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A geometric model of skyline thinning damage Ormerod, David William 1971

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A GEOMETRIC MODEL OF SKYLINE THINNING DAMAGE by DAVID WILLIAM ORMEROD B. S'; F., University of B r i t i s h Columbia, 1968 A Thesis Submitted i n P a r t i a l F u l f i l l m e n t of the Requirements for the Degree of MASTER of FORESTRY in the Faculty of FORESTRY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 19 71 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Forestry The University of British Columbia Vancouver 8, Canada Date A p r i l 26th, 1971. In thinning, physical damage to the residual trees may; r e s u l t from abrasion during f e l l i n g and yarding. The amount of damage i s primarily a function of the stand geometry, the thinning p r e s c r i p t i o n , and the f e l l i n g strategy, under a given extraction system. In terms of c o n t r o l l i n g the l e v e l of damage, there are two interdependent aspects to the problem. One i s to prescribe a desirable thinning^that i s compatible with the extraction system. The other i s to e f f i c i e n t l y engineer the extraction under the given s i l v i c u l t u r a l constraints. For skyline thinning, i t i s assumed that a geometric model: of the stand, and of the extraction system, provides a framework for examining p o t e n t i a l physical damage, in terms of t h i s interdependence. Based on a three-dimensional model, skyline thinning i s simulated, and indices of potential damage are enumerated. For a sample stand of Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), the potential damage i s studied as a function of the pr e s c r i p t i o n , and the system parameter that determines the f e l l i n g d i r e c t i o n s . Three d i f f e r e n t selection prescriptions are examined; a Low, a Crown, and a Graded. The synthetic data i s discussed i n terms of frequency d i s t r i b u t i o n s , and as a function of the parameter mentioned. It i s demonstrated that the system i s very sensitive to t h i s parameter. While the e f f e c t of the d i f f e r e n t prescribed thinnings might be thought to be i n t u i t i v e l y obvious, some enigmatic phenomena are apparent. It i s proposed that such a study i s a means of examining both the s i l v i c u l t u r a l and engineering aspects of the problem of physical damage i n the residual stand, for a skyline thinning system. It i s hoped that such deliberation w i l l provide a r a t i o n a l framework to determine the ef f e c t - o f t h i s damage upon thinning regimes.. Page ABSTRACT i i TABLE OF CONTENTS • • • 1 V LIST OF FIGURES v i ACKNOWLEDGEMENTS . . viai*? INTRODUCTION 1 GROWTH, THINNING, AND THE SKYLINE SYSTEM 3. Stand Development and Thinning Prescriptions . . 3 The,Thinning System and Damage to the Residual Stand . . . 4 The Component Thinning A c t i v i t i e s and the Skyline Set-up . 6 THE MODEL ' 7 The Geometric Framework 7 Selecting the Direction of F a l l 8 Interaction During F e l l i n g . . 15 Skyline Extraction . . . 23 The Computer Simulation Program , . 24 AN ILLUSTRATIVE EXAMPLE 26 Objectives . . . • 26 The Variables Discussed 2 6 The Stand Information 27 The Prescriptions 28 The Experimental T r i a l s 30 Comparing the Three Prescriptions 32 An Example of F e l l i n g Interaction 36 Page S e n s i t i v i t y of the System to 6; • . 38 J J mm F e l l i n g Direction 38 F e l l i n g Interaction . . . . . 41 Yarding Damage 43 SUMMARY 46 SUGGESTIONS FOR FUTURE WORK- . 49 BIBLIOGRAPHY 50 APPENDICES t. . 52 1. Stand and Pre s c r i p t i o n D e t a i l . . . 53 2. Stem maps 54 3. Parameter Values Used i n T r i a l s 58 4. The Fortran IV Program . .- 59 Figure. Page 1 -The Thinning Operation 6 2 The Thinning Process for an Individual Tree . . . . 6 3 The Thinning Area 8 4a-4c Examples of the Angle <5 9 5 The F e l l i n g Area . . 12 6 A Constrained F e l l i n g Area 13 7 The F e l l i n g Position . . . . . . . 15 8a-8c The Geometry of a P a r t i c u l a r F e l l i n g Situation . . 