt's collection of articles (1966) and his book (1968) contain worthwhile discussions on the art of model building. Watt (Chap. 1, 1966) suggested an experimental procedure of data collection, crude model building and sensitivity test-ing. The sequence was then recycled, each time refining or correcting the model. With growth models of forests this can be a very long term project. Holling (in Watt 1966, Chap. 8) proposed an approach termed "experimental components analysis," in which an i n i t i a l systems model is. b u i l t to include a few components basic to a l l examples of the process, and then the model improved to include components specific to a narrower and narrower spectrum. Each step would be taken only when the postulates of the previous step had proved adequate. This approach was taken in his analysis of the functional responses to prey and predator density, Holllng (1965)• This study took over ten years and i s s t i l l only partially completed. Simulation games have been used extensively since the mid 1950's in business and there is an increasing number of applications in resource sciences. Their objective is to provide a tool with which players can experiment and observe reactions, sometimes competing with one another. Babb and Eisgruber (1966) described some general ideas and presented four games developed at Purdue University. Newell and Newton (1968) described the conceptual stages of the development of game models to teach natural resource management. The com-ments of Forrester (i960, Chap. 20) are relevant here. Games il l u s t r a t e good practice as viewed by the designer and could perpetuate past cliches about what constitutes good management. Games should not be analogous to the "Link" cockpit simulator which trains a person to be a "robot" operator of a piece of machinery, rather they sould teach the manager to learn how to understand systems and perhaps design better systems. A game is useful i f It gives a manager a realization of the Interactions within a system and i f i t acts as an "opening wedge" to a true study of that system. 12 2.3 The Use of Simulation in Forest Management Bare (l97l) surveyed the use of operations research in forest management (263 refs.) and included a section on com-puter simulation (approx. 76 refs.). Somewhat peripheral to this thesis are studies on sampling techniques, e.g. O'Regan and Palley (1965) and Payandeh (1970), yard inventories e.g. Hamilton (1964), f i r e supression e.g. Shultz (1966) Kourtz and O'Regan (l97l), harvesting e.g. Newnham (1970) amongst many others. Worthy of note is the simulation of the U.S. plywood industry by Manetsch (l964a,b), of the timber market by Balderston and Hoggatt (1962) and Bonita's (1972) stochastic model of a logging division, validated with the help of spectral analysis. Central to forest management are those models which predict the growth of trees and which model decision-making over the whole of the forest estate. We can divide these into three classes? ecological models, forest estate models and stand models. Ecological models Walters and Bunnell (l97l) described a game to model plant production and succession, w i l d l i f e habitat and food selection, and the dynamics of wildlife population. Harvest-ing interactions were allowed. Like many biological models i t had a large number of parameters, but a lack of data to estimate them. It foresook accuracy of magnitude and degree of resolution in favour of generality and broad applicability. It was nonrandom and no attempt was made to validate i t . Botkin et a l . (1970) designed a model to simulate growth interactions of thirteen tree species as part of the Hubbard Brook Ecosystem study. This allows logging and modifications to s o i l and climatic variables. Recruitment and mortality are stochastic. Bledsoe and Van Dyne ( l 9 7 l ) modelled the succession of grass to forb to shortleaf pine to deciduous on an old-field ecosystem. Bunnell (1971) wrote a preliminary model of the Arctic tundra to explore hypothetical response patterns on nutrient cycling. Hatheway et a l . ( l 9 7 l ) mod-elled the uptake of water by a tree. Tumbull (personal comm.) at the University of Washington Is attempting to model stem c e l l growth within a tree (also see Bethel, 1 9 7 0 ) . These few references describe models which have a much finer resolution of detail than is concerned with here. How-ever they do show what can be done with very l i t t l e data and il l u s t r a t e techniques different from models of the socio-economic type. Much philosophy, especially when compared to Forrester ( i 9 6 0 ) is quite similar, but as in the forestry models that follow, the rigourous application of the design sequence of Naylor et a l . (1968) is lacking, especially the validation procedure. 14 F o r e s t e s t a t e models The model by Morgan and B j o r a ( l 9 7 l ) was used t o e v a l u a t e the c u t t i n g and p l a n t i n g p l a n s f o r a l a r g e e n t e r p r i s e , i n t h i s c a s e the B r i t i s h F o r e s t r y Commission. They a t t e m p t e d t o f o r e -c a s t p r o d u c t i o n , l a b o u r demand and revenue o v e r the n e x t twenty y e a r s f o r a l t e r n a t i v e p l a n s and changes i n l a b o u r p r o -d u c t i v i t y . T h i s model was i n i t s p r e l i m i n a r y s t a g e s of con -s t r u c t i o n and s u f f e r e d from l a c k o f computing f a c i l i t i e s . I t s im p o r t a n c e was i t s use i n a p r a c t i c a l n a t i o n a l s i t u a t i o n . The f i r s t f o r e s t management s i m u l a t i o n was t h a t o f Gould and O'Regan (1965), w i t h subsequent a r t i c l e s by O'Regan e t a l . (1966) and Howard e t a l . (1966). T h i s was the Ha r v a r d F o r e s t s i m u l a t o r . I t i n c l u d e d g r o w t h , h a r v e s t i n g , c o s t s , p r i c e s and s t o c h a s t i c e v e n t s such as f i r e and s t o r m s . Two c u t t i n g p o l i -c i e s were t e s t e d , one based on r i g i d s u s t a i n e d y i e l d , t he o t h e r f o l l o w i n g s u p p l y and demand c u r v e s . W a l ton (1965) extended the model. An i n t e r e s t i n g f e a t u r e was t h e a u t o m a t i c h a r v e s t i n g s c h e d u l e r . T h i s was done by s p e c i f y i n g a p o l i c y o f volume o b j e c t i v e s and c u t t i n g methods wh i c h i n t e r a c t e d w i t h p r i c e s t o a l l o c a t e t h e a r e a s t o be h a r v e s t e d . Myers (1968 and 1971) a l s o d e s c r i b e d a model of a f o r e s t e s t a t e s i m i l a r t o t h e Ha r v a r d F o r e s t s i m u l a t o r . Growth was performed by y i e l d t a b l e s I nput a t t h e b e g i n n i n g of a r u n . T h i n n i n g was p e r m i t t e d . C l u t t e r and Bamping (1965) s i m u l a t e d g r o w t h , c u t t i n g and h a r v e s t i n g p o l i c i e s on a l a r g e , s c a t t e r e d , h y p o t h e t i c a l f o r e s t . Costs and revenues were examined over two regimes, area regu-lation verses economic rotation. There were no thinnings, but planting was allowed i f required. The objective was to maxi-mize the net discounted revenue subject to constraints on fluctuations in annual cut. The project was subsequently-developed into the Max-Million system consisting of an appraisal phase, an unrestricted scheduling phase followed by the addition of constraints and optimizing by linear pro-gramming (Ware I969i Ware and Clutter 1971)• Sayers ( l 9 7 l ) described a model for the simulation of management on private estates in Scotland consisting of even-aged stands. Simulation was chosen over dynamic or linear programming because i t could provide annual accounts and analyses of growth, and because the significance of the output could easily be appreciated by the somewhat cynical Scottish l a i r d . The model was deterministic and growth derived from standard yield tables. Input of an annual schedule of oper-ations was required. Many alternatives were possible, some mandatory, including site preparation, planting, thinning, f e l l i n g , road-making, etc., and were specific to a particular s tand. The Purdue Forest Management game (Bare 19&9, l97°a, b) was different from the previous models, as It was primarily a teaching device in which three teams managed a forest competing with one another for their budgets. Sixty stands, even-aged of one species, composed the forest. Annual schedules of thinning, f e l l i n g , planting, site preparation and 1 6 protection had to be prepared within the tight restrictions of the budget and the annual allowable cut. These were checked by the "Chief Forester," a GEC computer, and rejected i f inaccurate - a frustrating experience for the careless. An inventory could be called at a cost, otherwise a team must predict growth and keep track of the forest through time. Growth response were rather crude, but the overall effect was r e a l i s t i c and stimulating. Another management game has been designed by Pierce (1971)• This is applicable to the Pacific Northwest (PNW). Single stand models Single stand models predict tree growth in more detail and hopefully more accurately than forest estate models. With two exceptions, dollar values have not been given to the output of the model. They can be subdivided into two typesi distance dependent models where the location of the indiv-idual tree is required; distance independent models where location is unnecessary. The lat t e r have much faster execu-tion times. D i £ t an c e _d ep e nde n t _m od el s_ Newnham (1964) constructed the f i r s t growth model of an even-aged plantation of Douglas-fir with square spacing. The objective was to study the effects of i n i t i a l mortality and subsequent thinning on the spatial distribution and growth of the stand. The i n i t i a l individual tree size/location was random, subsequent growth deterministic. The hypothetical maximum diameter at breast height (dbh) growth of a tree was reduced proportionally to the amount of "crown" competition i t received from neighbouring trees. The model was " c a l i -brated" by altering a crucial parameter, Redfac, the ratio of the calculated crown width to the competitive crown width, u n t i l the predicted values of the model f i t t e d published yield table data satisfactorily. The yield tables were the normal yield tables of McArdle et a l . (l96l), those of Duff (1956) from New Zealand and Barnes (195^) Interpreted from Britain. This procedure presupposed that Br i t i s h and New Zealand yield tables were comparable to conditions in B.C., and was dependent on the accuracy of the growth curves of the yield tables. Moreover, the model was calibrated for "normal density," unmanaged stands and was then assumed to be correct for a l l other conditions of I n i t i a l density and thinning regime with no further calibration. Validation was restricted to running the model for various i n i t i a l spacings, thinnings and patterns of I n i t i a l mortality and exhibiting the results. No tests were made comparing the model's predictions with data from Individual plots. Although crude, the model had many worthwhile features, i l l u s t r a t i n g what could be done with very l i t t l e data. Lee (1967) used essentially Newnham's model for Lodgepole pine, Plnus contorta Douglas. (Newnham and Smith, 1965) 18 increasing i t s value by calculating volumes and expected revenues. He attempted to place confidence limits on the results of the model but his method failed to take into account inter-tree competition effects, the errors due to calibration or the errors due to using the predictions from one time interval as the inputs for the next. The only other test of the model as a whole was the i l l u s t r a t i o n of the effects of four different I n i t i a l spaclngs on one s i t e . Bella (1970) modelled the growth of aspen and relaxed the regular spacing restraint of Newnham's model. He changed the competition index and incorporated an integrated indiv-idual tree height and dbh growth system. There was a logical error in his model (Page 84, Bella 1970). Having calculated the potential height growth AH for a tree of dbh d, height H, and the ratio of d/H from a + b.H, the model calculated the potential dbh Increment Ad as Ad = d/H .AH or ^d = (a + b.H).AH This was wrong. The correct formula was Ad = (a + 2.b.H).AH + b.(AH) 2 The true values of Ad would be half as much again as those calculated. The output s t a t i s t i c s of the model agreed with the yield table values because the model was calibrated to do so. Changing conditions of site index and i n i t i a l density required new calibration runs. The analysis of the model could be described as incomplete. Bella ( l 9 7 l ) showed that his distance dependent competition index was highly significant s t a t i s t i c a l l y in explaining dbh growth. A re-gression with only dbh terms had a multiple coefficient of determination (R ) value of 52#, while a regression including the competition index had an R2 value of 59%• The increase 2 in R value was not as much as would be expected when taking into account the great Increase in complexity. Lin (1969) developed a spatial competition index which appeared success-f u l in predicting dbh growth of Western Hemlock, Tsuga  heterophylla (Raf.) Sarg. (correlation coefficient . 8 0 7 ) * He tested his model against real data for only four two year growth periods, but his errors were small and did not appear to accumulate. The previous four models evaluate competition indices in terms of lntertree distance and tree dbh's. A different approach was taken by Mitchell (19&9) w n o constructed an inte-grated system of height, branch and dbh growth. Tree branches grew u n t i l in contact with branches of another tree. The resultant crown area determined height and dbh growth. The model was not tested or run very extensively. The same approach is being used to simulate not only tree competition but shrub and grass growth beneath the canopy (Quenet, C.F.S. Victoria, pers. comm.). Arney (1971) simulated the radial growth in each whorl of the stem for a l l trees in the stand, in order to Include the effects of competition on bole shape. Only five simulation runs were made using the completed model. Dress (1970) developed a system for the stochastic simulation of the growth and mortality of even-aged forest 20 stands w i t h a r t i f i c i a l stand generation. The model was very-t h e o r e t i c a l and designed f o r a high degree of f l e x i b i l i t y i n c o n t r o l l i n g the course of development of the stand. Only one run was i l l u s t r a t e d and there was l i t t l e e v a l u a t i o n of the model, f o r example, the c u r r e n t annual increment of volume, per acre had no recognizable p a t t e r n . There was no d i s c u s s i o n on methods of a n a l y s i n g the output of the model. The d i s t a n c e dependent models produce d e t a i l e d inform-a t i o n . They are bes t used t o examine the e f f e c t s of s p a t i a l d i s t r i b u t i o n of tre e s on growth, though only Newnham (1964) d i d t h i s t o any degree (Smith et a l . , 1965)* Unless the tr e e s are l o c a t e d at r e g u l a r spacings the execution times of the model are l o n g , p r o p o r t i o n a l e i t h e r t o the square of the number of tre e s or t o the area of the p l o t . L i n ' s model took 20 minutes to grow 120 t r e e s 8 years on an IBM 1130, M i t c h e l l ' s took \\ minutes f o r 40 years on l / 4 0 t h a c r e . Distance_1ndependen t_models Because the l o c a t i o n s of i n d i v i d u a l t r e e s are not re q u i r e d by d i s t a n c e independent models, i t i s unnecessary to use a lengthy search procedure to l o c a t e the competitors of a t r e e . Thus they can be q u i t e f a s t , but are u n l i k e l y t o pre-d i c t an i n d i v i d u a l t r e e ' s growth a c c u r a t e l y , nor the growth of h i g h l y I r r e g u l a r stands. Both Ople and P a l l l e used the s i n g l e t r e e as the b a s i c u n i t . Ople (1970) cons t r u c t e d a model t h a t would generate 21 and grow stands of Eucalyptus regnans on a y e a r l y c y c l e f o r up t o 300 years. Timing was i n the order of one second per year. B a s a l area growth and m o r t a l i t y per acre were c a l c u l -ated and then d i s t r i b u t e d t o the t r e e s . For short term runs, 20 - 30 y e a r s , e r r o r s i n volume/acre were of the order of 4$ f o r thinned stands, and \2% f o r unthinned stands, but t h i s was b e l i e v e d due t o the v a r i a b i l i t y i n m o r t a l i t y . P a l l l e (1970) c o n s t r u c t e d a growth model as p a r t of h i s t h e s i s on m o r t a l i t y of D o u g l a s - f i r . Diameter growth was independent of stand d e n s i t y , t r e e height f o r a l l ages was estimated from one height/dbh curve, and the p r o b a b i l i t y of a t r e e ' s death depended on i t s dbh/stand dbh r a t i o and age. E r r o r s on the only two p l o t s t e s t e d over 40 years p r e d i c t i o n were i n the order of 30 - 50$ f o r standing volume/acre; a c t u a l growth was three t o f o u r times g r e a t e r than p r e d i c t e d . Mauge (1970) modelled the growth and p r o f i t a b i l i t y of Plnus p i n a s t e r A l t o n , under v a r i o u s t h i n n i n g regimes and f e r t i l i z a t i o n t r i a l s . U n l i k e Opie or P a l l l e , h i s equations gave values f o r the whole stand or f o r average s i z e and were based on y i e l d t a b l e s . 2.4 Conclusions A l l the f o r e s t r y models discussed s u f f e r e d from f a i l u r e t o f o l l o w more than one c y c l e of Watt's (1966) procedure of data c o l l e c t i o n , model b u i l d i n g and r e v i s i o n . This i s under-standable, s i n c e many models were constructed f o r d i s s e r -22 tations. One pass of the cycle would be almost equivalent to following the design sequence of Naylor e_t a l . (1966). More serious faults were the lack of c r i t i c a l testing, validation and experimental design in the analysis of the models, some of which were at least partially stochastic. Many projects appeared to have as their objective solely the construction of a model to "grow" a stand or to "manage" a forest estate. Few were applied to specific goals to any large extent (Newnham's (1964) model was an exception), consequently there were no standards by which to assess v a l i d i t y . Few models were tested against the real world over a wide range of con-ditions of site, density or management regime. It was to attempt to overcome these faults that the design of the simulation model in this thesis closely followed the stages outlined by Naylor e_t a l . (1966) l i s t e d in section 2.2. The emphasis of the thesis was more In accordance with the philosophy of Forrester (i960) than with the more usual biological simulation models, in that the behaviour of the system as a whole and the values of the predictions were more of Interest than the underlying principles. General object-ives have been stated in Chapter one but examples of more specific objectives for the use of the model are given in Chapter six. Chapter three introduces the concept of black box design and discusses the various components of the forest system in the light of the objectives of Chapter one. In Chapter four the construction of the model is described, with the estimation and evaluation of the individual parameters. 23 The model was tested against actual sample plot data and run f a i r l y extensively. The discussion and results of the valid-ation procedure of the model are presented in Chapters five and six. 24 CHAPTER 3 THE FORMULATION OF THE MODEL 3 .1 The Level of Resolution of the Model In order to model a system It Is necessary to examine it s components and how they interact. It is possible to treat each component as a system and thus progressively break down the original system into Infinite d e t a i l . However such a procedure would either be too costly or impossible in the light of current knowledge and i t is necessary to stop at some level of resolution. The unknown details are grouped together into "black boxes" and the system examined as the Interactions of these black boxes. A black box can be defined provided that Its output is predictable from i t s input and i t has s t a b i l i t y of transformation. To quote an example from Van Court Hare (19&7)• i f a system is a dog wagging i t s t a i l i t is possible to explain this in terms of the nerve structure of the dog. However for the dog's owner the explanation is satisfactory in terms of the dog being offered a bone. It may be possible to refine this black box by correlating the amplitude of the t a i l wag with the length of the bone. This would be valid provided the relationship held under a l l conditions of use. A criticism of recent work, especially with multiple regressions, is that the functions used are not biologically meaningful. This sometimes appears to have been defined as "my functions are, his aren't." The black box concept circum-vents this problem provided a l l the components and inter-actions of the system are present in the model. To explain individual tree growth biologically i t is necessary to define ion exchange in the s o i l , water uptake by the roots, absorb-tion of li g h t by the leaves, etc., etc. However a reasonable stand model may be defined in terms of tree size, stand density and site index. Because of the relatively low level of resolution, linear regressions can be used to estimate the individual black box transformations. Regressions may be validly c r i t i c i s e d where not a l l the factors affecting the dependent variable have been specified, e.g. following elim-ination of non-significant variables due to high correlation between the independent variables. While the model may pre-dict the result correctly, within the domain of the data, the wrong Interpretation may be placed on any analysis where the inputs are systematically varied. For a discussion on the dangers of these hidden or latent variables see Box's (1966) a r t i c l e on the use and abuse of regression. Thus not only is i t important to attempt to Include a l l the components or the black boxes of a system In a model, i t is also important to ensure that a l l the variables affecting the output of a black box are represented. The level of resolution and the transformations used must be compatible with the primary objective of the thesis stated in Chapter one, yet not be so fine as to seriously Increase the execution time, or to make estimation of the 26 relevant parameters very d i f f i c u l t . The length of time re-quired for the distance dependent models precluded this approach. As information on product mix is desired and prices vary with tree size, a stand table of the main crop and any thinnings must be predicted. Hence a distance independent model with the tree as the basic unit was required. 3.2 The Components of the System Figure 3.1 shows the broad relationships of the systems model. The noticeable feature is that despite the Thinning Spacing Natural Regeneration Planting The Individual Tree Stan<* Density (Spatial Distribution) Competition Class Dbh Height Volume Mortality Site Improvement z Site F e r t i l i t y Climate Figure 3.1 The Influence Diagram of Stand Growth. 2? simplification there are a large number of interactions. Root spread and crown development are not e x p l i c i t l y shown other than interpreted from the competition class, which is an attempt to express the competitive status of the tree. Bole shape is the sum total of radial growth along the bole. Site f e r t i l i t y must of necessity include such factors as rooting depth, drainage, etc. as well as the nutritive status of the s i t e . The effect of climate is very marked but very unpredictable. Tree growth responds more to natural changes in the sap flow and therefore environmental water stress than to any other normal perennial factor in a forest (Zahner, 1 9 6 8 ) . This has long been recognized In dendrochronology which associates narrow diameter rings with droughts in semi-arid regions ( F r i t t s , 1 9 7 0 ) . In good sites specific limiting conditions are more d i f f i c u l t to distinguish and are probably caused by several factors alternating in their severity. Even with Douglas f i r in coastal B.C., growth reacts strongly with r a i n f a l l and climate in general (see G r i f f i t h , i 9 6 0 ) . The v a r i a b i l i t y of mortality and gross growth per year is shown by figure 3.2. The two plots shown are of pure, even-aged Douglas f i r In the PNW. Both plots are one acre in size. Smith (1971) suggested that mortality appears in waves pos-sibly triggered by climatic stress. A more detailed breakdown of the stand components is given in figure 3 « 3 « Several of the Interactions are implicit, e.g. change in site affects height growth, dbh growth and form factor which in turn affect volume increment. Q> O CD Q. 7-6-5-4-Plot I-Mortality cr p. ° i < o 0 CO o m '50 60 ~70 80* Age 4 i o 3 C C < .2 3^ O Q. *9o" Plot 2 -100 Gross Growth Mortality 0 50 60 i » 70 80 Age Mortality •Plot 2 Figure 3-2- The Variability of Periodic Annual Mortality and Growth-29 Levels Changes a f f e c t of Measures of Density and Size i n Volume . B.A. Stems Av. Dbh B cn • e rH < <D o • •p > CO > < < •p E O Q CD -P • H CO O -P O B O fe c o •p zt JO •p CO •3 .c as •ri •P a! co X X X X X X X X X X \ X X X X X X Development, Real and B i o l o g i c a l Age Dom. Height X X X X \ X X X X x x x x X. X X X Inherent f e r t i l i t y l i te x x x x Stem Shape form f a c t o r Stand Dbh d i s t r i b u t i o n Structure S p a t i a l X X X X X X X X X X v x 'x. ? = component only affected at extreme values. Figure 3 » 3 Stand Component Interaction Matrix. Radial growth takes place over the whole growing season and is directly related to conditions in the current season and to a lesser extent to those of the previous season. Height growth is dependent on the conditions of the previous season (Kozlowski, 1962). The radial growth of an individual tree is strongly dependent on the amount of space available to the tree, but the pattern of response varies between species, most markedly between light demanders and tolerant species. For example, with Pinus sylvestrls Linnaeus, the dbh growth of the top height trees (the largest 40 trees per acre) is f a i r l y independent of stand density, but this is not so with Plcea  ables (Linnaeus) Karst. (see Mackenzie, 1962) or Douglas f i r (see Curtis and Reukema, 1970). However for natural stands some of the largest trees may be almost free growing, at least for part of their l i v e s . This has considerable importance when the distribution of growth amongst the trees In a stand is carried out by the model. The relationship of radial growth along the stem varies, the region of maximum growth being at the widest part of the crown. Growth may fade away to zero at breast height (bh) in suppressed trees, but remain constant down the stem increasing at the butt for open grown trees (Duff and Nolan, 1 9 5 3 ) * Thinning can greatly Increase radial growth in the lower part of the stem without a corresponding increase in the upper portions (see Assman 1970, Farrar, 1961). Disregarding this can lead to over-enthusiasm for the volume growth properties of open grown or heavily thinned stands. Thus, ideally, an expression of form should be responsive to changes in the density of the stand. In summary, given a certain amount of photosynthate available to a tree, demands w i l l be made f i r s t l y for basic metabolism, then for storage and subsequent height growth, thirdly for upper stem radial growth and f i n a l l y for growth at b.h. Priorities for root and crown growth are uncertain, but w i l l probably be similar to height growth (Lyr and Hoffman, 1967). A r e a l i s t i c predictive model would have the a b i l i t y to produce different patterns of growth at different sites. It would respond to changes in density varying in i t s response with age and size of crop and tree. The following functions are required i 1. Individual tree volume, inside bark. (Total and merchantable to several u t i l i z a t i o n standards.) 2. Change in form and taper. 3« Height growth (either average or dominant). 4. Average growth at breast height (gross - before mortality). 5 . Mortality per acre. 6. A method to distribute growth at breast height amongst individual trees. 7. A method to distribute mortality amongst individual trees. 8 . A method of generating a r t i f i c i a l stands at some reference age. 3.3 The Justification of a Stochastic Component 32 The decision to Include stochastic varlates in a model Is not an easy one. To quote Walters and Bunnell (l97l) "Random errors In output data are l i k e l y to produce only additional confusion. It Is not the purpose of gaming to remind us of the normal state of a f f a i r s . " Undoubtedly the addition of randomness solely to terminal variables means l i t t l e in most models. Babb and Eisgrubber (1966) discussed the effects of too much va r i a b i l i t y on game players conclud-ing that i t makes evaluation of performances more d i f f i c u l t . However randomness may be a v i t a l part of the system. Con-sider a service f a c i l i t y . The expected service time may be shorter than the expected interarrival time but clearly this does not mean that there w i l l be no queues i f the system is stochastic. Similarly i f a biological system has a domain of s t a b i l i t y , the average pattern of events may indicate that the system is always stable, but the chance event may pert-urbate the system outside this domain. For example an over-dense stand may go into stagnation, but after a chance heavy mortality, may resume growth. Infrequent heavy mortality may interact with the growth function so that gross stand growth Is reduced below the average, whereas light mortality may not increase gross growth to compensate. Watt (1968) con-sidered the use of stochastic models in biology. He quoted Leslie (1958) to conclude that with large populations, deter-ministic models gave almost the same predictions of future 33 population size as do stochastic models. This does not infer that there w i l l be no effect on the future values of other variables of interest. Finally randomness may be Introduced solely for expediency. When an event has a certain probab-i l i t y of occurrence, generating a random variable can be used as a technique to decide whether or not the event should occur. Once a stochastic model has been created, there are considerably more problems in its analysis than i f i t were deterministic. Each result must be considered as one sample unit, and many runs made to obtain the average value within a desired confidence interval. Variances may be quite high and the model may be 20, 40 or 100 times more expensive to run. The problem of experimental design must be considered (see Naylor et a l . 1966, Naylor 1968). When sequences of sample units are taken in one run, e.g. annual profit over 5 0 years from a forest estate model, the problem of auto-correlation must be considered (Fishman and Kiviat, 1967)» Similarly the model may be influenced by the i n i t i a l starting conditions and the f i r s t samples must be discarded u n t i l steady state has been achieved (Conway, 1963)» In order to reduce computing costs one can use variance reducing tech-niques such as blocking by using the same pseudorandom number string for different treatments, or antithetic random va r i -ates (Kamraersley and Handscomb, 1964). In the previous section the considerable variation due to climate was noted, particularly with mortality. There is 34 a strong possibility that Interactions with stochastic vari-ables exist, especially when thinning is to be carried out. It was decided to include the effect of random variations when determining the annual mortality. Although growth is also subject to the same degree of va r i a b i l i t y , l i t t l e is known about the correlation between the va r i a b i l i t y of growth and that of mortality. Perhaps stochastic mortality w i l l explain most of the va r i a b i l i t y of growth. Also there is less chance of the f i n a l values being affected by random fluctuations in growth. Accordingly randomness in this study was restricted to calculations of mortality. 