18 9 An Example of Crown-on-bole Interaction . 19 10 Crown Shapes Given by Equation (4) 22 11 The Extraction Situation 2 3 12 Examples of Angle v . 27 13 Diameter Frequency Distributions . . . 30 14 Frequency Distributions, of v 33 15 Frequency Distributions of w . 33 16 Frequency Distributions of n 34 17 Three Attitudes During a F a l l 37 18 Interaction C o e f f i c i e n t Against Attitude of F a l l . 38 19 D i s t r i b u t i o n f o f v for Different Values of 6 . • . 40 mm 20 Expectation of v Against 6 40 . mm Figure Page 21 Expectation of w Against <5 . 41 c mm 22 Expected Max. Cc Interaction Against 6 for Low Thin 42 min 23 Percentage of Acceptable 6 Openings Against <5mj_n. • • 43 24 Expectation of n Against S • • 44 ACKNOWLEDGEMENTS I would l i k e to thank Dr. C. W. Boyd for suggesting the study, and for guidance and review throughout the work. Mr. L. Adamovich provided some, of the l i t e r a t u r e and participated i n discussions about the model. Drafts of the manuscript were reviewed and commented upon by Mr. J..Walters, Dr. J. Worrall, and Mr. G. G. Young. The comments provided by fellow graduate students and others have been much appreciated. I am esp e c i a l l y g r a t e f u l to my wife Rose Mary for her support and encouragement, and patience and assistance during the preparation of the manuscript. Despite several centuries of experience i n forest regulation, refined thinning practise i s not universal. In the P a c i f i c northwest of North America thinning i s rudimentary. In t h i s region thinning prescriptions are often acts of f a i t h , and incompatible ,with the extraction method. As a r e s u l t , the residual stands may be seriously damaged. There are two aspects to t h i s problem of incompatibility. One i s the d i f f i c u l t y of making and marking a p r e s c r i p t i o n so as to take into account the r e s u l t i n g extraction damage. The other i s the problem of engineering the extraction under the b i o l o g i c a l constraints. Hence, the s i l v i c u l t u r i s t and the harvesting engineer need common i n f o r -* mation (3) about the physical interdependence of the prescrip-t i o n and the extraction system. A framework i s needed within which t h i s information may be described and evaluated. Although t r a c t i v e methods for thinning extraction have seen tremendous development i n recent years (14), they s t i l l are not suitable for the rough t e r r a i n of the coastal region of the P a c i f i c northwest. In t h i s region skyline cable systems are being used for thinning. References to the l i t e r a t u r e are indicated by underlined and bracketed numbers. The bibliography i s numbered alphabetically. In t h i s study of skyline thinning damage, a model i s developed to describe the thinning process, from marking to . extraction. This model i s geometric and i s assumed as a framework for examining the physical interdependence of the pre s c r i p t i o n , the f e l l i n g , and the extraction of the logs. An enumeration of the physical i n t e r a c t i o n , i n terms of t h i s model, i s produced by a computer simulation of thinning. Such synthetic data i s assumed to be d i r e c t l y related to the actual physical damage that may r e s u l t i n the f i e l d . This data may be used i n conjunction with b i o l o g i c a l , mechanical, and economic theory, to provide a basis for choosing thinning prescriptions, and examining the technological parameters of extraction. It i s hoped that, i n addition.to the p r a c t i c a l framework that t h i s study may provide, the t h e o r e t i c a l deliberation w i l l expose the complexity of the problem, and.therefore indicate avenues for worthwhile research. It i s necessary to review some of the relevant b i o l o g i c a l and technical aspects of thinning. It i s hoped that t h i s discussion w i l l provide a clear perspective, and insight into the rationale used i n developing and tes t i n g the model. GROWTH, THINNING, AND THE SKYLINE SYSTEM Stand Development and