3.4 Several Approaches to the Prediction of Stand Growth The study of the growth of trees has received consider-able attention over a long period of time. No aspect of forestry has been as well mechanized as Czarnowski's "cemetery of numbers." Since the advent of the computer and convenient s t a t i s t i c a l packages, there have been a multitude of short term studies with limited data over restricted con-ditions. Comprehensive theories of growth are not lacking however, e.g. Czamowski (l96l), Crane (1962), Tadakl (1964), Gray (1966), Pienaar (1965) and Ando (1968). Usually they have not been subsequently refined and adapted for practical use * Several different approaches to stand development were examined a l l of which could have been suitable to form the 35 basis of a simulation model. Markov chains Markov chains and stand table projection are closely a l l i e d ; stand table projection can be formulated in terms of a Markov chain. If ^ Is a vector at time t whose elements x^ are the number of trees in dbh class 1, then "T^ is a stand table. Given the dbh increment Ad^ for each class of width w, the proportion of live trees moving one class is Ad^/w. If m^  is the probability of a tree in class i dying at time t, the number of trees moving up to class i+l is (l - m^ J.Adj^ /w . x^ By taking class 0 as dead trees, the probability transition matrix can be represented ast 1 0 0 0 0 . ni1 (l-n^Ml- dt/w) (l-n^) dx/w 0 m2 0 (l-m 2)(l- d2/w) (l-m 2) d2/w 0 0 0 etc etc . . . . . . . Thus ¥ T + 1 = T T T . P T The f i r s t order Markovian assumption is that any future value of ^ £ depends solely on the value of TT and is independent of values at times previous to this. In forestry terms, the increment of a stand depends only on its present composition and not on i t s past history. This is an implicit assumption not only of most stand table predictions, but also 36 of many multiple regression models. Moreover If the Indiv-idual transition probabilities are independent of time, i.e. P t = p Vt, then the process is called a stationary Markov chain and has a well developed theory, Karlin (1966), Kemeny and Snell (i960). Such a case would occur with diameter increment functions such as Ad * b D + b^.d, but not when age is included as an independent variable. Obviously station-arity does not apply well to even-aged stands, except over short periods of time. Rudra (1968) used this approach for 4" diameter classes over 8 years, but he found the elements of P t empirically rather than using an increment function as above. In fact this alternative method of determining the elements of P t directly by observing the number of individuals which move, is the method most commonly used in Markov chain theory. Usher (1966) adopted a related approach but his trans-it i o n matrix was a Leslie matrix. It was used to predict a stable structure of an uneven-aged stand (see also Usher, 1969). Suzuki (1967, 1970) developed continuous functions to describe the diameter distributions of a stand. By assuming certain functions to predict change in average diameter and the diameter variance, he showed that the dbh distribution function at any time was either normal or lognormal. The diameter distribution could be treated as a discrete process and the problem became a Markov chain. The average diameter growth function was of the form E>t+l = P + <l*Dt w n e r e D t w a s 37 the a verage dbh a t time t , p and q were c o n s t a n t s . The t r a n s i t i o n m a t r i x was d i f f i c u l t t o d e t e r m i n e . The d e f i c i e n c y o f Markov c h a i n s i s t h a t t he t r a n s i t i o n m a t r i x must be s t a t i o n a r y t o use d e v e l o p e d t h e o r y . However i n even-aged s t a n d s a s i x i n c h t r e e a t age 20 does n o t grow In t h e same way as a s i x i n c h t r e e a t age 50 ' The p r o c e s s c o u l d w e l l a p p l y t o uneven aged s t a n d s . C o m p a t i b l e growth and y i e l d models C l u t t e r (1963) d e f i n e d c o m p a t i b l e models as t h o s e where the y i e l d c o u l d be o b t a i n e d by t h e summation o f t h e p r e d i c t e d i n c r e m e n t s and where t h e growth c o u l d be o b t a i n e d by d i f f e r -e n t i a t i n g t he y i e l d f u n c t i o n . The method proceeds by formu-l a t i n g a y i e l d f u n c t i o n and d i f f e r e n t i a t i n g t h i s w i t h r e s p e c t t o time t o o b t a i n volume i n c r e m e n t as a f u n c t i o n of age, s i t e , b a s a l a r e a and b a s a l a r e a i n c r e m e n t ( C l u t t e r , 1 9 6 3 K B y o b t a i n i n g a f u n c t i o n t o p r e d i c t b a s a l a r e a i n c r e m e n t , and I n t e g r a t i n g t he r e s u l t a n t c o m p o s i t e f u n c t i o n t he p r o j e c t e d volume can be d e r i v e d . C l u t t e r o b t a i n e d 1 Volume y i e l d * InV = b Q + b t.S + b 2 . l n B - / A ]Proj © c t sci volume 1 InV = InV - b x . ( A " 1 -A" 1) + ( b 2 - b 3 , S ) ( l n A - l n A 0 ) -AQ.tbij, + b 5.S - b 6 . l n B 0 ) . ( A - l - A - l ) 3 .A. g r o w t h i B >= B . ( b 1 + b 2.S - l n B ) . A _ 1 where S = s i t e , B = b a s a l a r e a p e r a c r e , A = age, b^ a r e con -s t a n t s d i f f e r e n t f o r each f u n c t i o n , t h e s u b s c r i p t o = i n i t i a l . Buckman (1962) used a s i m i l a r t e c h n i q u e . Net b a s a l a r e a 38 increment was found to be unaffected by density, but volume increment increased with increase i n density. Curtis (196?) described a compatible model f o r Douglas f i r , p r e d i c t i n g gross values. His functions gave Increased growth f o r increase i n density. In these a r t i c l e s the form of the basal area increment function was a r b i t r a r y , the one explaining most v a r i a t i o n of growth being chosen. A l l neglected to account f o r height growth e x p l i c i t l y , allowing s i t e and age variables i n the functions to I m p l i c i t l y predict t h i s com-ponent. Thus i t could be that t h e i r conclusions about the e f f e c t s of density on volume y i e l d were determined as much by the empirical functions used, as by any true b i o l o g i c a l f a c t o r s . I t i s u s e f u l to pose the questions i what a l t e r n -a t i v e hypotheses could be formulated, what basal area and volume increment functions would be necessary to f u l f i l l these hypotheses, does one system of regressions explain more of the empirical v a r i a t i o n than another? Competition density models The Competition Density, C-D, models are the work of a Japanese school. In a series of a r t i c l e s , K l r a et a l . (1953• 1 9 5 4 ) , Ikusima et a l . (1955). Shinozaki and Klra (1956, l96l) developed the basic theory on a g r i c u l t u r a l plants. Origin-a l l y the theory was expressed as, the mean s i z e or weight of a plant, w^ , at time t, i s a function of the number of in d i v i d u a l s per u n i t area, p, or wt = K/ p a 39 where a = m.ln t + nj m, n, K were constants. wt attained a maximum value for some value of p which varied with time. The theory was subsequently developed into l/wt = A.p + B on the assumptions of l o g i s t i c growth, constancy of f i n a l yield and A, B being functions of time. De Wit (i960) obtained the same result by a different approach without the assumption of l o g i s t i c growth. Cooper (l96l) on Ponderosa pine, Plnus  ponderosa Laws., showed results substantiating the C-D effect, as did Black (1963) on Subtarranean clover, Trlfollum subtarr-aneum L. to quote two widely differing studies. Tadaki (1963, 1964) expanded the approach to apply to forests; height, representing the stage of biological development, was substi-tuted for time. Thus l/v = A.p + B where v = average volume per tree, p = number of trees per acre, A, B were functions of height. The relationship would also hold for thinning provided this was from below. Ando et a l . (1968) and Ando (1968) suggested that the coefficients A and B could be estimated from A = a^.H-^*; B = a2.H~*>2. The yield per acre Y could be obtained from l/Y = A + B/p. By defining a f u l l density curve (maximum density) and the course of a stand without thinning, a general stand density control diagram was obtained. Pro-vided thinning was light to moderate and from below, this diagram could be used for stand treatment. The C-D rule appeared a useful approach and was tested. The results and discussion are given in the next chapter. 40 Other growth models Plenaar (1965) u t i l i z e d the Chapman-Richards growth function in his quantitative theory of forest growth. The original function was described by Richards (1959) and is a generalization of Von Bertalanffy's (1957) function. Plenaar suggested that the forest be treated as an organism, whose biomass was most conveniently expressed by liv e basal area per acre. If basal area has an allometric relationship with photosynthetic area, and i f respiration is proportional to basal area then dB = ^.B m - v. B where n = anabolic growth rate, dt v = catabollc growth rate, m = the allometric constant, B = basal area per acre. Considering the stand as a biological unit may lead to d i f f i c u l t i e s , especially with Irregular spacing or widely spaced trees and has been c r i t i c i s e d by Bella (1970). Plenaar obtained good results using the model, both in thinned and natural stands but the coefficients were sensitive to changes in site and i n i t i a l stocking. Practical d i f f i c u l t i e s arise In determining these relationships. Moser and Hall (1967) illustrated some of these d i f f i c u l t i e s when they obtained a value of ^ that was negative. Leary (1970), partially basing his work on the clas s i c a l biomathematlcal theory of Lotka (1928), used non-linear ordinary d i f f e r e n t i a l equations to predict forest growth by size class. The system was interactive, growth in one class was affected by the amount of material in the classes above I t . The coefficients did not change with age and the system would appear best with uneven-aged stands. Clutter and Bennett (19&5) and Lenhart (1970) used beta distributions to describe the dbh distribution through time. The necessary two parameters were obtained as maximum and minimum dbh's from empirical regressions of site, age and trees per acre. It is not clear how the system could be mod-i f i e d to allow thinning. Conclusions A l l the systems examined have some defects which rule out their use in this thesis. Either they are time invariant, or they do not include a l l the interactions specified in figure 3 . 2 . A few of the methods are only in their i n i t i a l stages of development and are too vague arid complex to implement. It was necessary to revert to modelling the system using linear and nonlinear multiple regression techniques to estim-ate the black box transformations. Much work has been done, from Mackinney et a l . (1937) to Vuokila (1965)» which led to the comments at the beginning of this section. Many compli-cated variables have been created, most concerned with measuring that elusive index of stand density, from Reineke (1933) to Curtis ( 1 9 7 0 ) . However i t Is more important to ensure that a l l the components of the interaction matrix are represented than to find some ideal index or function, or to 42 obtain a marginal increase in the precision of one of the components. 3.5 Description of the Data The data used in this thesis were permanent sample plots from MacMlllan Bloedel Ltd. (MB) and the United States Forest Service (USFS). A total of 103 plots were selected in even-aged, unthinned, pure Douglas f i r located over the eastern part of Vancouver Island and in Washington State. The USFS plots consisted of some of the long-term "normal density" plots described by Williamson (1963), with plot sizes of 2 to 1. acre; the Wind River plantation, 4 acre plots, Eversole (1955) and Curtis and Reukema (1969)1 and the Volgt Creek unthinned control plots, 1/5 acre, Worthington et a l . (1962). Average size of MB plots was l/lO acre. A l l the plots were sited in areas that were occupied by trees, that Is there were no large, irregular gaps in the tree cover. Any plots suffer-ing from large-scale mortality were usually abandoned and were not used In this study. Plot data used consisted of the dbh of the individual trees, the age of measurement and the remeasurements and some height measurements. No distinction was made between other species and Douglas f i r as the proportion of the latter was always greater than Q0% by both basal area and stems per acre. Any ingrowth was minimal and was excluded. A certain amount of subjectivity was used in selecting most of the dominant heights used to calculate the average dominant height for site determination as crown class was often unrecorded. Site index, average height of dominant and codominant trees to base 100 years, was calculated using the tables of McArdle et a l . (I96l). Volume of the trees was calculated using the t a r i f number system of Turnbull et a l . (1963, 1965)* A l l increments were calculated as periodic annual increments and assumed to be a function of conditions at the start of the measurement period. Measurement intervals ranged from 3 to 10 years, the majority being 4, 5 or 6 years. A few plots had been period-i c a l l y remeasured for as long as 30 years or more. The sample plots were selected to give the most uniform and extensive coverage of stand conditions, thus some of the data was rejected because of over-representation of particular con-ditions, e.g. only four of the fifteen Voigt Creek plots were used. A total of 23^ increment periods were used from 337 measurements. The s t a t i s t i c s of the data points (increment periods) is given in Table 3*1• The distribution of the increment periods is shown by age and site and by stems per acre and site in figures 3«4 and 3«5« Despite care in selecting the data, there were some weaknesses. Densities less than 200 stems per acre were obtained only from the Williamson (1963) plots, and were biased towards higher ages and sites. There were few measurements with less than 100 stems per acre, whilst those with more than 1200 stems per acre were biased towards lower sites. The very high sites, greater than 165, had no 44 Table 3.1 Statistics of the Data Points. Mean Standard Deviation Minimum Maximum Coefficient of Variation Age 50.5 17.6 21.0 97.0 34.9 Site Index 128.6 28.9 66.0 19^. 22.5 Dominant Heiorht (ft.) 89.0 36.6 26.0 168. 4 l . l Stems per acre 5^3. 358. 104. 1980. 65.9 Average Dbh (inches) 8.86 ^.55 3.02 21.2 51.3 Basal Area per acre (sq.ft.) 181.4 62.0 ^5.3 322. 34.2 A l l s t a t i s t i c s are for the start of the Increment period. There is a total of 234 increment periods from 103 plots. *5 =-I C71 a COa. rvio CE LU LU CD a < E B -a si-X X X i X X3 X X 3 X X X X X X XX X X X X X X X X w x w x X X X X X X XX X X M . X X X X X X w X X X >c X X X X X X X X X . X X X X X X KK X X ^ X X X XX X x x X X X X X X X X X X XX X X X X X X X XX X X X X X X H< * x x x x x x £ X X X X*K X< x X X X X X xac xx x x x K x K XX X X X X X X X X X X X x K xx >** x *x x x x K X _ X X $ X< X< X X X _ x x x x x X XX XX X X X X X X X X X X X X I 1 i 1 1 1 50.0 75.0 100.0 125.0 150.0 175.0 200 SITE INOEX Figure 3-4- Distribution of Data by Increment Periods-Age vs-Site-rv 46 rv in a in. in 'rv. O X w a O <X \ CD z: coin. m i n . rv X X X X X X X X X X X # ~ X w x XX x K x x ** * *x x x X * X X X * X K x K X _ X X X * X X X * J X x * * * * * * x K X x x x *x * * X X 5 * *F*H * ^ * x x * K x * x A x x X X K -tf? X x x »W5 x x X X * X * * >? * K4**** **** X X x*x r 1 1 1 r 1 50.0 75.0 100.0 125.0 150.0 175.0 200 SITE INDEX Figure 3-5- Distribution of Data by Increment Periods-Stems per acre vs- Site-a measurements less than 45 years old. The domain of the data could be said to be a cube of 20 - 80 years, site index of 70 - 165 and densities of 200 - 1200 stems per acre, with local extensions beyond this domain. Unfortunately, there s t i l l remained the almost unavoidable high correlation between independent variables, see Table 3«2. Table 3.2 Correlation Matrix of the Data Points. Age Site Dom. Ht. Stems B.A. Av. Dbh Age 1.00 Site .520 1.00 Dom. Ht. .859 .866 1.00 Stems -.555 -.713 -.710 1.00 B.A. .789 .738 .888 -.472 1.00 Av. Dbh .784 .857 .951 -.753 .803 1.00 48 CHAPTER 4 CONSTRUCTION OF THE MODEL 4.1 Introduction To evaluate the components a "test-bed" model was con-structed. It was extremely crude and was constantly Improved as new components were added. This was a cyclic procedure, as one component was improved i t revealed defects in a com-ponent previously satisfactory at a lower level of precision. Several of the functions described below were the results of the second or third cycle. In a l l cases i t was deemed more important to include a l l the interactions of Figure 3 « 3 than to obtain a marginal improvement in an R2 value. Care had to be taken not to implicitly include an interaction effect twice. The model consists of three phases» stand i n i t i a l i z a t i o n where sample plot data is read into the model or where a r t i -f i c i a l data is generated; growth of a stand; thinning inter-ventions and output. Individual tree mortality and dbh growth are the two fundamental components of the growth phase. A stand is grown to any age by incrementing over short (six years or less) time periods, using the predictions of one period as the input for the next. An over-view of the major steps in the growth phase is shown in Figure 4.1. The des-cription and discussion of the components is given in the subsequent sections. 49 Loop over increment periods of A A veers Age of stand = t years Dominant Height = f(Age, Site Index) Expected Mortality (stems per acre) = f(Age, Site Index, Basal Area, Stems) Add Random Mortality per acre Find individual tree probabilities of mortality Loop over trees to determine liv e or dead 1 Average dbh increment (AD) = f(Site Index, Dominant Height, Av. dbh, Basal Area, Stems per acre) Smallest tree growing (ZZ) = f(Age, Site Index) Loop over l i v e trees * Tree dbh increment Ad = f (AD, ZZ, Av. dbh, tree dbh) Add A d to tree dbh J Calculate stand s t a t i s t i c s at Age t + A A Figure 4.1 Flow Diagram of the Calculations of Stand Growth 50 4.2 Site Index and Dominant Height The site index used in the model was that of McArdle et a l . (l96l), the average height of the dominant and codom-lnant trees to base 100 years. The dominant and codomlnant height - age curves were anamorphic, King (1966) proposed new polymorphic curves and demonstrated that the shape of the McArdle e_t a l . (1961) curves were erroneous at younger ages. However King based his sample of site index trees on the largest 20% of the standing trees, a proposal suggested by Weise as long ago as 1880 and dismissed by Assman in a terse sentence (Assman 1970, p. l44). Clearly the values obtained would be from a continuously decreasing sub-population and the very act of a low thinning would be to raise the apparent site index. A subsequent examination of Kind's work revealed that the majority of the samples were taken from stands of about 250 trees per acre and hence his trees were more nearly in accordance with European practise of defining site on the basis of the largest 100 trees per hectare. If the site sample trees were redefined this way, King's site indices would be much superior to those of McArdle et a l . (1961) used in this thesis. One advantage of McArdle et a l . (1961) was that there was a direct comparison between results predicted by the model and their yield tables, especially invaluable in the early stages of testing. Because the curves used were anamorphic, once one height-age curve was known for one sit e , the height - age curve for 51 any other site could be obtained from Height! = Site! . Height2 . S i t e 2 Values of dominant height for 10 year age intervals were obtained from McArdle et a l . (l96l) for site 130, and Aitkin's Interpolation method used in the model to obtain the height for any desired age. A polynomial of degree three was used. For a description of Aitkin's method, see Appendix 1. 4.3 Volume Calculations If volume equations such as V = a.DD.Hc (e.g. Browne, 1962) or V = a + b.D2.H (Spurr, 1952) were used, i t would be necessary to estimate the height of the individual trees within a stand or alternatively the average height weighted by basal area (Lorey's height). Spurr (1952) clearly warned against estimating volume over extensive ranges of age and site by using basal area alone, or by using one height -diameter equation. Newnham (1964) proposed the regression H = bo + bj^.D + b 2.D 2 + b^.B where H = individual tree height in feet, D = dbh in inches, B = basal area per acre in square feet and were regression coefficients. This was used by Lee (1967). It has the unfortunate property when used in a model, that after a thinning the residual trees would be reduced in height by 1.6 feet for every 10 square feet of basal area removed (for Douglas f i r ) . To define an integrated tree height and dbh growth system 52 would be too r e f i n e d f o r t h i s model. One s o l u t i o n would be t o o b t a i n volume equations as a f u n c t i o n of dominant h e i g h t . The problem was circumvented by u s i n g the comprehensive t a r i f t a b l e s of T u m b u l l et a l . (1963, 1965). With these i t was p o s s i b l e t o obtain not only t o t a l volume ( i n c l u d i n g top and stump), but a l s o merchantable volumes such as c l o s e u t i l i z -a t i o n volume to a 4-lnch top and intermediate u t i l i z a t i o n volume to an 8-inch top ( a l l volumes i n s i d e b a r k ) . T a r i f number i s d e f i n e d as the cubic f o o t volume to a 4-inch top of a t r e e w i t h a b a s a l area of one square f o o t . A l l the volume / b a s a l area l i n e s have a common i n t e r c e p t of .087 sq. f t . b a s a l area, hence once the t a r i f number ap p r o p r i a t e t o a stand i s known, the volume of any t r e e can be found from the r e -l a t i o n s h i p c v ^ = T ,(B - .087) (4.1) (1. - .087) where CV4 = Volume Inside bark to a 4-inch top (cu. f t . ) , B = b a s a l area of s i n g l e t r e e (sq. f t . ) and T = t a r i f number. Moreover i t has been found t h a t there i s a s t r o n g r e l a t i o n s h i p between the t a r i f number and the dominant height of a stand, F i n c h (1962). Thus knowing the s i t e Index and the age of a stand, the dominant height can be p r e d i c t e d from the dominant height - s i t e - age r e l a t i o n s h i p , the t a r i f number obtained from a t a r i f number - dominant height r e -l a t i o n s h i p and the l o c a l volume t a b l e then d e f i n e d by the t a r i f number. The t a r i f number f o r each remeasurement of the data was 53 calculated using the method suggested by Tumbull et a l . ( 1 9 6 5 ) » For each height sample tree with a dbh greater than 4 . 5 Inches, the volume to a 4-inch top was estimated using the tables of Browne ( 1 9 6 2 ) . Tarif numbers were calculated using equation 4 , 1 and averaged to give a value for the stand. They ranged from a value of 16.4 to one of 5 3 * 3 . with a mean of 3 3 . 0 6 and Standard Deviation of 8 . 1 . The plot of t a r i f number against dominant height is shown in figure 4 . 2 . It is quite linear. The regression obtained was T = 12.613 + .23222 . H s y x * = 2.340 R2 = .92 where T = t a r i f number, H is dominant height of the stand ( f t . ) . The t a r i f number system, from Turnbull et a l . ( 1 9 6 5 ) . is given below. Defining CVTS = total volume i.b. to a 4 -inch top, cu. ft.} CV8 = volume i.b. to an 8-inch top, cu. f t . Let a = T . 0 . 0 8 7 ; RTS4 = CVTS + 2.a ; R84 = CV8 0.913 c v l* + 2 .a CV4 then R T S 4 = 1 . 0 3 7 8 + 1.4967 . 0 . 0 i 3 4 D / l ° ( 4 . 2 ) R 8 4 = 0 . 9 8 3 . ( 1 . - 0 . 6 5 ( D ~ 8 , 6 ) ) ( 4 . 3 ) Hence given the t a r i f number of a stand, a l l required volumes can be obtained for a tree of dbh D inches. The coefficients of the volume equations were rearranged and * s V Y 1 Standard Error of the Regression surface. 54 a 2 - | 1 1 1 1 1 1 25.0 50.0 75.0 100.0 125.0 150.0 175 DOMINANT HEIGHT (FEET) Figure 4-2- Plot of Average Stand Tarif Number against Dominant Height-simplified for fast computer execution. Because of the form of the total volume function, i t s average stand form factor changed between stands of diffe r i n g densities. The individual tree form-height changed within a stand, as was desired. However the system is such that trees of the same dbh, age and site have the same volume irrespec-tive of stand densities. 4.4 T r i a l of Klra's C - D Rule From section 3.3 Klra's Competition - Density (C - D) rule when applied to average dbh as a measure of tree size, was 1^ = A . N + B where A = b^ . H b 2 ; B = bj . Hb^ ; D = average stand dbh (inches): H = dominant height (feet); N = number of trees per acre. Non-linear regression was used to f i t the inverse of the model to the data, i.e. D = 1' (4.4) A . N + B The values of the coefficients were = 0.33149 . 10"N b 2 = 0.2750?; b 3 = 12.652; b^ = -1.1677 S y x = 0.7143 inches. R2 = 0.98 with 344 d.f. This was an excellent f i t and the model was then tested for i t s a b i l i t y to predict net growth. Given i n i t i a l con-ditions, future values of stem number and dominant height 56 from the data were used to obtain a prediction of future average dbh of the stand and hence annual growth. Residuals were obtained from actual growth - predicted growth. An R2 value of .80 was obtained, a residual standard error (S.E.) of 0.0840 inches per year from a mean annual net dbh increm-ent of 0.17 inches per year. The S.E. was 46$ of the mean. The function was evaluated and is shown in Table 4.1. The most noticeable feature was the increase in dbh due to a decrease in stems per acre, while an increase in dominant height had a smaller effect. An explicit assumption of the model was independence of the path of development. For example a stand thinned regularly throughout i t s l i f e would have the same average dbh as one l e f t untended un t i l near the end when i t was reduced to the same numbers of stems per acre. It was f e l t that the C - D model was too formally structured for a highly dynamic system. A l l thinnings had to be light and from below. The assumption that decreasing the number of stems by thinning had the same effect on average dbh as had that by mortality was too r i g i d , preventing any crown thinnings or non-selective spacings. Although the C - D model was not used in the growth part of the model, i t s value for predicting the average tree size for a given level of density at a given dominant height made i t invaluable in the stand generation routine. 57 TA3L H 4 . 1 THE KIR A C - D FUNCTION TO PREDICT AVERAGE DS H DO MINA NT NO . STEMS PER ACRE HEIGHT 100 200 400 600 800 1000 15C0 2000 2 500 (FT.) AVERAGE DEH (IlJi CHES) 25 3. 3 3. 2 3. 1 2. 9 2.8 2. 7 2. 4 2. 2 2. 0 30 4. 1 3.9 3.7 3. 5 3.3 3. 1 2.7 2. 5 2.2 35 4. 8 4.6 4. 3 4. 0 3.7 3. 5 3. 0 2. 7 2.4 4 0 5. 6 5.3 4.8 4. 4 4. 1 3. 8 3. 2 2. 8 2.5 45 6. 3 6. 0 5. 4 4. 9 4.5 4. 1 3. 4 3.0 2. 6 5 0 7. 1 6.6 5.9 5. 3 4.8 4. 4 3. 6 3. 1 2. 7 55 7. 8 7. 3 6. 3 5. 6 5. 1 4. 6 3. 7 3. 1 2. 7 60 8. 6 7.9 6.8 6. 0 5. 3 4. 8 3. 8 3. 2 2. 8 65 9. 3 8. 5 7. 2 6. 3 5.5 5.0 3. 9 3. 3 2. 8 70 10. 1 9. 1 7. 6 6. 5 5.7 5. 1 4. 0 3. 3 2. 8 75 10. 8 9. 7 8.0 6. 8 5.9 5.2 4. 1 3. 3 2. 8 80 11.5 1 0. 2 8. 3 7. 0 6. 1 5. 3 4. 1 3. 4 2. 3 85 12. 2 10. 7 8. 6 7. 2 6.2 5. 4 4. 2 3. 4 2. 8 90 12. Q 11.2 8.9 7. 4 6. 3 5. 5 4. 2 3. 4 2. 8 95 1 3. 6 1 1.7 9. 2 7. 6 6.4 5. 6 4. 2 3. 4 2. 8 100 14.2 12. 2 9.5 7. 7 6.5 5. 7 4. 2 3. 4 2. 8 105 14. 9 1 2. 6 9.7 7. 9 6.6 5. 7 4.3 3. 4 2. 8 1 1 0 1 5. 5 13. 1 9.9 8. 0 6.7 5.8 4. 3 3. 4 2. 8 115 16. 1 1 3. 5 10. 1 8. 1 6.8 5. 8 4. 3 3. 4 2. 8 120 1 6. 8 13.9 10. 3 8. 2 6.8 5. 8 u . 3 3. 4 2. 8 125 17. 4 14. 3 10. 5 8. 3 6.9 5.9 4.3 3. 4 2. 8 13 0 17.9 14.6 10. 7 8. 4 6.9 5.9 4. 3 3. 4 2. 8 135 18. 5 1 5. 0 10.8 8. 5 7. 0 5.9 4. 3 3. 4 2. 8 14'0 19. 1 15.3 11.0 8. 5 7.0 5.9 4. 3 3. 3 2. e 145 19. 6 15. 6 1 1. 1 8. 6 7. 0 5.9 4. 3 3. 3 2. 7 150 20. 2 15. 9 1 1.2 8. 6 7.0 5.9 4. 3 3. 3 2. 7 4.5 Stand G e n e r a t i o n Stand g e n e r a t i o n f a l l s i n t o two p a r t s : p r e d i c t i o n of the average s t a n d dbh; p r e d i c t i o n of the d i s t r i b u t i o n o f the dbh's of the t r e e s . E f f o r t was c o n c e n t r a t e d on ages 20 t o 25 and i t was assumed t h a t t h e s i t e i n d e x and numbers of t r e e s would be g i v e n . The problem of r e g e n e r a t i o n and s u r v i v a l i n the e a r l y ages was l e f t t o a n o t h e r s t u d y . The K i r a C - D model was used t o p r e d i c t the average dbh of the s t a n d a f t e r 53 dominant h e i g h t was o b t a i n e d from the r e l a t i o n s h i p of s e c t i o n 4.2, I n t h e few t e s t s performed t h i s a p p r o a c h performed r e a s o n a b l y w e l l even i n p l a n t e d s t a n d s , though the aver a g e dbh appeared t o be u n d e r p r e d l c t e d by about 0.2 i n c h e s (see f o r example, T a b l e 4.2). The problem of g e n e r a t i n g a dbh d i s t r i b u t i o n has r e c e i v e d l i t t l e c o n s i d e r a t i o n i n the l i t e r a t u r e , a p a r t f rom C l u t t e r and B e n n e t t (1965) and L e n h a r t (1970) who used t h e B e t a d i s t r i b -u t i o n . Newnham (1968) and Newnham and M a l o l e y (1970) gener-a t e d p o p u l a t i o n s of t r e e l o c a t i o n s t o w h i c h were a t t a c h e d dbh's based on the p o l y g o n a r e a of occupancy from Brown (1965) i b u t t h i s method i s not r e l e v a n t h e r e . The dbh d i s t r i b u t i o n has been v a r i o u s l y d e s c r i b e d by the normal d i s t r i b u t i o n ( L e e , 1967), the gamma ( N e l s o n , 1964) o r the t h r e e p arameter l o g -normal ( B l i s s and R e i n k e r , 1964). Myer (1930) used C h a r l l e r ' s A and 3 d i s t r i b u t i o n s , b u t t h e r e was i n s u f f i c i e n t d a t a a t the young ages t o t e s t t h i s a p p r o a c h w h i c h r e q u i r e d the e s t i m a t i o n of f o u r p a r a m e t e r s . E i g h t p l o t s were a v a i l a b l e w i t h ages between 20 and 25 y e a r s . Two of t h e s e were from p l a n t a t i o n s . T h e i r s i t e , number of stems p e r a c r e and average dbh a r e g i v e n i n T a b l e 4.2, a l o n g w i t h t h e i r p r e d i c t e d average dbh from the K l r a C -D model. The dbh d i s t r i b u t i o n was t e s t e d a g a i n s t the n o r m a l , l o g -normal and gamma d i s t r i b u t i o n s u s i n g a C h i - s q u a r e d (X ) t e s t , e s t i m a t i n g the parameters from the p l o t d a t a . One d i f f i -c u l t y was the low number of de g r e e s o f freedom a v a i l a b l e when 59 T a b l e 4.2 P l o t Data f o r t h e Stand G e n e r a t i o n Component. N a t u r a l > Stands P l a n t a t i o n s S i t e 70 85 115 125 140 145 105 120 Stems p e r a c r e 630 610 760 770 900 730 580 4i5 Av. dbh i n . 3.6 3.7 4.6 4.7 4.8 4.7 3.3 4.5 P r e d i c t e d Av. dbh i n . 3.0 3.6 4.1 4.5 4.3 4.5 3.5 4.2 the range of the dbh's was d i v i d e d i n t o e q u a l i n t e r v a l s . A programme t o p e r f o r m the C h i - s q u a r e d t e s t was w r i t t e n w h i c h • maximized the degrees of freedom s u b j e c t t o the c o n s t r a i n t t h a t t h e ex p e c t e d number i n each c l a s s was g r e a t e r t h a n 5'0. A b i s e c t i o n s e a r c h p r o c e d u r e was u s e d . T h i s gave p a r t i c u l a r l y s e n s i t i v e C h i - s q u a r e d t e s t s , e s p e c i a l l y when t h e t r u e d i s t r i b -u t i o n was d i s c r e t e . The aver a g e d . f . a v a i l a b l e were from 6 t o l4 as opposed t o from 2 t o 6 u s i n g e q u a l c l a s s s i z e s . The normal d i s t r i b u t i o n was r e j e c t e d on a l l b u t one p l o t . T a b l e 4,3 g i v e s t h e r e s u l t s f o r t h e l o g n o r m a l and gamma d i s -t r i b u t i o n s . The two p l a n t a t i o n s were s i g n i f i c a n t l y d i f f e r e n t from b o t h the l o g n o r m a l and t h e gamma d i s t r i b u t i o n s , b u t n a t u r a l s t a n d s a t t h i s age c o u l d be a p p r o x i m a t e d by t h e l o g n o r m a l d i s t r i b u t i o n i n most c a s e s . However, t h e a s s u m p t i o n o f a l o g n o r m a l d i s t r i b u t i o n was n o t r o b u s t enough when the s t a n d a r d d e v i a t i o n and the mean were n o t e s t i m a t e d from t h e o r i g i n a l d a t a . I t f a i l e d t o a b s o r b t h e a d d i t i o n a l e r r o r s imposed by 60 Table 4.3 Xc Tests on the Dbh Distributions of 8 Plots at Ages 20 - 25. Number of Plots Non Significant Significant at $.% at 5% at 1% Lognormal 4 2 2 Gamma 1 4 3 estimating the average dbh from the Kira C - D model. An alternative approach using an empirical distribution was taken. It was assumed that there existed a transformation which when used reduced a l l the individual dbh distributions to a standard probability density function (p.d.f.), that could be estimated empirically. The standardized normal distribution transformation was the obvious i n i t i a l choice, Z = dj - D _ i where d^ Is the individual tree dbh, D = average dbh of stand, s = standard deviation. In view of the small amount of data a constant coefficient of variation (k) had to be assumed. Thus Z' = d^ - D where Z' = Z.k (4.5) 13 A method of generating an empirical p.d.f. is to estimate the inverse cumulative density function (c.d.f.) using a fourth or higher degree polynomial with additional terms such as ln(u) and l n ( l . ~ u ) to estimate lower and upper t a l i s of the distribution i f these extend to i n f i n i t y . By generating uni-form ( 0 , l ) random variates and substituting these Into the polynomial, random variables from the empirical p.d.f. can be generated (see Naylor et a l . , 1966 or Schmidt and Taylor, 6i 1970). In this thesis the term l . / ( l . - Tu+O.lE-05)* was used in place of the logarithmic terms as i t was found to be very-effective in the immediate neighbourhood of u = 1.0. For each plot, the dbh's were ordered in ascending size, the values of Z* calculated and the cumulative probabilities estimated. Three empirical inverse c.d.f.'s were obtained, for natural stands, plantations and for a l l stands. A l l the data were pooled within each group and non-significant terms in the regressions eliminated. The results are given in Table 4.4. Table 4.4 Regression Coefficients for Inverse Cumulative Density Functions of Dbh Distributions. No. of Plots bQ b^ b 2 b^ bj^ , b^ Natural 6 -.38811 N.S. 3.7248 -7.2762 4.7739 .17271E-05* Planted 2 -.58309 2.0934 -2.0994 N.S. 1.1527 .83877E-06 A l l 8 -.48077 1.1403 N.S. -2.2343 2.2995 .19351^ -05 Stands b^ are the coefficients in the regression Z' = b 0 + b 1.u + b 2.u 2 + bj.\x^ + b^.u^ + b ^ / ( l . - 7u+.iE-05) (4.6) Rd values were high from 0.94 to O.99. The three c.d.f.'s are illustrated in Figure 4,3, along with a histogram of the p.d.f. for the natural stand regression. The stand generation * .1E-01 = .1 x 10-1 z' ON 6 3 procedure was tested against the original data of the 8 plots. First the average stand dbh was predicted using the Kira C - D model, then the values of Z1 generated by substituting uniform random variates into the regressions of Table 4.4 and f i n a l l y the values of the individual tree dbh's obtained by back substitution into equation 4.5. Table 4.5 shows the detailed Chi-squared test on the dbh generator for a l l stands. The regression did not predict the • two planted stands overly well. Obviously separate dbh gen-erators were required for planted and natural stands as would be expected. Accordingly the function for natural stands al-one was used in the model, which was temporarily restricted to generating natural stands for the purpose of this thesis. The Chi-squared test was similar to Table 4.5 except that no X 2 values were significant at the 1.0$ level for natural stands. Figure 4.4 shows the distributions obtained for various sites and I n i t i a l densities at age 20. The standard deviation changes with the average dbh, with an apparent trend to a normal distribution as the average dbh increases. The maximum and minimum dbh's are about 2x and 0,6x the average dbh. It was f e l t that two plots were not sufficient to e s t i -mate the inverse c.d.f. for plantations with any r e l i a b i l i t y , but the model could easily be extended to generating planted stands given additional data. The model proved to be extremely sensitive to errors in the shape of the i n i t i a l dbh distribution. As an i n i t i a l Table 4.5 Test on the Inverse C.D.F. ( a l l stands) for Dbh Distributions for Eight Plots. Numbers of trees in the plot Site 2 3 4 Dbh 5 inches 6 7 8 9 10 df X 2 71 Real 34 21 6 1 1 2 Sim. 19 28 11 5 10.9 ** 86 Real 3 31 11 11 5 3 Sim. 9.5 21.5 18 8 4 5.7 NS 113 Real 19 21 16 7 12 1 4 Sim. 5-5 20 26.5 13 7.5 3.5 11.2 * 125 Real 17 25 19 6 4 1 2 2 5 Sim. 3 17 25.5 16.5 8.5 6 5 3.1 NS 138 Real 17 30 17 10 11 1 2 2 5 Sim. 5 21 31 17.5 9.5 6 11.1 NS 144 Real 18 18 19 8 3 5 1 1 4 Sim. 2 15-5 23 17 8.5 5.5 1.5 2.5 NS 105 Real 17.5 25 58 25 10 4 Sim. 25 51 34 16 5 46.0 ** 119 Real 7 12 29 26 16 5 5 Sim. 8 25 33 16 8 4 2i.7 ** The last two plots are plantations. NS Not significant * Significant at $% ** Significant at \% 0-8 0-7 0-6 0-5 0-4 0-3 0-2 01 0 0 2000/acre « 2 3 4 5 DBH Site 90 0-4 0-3 0-2 0 - 0 ^ 300/acre 2 3 4 ' 5 6 DBH >0-5 o UJ 0-4 a U J 0-3-1 oc u. 0-2 01 0 0 1600/acre <2 0-4 0-3-1 0-2 01 0 0 3 4 DBH 5 6 1200/acre « 2 3 4 5 DBH Site 120 0-4T 0-3 0-2 01 0 0 300/acre « 2 Site 150 0-4i 0-3 0-2 01 4 5 \ DBH 300/acre « 2 3 4 5 6 DBH 8 8 Figure 4-4- DBH Distribution at Age 20 Generated by the Model-66 approximation while testing the other components, the normal distribution had been used in the test-bed model. 3y age ?0, later in lower sites, annual mortality in terms of volume per acre was very high, although not in number of trees per acre. The standing average dbh was lower than expected. What appeared to be happening was that by these older ages, the trees that were being suppressed were f a i r sized trees with a subsequently large volume. They were originally the large number of average sized trees which for a long time had been growing from adequately to slowly. With the lognormal or the empirical distribution the majority of the trees were smaller than average and volume growth was concentrated on fewer, larger trees not susceptible to mortality even at older ages. 4.6 The Diameter Increment Function The dbh increment of trees within a stand was found to be very variable. Figure 4.5 shows the periodic annual dbh growth plotted against dbh of the Individual tree for four plots of different stands. Paille (1970) attempted to pre-dict individual tree increment as a function of tree size and stand variables. He met with very l i t t l e success, stand density failed to be significant and the regressions had very low precision, clearly due to the high v a r i a b i l i t y within stands masking trends between stands. It was therefore decided to predict average dbh growth of a stand as a function of stand variables and then to distribute this growth amongst S ITE 106,PCS. 21, Sam CTCrtS/flCKE 6? 6 8 the trees within a stand. It was necessary to distinguish between net average dbh . increment of the stand and the average dbh increment of the li v e trees. The former was confounded with the mechanical increase due to mortality in the smaller tree sizes or due to thinning. The li v e tree growth was more appropriate in this study as i t allows the effects of mortality and thinning to be calculated separately. A l l subsequent discussion in this section refers to this l i v e increment. A preliminary plot of average annual dbh increment (AD) revealed considerable variation, even when st r a t i f i e d by s i t e . Regressions involving percentage dbh increment were abandoned when variances were found to be non-homogenous. The correl-ation coefficients of AD with independent stand variables are shown in Table 4 . 6 . The percentage AD varied from 0 . 3 $ to 7.9$ with a mean of 1 . 6 $ . Table 4 . 6 Correlation Coefficients of Average Annual Dbh Increment with Independent Stand Variables. Site Dom. Ht Stems Bsq?^ftTe& A v < Age Index (ft.) per acre p e r acre Inches AD -.508 -.003 -.323 - . 1 7 2 -.501 - . 1 7 3 Mean AD = 0.109 inches per year. Standard Deviation (S.D.) = 0 . 0 4 7 8 Minimum = 0 . 0 2 8 7 " " " • Maximum = 0.272 Inches per year 69 F o l l o w i n g extensive t r i a l s s e v e r a l f u n c t i o n s were obtained, a l l w i t h R 2 values of about ?0% and s y x of about 0.026 inches per year. P l o t t i n g the r e s i d u a l s a g a i n s t pre-d i c t e d values and a g a i n s t Independent v a r i a b l e s revealed no t r e n d s . I t became c l e a r t h a t i n order t o increase the pre-c i s i o n of the r e g r e s s i o n a small amount, a l a r g e number of v a r i a b l e s had t o be i n c l u d e d . This gave f u n c t i o n s which were unstable or erroneous at the borders of the data's domain. The f i n a l r e g r e s s i o n chosen was D = 0.08204 + 0.0014062.S - 0.00l3l3.H + 4.3007/B - 0.l8l42.N.D.l0 -^ (4.7) s y x = 0.2593 inches per year R 2 = 0.71 w i t h 4 and 229 d.f. where S = S i t e Index, H = Dominant Height ( f t . ) , B = B a s a l Area per acre (sq. f t . ) , N = stems per a c r e , D = Average Stand Dbh ( i n c h e s ) . I t was necessary t o impose a n o n - n e g a t i v i t y c o n s t r a i n t on (4.7) though there were i n d i c a t i o n s that t h i s would only apply w e l l outside the data's domain. S i m i l a r l y an upper l i m i t f o r open grown t r e e s was set a t 0.6 inches per year (from G r i f f i t h , i960). The l a c k of an age term was not considered important because stand development was represented by the dominant height term. With the average dbh increment f o r the stand r e a d i l y a v a i l a b l e , i t remained to determine a method t o a l l o c a t e the dbh growth among the i n d i v i d u a l t r e e s . Examination of many p l o t s s i m i l a r to Figure 4.5 revealed t h a t the i n d i v i d u a l t r e e p e r i o d i c annual dbh increment (Ad) was approximately l i n e a r l y 70 correlated to the tree dbh (d) or Ad = b Q + b x.d (4.8) where b G and b^ are coefficients which vary from stand to stand with time. The average dbh increment (AD) can be predicted from equation ( 4 . 7 ) . Hence the point (AD, D) on equation (4.8) is known and only one other point need be obtained to f u l l y de-fine the equation at a particular time. The x-intercept ZZ {=bQ/b^) represents the dbh of the tree with zero increment and appeared an appropriate variable to predict as a function of stand variables. With increase in age, site index and density, the value of ZZ is expected to increase, whereas the reaction of the value of b c is not so obvious. The regression coefficients of equation (4.8) were c a l -culated for each increment period of a sample of plots from the data (103 regressions). A l l but one of the regressions were highly significant. R2 values ranged from 35$ to 75$. indicating the considerable within stand v a r i a b i l i t y . Values of ZZ were calculated and their related stand variables obtained. The st a t i s t i c s for ZZ were Mean S.D. Minimum Maximum ZZ 2.26 2.231 -3.24 8.89 Several regressions were obtained from an analysis of ZZ on stand variables. Some caused the model to perform badly. The f i n a l one chosen was ZZ = -8.7310 + 0.034434.S + 0.24514.A - 0.0017599.A 2 (4.9) where S = site index A = age. s y x = l«'4'3l i n c h e s . = .60 w i t h 3 a n d 99 d . f . A l l c o e f f i c i e n t s i n t h e r e g r e s s i o n w e r e h i g h l y s i g n i f i -c a n t . T h e v a l u e o f ZZ c u l m i n a t e s a t a b o u t a g e 70. S u b s e q u e n t e x a m i n a t i o n o f t h e f u n c t i o n r e v e a l e d t h a t t h i s p a r t i c u l a r s a m p l e o f t h e d a t a was weak a t t h e s e h i g h e r a g e s . T h e f u n c -. t i o n a b o v e a g e 60 was amended b y t a k i n g t h e t a n g e n t t o t h e c u r v e a t a g e 60, T h u s a b o v e a g e 60 Z Z = E q u a t i o n 4.9 e v a l u a t e d a t a g e 60 + (Age-60).0.04 (4.9a) I n summary, t o o b t a i n t h e i n d i v i d u a l t r e e d b h i n c r e m e n t f o r a s t a n d a t a p a r t i c u l a r t i m e , a n d ZZ a r e p r e d i c t e d f r o m t h e e q u a t i o n s (4.7) a n d (4.9). T h e n t r e e g r o w t h i s o b t a i n e d f r o m A d = A D . ( d - Z Z ) (D - Z Z ) I n t e r m s o f t h e c o e f f i c i e n t s o f e q u a t i o n 4.8, t h i s i s b 0 = ZZ.AD ; b 1 = A D ( Z Z - D ) ( D - Z Z ) The f u n c t i o n o f ZZ was more i m p o r t a n t t h a n p r e d i c t i n g t h e s i z e o f t h e s m a l l e s t g r o w i n g t r e e . I t d e t e r m i n e d how f a s t t h e l a r g e t r e e s g r e w i n r e l a t i o n t o t h e s m a l l t r e e s . A s m o r t -a l i t y r e m o v e s t h e s m a l l e r t r e e s , t h e v a l u e o f ZZ u l t i m a t e l y p l a y s a n i m p o r t a n t p a r t i n p r e d i c t i n g t h e s i z e o f t h e c r o p i n l a t e r y e a r s . T h e l i n e a r r e l a t i o n s h i p o f t r e e d b h g r o w t h t o d b h w i t h i n a s t a n d was s u i t a b l e f o r D o u g l a s f i r b e c a u s e o f t h e a b i l i t y o f t h e l a r g e r t r e e s t o r e s p o n d t o i n c r e a s e d g r o w i n g s p a c e , b u t i n some o t h e r s p e c i e s w i t h o u t t h i s a b i l i t y a n u p p e r 72 lim i t or a curvilinear relationship might be required. It should be noted that trees with a dbh smaller than ZZ actually have negative growth. A non-negativity con-straint could easily have been imposed but was not done in order to decrease execution time. The negative growth was small and as such trees had a high probability of mortality the resultant errors were negligible. Moreover in permanent sample plot data some trees have been observed to shrink a small amount some five years before death. 4.7 Mortality Staebler (1953) c l a s s i f i e d mortality as regular or irregular. Irregular mortality was defined as widespread or catastrophic mortality such as wlndblow or f i r e . Regular mortality occurred throughout the l i f e of the crop caused by suppression or disease though even this was highly variable as was suggested in Chapter 3« It was necessary to predict this regular mortality. A l l the data was from permanent sample plots selected to exclude catastrophic mortality. Some of this data had been used by Pallle (1970) to obtain probabilities of Individual tree death which were age and com' petition class dependent. However in his model for a given acre, a tree 3/4 the size of the average dbh, for example, had the same probability of dying in a stand at site 150 with 1000 stems per acre as an equivalent tree In a stand of site 90 with 400 stems per acre. 73 A three stage approach was adopted. 1. Predict the number of trees dying as a function of stand variables. 2. Predict the component due to v a r i a b i l i t y , add this to 1. 3. Allocate mortality among the Individual trees u n t i l the predicted mortality per acre is attained. An attempt to combine a l l three stages using a discrim-inant function failed, because although a significant function was obtained, the low ratio of Number dead combined with the Number l i v i n g high probability of misclasslfication by the discriminant function (20$) rendered the results grossly unreliable. An extensive multiple regression analysis was performed to predict the number of trees dying per acre per year. In order to homogenize the variances as much as possible, mort-a l i t y expressed as percentage of number of trees alive was used as the independent variable. I n i t i a l plotting revealed no trends but 10 values had extremely high mortality (greater than 6% per year). These were omitted when determining the regression coefficients. The correlation between the inde-pendent variables proved very troublesome. A great deal of effort was expended in ensuring that basal area per acre was represented In the regression. The average dbh was deliber-ately excluded to reduce the correlation between the inde-pendent variables. Again several regressions were obtained, a l l with about the same precision. The one selected was M = -.0073544 - 0.0010093.N/A + 0.50128.N.S. +0.25057.B.S.10"6 (4.10) S y x = 0.94397.10-^ R2 = .45 with 3 and 220 d.f. where M = percentage trees dying per year, A = age, N = number of stems per acre, B = basal area per acre (sq. f t . ) , S = site index. A non-negativity constraint was placed on the regression. When re-evaluated in terms of numbers of stems dying, the R2 value was .75. with s y x = 6.5 stems per acre per year com-pared to the original S.D. of l2.8. The residuals (real -predicted) were plotted against the predicted values, Figure 4.6. The line in the lower l e f t comer was caused by the fact that mortality can never be negative. Excepting this, and the largest 10 residuals, the scatter shows a f a i r degree of homoscedaclcity. It was decided to obtain the stochastic component of mortality by randomly selecting one of the residuals of the regression. One method would be to generate a random number normally distributed with mean 0 and standard deviation equal to that of the residual standard error, assuming that the underlying regression hypotheses had been met. This would have been inappropriate here after consideration of Figure 4.6. By dividing the range of the predicted values Into four "risk" classes the distribution of residuals within each class became f a i r l y homogenous. The classes were 0. - 0.67; 0.67 - 1.34} 1.34 - 2.0i, >2.0l percent annual mortality, a o 10 ac o x; a _J CE ZD a • — i tip.. I CLASS I x x x X * X X X X * * * € X X X I X X X X X X « x 1 is: • X X *x * x x | X i * X x X X X X X X X X X X I l " " X X X x x x x X X x * X t. 2. P R E D I C T E D % M O R T A L I T Y Figure 4 - 6 - Plot of Mortality Residuals and the Four "Risk" Classes-T " 4. 1. 7 6 with 30» 5 6 , 8 5 and 6 3 observations respectively, and are shown in Figure 4 . 6 . The assumptions were made that the dis-tribution of the residuals within one class could be approx-imated by one pdf which was related to the predicted value, not to any independent variables. This had been partially tested as the residuals had been plotted against the inde-pendent variables with no observable trends. For each class the empirical inverse cumulative density function was obtained using a polynomial of the form used in section 4 . 5 , equation ( 4 . 6 ) , I.e. X = b 0 + bj^.u + b 2.u 2 + b3«u3 + b^.u^ + b^/d.-Zu + . 1 E - 0 5 ) where X = residual percent mortality, u = P(x<X), b A = re-gression coefficients. A l l non-significant terms were elim-inated. The coefficients are shown in Table 4 . 7 . Table 4 . 7 Coefficients of the Inverse C.D.F.'s of Residual Mortality. Class b 0 b l b 2 b 3 b 4 b 5 1 . O-O.67 - . 0 0 8 6 0 5 2 . 0 5 6 8 1 0 - . 1 7 7 0 3 .25553 -.11808 •19352E-07 2 . . 6 7 - 1 . 3 ^ -.011722 N.S. . 1 2 9 8 8 - . 2 7 3 4 5 . 1 8 3 0 4 - . 1 5 3 6 6 E - 0 8 3 . 1 . 3 4 - 2 . 0 1 -.013395 N.S. .22096 _ . 4 9 4 4 9 . 3 1 9 0 2 •31733E-07 4 . 2 . 0 1 -.0179^ 8 -.064547 . 6 3 6 3 2 - 1 . 2 4 2 9 .76178 .73961E-08 N.S. = Non Significant. Histograms of the distribution of the residuals obtained 7 7 f r o m t h e r e g r e s s i o n s a r e shown i n F i g u r e 4.7. T h e y s u g g e s t t h a t when t h e p r e d i c t e d m o r t a l i t y i s l o w , t h e r e i s l i t t l e v a r i a t i o n o r t h e s t a n d i s i n a " l o w r i s k " s t a t u s . I n t h e f o u r t h c l a s s h o w e v e r , e v e n t h o u g h p r e d i c t e d m o r t a l i t y i s g r e a t e r t h a n 2% p e r y e a r , t h e r e i s a c h a n c e o f h i g h r e s i d u a l m o r t a l i t y b e i n g a d d e d t o t h e p r e d i c t e d m o r t a l i t y , o r t h e s t a n d i s i n a " h i g h r i s k " s t a t u s . I n o r d e r t o o b t a i n t h e a n n u a l p e r c e n t m o r t a l i t y t h e p r e -d i c t e d v a l u e was o b t a i n e d f r o m e q u a t i o n ( 4 . 1 0 ) . T h e n a u n i f o r m r a n d o m number was g e n e r a t e d b e t w e e n 0. - 1. T h i s v a l u e was s u b s t i t u t e d f o r u i n t h e a p p r o p r i a t e i n v e r s e c . d . f . d e p e n d e n t on t h e v a l u e o f t h e p r e d i c t e d m o r t a l i t y . The f i n a l p e r c e n t m o r t a l i t y was t h e n t h e p r e d i c t e d v a l u e p l u s t h e r a n d o m r e s i d u a l m o r t a l i t y . T o i l l u s t r a t e t h e e f f e c t o f t h e r a n d o m r e s i d u a l , a t i m e p a t h f r o m t h e c o m p l e t e d m o d e l f o r a p l o t w i t h l 6 0 0 s t e m s p e r a c r e a t a g e 20, s i t e 120 i s shown i n T a b l e 4.8. L e e ( 1 9 7 1 ) c a l c u l a t e d t h e p e r c e n t a g e m o r t a l i t y f r o m y i e l d t a b l e s o f L o d g e p o l e p i n e a n d d i s t r i b u t e d t h i s a m o n g s t t r e e s o f t h e s t a n d . He a s s u m e d t h a t m o r t a l i t y a m o n g s t d b h i n c h c l a s s e s was n o r m a l l y d i s t r i b u t e d w i t h a mean d b h 2" l e s s t h a n t h e a v e r a g e s t a n d d b h . He c a l c u l a t e d t h e s t a n d a r d d e v i -a t i o n a s a f u n c t i o n o f ( a v e r a g e d b h ) . I n X t e s t s o f t h i s h y p o t h e s i s , o n l y s t a n d s o f 5 - i n c h a v e r a g e d b h h a d a m o r t a l i t y d i s t r i b u t i o n s i g n i f i c a n t l y d i f f e r e n t f r o m n o r m a l , b u t 3 - a n d 4 - l n c h s t a n d s w e r e p o o r l y r e p r e s e n t e d (2 a n d 8 p l o t s o n l y ) a n d h a d few d e g r e e s o f f r e e d o m f o r a s e n s i t i v e X t e s t . I t was d e c i d e d t o u s e a d i f f e r e n t a p p r o a c h u t i l i z i n g t h e 78 0-4 0-3 0-2 01 0 0 Class 1 Predicted Mortality • 1 1 0 0 - 0 - 6 7 % 1 1 1 1 i >-o z L U Z) a u or Class 2 Predicted Mortality 0 6 7 - I 33 % Class 3 Predicted Mortality I 33 - 2 0 % 0-3i 0-2-01-0 0 Class 4 Predicted Mortality > 2 0 % • 2 - 1 0 I 2 3 4 5 6 STOCHASTIC MORTALITY (%) 8 Figure 4-7- Distribution of Random Residuals of Mortality Predicted by the Model for 4 Classes-Table 4.8 The E f f e c t of the Random Residual M o r t a l i t y on a P l o t , S i t e 120, 16OO stems per acre at age 20. Age 20 25 30 35 40 45 50 Stems per acre 1600 1472 1240 1160 608 608 512 C a l c u l a t e d * % M o r t a l i t y .010 .026 .030 .035 .018 .021 .019 " R i s k " C l a s s 2 4 4 4 3 4 3 Random % * M o r t a l i t y .006 .007 -.017 .072 -.012 .009 -.013 T o t a l % * M o r t a l i t y .016 .033 .013 .107 .006 .030 .006 * annual p r o b a b i l i t i e s of i n d i v i d u a l m o r t a l i t y c a l c u l a t e d by P a l l l e (1970). Let E(x) denote the expected number of t r e e s d y i n g w i t h i n the stand, the p r o b a b i l i t y of the death of a t r e e i n the i ' t h c o m p e t i t i o n c l a s s f o r m c l a s s e s , the number of t r e e s i n the i ' t h c o m p e t i t i o n c l a s s . Let R i = Pj^/Pi i = 1,... ,m Then E(x) = E P,.N* Z R* .N. Thus i f E(x) i s assumed t o be the m o r t a l i t y per acre c a l c u l a t e d above, the Rj^ obtained from P a i l l e (1970) and the Nj_ calculated from the stand, the probabilities of death (P^) can be determined. The interpretation of R^  was; the l i k e -lihood of mortality of a tree in the i'th class relative to a tree in the f i r s t class. The competition classes used by Pallle (1970) were quarter classes of the ratio of tree dbh to average stand dbh. He assumed that the Pj, depended solely on age. The i n i t i a l class was taken as 0.25, i.e. C«l25, «375) and the values of R^  calculated for each age. A plot of the R± against age, see Figure 4.8, revealed that the values were almost equal for a l l ages with the exception of age 20, which had been derived from scanty data. As most of the mortality was concentrated in classes 1 -3 the class widths were halved. The additional values of R^  were obtained by interpolation from a smooth curve. The values used in the model are given in Table 4.9. Table 4.9 Relative Probabilities of Mortality by Competition Class for Individual Trees. Tree dbh Av. dbh 1/8 1/4 3/8 1/2 5/8 3/4 7/8 Ri=V pi 1. 1. 1. .9263 .6686 .3755 .1890 Tree dbh Av. dbh 1 9/8 5A 11/8 3/2 13/8 Ri=V Pl .1009 .0640 .0566 .0420 .0274 .0195 0. The assumption that the chances of a tree in one 81 1-0-os-os-0-7-_ 0 - 6 -o_ £-0.5-O Q.4. o 1 0-3-0-2 01 0 0 o 2 0 Dbh/Av Dbh class X 0-5 i = 2 o 0-75 3 • 10 4 A 1-25 5 P- = probability of dying for a tree in i t n class o o o e o & + + • • 2 A 30 4 0 50 Age 60 70 8 0 Figure 4-8- Ratios of Probabilities of Mortality Plotted Against Age-82 com p e t i t i o n c l a s s d y i n g r e l a t i v e t o another i n another c l a s s are the same f o r every age and s i t e must be viewed w i t h some s u s p i c i o n . This a f f e c t s the d i s t r i b u t i o n not the q u a n t i t y of m o r t a l i t y w i t h i n a stand, and must s u f f i c e as a f i r s t approximation i n t h i s model. Once the p r o b a b i l i t i e s of m o r t a l i t y f o r each c l a s s have been determined f o r a stand at a p a r t i c u l a r time, the d e c i s i o n whether an i n d i v i d u a l t r e e should d i e or not i s made by generating a random number. I f the number i s l e s s than the appropriate p r o b a b i l i t y , the t r e e d i e s , i f g r e a t e r the t r e e l i v e s . By ensuring t h a t the number of i n d i v i d u a l t r e e s w i t h i n the simulated p l o t at any one time remains f a i r l y h i g h , the v a r i a b i l i t y due t o t h i s method can be minimized, l e a v i n g the a d d i t i o n of the random m o r t a l i t y as the c h i e f component of v a r i a b i l i t y . 4.8 The S i m u l a t i o n Programme Once the components of the model had been obtained and p a r t i a l l y t e s t e d on the "test-bed" model, w r i t i n g the a c t u a l programme was f a i r l y s t r a i g h t f o r w a r d . The main programme i s a c a l l i n g programme c o n t r o l l i n g the three phases of stand i n i t i a l i z a t i o n , growth and i n t e r v e n t i o n s . The b a s i c u n i t of the s i m u l a t o r i s an arr a y of dbh's r e p r e s e n t i n g the t r e e s on a sample p l o t of known area, s i t e and age. A l l s t a t i s t i c s are then converted t o a per acre b a s i s . The sequence of con-t r o l i s shown by the flow diagram, Figure 4.9. The sub-r o u t i n e s are defined, i n Table 4.10 and the programme l i s t e d 83 Read Plot I n i t i a l i z a t i o n Card 5E Rewind I F i l e _ L U j <(EMDFILE " E L -»{ STOP -^~(jitewind Option"?^-^ S t a n d Generation~?^>ii. C a l l GIGNO to c a l c u l a t e av. dbh & generate dbh l i s t [Read Data Format | C a l l LEGEO to read dbh l i s t C a l l SORT to sort dbh l i s t i n to ascending order Calculate i n i t i a l stand s t a t i s t i c s Write T i t l e s Read Option Card 3E -£<ENDFILE iS Is stand at desired Age «. / j Determine increment period (DELA) C a l l MO?£RE to ca l c u l a t e no. trees dying C a l l NECO to a l l o c a t e m o r t a l i t y C a l l CRE3C0 to update stand to AGE + DELA AGE = AGE + DELAfc-<^No. of trees i n the p l o t S 50 , TS i C a l l T RIPLE -il<^Th inning Required C a l l THIN to remove BA from below or i n s p e c i f i e d amounts from s i z e classes Thinning Stand Table Required ? C a l l METIOR f o r thinnings to c a l c u l a t e t o t a l & merch. volumes & to c a l l SCRIBO to p r i n t a l i n e of stand table ? : Calculate t o t a l volume of Thinnings r-<^Main Crop Stand Table Required C a l l METIOR f o r Main Crop to c a l c u l a t e t o t a l & merch. volumes & to c a l l SCRIEO to p r i n t a l i n e of stand table I Calculate t o t a l volume of Main Crop 3  JL-Write a l i n e of Yield Table Figure 4 - 9 - Abbreviated Flow Diagram of the Simulation Model-84 Table 4.10 D e f i n i t i o n of the Subroutines used i n the Programme. CRESCO - grows the sample p l o t f o r DELA yea r s . (DELA 6 years) GIGNO - generates a stand f o r ages 20 - 25 y e a r s . LEGEO - reads i n sample p l o t d a t a . METIOR - c a l c u l a t e s a stand t a b l e f o r e i t h e r main crop or t h i n n i n g . MORERE - c a l c u l a t e s the number of t r e e s d y i n g i n DELA years. NECO - d i s t r i b u t e s the m o r t a l i t y amongst the t r e e s . SCRIBO - w r i t e s a l i n e of a stand t a b l e . SORT - s o r t s an array i n t o ascending order. THIN - performs a low or low/crown t h i n n i n g . TRIPLE - increases area of p l o t by 3 times. i n Appendix 6. P l o t i n i t i a l i z a t i o n can be e i t h e r by re a d i n g i n a c t u a l data of a sample p l o t a t any age, or by u s i n g the stand generation option d e s c r i b e d i n s e c t i o n 4.5, subroutine GIGNO. The dbh's are then sorted i n t o ascending order by s i z e , subroutine SORT. A l i s t s o r t was used. I t i s f a i r l y f a s t and i s e f f i c i e n t when the arr a y i s already p a r t i a l l y s o r t e d . S o r t i n g speeds execution i n the m o r t a l i t y and t h i n n i n g r o u t i n e s . There are two methods of keeping t r a c k of time i n simu-l a t i o n models; f i x e d time increments or next-event formu-l a t i o n . I t was assumed th a t a f o r e s t e s t a t e model would operate on a f i x e d time Increment of one year because each year at l e a s t some stands would r e q u i r e f e l l i n g , t h i n n i n g , 85 etc. The development of an Individual stand however would be governed by next-event formulation. The stand would be Ignored u n t i l some event concerning i t occurred, e.g. thinning, at which time the stand would be updated (grown) and the Intervention carried out. This would conserve exe-cution time. Accordingly the philosophy of next event formu-lation was used in this model. Following plot i n i t i a l i z a t i o n , an option card is read in, the stand updated to the specified age and then the interventions carried out on the stand. Although a l l the increment functions were on an annual basis, i t would have been erroneous to have evaluated them each year, adding the resultant increment to the stand's values to obtain the input values for the next year. This was because the Increments used in obtaining the regressions were periodic rather than current. Moreover such a procedure would be f a i r l y slow. To adopt a fixed number of years for an increment period would have been too r i g i d . Therefore i t was decided to adopt an average period of five years but allow the period to fluctuate from one to six years i f necessary. Thus i f the next event was scheduled for two years ahead the annual increments were multiplied by two, If the next event was 10 years ahead the stand was updated over two 5 year periods, i f 19 years ahead over three 5 year and one 4 year periods. Although this has the disadvantage of Inconsistency and perhaps could be open to abuse, this is more than outweighed by the time-savings and the f l e x i b i l i t y . Also, the original 86 data i t s e l f was composed of many d i f f e r e n t increment i n t e r v a l s . The number of t r e e s d y i n g i s c a l c u l a t e d a t the b e g i n n i n g of a p e r i o d , subroutine MORERE, the m o r t a l i t y a l l o c a t e d and dead t r e e s removed, subroutine NECO, and the remaining t r e e s grown on the b a s i s of the stand s t a t i s t i c s before m o r t a l i t y was c a l c u l a t e d , subroutine CRESCO. Thus t r e e s d y i n g w i t h i n a p e r i o d are assumed t o have no growth. The volume of t r e e s d y i n g i s c a l c u l a t e d and accumulated i n the gross volume t o t a l . The dbh l i s t i s maintained as a deck (Knuth, 1968), t h a t i s t r e e s which d i e or were thinned are removed and the r e -maining t r e e s moved up so t h a t no gaps remain i n the l i s t . The use of l i n k e d l i s t s was considered but the savings i n time were q u i t e small and not worth the a d d i t i o n a l complexity. To minimize the a r t i f i c i a l v a r i a b i l i t y introduced by the method of k i l l i n g each tr e e i t was f e l t t h a t a minimum of f i f t y t r e e s should be maintained i n the l i s t . When the number f a l l s below t h i s , the area of the p l o t i s t r i p l e d by the simple, crude method of t r i p l i c a t i n g each t r e e , sub-r o u t i n e TRIPLE. The maximum s i z e of the l i s t was set at 800, but between 100 and 200 t r e e s appeared the optimum balance between speed of execution and too few t r e e s . The t h i n n i n g subroutine a l l o w s two types of t h i n n i n g , c o n t r o l l e d by b a s a l area removed. The f i r s t , termed a low t h i n n i n g , removes t r e e s i n ascending order from a s p e c i f i e d minimum dbh u n t i l the d e s i r e d amount of b a s a l area i s removed. The second, l o o s e l y termed a crown t h i n n i n g , d i v i d e s the crop 87 into five classes, based on the dbh/average stand dbh ratio. The classes are <3/4; 3 A - l l 1 - iii l i - l $ i * l£. The user specifies the amount of basal area per acre to be removed in each class and can completely remove a l l the trees in one class by specifying a large value for that class. Within each class, trees are removed in ascending order of size. The THIN subroutine stores thinned trees in an array and calculates thinning and main crop after thinning stati s -t i c s . It has the disadvantage that i t is unable to salvage any mortality that has occurred at ages previous to the thinning. Moreover in a selective thinning performed by skilled markers, trees which are l i k e l y to die in the near future are removed in preference to the others, reducing the immediate mortality far more than the level of stocking would indicate. This does not occur to an equal extent in a l l types of thinning. Because of this d i f f i c u l t y i t was not simulated by the model originally, except insofar as determined by the mortality function. The model prints two types of output; a yield table (in the main programme) and stand tables for both thinnings and the main crop (subroutines METIOR and SCRIBO). Stand tables are produced only on request. Merchantable volumes are c a l -culated for close and intermediate u t i l i z a t i o n , 4-inch and 8-inch top. The minimum tree size can be specified, other-wise default values of 7.0 and 13.0 inches are assumed. There is a rewind option for replicating the cards on a f i l e . When comparing two or more treatments, running a 88 s t o c h a s t i c model once f o r each treatment i s v a l u e l e s s . How-ever a b l o c k i n g e f f e c t s i m i l a r to that i n experimental design can be achieved by u s i n g the same random number s t r i n g f o r each treatment. In the model t h i s i s done by a l l o w i n g the i n i t i a l i z a t i o n of the random number generator t o be s p e c i f i e d by the user f o r the growth of each p l o t . This a p p l i e s only to the s t r i n g used i n the c a l c u l a t i o n of the random r e s i d u a l m o r t a l i t y not t o the numbers used i n d e c i d i n g whether i n d i v -i d u a l t r e e s w i l l d i e or not. This means tha t the same p e r t u r b a t i o n s can be given t o the c a l c u l a t i o n s of m o r t a l i t y f o r d i f f e r e n t treatments. I t could not be done f o r the other random number s t r i n g because t h e i r sequence depends on the number of t r e e s i n a p l o t which may vary from treatment t o treatment. One advantage of the next event f o r m u l a t i o n i s the f l e x i b i l i t y of c o n t r o l . On a t e r m i n a l i t i s p o s s i b l e t o grow a stand t o a p a r t i c u l a r age and p r i n t a l i n e of y i e l d t a b l e ; t h i n the stand and p r i n t a f t e r t h i n n i n g s t a t i s t i c s ; then r e - t h i n again a t the same age i f d e s i r e d , observing the r e s u l t s a t each stage before making the d e c i s i o n s f o r the next. An example of a t y p i c a l run i s given i n Appendix 5« S i m i l a r l y i t i s p o s s i b l e t o grow the stand w i t h no i n t e r -mediate output t o any d e s i r e d age. Execution time f o r an 80 year p e r i o d v a r i e d from 0.6 seconds w i t h no intermediate output t o about 1.5 seconds w i t h 10 t h i n n i n g s and output every 5 years. By v a r y i n g the p l o t s i z e , any d e n s i t y of stems per acre can be obtained, but relevancy i s only 8 9 guaranteed within the domain of the data. Naming the model presented no d i f f i c u l t y . As i t was stochastic, a g i r l ' s name was appropriate. The f i r s t "test bed" model was only some 30 lines long, but this grew in stages to some 650 lines in the f i n a l model. Thus the name TOPSY was the obvious choice. 90 CHAPTER 5 VALIDATION 5.1 G e n e r a l C o n s i d e r a t i o n s A l t h o u g h C h a p t e r f o u r d e a l t w i t h the components of t h e model and w i t h the c o r r e c t e s t i m a t i o n o f the p a r a m e t e r s , th e problem of v a l i d a t i n g the model as a whole must s t i l l be con-s i d e r e d ( s t a g e seven of t h e d e s i g n sequence of N a y l o r e t a l . , 1966). As t h e i n t e r a c t i o n between the components i s one o f t h e most i m p o r t a n t f e a t u r e s of a systems model, e s t i m a t i n g t h e i n d i v i d u a l components as c o r r e c t l y as p o s s i b l e may n o t n e c e s s a r i l y i m p l y t h a t the model as a whole i s c o r r e c t . The problem of v a l i d i t y i s t h a t i f the r e a l system was known e x a c t l y so t h a t the model can be compared, t h e r e would have been l i t t l e p o i n t In c r e a t i n g a s i m u l a t i o n model. Van Horn (1968) d e f i n e d v a l i d a t i o n as "the p r o c e s s of b u i l d i n g an a c c e p t a b l e l e v e l of c o n f i d e n c e t h a t an i n f e r e n c e about a s i m u l a t e d p r o c e s s i s a c o r r e c t o r v a l i d i n f e r e n c e o f the a c t u a l p r o c e s s . " He s u g g ested t h a t seldom w i l l a p r o o f t h a t the s i m u l a t o r i s c o r r e c t be o b t a i n e d . To quote from F o r r e s t e r (i960), "our c o n f i d e n c e i n the model as an e x p e r i -m e n t a l t o o l f o r s t u d y i n g t h e e f f e c t of s t r u c t u r e and p o l i c y changes i n the a c t u a l system i s based on our c o n f i d e n c e i n t h e model's components s e p a r a t e l y and i n d i v i d u a l l y and i n t h e f a c t t h a t i n c o n c e r t they produce the b e h a v i o u r t h a t i n t e r e s t s us i n the a c t u a l system." The l e v e l of c o n f i d e n c e and t h e 91 v a l i d i t y of a model can not be divorc e d from the purpose of the study; i f i t i s then the v a l i d i t y of a model has no mean-i n g . A l s o , F o r r e s t e r (19^8) suggested that a model should not be Judged a g a i n s t imaginary p e r f e c t i o n , but i n comparison w i t h other models. W h i l s t t h i s may be t r u e , i t i s not a severe t e s t when there are only a few crude a l t e r n a t i v e models. F o r r e s t e r (i960) suggested that two questions must be answered when v a l i d a t i n g a model i 1. How acceptable i s the model as a r e p r e s e n t a t i o n of the o r g a n i z a t i o n of the system? 2. How c l o s e i s the model behaviour t o r e a l i t y ? I t i s important t o d i s t i n g u i s h between behaviour and t e s t s of p r e d i c t i v e a b i l i t y . F igure 5*1• from F o r r e s t e r (i960) shows the time paths of some v a r i a b l e of i n t e r e s t as p r e d i c t e d by two models and as i n r e a l i t y . Although the sum of ( e r r o r ) 2 Time Figur e 5»1 I l l u s t r a t i o n of the D i f f e r e n c e between Model Behaviour and P r e d i c t i o n . 92 Is smaller for the nondynamic model I over that of the o s c i l -lating model II, clearly model II gives a better insight into the behaviour of the variable of interest. Naylor and Finger (196?) proposed a three stage approach. 1. Construct a set of hypotheses for the process using a l l available knowledge. Hence models Incorporating these w i l l have a degree of a p r i o r i confidence. 2. Attempt to verify any assumptions of the model by empirical testing. 