Thinning Prescriptions Trees and stands grow i n a complex and sensitive b i o l o g i c a l system. The development of individuals i s dependent upon t h e i r vigour, and their. position i n the stand geometry. In terms of economic -regulation of the forest i t may be desirable to intervene i n stand development by thinning. The v a r i a t i o n of s i t e carrying capacity throughout the stand i s the prime cause of v a r i a t i o n i n vigour, and t h i s e f f e c t i s most pronounced i n the early period of growth. However, the growth history of the individuals affects future growth. Thinning seeks to maintain a high rate of increment, and therefore improve the y i e l d of the stand. It has been suggested by Day (5_) that high increment can be continuously effected by a regular and proportionate development of the tree components. Thinning regimes should therefore be r a t i o n a l manipulation of the stand with the aim of increasing the y i e l d . The more s k i l l f u l l y applied the prescriptions and marking, the greater may-be this increase. Thinning pr e s c r i p t i o n and marking are s k i l l s based on the theory of growth, and honed to an art by f i e l d experience. The manipulation of growth i s j u s t i f i a b l e p h y s i o l o g i c a l l y , as has been substantiated by f i e l s ; .iplot research (13_) . The th e o r e t i c a l regime may be managed within the framework of a mathematical growth model, such as that discussed by Pienaar (12). The Thinning System and Damage to the Residual Stand Thinning, as a logging system, has been described t r a d i t i o n a l l y by the amount of conversion that occurs at the stump. This c l a s s i f i c a t i o n , and the evolution of some logging methods, has been described by S i l v e r s i d e s (14). In thinning, the transportation of the converted trees out of the stand has been achieved through t r a c t i v e power, or by winching. The trend of t r a c t i v e system development i s towarcfs a highly mechanized ' f u l l - t r e e ' system. These machines are not being developed primarily for thinning. However, there have been studies into the mechanics (9), and the harvesting productivity (10), of thinning systems using these machines. Yet t h e i r u t i l i t y for thinning i s limited by t h e i r s i z e . Some of these machines exceed t h i r t y tons i n weight, and the p o t e n t i a l for damage they can cause i s obvious. The thinning experience i n North America has usually been on areas of f l a t t e r r a i n •, and therefore cable systems Have not been common. However, with the ever-increasing demands upon the timber resources of the P a c i f i c northwest, forest regulation has become viable. The l a s t decade has seen the evolution of cable thinning systems suitable for the rough t e r r a i n of t h i s region. It i s now generally conceded that skylines are the most useful of these systems. The problems of thinning i n t h i s region, and some of the developments, have been discussed by Adamovich (1, 2 ) . The residual stand may be d e b i l i t a t e d i n several ways. Indiscriminate f e l l i n g may damage the crowns., predisposing the stand to insect and fungal attack. The,stems and roots may be injured during extraction, or whole trees may be uprooted or broken o f f . After thinning, windthrow, snowbreak, and sunscald may occur, and disease and decay spread. The f e r t i l i t y of the s i t e may be decreased by removing nutrients in the slash, and by exposure and s c a r i f i c a t i o n (15_) . In t h i s study, i t i s only the physical damage to the residual trees that i s considered. Aside from damage to the s i t e , most of the losses i n y i e l d are due to the decrease i n growth po t e n t i a l caused by physical damage to the residual trees. Most of t h i s damage occurs above the ground, i n skyline thinning. Hence the model i s r e s t r i c t e d to the space above the ground. The prediction of the amounts and d i s t r i b u t i o n of damage i s e s s e n t i a l for the planning and control