3. Test the model's a b i l i t y to predict the behaviour of the real system. In addition to basic s t a t i s t i c a l tests, Van Horn (1968) suggested that sensitivity testing would be appropriate for stage two. The use of time-series analysis may be appropriate for stage three (see Fishman and Kiviat, 1967). while Cohen and Cyert (l96l) suggested regressing simulated series on real series and testing whether the coefficient was s i g n i f i -cantly different from one, the intercept significantly d i f f e r -ent from zero. Van Horn suggested a "Turing" test for stage three to partially overcome a lack of material for testing or d i f f i c u l t i e s in the underlying assumptions In the s t a t i s t i c a l tests. The test appeared simple. People directly involved with the actual process were asked to compare real and simu-lated output, perhaps without knowing which was which. The reasons for any discrepancies found were then thoroughly discussed. A rough order of decreasing value-cost ratio for some 93 validation actions were (from Van Horn, 1968)J 1. Find models with a high face v a l i d i t y . 2. Make use of existing research and experience to supplement models. 3. Conduct simple empirical tests of means, variances and distributions using basic s t a t i s t i c a l tests. 4. Run "Turing" type tests. 5. Apply complex s t a t i s t i c a l tests, e.g. spectral analysis. 6. Engage in special data collection. 7. Perform f i e l d tests. 8. Implement the results with no validation. Finally i t must be stressed that the validation proced-ures are really sequences of tests of null hypotheses. Any test either shows that the model is incorrect or f a i l s to show that i t is Incorrect. In fact i t is almost certain that a model is an imperfect representation of a system and w i l l f a i l i f the testing procedure can be refined enough. 5.2 Specific Problems in Validating TOPSY Despite the emphasis on different points, i t was believed that the authors above were essentially in agreement. As far as possible their Ideas have been used in this chapter. The f i r s t two stages of Naylor and Finger's (1967) approach to validation were believed to have been adequately covered in Chapters three and four. By limiting the use of the model 94 t o s t a n d s w i t h no l a r g e i r r e g u l a r g a p s i n t r e e c o v e r , a l l t h e i n t e r a c t i o n s shown i n F i g u r e 3*3 h a v e b e e n i n c o r p o r a t e d i n t h e i n d i v i d u a l c o m p o n e n t s , w i t h t h e e x c e p t i o n o f c h a n g e s I n f o r m -f a c t o r d u e t o t h i n n i n g . Some d e f e c t s w e r e n o t e d i n C h a p t e r f o u r , b u t t h e r e was a h i g h d e g r e e o f c o n f i d e n c e i n n e a r l y a l l o f t h e i n d i v i d u a l c o m p o n e n t s , a t l e a s t a t t h e l e v e l o f r e s o l -u t i o n o f t h i s m o d e l . The r e m a i n d e r o f t h e v a l i d a t i o n o f t h e m o d e l h a s b e e n d i v i d e d i n t o two p a r t s : c o n d u c t i n g s i m p l e t e s t s on t h e i n p u t - o u t p u t r e l a t i o n s h i p s o f t h e m o d e l a s a w h o l e ; e x h i b i t i n g t h e b e h a v i o u r o f t h e m o d e l a n d c o m p a r i n g I t w i t h s u g g e s t e d t h e o r y . B e c a u s e t h e r e i s s o much c o n t r o v e r s y i n t h e l i t e r a t u r e o v e r t h e b e h a v i o u r o f t h e f o r e s t s t a n d a s a s y s t e m , r u n n i n g t h e m o d e l t o f i n d t h e r e s u l t s o f d i f f e r e n t i n i t i a l d e n s i t i e s , t r e a t m e n t s a n d s i t e s o v e r l a p s c o n s i d e r a b l y w i t h a c h a p t e r on r e s u l t s . C o n s i d e r a t i o n must b e g i v e n t o t h e d e s i g n o f c o m p u t e r s i m u l a t i o n e x p e r i m e n t s t o o b t a i n t h e s e r e s u l t s , a c c o r d i n g l y t h e s e c o n d p a r t h a s b e e n l e f t t o t h e n e x t c h a p t e r . T h e r e m a i n d e r o f t h i s c h a p t e r d e s c r i b e s t e s t s on t h e p r e d i c t i v e a b i l i t y o f t h e m o d e l . T o t e s t t h e s i m u l a t i o n m o d e l , some o f t h e o r i g i n a l d a t a w e r e u s e d t h a t h a d b e e n e m p l o y e d i n e s t i m a t i n g t h e p a r a m e t e r s o f t h e t r a n s f o r m a t i o n s . I n a m o d e l c o n s i s t i n g o f one r e g r e s -s i o n o n l y , t h i s w o u l d p r o b a b l y n o t h a v e b e e n a s e v e r e t e s t . The m o d e l i s a s s u m e d t o f i t t h e d a t a r e a s o n a b l y w e l l ; i t h a s b e e n e s t i m a t e d t h a t way. W h e t h e r i t m o d e l l e d t h e r e a l w o r l d a c c u r a t e l y i s a p o i n t f o r d e b a t e . H o l t (1965) d e f e n d e d t h e u s e o f d a t a f o r b o t h e s t i m a t i n g t h e p a r a m e t e r s a n d t e s t i n g 95 the whole model. He suggested that the Individual equations of a complex simulation model may f i t data reasonably well but due to the interaction effects the model as a whole may be in error. Also, in this thesis, the individual regres-sions were estimated from data consisting of forecasts one time period ahead, while the model is being used to predict over many time periods. The use of the output of one fore-cast as the input of the next can be expected to accumulate and magnify errors. One of the objectives of Chapter one was that the model should be reasonably accurate in prediction. This is a severe test, even against the original data. The model ought to be accurate over a l l the stand parameters and the dbh distribution. This gave rise to the problem of multiple responses. The data available for testing the model presented many problems. Each sample plot represented one time path from a population which is highly variable but which is s t i l l dependent on variables such as site and density. The time path is dependent on age, hence is non-stationary. Successive measurements on the same plot suffer from the autocorrelation problem. Finally few sample plots have been remeasured over a long period of time, though four had been measured over a ^5-year period. As i t was important to ensure that the growth curves of the model were r e a l i s t i c , as opposed solely to the accuracy of the predicted value at some arbitrary date, time-series analysis appeared to offer most hope of a solution. However two requirements must be met before the main body of established theory can be used» the time-series must be stationary (or capable of transformation to a stationary time-series); a large number of samples must be available for each curve. Neither of these was f u l f i l l e d . The data in fact consists of many short segments from non-stationary time-series with a high "noise" content, each of which is highly dependent on independent variables. The validation problem is caught between two hornsi a valid comprehensive test does not exist; even i f i t did, the data was not adequate for such a test. Three specific tests were made corresponding to actions three and four of Van Horn in section 5.1. A paired " t " test was made on long range predictions; a paired " t " test and a regression approach (modified from Cohen and Cyert, l96l) made on short term predictions; a "Turing" test carried out on four plots. 5.3 Validation of Long Term Predictions Fourteen plots were selected that had been periodically remeasured over a period longer than twenty years. Ten had been remeasured over longer than 30 years of which 4 had periods longer than 40 years. Unfortunately, apart from two plots, a l l were from the Williamson (1963) data, higher sites Classes II and III, i n i t i a l ages of 40 to 50 years and mod-erate to high densities. The sample plot data were read into the model (the average site index over the plot's lifetime was u s ed) and the growth of the p l o t s i m u l a t e d t i l l t h e f i n a l remeasurement d a t e . Ten runs were made f o r each p l o t and the a v e r a g e s t a k e n i n o r d e r t o reduce th e model's v a r i a b i l i t y . D i f f e r e n c e s ( s i m u l a t e d - r e a l ) i n the growth of average dbh, s t a n d i n g b a s a l a r e a p e r a c r e , n e t volume p e r a c r e , and the change i n numbers of stems p e r a c r e were o b t a i n e d . The r e s u l t s a r e g i v e n i n T a b l e s 5.1, 5.2, 5.3 and 5.4. The a v e r a g e l e n g t h of s i m u l a t i o n was 35 y e a r s . The a v e r a g e d i f f e r e n c e s ( s i m u l a t e d - r e a l ) f o r the growth o f the l4 p l o t s o v e r the whole s i m u l a t i o n p e r i o d were» n e t t o t a l volume 10.1 c u . f t . per a c r e n e t b a s a l a r e a 5.86 s q . f t . p e r a c r e a verage dbh .307 i n c h e s numbers of stems -6.9 p e r a c r e The f o u r p a i r e d " t " t e s t s were n o n - s i g n i f i c a n t . The r e s u l t s appeared r e a s o n a b l e b u t t h e r e were c o n s i d e r a b l e d e v i a t i o n s i n the i n d i v i d u a l c o m p a r i s o n s . T h i s was t o be e x p e c t e d w i t h a h i g h l y s t o c h a s t i c system such as a f o r e s t . There were d i f f e r e n c e s i n c a l c u l a t i n g s t a n d i n g volume p e r a c r e ? i n t h e r e a l p l o t d a t a , t a r i f number was e s t i m a t e d d i -r e c t l y , i n s i m u l a t e d p l o t d a t a , t a r i f number was e s t i m a t e d from t h e t a r i f number/dominant h e i g h t / s i t e i ndex-age r e l a t i o n -s h i p s . The average d i f f e r e n c e i n i n i t i a l volume p e r a c r e was 130 c u . f t . per a c r e , from an average volume of 6720 c u . f t . p e r a c r e . These p a i r e d " t " t e s t s d i d n o t a c c o u n t f o r d i f f e r e n c e s i n t h e shape of growth c u r v e s . T h i s c o u l d p o s s i b l y l e a d t o e r r o r s i f t h e r e a l and s i m u l a t e d c u r v e s had by chance c r o s s e d a t the p o i n t o f c o m p a r i s o n . T h i s was u n l i k e l y here due t o Table 5*1 Paired " t " Tests on Long-term Predictions Net Volume per acre cu. f t . Site Index Ages I n i t i a l Real Values Final Growth Deviation* 107 21-1+1 589 3337 2748 805 123 2l-4i 581 444i 3860 -63 163 54-100 7234 15^70 8236 9 169 54-100 9225 15600 6375 64 165 54-100 . 8830 16220 7390 -432 165 50-90 9700 16200 6500 -1168 129 51-81 6265 10547 4282 1127 132 51-81 5925 9292 3367 1394 122 42-72 5920 8780 2860 4 i 6 177 50-81 11153 18972 7819 -3402 l 6 4 50-81 8520 16325 7805 -2010 132 58-88 7940 13001 5061 2 l 4 l45 42-72 5010 10166 5156 4 l 6 l 127 45-67 7144 10607 3463 -974 Average Deviation 10.1 Average Absolute ll60. Deviation Average Increment 5351• s_ of Deviations** 465. x Calculated " t " .02 N.S. *Deviation = Simulated - Real Growth **s_ = Standard Error of the mean x 99 Table 5.2 Paired " t " Tests on Long -term Predictions Stems per acre Site Index Ages I n i t i a l Real Values Final Mortality Deviation* 107 2l-4l 548 484 64 4 123 2l-4l 380 356 24 -18 163 54-100 216 109 107 9 169 54-100 222 99 123 7 165 54-100 202 102 100 4 165 50-90 382 184 198 -23 129 51-81 454 264 190 -16 132 51-81 387 238 149 -13 122 42-72 951 388 563 -16 177 50-81 227 140 87 -14 l 6 4 50-81 295 163 132 -4 132 58-88 4i3 256 157 -23 1*5 42-72 321 169 152 37 127 45-67 494 324 170 -31 Average Deviation -6.9 Average Absolute 15.6 Deviation Average Increment -153*2 s_ of Deviations 4.74 x Calculated " t " 1.45 N.S. ^Deviation = Simulated - Real Growth s— = Standard Error of the mean x 100 Table 5.3 Paired " t " Tests on Long -term Predictions Basal Area per acre sq. f t . Site Index Ages Real I n i t i a l Values Final Growth Deviation* 107 21-41 46 132 86 28 123 21-41 ^5 1^5 100 8 163 54-100 188 276 88 16 169 54-100 2l4 262 48 14 165 54-100 210 282 72 3 165 50-90 231 300 69 -29 129 51-81 194 257 63 -18 132 51-81 174 220 46 27 122 42-72 194 225 31 0 177 50-81 257 326 69 -40 l 6 4 50-81 222 303 81 -16 132 58-88 224 289 65 11 1^5 42-72 166 234 68 97 127 45-67 229 270 4i -19 Average Deviation 5«86 Average Absolute 23•3 Deviation Average Increment 66.2 s_ of Deviations 8.92 x Calculated " t " .66 N.S *Deviation = Simulated - Real Growth 101 Table 5 .4 Paired "t" Tests on Long-term Predictions Average Dbh inches Site Index •Ages Real Values I n i t i a l Final Growth Deviation* 107 2l-4l 3.8 6.8 3.0 .7 123 2l-4l 4.5 3.4 3.9 .4 163 54-100 11.6 20.7 9.1 -.5 169 54-100 12.5 21.3 8.8 -.1 165 54-100 12.9 21.8 8.9 -.2 165 50-90 10.0 16.5 6.5 .5 129 51-81 8.0 12.3 4.3 .9 132 51-81 8.5 12.2 3-7 1.2 122 42-72 5.8 9.7 3.9 .5 177 50-81 13.8 20.1 6.3 -.1 l 6 4 50-81 10.9 17-6 6.7 -.4 132 58-88 8.9 13.3 4.4 1.0 1*5 42-72 8.2 14.3 6.1 .3 127 45^67 8.8 11.8 3.0 .4 Average Deviation .307 Average Absolute Deviation .514 Average Increment 5.6 s_ of Deviations X .457 Calculated M t " .67 N.S. ^Deviation = Simulated - Real Growth 102 the d i f f e r i n g i n t e r v a l s between f i r s t and l a s t measurements. The time p a t h s of f o u r o f the l o n g r u n n i n g s i m u l a t i o n and r e a l p l o t s a r e shown i n F i g u r e s 5*2 and 5«3« A g a i n t h e simu-l a t e d v a l u e s a r e the av e r a g e s of 10 r u n s . Any d i f f e r e n c e s between r e a l and s i m u l a t e d v a l u e s i n c r e a s e d o n l y s l i g h t l y w i t h l e n g t h o f the s i m u l a t i o n . There was no n o t i c e a b l e c o n -s i s t e n t d i f f e r e n c e i n t h e shape o f the c u r v e s . A.n a t t e m p t was made t o t e s t the shape o f t h e c u r v e o f t h e average d e v i a t i o n s of n e t volume growth t h r o u g h time of simu-l a t i o n f o r a l l l 4 p l o t s . Three p a i r e d " t " t e s t s were made a f t e r 10, 20 and JO y e a r s o f growth. However t h e r e was c o n s i d -e r a b l e v a r i a b i l i t y i n the remeasurement p e r i o d s . Three p l o t s were n o t measured over t h e whole 30 y e a r s . Some p l o t s were measured a f t e r 25 y e a r s b u t n o t a f t e r 20. These d i f f e r e n c e s c o n s i d e r a b l y weaken the t e s t s . The p l o t o f the average d e v i -a t i o n and c o n f i d e n c e l i m i t s a g a i n s t time a f t e r the f i r s t meas-urement i s shown i n F i g u r e 5*4. The average d e v i a t i o n of f o u r p l o t s ( s i t e 165) a l l measured a f t e r 10, 20, 30 and 40 y e a r s i s a l s o shown f o r c o m p a r i s o n . None o f the d e v i a t i o n s were s i g n i f -i c a n t a t 5%. The c o n f i d e n c e l i m i t s f o r the average d e v i a t i o n I n c r e a s e w i t h l e n g t h of s i m u l a t i o n i n d i c a t i n g an i n c r e a s e i n v a r i a b i l i t y o f the e r r o r s between i n d i v i d u a l p l o t s . However the t r e n d of the change i n the average d e v i a t i o n was n o t con-s i s t e n t i n d i c a t i n g t h a t , a t l e a s t o ver the l e n g t h of ti m e t e s t e d , the e r r o r s d i d n o t a c c u m u l a t e . The c r i t i c i s m of t h e above t e s t s was t h a t due t o t h e l a c k of d a t a no a t t e m p t was t o a s s e s s the model's 103 er u> <u k. u o < 3 0 0 280 260 2 4 0 220 2 0 0 180 Site x 163 o 165 • 132 + 122 Real Simulated ^ # ^ " " " 4 0 16,000 I5,000|-14,000 13,000 r 1 2 , 0 0 0 1 11,000 u ° 10,000 fl> 1 E 2 9 0 0 0 o > 8 0 0 0 7 0 0 0 6 0 0 0 5 0 0 0 50 Site x 163 o 165 • 132 + 122 60 70 80 Age 90 100 Real * Simulated .S X-"" .X . X • yy X ' X ' / / X / o o' + 4^ • ' 40 50 60 70 Age 80 90 100 Figure 5-2- Comparison of real and simulated plots,basal area/ acre and volume/acre-1 C 4 9 0 0 -8 0 0 • CP 7 0 0 -u o -V 6 0 0 -cn c a> 5 0 0 -to H-o 4 0 0 -w. E 3 0 0 -z 2 0 0 -100 -oL 4 0 Real Simulated 50 6 0 70 Age 8 0 9 0 100 to c a a> o> o i_ > < 22 -Site 2 0 - x 163 o 165 18 • 132 • 122 16 -14 12 10 -8 «' ^+— 6 i 4 0 5 0 Rea Simulated 6 0 70 Age 8 0 9 0 100 Figure 5-3- Comparison of real and simulated plots, number of stems/acre and average dbh-105 u o 3 u e _3 O > "5 «-|2 o> c '•5 c o to c o LU 1600 1400 1200 1000 800 600 400 200 0 -200 -400 - 6 0 0 -800 -1000 * Upper Confidence Limit / Average of 14 Plots / \ Average of / ^ ' 3 — M \ 4 p|0ts Lower Confidence Limit 10 20 30 Years of Simulation 40 Figure 5-4- Results of paired "t" test over 10,20 and 30 years of simulation-106 d e f i c i e n c i e s i n any p a r t i c u l a r r e g i o n of s i t e , age or d e n s i t y . I t was t o remedy t h i s that s i m i l a r t e s t s were made on c o n s i d -e r a b l y more data simulated over the r e g r e t a b l y short p e r i o d of 10 y e a r s . Even here however, the model used the output of one f i v e year p r e d i c t i o n as the input of the next. 5.4 V a l i d a t i o n of Short-Term (lO years) P r e d i c t i o n s A t o t a l of 32 p l o t s were s e l e c t e d t h a t had been remeas-ured twice over 10 y e a r s . They were s e l e c t e d t o cover the range of age and s i t e , i . e . 8 p l o t s i n each of the high/low combinations. Again the p l o t data was read i n t o the model. The averages of 10 s i m u l a t i o n runs f o r each p l o t were com-pared w i t h the r e a l development of the p l o t s a f t e r 10 years of growth. One p l o t was r e j e c t e d f o r t h i s s e r i e s of t e s t s as i t had s u f f e r e d very heavy m o r t a l i t y . In f a c t i t s net volume growth was -766 cu. f t . per acre a f t e r 10 years and the d i f -ference between r e a l and simulated values was 2240 cu. f t . per a c r e , very much higher than the others. While t h i s i l l u s t r a t e d the s t o c h a s t i c nature of m o r t a l i t y and the r a r e event nature of extreme m o r t a l i t y i n p a r t i c u l a r , i t would have obscured the r e s u l t s of the t e s t s . Table 5«5 shows s t a t i s t i c s f o r p a i r e d " t " t e s t s on the remainder of the p l o t s f o r numbers of stems per a c r e , net volume growth per a c r e , b a s a l area per acre and the average dbh of the stand. Three of the t e s t s were n o n - s i g n i f i c a n t , but the average d e v i a t i o n of the stand dbh of..268 Inches a f t e r 10 years was 107 highly significant. Table 5.5 Summary of Results of Paired " t " Tests on Short term Predictions. I n i t i a l Values Mean Min. Max. Average Deviation in Growth s- of Deviations Calculated t Site Index 123 78 178 Age 49 22 97 Stems per acre 640 119 1648 -17.3 14.60 1.19 NS Volume per acre (cu. f t . ) 5555 548 14310 36.9 113.4 .32 NS Basal Area per acre (sq. f t . ) 171 47 281 3.03 3.106 .98 NS Av. dbh (ins.) 7.6 3.6 19.2 .268 .066 4.06 The deviations of the four variables were individually regressed against site, age, numbers of stems per acre and volume per acre at the start of the growth period to test for any trends. None of the 16 regressions were significant, the largest "F" value being 2.07 with 1 and 29 d.f. A f i r s t order model of site and age was tried, again with no significant regression coefficients. There Is l i t t l e point in reproduc-ing the values of the regression coefficients here. Clearly the errors exhibited no trends with the variables tested. Of the eleven paired " t " tests and the 20 regressions, 108 only one was found t o question the model. Even here the over-p r e d i c t i o n of the average dbh a f t e r 10 years of .268 Inch was sma l l by p r a c t i c a l standards. The f a c t t h a t the lon g term t e s t showed no s i g n i f i c a n t d i f f e r e n c e suggested t h a t t h i s was one of those occasions when a type I e r r o r had occurred ( r e j e c t i n g the n u l l hypothesis when i t was t r u e ) . 5 . 5 The "Turing" Test An attempt was made to perform a "Turing" t e s t (see Van Horn, 1968) w i t h three f a c u l t y members of the F o r e s t r y Department i n the areas of s i l v i c u l t u r e and management. Four sample p l o t s were s e l e c t e d , one from each of fo u r d i f f e r e n t s i t e c l a s s e s . One was a p l a n t a t i o n , one was a long term sample p l o t , one had f o u r successive measurements a t short I n t e r v a l s (every three y e a r s ) , and one was r e p r e s e n t a t i v e of the b u l k of the dat a . The p l o t s had been s e l e c t e d randomly from w i t h i n the p a r t i c u l a r narrow groups r e q u i r e d . Two s i m u l a t i o n runs were made on each p l o t and the r e a l d ata presented i n e x a c t l y the same format as the output from the model. The o b j e c t i v e was f o r each person independently t o decide on the b a s i s of t h e i r experience which was the r e a l data f o r each of the p l o t s . More Important was the d i s c u s s i o n of what d e f e c t s , i f any, were shown up i n the simulated runs. The t e s t was not as simple t o perform or a d m i n i s t e r as Van Horn had claimed. The problem of sm a l l spans of measure-ment which a f f e c t e d the s t a t i s t i c a l t e s t s a l s o a f f e c t e d the 109 "Turing" test. In one case the conditions of the test were misunderstood. In the other two, familiarity with traditional methods of yield tables led to erroneous c r i t e r i a being used, e.g. i t was assumed that the run with most v a r i a b i l i t y was the real sample plot data. Only one person was able to correctly identify any of the real plots, in this case two out of four. The probability for an individual of correctly selecting two or more plots by chance was about .4. One plot was identified by using the criterion given above. The other plot was the plantation (from the Wind River data) and here the close u t i l i z a t i o n volume predicted by both simulation runs overestimated that of the real data. The test as conducted in this manner was not successful in identifying any structural deficiencies in the model. Naturally this could have been due to the fact that there weren't any, or that the test was not severe enough. Whether extensive running of the model by the experienced c r i t i c s without reference to real data would have been a more valid test is open to criticism. Correct results predicted by the model could have clashed with preconceived theories. Correct running and analysis of the model is also d i f f i c u l t and open to misconception. However the amount of Insight obtained from the test as performed was not as much as had been hoped. 5.6 Conclusions The previous tests in conjunction with the construction 110 of the individual components failed to reveal any major errors in the model. Even over as many as eight input-output, transformations (40 years simulation) errors appeared small and did not accumulate. However there were several l i m i t -ations to the validation procedure. The length of the long term runs was only 35 years on average, i t was hoped that the model would be valid for runs from age 20 to 100 years. The long term runs were conducted only on higher sites and higher ages, though short term predictions revealed no trends with age, site or density. The interaction between stand gener-ation and stand growth had not been tested, though each part appeared to perform well alone. The thinning mode had not been tested. The dbh distribution was only tested implicitly. If a l l four responses (numbers of stems, basal area per acre, volume per acre, and average dbh) performed well, then the errors in the dbh distribution should be small. No major errors were noted in the "Turing" test. It was to overcome these limitations that the third part of the validation procedure was carried out. This was to compare the results of the model with traditional theory and is described in the next chapter. It is inextricably com-bined with the design and analysis of experiments on the simulation model. To conclude! a f a i r degree of confidence has been b u i l t up that the unthinned model is a f a i r repre-sentation of the growth characteristics of the forest system, that i t s behaviour is close to reality and that i t can predict long term growth reasonably well on the average. I l l CHAPTER 6 THE DESIGN, ANALYSIS AND RESULTS OF SOKE SIMULATION EXPERIMENTS 6,1 Introduction This chapter examines the behaviour of the model and compares i t with the published r e s u l t s on forest growth and y i e l d i n order to increase confidence i n the inferences obtained from the simulation model. In the previous chapter i t was stated that the v a l i d i t y of the model could not be divorced from the objectives of the study. These objectives were stated i n Chapter one. The model must be suitable f o r use as the growth module of a l a r g e r simulation model and must predict y i e l d and stand tables i n response to various treatments. Many treatments and hence d e t a i l e d objectives are possible. Two such objectives were s p e c i f i e d and two experiments designed to i l l u s t r a t e some methods used in computer simulation experiments. The two objectives were* 1 . To observe the response of the unthinned model to conditions of s i t e and i n i t i a l density (number of stems per acre at age 2 0 ) . 2. To observe the response of the model thinned to a s p e c i f i c regime as a function of s i t e , i n i t i a l density and thinning i n t e n s i t y . A random unplanned approach to a t t a i n i n g the objectives of any simulation model i s l i k e l y to be unsuccessful or to be very c o s t l y i n terms of computer and r e a l time. With a 112 stochastic model the problems of estimating the r e s u l t s and comparing treatment means are a l l i e d to those of s t a t i s t i c a l analysis with some features peculiar to simulation models. Before It i s possible to discuss experimental design some preliminary knowledge of the behaviour of the system under study i s required. This was obtained by running the model in the unthinned mode f o r several s i t e and i n i t i a l density combinations i n order to determine the stochastic nature of the model and to observe the pattern of growth and mortality. These preliminary r e s u l t s were compared to t r a d i t i o n a l y i e l d t a b l e s . Afterwards i t was possible to design and perform the two experiments and to c r i t i c i s e the r e s u l t s i n the l i g h t of published research. This was by no means a complete analysis of the model, rather a demonstration of i t s poten-t i a l . 6.2 The V a r i a b i l i t y of the Model and i t s Behaviour when Compared to Yi e l d Tables To obtain the re s u l t s of t h i s section the model was run at three l e v e l s of s i t e and three l e v e l s of i n i t i a l density. One of the l e v e l s of I n i t i a l density was chosen to be com-parable to values from published y i e l d tables. These tables were those of McArdle et a l . (l96l) and the gross volume estimates of Curtis (1967), the l a t t e r being more recent and presumably more accurate than those of Staebler (1955)• The y i e l d tables are tables of the supposed development of f u l l y stocked stands at "normal" d e n s i t i e s . The o r i g i n a l data were temporary sample plots and the compilation dates from 1931. The data were weak in low sites and low ages. For example, there were no stands younger than 40 years and less than site index 120. At age 20, the numbers of stems per acre predicted by the tables were believed to be unrealisticly high in the low sites. The estimates of Curtis (1967) were obtained from compatible growth functions with the relative density term fixed at the value 1.0, equivalent to the curves of the normal density yield tables. The data was derived from.temporary sample plots and some permanent sample plots, the l a t t e r treated as though they were temporary plots. In order to run the model, the numbers of stems per acre were redefined for normal densities to be 2000, l600 and 1200 stems per acre for sites 90, 120 and 150 respectively. These values were chosen subjectively. Two other i n i t i a l densities were specified at 800 and 300 stems per acre. A total of 20 replications of each site - i n i t i a l density combination were generated at age 20 and grown t i l l age 100. The average simulated yield and stand tables were calculated along with an estimate of the standard deviation of the population of each simulated value of the model. These are given in Appendix 2. There are two random components in the model; that due to the purposeful generation of a random additional percent mortality and that due to the method of selecting individual trees to die once the probabilities of mortality have been determined. Provided the number of individuals in the sample 11* p l o t remains high, the v a r i a b i l i t y due to the f i r s t component should be the most important. Two questions were posed; how variable was the complete model and were i t s expected values equal to those of the model with the f i r s t component removed. Eight Individual runs, d i f f e r i n g only i n the pseudo-random number s t r i n g were taken from the "normal" density series of the complete model. The number of stems, standing t o t a l volume and gross t o t a l volume per acre on the three s i t e s were plotted to i l l u s t r a t e t y p i c a l time paths of i n d i v -idual stands (Appendix 4 ) . The considerable v a r i a t i o n i n numbers of stems per acre i s evident. The v a r i a b i l i t y of standing volume per acre increased with s i t e so that i t was quite large i n p r a c t i c a l terms. There was l i t t l e v a r i a t i o n in gross production. The standard deviations (Appendix 2) followed a d e f i n i t e pattern. They were very low f o r i n i t i a l density of 300 stems per acre. With the exception of stems per acre they increased r a p i d l y t i l l age 50 thereafter increasing slowly. The standard deviation (S.D.) of stems per acre decreased with age with high i n i t i a l d e n s i t i e s , the reverse was true with low i n i t i a l d e n s i t i e s . In general the S.D.'s of standing basal area and volume per acre were l e s s than 10 - 15^ of the mean. From a s t a t i s t i c a l viewpoint these were low but s t i l l implied a range of some 4000 cu. f t . per acre standing volume i n s i t e 150 from age 60 onwards. The S.D. of gross volume production was very low, les s than 5% of the mean. The random re s i d u a l mortality was removed from the model 115 temporarily and the average of 10 runs obtained for the normal d e n s i t i e s . The gross and net volumes were plotted and shown i n Figures 6.1, 6.2 and 6.3 f o r s i t e s 90, 120 and 150 respect-i v e l y . The ranges of the values f o r the complete model, taken as four times the S.D., are also shown. The addition of the r e s i d u a l random mortality depressed the values f o r the higher s i t e s . Though the difference was c l e a r , i n p r a c t i c a l terms i t was very small, l e s s than 5%. The v a r i a b i l i t y of the model without the random re s i d u a l mortality was quite small with S.D.'s only * to l/3rd of those of the complete model and i s not shown. In one of the test-bed models, where the method of s e l e c t i n g i n d i v i d u a l trees to die was d e t e r m i n i s t i c , the difference due to the addition of randomness was more marked, i n the order of 7 or 8$. Thus the addition of randomness had a c l e a r e f f e c t on the model, having an i n t e r a c t i o n with the other components that resulted i n a decrease i n y i e l d . How-ever i n p r a c t i c a l terms the difference between a stochastic model such as t h i s and a deterministic one would be n e g l i g -i b l e when used s o l e l y to obtain average values. The stochastic component did mimic the r e a l development more f a i t h f u l l y and provided some insight into the l i k e l y behaviour of the system. The i n t e r a c t i o n e f f e c t when Incorporated into a l a r g e r economic model or when used f o r teaching i s unknown. Because the second component of v a r i a b i l i t y was an i n t e g r a l part of the model that could not be removed, removal of the random r e s i d u a l mortality would not have given a deter-m i n i s t i c model. The problem of stochastic r e s u l t s would s t i l l 116 NORMAL DENSITY SITE SO a §_ A WiFK random residual morfcdihj * W i t W t « " - fcary (4 x S.D.) a a a f M _ f M O a «—i • 0.0 20.0 40.0 60.0 80.0 100.0 AGE Figure 61- The Effect of the Stochastic Component-Volume per acre for Site 90-117 0.0 NORMAL DENSITY SITE 120 A Wihh random residual morr/aliru x Wirhour " " Ranje (4 x S.D.) 20.0 I 40.0 AGE 63.0 80.0 100.0 Figure 6-2- The Effect of the Stochastic Component- Volume per acre for Site 120-118 a a a , OJ in o i n . 01 O o CC U J O i n . in i n . 01 0.0 NORMAL DENSITY SITE 150 - WiH\ random resi iusl m o r U t i t j * Without * v " Crro« N e r T'OSS 20.0 "1 40.0 AGE 60.0 80.0 1 100.0 Figure 6-3- The Effect of the Stochastic Component- Volume per acre for Site 150-119 have t o be considered, hence the complete model was used i n subsequent work. The values of the 9 s i t e / i n i t i a l d e n s i t y combinations plus the values f o r the t r a d i t i o n a l y i e l d t a b l e s were p l o t t e d i n Figures 6.4, 6.5» 6.6 f o r b a s a l area per ac r e , and Figures 6.7, 6.8, 6.9 and 6.10 f o r the average dbh, stems per acre s t a n d i n g volume per acre and gross volume per acre r e s p e c t -i v e l y . Dbh frequency histograms were shown f o r s i t e s 90 and 150, i n i t i a l d e n s i t y 800 stems per ac r e . Figures 6.11 and 6 .12. The comparisons between the path of the normal d e n s i t y y i e l d t a b l e s and the simulated values were q u i t e c l o s e f o r stems and gross volume per a c r e . However there were some di v e r g e n c i e s i n the shape of the curves f o r the other v a r i -a b l e s . The o v e r - p r e d i c t i o n of standing volume per acre a t e a r l y ages compared t o the y i e l d t a b l e s of McArdle et a l . (l96l) could p a r t i a l l y have been due t o the erroneous dominant height curves used i n the model which over-predicted the dom-ina n t height f o r a given s i t e . The t a r i f number would then have been o v e r p r e d i c t e d . The growth i n the average dbh of the normal d e n s i t i e s was not as great as the y i e l d t a b l e s . The curves of standing b a s a l area and volume per acre from the s i m u l a t i o n model show th a t the highest values i n the e a r l y ages were obtained from the curve of normal d e n s i t y i n i t i a l s t o c k i n g , but t h i s curve f e l l away from the maximum, r a p i d l y i n low s i t e s , not so r a p i d l y i n high s i t e s , l e a v i n g the highest values t o 800 stems per acre i n i t i a l d e n s i t y . a a co. a O a . L U a: <_> a: \<=> # •o CCOJ . ad CO U N T H I N N E D S T A N D S S I T E Y.eU TaUts. A "Normal DenSihj X $00 &vt\ijft. *T o^tl 20 * 360 * . . . . . . .. 10 O.D 20. D I 40.0 A G E 60.0 80.0 100 Figure 6-4- The Growth Curves of the Unthinned Model- Basal Area per acre for Site 9 0 -CJ o O f . o o C J o a. P J UJ cn CD 9' 0 . 0 U N T H I N N E D S T A N D S S I T E 120 — Yield Tables A "Normal" Detvsirij ^ 3 0 0 *• * u ** •• 2 0 . 0 4 0 . 0 A G E 6 0 . 0 8 0 . 0 100 Figure 6-5- The Growth Curves of the Unthinned Model -Basal Area per acre for Site 120-a O i . a co. O a a a . QTru . CD co UNTHINNED STRNDS S I T E 1 5 0 - — Yield TobUs * 8 0 0 Sr«ms/*.cre or a<jf> ZO * 3 0 0 *• " *" •* ** 0.0 20.0 40.0 A G E 60.0 80.0 100 Figure 6-6- The Growth Curves of the Unthinned Model- Basal Area per acre for Site 150-125 a (P. a a o* O 3 CJ UJ C . a C J „ -CES • U J 3 1 .10 U J 9-a a . 0.0 UNTHINNED STANDS Yield Tables A "Normal" densirij x 800 jK<ns/«.cre at 2.0 * 300 •• " " S I T E 20.0 40.0 AGE 60.0 80.0 100.0 Figure 6-9-The Growth Curves of the Unthinned Model-Standing Volume per acre-0 • a m o o m. n O « — « m »- m. UJ U J 2 -C O a . com. CD 1 -or in m. UNTHINNED STANDS o.o x Curhs Ou7) Normal derviiru 3 0 0 sVtms/acre o.r aeje 2 0 300 20.0 T 40 0 AGE 60.0 eo.o 100.0 Figure 6-10- The Growth Curves of the Unthinned Model- Gross Volume per acre-127 400-1 350 300-250-200 150-100-50-0 0) o O \ 120-E 100-CO o 80-.o E 60-3 z 40-20-0 60-i 40-20-0 Age 20 800 stems /acre Age 60 550 stems /acre Age 4 0 720 stems/acre 8 10 Age 8 0 4 2 0 stems /acre 4 6 8 10 12 14 16 Age 100 330 stems /acre Figure 611-6 8 10 12 14 16 18 DIAMETER CLASS (inches) Diameter Frequency Distribution for the Unthinned Model 1 Average Values for Site 9 0 , Initial Density 8 0 0 stems per acre-128 3 0 0 250 2 0 0 I50H 100 50-1 0) O o E a> to a> E z Age 20 8 0 0 stems/acre Age 4 0 480 stems /acre i i 4 8 8 10 12 14 16 Age 6 0 2 8 0 stems /acre 8 10 12 14 16 18 20 Age 8 0 190 stems/acre 8 10 12 14 16 18 20 22 24 Age 100 140 stems/acre 8 10 12 14 16 18 20 22 24 26 DIAMETER C L A S S (inches) Figure 6-12- Diameter Frequency Distributions for the Unthinned Model : Average Values for Site 150,Initial Density 8 0 0 stems per acre-129 A f t e r age 80, the standing values were greatest at the low i n i t i a l density of 300 stems per acre. The o r i g i n a l data f o r the t r a d i t i o n a l y i e l d tables had been subjectively selected from "normal" stands at a l l ages. It i s suggested that at the older ages the stands selected were those equivalent to the low i n i t i a l d e n sities which were now i n excellent appearance in terms of health and standing volumes. Thus the y i e l d tables were not growth curves but rather "envelopes" repre-senting an average upper l i m i t to standing volume. Supporting evidence to t h i s hypothesis comes from con-si d e r a t i o n of the permanent sample plots described by Williamson (1963) which had o r i g i n a l l y been part of the y i e l d table data. Of the 26 "normal" plots established before 1930 at l e a s t 10, or 40$, have suffered badly from a v a r i e t y of causes with heavy mortality and decreasing values r e l a t i v e to normality. Two plots have i n f a c t l e s s standing volume per acre than when established twenty-five years e a r l i e r . Several plots showed disappointingly low standing volume increments. The small net growth a f t e r about ages 60 - 70, or rather the extremely i r r e g u l a r path, was very obvious i n basal area per acre. This was i l l u s t r a t e d by the model. It was f e l t , however, that the net basal area growth of the model was too slow i n the older ages, higher s i t e s , possibly because the model selected too large a tree f o r mortality at those ages. This was a subjective f e e l i n g and i t should be noted that Newnham's (1964) model showed the same character-i s t i c s (Figure 15, p. 71)• 130 6.3 The Design of Computer Simulation Experiments The v a r i a b i l i t y of the model has been shown to be quite small i n s t a t i s t i c a l terms, the c o e f f i c i e n t of v a r i a t i o n being In the order of 10^. The question of i n i t i a l s t a r t i n g con-d i t i o n s and a t t a i n i n g steady state (Conway, 1963) w a s not pertinent i n thi s model. Thus the problem of stochastic convergence - the convergence of averages of samples to the true expected value of the model - was not as c r i t i c a l as i n many simulation models, f o r example, multiprocess queuing models. The important problems were those of multiple response and experiment size when related to motive. The motive behind each experiment was to explore the response of desired variables as a function of selected independent variables rather than to predict an optimum point. Were costs and revenues allocated to the model, the determin-ation of an optimum treatment combination would have been important, though even there some knowledge of the behaviour of the response surface would have been required. Such a response surface would p a r t i a l l y avoid the multiple response problem of t h i s a n a l y s i s . There were several responses which i n t e r e s t the user, f o r example net t o t a l and merchantable volumes per acre, average tree s i z e . Gross values were important b i o l o g i c a l l y . These were a l l functions of age which could d i f f e r f o r various treatments. To remove the e f f e c t of age, the maximum mean annual increment was used i n many cases to provide a single response with which to evaluate treatment 131 e f f e c t . Even th i s was not e n t i r e l y s a t i s f a c t o r y as i t pro-vided no information on the slope of the Volume/age curve nor on the age of the culmination of mean increment. The problem of size was not very c r i t i c a l i n the un-thinned model but became very important i n the experiment on thinning. It was desirable to obtain the most amount of information from a given number of runs. Computer time i s expensive and even the simple unthinned experiment involved some ?6 runs. Because a l l the responses and treatment means were quantitative, response surface designs were appropriate to the a n a l y s i s . Response surface designs were described by Box ( 1 9 5 4 ) , Box and Hunter ( 1 9 5 7 ) , Box and Draper (1963) amongst other papers. Cochran and Cox (1957) described some designs, while Myers ( l 9 7 l ) presented a complete d e s c r i p t i o n of theory and design f o r f i r s t and second order designs. H i l l and Hunter (1966) provided a l i t e r a t u r e survey of methodology. The designs are applicable to both optimization and exploration problems. A design assumes that the unknown response function can be approximated by a polynomial of known degree such that the errors i n estimating points over the domain of i n t e r e s t remain small. If the equation were to prove inadequate, biasses would be small and provision i s made to estimate higher order polynomials. An important concept i s that of r o t a t a b i l i t y . It i s necessary f i r s t to standardize or code treatment l e v e l s so that the means of the coded l e v e l s (x^) are a l l zero, the centre of the design. A design i s said to 132 b e r o t a t a b l e when t h e v a r i a n c e o f t h e e s t i m a t e d r e s p o n s e i s a f u n c t i o n o n l y o f t h e d i s t a n c e (^ >) f r o m t h e c e n t r e o f t h e d e s i g n , i . e . , = I x ^ s o t h a t t h e v a r i a n c e s a r e c i r c l e s , s p h e r e s o r h y p e r s p h e r e s a b o u t t h e c e n t r e . T h e d e s i g n t h e n h a s a s p h e r i c a l v a r i a n c e f u n c t i o n , w i t h t h e p r e c i s i o n g r e a t e s t a t t h e c e n t r e , b e c o m i n g s m a l l e r i n c o n c e n t r i c c i r c l e s away f r o m t h e c e n t r e . R o t a t a b l e d e s i g n s a r e s a i d t o b e u n i f o r m p r e c i s i o n d e s i g n s when t h e p r e c i s i o n w i t h i n t h e r e g i o n b o u n d e d b y f> = 1. i s a p p r o x i m a t e l y e q u a l . Of t h e c l a s s o f r o t a t a b l e d e s i g n s , s e c o n d o r d e r c e n t r a l c o m p o s i t e d e s i g n s a r e v e r y commonly u s e d . T h e s e c o n s i s t o f a b a s i c 2 k f a c t o r i a l d e s i g n s u p p l e m e n t e d b y a x i a l p o i n t s a t +( <* , 0,0,..., 0) t o +(0,...0 , « x ) a n d c e n t r e p o i n t s ( 0 , . . . , 0 ) . T h e c o n d i t i o n f o r r o t a t a b i l l t y i s t h a t «< = 2 ^ / ^ , w h i l e t h a t f o r u n i f o r m p r e c i s i o n d e p e n d s on t h e m i x e d f o u r t h moments o f t h e d e s i g n m a t r i x ( X ^ ) , i t s e l f a f u n c t i o n o f t h e number o f p o i n t s a t t h e c e n t r e . F o r a t h r e e d i m e n s i o n a l f a c t o r i a l , t h e number o f p o i n t s i n t h e f a c t o r i a l d e s i g n i s 8 , t h e number o f a x i a l p o i n t s 6 w i t h o< = 1 . 6 8 2 , a n d t h e number o f c e n t r a l p o i n t s f o r u n i f o r m p r e c i s i o n i s 6 ( Xij , = . 8 6 ) . By t a k i n g 9 c e n t r a l p o i n t s a n o r t h o g o n a l d e s i g n c a n b e o b t a i n e d . Some a d v a n t a g e s o f r o t a t a b l e a n d u n i f o r m p r e c i s i o n d e s i g n s a r e o b v i o u s a s t h e y a l l o w t h e d e s i g n t o b e c e n t r e d o v e r a p o i n t o r r e g i o n o f i n t e r e s t a n d a h i g h p r e c i s i o n o f t h e r e s p o n s e e s t i m a t e s o b t a i n e d t h e r e . T h e y a r e o f p a r t i c u l a r v a l u e when u s e d t o s e a r c h f o r a n o p t i m u m p o i n t , b u t a r e a l s o v a l u a b l e when e x p l o r i n g t h e r e g i o n o f i n t e r e s t . A c e n t r a l 133 composite design was used i n the thinning experiment. It was not used f o r the unthinned experiment as 9 treatment combin-ations had already been obtained i n section 6.2. It was more economical to supplement these with more runs than to design a new experiment. 6.4 Experiment 1 - Unthinned Stands The objective of the experiment was to determine the e f f e c t on y i e l d of the numbers of trees per acre at age 20 and the s i t e index. The model was run to obtain four r e p l i - . cates of the following treatment combinations t numbers of trees per acre at age 20; 200, 300, 500, 800, 1200, and l600; s i t e ; 90, 120 and 150; plus I n i t i a l density 2000, s i t e 90. Runs made i n section 6.2 were used where appropriate. Four r e p l i c a t e s would b r i n g the S.E. of the mean of the i n d i v i d u a l values of the y i e l d tables to within about ±5% of the mean. The stands were generated at age 20 and output obtained every 5 years u n t i l age 100. The behaviour of the model can now be examined i n d e t a i l . To i l l u s t r a t e both the b i o l o g i c a l and commercial responses of such treatments, two functions were f i t t e d i the maximum mean annual increment of gross volume per acre as a function of i n i t i a l density and s i t e index; the close u t i l i z a t i o n volume standing per acre as a function of i n i t i a l density, s i t e index and age. Each Individual point from the output was used, rather than treatment averages. Polynomials were used 134 which were 2nd order In terms of s i t e and age, 3rd order i n terms of i n i t i a l d e n s i t y . N o n - s i g n i f i c a n t terms were e l i m i n -ated. The f u n c t i o n s were evaluated, Tables 6 . 1 and 6 . 2 and i s o m e t r i c p r o j e c t i o n s made, see Figures 6.13 and 6.l4, 6.15 and 6.l6. For maximum mean annual increment (Max. MAI), gross . t o t a l volume per a c r e , the f u n c t i o n wasi Max. MAI = 30.005 + 0 . 0 2 5 7 8 0.N - 0 . 3 0 7 7 4 E-04.N 2 + . O O 5 7 6 5 8.S 2 + 0.45420E-03.N.S s y x = 3.7397 cu. f t . per acre per annum R2 = , 9 7 w l t h 4 and 71 d.f. where N = I n i t i a l number of stems per acre a t age 2 0 , S = S i t e Index When i n t e r p r e t i n g the f i g u r e s i t should be noted t h a t they have been sc a l e d t o o b t a i n the maximum amount of u s e f u l s u r f a c e and hence the d i f f e r e n c e s i n the l e v e l s of the response may be exaggerated. Thus f o r Max. MAI the maximum d i f f e r e n c e i n s i t e 9 0 was ~25 cu. f t . per acre per annum, i n s i t e 150 "55 cu. f t . per acre per annum or about 25$ of the mean values i n each case. C l e a r l y the i n i t i a l d e n s i t y a f f e c t e d t o t a l production of the model. On low s i t e s there was a d e f i n i t e optimum, s u r p r i s i n g l y low, a t about 1 0 0 0 stems per acre a t age 2 0 ; at medium s i t e s the optimum was a t t a i n e d w i t h a higher i n i t i a l d e n s i t y , but the surface became more of a p l a t e a u . The optimum f o r s i t e 150 was about l5°0 stems per acr e . At l e a s t 95% of the maximum production was a t t a i n e d over a range of 6 0 0 to l400 stems at s i t e 90, above 900 stems 135 Figure 6-13- Response Surface of Gross Volume Maximum MAI to Initial Density and Site-136 TABLE 6 .1 MAXIMUM MAI OF GROSS VOLUME/ACRE (CU. FT./ACRE/YEAR) AS A FUNCTION OF SITE AND INITIAL EENSITY FOR UNTHINNED STANDS STEMS/ACRE 80 1800. 79 1700. 83 1600. 87 1500. 90 1400. 93 1300. 95 1200. 97 1 1C0. 97 1000. 98 900. 97 800. 96 700. 95 600. 93 500. 90 400. 86 300. 82 200. 78 SITE 90 100 110 120 96 1 16 136 157 10 V 1 19 139 160 104 1 22 142 162 107 125 144 164 109 127 145 165 1 1 1 128 146 165 112 1 28 146 165 112 128 145 164 112 128 144 162 1 1 1 126 143 160 110 124 140 157 108 122 137 154 105 1 19 134 150 102 115 129 145 98 111 125 140 93 106 119 134 88 100 113 127 130 140 150 160 180 204 229 255 182 206 230 256 184 207 2 3 1 256 185 207 231 256 185 207 230 255 185 207 229 253 184 205 228 251 18 3 204 225 248 18 1 201 222 245 178 198 2 19 241 175 194 2 15 236 17 1 190 210 231 167 185 205 225 162 180 1 98 219 156 173 1 92 212 150 167 185 204 143 159 177 196 137 at s i t e 150. The net close u t i l i z a t i o n volume per acre (CV4) function was CV4 = -4l47.l - 54.438.S + 8.3627.N - 0.013346.N2 + 0.035977.S.N + 2.9125.S.A + 0.15786E-04.S.N2 - 0.014244.S.A2 + 0.16372E-04.A.N2 - 0.46447E-03.N.S.A + 0.30916E-05.N3 + 0.24073E-02.A3 s v i r = 685.19 cu. f t . per acre R 2 = .96 with 11 and 444 7 d.f. where A = Age, N = i n i t i a l number of stems per acre at age 20, S = S i t e Index The function i s shown i n Figures 6.l4, 6.15 and 6.16 and evaluated i n Table 6.2. In the high s i t e s the t h i r d order terms caused the function to " c u r l up at the edges" but the size of t h i s was n e g l i g i b l e and should be Ignored. The shape of the surface was d i f f e r e n t for the three s i t e s . For low s i t e s , high i n i t i a l d e n s i t i e s had a deleterious e f f e c t on merchantable volume at a l l ages. A maximum amount at any age was obtained with i n i t i a l d e n s i t i e s of 500 - 600 stems per acre. In the high s i t e s the s i t u a t i o n was more complex. At younger ages, 50 years, high i n i t i a l stockings of 800 - 1000 stems per acre yielded maximum close u t i l i z a t i o n volumes. With Increase i n age t h i s quickly changed, the maximum point attained with decreasing i n i t i a l d e n s i t i e s so that by age 90 - 100 the maximum was attained at 400 - 500 stems per acre at age 20. The behaviour of the model can now be compared with hypotheses derived from spacing t r i a l s . Many studies on 8£I Figure 6-15- Response Surface of Close Utilization Volume per acre to Age and Initial Density,Site 120-SITE 150 Figure 6-16- response Surface of Close Utilization Volume per acre to Age and Initial Density, Site 150-1>1 T A B L E 6 . 2 . C L O S E U T I L I Z A T I O N VO IU K E / AC H E ( C U . F T . / ACR E T O 4" T O P ) AS A F U N C T I O N OF A G E , S IT E A N ! IN I T I A L D E N S I T Y S I T E 9 0 • AGE S T E M S / A C R E 40 5 0 6 0 7 0 8 0 9 0 1 0 0 1 5 C 0 . - 4 8 9 8 6 6 2 0 3 7 3 0 3 9 3 8 8 5 4 5 9 1 5 1 7 0 1 4 C C . - 1 6 4 1 1 8 5 2 3 5 1 3 3 4 7 4 1 8 7 4 8 8 7 5 4 6 1 1 3 C 0 . 1 9 4 1 5 4 1 2 7 0 5 3 6 9 8 4 5 3 7 5 2 3 4 5 8 0 6 1 2 C C . 5 6 9 1 9 17 3 0 8 1 4 0 7 6 4 9 1 5 5 6 1 4 6 1 8 6 1 1 C 0 . 9 4 1 2 2 9 3 3 4 6 2 4 4 6 C 5 3 0 4 6 C C 6 6 5 8 2 1 C C C . 1 2 9 1 2 6 5 1 3 8 2 7 4 8 3 3 5 6 8 4 6 3 9 4 6 9 7 8 9 C 0 . 1 6 0 2 2 9 7 3 4 1 5 9 5 176 6 0 3 7 6 7 5 8 7 3 5 2 8 C C . 1 8 5 4 3 2 3 9 4 4 3 9 5 4 7 C 6 3 4 6 7 0 8 C 7 6 8 9 7 C 0 . 2 0 2 9 3 4 3 1 4 6 4 9 5 6 9 7 6 5 9 0 7 3 4 2 7 9 6 7 6 C G . 2 1 0 9 3 5 3 1 4 7 7 0 5 8 3 8 6 7 5 2 7 5 2 4 8 1 7 0 5 C 0 . 2 0 7 5 3 5 2 1 4 7 8 3 5 8 7 5 6 8 1 2 7 6 C 8 8 2 7 8 4 c c . 1 9 0 7 3 3 8 1 4 6 7 0 5 7 8 9 6 7 5 3 7 5 7 7 8 2 7 4 3 C 0 . 1 5 8 9 3 0 9 3 4 4 1 2 5 5 6 2 6 5 5 6 74 1C 8 1 3 7 2 C C . 1 1 0 1 2 6 3 6 3 9 9 1 5 1 7 5 6 2 0 3 7 0 9 0 7 8 5 1 S I T E 1 2 0 • A G E S T E P S / A C B E 4 0 5 0 6 0 7 0 8 0 9C 1 0 0 1 5 C C . 2 5 3 7 4 1 7 2 5 5 3 9 6 6 5 C 7 5 2 0 8 1 6 4 8 59 6 m c o . 2 6 7 2 4 3 16 5 6 9 0 6 8 1C 7 6 8 8 6 3 4 C 8 7 8 1 1 3 C C . 2 8 5 1 4 5 C 6 5 8 9 2 7 0 2 3 7 9 1 3 8 5 7 7 9 0 2 9 1 2 C 0 . 3 0 5 5 4 7 2 5 6 1 2 6 7 2 7 2 8 1 7 7 6 6 5 5 9 3 2 2 1 1 C C . 3 2 6 6 4 9 5 4 6 3 7 3 7 5 3 7 8 4 6 0 9 1 5 6 9 6 4 1 1 0 C 0 . 3 4 6 5 5 1 7 5 6 6 1 5 7 8 C C 8 7 4 4 S 4 6 2 9 9 6 8 9 C C . 3 6 3 3 5 3 6 8 6 8 3 3 8 0 4 2 9 0 1 1 9 7 5 4 1 0 2 8 5 8 C 0 . 3 7 5 3 5 5 15 7 0 0 8 8 2 4 6 9 2 4 2 1 C 0 13 1 C 5 7 1 7 C C . 3 8 0 5 5 5 9 8 7 1 2 2 8 3 9 1 9 4 1 9 1 0 2 2 1 1 0 8 1 1 6 C 0 . 3 7 7 1 5 5 9 9 7 1 5 7 84 6 C 9 5 2 3 1 0 3 5 9 1 0 9 8 3 5 C C . 3 6 3 2 5 4 9 8 7 0 9 4 8 4 3 5 9 5 3 5 1 0 4 0 9 1 1 0 7 1 4 C 0 . 3 3 7 0 5 2 7 7 69 14 8 2 9 6 9 4 3 7 1 0 3 5 2 1 1 0 5 5 3 C C . 2 9 6 6 4 9 1 7 6 5 9 9 8 0 2 5 9 2 1 0 1 0 1 7 0 1 0 9 1 7 2 C 0 . 2 4 0 3 4 4 0 1 6 1 3 0 7 6 0 4 8 8 3 7 S 6 4 3 1 0 6 3 8 S I T E 1 5 0 • AGE S T E M S / A C R E 40 5 C 6 0 7 0 8 0 9 0 1 0 0 1 5 C 0 . 5 5 6 3 7 4 7 9 9 0 4 0 1 C 2 6 C 1 1 1 5 4 1 1 7 3 7 1 2 0 2 1 1 4 C C . 5 5 0 9 7 4 4 7 9 0 3 0 1 C 2 7 3 1 1 1 8 9 1 1 7 9 3 1 2 1 0 0 1 3 C 0 . 5 5 0 8 7 4 7 1 9 0 8 0 1 0 3 4 e 1 1 2 9 0 1 1 9 1 9 1 2 2 5 2 1 2 C C . 5 5 4 1 7 5 3 4 9 1 7 1 1 0 4 6 8 1 1 4 3 8 1 2 0 9 7 1 2 4 5 8 1 1 C 0 . 5 5 9 1 7 6 16 9 2 8 5 1 0 6 13 1 1 6 1 6 1 2 3 C 7 1 2 7 0 0 1 C C C . 5 6 3 8 7 6 9 6 9 4 0 3 1 0 7 6 7 1 1 8 0 4 1 2 5 3 0 1 2 9 5 9 9 C 0 . 5 6 6 5 7 7 6 3 9 50 6 1 0 9 C 9 1 1 9 8 5 1 2 7 4 9 1 3 2 1 7 e c c . 5 6 5 2 7 7 9 2 9 5 7 7 1 1 C 2 1 1 2 1 3 9 1 2 9 4 5 1 3 4 5 4 7 C 0 . 5 5 8 0 7 7 6 6 9 5 9 6 1 1 0 8 5 1 2 2 4 8 1 3 1 C C 1 3 6 5 4 6 C C . 5 4 3 3 7 6 6 6 9 5 4 5 1 1 C 8 2 1 2 2 9 4 1 3 1 9 4 1 3 7 9 6 5 C O . 5 1 9 0 7 4 7 5 9 4 0 5 1 0 9 9 4 1 2 2 5 8 1 3 2 C 9 1 3 8 6 3 4 C C . 4 8 3 3 7 1 7 3 9 1 58 1 0 8 C 2 1 2 1 2 1 1 3 1 2 7 1 3 8 3 6 3 C 0 . 4 3 4 4 6 7 4 2 8 7 8 5 1 0 4 8 8 1 1 8 6 4 1 2 9 2 9 1 3 6 9 6 2 C C . 3 7 0 4 6 1 6 4 8 2 6 9 1 0 0 3 3 1 1 4 7 1 1 2 5 9 7 1 3 4 2 6 1>2 I n i t i a l spacings have been made, most over l i m i t e d ranges of s i t e s . S j o l t e Jorgensen (1967) wrote a comprehensive summary on the e f f e c t s of spacing, Evert (1971) summarized Canadian experience and Assman (1970) included a discussion i n his book, mainly from European examples. Comments are mainly l i m i t e d to these three authors as l i t t l e i s to be gained by repeating t h e i r l i t e r a t u r e surveys. To quote S j o l t e Jorgensen, "the t o t a l volume production i s In most cases reduced with increased spacing i n such a way, furthermore, that the reduction i s accelerated with the widening of the spacing." However there i s a site/spacing i n t e r a c t i o n stated by Assman as - low s i t e s with high i n i t i a l d e n s i t i e s s u f f e r a loss i n gross production due to d i f f i c u l t i e s i n s e l f thinning, high s i t e s require high i n i t i a l d e n s i t i e s to produce maximum gross volume. These are exactly the re l a t i o n s h i p s shown i n Figure 6.13. If i t i s remembered that i n d i v i d u a l spacing studies are usually located on one s i t e and have been measured over a few years only, the considerable confusion noted by Evert (l97l) i n i n t e r p r e t i n g the e f f e c t s of spacing on merchantable volume i s understandable when considering the complex response surface of Figures 6.i4 - 6.l6. Part of the r e s u l t s of the Wind River spacing t r i a l s (Reukema, 1969) with Douglas f i r on a poor, dry s i t e planted at various spacings showed an Increase i n gross t o t a l volume y i e l d of 4775. cu. f t . per acre at 12 x 12 spacing (3^0 per acre) over 4060 cu. f t . per acre on 4 x 4 spacing (2700 per acre) at age 4 i . The differences in net merchantable volume t o a 4 - i n c h t o p were v e r y marked; from 1500 c u . f t . p e r a c r e a t 4 x 4 s p a c i n g t o 4350 c u . f t . p e r a c r e a t 12 x 12 s p a c i n g . The r e s u l t s of the model were c l e a r l y i n a c c o r d a n c e w i t h t h e s e r e s u l t s o f Wind R i v e r . 6.5 E x p e r i m e n t 2 - Thinned Stands I n o r d e r t o a n a l y s e t h e t h i n n i n g mode of the model i t i s f i r s t n e c e s s a r y t o d i s c u s s the f a c t o r s w h i c h c o n t r o l a t h i n n i n g . There a r e two extremes t o t h i n n i n g c o n t r o l . I n the f i r s t , a f t e r s p e c i f y i n g the type o f t h i n n i n g r e q u i r e d , the amount and t h e s e l e c t i o n o f the t r e e s i s l e f t e n t i r e l y t o a s k i l l e d marker who removes t r e e s on a s i l v i c u l t u r a l " b a s i s . I n t h e second, c o n t r o l i s t o some s p e c i f i c q u a n t i t a t i v e l e v e l p r e d e t e r m i n e d i n t h e o f f i c e and the marker has l i t t l e leeway i n s i l v i c u l t u r a l c o n s i d e r a t i o n s . Modern management r e q u i r e s p r e c i s e y i e l d c o n t r o l w i t h a c c u r a t e f o r e c a s t s o f y i e l d t o use e x p e n s i v e p l a n t most e c o n o m i c a l l y . The tendency i s thus t o v e e r more t o the form o f t h i n n i n g c o n t r o l where economic c o n s i d e r a t i o n s a r e more i m p o r t a n t ( J o h n s t o n e t a l _ . , 196?). I d e a l l y t h e p r e s c r i p t i o n s a r e such t h a t the c r o p i s s t i l l a b l e t o f o l l o w the most s l l v i c u l t u r a l l y d e s i r a b l e development. I n t h i s model, t h i n n i n g c o n t r o l was s t r i c t l y n u m e r i c , t h e r e was v e r y l i t t l e scope f o r s i l v i c u l t u r a l c o n t r o l . T h i s i s u n d e r s t a n d a b l e as t h e r e was no I n f o r m a t i o n on t h e q u a l i t a t i v e a s p e c t s o f t h e s t a n d such as crown form, h e a l t h i n e s s of n e e d l e s , e t c . The method of s p e c i f y i n g b a s a l a r e a per a c r e t o be removed used by the model i s t h e one most l i k e l y t o be used i n p r a c t i s e . The methods o f removing t r e e s a r e however o n l y c r u d e a p p r o x i m a t i o n s . The problem o f d e c i d i n g how much b a s a l a r e a p e r a c r e t o remove i s t h e problem o f t h i n n i n g c o n t r o l and I s d i s c u s s e d b e l o w . V e z i n a (1963) summarized some o b j e c t i v e measures of t h i n n i n g grades and methods. Hummel's (195*) s t a n d d e n s i t y i n d e x has l o n g been used as i t q u a n t i f i e d t he s i l v i c u l t u r a l g rades of A, B, C and D. He d e f i n e d the s t a n d a r d d e n s i t y as c o r r e s p o n d i n g t o a s p a c i n g o f 20% of t o p h e i g h t and o t h e r d e n s i t i e s were e x p r e s s e d by r e l a t i n g t h e number of t r e e s p e r a c r e t o t h e s t a n d a r d . Assman (1970) c l a s s i f i e d d i f f e r e n t s t a n d d e n s i t i e s u s i n g r e l a t i v e b a s a l a r e a p e r a c r e , where the v e r y l i g h t l y t h i n n e d s t a n d was assumed t o have t h e maximum b a s a l a r e a and g i v e n an i n d e x of 1.0. Lexen's (19*3) b o l e a r e a i n d e x ( h e i g h t . number of stems . average d b h ) ; crown w i d t h - average dbh r a t i o s ( S m i t h , Ker and C s l z m a z i a , 1961); M i t c h e l l ' s r u l e , average space p e r t r e e = (dbh + X) , where X i s a s p e c i e s dependent f a c t o r (Lemmon and Schumacher, 1963) have a l s o been s u g g e s t e d . Myers (1968) t h i n n e d t o a c e r t a i n l e v e l of b a s a l a r e a per a c r e i n t h e r e s i d u a l c r o p . These a l l emphasized t h e s i l v i c u l t u r a l a s p e c t s of t h i n n i n g c o n t r o l i n so f a r as they were concerned w i t h the l e v e l of the main c r o p and i n some way c o u l d be r e l a t e d t o q u a l i t a t i v e t h i n n i n g g r a d e s . In t h e f i n a l a n a l y s i s c o m m e r c i a l t h i n n i n g s h o u l d be c o n t r o l l e d by volume p e r a c r e removed ( t h i s c an be t r a n s f o r m e d t o b a s a l a r e a p e r a c r e f o r f i e l d c o n t r o l ) ' . A l l c o s t s and revenues a r e r e l a t e d t o t h i s . J o h n s t o n and Waters (1961) p o i n t e d out t h a t a D grade t h i n n i n g d i d n o t n e c e s s a r i l y remove more volume p e r a c r e a t each t h i n n i n g t h a n a C grade i f t he d e s i r e d s p a c i n g had been a c h i e v e d e a r l y i n the c r o p ' s l i f e . A s i m p l e a b s o l u t e v a l u e o f volume p e r a c r e removed does n o t g i v e t h e I n f o r m a t i o n about the s e v e r i t y o f t h e t h i n n i n g when compared o v e r v a r i o u s s i t e s . I t was proposed t o adopt t h e t h i n n i n g c o n t r o l method of the B r i t i s h F o r e s t r y Commission, expounded i n a s e r i e s of a r t i c l e s by J o h n s t o n and Waters (1961), B r a d l e y (1963) and J o h n s t o n and B r a d l e y (196*), and c u l m i n a t i n g i n t h e F o r e s t Management T a b l e s ( B r a d l e y e t a l . , 1966) and T h i n n i n g C o n t r o l ( B r a d l e y , 19^7) d e s c r i b e d by J o h n s t o n and B r a d l e y (19&3)• T h i n n i n g i n t e n s i t y , t he a n n u a l r a t e of removal of volume o b t a i n e d by d i v i d i n g t h e volume removed by t h e t h i n n i n g c y c l e , was d e f i n e d i n terms of t h e pe r c e n t a g e maximum mean a n n u a l i n c r e m e n t , g r o s s volume p e r a c r e . I n B r i t a i n t he v a l u e of 70% o f max. MAI was used i n the s t a n d a r d regime as t h i s a p p r o x i m a t e d t h e m a r g i n a l t h i n n i n g i n t e n s i t y , t he maximum i n t e n s i t y t h a t can be a p p l i e d w i t h o u t s e r i o u s l y r e d u c i n g the volume p r o d u c t i o n . I n t h i s t h e s i s t h e pe r c e n t a g e of max. MAI g r o s s volume p e r a c r e was used as a v a r i a b l e t o c o n t r o l t h i n n i n g I n t e n s i t y . To f u l l y d e f i n e a t h i n n i n g , o t h e r f a c t o r s b e s i d e s t h e t h i n n i n g i n t e n s i t y need t o be s p e c i f i e d , t h e t h i n n i n g t y p e -e.g. l o w , f r e e o r crown, the i n i t i a l age o f t h i n n i n g , the t h l n n i n c c y c l e and the number of t h i n n i n g s . C l e a r l y , d e s p i t e 1*6 it s advantages from a management viewpoint, thinning intens-ity does not provide an absolute index of the severity of a thinning. For the same intensity, the longer the thinning cycle, the lower the i n i t i a l density at age 20, the earlier the i n i t i a l thinning age, the more severe w i l l be the effect of the thinning. These controlling factors are interrelated somewhat. Certain combinations can cause the thinning to be so heavy that the stand structure Is destroyed and the model ceases to function. Others would be so light as to have no effect. The large number of controlling factors, combined with the problem of multiple response would make the results of one large experiment incomprehensible especially in the preliminary stages of the investigation. Therefore one regime was proposed and the experiment designed to find the response of the model to site, i n i t i a l density and thinning intensity. A low thinning was carried out with no minimum dbh l i m i t . Max. MAI gross volume per acre has been shown to be a function of site and i n i t i a l density in the model. For simplicity the absolute max, MAI for the particular site was used. Three thinnings with a cycle of 10 years were executed. I n i t i a l age varied from *5 years in low sites to 30 years in high sites. A 2nd order central composite rotatable design was used with enough central points to obtain a uniform precision model. From section 6.3 this meant 6 central points and a value for ©< = 1.682 were to be used. The independent values were coded so that the centre of the design was site 120, 1*7 i n i t i a l d e n s i t y 800 stems p e r a c r e a t age 20, t h i n n i n g i n t e n -s i t y 50%. The d e s i g n wasi Coded V a l u e 0 l - l *< I n i t i a l D e n s i t y 800 1100 500 1300 300 T h i n n i n g I n t e n s i t y % 50. 70. 30. 83.6 l 6 . * S i t e Index 120 150 90 170 70 The max. MAI g r o s s volume p e r a c r e and t h e i n i t i a l t h i n n i n g age f o r the p a r t i c u l a r s i t e i n d e x w e r e i S i t e Index 120 150 90 170 70 Max. MAI c u . f t . p e r 165 230 H O 285 85 a c r e I n i t i a l T h i n n i n g Age 35 35 *0 30 *5 The volume t o be removed a t any one t h i n n i n g was s i m p l y Max. MAI f o r the s i t e x T h i n n i n g i n t e n s i t y % x 10. T h i s volume had t o be t r a n s f o r m e d t o b a s a l a r e a p e r a c r e . Because a low t h i n n i n g was c a r r i e d o u t , the average f o r m - h e i g h t o f the main c r o p was l a r g e r t h a n t h a t o f t h e t h i n n i n g , so c a r e had t o be t a k e n when c a l c u l a t i n g t h e b a s a l a r e a t o be removed. A l t h o u g h v a l u e s were c l o s e t o the r e q u i r e d volume p e r a c r e t h e r e was a g e n e r a l tendency t o u n d e r e s t i m a t e . I t was f e l t t h a t i n t h i s r e s p e c t the model was a good r e p r e s e n t a t i o n o f some o f the d i f f i c u l t i e s o f t h i n n i n g c o n t r o l i n p r a c t i s e . Four r e p l i c a t i o n s of each d e s i g n p o i n t were made. The s t a n d s were g e n e r a t e d a t age 20 and grown t i l l age 100 w i t h t h e t h i n n i n g regime a p p l i e d a t the p a r t i c u l a r i n t e n s i t y . Some of t h e t h i n n i n g s were q u i t e heavy, e s p e c i a l l y i n t h e h i g h e r s i t e s . F o r example, a f i r s t t h i n n i n g a t s i t e i n d e x 150 i n t e n s i t y 70% would e x p e c t t o remove 161O c u . f t . p e r a c r e from a s t a n d i n g volume o f 5500 c u . f t . p e r a c r e , a b a s a l a r e a of 60 s q . f t . p e r a c r e from 180 s q . f t . p e r a c r e and about l 4 8 h a l f o f the stems per a c r e (depending on p r i o r m o r t a l i t y ) . Note t h a t no a l l o w a n c e was made t o a l t e r the w e i g h t of t h e t h i n n i n g f o r a p a r t i c u l a r s t a n d w h i c h had a d i f f e r e n t t h a n e x p e c t e d amount of m o r t a l i t y . W i t h o u t c o s t s and r e v e n u e s , t h e m u l t i p l e r e s p o n s e problem becomes p a r t i c u l a r l y a c u t e w i t h t h i n n i n g . Average s i z e and volume of the f i n a l c r o p , amount and d i s t r i b u t i o n o f t h i n n i n g s , max. MAI f o r t o t a l and me r c h a n t a b l e volumes, g r o s s and n e t , a r e a l l of i n t e r e s t . I t was n o t e d t h a t i n t h e h e a v i l y t h i n n e d h i g h s i t e s , g r o s s volume MAI had c u l m i n a t e d b e f o r e t h e l a s t t h i n n i n g and i n g e n e r a l t h e shape of t h e g r o s s volume MAI c u r v e d i f f e r e d between t h i n n i n g s and w i t h t h e u n t h i n n e d s t a n d s . D e s p i t e t h i s i t was d e c i d e d t o l o o k a t t h e max. MAI of n e t and g r o s s t o t a l volume p e r a c r e o n l y , t h a t i s t o examine th e b i o l o g i c a l a s p e c t o f the t h i n n i n g , as shown by the model. Second o r d e r p o l y n o m i a l s were f i t t e d t o t h e v a l u e s o b t a i n e d from each r u n , a t o t a l o f 80 o b s e r v -a t i o n s . The a c t u a l average t h i n n i n g i n t e n s i t y was used r a t h e r t h a n the i n t e n d e d v a l u e , a l t h o u g h as mentioned, d i f -f e r e n c e s were s m a l l . A f t e r e l i m i n a t i o n o f n o n - s i g n i f i c a n t v a r i a b l e s , t h e maximum MAI of n e t volume p e r a c r e was = -63.795 + 1.132.S + . 14-189.N - .71871.N 2 - .12637-S.T s y x = 9.29 c u . f t . per a c r e p e r annum R2 - ,9! w l t h 4 and 75 d . f . where N = i n i t i a l number of t r e e s p e r a c r e a t age 20; S = s i t e i n d e x ; T = t h i n n i n g i n t e n s i t y . 1 * 9 From e x a m i n a t i o n of the f u n c t i o n and t h e i n d i v i d u a l r uns i t became c l e a r t h a t t h e model was n o t s a l v a g i n g m o r t a l i t y . As e x p l a i n e d i n C h a p t e r 4 . 8 , the model was u n a b l e t o s a l v a g e p r e v i o u s m o r t a l i t y and was l i k e l y t o o v e r p r e d l c t m o r t a l i t y a f t e r a t h i n n i n g as s k i l l e d markers would be a b l e t o remove a l l t r e e s l i k e l y t o d i e i n the n e a r f u t u r e i n p r e f e r e n c e t o h e a l t h y t r e e s o f the same s i z e . The model c a l c u l a t e d number of t r e e s t o d i e from average a f t e r - t h i n n i n g c r o p s t a t i s t i c s o n l y and because a l l the s m a l l t r e e s were removed by t h e low t h i n n i n g the s t r u c t u r e o f t h e model was such t h a t i t was f o r c e d t o s e l e c t l a r g e r t r e e s f o r m o r t a l i t y t h a n i t would have done even when u n t h i n n e d . T h i s d e f e c t was g r e a t e r t h a n e x p e c t e d . I t i s p o s s i b l e t o make a c r u d e a s s u m p t i o n t h a t a l l m o r t a l i t y a few y e a r s a f t e r a t h i n n i n g can be a n t i c i p a t e d and t h e model amended so t h a t no t r e e s d i e d u r i n g t h a t p e r i o d . The d i s c u s s i o n and r e l e v e n t programming changes a r e g i v e n i n A p p e n d i x 7, though t h e y were n o t e f f e c t e d f o r the e x p e r i m e n t s i n t h i s c h a p t e r . I n t h i s e x p e r i m e n t the e f f e c t i v e t h i n n i n g i n t e n s i t y was t h u s a c t u a l l y h e a v i e r t h a n s t a t e d , though n o t d i r e c t l y p r o -p o r t i o n a l o v er a l l i n t e n s i t i e s . Net y i e l d s i n c l u d i n g t h i n n i n g s were l o w e r t h a n i n p r a c t i s e . I f i t were assumed t h a t n e a r l y a l l the m o r t a l i t y had been s a l v a g e d , t h e n e t max. MAI volume p e r a c r e would be v e r y c l o s e t o the g r o s s . A f t e r e l i m i n a t i o n o f n o n - s i g n i f i c a n t v a r i a b l e s , the e q u a t i o n f o r maximum MAI g r o s s volume p e r a c r e was 150 = -37.029 + 0.97910.S + 0.092195.N - 0.70193E-04.N 2 + O.53735E-O3.S.N - 0.28242.S.T s y x = 8.71 cu. f t . per acre per annum R2 - ,9^ w l t h 5 and 74 d.f. where N = i n i t i a l number of tre e s per acre a t age 20; S = s i t e index; T = t h i n n i n g i n t e n s i t y . The f u n c t i o n Is evaluated i n Table 6.3 and an i s o m e t r i c p r o j e c t i o n shown i n Figure 6.17. E l i m i n a t i o n of non-s i g n i f i c a n t v a r i a b l e s has removed the t h i n n i n g I n t e n s i t y / I n i t i a l d e n s i t y i n t e r a c t i o n that was expected to be present. I n t e r p r e t i n g the f u n c t i o n suggested t h a t i n i t i a l d e n s i t y had a g r e a t e r e f f e c t than t h i n n i n g i n t e n s i t y . On low s i t e s and high i n i t i a l d e n s i t i e s , heavy t h i n n i n g d i d not reduce gross max. MAI volume much when compared t o unthinned stands, l i g h t t h i n n i n g increased i t s l i g h t l y . With low i n i t i a l dens-i t i e s , heavy t h i n n i n g reduced production by about 15$ when compared t o unthinned stands. On high s i t e s a l l t h i n n i n g s reduced gross max. MAI. The response surface was such t h a t on s i t e index 150 i n c r e a s i n g the i n t e n s i t y from 20 - 70% r e s u l t e d i n a drop of about 20 cu. f t . per acre per annum f o r a l l i n i t i a l d e n s i t i e s . Percentage-wise t h i s was much gre a t e r f o r low i n i t i a l d e n s i t i e s . When the heavy t h i n n i n g on a low i n i t i a l d e n s i t y was compared to the absolute max. MAI f o r s i t e index 150 the drop was q u i t e severe, l49 cu. f t . per acre per annum versus 230 cu. f t . per acre per annum, or some 75% of the absolute maximum. Table 6.4 summarizes the r e s u l t s . 