of operations. Yet there has been very l i t t l e research. For some of the thinning methods of Finland, Karkkainen (7_) has used multiple regression to develop equations that estimate the numbers of, injured trees (in j u r i e s to the stem and roots) per unit area. As yet there has not been any s i g n i f i c a n t experimentation i n regards to such damage, nor exposition of the physics involved i n thinning damage. The Component T h i n n i n g A c t i v i t i e s and the S k y l i n e Set-up There are three p r i n c i p a l phases to the t h i n n i n g o p e r a t i o n ; marking the p r e s c r i b e d c u t , f e l l i n g the t r e e s under a g i v e n s t r a t e g y , and e x t r a c t i n g the logs i n a s e t method. F i g u r e 1 r e p r e s e n t s the o v e r a l l o p e r a t i o n , and F i g u r e 2 the t h i n n i n g process f o r an i n d i v i d u a l t r e e . Although the f i g u r e s r e f e r to a s p e c i f i c s k y l i n e system, any s e l e c t i v e t h i n n i n g o p e r a t i o n may be r e p r e s e n t e d i n a s i m i l a r way.• I n i t i a l Managerial D e c i s i o n S i l v i c u l t u r a l P r e s c r i p t i o n and E n g i n e e r i n g P l a n Marking * F e l l Ling S k y l i n e Set-up E x t r a c t i o n F.igure 1 The T h i n n i n g Operation S e l e c t the F e l l i n g D i r e c t i o n -*-. F e l l the Tree -> E x t r a c t the Logs F i g u r e 2 The T h i n n i n g Process f o r an I n d i v i d u a l Tree The computer program prepared f o r t h i s study r e p e a t s the t h i n n i n g process of F i g u r e 2, f o r each marked t r e e , u n t i l the p r e s c r i b e d h a r v e s t i s complete. F i g u r e 2 i s the modelJstu'died. THE MODEL The Geometric Framework. In the context of t h i s study a stand of trees i s described i n three dimensional Cartesian space, with the major coordinates denoted by x', y', and z 1 . The trees are rooted i n the x'-y' plane and grow upward i n the z' d i r e c t i o n . The trees i n the area are denoted by t ^ for i = 1, ... , n. Each tree's location can then be described by the location of i t s base at (x|, y|, 0 ) . Whenever a temporary Cartesian system i s required within the major system, the coordinates are denoted x, y, and z. It i s also convenient to describe points i n the space by other coordinate systems, namely the polar, c y l i n d r i c a l , and spherical systems. Hence the one-to-one mappings used i n th i s study, with transformed ori g i n s always set to a tree base, are as follows; (x|, y [ , z| = 0) : (x = 0, y = 0, z = a) (xj_, y|) : (r = 0, e =GQ) (x, y, z) : (r, 6, z) (x, y, z) : (p• ; 8 , \p) The area to be thinned i s assumed to be rectangular ; with the skyline cable running along the x' axis (Figure 3 ) . The skyline road (the cleared area underneath the cable) i s assumed to be exterior to the thinning area. The boundaries of the thinning area are the l i n e s x' = 0, x' = x' , y' = 0 , and y 1 = y' • •* max !max yarding d i r e c t i o n • « x' Skyline road x. max Figure 3 The Thinning Area Selecting the Direction of F a l l The marked trees should be f e l l e d i n such a way that the r e s i d u a l crop i s not unduly damaged. To t h i s end, the s i l v i c u l t u r i s t and the harvesting engineer must e s t a b l i s h simple guidelines for the workmen f e l l i n g the trees. These guidelines w i l l specify a preferred f e l l i n g d i r e c t i o n , and a minimum acceptable opening size for any f e l l i n g d i r e c t i o n . In t h i s model i t i s assumed that the l a t e r a l yarding distance should be minimized, arid hence the preferred f e l l i n g d i r e c t i o n i s towards the skyline (negative y' d i r e c t i o n i n Figure 3 ) . In following the guidelines the workmem must i n t u i t i v e l y assess each a l t e r n a t i v e f e l l i n g opening. The parameters considered are the r e l a t i v e sizes of the trees and the i n t e r - t r e e distances. For the purposes of simulating t h i s decision process i t i s necessary to quantify the i n t u i t i v e assessment i n terms of the geometry of the stand. In t h i s model the angle 6, which i s a function of basal tree diameter and horizontal i n t e r - t r e e distance, i s used:to describe the f e l l i n g openings. This angle i s defined by the tangent l i n e s connecting the bases of.the trees, as * shown i n Figure 4a. The tree to be f e l l e d is. located at A. Figures ^b and 4c are examples of, other configurations. Figures 4a to 4c 'Examples of the Angle 6 * An angle approximating <5 was used by M. McGreeyy at the University of B r i t i s h Columbia, as a possible growth parameter. The d e r i v a t i o n of <5 > i n terms of the geometry of the three t r e e s (Figure 4a), i s as f o l l o w s ; To f i n d angle PRQ, denoted <5 V e c t o r s AB and AC, with lengths AB and AC, have the r e l a t i o n s h i p ; c o s i n e Y = AB AC AB AC The r a d i i of the t r e e s l o c a t e d at A, B, and C are denoted r , r, , and r . a b' c Now t r i a n g l e s ASP and BUP are s i m i l a r , as are t r i a n g l e s ATQ and CVQ, r r a AP j a — = ==- and — r, PB r b c Apr QC /. AP = AB r r + r, a b .'. a = a r c s i n e and s i m i l a r l y , 3 = a r c s i n e j AP AQ Now i n t r i a n g l e s PRQ and PQA; a + 3 + <S + (TT - y) = TT .'. a + 3 + 6 =: Y 6 = y - a - 3 Note: 6 has s i g n i f i c a n c e only i n the range (-TT,,TT). I n a s s e s s i n g t h e a l t e r n a t i v e f e l l i n g o p e n i n g s t h e workman must d e c i d e i f a p a r t i c u l a r o p e n i n g i s as l e a s t as l a r g e as t h e minimum s i z e i n d i c a t e d i n t h e f e l l i n g g u i d e l i n e s . I n t h i s s i m u l a t i o n m odel i t i s assumed t h a t a p a r t i c u l a r v a l u e o f 6 c a n be u s e d as a s y s t e m p a r a m e t e r t o a p p r o x i m a t e t h i s minimum s i z e o p e n i n g . T h i s p a r a m e t e r i s d e n o t e d 6 m ; j _ n -As t h e s i z e o f t h e c o n s t r a i n t t r e e s ( a t B and C i n F i g u r e 4a) i n c r e a s e s , 6 w i l l d e c r e a s e t o r e f l e c t t h e s m a l l e r o p e n i n g ; as t h e d i s t a n c e between t h e c o n s t r a i n t t r e e s and t h e marked t r e e i n c r e a s e s , t h e ' t a r g e t ' w i l l g e t s m a l l e r , and 6 w i l l d e c r e a s e . I n s h o r t , 6 may be a s e n s i t i v e measurement o f t h e f a l l i n g ' c h a n c e 1 . I t i s assumed t h a t a l l t h e marked t r e e s a r e f e l l e d w i t h i n t h e b o u n d a r i e s o f t h e t h i n n i n g a r e a , e x c e p t where t h e y may be a l l o w e d t o f a l l a s p e c i f i e d d i s t a n c e i n t o t h e . s k y l i n e r o a d . I t i s a l s o assumed t h a t f a l l i n g p r o c e e d s away f r o m t h e s k y l i n e , and h e n c e i t i s o n l y t h e r e s i d u a l s t a n d t h a t m i g h t o b s t r u c t t h e f a l l o f a t r e e when f e l l e d t o w a r d s t h e s k y l i n e . A r o u n d any marked t r e e t h e r e a r e p o s s i b l y o t h e r s t h a t m i g h t o b s t r u c t t h e f a l l . I f t h e t r e e i s assumed t o p i v o t on t h e stump w h i l e f a l l i n g , t h e n any o t h e r t r e e f u r t h e r away t h a n a h o r i z o n t a l d i s t a n c e e q u a l t o t h e h e i g h t o f t h e marked t r e e , c a n n o t be damaged. However, t h e t o p o f t h e marked t r e e may be t o o f l e x i b l e t o damage o t h e r t r e e s d u r i n g t h e f a l l , and t h e r e f o r e t h i s d i s t a n c e may b e . l e s s . I n t h i s model i t i s assumed t h a t t h i s i n e f f e c t i v e t o p i s a c o n s t a n t l e n g t h f o r a l l marked.trees, and i s denoted 1 top' The radius of the c i r c u l a r area containing the constraint trees i s then h - l t 0 p / for any-marked tree of height .h. Figure 5 represents the s i t u a t i o n for a p a r t i c u l a r tree. Within the c i r c l e described by radius = h - l t 0 p the m constraint trees are denoted t. for j = 1,. ... , m. In the 3 figure the marked tree i s denoted,, t Q , and i t s base i