152 TABLE 6.3 MAXIMUM MAI OF GROSS VOLUME/ACRE (CU. FT./ACRE/YEAR) AS A FUNCTION OP THINNING INTENSITY, SITE AMD INITIAL DENSITY FOR A SPECIFIC REGIME SITE 9 0. THINNIN G IN TEHSIT Y (% A B SOLUTE MAX STE MS/A CRE nNTHINNED 20 30 4 0 50 60 70 1400 110 105 103 100 98 95 92 1300 1 1 1 110 108 1 05 102 100 97 1200 112 114 1 1 1 1 0 9 105 103 10 1 1100 1 1.3 116 1 1 3 1 1 1 108 106 103 1000 113 1 16 1 1 4 1 1 1 109 106 104 900 1 12 116 1 1 3 1 1 1 108 105 103 800 1 10 114 11 1 1 08 105 103 101 700 108 110 107 105 102 100 97 600 106 105 103 100 97 95 92 500 102 99 96 94 91 89 86 400 98 9 1 88 86 83 81 78 SITE 120. THINNIN G INTEN SIT Y (% k BSO LUTE MAX STEMS/ACR E UNTHINNED 20 30 4 0 50 60 70 1400 1 65 155 152 149 145 142 139 1300 165 159 155 152 149 1 45 142 1200 1 65 15 1 1 57 154 150 147 144 1 100 164 16 1 158 154 15 1 148 144 1000 1 63 150 1 57 153 150 147 143 90 0 160 158 154 151 148 144 14 1 800 1 58 154 1 51 147 144 141 137 70 0 1 54 149 146 142 1 39 1 35 132 600 1 50 142 1 39 136 132 129 125 500 145 134 131 128 124 121 118 400 1 40 125 122 1 1 8 115 112 108 SITE 150. THINNING i INT ENS ITY {% AB SOLUTE MAX STEMS/ACR E U NTH INN'ED 20 30 40 50 60 70 1400 231 206 20 1 1 97 193 189 185 1300 230 207 20 3 199 195 190 186 1200 228 208 20 3 1 9 9 19 5 19 1 186 1 100 226 207 202 1 98 194 190 185 1000 223 204 200 1 95 19 1 187 183 900 219 200 196 1 92 187 1 83 179 800 215 195 190 1 86 182 1 78 173 700 210 188 184 1 79 175 1 71 167 600 205 180 176 1 7 1 167 163 159 500 199 170 166 1 62 158 1 53 149 400 192 159 1 55 151 147 142 138 153 T a b l e 6.4 Summary of t h e E f f e c t o f the T h i n n i n g Regime on Maximum MAI Gross T o t a l Volume p e r a c r e . S i t e I n i t i a l Index D e n s i t y max. MAI g r o s s volume c u . f t . p e r a c r e p e r annum No T h i n T h i n n i n g I n t e n s i t y 20$ 70% 90 150 1100 500 1100 500 113 102 225 200 116 (102) 103 (93) 99 ( 98) 86 (85) 206 ( 91) 185 (82) 170 ( 85) 14-9 (74) f i g u r e s i n b r a c k e t s a r e the p e r c e n t a g e o f max. MAI f o r t h e u n t h i n n e d s t a n d w i t h the same i n i t i a l d e n s i t y . 6.6 The V a l i d i t y of the Model when Thinned I n a d d i t i o n t o t h e p r e v i o u s e x p e r i m e n t , a low t h i n n i n g and a crown t h i n n i n g were performed on s i t e s 90, 120 and 150 f o r 800 stems p e r a c r e a t age 20. The low t h i n n i n g had an i n t e n s i t y o f 50% max. MAI on a f i v e - y e a r c y c l e commencing a t age 40. The number of t h i n n i n g s were 8, 7 and 6 w i t h , minimum dbh l i m i t s of 3«5, 4,5 and 5«5 i n c h e s f o r the t h r e e r e s p e c t i v e s i t e s . The amount of volume removed was i n t e n d e d t o be 270, 400 and 570 c u . f t . p e r a c r e i n each t h i n n i n g . The crown t h i n n i n g removed a c o n s t a n t b a s a l a r e a p e r a c r e of 40 s q . f t . a t ages 40, 50, 60 and 70 f o r each s i t e . I t was d i s t r i b u t e d o v e r the c r o p by removing 5>t 13*. 12., and 10. s q . f t . p e r a c r e i n the <3/4, 3/4 - 1, 1 - I t , I f - l l dbh/average dbh 154 c l a s s e s r e s p e c t i v e l y . Twenty r e p l i c a t i o n s of each t h i n n i n g were performed and the averages shown i n Appendix 3« The volume r e l a t i o n s h i p s between the low t h i n n i n g and the unthinned stand are shown i n Figur e s 6.18, 6.19. and 6,20. Dbh frequency histograms are shown i n Figure 6.2i and 6.22 f o r s i t e 90. low t h i n n i n g and s i t e 150, crown t h i n n i n g , and should be compared w i t h the ones f o r the unthinned stands (Figures 6 . H and 6.12, pages 127 and 128). With t h i n n i n g regimes such as these a l l the m o r t a l i t y o c c u r r i n g d u r i n g the period of t h i n n i n g should have been salvaged w i t h the exception of the non-merchantable t r e e s i n the low t h i n n i n g . The age of maximum MAI f o r net t o t a l volume per acre of the unthinned stands was 55. 55 and 50 f o r s i t e indexes 90, 120 and 150 r e s p e c t i v e l y , much lower than th a t of the gross volume max. MAI c u l m i n a t i o n . Standing t o t a l volumes were then 5130, 7720 and 8900 cu. f t . per acre f o r the three s i t e s . These should be compared w i t h the summary of volume r e l a t i o n s h i p s between the two t h i n n i n g regimes and unthinned stands i n Table 6.5. The t h i n n i n g reduced the age of c u l m i n a t i o n of MAI of gross volume per acre and from the f i g u r e s i t can be seen th a t the decrease of gross volume increment of the thinned stands r e l a t i v e t o the unthinned stands occurred i n l a t e r l i f e , c o n t i n u i n g a f t e r t h i n n i n g had stopped. This c o n t r a s t s w i t h decreasing the i n i t i a l d e n s i t y which had the e f f e c t of i n c r e a s i n g the age of c u l m i n a t i o n . These were the r e l a t i o n -ships suggested by Assman (1970). The amount of m o r t a l i t y i n 1 5 5 I2,000«i thinned SITE 90 unthinned 1 0 , 0 0 0 4 gross main crop-main crop plus thinnings o o 3 U 8000H E I 6000-> o 4000-//''' 2000-^ 0 ' i ' i i i i i ' 0 20 40 60 80 100 Age Figure 6-18- Comparison of Net and Gross Volumes per acre for Unthinned and Low Thinned Model-Site 9 0 -16,0001 14,000 12,000-1 10,000-^  u o § 8000-1 © E zt o > | 6000 4000H 2000-1 gross main crop main crop plus thinnings 20 SITE 120 thinned unthinned i i i 40 60 Age 80 100 Figure 6-19- Comparison of Net and Gross Volumes per acre for Unthinned and Low Thinned Model- Site 120-157 21,000-18,000 15,000^ _ o o 12,000-1 3 U a> E £ 9000 6000H 3000 gross main crop main crop plus thinnings SITE 150 thinned unthinned ~ZO 80 100 40 60 Age Figure 6 -20 - Comparison of Net and Gross Volumes per acre for Unthinned and Low Thinned Model- Site 150-158 Main Crop (after low thinning) 2 0 O 150-\00] 5 0 -o o ^ 0 1 e • £ 100-E z 8 0 -6 0 -4 0 -2 0 -0 ' Age 4 0 571 stems/acre 8 10 Age 6 0 2 9 0 stems /acre Age 50 389 stems/acre 8 10 12 Age 80 202 stems/acre 8 10 8 10 12 14 16 Age 100 180 stems /acre 10 12 14 16 18 20 D IAMETER C L A S S (inches) Figure 6-21- Diameter Frequency Distribution for the Low Thinning, Site 9 0 , Initial Density 8 0 0 stems per acre-159 Main Crop (after crown thinning) Age 40, 349 stems/acre 4 6 8 10 12 14 16 30 20 10 Q> O 0 O ^» V) E CD 20-j <fi «*-O \ 10 CD X> E 0 z 10-0 Age 50, 207 stems/acre 6 8 10 12 14 16 18 Age 60, 130 stems /acre 8 10 12 14 16 18 20 Age 80,80 stems/acre 10 12 14 16 18 20 22 24 Age 100, 71 stems/acre ^ - ^ r — > i i 10 12 14 16 18 20 22 24 26 28 DIAMETER CLASS (inches) Figure 6-22- Diameter Frequency Distribution for Crown Thinning, Site 150, Initial Density 8 0 0 stems per acre-Table 6.5 Summary of the Comparison between Unthinned, Low and Crown Thinning. I n i t i a l Density at Age 20, 800 stems per acre Volumes are cu. f t . per acre Site 90 Un-thinned Low Crown Site 120 Un-thinned Low Crown Site 150 Un-thinned Low Crown Age of max. MAI gross volume Max. MAI gross v o l . cu. f t . per acre per annum 80 80 65 108 106 103 75 70 60 l6l 155 151 70 55 55 224 212 212 At Age of Max. MAI Total Thin volume Main Crop volume Mortality during thinning Volume y i e l d If mortality salvaged C U . Volume y i e l d * Average Dbh Main Crop Stems per acre 2070 4i89 6980 5600 3425 730 286 6980 8*00 7900 616O 6 l 8 0 6070 9.6 12.3 13.5 396 202 99 253* 2300 9*25 7103 5960 8*0 *10 9425 10*80 8670 8690 8180 8110 11.6 13-3 11.1 312 186 235 1660 2830 11650 8260 7330 870 730 11650 10750 10890 11070 8780 8970 i * . o 13.0 12.7 233 2i4 193 At Age 100 Gross volume Total Net Y i e l d * C U . Volume Y i e l d * Average Dbh Main Crop ^excluding mortality 10600 10*10 9560 7870 9010 9080 7300 7*90 7*80 11.0 13.9 15-8 r correction 15600 1*710 13610 11100 12*10 12360 10570 10900 10880 l * . l 17.0 18.1 20980 19100 18390 13300 14830 15780 12760 13570 14430 18.2 20.1 21 .1 161 terms of volume a v a i l a b l e f o r s a l v a g e i s q u i t e s m a l l u n t i l o l d e r a g e s . T h i s would be n o t i c e a b l e on s h o r t r o t a t i o n s , B r a a t h e (1957) s u g g e s t e d 18% o f the g r o s s p r o d u c t i o n by age 53 i s a l l t h a t c o u l d be s a l v a g e d . The q u e s t i o n of whether t h i n n i n g reduced y i e l d has l o n g been d e b a t e d . The r e l a t i o n s h i p between i n i t i a l d e n s i t y and t h i n n i n g i n t e n s i t y has n o t been e x p l o r e d and most d i s c u s s i o n s i m p l i e d a h i g h i n i t i a l d e n s i t y u s u a l l y i n s t a n d s p l a n t e d a t 6 x 6 s p a c i n g o r l e s s . B r a a t h e (1957) gave an e x c e l l e n t summary of European e x p e r i e n c e . The M a r - M o l l e r t h e o r y (195*) was w i d e l y h e l d , f o r example i n the a r t i c l e s on y i e l d c o n t r o l from B r i t a i n . T o t a l g r o s s volume p r o d u c t i o n a t the end o f t h e r o t a t i o n was thought t o be about e q u a l o v e r a wide range of r e l a t i v e b a s a l a r e a s . H e i b e r g and Haddock (1955) used t h i s h y p o t h e s i s t o f o r m u l a t e t h i n n i n g regimes f o r Douglas f i r i n the P a c i f i c North-West. However C r a i b ' s p r e s c r i p t i o n s f o r South A f r i c a n p i n e p l a n t a t i o n s assumed a l o s s i n p r o d u c t i o n w i t h t h i n n i n g (see H i l e y , 1 9 5 9). Assman (1970) produced a more comprehensive t h e o r y i n w h i c h the r e l a t i o n s h i p of p e r i o d i c a n n u a l i n c r e m e n t w i t h r e l a t i v e b a s a l a r e a v a r i e d w i t h s i t e and age. I n younger ages, t h e removal of s m a l l t r e e s on p o o r e r s i t e s would enhance the i n c r e m e n t , w h i l e i n o l d e r ages t h i n n i n g must c u t i n t o the more e f f i c i e n t t r e e s and reduce i n c r e m e n t . Cn h i g h s i t e s t h i s would o c c u r q u i t e q u i c k l y . He d e f i n e d t h e c r i t i c a l r e l a t i v e b a s a l a r e a as t h a t w h i c h p r o -duced 95% of the maximum i n c r e m e n t and sug g e s t e d t h a t t h i s was q u i t e low a t young ages b u t became h i g h (75 - 78%) a f t e r 1 6 2 age 5°« The results of the Bovrmont Spruce plots, MacKenzie (1962), (see also Bradley, 1963) on a low s i t e with about 3000 steins per acre at age 20 indicated that heavy i n i t i a l thinning gave improved increment, but that with older ages l i g h t e r thinnings were best. Similar r e s u l t s were obtained by Carbonnier on Norway Spruce i n Sweden (see Holmsgaard, 1958). It was thought that the relationships postulated f o r Norway Spruce would apply to Douglas f i r i n the P a c i f i c North-West, c e r t a i n l y more so than would those of the pines. The behaviour of the model when thinned as shown by t h i s and the previous section mainly conformed with Assman's hypotheses. If the model's behaviour i s correct, and i f the d i f f i c u l t i e s and l i m i t a t i o n of f i e l d t r i a l s are considered, i t i s c l e a r how the Mar-Moller theory could have been formul-ated f o r the equivalent lower s i t e s and higher i n i t i a l den-s i t i e s common in Europe. On the basis of the estimates of the model's predictions f o r the p a r t i c u l a r regimes tested, the difference In gross production between the unthinned stands and the 50% i n t e n s i t y thinning i s l e s s than 10% on any stand of s i t e index l e s s than 120 and I n i t i a l density more than 1000 stems per acre at age 20. The growth relationships of the model appeared to be sound and i t i s suggested that the model i s a f a i r approximation to r e a l i t y . How accurate were the predictions of growth a f t e r thinning can only be determined by t e s t i n g against r e a l sample plot data. 163 6.7 Summary of Results The s t o c h a s t i c component of the model was shown to have a small but s i g n i f i c a n t e f f e c t on the expected time path of. the unthinned stand, reducing the values by not more than 8$ from a d e t e r m i n i s t i c model, not more than 5$ from the model w i t h the random r e s i d u a l m o r t a l i t y omitted. The v a r i a b i l i t y of the model was s m a l l i n s t a t i s t i c a l terms, the c o e f f i c i e n t of v a r i a t i o n was i n the order of 10$. This gave a range of about 4000 cu. f t . per acre f o r standing volume a t ages 80 -100 on s i t e 150. The v a r i a b i l i t y increased r a p i d l y w i t h l e n g t h of run, then remained r e l a t i v e l y constant. When "normal" d e n s i t y s i m u l a t i o n runs were compared t o pu b l i s h e d y i e l d t a b l e s , numbers of stems per acre and gross volume per acre appeared i n accordance w i t h the published r e -s u l t s . However i n o l d e r ages, standing b a s a l area and volume per acre decreased r e l a t i v e t o the y i e l d t a b l e values which were more n e a r l y approximated by the values from lower i n i t i a l d e n s i t i e s . I t was suggested t h a t the y i e l d t a b l e s were "envelopes" r e p r e s e n t i n g an average upper l i m i t t o w e l l stocked stands. The response surface of maximum MAI gross volume per acre f o r unthinned stands showed the e f f e c t of i n i t i a l den-s i t i e s a t age 20 i n t e r a c t i n g w i t h s i t e . Maximum production occurred a t about 1000 stems per acre at age 20 on s i t e 90 but w i t h about 1500 stems per acre on s i t e 150. The response of c l o s e u t i l i z a t i o n volume t o age, s i t e and i n i t i a l 164 d e n s i t y was complex and i s b e s t d e s c r i b e d by F i g u r e 6.14-. The r e l a t i o n s h i p s were s i m i l a r t o t h o s e p r e d i c t e d i n t h e l i t e r a t u r e . The model had d i f f i c u l t y i n s a l v a g i n g m o r t a l i t y when a low t h i n n i n g removing s m a l l e s t t r e e s f i r s t was c a r r i e d o u t . The h y p o t h e s i s assumed i n t h e model t h a t a t h i n n e d s t a n d would have t h e same m o r t a l i t y as an u n t h i n n e d s t a n d w i t h t h e same c r o p s t a t i s t i c s was f a l s e w i t h a s e l e c t i v e t h i n n i n g . T h i s e r r o r was n o t l a r g e w i t h a crown t h i n n i n g o r on a low t h i n n i n g w i t h a dbh l i m i t and an a p p r o x i m a t e c o r r e c t i o n i s g i v e n i n Appendix ?. T h i n n i n g i n t e n s i t y was d e f i n e d i n terms of maximum MAI g r o s s volume p e r a c r e f o r a g i v e n s i t e . F o r a s p e c i f i c regime the r e s p o n s e s u r f a c e o f max. MAI g r o s s volume showed t h a t p r o d u c t i o n was d e c r e a s e d by d e c r e a s i n g i n i t i a l d e n s i t y more t h a n by i n c r e a s i n g t h i n n i n g I n t e n s i t y . The e f f e c t o f t h i n n i n g i n t e n s i t y was more marked i n h i g h s i t e s t h a n l o w . T h i n n i n g p o s s i b l y a f f e c t e d p r o d u c t i o n more t h a n was suggested i n t h e l i t e r a t u r e b u t the re s p o n s e o f the model f o l l o w e d t h e t h e o r i e s o f Assman (1970) i n many r e s p e c t s . There was an i n t u i t i v e f e e l i n g t h a t the dbh growth was n o t r e s p o n d i n g enough t o t h i n n i n g s and t h a t m o r t a l i t y a t h i g h e r ages was o c c u r r i n g on t o o l a r g e a t r e e i n b o t h the t h i n n i n g and u n t h i n n e d modes. However the r e s p o n s e s of growth and volume p r o d u c t i o n due t o s i t e , i n i t i a l d e n s i t y and t h i n n i n g i n t e n s i t y l a r g e l y conformed t o p u b l i s h e d r e s u l t s . T a k i n g i n t o a c c o u n t the s u c c e s s o f the v a l i d a t i o n p r o c e d u r e s of C h a p t e r 5 , the r e s u l t s o f t h i s c h a p t e r i n d i c a t e d t h a t the 165 c o n f i d e n c e I n the v a l i d i t y of the model was f a i r . 166 CHAPTER 7 SUGGESTIONS FOR FUTURE WORK AND CONCLUSIONS 7.1 F u t u r e Work Once a s a t i s f a c t o r y c o r r e c t i o n f o r t h e t h i n n i n g m o r t a l -i t y has been made, t h e most p r e s s i n g need would be t o o b t a i n l o n g term sample p l o t s of Douglas f i r i n t h e P a c i f i c N o r t h -West t h a t have been t h i n n e d . T e s t s s i m i l a r t o t h o s e of C h a p t e r 5 would t h e n be p e r f o r m e d . Assuming t h a t t h e s e were s a t i s f a c t o r y an e x t e n s i v e a n a l y s i s o f t h e t h i n n i n g mode would be t h e n e x t l o g i c a l s t e p . T h i s would r e s u l t i n some i n f o r m -a t i o n o f optimum t r e a t m e n t s f o r c e r t a i n o b j e c t i v e s . The f u t u r e development o f t h e model c o u l d t h e n t a k e one o f t h r e e p a t h s . The f i r s t would be t o f o l l o w the s t a g e s o u t -l i n e d i n C h a p t e r l by a t t a c h i n g c o s t s and revenues t o t h e model, t h e n b u i l d i n g a l a r g e f o r e s t e s t a t e model. The d e s i g n of such a model would f o l l o w t h e same p a t t e r n l a i d out i n C h a p t e r 2 and would have much i n common w i t h t h e s o c i o -economic models. I t i s hoped t h a t some of t h e t e c h n i q u e s a p p l i e d t o t h o s e models c o u l d be u s e d , f o r example s p e c t r a l a n a l y s i s o r v a r i a n c e r e d u c t i o n t e c h n i q u e s . A second development would be i n the use of t h e model f o r t e a c h i n g p u r p o s e s . T h i s i s n o t o b v i o u s l y s i m p l e . P l a y -i n g w i t h t h e model w i t h o u t aim o r p r e p a r a t i o n would be l i t t l e more t h a n amusing. A l t h o u g h i l l u s t r a t i n g t h e d i f f i c u l t i e s o f growth p r e d i c t i o n , the s t o c h a s t i c n a t u r e o f the model c o u l d be m i s l e a d i n g o r f r u s t r a t i n g t o the s t u d e n t . F o r r e s t e r ' s (i960) comments on the dangers of s i m u l a t i o n games s h o u l d be remembered. However, p r o p e r l y r u n , a s i m u l a t i o n game can be b o t h e d u c a t i o n a l and i n s p i r a t i o n a l , the l a t t e r p r o b a b l y b e i n g the most i m p o r t a n t a t t r i b u t e . I f the s t u d e n t l e a v e s the game w i t h a d e s i r e t o know why c e r t a i n r e s u l t s a r e o b t a i n e d and why the model i s d e f i c i e n t , t h e n the model w i l l have s e r v e d a u s e f u l p u r p o s e . F i n a l l y t h e model needs r e v i s i o n . T h i s t h e s i s i s o n l y e q u i v a l e n t t o one pass of Watt's (1966) c y c l e i n model b u i l d -i n g . Kore d a t a s h o u l d be c o l l e c t e d , e s p e c i a l l y i n t h i n n e d s t a n d s and i n low d e n s i t y , e a r l y a g e s . A t t h e moment t h e r e s u l t s i n t h o s e a r e a s a r e an e x t r a p o l a t i o n of t h e model w i t h a l l t h e i n h e r e n t dangers o f e x t r a p o l a t i o n i n m u l t i p l e r e g r e s -s i o n . The s i t e - age r e l a t i o n s h i p s h o u l d be c o n v e r t e d f r o m t h a t o f K c A r d l e e t a l . (1961) t o t h a t of K i n g (1966). T h i s means the complete r e - a n a l y s i s of n e a r l y a l l t h e r e l a t i o n -s h i p s . Even w i t h o u t t h i s , t he r e g r e s s i o n of ZZ ( t h e s m a l l e s t g r o w i n g t r e e ) r e q u i r e s r e - e s t i m a t i n g . The a s s u m p t i o n d e r i v e d f r om P a i l l e (1970), t h a t t h e r e l a t i v e p r o b a b i l i t i e s of d e a t h f o r i n d i v i d u a l t r e e s w i t h i n a s t a n d a r e t h e same f o r a l l s i t e s and ages i s l i k e l y t o be wrong. T h i s r e q u i r e s t e s t i n g b u t u n f o r t u n a t e l y r e q u i r e s a l a r g e amount o f d a t a t o do s o . Only by c o l l e c t i n g more h i g h q u a l i t y d a t a , e n s u r i n g t h a t t h i s a d e q u a t e l y c o v e r s t h e domain of the model, and r e p e a t i n g t h e whole p r o c e d u r e can a s i g n i f i c a n t improvement be made I n t h e growth and y i e l d f o r e c a s t s o f Douglas f i r . 168 I t i s p o s s i b l e t h a t a much f a s t e r and as a c c u r a t e a model c o u l d be b u i l t u s i n g many o f the r e l a t i o n s h i p s here b u t i n t h e f o r m u l a t i o n of s t a n d t a b l e p r o j e c t i o n . 7.2 Summary and C o n c l u s i o n s The o b j e c t i v e s of t h i s t h e s i s were t o b u i l d , v a l i d a t e and p a r t i a l l y a n a l y s e a growth model osf Douglas f i r s u i t a b l e f o r use as the growth module o f a l a r g e f o r e s t e s t a t e model. The d e s i g n sequence f o l l o w e d t h a t o f N a y l o r e t a l . (1966). By u s i n g the s i n g l e t r e e as t h e b a s i c u n i t b u t n o t r e q u i r i n g i n f o r m a t i o n as t o t h e l o c a t i o n o f the t r e e s on the ground, i t was hoped t o o b t a i n r e l a t i v e l y a c c u r a t e i n f o r m a t i o n , n o t o n l y on average s t a n d g r o w t h , b u t a l s o on s t a n d t a b l e d e v e l o p -ment. Two b a s i c a s s u m p t i o n s were t h a t t h e s t a n d was f u l l y o c c u p i e d , t h a t i s t h e r e were no l a r g e i r r e g u l a r gaps i n t r e e c o v e r , and t h a t growth i n t h i n n e d s t a n d s was I d e n t i c a l t o growth i n u n t h i n n e d s t a n d s w i t h i d e n t i c a l s t a n d s t a t i s t i c s . Because of the e x t r e m e l y v a r i a b l e n a t u r e of s t a n d development, e s p e c i a l l y r e g a r d i n g m o r t a l i t y , a s t o c h a s t i c component was d e s i g n e d f o r m o r t a l i t y p r e d i c t i o n and i t s e f f e c t t e s t e d . R e p l i c a t i o n o f ru n s was r e q u i r e d t o o b t a i n a v e r a g e s . T h i s was e a s i l y performed as t h e model c o u l d s i m u l a t e t h e s t a n d gener-a t i o n a t age 20, t h i n n i n g and growth t i l l age 100 w i t h d e t a i l e d o u t p u t i n about l i seconds. The s t o c h a s t i c com-ponent had an i n t e r a c t i o n w i t h growth so t h a t the complete model had s m a l l e r v a l u e s o f e x p e c t e d volume p r o d u c t i o n t h a n a 169 nodel with the random mortality omitted. This difference was small and a deterministic model should give values which were acceptable from a p r a c t i c a l standpoint. The v a l i d i t y of the model was tested extensively against r e a l data i n the unthinned mode. However an average of 35 years simulation was the maximum period tested. No large scale differences were noted. The t e s t i n g of the v a l i d i t y of the thinning mode was by inference only. More extensive t e s t -ing against r e a l data i s necessary. In general inferences derived from the model appeared to conform with the r e s u l t s of spacing t r i a l s noted by S j o l t e Jorgensen (1967) a " d i n p a r t i c -u l a r with the theories of Assman (1970), (the hypothesis of the development of Assman's c r i t i c a l basal area was not examined). These were that i n the unthinned stand, gross production of volume was affected by i n i t i a l density. In low s i t e s , trees needed more space f o r optimal development, so that lower i n i t i a l d e n s i t i e s gave maximum production. In high s i t e s higher i n i t i a l d e n s i t i e s were required. Thinning i n t e n s i t y was defined i n terms of maximum MAI gross volume per acre f o r a s i t e , but actual thinning was controlled by basal area. The model showed that high s i t e s l o s t more increment due to Increasing thinning i n t e n s i t y than low s i t e s . In f a c t on low s i t e s with high i n i t i a l d e n s i t i e s thinning could increase gross production. The model had some d i f f i -c u l t y i n salvaging both past and the near future mortality, but a suggestion for p a r t i a l l y eliminating t h i s i s made i n Appendix 7» 170 The problems of size and multiple responses i n experi-ments to analyse the model were discussed. The technique of response surface analysis was used to overcome these d i f f i -c u l t i e s . Care should be taken to code the variables c o r r e c t l y to ensure that the domain of i n t e r e s t l i e s within the area of high p r e c i s i o n . With the modification to mortality following thinning suggested i n Appendix 7, the model appeared s a t i s f a c t o r y f o r Incorporation In a large forest estate model or f o r teaching purposes. The model could be used as a variable density y i e l d table to produce tables s i m i l a r to those given i n Appendices 2 and 3» Although unable to simulate i r r e g u l a r l y spaced, p a r t i a l l y stocked stands, the model has the advantage that i t does not require tree-locations, information r a r e l y a v a i l a b l e i n research plots l e t alone commercial f o r e s t s . It i s possible to use the model as a simple short term (10 -30 years) growth predictor by inputting plot data c o n s i s t i n g of age, s i t e , plot size and a dbh l i s t to give expected values and a crude i n d i c a t i o n of the v a r i a b i l i t y of future development. In f a c t the model i s l i k e l y to be at i t s most accurate f o r such a use. 171 BIBLIOGRAPHY Ando, T. 1968. 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USDA F o r . S e r v . , P a c i f i c NthWest F o r . and Range S t a . , Res. Pap. PNW-4. 24 pp. W o r t h i n g t o n , R.P.; D.L. Reukema; and G.R. S t a e b l e r . 1962. Some e f f e c t s o f t h i n n i n g and in c r e m e n t i n Douglas f i r i n Western Washington. J o u r . F o r . 60:115-119-183 V e z l n a , P.E. 1963. O b j e c t i v e measures of t h i n n i n g grades and methods. F o r . Chron. 39(3)»290-300. Zahner, R. 1968. Water d e f i c i t s and growth o f t r e e s . I n "Water d e f i c i t s and p l a n t growth." V o l . 2. pp. 191-254. Ed. T.T. K o z l o w s k l . New Y o r k i Acedemlc P r e s s . 184 APPENDIX 1 AITKIN'S INTERPOLATION The problem of interpolation is that of determining an approximation to an unknown function f(x) defined on an Interval [a,b]. Aitkin's interpolation is a method of find-ing an Interpolating polynomial which has the properties that i t approximates f(x) and is equal to i t at known points in the Interval (the Lagrange interpolating polynomial). Suppose that f(x) Is given on x^ , X2. • • • » x m distinct points. If n^ , n2» » • • , n k denote k distinct integers in \l ,m\ then l e t denote the Lagrange interpolation polynomial of degree ^ k-l to f(x) on x n , x n , . . . . x n , For example P l f2 ij.(x) denotes a polynomial of degree ^ 2 which Interpolates f(x) on x^ , X £ and x/j,. Aitkin's interpolation method rests on the theorem that l,2,...,k,k+l (x) = Lk+i" xk x-x k l,2,...k-l,k x" xk+l P l , 2 k - i , k + l ( x ) (1) To find the value of the interpolating polynomial of degree 3 at x, the four nearest points to x are found from x^ , X2, • • • »x m (denote these by x^,x 2 iX-^ and x^), and the value of p i , 2 , 3 , 4 e v a l u a t e d . This is most conveniently executed by repeated row by row iteration of (l) in the work table below. 185 x l x-x^ Pj.(x) x2 x - x 2 P 1 > 2 ( x ) x3 x - x 3 P 3 ( x ) P 1 ( 3 ( x ) P l ,2,3(x) X-Xlj, Pi4.(x) P l t ^ ( x ) P l ,2Mx) P1.2,3,4(*) F o r example P1.2, 3 ( x ) = 1 x2 x"x2 Pl,2<x) x - x 3 P 1 ( 3 ( x ) and P i ( x ) = f(Xi) i = 1,2,,,,,4 By i n c r e a s i n g the number of rows , a l l i n t e r p o l a t i n g p o l y n o m i a l s t o h i g h e r d e g r e e s can be f o u n d . R e f e r e n c e s Moursund, D.G., and C S . D u r l s . 19^7• E l e m e n t a r y t h e o r y and a p p l i c a t i o n of n u m e r i c a l a n a l y s i s . New Y o r k i McGraw H i l l . 297 pp. 186 APPENDIX 2 YIELD AND STAND TABLES - UNTHINNED Yi e l d and stand tables are provided f o r three i n i t i a l d e n s i t i e s , normal, 800 and 300 stems per acre at age 20, f o r three s i t e Indexes, 90, 120 and 150. A l l values are per acre. The headings f o r the y i e l d tables are obvious. The headings f o r the stand tables read (from l e f t to r i g h t ) - Age, T a r i f Number, Close U t i l i z a t i o n Volume, Intermediate U t i l i z a t i o n Volume, Numbers of trees i n each dbh inch c l a s s . 187 MAIN CROP SI 90 AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL THINNINGS * GROSS * * PRODUCTION * * T O T A L * iV. * VOL MAI * STEMS DBH TOTAL EA VOL 20 2000 2. 1 52. 471. * 471. 24 * 25 19 70 3. 0 105. 1 236. * 1240. 50 * 30 1908 3. 5 138. 1994. * 20 21. 67 * 35 1777 3. 9 156. 2590. 26 9 3. 77 * 40 1 568 4. 2 163. 3004. * 3268. 82 * 45 1288 4. 7 16 0. 3235. 3798. 84 * 50 1075 5. 1 159. 3495. * 4356. 87 * 55 907 5. 5 158. 3 72 5. * 4923. 89 * 60 81 6 5. 9 164. 4094. * 55 17. 92 * 6 5 722 6. 4 167. 4394. * 6094. 94 * 70 655 6. 8 172. 4725. * 6662. 95 * 75 600 7. 2 176. 50 15. * 7209. 96 * 80 539 7. 6 176 . 5234. * 77 33. 97 * 35 498 8. 0 180 . 551 1. * 8252. 97 * 90 462 8. 4 184. 5786. * 8753. 97 * 95 439 8. 7 189. 6100. * 9228. 97 * 100 415 9. 0 193. 6341. * 9677. 97 * STANDARD DEVIATIONS OF POPULATION 20 RUMS 25 38. 6 0. 02 1 .3 11. * 5. 0. 44 30 55. 5 0. 04 2 .6 33. * 13. 0. 60 35 69. 0 0. 05 3 .9 58. * 19. 0. 56 40 1 30. 4 0. 10 8 .3 126. * 35. 0. 94 45 1 70. 3 0. 15 13 . 2 235. * 7 1. 1 . 57 50 121. 1 0. 17 1 1 .0 218. * 92. 1 . 92 55 117. 6 0. 22 1 1 .5 242. * 108. 1. 88 60 104. 6 0. 25 10 .6 236. * 1 20. 2. 00 55 120. 2 0. 35 12 .9 290. 1 20. 2. 00 70 112. 0 0.4 1 11 .8 269. * 1 31. 1 . 88 75 1 1 6. 9 0. 5 1 14 .9 377. * 154. 1 . 99 80 101. 2 0.49 16 .2 440. * 181. 2. 31 35 83. 6 0. 5 1 14 . 3 410. * 192. 2. 27 90 80. 1 0.55 13 .2 381. * 192. 2. 14 95 71. 1 0.60 1 1 . 3 340. * 202. 2. 15 100 69. 4 0.64 12 .7 401. * 216. 2. 17 188 MAI AFT AGE STEMS ********* N CROP SI 9 0 ?,R THINNING AV. TOTAL D 3 H BA VOL ***************** * GROSS * THINNINGS * PRODUCTION * * TOTAL * AV. TOTAL * VOL MAI * STEMS DBH BA VCL *********************************** 20 800 2.6 32. 315. * 3 15. 1 6 * 25 738 4.0 73. 1052. * 1054. 42 * 30 779 4.8 105. 1 857. * 1366. 62 * 35 753 5.5 132. 2652. * 2693. 77 * 40 721 6. 1 154. 3392. * 3499. 88 * 45 677 6.6 170. 4033. * 4262. 95 * 50 633 7. 1 183. 4 601. * 4983. 100 * 55 59 6 7.5 194. 5132. * 5668. 103 * 50 549 a. o 202. 5553. * 6313. 105 * 55 51 5 8.4 209. 59 75. 6941. 107 * 70 487 8.7 216 . 639 3. * 7544. 108 * 75 451 9. 2 219. 6716. * 8116. 108 * 80 419 9.6 222. 6979. * 8663. 108 * 35 396 9. 9 225. 7276. * 9192. 108 * 90 3 71 10.3 226 . 7493. 9690. 108 * 95 347 10.7 228. 7680. * 10152. 107 * 100 327 11.0 229. 7873. * 10599. 106 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 12. 8 0. 0 1. 0 8. * 5. 0. 0 30 19. 8 0. 04 1 . 5 22. * 13. 0. 55 35 28. 2 0. 07 2. 8 4 6. * 20. 0. 69 40 43. 7 0. 10 5. 5 104. * 30. 0. 83 45 53. 1 0. 13 8. 1 171. * 39. 0. 88 50 60.2 0. 18 10. 5 2 3 7. * 45. 0. 81 55 55. 7 0. 20 10. 4 250. * 40. 0. 79 60 59. 3 0. 26 1 1 . 7 295. * 34. 0. 72 65 66. 3 0. 33 13. 3 339. * 37. 0. 62 70 60. C 0. 32 13. 1 349. * 30. 0. 55 75 53. 4 0. 36 11. 2 296. * 25. 0. 53 60 59. 7 0. 43 13. 7 3 80. * 38. 0. 66 35 60. 7 0. 47 15. 6 459. * 58. 0. 73 90 50. 8 0. 48 12. 8 392. * 85. 1 . 03 95 39. 7 0. 42 14. 5 473. * 1 13. 1. 28 100 3 5. 9 0. 43 15. 4 515. * 121. 1. 26 1 8 9 MAIN CROP SI 90 * GROSS * THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS ********************************************** * PRODUCTION * T O T A L ft V. DB H BA TOTAL VCL 20 300 2.9 15. 1 59. * 159. 8 * 25 296 5. 1 46. 761. * 762. 31 * 30 293 6.3 68 . 13 4 3. * 13 48. 45 * 35 290 7. 3 88. 1962. * 1976. 56 * 40 288 8. 0 108. 2603. * 25 28. 66 * 45 284 8.7 125. 3243. * 3290. 73 * 50 277 9.4 142. 3847. * 3947. 79 * 55 274 9. 9 158. 4479. * 4602. 84 * 60 268 1 0.5 17 3. 50 6 8. * 5245. 87 * 55 260 1 1.0 185. 5607. * 5880. 9 0 * 70 253 11.5 196 . 6152. * 6503. 93 * 75 245 12. 0 207. 6654. * 7109. 95 * 80 238 12.4 215. 7124. * 7695. 96 * 35 229 12. 9 223. 7540. * 8260. 97 * 90 222 13.3 230 . 7937. * 8799. 98 * 95 214 13.7 235. 8263. * 9298. 98 * 100 208 14. 1 242. 8626. * 9778. 98 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 6. 0 0. 05 0.6 8. * 6. 0. 51 30 7. 7 0. 05 1.2 1 9. * 13. 0. 41 35 9. 1 0. 07 1.6 34. * 22. 0. 69 40 9. 6 0. 07 2.2 49. * 30. 0. 92 45 10. 8 0. 08 2.9 70. * 40. 0. 82 50 10. 6 0. 09 3.4 84. * 47. 0. 99 5 5 10. 6 0. 1 1 3.4 38. * 54. 1. 04 50 1 1 . 8 0. 12 4.2 115. * 63. 1 . 23 65 12. 4 0. 14 4.5 127. * 69. 1 . 10 70 12. 9 0. 16 5.0 145. * 75. 1 . 10 75 12. 2 0. 17 5. 3 159. * 80. 1 . 07 80 13. 3 0. 18 6 . 5 200. * 86. 1 . 21 35 13. ~\ 0. 2 1 6.5 20 2. * 89. 1. 12 90 12. 8 0. 24 6 . 1 19 8. * 9 1. 1 . 03 95 14. 1 0. 27 7.2 239. * 94. 0. 91 100 1 5. 1 0.3 2 8.2 277. * 97. 0. 91 190 MAIN CROP SI 120 * GROSS * . THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * PRODUCTION * TOTAL * V L AI A V . DBH BA TOTAL VCL 20 1 600 2. 7 71 . 782. * 7 8 2 . 3 9 * 25 1 480 3. 7 123. 19 37. * 1964. 79 * 30 123 6 4. 6 152. 2992. * 3169. 106 * 35 1048 5. 3 172. 3936. * 4343. 124 * 40 862 6. 0 181 . 4651. * 5440. 136 * 45 710 6. 8 187. 5239. * 6484. 144 * 50 600 7. 5 192. 5770. * 7480. 1 50 * 55 519 8. 2 196. 6247. * 84 23. 1 53 * 60 442 8. 8 197. 6590. * 9306. 155 * 65 404 9. 4 204. 7127. * 10186. 157 * 70 355 10. 0 203. 7399. * 1 1005. 157 * 75 322 10. 6 207. 7810. * 1 1803. 157 * 80 296 11. 2 210. 8168. * 12560. 157 * 35 274 11. 7 213. 8510. * 13281. 1 56 * 90 256 12. 2 215. 8823. * 13960. 155 * 95 240 12. 7 218. 9096. * 14582. 1 54 * 100 228 13. 1 221 . 9400. * 15176. 152 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 93. 9 0. 06 4 . 4 55. * 34. 1. 47 30 188.3 0. 2 1 1 3 . 1 211. * 94. 3 . 15 35 1 37. 2 0. 23 10 . 2 183. * 8 8 . 2. 47 40 126.5 0 .30 12 .9 289. * 9 2 . 2. 28 45 10 6. 6 0. 36 13 . 