s the origin, of a temporary x and y coordinate system. Associated with t ^ and any neighbouring pair of constraint trees, say tg and t^g, there i s an angle 6^. There are m--- 1 such angles that describe the f e l l i n g openings. The po s i t i o n of each 6 k , with respect to i t s deviation from the y axis, can be described by the smallest angle a^. This angle i s measured to the l i n e that bisects the relevant 6, . \ \ \ / 0 / / / / Figure 5 The F e l l i n g Area Whenever a thinning area boundary intersects the c i r c l e containing the constraint trees, then the tree cannot be f e l l e d i n a d i r e c t i o n that would cross the l i n e s of i n t e r -section (because of the assumption stated e a r l i e r ) . . In Figure 6, the c i r c l e associated with a p a r t i c u l a r tree i s bisected by the l i n e s x' = x' and y 1 = y' The i n f e a s i b l e J max. J * max. f e l l i n g area i s shaded i n , and the boundaries are denoted b g, for e = 1, ... , 4. : i x ' i max -*x t Figure 6 A Constrained F e l l i n g Area If there i s a space between boundaries, devoid of constraint trees, as for b^ and i n Figure 6, or.between a boundary and the closest constraint tree, then such openings . . , p. The-positions of these openings can be described s i m i l a r l y can be described, by the the angle J f r , for g = 1, to the d e s c r i p t i o n s f o r the 6 openings, by an angle.denoted 8 . I t should be noted when there i s i n t e r s e c t i o n of the f e l l i n g c i r c l e by the t h i n n i n g area boundaries, t h e r e may be l e s s than m - 1 6-openings f o r m c o n s t r a i n t .trees. The d e s c r i b e d a l t e r n a t i v e openings f o r any marked t r e e are t h e r e f o r e the s e t s { (<5, , a , ) , ... , (6 , a )} and { (e, , 8-,), . . . , . feD , 8 )} f o r m c o n s t r a i n t t r e e s where j . J. p p q < m - 1. The parameters used i n s e l e c t i n g the f i n a l f e l l i n g d i r e c t i o n are 6 • , e x p l a i n e d p r e v i o u s l y , a n d © . , the minimum mm' e * 1 1 ;mm' e opening i n t o which a t r e e may be f e l l e d without c o n s t r a i n t . The d e c i s i o n process i s as f o l l o w s ; 1. Scan the e openings f o r a e > e . wit h minimum 8. ^ ^ g = min M I f such an e e x i s t s proceed t o step 4. 2. Scan the 6 /)openings f o r a 6 > 6 . with minimum a. * ^ q = mm I f such e x i s t s proceed t o step 4. 3. F i n d .6' w i t h minimum d i f f e r e n c e 6 - 6. q mm 4. Describe the s e l e c t e d p o s i t i o n . The s e l e c t e d p o s i t i o n f o r any marked t r e e can be d e s c r i b e d i n p o l a r c o o r d i n a t e s ( r , 6 ) , where the merchantable l e n g t h o f the t r e e equals r . In F i g u r e 7 the f e l l i n g p o s i t i o n f o r a t r e e f e l l e d i n t o a 6 opening i s d e s c r i b e d . The ' f e l l i n g c o r r i d o r ' i s r e p r e s e n t e d by the dashed p a r a l l e l l i n e s , and i t s width i s double .the l e n g t h s. The f e l l i n g d i r e c t i o n d e f i n e d by 9 b i s e c t s the l i n e j o i n i n g the ce n t r e s of the two c o n s t r a i n t t r e e s . The d o t t e d l i n e s r e p r e s e n t the merchantable t r e e l e n g t h . The v a r i a b l e s i s used i n the f o l l o w i n g s e c t i o n t o d e s c r i b e i n t e r a c t i o n d u r i n g f e l l i n g . \ F i g u r e 7 T h e F e l l i n g P o s i t i o n I f t h e s e l e c t e d p o s i t i o n i s i n a 6 o p e n i n g t h e n t h e r e may b e p h y s i c a l i n t e r a c t i o n d u r i n g f e l l i n g , w h i c h c o u l d d a m a g e t h e r e s i d u a l t r e e s . I f n o t , t h e n i n t e r a c t i o n i s n o t p o s s i b l e , a n d t h e t h i n n i n g s i m u l a t i o n w i l l p r o c e e d d i r e c t l y t o t h e y a r d i n g p h a s e . I n t e r a c t i o n P o u r i n g F e l l i n g I t i s a s s u m e d t h a t t h e t w o c o n s t r a i n t t r e e s t h a t d e f i n e t h e a n g l e .6 a r e t h e o n l y t r e e s t h a t o b s t r u c t f e l l i n g , a l t h o u g h t h e r e may b e m o r e i n v o l v e d i n t h e f i e l d . H o w e v e r , t h e t w o t r e e s c l o s e s t t o t h e f e l l i n g p a t h w i l l i n t e r a c t t h e m o s t , i f t h e t r e e s i n t h e s t a n d a r e o f s i m i l a r s i z e . I t i s a l s o assumed t h a t i n the f i e l d the workmen are able to d i r e c t the f e l l e d t r e e s down the middle of the f e l l i n g c o r r i d o r . The p h y s i c a l l i m i t a t i o n imposed by the two c o n s t r a i n t t r e e s may be measured by c o e f f i c i e n t s t h a t r e f l e c t the p o t e n t i a l i n t e r a c t i o n . These c o e f f i c i e n t s are c a l c u l a t e d f o r each c o n s t r a i n t t r e e , from the t h e o r e t i c a l amounts of o v e r l a p t h a t * can occur f o r any g i v e n p o s i t i o n i n the f a l l of a t r e e . The o v e r l a p of the peripheral twigs of the crown i s o b v i o u s l y of l i t t l e s i g n i f i c a n c e f o r p o t e n t i a l damage, and therefore, r e f e r e n c e i s always to the ' e f f e c t i v e ' crown. Three types of i n t e r a c t i o n are d e f i n e d ; 1. The crown of the f a l l i n g t r e e a g a i n s t the crown of the s t a n d i n g trees.(crown-on-crown). 2. The crown of the f a l l i n g t r e e a g a i n s t the c l e a r . b o l e of the s t a n d i n g t r e e s (crown-on-bole). 3. The c l e a r bole of the f a l l i n g t r e e a g a i n s t the crown of the s t a n d i n g t r e e s (bole-on-crown). A f e l l i n g s i t u a t i o n i s r e p r e s e n t e d i n F i g u r e 8a, with the r e l e v a n t geometry shown i n F i g u r e s 8b and 8c. For any p o s i t i o n i n the f a l l of a t r e e a c o e f f i c i e n t may be c a l c u l a t e d to r e f l e c t the p o t e n t i a l amount of i n t e r a c t i o n w i t h a c o n s t r a i n t t r e e . Two c o e f f i c i e n t s have been d e r i v e d . * S i m i l a r c o e f f i c i e n t s were used i n He s t u d i e d the r e s i s t a n c e of t r e e s p o s i t i o n . experiments of Myrhman l(.8) . to e x t r a c t i o n i n a v e r t i c a l If the i n t e r a c t i o n i s crown-on-crown then the amount-of overlap can be described as; Cc < 0 when no overlap Cc = 1 - \c + c f s 0 < Cc < 1 when overlap ^ where s i s the separation of the two trees (as shown i n Figure 7 ) , and c^ and c are the l a t e r a l crown extensions f o r , respectively, the f a l l i n g tree and the standing tree. When the in t e r a c t i o n i s crown-on-bole or bole-on-crown then the c o e f f i c i e n t i s given by; , _ , s + d/2 jCb < 0 when no overlap ( r ). L c + d ' \0 <"Cb < 1 when overlap K ' where s i s as i n Equation (1), d the diameter of the clear bole, and c the l a t e r a l crown extension. This c o e f f i c i e n t r e f l e c t s the.amount the clear bole i s embedded into the crown, whereas the c o e f f i c i e n t Cc r e f l e c t s the amount of crown overlap. The po s i t i o n of the f a l l i n g tree can be described by the angle . The c o e f f i c i e n t s of i n t e r a c t i o n with each constraint tree may then be calculated for any value of i p . In Figure 8b the f a l l i n g tree may in t e r a c t with the constraint trees across the li n e s RQ and VW. This i n t e r a c t i o n w i l l be crown-on-crown for the p a r t i c u l a r position shown. Figure 8c i s a view i n the x-y plane passing through z of Figure 8b, and demonstrates the crown overlap for the constraint tree at B. The s o l i d l i n e s i n Figure 8c represent the perimeters of the crowns i n the x-y plane, and the dotted l i n e s are the p r o f i l e projections. T h e : c o e f f i c i e n t Cc that may be calculated Figures 8a to 8c The Geometry of a P a r t i c u l a r F e l l i n g Situation along the l i n e VW w i l l therefore be less than zero, because there i s no overlap. In Figure 8c the projection of the crown of the tree at C i s . given for the x-y plane passing through z__. It i s obvious from the projection of the p r o f i l e of the