2 320. * 10 1. 2. 32 50 103.2 0.4 1 17 .2 4 72. * 1 22. 2. 30 55 110.7 0. 48 23 .0 674. * 155. 2. 82 60 81 .9 0 .52 18 .0 551. * 1 58. -> Z. • 62 55 70. 6 0. 52 17 .0 547. * 170. 2. 68 70 58.4 0 .47 19 . 6 682. * 179. 2 . 56 75 47. 5 0.44 18 .7 681. * 187. 2. 39 60 46. 8 0 .47 20 . 6 775. * 202. 2. 51 35 46. 1 0 .53 22 .7 876. * 2 17. 2. 51 90 41.1 0 .50 22 . 5 89Q. * 226. 2. 43 95 37. 2 0. 50 21 . 4 864. * . 2 2 7 . 2. 42 100 3 1 . 6 0. 48 20 . 4 844. * 2 19. 2. 19 1 9 1 MAIN CROP SI 120 * GROSS * THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS * PRODUCTION * TOTAL AI AV. DBH TOTAL BA VOL 20 800 3.4 5 4 . 6 8 7. * 6 8 7 . 34 * 25 787 4 . 6 9 8 . 1 743. * 1748. 70 * 30 741 5 .5 134. 2888. * 29 39. 98 * 35 681 6 .3 161 . 3979. * 4152. 1 1 9 * 40 612 7. 1 182. 4949. * 53 31. 133 * 45 547 7. 8 197. 5791. * 6455. 143 * 50 482 8.6 208. 6527. * 75 15. 150 * 55 443 9.2 220. 7272. * 8527. 155 * 50 403 9. 9 229 . 7893. * 9482. 1 58 * 6 5 3 64 10.5 235. 841 9. * 10397. 160 * 70 338 11.1 24 1. 8973. * 1 1274. 161 * 75 312 1 1 . 6 245. 9425. * 12105. 161 * 30 282 12.2 244. 9662. * 1288 1. 161 * 85 260 12. 8 246. 10002. * 136 30. 1 60 * 90 248 13. 2 252. 10497. * 14344. 1 59 * 95 236 13.7 256. 10842. * 14990. 158 * 100 223 14. 1 258. 11103. * 15598. 1 56 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 1 6. 1 0. 02 1 . 2 16. * 10. 0 . 42 30 4 7. 1 0. 10 4 .7 88. * 39. 1. 3 4 35 58. 6 0. 16 7 .7 16 8. * 7 1 . 2. 06 40 69. 7 0. 20 1 1 . 3 271. * 107. 2. 66 45 9 3 . 6 0. 3 8 1b . 6 434. * 1 40. 3. 15 50 73. 8 0. 39 15 . 3 429. * 159. 3. 26 55 7 7. 6 0. 4 7 18 .8 559. * 192. 3. 58 50 68.0 0. 49 18 .0 562. * 207. 3. 60 65 62. 7 0. 54 17 . 5 561. * 2 17. 3. 48 70 59. 5 0. 60 13 .3 6 3 7. * 2 27. 3. 24 75 53. 6 0. 63 18 .9 671 . * 237. 3. 22 30 42. 9 0. 64 16 . 6 618. * 231. 2. 89 35 3 4 . 8 0. 60 16 . 1 62 8. * 2 29. 2. 78 90 30. 9 0. 62 15 . 1 606. * 229. 2. 56 95 30. 8 0. 64 15 . 3 61 7. * 2 35. 2. 59 100 29. 5 0. 68 14 .8 602. * 235. 2. 37 \ 1 9 2 MAIN CROP SI 120 * GROSS * THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS   * PRODUCTION * TOTAL   AI AV. DBH B A TOTAL VOL 20 300 3. 9 28 . 392. * 392. 20 * 25 295 5. 6 5 6 . 1107. * 11 10. 45 * 30 29 3 6. 9 8 2 . 1926. * 19 34. 65 * 35 287 7. 9 106 . 2804. * 28 33. 81 * 40 279 8. 8 128. 3 70 9. * 37 7 3. 94 * 45 271 9. 6 149. 4601. * 4728. 105 * 50 258 10. 3 166. 5439. * 5674. 113 * 55 242 11. 1 181 . 6207. * 6595. 120 * 50 232 1 1. 8 195. 6971. * 7495. 125 * 65 224 12. 4 209. 7742. * 8386. 129 * 70 213 13. 1 220 . 8423. * 9251. 1 32 * 75 204 13. 7 230. 9074. * 10086. 134 * 30 198 14. 2 24 1. 9770. * 10894. 136 * 85 189 14. 8 249 . 10337. * 1 1669. 137 * 90 180 15. 4 255. 10823. * 123 97 . 138 * 95 1 71 16. 0 259. 1 1195. * 13058. 138 * 100 1 63 16. 5 26 4 . 11581. * 135 8 6 . 137 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 6.0 0. 05 0 . 9 12. * 8 . 0 . 51 30 7. 5 0. 06 1 . 2 26. 18. 0 . 61 35 8. 3 0. 07 1 .8 41 . 28. 0 . 83 40 10. 9 0. 08 2 .8 74. * 43. 1. 1 8 45 12.2 0. 1 1 3 .9 112. * 6 1 . 1 . 40 50 1 7. 0 0. 19 5 . 7 1 68. * 7 4 . 1 . 46 55 1 8. 2 0. 27 6 . 8 213. * 84. 1 . 60 50 1 6. 9 0. 26 7 . 1 240. * 100. 1. 62 65 19.2 0. 3 1 9 . 0 317. 1 20. 1 . 79 70 22. 7 0. 40 12 . 1 44 1. * 142. 2. 04 75 24. 6 0. 50 1 3 .7 516. * 166. 2. 21 30 23. 1 0. 5 1 13 . 3 50 6. * 187 . 2 , 37 85 21 .3 0. 5 1 1 3 .8 544. * 2 12. 2. 58 90 25. 2 0. 65 17 . 5 713. * 24 1. 2. 86 95 25.2 0. 74 13 . 4 755. * 26 3. 2. 74 100 24. 6 0. 76 18 . 4 767. * 28 1. 2. 85 193 MAIN CROP SI 150 * GROSS * THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS ******************************************** * PRODUCTION * TOTAL A V . DBH TOTAL BA VCL 20 1200 3. 5 87. 1249. * 1249. 62 * 25 1003 4. 8 137. 2785. * 2873. 1 1 5 * 30 820 5. 9 170. 4278. * 46 12. 154 * 35 680 7. 0 196. 5664. * 6 3 4 4. 181 * HO 580 8. 0 215. 6916. * 8007. 200 * 45 489 9. 0 227 . 7920. * 9559. 212 * 50 40 1 10. 0 230. 8584. * 10970. 219 * 55 359 10. 8 239 . 9377. * 12295. 224 * 60 321 1 1. 6 246 . 10072. * 135 37. 226 * 55 285 12. 4 248. 10609. * 14713. 226 * 70 254 13. 2 252. 1 1 155. * 158 30. 226 * 75 226 14. 0 252. 11519. * 16868. 225 * 80 207 14. 7 253. 11935. * 17847. 223 * 3 5 195 15. 3 255. 12359. * 1 87 7 3 . 221 * 90 177 15. 9 252. 12493. * 196 17. 218 * 95 1 67 16. 4 253. 12788. * 20385. 21 5 * 100 1 56 1 7. 0 25 3. 12976. * 21106. 21 1 * STANDARD DE VIATIONS OF POPULATION 20 25 99. 8 0. 16 6.9 110. * 65. 2. 71 30 1 17.8 0. 28 12.6 266. * 1 48. 4. 82 35 105. 5 0. 35 14.9 371. * 2 12. 6. 21 uo 100.2 0. 44 20.2 590. * 290. 7.27 45 10 1.1 0. 57 24. 4 774. * 367. 8.20 50 88.3 0. 65 27.5 948. * 4 31. 8. 57 55 86. 4 0. 6 7 32.3 1 190. * 5 11. 9.23 50 77. 7 0. 69 33.2 1289. * 591. 9. 76 55 68. 6 0. 74 .3 3.6 1.3 65. * 664. 10.17 70 58..1 0. 74 31.0 1300. * 7 25. 10.31 75 41.4 0. 65 25. 3 1113. * 753. 9.95 80 3 6. 5 0. 58 25.2 113 7. * 784. 9. 81 35 39. 9 0. 70 29 .0 1 340. * 8 12. 9. 60 90 34. 6 0. 69 27.5 1317. * 8 27. 9. 13 95 27. 1 0. 64 22. 3 10 8 7. * 8.32. 8. 91 100 24.4 0.64 23.0 1 1 53. * 8 38. 8. 34 194 MM AFT AGS STEMS ********* N CROP SI 150 * GROSS * THINNINGS ER THINNING * PRODUCTION * AV. TOTAL * TOP AL * AV. TOTAL DBH BA VOL * VOL MAI * STEMS DBH BA VOL ***************>************************************ 20 800 4. 0 7 6 . 1198. * 1198. 60 * 25 749 5. 3 125. 2681. * 27 15. 109 * 30 63 7 6. 5 16 1 . 41 76. * 4389. 146 * 35 562 7. 6 191 . 5669. * 6 104. 174 * 40 479 8. 7 210. 6912. * 7747. 194 45 420 9. 6 226 . 8022. * 9298. 207 * 50 359 1 0. 6 236 . 8899. * 10745. 215 * 55 316 1 1. 5 244. 970 1. * 12089. 220 * 50 276 12. 4 247. 10268. * 133 33. 222 * 55 253 13. 2 254. 10979. * 145 30. 224 * 70 233 14. 0 26 1 . 11650. * 15666. 224 * 75 210 14. 8 262. 12107. * 167 30. 223 * 80 1 86 15. 6 259. 12326. * 17712. 221 * 35 1 72 16. 3 26 1 . 12720. * 18643. 219 * 90 1 56 1 7. 0 257. 12822. * 19490. 217 * 95 149 17. 6 259. 13195. * 20261. 21 3 * 100 138 1 8. 2 257. 13305. * 20976. 210 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 33. 1 0. 08 2 .9 50. * 29. 1. 14 30 85. 1 0. 29 10 . 4 229. * 1 10. 3 . 71 35 85. 5 0. 35 13 . 6 354. * 204. 5. 71 4 0 81. 1 0. 45 19 . 6 590. * 3 0 2. 7. 53 45 71. 9 0. 49 20 . 7 684. * 386. 8. 56 50 68. 4 0. 58 23 .3 S1 8. * 4 6 1 . 9 . 26 55 56. 8 0. 57 22 .8 849. * 520 . 9 . 40 60 53. 6 0. 68 24 . 6 9 5 8. 556. 9. 3 7 55 54. 0 0. 79 29 . 3 1 203. * 6 17. 9 . 57 70 4 6. 9 0. 82 28 . 4 1213. * 680 . 9 . 63 75 4 1 . 9 0. 89 29 . 1 1 289. * 7 37. 9 . 77 80 37. 0 0. 90 28 .7 1317. * 780 . 9 . 81 35 32. 0 0. 82 29 .7 1 408. * 8 2 3 . 9 . 58 90 31. 0 0. 88 31 .6 1 539. * 365. 9 . 62 95 31. 4 0. 99 3 3 .9 1 679. * 904. 9 . 44 100 2 6. 7 0. 97 29 . 7 1 502. * 9 40. 9 . 46 195 AGE **** MAIM CROP SI 150 * GROSS * THINNINGS AFTER THINNING * PRODUCTION * AV. TOTAL * TOTAL * AV. TOTAL STEMS DBH BA VOL * VOL MAI * ST!!MS DBH 3A VOL 20 30 0 4. 9 4 3 . 754. * 7 54. 3 8 * 25 298 6. 5 7 5 . 1740. * 1743. 70 30 288 7. 8 105. 2885. * 29 14. 97 * 35 278 8. 9 133. 4119. * 4199. 120 * 40 266 9. 9 158. 5369. * 5534. 138 * 45 251 10. 8 18 1. 6598. * 6.8 85 . 153 * 50 233 11. 8 200 . 7730. * 8207. 1 64 * 55 217 12. 8 216. 8778. * 9482. 172 * 60 20 6 13. 6 232. 9806. * 10714. 1 79 * 55 193 14. 4 245. 10741. * 119 18. 183 * 70 1 76 15. 4 254. 11504. * 13066. 1 87 * 75 1 67 16. 1 264. 12334. * 14160. 189 * BO 1 54 17. 0 27 0 . 12973. * 15193. 190 * 35 146 17. 7 276 . 13627. * 16 174. 190 * 90 136 1 8. 5 279 . 14065. * 17080. 190 * 95 127 19. 2 28 1. 14444. * 17895. 188 * 100 1 1 9 20 . 0 283. 14751. * 18655. 1 87 * STANDARD DEVIATIONS OF POPULATION 20 RUNS 25 2. 7 0. 0 0 . 7 8. * 4. 0 . 31 30 6. 1 0. 06 1.2 30. * 15. 0 . 59 35 11.6 0. 1 1 2.8 80. * 3 3. 1 . 03 40 13 .8 0. 14 4 .4 135. * 63 . 1. 68 45 1 6. 3 0. 2 1 6 . 1 208. * 101 . 2. 27 50 1 6. 4 0. 28 6 . 4 236. * 1 29. 2. 49 55 1 6. 1 0. 32 7 . 5 283. * 155. 2. 91 60 1 7. 8 0. 39 9 . 6 382. * 182. 3 . 05 55 17. 2 0. 42 10.9 452. * 2 12. 3. 42 70 18. 8 0. 5 3 13.1 5.6 8. * 247. 3 . 69 75 1 5. 8 0. 48 12.5 567. * 284. 3. 77 80 1 5 .5 0. 50 13.7 62 8. * 3 14. 3 . 95 35 1 5. 4 0. 57 14.9 712. 344. 4. 09 90 14. 8 0. 57 16.8 824. * 375. 4. 21 95 1 4. 5 0. 69 16 . 2 607. * 396. 4. 14 100 1 5.4 0. 86 18.6 944. * 4 19. 4. 27 SITE 90 2000 STEBS/ACRE AT AGE 20 AGE TARIF CU VOL 10 > 7 . > 13. 2 3 14 20 1 8 . 1 0 01550 370 80 30 22. 3 0 0 2 30 86"4 513 t o 2 5 . 2 35 0 0 "428 621 50 2 7 . 2 699 0 0 71 350 60 2 8 . 9 1643 0 0 10 1 214 70 3 0 . 3 2688 0 0 II 36 tiO 3 1 . 6 3624 0 0 2 12 90 3 2 . 7 45S6 13 0 2 6 100 3 3 . 5 5340 218 0 1 3 AGE TARIF CU VOL 10 > 7 . > 13. 2 3 14 20 1 8 . 1 0 0 420 260 96 30 2 2 . 3 360 0 0 116 248 140 2 5 . 2 1UU0 0 - 128 50 2 7 . 2 2780 0 28 60 2H.9 4149 152 0 70 3 0 . 3 5313 860 0 80 3 1 . 6 6159 16 26 0 90 3 2 . 7 6811 2118 0 100 3 3 . S 7302 3223 0 KG E TARIF CU VOL 10 > 7 . > 13 . 2 3 It 20 18 . 1 0 0 l i t 120 43 30 2 2 . 3 616 0 33 10 2 5 . 2 1959 166 0 50 2 7 . 2 3300 881 60 2 8 . 9 1600 1769 70 3 0 . 3 57,86 2756 80 3 1 . 6 67H0 3811 90 3 2 . 7 . 7539 4867 100 3 3 . 5 8216 5877 *• BAIN 6 7 8 CROP 9 10 DBH CLASS 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 218 281 3 1 « 251 129 56 27 13 8<4 167 69 168 123 173 117 163 112 117 100 70 81 146 6<4 50 94 92 81 71 58 <48 1 7 3 145 2 70 59 38 3 66 55 50 30 61 5 5 U 7 (43 3 22 S I T E 90 800 STEBS/ACRE AT AGE 20 *• NAIN CHOP » • 5 2 « 219 175 111 67 31 12 6 1 DBH CLASS 10 11 12 13 14 15 16 100 60 3b 189 9 7 59 113 30 1H8 137 75 51 39 31 12 80 126 914 58 143 3 3 28 18 149 88 101 64 46 3 5 28 25 18 1 26 52 83 68 U7 3 5 30 25 22 16 13 31 64 6 3 48 36 20 26 22 20 7 16 42 56 44 36 29 24 22 19 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 2 13 2 18 10 S ITE 90 300 STEBS/ACRE AT AGE 20 ** WAIN CROP •* DBH CLASS 71 19 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 80 47 26 18 14 4 58 58 58 33 21 1 6 13 10 3 30 42 45 50 34 22 16 13 10 9 6 0 42 34 40 43 31 21 15 12 10 9 8 5 0 19 28 28 36 37 27 19 14 11 9 8 8 6 3 0 2 26 22 27 32 31 22 16 12 10 9 8 7 7 5 17 18 20 27 29 25 18 14 11 9 8 7 6 6 5 2 10 12 16 21 25 26 20 15 11 10 9 7 7 6 6 5 3 CTs SITE 120 1600 STENS/ACR E AT AGE 20 AGE TARIF CU VOL IU * * MAIN CROP ** DBH CLASS > 7 . > 13. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 .23 21 22 23 24 20 1 9 . 9 0 0 744 576 2 08 72 30 2 5 . 5 483 0 0 266 429 265 140 101 34 no 2 9 . 3 2090 0 0 27 126 245 180 111 82 68 24 50 3 2 . 1 3806 0 0 2 15 78 136 114 87 60 54 49 6 60 3 4 . 3 5U56 218 3 16 57 78 69 6 3 46 45 40 25 1 70 3 6 . 2 6629 17 32 0 6 17 40 54 50 46 3b 36 33 29 6 ao 3 7 . 9 7579 3242 0 1 7 16 34 32 42 35 30 31 28 28 11 90 3 9 . a B309 4 6 27 0 1 2 8 15 27 28 34 26 29 24 24 26 11 1 100 « 0 . 5 8908 5798 1 4 8 17 19 27 27 22 26 22 22 22 10 1 S ITE 120 800 STEMS/ACRE AT AGE 20 AGE TARIF CU VOL IU *» MAIN CROP ** DBH CLASS > 7 . > 13. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23 21 22 23 24 20 1 9 . 9 0 0 188 324 168 80 40 • 30 2 5 . 5 1 100 0 0 53 173 205 135 71 50 37 16 40 2 9 . 3 3133 0 57 90 139 115 69 46 36 30 26 2 50 3 2 . 1 52 83 1076 12 30 60 98 81 54 41 31 26 24 21 4 60 3 4 . 3 7014 2545 4 9 24 51 75 56 42 34 27 23 21 19 16 1 70 3 6 . 2 8289 3996 1 4 9 24 46 48 4 3 33 28 23 21 18 17 17 7 bO 3 7 . 9 90 83 5325 0 1 3 1 1 20 39 33 32 25 23 21 18 16 15 14 9 1 90 3 9 . U 9949 6724 1 5 11 2 3 25 29 24 22 21 17 16 14 14 14 13 2 100 UO .5 10565 7804 1 2 7 12 20 23 22 19 19 18 15 14 13 13 11 9 3 S ITE 120 300 STEMS/ACRE AT AGE 23 AGE TAB IF 1 CD VOL III • * MAIN CROP »* DDI! CLASS > 7 . > 13. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23 21 22 23 24 20 1 9 . 9 21 0 22 105 95 40 24 14 30 2 5 . 5 1 164 0 30 56 62 56 3 1 21 15 12 10 1 ao 2 9 . 3 3060 874 0 26 38 40 47 39 25 17 13 11 9 8 6 50 3 2 . 1 4867 2212 30 26 30 36 35 25 17 13 11 9 8 7 6 5 bO 3 4 . 3 6456 3716 12 17 19 23 29 30 22 16 12 10 9 8 7 6 6 5 2 70 3 6 . 2 7924 5337 3 15 12 15 21 24 24 19 14 11 10 8 7 6 6 5 5 5 2 bO 3 7 . 9 9264 6939 0 11 8 11 14 19 22 20 16 13 10 9 8 7 6 6 S 5 4 90 3 9 . 4 10314 8366 0 6 5 8 9 14 16 19 16 13 11 9 8 7 6 6 5 5 5 100 <t0.5 11070 95 37 0 4 3 5 6 10 12 15 15 14 11 9 8 7 7 6 5 5 5 28 29 30 31 32 33 30 31 32 33 29 30 31 32 33 r i ro ro ro CM IN ro ro ro ro o O rt ro CP- o CM CN 00 CO CN IN p* r* n CN vO v0 IN <N m m Ol CN CT CT CN CH f» in ro CN IN CN m — IN <N * * fN r» CT fNJ st r» CN «- «- CN O r- ro in cr O IN *~ *~ CN CT* (N vo *n fN CT-*r~ r- r- «- r- r-CO ro co co cr cr 00 *"~ *~ p- C 4 O yj "rt J» p-r" «- CM »- «- t- r* \D 3 (sj o» I s in cr •O (N •-»-•- *-o o CN m lT» ^ * O CD P» fN in cn «- (N fsi (N »- r- f- «-u to CO VI U -a ^ vO M r- O ^ <J cr -c . j *- CN IN CN IN «~ ^ -e C) CJ) r- ro N O v 0 3 r N r > - ~ f-« -e ~ *~ CN ro CN oj rsj •< a: *~ CO co UJ CJ <N m r i o r* CP fN CP DJ Q fN 20 rx n M rN r f CC CJ ( J <* CO <C m uO >D O CT *S CO N. ro fO fO fN r» i — \ CN t-~ rz • O vo »— P- cr cr CO ro fZ • O in CO • «- =r CT CT fO rN «- C-J • ro fr« VI & ro U5 fN o (N (O r V) CU fN r» er- in m in cr IN CO fr-o as o ee o t> o t» rg vO CC r- O « ry n N CO r-CO vO i> vO m r- 00 in X z w M o -t cr o -c i» ro 1/1 cr* o m rM m » * w • o o •fl CO r* r- o u * f- vO CD vo «- IN e-« vO 3 n M o ro in oo f» \Q CN in ro (N m in — fM *~ fN *~ *" in 0" r» «- O O sO Cf cr CO CN *— 3- in o fN *" CN O fN in CC ro P- i n ro CO =x fM in o o o CM IN ro IN ro tN o »- iD 3 i/i ip vo rs! • O O co O \o rt cr* P* r— — ro »-« *~ rsi x so o r- »- -^ *~ rs a- r- co o «-»J A -3 A O O > • o CT ro in co fM un co o > • ro O o r-, O vA r- CC J n P- cr- o » o r- '-O n O n t? rr ro u (N in P* o* o »- IN u fM A Ow p* in o r» fN ^  ct M OS • • • • • • t • • c: •* CO r* o> fN ^ %o *c t- rM fN H fM fM w O O o o o o o o o u O O u IN ro cr in ^ r— 3> ct o o rM ro «c •< ro ro ro fM ro r- ro ro ro ro o ro CM ro ro CP cT ro ro CM X> (N cf cf CN r> cf cf cf ct fM CN *- =r cr sr cr CN r- in cr cr cr cr cf (N CO cr> St cj cf cf cf cj CN vO <T-n* ro in in in in m IN O O o fM in tn in <n in in fM o o in in m in in i n CN 3- O vO vO vO vO vO vO w~~ ^ * *** fM fM cr CN o o CP cT vO vO O P* P* P» r- •* r- r~ rn ZT ro CN CN Ct 00 %o P* p- 00 CO 50 P* *~ r " r— in -O cr CT P* r* cc CO Cf* Cr* CO CO *~ *~ ^* r- M3 fM vO P* CO o o o o co r> ^ * r- r- r— T~ O m cr in IN CP* fN CN o r- CN CN O CP tA W irt o o> X CO ^n fM r— U cT CP o fM cf cf cf t- CO * : »— r— ^* r- T— ^ -CJ ro a- CC vO ro F-* ro cjr* fM in co r* cr «- p* fM fM CM ^~ « £ £ «~ r- r- r- r— CO r» vo in ro O vC CO Q fN fM vO O fM CO fM CO cf fM fM fM fM f-» CC fM f j r r o «- ro in cr cj-' *— ro r— vO CM cr co in CM ro ro ro fM •— \ fM fM fM to r* o >n ro r-tr »- ro in o c» ro n * CS CO O CC ro ro CN CJ * ro CN t< ro p* r** fM *~ tn CC P- P- ro ro p. cr IN *-in vn ro t— o CT» fM ro CN ^ * o CS o f) in i * O GO ro O O ro r* ro fM r» O I T I N P O r- in <N CO CT ro 9— 3S M co a- cr» o «^ fM rM CO in C st <N ** *-co ro in r- fM in fM # CN a ro o w • P* ro CO o O cf fN r- O vO H vO ro cT fM fM M to a co o P» CO ro in r» sr ro r. o o *~ cT CO CO ro m o fN ro ro cr- ST r- o o o ro P- o >n fM =r o X =» o •o fM ro o ro in O* fN \0 CN |Ti r~ r> cr> c vO CO M r- CT »- fO vO cr* co ro ro r- ro in CO cr. O fM cT -O CO o fN ro A o cc cr> o o r* o > • sTJ O vO in 03 P* cT co p» fN O vO r- r- r- tn r- in CT ro cf o f-* in <C »- vO c r- fM fN p- CN cr* cf =f »-in oc Cr* fN fM u fM c; P- CP o fM rt cr A IT* O r- fM f» in O P- fM Cf CC • t • • » • • • ro j-.. rj> rN c3" vO < *— CO ro. CP fM cr \o p» ro ro ro cj CT cj CT fM fM ro ro ro cf cf cf cf o o o o o o o O O O O O O O O O a -.n vo r* co cy o fM ro c» in \0 P* CO Cf* o 4 199 APPENDIX 3 YIELD AND STAND TABLES - THINNED Y i e l d and stand t a b l e s are provided f o r a low and a crown t h i n n i n g on each of three s i t e indexes, 9°, 120 and 150, The i n i t i a l d e n s i t y was 800 stems per acre a t age 20. The low t h i n n i n g was a t a t h i n n i n g i n t e n s i t y of 50% maximum MAI on a f i v e y e a r l y c y c l e commencing a t age 40. The number of t h i n n i n g s were 8, 7 and 6 w i t h minimum dbh l i m i t s of 3.5. 4,5 and 5.5 inches f o r the three r e s p e c t i v e s i t e s . The amount of volume removed was intended t o be 270, 400 and 570 cu. f t . per acre i n each t h i n n i n g . The crown t h i n n i n g removed a constant b a s a l area of 40 sq. f t , per acre at ages 40, 50, 60 and 70 f o r each s i t e index. I t was d i s t r i b u t e d over the crop by removing 5»» 13•» 12., and 10. sq. f t . per acre i n the 3/*, 3A - 1. 1 - l i , It - li dbh/average dbh c l a s s e s r e s p e c t i v e l y . For a d e s c r i p t i o n of the heading of the t a b l e s , see Appendix 2. 200 MAIN CROP SI 90 * GROSS * THINNINGS AFTER THINNING AV. TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  * PRODUCTION * TOTAL * AV. DB H BA TOTAL VCL ****** 20 800 2.6 32 . 315. * 315. 1 6 * 25 790 4 . 0 7 3 . 1051. * 1054. 42 * 30 780 4 . 8 105. 1857. * 1366. 62 * 35 765 5 . 5 133. 2666. * 2698. 77 * 40 731 6 . 0 155. 3406. * 3504. 87 * 40 571 6. 5 139. 3127. * 3504. 87 * 160 4.2 1 6. 278. 45 551 7. 1 158. 3802. * 4255. 95 * 45 461 7.4 144. 351 1. * 4255. 95 * 90 5 . 3 14. 290. 50 446 7. 9 160 . 4129. * 4957. 99 * 50 389 8. 2 143. 3855. * 4957. 99 * 58 6. 1 12. 275. 55 371 8 .7 160. 4 3 79. * 56 12. 102 * 55 329 9. 0 150. 4113. 56 12. 102 * 42 6 .8 11. 266. 60 322 9.5 164. 4687. * 6243. 104 * 50 290 9.7 154. 4433. * 6243. 104 * 3 2 7 . 4 10. 254. 65 279 1 0. 2 165. 4902. * 68 43. 105 * 55 254 10.5 156. 4665. * 68 4 3. 105 * 25 8.0 8. 237. 70 249 10 .9 167. 51 54. * 74 19. 106 * 70 227 1 1.2 159. 4907. * 74 19. 106 * 21 3 . 5 8. 246 . 75 222 11.7 168. 5359. * 7969. 106 * 75 205 11 .9 16 1. 5137. * 7969 . 106 * 16 9 .1 7. 221 . 80 202 12.3 171 . 5604. * 8498. 106 * 35 199 12.7 180. 6041. * 90 14. 106 * 90 194 13.2 187. 6404. * 9508. 105 * 95 189 13.5 193. 6729. * 9971. 105 * 100 1 80 13 .9 196. 69 3 9. * 104 11. 104 * MAIN CROP SI 90 * GROSS * THINNINGS AFTER THINNING * PRODUCTION AV. TOTAL * TOTAL AGE STEMS DBH BA VOL * VOL MAI * STEMS * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * AV. DBH BA TOTAL VOL 20 800 2.6 3 2 . 315. * 3 15. 1 6 * 25 792 4 . 0 7 3 . 1055. * 1056. 42 * 30 787 4 . 8 106 . 1867. * 1372. 63 * 35 769 5 . 5 134. 2673. * 2704. 77 * 40 730 6. 0 155. 3398. * 35 0 7 . 88 * 40 479 6.4 114. 2557. * 3507. 88 * 25 1 5 . 3 41. 84 1. 45 465 7. 1 134. 3243. * 4 2 39. 94 * 50 444 7.7 15 1 . 3879. * 4954. 99 * 50 2 83 8. 2 110. 2872. * 4954. 99 * 16 1 6 . 7 41. 1008. 55 276 8. 9 125. 3430. * 5566. 101 * 60 269 9.5 139. 3999. * 6177. 103 * 50 1 63 10. 2 100. 2906. * 6 177. 103 * 101 8 . 3 39. 1 093. 65 167 10.9 112. 3394. * 65 84. 103 * 70 166 11.5 125. 3894. * 7 196. 103 * 70 100 12. 2 8 4 . 2 64 7. * 7 196. 103 * 66 10. 4 41. 1247. 75 100 12. 9 9 4 . 30 4 4. * 7599. 101 * 80 99 13.5 103. 3425. * 8002. 100 * 35 98 14.2 111. 380 1. * 8405. 99 * 90 97 1 " . 7 120. 4184. * 8303. 98 * 95 96 15. 3 128. 4540. * 9186. 97 * 100 95 15 .8 136. 4886. * 9564. 96 * 201 MAIN CROP SI 120 * GROSS * THINNINGS AFTER THINI! TNG * PRODUCTION * AV. TOTAL * TOTAL * AV. TOTAL AGS STEMS DBH 3A VOL * VOL MAI * STEMS DBH BA VOL ************************** *********************************** 20 800 3.4 54. 687. * 687. 34 * 25 790 4.6 98. 1 747. * 1750. 70 * 30 73 9 5.5 133. 28 32. * 29 38. 98 * 35 682 5. 3 162. 3 9 9 3. * 4 1 58. 119 * UO 61 5 7. 1 182. 49 78. * 5 3 4 3. 134 * u o 483 7.6 16 3. 4 514. * 5343. 134 * 13 1 5. 2 20. 464. 4 5 4 50 3. 3 132. 5448. * 64 36. 143 * 45 3 79 8.7 166 . 4993. * 6 4 36. 143 * 72 6.6 17. 455. 50 361 9.4 185. 5909. * 7454. 149 * 50 314 9.7 170. 5478. * 7454. 149 * 48 7.5 15. 43 1. 55 292 1 0. 5 185. 6230. * 8386. 1 53 * 55 258 1 0. 8 172. 5831. * 8386. 1 53 * 3 3 8.3 12. 398. 50 246 1 1.5 185. 6529. * 9246. 1 54 * 60 221 11.7 173. 6145. * 9246. 1 54 * 25 9.2 11. 384. 55 214 12. 4 187. 6.R6 7. * 10 066. 155 * 55 . 193 12.6 175. 64 65. * 10066. 1 55 * 21 10.0 11. 402. 70 1 86 13.3 186. 7103. * 108 34. 155 * 70 1 70 13.6 176 . 6723. * 108 34. 155 * 16 10.9 10. 38 0. 75 1 64 14. 1 185. 7284. * 1 1551. 1 54 * 80 1 59 14.7 195. 7878. * 12252. 1 53 * 35 1 54 15. 3 203. 8412. * 1 29 29. 152 * 90 148 15. 9 210 . 8887. * 13571. 1 51 * 9 5 1 42 16.4 214. 9230. * 14157. 149 * 100 135 1 7. 0 217. 9496. * 14712. 147 * MAIN CROP SI 120 * GROSS * THINNINGS AFTER THINNING * PRO DUG TIOK * AV. TOTAL * TOTAL * A V. TOTAL AGE STEMS DBH BA VOL * VOL 1 AI * ST EMS DBH BA VCL ************************************************************* 20 300 3.4 54. 687. * 687. 34 * 25 792 4.6 98 . 1 748. * 1751. 70 * 30 741 5.5 133. 2883. * 29"0. 98 * 35 689 6. 3 162. 4000. * 4 16 1. 1 1 9 * 40 628 7. 0 184. 500 1. * 5351. 134 * 40 429 7.5 143. 3955. * 535 1. 134 * 198 6.0 41. 1 045. 45 39 8 8.3 162. 4830. * 6389. 142 * 50 373 9. 0 179 . 5690. * 7404. 148 * 50 249 9. 8 138 . 4437. * 7 4 0 4. 148 * 124 7.6 41. 1253. 55 241 10.5 154. 5194. * 8 2 5 3. 1 50 * 60 235 11.1 169 . 5955. * 9094. 1 51 * 50 1 53 12.0 127. 4519. * 9094. 1 51 * 82 9.5 42. 1436 . 55 1 51 12.7 140 . 51 75. * 9788. 151 * 70 146 13.4 15 1. 5763. * 10475. 1 50 * 70 95 14.3 110. 422 1. * 10U75. 1 50 * 5 1 11.9 41. 1542. 75 94 15.0 1 19. 4733. * 11021. 147 * 80 93 15.6 1 29 . 5257. * 1 1566. 145 .* 35 9 1 16. 3 137. 5732. * 12103. 142 * 90 89 16.9 146. 621 8. * 125 28. 140 * 95 88 1 7. 5 154. 6673. * 13127. 138 * 100 86 1 8. 1 16 1 . 7083. * 135 10. 136 * * n e t * l b £ 8 l * "t/Ofi 6 * 0 8 l I " 1 2 IL 0 0 1 * 8 8 1 " 9 1 6Z.I * * 5 / . 6 8 ' n a n * 0 2 IL 5 6 * 1 6 1 ' 9 1 Z L l * "8Z .U8 * 8 9 l 8 " 6 I 5/L 0 6 * 5 6 1 " 2 Z . 5 9 1 * 'ZLbL * 191 I " 6 I 9L 5 8 * 6 6 1 ' 2 0 6 5 1 * •6£nz. * n 5 i n " 8 1 0 8 o e * £ 0 2 ' 9 1 2 5 1 * * n n 8 9 " 9 t j l L ' L l 18 5£ "Z . f i8L * l t 7 L ' i l 8 e * 8 0 2 * LI St i I * ' 0 1 2 9 * 9 £ l 0 'LI £8 QL * 8 02 'Li5nI * ' 9 5 0 8 "8Z. 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' 8 8 1 9 " n i Z.5 I 0 9 * I 12 * 5 8 9 2 l * * 9 6 n e * 1 0 2 2 " n i LLl 0 9 • 9 5 5 * 5 l b " 6 LZ * 2 1 2 *5E 91 I * 'ZOLL ' 0 6 1 n " £ i LSI 5 5 * 2 1 2 *5E 91 I * • 8 5 2 8 * 5 0 2 0 " £ l n i 2 5 5 " 8 t i S " 5 1 8 * 8 L£ * 6 0 2 * o / , n o i * •982Z. * 6 8 l l " 2 1 L 2 2 0 5 * 6 0 2 "OZ . t iO l * "t i£8Z. ' 5 0 2 L * I I n 9 2 0 5 * l tiS " 9 1 L'L I 5 * n o 2 * 5 8 1 6 * "Z .099 " 8 8 1 8 " 0 I £ 8 2 5 n * n o 2 * 5 8 1 6 * "Z-tjfZ. " 5 0 2 t " " 0 L n££ 5 n * e / : 5 " 0 2 E * 9 16 * tiG t *£5 Z.Z. * •6££9 * I 61 £ * 6 £8£ o n * t i 6 l 'ZSLL * * 2 l 69 ' 0 1 2 L'Q s z . n o n * SZ.I * 8 0 l 9 * *£895 ' 161 5"Z. 5 9 5 5£ * 9 t j l * 6 8 E n * * £ 6 l n "L 9 1 5 *9 2 n 9 OE * 8 C I * LQLl * * 8 9 9 2 ' 5 2 1 £ " 5 IUL 5 2 * 0 9 * 8 6 11 * * 8 6 l I •9L 0 "ft 0 0 8 02 ********************************************* 1 0 A v a n e a Sk 'S J . S * I V K 1 0 A * 1 C A V 8 H a a S W 3 J . S 3 D V 1 V . L 0 I * A V * T V 1 0 1 * I V i O l •AV * N O i i D n a c H d * O H I N K I H I a s d , j v 5 0 K I SKI H i * S S 0 H 3 * 0 5 I I S d O H 3 NIVw 202 LOW THINNING 800 STEMS PER ACRE AT ACE 20 A G E T A R I F CU VOL I U *• M A I N CHOP S I T E 9 0 ** DBH ! C L A S S I > 7 . > 1 3 . 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 1 6 1 7 2 0 1 8 . 1 0 0 4 2 0 2 6 0 96 24 3 0 2 2 . 3 3 6 0 0 0 1 1 4 2 5 3 2 1 9 9 8 6 0 36 UO 2 5 . 2 1 4 ) 8 0 1 3 2 184 1 8 8 9 6 5 9 4 3 30 10 2 5 . 2 1 4 3 8 0 0 156 1 8 8 9 6 5 9 4 3 30 4 5 2 6 . 3 2 1 1 8 0 0 7 5 181 121 6 8 4 7 37 2 3 4 5 2 6 . 3 2 1 1 8 0 0 3 1 6 3 1 2 1 6 8 4 7 37 2 3 5 0 2 7 . 2 2 8 9 9 0 9 2 1 3 8 7 8 5 1 39 32 16 5 0 2 7 . 2 2 8 9 9 0 3 4 1 3 8 7 8 5 1 39 32 16 5 5 2 8 . 1 3 6 9 5 0 2 1 1 5 87 5 6 40 33 2 8 10 5 5 2 8 . 1 3 6 6 2 0 0 74 8 7 5 6 40 3 3 2 8 10 6 0 2 8 . 9 4 2 9 9 37 1 0 2 9 96 6 1 44 3 4 2 9 2 5 3 6 0 2 8 . 9 40 8 5 3 7 1 0 7 86 6 1 44 34 2 9 2 5 3 6 5 2 9 . 6 4 5 5 3 8 4 4 6 0 6 3 46 3 5 3 0 2 5 2 0 6 5 2 9 . 6 4 3 4 6 84 4 3 6 6 3 46 3 5 3 0 2 5 2 0 7 0 3 0 . 3 4 8 2 7 1 3 4 8 12 6 3 4 7 3 6 3 0 2 5 2 3 12 7 0 3 0 . 3 4 6 0 6 134 8 2 5 2 47 3 6 3 0 2 5 2 3 12 75 31.0 5 0 4 9 1 8 9 1 0 3 3 49 3 8 2 8 26 2 3 20 5 7 5 31.0 4 8 4 8 1 8 9 1 0 18 47 3 8 2 8 26 2 3 20 5 8 0 3 1 . 6 5 3 0 3 2 4 5 7 0 7 4 2 3 8 31 2 5 2 3 21 1 6 9 0 3 2 . 7 6 0 8 5 3 4 8 8 2 1 3 8 31 2 5 2 3 2 1 18 16 1 0 0 3 3 . 5 6 6 1 1 4 4 5 3 5 2 7 3 0 2 5 2 2 20 1 8 1 7 CROWN THINNING 800 STEMS PER ACRE AT ACE 20 A G E T A R I F CU VOL I U *• B A I N CROP S H E 90 1 •* EBH C L A S S > 7 . > 1 3 . 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 1 6 1 7 2 0 1 8 . 1 0 0 4 2 0 2 6 0 96 24 3 0 2 2 . 3 3 6 0 0 0 1 1 7 2 5 4 2 2 0 100 60 36 4 0 2 5 . 2 1 4 3 1 0 1 3 3 183 1 8 5 9 7 5 9 4 3 30 4 0 2 5 . 2 1 2 2 4 0 7 1 8 4 129 9 2 31 4 3 30 4 5 2 6 . 3 1 9 7 8 0 15 5 9 161 74 5 8 3 2 38 2 8 5 0 2 7 . 2 2 5 8 7 0 0 6 1 8 5 9 7 7 9 3 0 36 33 24 5 0 2 7 . 2 2 0 6 1 0 0 20 2 5 91 4 4 27 19 33 24 5 5 2 8 . 1 2 8 9 8 2 3 0 . 0 15 3 9 4 30 5 0 9 2 3 2 9 2 2 1 6 0 2 8 . 9 3 6 1 0 8 2 7 0 5 12 4 1 66 3 2 36 7 2 5 26 19 6 0 2 8 . 9 2 6 9 4 8 2 7 0 6 5 3 24 20 5 16 24 19 6 5 2 9 . 6 31 72 14 4 5 4 7 2 2 2 5 10 6 17 2 2 17 1 7 0 3 0 . 3 3 6 5 9 1 9 7 2 2 2 3 7 19 21 5 9 17 20 14 1 7 0 3 0 . 3 2 5 0 0 1 5 0 1 7 1 5 1 5 1 8 3 2 10 16 14 1 7 5 31.0 2 8 8 7 1 7 8 0 0 16 11 19 10 1 3 11 1 6 13 8 0 3 1 . 6 3 2 5 6 2 0 8 1 0 10 11 13 18 3 1 5 10 1 5 9 0 3 2 . 7 3 9 9 2 2 8 9 1 0 1 14 8 13 16 4 0 3 8 1 0 0 3 3 - 5 4 6 7 2 3 7 9 2 6 1 0 8 12 1 3 3 1 2 18 19 20 2 1 2 2 2 3 24 2 5 2 6 2 7 28 29 30 1 15 1 11 1 11 14 o LOW THINNING 800 STEMS PER ACRE AT A C E BO AGR T A R I F C U VOL I U ** •IAIN C R O P S I T E 1 2 0 • > 7 . > 1 3 . 2 3 it 5 6 7 8 9 10 11 2 0 1 9 . 9 0 0 1 8 8 3 2 4 1 6 8 8 0 10 3 0 2 5 . 5 1 1 0 2 0 0 5 3 17U 2 0 2 1 3 5 71 5 0 3 8 16 UO 2 9 . 3 3 1 6 6 0 6 3 9 1 1 3 3 1 1 7 6 8 1 8 37 30 u o 2 9 . 3 3 1 6 6 0 6 3 0 9 3 1 1 7 6 8 U 8 37 30 45 3 0 . 8 U 3 U U U 0 6 33 IU 3 2 n u 7 8 5 1 U l 32 4 5 3 0 . 8 U32 1 1 0 6 3 3 11 0 7<l 7 8 5 1 U l 32 5 0 3 2 . 1 5 3 6 0 1 116 1 9 18 0 3 0 8 5 5 7 U l 3U 5 0 3 2 . 1 5 0 0 2 1 146 1 9 18 0 2 6 5 5 7 U l 3U 55 3 3 . 2 5 7 6 8 1 9 U 0 8 16 0 0 3 5 5 8 U2 3 5 55 3 3 . 2 5 1 1 5 1 9 1 0 8 16 0 0 9 5 1 U l 3 5 6 0 3 4 . 3 6 1 0 7 2 7 8 2 5 IS 0 0 1 3 6 uo 3U 6 0 3 1 . 1 5 7 5 7 2 7 8 2 5 15 0 0 0 15 37 3 3 6 5 3 5 . 3 6 4 6 6 3 6 06 3 13 0 0 0 4 32 3 2 6 5 3 5 . 3 6 0 9 5 3 6 06 3 13 0 0 0 0 17 30 7 0 3 6 . 2 6 7 2 1 U U 7 2 1 11 0 0 0 0 8 2 6 7 0 3 6 . 2 6 3 6 5 U 4 7 2 1 13 0 0 0 0 2 1 9 7 5 3 7 . 1 6 9 1 5 5 3 0 9 0 11 0 0 0 0 0 11 8 0 3 7 . 9 7 U 9 S 6 0 8 1 0 9 0 0 0 0 0 6 9 0 3 9 . U 8 4 8 7 7 U 7 9 0 5 0 0 0 0 0 1 1 0 0 110 .5 9 0 9 2 8 4 2 0 0 2 0 0 0 0 0 0 DBH : L A S S 12 13 14 I S 1 6 1 7 18 19 20 21 22 2 3 24 2 5 2 6 2 7 2 8 29 30 2 6 2 2 6 2 2 7 2 3 5 2 7 2 3 5 2 8 24 23 6 2 8 24 20 6 2 8 2 5 21 19 5 . 2 8 2 5 21 19 5 31 24 20 19 1 8 U 11 24 20 19 1 8 U 3 0 2 5 2 2 19 17 1 5 2 2 9 2 5 2 2 19 1 7 15 2 2 6 2 5 2 2 18 17 16 13 2 5 2 5 2 2 18 1 7 1 6 13 2 2 2 3 2 2 18 1 7 1 6 IU 9 1 8 2 1 21 2 0 1 7 1 5 15 13 5 9 17 13 19 1 7 1 5 13 14 1 2 8 4 11 16 1 5 1 7 1 5 12 12 13 11 8 CROWN THINNING 800 STEMS PER A C R E AT A C E 20 A G E T A R I P C U VOL I U «« M A I N C R O P S I T E 120 * * EBI C L A S S > 7. > 13. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 6 1 7 18 19 23 2 1 20 11.9 0 0 1 8 8 3 2 4 168 80 40 30 2 5 . 5 1092 0 0 55 173 205 135 71 50 3 7 16 40 2 9 . 3 31U9 0 68 93 140 1 1 6 70 47 37 30 25 2 40 2 9 . 3 2 6U 7 0 17 62 90 79 6 3 26 35 30 25 2 45 30.8 37U5 U 78 3 48 31 112 45 50 2 0 31 27 2 2 8 50 32 . 1 4U72 1254 0 29 21 83 59 47 33 20 2 8 24 20 9 50 32 . 1 39 9 U 1254 0 5 12 3 6 56 29 25 13 21 24 20 9 55 3 3 . 2 U762 2147 0 2 12 12 6 1 27 26 1 7 13 19 21 18 12 60 34. 3 5520 2971 0 2 11 5 44 37 23 22 11 14 18 19 17 12 60 34 .3 U256 2629 2 1 2 1 23 20 1 3 6 9 14 16 17 12 65 3 5 . 3 4 09 3 3226 1 1 8 2 9 16 16 11 5 10 14 1 5 15 11 70 36 .2 54 6 7 3779 1 1 3 2 5 13 1 7 12 7 5 11 14 14 14 11 70 3 6 . 2 4021 3 000 1 1 6 6 8 9 4 1 b 11 11 12 11 75 37.1 4517 3519 0 8 13 4 8 9 1 2 8 9 12 11 10 80 3 7 . 9 5023 4 128 0 2 17 4 7 8 U 1 3 9 9 10 11 8 9 0 39.4 5953 5175 10 8 3 6 8 3 1 2 7 8 10 9 1 0 0 4 0 . 5 6789 6085 2 1 2 3 4 6 7 1 1 2 6 8 8 2 2 2 3 2U 2 5 2 6 2 7 28 29 30 11 9 ro o LOW THINNING 800 STEMS PER ACRE AT ACE20 AGE TARTP CU VOL IU • « fl A IN CROP SITE 150 • > 7. > 13. 2 3 u 5 6 7 8 9 10 11 20 21.7 93 0 32 288 256 116 64 44 30 28.7 2358 0 108 126 147 91 57 42 35 28 10 33.5 5635 1388 15 30 60 89 76 52 39 31 UO 33.5 5606 1388 15 30 1 58 76 52 39 31 45 35.4 6756 2525 6 12 6 16 67 5 3 38 12 U5 35.4 6304 2525 6 12 6 1 32 52 38 32 50 37.0 7314 365 3 4 8 U 0 10 45 39 31 50 37.0 6840 3653 U 8 4 0 2 19 38 31 55 38.4 780 3 4 834 1 5 4 0 0 9 30 29 55 38.4 7290 4834 1 5 4 0 0 2 12 26 60 39.7 8074 5965 1 4 3 0 0 1 5 21 60 39.7 7555 5965 1 U 3 0 0 0 2 8 65 40.9 8264 7000 1 3 2 0 0 0 1 4 65 40.9 7761 6935 1 3 2 0 0 0 0 1 70 4 2. 1 8 166 7770 I 2 2 0 0 0 0 1 75 4 3.2 8969 846 1 0 2 1 0 0 0 0 0 60 44.2 9555 9121 0 1 1 0 0 0 0 0 90 46. 1 10489 10129 0 1 0 0 0 0 0 0 100 47.4 11061 10739 DBH CLASS 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 8 26 23 21 12 26 23 21 12 27 23 20 18 16 27 23 20 18 16 27 23 20 18 17 16 2 27 23 20 16 17 16 2 25 24 20 18 16 16 15 3 25 24 20 18 16 16 15 3 23 22 20 17 16 15 14 1 3 3 19 22 20 17 1 6 15 IU 13 3 12 19 19 18 15 15 14 12 12 2 5 15 19 18 15 15 1U 12 12 2 3 9 16 17 15 14 13 12 12 11 1 2 5 12 16 15 13 13 12 11 11 9 1 3 9 13 15 13 12 12 11 10 10 0 1 4 8 12 12 11 11 11 10 10 0 1 1 5 8 11 10 9 10 9 10 CROWN THINNING 800 STEMS PER ACRE AT ACE 20 AGS TARIP CU VOL IU *» HA IN CROP SITE 150 ** DBH : .A SS > 7. > 13. 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 20 21.7 93 0 32 288 256 116 64 uu 30 28.7 2345 0 112 125 1U7 92 55 42 35 28 fi UO 33.5 5663 1399 19 37 63 9U 75 54 39 32 27 23 20 12 UO 33.5 49 18 1399 0 16 45 54 67 33 14 20 26 23 20 12 U5 35.4 6101 2659 0 5 32 22 67 36 12 27 18 22 21 19 1 8 1 50 37.0 7331 39U 1 0 2 19 14 43 49 21 29 20 17 20 18 18 16 5 50 37.0 602 1 3591 U 8 27 30 21 18 IS 14 14 16 1 8 16 5 55 38.4 69 1U U5U2 0 8 13 2 3 21 19 15 12 14 12 14 16 15 8 60 39.7 7831 55116 0 6 6 19 23 1 6 15 14 10 13 11 14 14 15 9 60 39.7 624 7 4 74 9 2 14 11 8 12 1 1 5 10 11 9 13 14 9 65 40.9 70U0 5681 1 10 14 6 9 1 1 7 4 9 1 1 9 13 13 9 70 42. 1 770 4 6512 0 5 12 8 6 10 9 4 7 9 10 9 12 12 9 70 42. 1 594 7 5328 0 1 8 6 2 2 7 4 2 6 9 8 9 11 9 75 4 3.2 6558 5955 5 8 2 1 5 6 2 3 7 8 8 a 11 8 80 44.2 7134 652U 3 8 3 1 2 7 U 2 3 9 6 8 8 11 6 90 46. 1 8138 7575 1 5 4 2 1 2 6 3 2 3 7 6 7 7 9 100 47.4 9032 8531 0 2 s 2 1 1 3 5 2 2 3 7 6 6 6 25 26 27 28 29 30 9 2 8 9 ro O vo-x I AGE TARIP CU VOL IU > 7. > 13. 10 2 5 . 2 0 0 45 2 6 . 3 0 0 50 2 7 . 2 0 0 55 28 . 1 33 0 60 2 8 . 9 215 0 65 2 9 . 6 207 0 70 3 0 . 3 220 0 75 3 1 . 0 201 0 AGE TARIP CU VOL IU > 7. > 13. 40 2 5 . 2 207 0 50 2 7 . 2 52 7 0 60 28 .9 916 0 70 3 0 . 3 1159 470 AGE TARIF CU VOL IU > 7. > 13. UO 29 .3 0 0 45 30.8 23 0 50 32. 1 358 0 55 3 3 . 2 354 0 60 3 4 . 3 350 0 65 35. 3 372 0 70 36 .2 356 0 AGE TARIF CU VOL IU > 7. > 13. <I0 2 9 . 3 502 0 50 32.1 878 0 60 3 4 . 3 1264 342 70 3 6 . 2 1446 779 LOW THINNING 600 STEMS PER ACRE AT AGE 20 ** THINNINGS SITE 90 * » DBH CLASS 4 5 6 7 8 9 10 1 1 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 132 28 0 72    IB 57 1 2 41 0 22 10 24 10 0 11 15 2 CROWN THINNING 800 STEMS PER ACRE AT ACE 20 ** THINNINGS SITE 90 *• DBH : L A S S 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 62 99 55 5 28 0 41 60 6 34 3 16 0 5 11 35 13 8 16 1 10 2 15 23 4 3 2 8 7 4 LOW THINNING 800 STEMS PER ACRE AT ACE 20 • » THINNINGS SITE 120 ** DBH CLASS 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 0 91 41 32 40 0 28 20 26 7 1 1 21 3 1 0 4 15 2 1 6 8 1 1 CROWN THINNING 800 STEMS PER ACRE AT ACE 20 ** THINNINGS SITE 120 •* DBH CLASS 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 51 32 0 24 0 2              49 37 7 21 1 10 47 2 18 8 8 7 9 4 23 14 3 9 5 5 4 3 1 1 3 9 8 9 3 3 4 5 3 3 2 ro o ON LOW THINNING 800 STEMS PER ACRE AT ACE 20 AGE TARIP CU VOL IU * » THINNINGS S I T E 150 * • CDH CLASS > 7 . > 13. 2 3 U 5 6 7 8 9 10 11 12 13 1U 15 16 17 18 19 20 21 UO 3 3 . 5 29 0 60 31 U5 3 5 . U U52 0 0 16 35 1 50 3 7 . 0 U9U 0 9 26 2 55 3 8 . U 51 U 0 0 7 17 3 60 3 9 . 7 519 0 0 1 3 12 U 1 65 U 0 . 9 50 3 6U 1 3 7 5 1 CROWD THINNINC 800 STEMS PER ACRE AT ACE 20 AGE TARIP CU VOL IU * * THINNINGS S I T E 150 •* DBH CLASS > 7. > 13. 2 3 u 5 6 7 8 9 10 11 12 13 IU 15 16 17 18 19 20 21 UO 3 3 . 5 7UU 0 19 21 18 UO 8 21 5 12 1 50 3 7 . 0 1310 3U8 0 2 15 6 16 18 2 11 5 3 5 3 60 3 9 . 7 158U 837 0 6 U 5 10 8 3 3 5 « 1 5 1 70 U 2 . 1 1757 1183 0 5 U 1 5 8 2 0 U 3 1 1 3 1 ro O -o 208 APPENDIX 4 PLOTS OF INDIVIDUAL RUNS FOR NORMAL DENSITY STANDS The plots of stems per acre, standing t o t a l volume per acre and gross volume per acre are shown for 8 i n d i v i d u a l runs for each of three s i t e indexes. 90, 120 and 150. Each set of runs had i d e n t i c a l s t a r t i n g conditions f o r the respective s i t e indexes. The i n i t i a l d e n s i t i e s were 2000, l600 and 1200 stems per acre at age 20, corresponding to normal den s i t i e s f o r the three respective s i t e indexes. The only differences between runs i n a set were due to d i f f e r e n t random number s t r i n g s . 2 l 4 40.0 AGE 215 2 i 6 I I I ! i ; M M 1 i > i i X P 1 TON ! ' M M j ! | ' t j - „ ' M_. M l ! M M ; M M Y ST i l : : » v U n n n L U l ! | 1 1 i -_ h ~ M ~ - 1 . : M ' i M i i M I i i . < M M -M-r-1 i I . c 1 M M M l M M n , i M j | ' ' : M l . i ! 1 1 : 1 n 1 ! M M M ! i M M | 1 M i ; ! ; M M i ! ' _Li_ : 1 — i i M in 1 • ! : ' ' M M : . i i ' 1 • i 1 : i i . — - : i M i l , . , r • _L . : 1 1 ! 1 • • i ' ! • i ; 1 i 1 M M M M M M i M l M M M i : i i i - j -~r h i ' * I i M M ! . -! 1 1 1 - M M ! 1 * ' ' i i -M i l - - i s ; 1 i : i ; i 1 ! • ! • i i 1 • « : 1 : 1 _ i — . J — ; — M l ! l M M ' I ' 1 ! : ! . . . M M M M M i . - i ; ; t : M ' | 1 i ' M M M i l i i i r i M i i i I 1 i | , 1 1 ! M M i 1 ' M M M i l M M i i 1 M I i I ' rr> - M M 1 i ! M l | i M ! M M M M { ! 1 • i I M M 1 : I M M 1 i I ' l l M l ' M M M M . . . . 1 ' ' . : - ; , - : : 1 M M M . . i i : t , i 1 i M i l M i l l , -M M M M i i : i i ; 1 M i i r r r 1 ' • ' 1 . . 1 1 1 M M M M M M ! ' • i I 1 1 M M M M 1 : j ; ! ! I - _ . : _ . M _ _ , • i 1 • M 1 : « ' ! 1 I i i M M 1 M ' i i i l i t i i i ' ! M 1 - ; ; i 1 : I M l M M • '• I M M ' • . ' ' 1 M i i i : ! 1 1 ' 1 M M M M . , I . i 1 1 1 i ! 1 1 1 I M i l 1 l 1 M M M M i . : M M i I A ! 1 1 1 i M n M M i i i ! M i l 1 i ; j | M M M ; : , M M M M P J ' . 1 1 M M t t i i 1 i i M i i M M * < i ! I i ;. —* i ' 1 . 1 M M ' 1 1 1 i 1 ' i M M ! i i : i A i i i . 1 - ! 1 1 i 1 M " 1 M M 1 i M ! I I 1 i i 1 1 I M l j j i i i ajr M M ! 1 i 1 i 1 1 ! M i l i i 1 M M 1 i 1 I 1 I i i ! i t i i ! M M M M M 1 i ] 1—» 1 M M I I I ! 1 1 L 1 I M l M M i i i M M M , I 1 ! ' 1 l M M M M M i l ! M i i M M ! I i J7 I i M M : : i 1 1 i OJ M 1 M M ' 1 M I 1 I 1 I i l i 1 I M i ! 1 M j i 1 ' : M M : i 1 : i : I i ' f i r - i M M i M ! i 1 1 1 I 1 | 1 M i l 1 i Js/ i ; i M M i ! • 1 • 1 j | * — • * C J M i l ! I i 1 1 M M 1 1 i i II 1 l i I M M..M 1 . 1 i 1 - . r~» M M 1 M 1 I I 1 1 M i i 1 i I I M l ! ! 1 1 M M ' ' I i ; 1 i M 1 i M M i ! i i i M ' i : '• ' 1 > I 1 I / —*, M j ! M M M M 1 i I 1 | M M i l i i M M . M M 1 i I M M M j i M M M i ! i i 1 M M M l ! Mf ! ! I i ' M M M M M M M i l 1 1 i 1 M I I I 1 I M M M M Iff ' 1 ! 1 I | ' ! 1 1 1 1 M M i M 1 I I I ! I M ! i 1 1 I i i I ! -B\ • I i ! j M i : ; : i 1 i i i 1 i ! 1 M M M M ! i l I M l I I i & ' M M 1 i 1 i ; ' M i 1 II 1 UJ ! i : J ! i M M II 1 1 i i I 1 1 1 1 1 1 i ! M. M i M ' I I I cr i i M j i 1 I i M M 1 ! I 1 1 ! ! i AW I i 1 I i i 1 i M M i • 1 1 1 1 ; M ' ! i 1 M i l 1 1 1 i ! i i Sr i I i M I I i i I M M ' • I I I ! -GET - ' M M 1 1 1 M M 1 1 i l 1 ! <M 1 I M 1 i 1 I 1 1 i ! M i l l ! : , . 1 r M i M i l I I i ' 1 ! M M i • I II ! . 1 . _ ] . _ ! . . ! . ! . ! L. - 4 - _.!_!_!_ i 1 M i 1 i 1 i l l ! 1 M i i 1 1 1 1 1 i SI i i M I I i 1 | I i , M M M l ' M M a: M i i ! j 1 i 1 1 | 1 ! iff I I M i l I ! 1 1 i M ' ' - r r r r 1 : j 1 ! 1 1 M M i 1 1 1 1 1 i 1 M l ; 1 1 1 • i . M i l ' ! ' M M 1 i ' M M M M 1 ' 1 I 1 1 1 M M ! ! i : ' ! ' 1 l I i n ! . • ', i i i ; | 1 i M I I 1 B i I 1 M . . 1 I i i i i U l | M M I i ! i i M i l 1 ! ; M ' | ! i I I 1 I UJ, <TJC M i ! i ! 1 i 1 I 1 I 1 ! i 1 M M 1 I 1 ! i 1 ! ' -1 i i , I 1 • I i M M i 1 I M 1 I I M M 1 ! 1 ! i 1 i ' 1 i M M i 1 i AT i I ' M M ' • 1 . I i M i l i i : I ; , M M 1 M I I M M i . , , M M ; i 1 ! I ! M i l 1 1 1 ' i 1 M l ! I ' M 1 1 ! : : 1 ' ! 1 i i M M ! 1 ! : i i M M i I I 1 M M M M I I I M M M M I i 1 i i ; M M i 1 ! I | I M M l : j | i ! i ; j , M M i i i ] j 1 I n M M i i M M i • i 1 , ' 1 M i l ! II 1 1 1 1 I i ' 1 1 I 1 1 M M M M i l i 1 . i , , M M M M 1 1 ! ! 1 1 i I 1 1 ! i i M M M M I 1 1 i i 1 1 1 M M M 1 1 1 i i i M M i mf '< i i i M M M M | j M M 1 M ' M M i r ' ! i 1 1 1 M M I i i M M , i 1 j 1 1 M M M M M M c 1 i i i • i ' rr i i i M M 1 ' ' . i ! M M I 1 M 1 1 1 M M ' . i i i i i 1 II i . 1 M : M M M M M i l ! I ! M i i I 1 I M M i i M 1 i i l l . J i ' M M i 1 M M M 1 I i M M M M i I ! M M M M j j I ! i : . ' M M M ! - 1 I i 1 ! 1 M M . i ! 1 1 1 I M M M M j ; 1 t i • i : M M ! ! 1 ! ' 9 i I i 1 1 l M M ' ! M M i ; ' ' : . 1 ; ' . \ . . f- ' I I i i M M ! I ! ; ! I I I ! ' M 'i ' M l ; : ; M ' . i l l—1 i M M i . . • M M 1 I Q ' • i M i • • i i I i . i i i r M M ! ; -! M 1 I 1 : • • ' / . ! - 1 ' ' M M i ' ' i ' • a 1 . ' 1 -N : ! . M M M M M i l | ! ; \ ' i 1 1 1 . M . : i j i i M M M M M M I i ! 1 1 1 I M l ! I ' ; ! I i i •  M i l M i l M i l 1 1 , / 1 1 i i i I M M ' - • : ! 1 i 1 i 1 i i M M M M 1 | ! / M 1 ( ! M M I ' M i j i 1 ; 1 I ' I ] ; ; 1 1 f i l l . ,' . i 1 i l M I : i , ! 1 1 1 M M i • i 1 1 M i l M l ! 1 1 ! ' M M ! • 1 M M ! M 1 I i 1 i ; 1 1 : • M M 1 1 l i l l ! i M 1 ! I 1 1 1 I I I i I M M i i 1 i I ' M I 1 1 U i l i i l a i M M i 1 1 ! 1 I i ! 1 i M M i M i \ M I i i i I ' M i i i 0 . 0 20.0 40.0 60.0 80.0 JdJ.O AGE 218 APPENDIX 5 EXAMPLE OF CONTROL AND OUTPUT AT A REMOTE TERMINAL Two runs on a remote t e r m i n a l are given as an example of the c o n t r o l cards and the model's c a p a b i l i t i e s . Output i s given before the next command i s r e q u i r e d . The main crop and t h i n n i n g stand t a b l e s are p r i n t e d on s c r a t c h f i l e s -A and -B. The f i r s t p l o t shows c o n t r o l cards f o r i 1. the generation of a r t i f i c i a l data at age 20, s i t e 120, 200 t r e e s on a .25 acre p l o t . 2. 2 l i n e s of y i e l d t a b l e a t age 20 and 26 (unthinned). 3 . A low t h i n n i n g (the L) a t age 26, min. t r e e s i z e 4.0", B.A. removed 25 sq. f t . per a c r e ; a stand t a b l e f o r both the main crop and the t h i n n i n g (the D's). 4. A repeat low t h i n n i n g at age 26 removing an a d d i t i o n a l 5 sq. f t . per acre and a stand t a b l e f o r the t h i n n i n g . 5. A l i n e of y i e l d t a b l e and a stand t a b l e a t age 3 0 . 6. A l i n e of y i e l d t a b l e f o l l o w e d by a "Crown" t h i n n i n g (the C) at age 40, removing 5 . lOt 10. 5 sq. f t . per acre from < 3 A > It l i . \\ dbh/Av. dbh c l a s s e s r e s p e c t i v e l y . A t h i n n i n g stand t a b l e i s a l s o requested. 7. Lin e s of y i e l d t a b l e f o r ages 4 5 , 77, and 80 years. The second p l o t shows the generation of data f o r a stand a t age 20, s i t e 120, 200 tre e s on a .5 acre p l o t and output a t age 20, 40 and 80. A dump of the t h i n n i n g stand t a b l e i s shown f o r p l o t 1. 219 @ S i g n i f i e s a c o n t r o l l i n e e n t e r e d by the o p e r a t o r l @ 20120200 . 2 5 I1AIN CROP SI 120 * GROSS * THINNINGS AFTER THINNING * PRODUCTION * AV . TOTAL * TOTAL * AV . TOTAL AGE STEMS DJH BA VOL *. VOL MAI * STEMS DBH BA VOL ************************************************************* @ 20 6 8 7 . 34 * 800 3 . 4 5 4 . 1 9 8 9 . 1 0 8 9 . 1 9 8 9 . "2776. 76 * 76 * 76 * 9 3 ~ * *. 20 @ 26 ' 26 740 4 . 9 1 0 4 . 3 . 3 2 6 d l d 4 0 2 5 . 26 512 5 . 1 7 9 . *@ 26 Id 40 5 . 26 476 5 . 1 7 4 . S.@ 30d 30 4 4 4 6 . 0 9 7 . a 40 *• 40 440 7. 6 1 5 7 . @ 40 cd 5 . 1 0 . 1 0 . 5 . 40 340 8 . 1 1 3 5 . 1® 45 45 328 8 . 9 1 5 8 . @ 77 77 248 1 3 . 4 2 6 7 . 1 0 5 8 5 . * 1 2 3 0 3 . 160 * © 8 0 80 236 1 3 . 6 2 6 3 . 1 0 6 1 7 . * 1 2 7 8 0 . 160 * © S e n d f l i e •g 20120200 . 5 " " 6 8 7 . * 1 9 6 4 . * 1 5 3 4 . * 1 4 4 1 . * 2189." * 4 4 1 7 . * 3 3 3 3 . * 4 8 4 6 . * 5 0 1 4 . 125 * 5 0 1 4 . 125 * G0F.1. 135 * 228 36 4 . 5 5.1 25, 5, 4 3 0 . 9 3 . 100 6 . 1 ' 2 2 . 5 8 4 . THINNINGS MAIN CROP SI 120 * GROSS * AFTER THINNING * PRODUCTION * AV. TOTAL * TOTAL * AGE STEMS DBH BA VOL * VOL MAI * STEMS DDH BA VOL ************************************************************* AV. TOTAL @ 20 20 400 3 . 8 3 4 . @ 40 40 370 8 . 2 1 4 9 . 0 80 80 178 1 4 . 5 2 2 2 . @ $ e n d f I l e © S e n d f I l e STOP 0 #EXECUTION TERMINATED #$copy - b >AGE TARIF CU VOL IU > > 7. > 1 3 . > 26 2 3 . 5 0 0 > 26 2 3 . 5 0 0 > 40 2 9 . 3 408 0 4 7 8 . * 4 7 8 . 24 * 42 2 3 . * 4 3 0 5 . 108 * 8 9 9 4 . * 1 1 3 4 1 . 142 * ** THINNINGS S ITE 120 ** 4 5 6 7 8 9 10 11 12 116 112 0 36 52 0 4 0 32 0 12 D3H i 13 ! 220 The programme i s w r i t t e n I n F o r t r a n IV, l e v e l G, f o r use on an IBM 360/6? computer. APPENDIX 6 PROGRAMME LISTING FOR TOPSY The major v a r i a b l e names used i n the programme a r e d e f i n e d b e l o w . ADD Random a d d i t i o n a l p e r c e n t a g e m o r t a l i t y . AGE T o t a l age of the s t a n d . AREA A r e a of the p l o t . AVD3H Average dbh of t h e main c r o p . AVINC Average dbh i n c r e m e n t . AVTHN Average dbh of the t h i n n i n g . BA B a s a l a r e a p e r a c r e ( s q . f t . ) of the main c r o p . BATHN B a s a l a r e a p e r a c r e ( s q . f t . ) of the t h i n n i n g . D3CU C l o s e u t i l i z a t i o n minimum dbh. DBH A r r a y of dbh's f o r t h e t r e e s on the p l o t . DBIU I n t e r m e d i a t e u t i l i z a t i o n minimum dbh. DELA Increment p e r i o d ( y e a r s ) . DCMHT Dominant h e i g h t of the c r o p . DTHN A r r a y of dbh's f o r the t r e e s t h i n n e d . DYING Numbers of t r e e s d y i n g i n DELA y e a r s . FORM Format f o r the i n p u t of the dbh a r r a y . FSA A r r a y of ages f o r A i t k i n ' s i n t e r p o l a t i o n . FSH A r r a y of dominant h e i g h t s f o r A i t k i n ' s i n t e r p o l a t i o n . FUT The age t h a t t h e s t a n d i s t o be updated t o . IRNDM P o s i t i o n of the n e x t random number. ISTHN Numbers of t r e e s t h i n n e d p e r a c r e . KIL Numbers of t r e e s t h i n n e d on the p l o t . LU L o g i c a l u n i t f o r i n p u t of the dbh a r r a y . LUMC L o g i c a l u n i t f o r o u t p u t o f the main c r o p s t a n d t a b l e . LUTHN L o g i c a l u n i t f o r o u t p u t o f the t h i n n i n g s t a n d t a b l e . MAI Mean a n n u a l i n c r e m e n t , g r o s s t o t a l volume p e r a c r e . NCC A r r a y of number o f t r e e s i n each c o m p e t i t i o n c l a s s . NTR Number of t r e e s on the p l o t . CUTMC D e c i s i o n v a r i a b l e f o r main c r o p s t a n d t a b l e . OUTHN D e c i s i o n v a r i a b l e f o r t h i n n i n g s t a n d t a b l e . PERC C a l c u l a t e d p e r c e n t a g e a n n u a l m o r t a l i t y . RANDOM A r r a y of random numbers f o r use i n the m o r t a l i t y s u b r o u t i n e . RAT A r r a y of r e l a t i v e p r o b a b i l i t i e s of m o r t a l i t y by c o m p e t i t i o n c l a s s e s . SITE S i t e Index. STM Number of stems p e r a c r e . TARIF T a r i f Number. TEN A r r a y o f t h i n n i n g c o n t r o l v a r i a b l e s . 221 TYPE Type of t h i n n i n g (L f o r Low, C f o r Crown). VCU Volume p e r a c r e - C l o s e u t i l i z a t i o n . VIU Volume p e r a c r e - I n t e r m e d i a t e u t i l i z a t i o n . VMORT Volume p e r a c r e ( t o t a l ) of m o r t a l i t y . VTHN Volume p e r a c r e ( t o t a l ) of the t h i n n i n g . VTOT Volume p e r a c r e ( t o t a l ) of the main c r o p . XXP A r r a y of c a l c u l a t e d p r o b a b i l i t i e s o f i n d i v i d u a l t r e e m o r t a l i t y . 222 1 c ••*••*••»•****•*»»»••*••»•»»*»•»•••»**»»»*••«*»»*«•»•••*•*»*•»*••*• 2 C 3 C TOPS* 1 C 5 C A STOCHASTIC MODEL OP THE GROWTH OP 6 C PUKE, EVEN-AGED DOUGLAS Fin STANDS 7 C 8 C M I > t < ) > M l t l M I > > l l l l l t t l M M t > t < l l l l l l t l l < l l t M I 4 l l t t ( l l > I M M « 9 C 10 C *»KAIN PROGRAMME 1 1 C 12 DIMENSION 03l! (800) ,DTHN (000) ,THN (5) , FSA ( 1 1 ) , FSH (I'l) ,QQ (OJ 13 DIMENSION !t AKDOM (SO) 11 COMMON /cr.op/ AGE,DOHHT,AVDOH,B.\,STM, S I T S , T A R I F , A S S A , D D H , NTR 15 COMMON / St/ P3A,FSH,QQ 16 COMMON /THINS/ CT HM , B ATMS , ISTH N ,K II. , A VT H H, T Y P E,T HN 17 COMMON /RIIOM/ RANDOM, IRNDil 18 DATA BLANK/' •/ 19 C 20 NUMREW=0 21 LUTHS=8 22 LUMC=7 23 B=RAND(3.918825) 21 C 25 C *»BEGIN THE LOOP FOR A PLOT 26 C READ IN PLOT PARAMETERS 27 C 28 888 CONTINUE 29 READ(S,S1,END=999) A GS, S I T S , NTR , AR EA , LU, D8C0, DBtU , a 30 51 FORSAT(2F3.0,I3,F5.3,I1,2F3.1,F9.6) 3 1 C R INITIALIZES T H ? RANDOM NUMilER GENERATOR 32 C ALL OTHER PARAMETERS ARE AS DEFINED 33 C IF LU I S NOT = 0 THE DBH LIST WILL BE READ F80M LOGICAL UNIT 31 C LU 35 C I P AGE = 999 LOGICAL UNIT Lll WILL BE REWOUND I F THE NUMBER OP 36 C TIMES IT HAS BEEN REWOUND ALREADY IS LESS THAN THE VALUE GIVEN 37 C TO MTU 38 C 39 IP(AGF..L?.2O0.) GO TO 25 UO IF(AGE.GI.918.) GO TO 26 l»1 WRITE(6,61) AGE 92 61 FORMAT ( *0 ERROR: AGE'.FS.O,' TCO LARGE'/ 1.3 1 1 TO CONTINUE - INPUT A JRNDPILE FOLLOWED BY ANOTHER PLOT'/ 11 2 'INITIALIZATION CARD') 15 DO 27 1=1,21 16 27 READ(5,S1,END = 888) 17 STOP 18 26 N0MREW=NUHRE9* 1 19 IFfNUMREW.GT. NTR) STOP 50 REWIND LO 51 GO TO 888 52 C 53 25 X=RAND(R) 51 DO 28 1=1,50 55 28 RA:iDOM(I) = FaAND(0.) 56 IRND.1=0 57 IF(D3C0.LT.U. 1) DBCU»7. 58 IF(D3IU.LT.8.6) DBIU = 13. 59 STM=FLOAT (NTR) /AREA 60 C AITKINS INTERPOIATICN IS PERFORMED BT A SUPPLIEC SUBROUTINE S A I N T 61 DO.«,Hr= SITE/110. »S A I NT < 11, FSA, FSH. ACE, 3,33) 62 TARIF=12. 613 *0.2322«DOMHT 63 IF (LU. EQ.O) GO TO 21 61 C *INPUT DBH LIST FROM LOGICAL UNIT LU 65 CALL LEGEO(0F.H.NTR.LU) 66 GO TO 22 67 C *3ENEHATE THE INITIAL DBH DISTRIBUTION 68 21 CALL GIGNO 69 C *SOitT T HESS INTO ASCENDING ORDER BT DBH 70 22 CALL SORT 71 C 'CALCULATE INITIAL STAND STATISTICS 72 SUM=0. 73 BASUS=0. 71 DO 23 1=1,HTR 75 D=DtfH (I) 76 SUM=SUM*D 77 23 bASJM=UASCB*D*D 78 AV DuK = SUM/NTR 79 BA=bASUn»0.005U51/A8EA 80 VTOT=0. 81 VGROS*0. 1 c t*****************.**********************-***********************.** 2 C 3 C TOPSY a c 5 C A STOCHASTIC MODEL OP THE GROWTH OP 6 C PURE, EVE'.I-AGED DOUGLAS FIR STANDS 7 C 9 C 10 C "MAIN PROGRAMME 1 1 C 12 DIMENSION DBH (800) , DTHN (800) , TH N (S ) , FSA (11) , FSH (11) , QQ (0) 13 DIMENSION RANDOM (50) 19 COMMON /CSOP/ AGE,DOMHT,AVD&H,B.,.,STM, S IT E, T AR 1 F , A R EA , D3H, NTR 15 COMMON / SI/ F5A,FSI1,QQ. 16 COMMON /THINS/ CTHN ,3ATHN, ISTHN ,K II., A VTHN,T YPE.THN 17 COMMON /RH DM/ RANDOM , ISNDfl 18 DATA BLANK/' •/ 19 C 20 NUHREW=0 21 LUTHH=8 22 LUMC=7 23 R = RAND(D.918825) 24 C 25 C *»BEGIN THE LOOP FOR A PLOT 26 C READ IN PLOT PARAMETERS 27 C 28 888 CONTINUE 29 READ (5, SI , END=999) A GS, S ITE. NTR , AR EA . LU, DBCU, D8I0 , R 30 51 PORMAT(2F3.0.I3,FS.3,I1,2F3. 1.F9.6) 31 C R INITIALIZES THE RANDOM NUMBER 3RNEBATOR 32 C ALL OTHER PARAMETERS ARE AS DEFINED 33 C. IF LU IS NOT = 0 THE DBH LIST WILL BE READ FROM LOGICAL UNIT 39 C LU 35 C IF AGE = 999 LOGICAL UNIT LU WILL BE REWOUND IF THE NUMBER OF 36 C TIMES IT HAS BEEN REWOUND ALREADY IS LESS THAN THE VALUE GIVEN 37 C TO NTU 38 C 39 I P (AGE- LT. 200. ) GO TO 25 10 IF (AGE.GT.998. ) GO TO 26 HI WRITE(6,61) AGE <»2 61 FORMA T ( '0 ERROR: AGE'.FS.O,' TCO LARGE'/ 13 1 'TO CONTINUE - INPUT A tENDPILE FOLLOWED BY ANOTHER PLOT'/ 11 2 ' INITIALIZATION CARD') 15 DO 27 1=1,21 16 27 READ(5,51,ENC = 888) 17 STOP 18 26 N0MREW=NUHRES*1 11 IF (NUMREW.GT. NTR) STOP 50 REWIND LU 51 GO TO 883 52 C 53 25 X=RAND(fl) 59 DO 28 1=1,50 55 28 RANDOM (I) =FRAND (0.) 56 IRNDM=0 57 IF (D3C0.LT.1. 1) DBC*J=7. 58 IF(D9IU.LT.8.6) CBID = 13. 59 STM=FLOAT (NTR)/AREA 60 C AITKINS INTERPOLATION IS PERFORMED BT A SUPPLIEC SUBROUTINE SAINT 6 1 DOMHT = SITE/1 3 0. *SA I NT(11,PS A, PSH.ACE, 3 , 0 3 ) 62 TARIF=12.613«0.2322»DOMHT 63 IF (LO. EQ.O) GO TO 21 61 C 'INPUT DBH LIST FROM LOGICAL UNIT LO 65 CALL LSGEO(DEH,NTR.LU) 66 GO TO 22 67 C *3BNEHATE THE INITIAL DBH DISTRIBUTION 68 21 CALL GIGNO 69 C »SOitT TREES INTO ASCENDING ORDER BY DBH 70 22 CALL SORT 71 C 'CALCULATE INITIAL STAND STATISTICS 72 SUM=0. 73 BASUB=0. 79 DO 23 1=1,NTR 75 D=DBH(I) 76 SUM=SUH*D 77 23 BASUN=UASCM*D*D 78 AV Do H = SUM/NTR 79 BA = BASUM»0.OO5l451/AREA 80 VTOT=0. 81 VGROS'O. 224 8 2 C 8 3 C " W R I T E T I T L E S 8<4 C Y I K L C T A B L E 8 5 I S I T E = S I T E * . 5 8b W R I T E ( 6 , 6 0 ) I S I T F 8 7 6 0 F O R M A T ( 1 H O / 7 X , ' K A I N CROP S I • , 14 , 4 X , • • ' , • G R O S S • • , 5 X , ' T H t N H I N 8 8 1 G S V / X , " AFTER T H I N N I N G 1 , 6 X , • , • P R O D U C T I O N * • / 1 1X , • A V . • , 6 X . ' T O T A L 8 9 2 * TOTAL *, 6 X , ' » ' , 7 X , ' A V . 5 X , • TOTAL ' / ' A G E S T E M S D3H BA VOL * 9 0 3 VOL MA I » S T E M S DBH BA V C L ' ) 9 1 W R I T E ( 6 , 6 6 ) 9 2 66 F O R M A T ( 1 • " » • " " " • » " • " » • " " • • • " • " • " • " " " • " • • • • " • " • " 9 3 ) •»••• ) S 94 C 9 5 C H A I N CROP STAND T A B L E 96 W R I T E ( L U M C , 70) I S I T E 97 70 FORMAT (1 HO, • AGE T A R I F CU VOL I U ' , 1 6 X , ' " a * I U c 8 0 p S I T E ' , 1 4 , 9 8 1 • » « ' , 3 X , ' D t H C L A S S ' ) 9 9 W R i r E ( L U H C , 7 1 ) D 3 C U , D B I U .100 71 F O R M A T ( 1 H , 9 X , • > • , F 4 . 0 , • > ' , F 4 . C , • 2 3 9 5 6 7 8 9 101 1 10 11 12 13 19 15 16 17 18 19 2 0 21 22 2 3 24 25 26 27 2 8 2 9 3 0 31 1 0 2 2 3 2 3 3 ' ) 1 0 3 C 109 C T H I N N I N G S STAND T A B L E 105 W R I T S ( L U T H N , 8 0 ) I S I T E 106 80 F O R M A T ) 1 H 0 , ' AGE T A R I F CU VOL I U ' , 1 6 X , ' * * T H I N N I N G S S I T E ' , 1 4 , 107 1 ' * « ' , 3 X , ' D B H C L A S S ' ) 108 W R I T E ( L U T H N , 7 1 ) C B C U . D B I U 109 C 110 C " S T A R T OF S I M U L A T I O N OF GROWTH 111 C. 1 1 2 C " R E A D O P T I O N S - J E H D F I L E S T A R T S A NEW PLOT 1 1 3 C 119 1 5 R E A D ( 5 , 5 0 , E N C = 8 8 8 ) F U T , O U T H C , T Y P E , C O T H H , T H H 1 1 5 50 FORMAT ( F 3 . 0 , 3 A 1 , 5 F 3 . 0 ) 116 C 117 I F ( P U T . L E . 2 0 0 . ) GO TO 6 118 c AGE OF S I M U L A T I O N TOO H I G H 119 K 3 I T E ( 6 , 6 S ) FUT 120 6 5 FORM AT ( ' * * * E R R 0 3 : F U T U R E AGE ( ' , F 5 . 0 , ' ) T C O H I G H ' ) 121 GO TO 1 5 1 2 2 C • • L O O P TO GROW P L O T FROM AGE TO P U T ( Y R S ) BY P E R I C E S OF D E L A ( Y R S ) 123 6 I T I M E = P U T - A S E + . 0 0 1 129 I F ( i r i M E - 7 ) 2 , 3 , 9 125 2 I F ( I TIM E. L E . O ) GO TO 1 126 DELA = I T I M E 127 GO TO 5 128 3 DELA = 9 . 129 GO TO 5 130 4 DELA = 5 . 131 C 1 3 2 5 C A L L M O R S H E ( D E L A , D Y I N G ) 1 3 3 I F ( D Y I N G . G T . 0 . 6 ) C A L L N ECO (DT I N3 , V GROS) 1 3 1 C A L L C B Z S C O ( C S L A ) 135 I F ( N T R . ' L E . 5 0 ) C A L L T R I P L E 136 GO TO 6 137 C 1 3 8 c • •GROWTH C O M P L E T E D . T H I N N I N G AND S T A N D T A B L E S O P T I O N S 139 c 190 1 I F ( T Y P E . E Q . B L A N K ) GO TO 7 141 c • T H I N N I N G R E Q U I R E D 1 4 2 C A L L THIN<(<-.7,R888) 1 9 3 I F ( O J T H i l . E Q . E L A N K ) GO TO 8 144 C A L L MET IOR (CTH N , VTBN , T AR I F , AREA ,K I L . D B C U , D B I U , L U T H N ) 195 3 0 TO 9 146 c C A L C U L A T E S V O L . OF T H I N . 147 8 VTHS=-21 . 31569»K I L 148 DO 17 1 = 1 , K I L 149 D=DTHN ( I ) 150 17 VTHS=VTHH • (. 6 9 1 3 9 2 1 * EXP ( - . 4 3 1 25 •€ ) ) • ( D ' 0 * 1 5 . 9 5 1 6 ) 151 VTHN = VTHN/ARSA«TARIP^ .8940856E -02 1 5 2 9 V G R O S = V G R O S * V T H N 1 5 3 •NO T H I N N I N G 159 7 V G R O S = V G R O S - V T O T 155 I F ( O ' JTMC. E O . B L A N K ) GO TO 10 156 C A L L M E T I O R ( C D H , V T O T , TAR I F , A R E A , N T R , D B C U , D B I U , L U H C ) 157 GO TO 11 158 C C A L C U L A T E T O T A L VOLUME FOR MAIN CROP 159 10 V T O T = - 2 1 . 3 1 5 6 9 ' N T R 1 6 0 DO 16 1 = 1 , N T R 161 D=DBH ( I ) 162 16 VTOT = V T 0 T * (. 6 9 3 3 9 2 1 * EXP ( - . 4 3 125 •D) ) • ( D ' D * 1 5 . 1 5 16) 1 6 3 V T o r = Y T O T/ARKA » r A H I P • . 8 9 4 0 8 5 6 E - 0 2 16« 11 V G R 0 S = V G R 0 S * 7 T O T 165 C W R I T E Y I E L D T A D L S 166 MAI=VGROS/A<;H».5 1 6 7 I A r . ; ^ A G E * . 0 0 1 1 6 8 I STM=STM«0.5 1 6 9 I F ( T Y P E . EO . B L A N K ) GO TO 12 1 7 0 WR ITE ( 6 , 62) I AG 2, I S T M . A V U B I I , DA , VTOT , V G R O S . M A 1 , 1ST HN , AVT [ill, B A T H S , 171 1 VTHN 1 7 2 62 F 0 R M A T ( I 4 , I 5 , P S . 1 , F 5 . 0 , F 7 . 0 . ' • • , F 7 . 0 , I 4 . » • • , 15 , F5. 1, r 5. 0 , F 6 . 0 ) 1 7 3 3 0 TO 1 5 1 7 9 12 W R I T E ( 6 , 6 3 ) I A G E . I S T M , A V D B H , BA , V T O T , ' / G R O S , MA I 1 7 5 63 FORMAT (14, 15, F5. 1 , F 5 . 0 , F 7 . 0 , • « ' , F 7 . 0 , I 4 , ' • •) 176 C * * G 0 BACK AND R E A D ANOTHER O P T I O N CARD 1 7 7 GO TO 1 5 1 7 8 C 1 7 9 9 9 9 S T O P 1 8 0 END 1 8 1 BLOCK DATA 1 8 2 D I M E N S I O N P S A ( 1 4 ) , F S H ( 1 4 ) , Q 3 ( 8 ) ' 1 8 3 COMMON / S I / F S A , F S H , Q Q 184 DATA F S A / 2 0 . , 3 0 . , 4 0 . , 5 0 . , 6 0 . , 7 0 . , 8 0 . , 9 0 . , 1 0 0 . , 1 1 0 . , 1 2 0 . , 1 3 0 . , 1 4 0 . 1 8 5 1 15 0 . / , F S H / 3 4 . , 6 0 . , 7 8 . , 9 1. , 10 1. , 1 1 0 . , 1 1 8 . , 1 2 5 . , 1 3 0 . , 1 3 5 . , 1 3 3 . , 141 186 2 , 1 4 4 . , 1 4 7 . / 1 3 7 END 1 3 8 C * **• »»« * 1 8 9 S U B R O U T I N E CR ESCO ( D E L A ) 1 9 0 c 191 c * * T 0 GROW DBH FROM A G E TO A G E + D E L A , G I V E N A L L S T A T I S T I C S AT A G E . 1 9 2 c C A L C U L A T E S NEW S T A T I S T I C S » * * • 1 9 3 c 1 9 4 D I M E N S I O N DBH ( 8 0 0 ) , C.Q (8) , TS A (14 ) , F S H (1 4 ) 1 9 5 COMMON / C R O P / A 3 Z , D O B H T , A V D B H , B A , S T M , S I T E , T A R I F , A R E A , 0 B H , N T B 196 COMMON / S I / F S A , F S H , Q Q 1 9 7 c 1 9 8 c • C A L C U L A T E A V E R A G E EBH GROWTH (0< A V I NC/Y R < . 6 ) 1 9 9 A V I N C = . D 8 2 0 4 + . 0 0 14 06 2 ' S I T 2 - . 0 0 1 3 13•DOM HT » 9 . 3 0 0 7 / B A -. 18 1 4 2 E - 04 •STB 2 0 0 1 • A V D B H 2 0 1 I P ( A V I N C . L T . 0 . ) A V I N C = 0 . 2 0 2 I F ( A V I N C . G T . 0 . 6 ) A V I N C = 0 . 6 2 0 3 A V I N C = A V I N C ^ D B L A 2 0 9 c 2 0 5 X = A H I N 1 ( A G E , 6 0 . ) 2 0 6 Y = ( A M A X 1 ( A G E , 6 0 . ) - 6 0 . ) * . O t 1 2 0 7 Z Z = - 3 . 7 3* (. 2 4 S 1 4 - . 0 0 17 6 ^ X ) ^ X » . 0 3 4 4 3 ' S I T E » t 2 0 8 U P L I M = . 6 ^ A V D E H 2 0 9 I P ( Z Z . G T . U P L I M ) Z Z = U P L I B 2 1 0 X = A V I N C / ( A V D E H - Z Z ) 2 1 1 S U B T = Z Z ^ X 2 1 2 R A T = X * t . 2 1 3 c 2 1 4 • c • C A L C U L A T E NEW DBH ' S FOR A L L L I V E T R E E S . A L S O F I N E A V D B H 6 BA 2 1 5 s n M = 0 . 2 1 6 B A S U M ' O . 2 1 7 DO 1 1= 1 , N T R 2 1 8 D= D B H ( I ) • R A T - S U B T 2 1 9 SUM=SUM»D 2 2 0 3 A S U M = B A S U H * C # D 2 2 1 1 D B H ( I ) = D 2 2 2 I F ( D 3 H ( 1 ) . L T . O . ) D B H ( 1 ) = . S 2 2 3 c t 2 2 4 AGE=\GE»DELA 2 2 5 A V D 3 H = SI!M/NTR 2 2 6 BA=bA S U M • . 1 0 5 4 5 4 / A R E A 2 2 7 D O M H T = S I T E / 1 3 0 . • S A T N T ( 1 4 , F S A , F S H , A G E , 3 , Q 2 ) 2 2 8 T A R I F = ! 2 . 6 1 1 * . 2 3 2 22^DOMHT 2 2 9 STM= F L O A T ( N T H ) / A R E A 2 3 0 RETURN 2 3 1 END 2 3 2 c * « »•»«* * 2 3 3 S U B R O U T I N E M O R E R E ( D E L A , D Y I N G ) 2 3 4 c 2 3 5 c • • P R E D I C T S NO. OF T R E E S D Y I N G IN P L O T D U R I N G T I M E D E L A , G I V E N T R E E S 2 3 6 c (RNTR) , A G " ( A G E ) , S I T E (SI) A N D E A / A C R E (BA) . P E R C = * CF T R E E S D Y I N G 2 3 7 c 2 3 8 D I M E N S I O N DBH ( 8 0 0 ) 2 3 9 D I M E N S I O N RANDOM (50 ) 2 4 0 COMMON / C R O P / A G E , DOMHT, A V D B H , BA , R N T R , S I , T A R I F , A R E A , D B H , NTR 2 4 1 COMMON /RNDM/ R A N D O M , I R N D H 2 4 2 c 2 4 3 P E R C = - . 0 D 7 3 5 4 4 » R N T R > (. 5 P 1 2 P E - 0 6 ^ S I - . 0 0 1 0 0 9 3 / A G E ) •. 2 5 0 5 7 E - O 6 • 24H 1 SI*BA 245 IF (PERC. L 7 . 0. ) FERC=0. 246 C • 3ENFRATE A RANDOM RESIDUAL 2*7 IRS'DM=IaN0M* 1 248 RR = R^D3.1 (IRHDM) 249 KLAS=PSnC/.0067* 1. 2S0 IF(KLAS.GE.4) CO TO 4 251 3 0 TO (1,2,3), KLAS 252 C 253 1 ADD =-. 36052E-02* RR *(.05681* RR *(-. 17703* RR *(.25553- R8 * 254 1 . 1 1308)) •. 19352"S-07/(1.-SORT ( RR )*.1E-05) 255 30 TO 5 256 2 ADD =-.0 1 1 722* ( ( RR • . 1 8 3 0 U - . 2 7 37 5 ) • RR •. 12988)* RB * RR-257 1 . 15366E-08/ (1.-SQKT ( RR J+.1E-05) 258 GO TO 5 259 3 ADD = -. 3 1 3395* ( (. 3 1902* PR -. 49449)* RR +.22096)* RH * BR + 260 1 . 1 17 332-07/(1.-SQRT ( RR )*.1E-05) 26 1 30 TO 5 262 4 ADD =-.01 794** ( ( (.76 173* RR - 1.2429)* RH •. 63632)* RR -.064547)* 263 1 BR *.739606E-08/(1.-SQRT( RS )•. 12-05) 264 C 265 5 PSRC= P E RC * AD D 266 DYLNG=?ERC*3NTR*CELA*AREA 267 RETU8S 268 END 269 C * *** *** * 270 SUBROUTINE NECO(CYING.VGROS) 271 c 272 c **T0 DISTRIBUTE MORTALITY AMONGST TREES IN THE PLOT. 273 c USES RANDOM NUMBER ROUTINE 279 c 275 REAL DBH (300) , NDUM (2) , NCC (40),RAT(16),XXDUM(2),XX?(90) 276 COMMON /CROP/ AG E, DOM HT, AVD3H, BA ,5TM , S ITE, TARIF, AREA, DBH, NTR 277 DATA RAT/2. ,3*1. ,.926 3, .6686, . 3755 5, .189, .1009, .064, 278 1 .05659,.042,.0274,.0195,0.,0./ 279 DATA XXDUM,XXP/2*2.,40*0./ 280 c 281 c **FIND NO. OF TREES /RATIO CLASS, THEN CALCULATE PROBABILITIES OP 282 c DYIMG. 283 VMORT=0. 284 DO 1 1=1,40 285 1 NCC (I) =0. 286 X=3./AVDBH 287 DO 2 1=1,NTR 288 K = DBH (I) *X + 1 .5 239 2 NCC (K) =NCC (K) • 1. 290 DENOM = NCC (U) *NCC (3) *NCC (2) *NCC (1) *RAT(1) 291 DO 8 1=5,14 292 8 DSNOM=DENOM + NCC(I) *RAT(I) 293 PROB=DY I.NG/DEMOM 294 DO 3 1=3,14 295 3 XXP(I) =PROB*RAT (I) 296 XXP (2) =PHOB 297 XXP(I) =XXP (2) 298 DBH (UTR*1) =9999. 299 UPLIS=1.6875*AVDBH 300 C 301 c **LOOP OVER TREES TO DECIDE DEATHS 1 302 1 = 0 303 JJ=0 304 5 1=1*1 3 05 D=BB!(I) 306 IF(D.GE.UPLIH) GO TO 6 307 K=D*X*1.5 308 IF(FRAND(0.) .GT.XXP(K) ) GO TO 9 309 C TREE DISS 310 VMORT = VMORT* (. 693392 1 • EXP (-. 4 3 125« C) ) * (D*D• 15.95 16) 311 3 0 TO 5 3 12 c TREE LIVES 313 4 JJ=JJ*1 314 DBH (JJ) =D 315 30.TO 5 316 6 IF(D.GT.8888.) GO TO 7 317 JJ=JJ*1 318 DBH (JJ) =D 319 1 = 1* 1 320 C = DBH(I) 321 CO TO 6 322 C 323 7 CONTINUE 324 C ADD VOLUME OF MORTALITY TO GROSS VOLUME 227. 3 2S VCROS=VGR05» (V.IORT- (MTR-JJ) *21. 3 1569) /A2 E A *TA R I F * . 8 9 40 U 56E-02 3 26 NTR=J  327 RETURN 3 28 END 329 c * *«* * 3 39 SUBROUTINE THIM(«,«) 331 c 332 c • »TO RE.10VE TREES USING EIT'IER A LO." THIN WITH HIM DBH OR A T.OW/CR0WN 333 c THIN OVER AVDBH RATIO CLASSES. THINNED TSSSS STORED 1,  DTHN 334 c 335 DIMENSION DBH (H00) ,DIV (5) ,THN (5) ,DTH1 (800) 336 COMMON /CROP/ AGE.DOMIT.AyDBH.BA.STM.SITE.TARIF.AREA.DBH, ST3 337 COMMON /THINS/ LTRN,DATHN, ISTHN,SIL,AVTHN,TYPS,TH* 338 DATA RLOWjCROWl/'L', 'C •/ .3 39 c 340 SUM=0. 341 BASDM=0. 342 SMTHN=0. 343 BATHN=0. 344 RMUL P = A 3 EA/.005459 345 DBH(NTR*1)=4500. 346 c 347 IF (TYPS.NE. RLOW) GO TO 9 348 c •LOW THINNING 349 DMIN = THN (1) *0. 1 350 TEST = THN (2) *R MULT-.5*AVD3H*AVDBH 351 I F (THN (2) . GT. 0. 85«BA) GO TO 99 352 IF(DMIN.GS.DEH (NTR) GO TO 98 353 JJ=1 359 2 D = DBH(JJ) 355 IF (D.GE. DM IN) SOTO 1 356 SUM=SUM*D 357 BASt)M = 8ASnM*D*D 358 JJ=JJ*1 359 GO TO 2 360 c TREES REMOVED 361 1 I=JJ 362 KIL=0 363 5 KIL=KIL*1 364 DTHN (K IL) = DBH (I) 365 SHTHN=SMTHN*DBH (I) 366 BATHN=BATHN*DBH (I) *DBH(I) 367 1=1*1 368 IF (BATHN .LT. TEST) GO TO 5 369 IF (I. GT. NTR) GO TO 97 370 C TREES LEFT 371 DO 6 K=I,NTR 372 D=D8H(K) 373 BASUS=BASUM*D*D 374 S0M=SUM*D 375 DBH (JJ) = D 376 6 JJ=JJ+1 377 NT3=JJ-1 378 GO TO 20 • 379 C 380 9 IF (TYPE. NE. CROWN) GO TO 100 381 •CROWN THINNING - BY B.A. IN EACH RATIO CLASS 382 KLAS=0 383 ADD=.5 384 TEST=THN(5) 385 DO 7 1=1,4 386 T EST=TEST*THN (I) 387 ADD=ADD*.25 388 7 DIV (I) =AVDBH*ACB 389 IP (TEST.GT.O. 85*BA) GO TO 99 390 DIV (5) =4000. 391 JJ=1 392 1 = 1 393 KIL = 0 399 13 TEST=0. 395 8 KLAS = KLAS *1 396 IF (KLAS.GE.6) GO TO 11 397 I F (T HN (XL AS) . LE. 0. )' GO TO 12 398 T EG T = TSST*TH N(KLAS)*R«ULT 399 14 D = DBH (I) 400 IF(D.GE.DIV(KLAS)) GO TO 13 401 c TREES REMOVED 002 KIL=KIL*1 403 DTHN (KIL) *D 404 SMT HN =SMTHN•D 4 05 1 = 1*1 228 006 X = D»D 007 BArH:l-BAT!IN»I 008 TEST = fEST-X 009 I F (TEST.GT.O. ) CO TO 10 110 C TRESS LEFT U 1 1 12 D = D3H (I) 012 I F (D.GE.DIV (XLAS)) GO TO 8 0 13 DrtH (JJ) =D 0 1 0 SI1M=SUM*D 1 IS BA50« = 3ASU.M + D*D U16 JJ=JJ»1 « n 1=1*1 U 18 GO TO 12 119 11 NTR=JJ-1 120 C 021 20 AVD3H = S'Jr!/!«TR 0 22 BA = BAS0.1/.1KULT 023 EATHS=BAT!!!I/aHI1LT 020 ISTHS=KIL/ARSA*0.5 t!25 AVTHH = SMTHN/KIl. 426 5TM = !lTR / \ a S A 027 RETURN 0 28 C 029 100 WRITE(6,60) TYPE 030 60 FORMAT ('0 ERROR: THINS ING SPECIFIED AS ', AO) 03 1 RETURN 1 032 99 WRTTE(6,61) BA 033 61 FORMAT ( '0 . ERROR: THINNING GREATER THAN 85% EA = ',F6.1) 030 RETURH 1 035 98 URITE(6,61) DMIS 036 63 FORMAT ('0 ERROR: MINIMUM S U E ' , F6. 1,' GREATER THAN LARGEST TREE') 037 RETURH 1 038 97 WRITS (6,62) 039 62 FOR»AT('0 THIHNING HAS REMOVED THE LARGEST T R E E S 1 / ' GROWTH OF T 000 111E PLOT IS TEH HI MAT ED ') 00 1 102 READ(5.50,EHD=101) X 002 50 FORMAT (F3.0) 003 GO TO 102 000 101 RETURH 2 0 0 5 END 006 C * * * * * * * * 007 SUBROUTINE METIOR(DBH,VTOT,TARIF,AREA,NTR,DECU,CBIU,LH) 008 C 009 C **TO CALCULATE A STAND TABLE AND HERCH. VOLUMES FOR EITHER ti. C. CB 050 C THINNINGS. 051 DIMENSION DBH (800) ,ND0N (2) , NCL (70) 052 C 053 VCU=0. 050 VIU=0. 055 VTOT=-21.31569*STR 056 DBH (NTR + 1) =9999. 057 !U=(D3H(1) »0.S)/2 058 IF(KX.EQ.O) MX=1 059 MIN=MX*HX 060 KLAS=MX 061 f AX = DB K (NTR) »1.5001 062 DO 13 1=1,MAX 063 10 NCL(I)=0 060 C TREES LESS THAN CU LIMIT 065 1=0 066 1 1=1+1 067 D=D3H(I) 068 IP(D.uT.D3CU) GO TO 2 069 J = DM.5 070 NCL (J) =NCL (J) • 1 071 VTOT=VT0T* (- 6 93 39 21 » EXP (-. 3 3 1 25 *D)) « (D* D »15. 95 16) 072 SO TO 1 073 C TREES GREATER THAN CU LIMIT 070 2 K=I 075 3 I F (D.GT.DBIU) GO TO 0 0 7 6 J=D*1.5 077 NCL (J) = NCL(J) • 1 078 X=D«D 079 VC0=VCU*X 0 80 VTOT = VTOT* (. 6913921*EXP (-.0 3 125*D) ) * (X»1 5. 9516) 081 K=K»1 082 D=DBH (K) 0 8 3 GO TO 3 0 8 0 0 V C U = V C U - ( K - I ) » 1 5 . 9 5 1 6 085 C TREES GREATER THAN IU LIMIT 086 5 IF(D.GT.90O0.) GO TO 6 487 J = D» 1.5 083 NCL (J) = NCL (J) » 1 1 89 X = 0*D 990 V0L=X-1S.9S16 491 VC'l = VCU*VOL 492 VTOT = VT3T* (. 6913921*EXP (-. 4 3 1 25 » 0) ) * (X *1 5.95 16) 493 VIH = V ID* VOL- VOL/FX P ( (D-8. 6) » . 4 3078 29) 494 K =K • 1 495 D= DBH (K) 996 GO TO 5 497 C 498 6 NCL (3) = NCL(3) *NCL(2) *HCL(1) 499 JST=HIN*1 500 DO 7 I=JST,MAX 501 7 NCL (I) = SCL(I)/AREA».5 502 X-TA RIF/ARRA 503 VTOT = VTOT«X*. 8940856E-02 504 X=X*.0059737 505 VCU=VCU*X 506 VIU=VIU«"X».983 507 NOC=MAX-NIN 508 IF(MAX.LE.34) GO TO 8 509 NSUH = 0 5 10 DO 9 1=35,MAX 511 9 HSUM = NSUM*:iCL (I) 512 NCL(35)=NSUM 513 SCL (36) =MAX-1 519 HOC=36-HIH 515 C 5 16 8 CALL SCRIEO(NCL(JST),NOC,VCU,VIU,KLAS,LU) 517 RETURS 518 END 519 C • *»* *** • 520 SUBROUTINE SCRIBO (NCL , NOC , VCU , V IU , KL AS , LU) 521 c •• WRITE A STAND TABLE FOR EITHER A THINNING OS THE MAIN C HOP S22 c 523 DIMENSION DBH (800) 529 COMMON /CROP/ AGE.OOMHT/AVDBH.EA.STM, SITE,TARIF,AREA, DBH, NTH 525 DIMENSION NCL(NOC) 526 DIMENSION BLANK (10) ,FR5T(10) ,SECND(10) , FMT (10) 527 DATA BLANK/* •,' 8 X , ' , • 1 SX, », ' 24 X, • , • 32X , ' , • 3 BX , •, • 4 4X * i , , 50X,', 528 1 ' 5 6 X , ' 6 2 X , ' / 529 DATA FRST/'3I4, • .'614, ' , '4IU, •, • 214, ' , 6«« •/ 530 DATA SECND/5"24I3' . ^ I I ' . ^ O I l'.MSn'."^^.'^^ / 531 DATA FMT/' (1X, ' , '13,F\ '5. 1, •, • 216, ' , 3*' • , • , I 3 , ' , 4H • < t 12)*/ 532 c 533 FMT (5) =BLASK (KLAS) 534 FMT (6) = FRST (KLAS) 535 FMT (7) =SSC:iD (KLAS) 536 NVCU=VCU*.5 537 NVIU = VIU*. 5 538 IAGE=AGE*.S 539 c 590 WRITE (LU, FMT) IAGE,TARIF,HVCU, NVIO ,NCL 59 1 a ETURS 542 END 54 3 c * •** »»* • 544 SUBROUTINE GIGNO 545 c 546 c • •GENERATES THE TREE DBHS FOR A3E 20 - 25. 547 c 548 DIMENSION D3H (300) 549 COMMON /CROP/ AGE,DOMHT,AVDBH,3A.STM, SITE,TARIF,AREA, DBH, NTR 550 c 551 c •CALCULATES AVDBH (3Y KIRA'S EQUATION) 552 X = . 33149E-04 »COMHT".2750 7 553 Y=12.6 52/DOKi!T"1. 1677 554 A V D3 H = 1 ./ (X^ STM* Y) 555 ADD=1./NTR 556 c • CALCULATE TREE CBH USING INVERSE C. D. F. FUNCTION 557 R=.5^ ADD 558 DO 1 1=1,NTR 559 X = - . 38 3 11 • ( (4. 77 39•R-7. 2 762) •R* 3. 7 248) *R"R*. 1727 1E-05 560 1 /(l.-SQRT(R) •. 1E-04) 561 DBH (I) =X^ AVDEH*AVDBH 562 R=R*ADD 563 1 CONTINUE 569 RETURN 565 END 566 c • «»••*• » 567 SUBROUTINE LEGEO(DBH,NTR,LU) 568 C ••TO PEAO IN DBH LIST 569 C 570 DIMENSION CBH (NTit) , FORM (20) 571 READ (5, S2) FORM 572 52 FORMAT (20A0) 573 READ (LIJ, FORM) DBH 570 RETURN 575 END 576 C • ••• ••• * 577 SUBROUTINE TRIPLE 578 C ••INCREASES AREA OF PLOT BY 3 TIMES 579 C 580 DIMENSION DBH (800) 581 COMMON /CROP/A" E, DOM HT, AVDBH, BA,STM,SITE,TA9IF,AREA,DBH,NTS 582 KTHRE=:iTR*NTR*NTR 583 K=KTHRE 580 J = NT3 585 2 D=DBH(J) 586 DO 3 1=1,3 587 DBIt(K)=D 588 3 K = K-1 589 J=J-1 590 IF(J.GT.O) GO TO 2 591 NT R=KTHRE 592 AREA = 3.*Aa3» 59 3 RETURN 591 END 595 c * • • * •** * 596 SUBROUTINE SORT 597 C SORTS A INTO AN ASCENDING ORDEB. GOOD WHEN A I S PARTIALLY SORTED 598 C 599 DIMENSION A(800) 600 COMMON /CROP/ AG E , DOM RT, A VDBH , EA ,STM , S ITS, T ARIF, A REA, A, II 601 DIMENSION MIN (20) ,MAX (20) 602 C HIN AND SAX ADEQUATE FOR N OP TO 1,000,000 603 L=0 600 HN=1 605 MX = N 606 10 IF(MX.LE.MN)GO TO 09 607 11 MD=KUT(MN,MX) 608 C NOW ICE'ITIFY LONGER AND SHORTER SUBLISTS 609 I F (MX-MD.GT. MD-MN) GO TO 25 6 10 IF(MX-MD.GE.2)GO TO 13 611 MX=MD-1 612 GO TO 10 613 13 L = L*1 6 10 C PUT AWAY RECORD OF LONGER L I S T 615 MIN(L)=MN 6 16 MAX (L) =MD-1 617 BN = MDH 618 GO TO 11 619 25 IF(MD-MN.GE.2)SO TO 26 620 MN=MD»1 621 GO TO 10 622 26 L = L H 623 C PUT AWAY RECORD OF LONGER L I S T 620 MIN(L)=MD*1 625 MAX(L)=SX 626 KX=MD-1 627 GO TO 11 628 '19 I F (L."0.0) RETURN 629 C SORTING COMPLETE IF HO DEFERRED TASKS 630 C OTHERWISE SELECT A DEFERRED TASK 631 MN = MIN(L) 632 MX = MAX(L) 633 L = L-1 630 GO TO 11 635 END 635 C 637 FO'ICTION KUT(L.B) 638 DIMENSION A(800) 639 COMMON /CSO?/ AGE,DOMHT, AVDCH, BA.STM,SITE,TARIF,AREA,A,H 600 C THIS FUNCTION WORKS CN SUBLIST RUSHING FaOM A (L) TO A(B) 60 1 K= ( L * f l ) / 2 602 X = A(K) 603 A(K)=A(L) 600 I=L 605 J=M»1 606 C NEXT 3 LINES LOOK BACKWARDS FROM END FCR SHALL ENTRIES 607 1 J=J-1 608 I F (I . Etj. J) GO TO 99 649 I F (A ( J ) . c n . x) no ro t 650 C MOVE A (J ) IH TO T H E G A P 651 A (t) =A (J) 652 C NOW S E A R C H FORWARD FOR L A R G E E N T R I E S 653 5 ! = !• 1 651 I F ( I . E Q . J ) G O T O 99 655 I F (A (I ) . L E . X) GO T O 5 656 A ( J ) =A (I) 657 C HOVE A (I ) INTO T H E S A P AND GO TO 1 658 GO T O 1 659 99 XUT = I 660 A(XUr)="X 661 R E T U R N 662 END E N D O F F I L E 232 APPENDIX 7 MODIFICATIONS TO TOPSY FOR MORTALITY AFTER THINNING To c o r r e c t l y s i m u l a t e the a b i l i t y of a t h i n n i n g t o s a l v a g e m o r t a l i t y and t o remove a l l t r e e s l i k e l y t o d i e i n the n e a r f u t u r e was n o t s i m p l e i n a model such as t h i s . How l o n g m o r t a l i t y i n Douglas f i r would s t i l l be a v a i l a b l e f o r s a l v a g e was n o t known e x p l i c i t l y . However S m i t h e_t a l . (1970) i n a s t u d y on f u n g a l d e t e r i o r a t i o n o f c u t l o g s showed t h a t decay a f t e r 2 y e a r s was o n l y 10$ by volume b u t a f t e r 6 y e a r s had r i s e n t o ^7%> On t h e o t h e r hand t h i n n i n g and e x t r a c t i o n c o u l d i n c r e a s e damage t o t h e r e s i d u a l s t a n d , f o r example W o r t h i n g t o n (19^1) found up t o 17^ of t h e l a r g e r c l a s s o f t r e e had been i n j u r e d , w h i l e a n y t h i n g from 2 - 28$ o f t h e s m a l l e r t r e e s had been k i l l e d . In a d i s t a n c e dependent model, such as Newnham's (196*0, the c o m p e t i t i o n i n d e x of an i n d i v i d u a l t r e e i n c o n j u n c t i o n w i t h i t s r e l a t i v e s i z e w i t h i n a s t a n d gave an i n d i c a t i o n o f t h e l i k e l i h o o d of m o r t a l i t y and c o u l d be used as a g u i d e f o r t h i n n i n g . In v i e w of the r e q u i r e m e n t s of m i n i m a l e x e c u t i o n t i me even an a p p r o x i m a t i o n t o t h i s was n o t p o s s i b l e h e r e . P r o v i d e d some crud e a s s u m p t i o n s were made, c o r r e c t i n g t h e c a l c u l a t i o n o f f u t u r e - m o r t a l i t y a f t e r t h i n n i n g can be accomp-l i s h e d . These a s s u m p t i o n s were t h a t t h e markers o f t h e t h i n n i n g c o u l d p r e d i c t m o r t a l i t y and remove a l l the t r e e s l i k e l y t o d i e w i t h i n a c e r t a i n t i m e , say 5 y e a r s , p r o v i d e d t h a t a c e r t a i n p r o p o r t i o n of t r e e s were removed, say one 233 q u a r t e r by number. I f the y removed a s m a l l e r number than t h i s , t h e i r p r e d i c t i v e a b i l i t y c o u l d be reduced p r o p o r t i o n -a l l y . The number of t r e e s d y i n g c a l c u l a t e d by s u b r o u t i n e KORERE c o u l d be reduced by t h i s p r o p o r t i o n . T h i s would n o t work so w e l l when the t h i n n i n g ' s main aim was n o t t o reduce m o r t a l i t y , e.g. a low t h i n n i n g w i t h a m e r c h a n t a b l e l i m i t . I n t h i s c a s e the markers would n o t be a b l e t o p r e v e n t m o r t a l -i t y i n t h e t r e e s b e l o w t h e dbh l i m i t . The r e l e v e n t program-ming changes t o the model a r e g i v e n b e l o w . EFFECT = L e n g t h of ti m e t h i n n i n g i s e f f e c t i v e SALVGE = P r o p o r t i o n of f u t u r e m o r t a l i t y s a l v a g e d Column number 1 7 16.1, common /thins2/ s a l v g e . e f f e c t 56.1, 56.2, e f f e c t = 0 . salvge=0. 132.5, 29 132.6, 132.7, 30 132.1, 132.2, 132.3, 132.4, i f ( d e l a . l t . e f f e c t ) go t o 29 x = e f f e c t / d e l a e f f e c t = 0 . go t o 30 e f f e c t = e f f e c t - d e l a x«=l. d y i n g = d y i n g - s a l v g e * x * d y l n g 337.1. common /thins2/ s a l v g e , e f f e c t 344.1, 344.2, 3 6 l .1, xntr=0.25*ntr resid=0. r e s i d = j j - l . 426.1, 426.2, 426.3, s a l v g e = a m i n l ( k l l / x n t r , 1 . ) - a m i n l ( r e s i d / x n t r , 1 . ) i f ( s a l v g e . ltX>.) salve;e=0. e f f e c t = 5 . R e f e r e n c e s S m i t h , R.B.; H.M. C r a i g ; and D. Chu. 1970. F u n g a l d e t e r i o r -a t i o n of second growth Douglas f i r l o g s on c o a s t a l B r i t i s h C o l u m b i a . Can. J o u r . Botany 4 8 ( 9 ) » I 5 l * * - l 5 5 l • W o r t h i n g t o n , R.P. 1961. Tree damage r e s u l t i n g f r om t h i n n i n g i n young growth Douglas f i r and Western hemlock. USDA F o r . S e r v . , P a c i f i c NthWest. F o r . and Range Exp. S t a . , Res. Note 202. 7 pp. 

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