f a l l i n g tree.that a p o s i t i v e Cc c o e f f i c i e n t may be calculated for the tree at C. As there i s no crown-on-bole or bole-on-crown in t e r a c t i o n evident i n Figure 8a, an example of the former type i s given i n Figure 9. In-this figure the s o l i d l i n e s depict the dimensions of the two trees i n an x-y plane crossing the/ z axis,, and the dotted l i n e s the projection of the p r o f i l e of the f a l l i n g tree onto t h i s plane. i / i ul »x - I , . L _ J Figure 9 An Example of Crown-on-bole Interaction In simulating the f a l l of the tree i t i s assumed that the interactions with the constraint trees do not cause de f l e c t i o n out of the v e r t i c a l plane that divides the f e l l i n g corridor; that i s , the distances RQ and VW (in Figure 8b) are equal, for any value o f T h e assumption i s necessary, because although something i s known about the dynamics of f a l l i n g trees, it.has not been shown how to describe the s i t u a t i o n i n functional terms. However, i t i s useful to describe the interactions possible i n the fixed plane. Simulation of the f a l l of the tree i n the fixed plane may then be achieved by incrementing ^ and c a l c u l a t i n g the appropriate c o e f f i c i e n t s for each position. for respective Equations (1) and (2), must be provided for each increment position of i\>. Simple continuous and d i f f e r e n t i a b l e functions are needed to describe the longitudinal shapes of the trees. The l a t e r a l extensions of the tree compon-ents (from the central axis) should be described with these functions. The points along the central axes where the value of these extensions are needed can be calculated from the geometry shown i n Figure 8b. The point on the f a l l i n g tree, for example, that w i l l interact with the tree at C, i s given by the spherical coordinates (p , 9 , ty) where 6 = 8 , + arcsine (v) , q q q c where v = CP/r c. Two functions are needed; one to describe the bole, and the other to describe the crown. The equation that provides an estimate of bole diameter d at height h i s given by; where H i s the t o t a l height of the tree, and D i s a diameter measurement taken at h = h . If p >v 1 the taper i s n e i l o i d a l , The apprpp'-rviate .values of .c f ahd-,c ,- or> c and.d, (3) but i f p < 1 i t i s p a r a b o l i c . The equation i s c o n d i t i o n e d so t h a t when h = n D - d = D, and when h = H, d = 0. T h i s p a r t i c u l a r f u n c t i o n was chosen because i t i s simple, f l e x i b l e , and can be d i f f e r e n t i a t e d and i n t e g r a t e d . I f Equation (3) f a i l s t o s a t i s f a c t o r i l y d e s c r i b e a p a r t i c u l a r b o l e shape, then i t can be used as a step f u n c t i o n to d e s c r i b e s e c t i o n s o f the b o l e . In t h i s case, diameter measurements, or e s t i m a t e s , are needed f o r each s e c t i o n . The e q u a t i o n used to ; d e s c r i b e the ' e f f e c t i v e ' crown shape i s ; cw = A x - g ( p 2 . p i , IJ where cw i s the estimated crown width a t a h e i g h t x = h",4f L - H f o r a crown l e n g t h L. H and h are as f o r Equation (3). The two exponents p and p 9 , and the c o e f f i c i e n t A, are the I • ^ parameters t h a t d e s c r i b e a p a r t i c u l a r shape. The equation i s c o n d i t i o n e d so t h a t when x = 0 (at , the base of the crown), cw = 0, and when x = L; cw = 0..- The equation i s l i m i t e d because as p^ approaches zero, the f i r s t term i n the b r a c k e t s approaches one,, and the shape d e s c r i b e d becomes u n r e a l i s t i c . However, the equ a t i o n i s simple and describes^varxgus^sjiap^,^ as shown i n F i g u r e 10. I t has t h e r e f o r e been i n c o r p o r a t e d i n the model f o r l a c k of b e t t e r equations i n the a v a i l a b l e l i t e r a t u r e . The h e i g h t of maximum crown width can be determined by d i f f e r e n t i a t i n g E q u a t i o n ( 4 ) , e quating the r i g h t - h a n d s i d e JO < p, < 1 U i p; to zero, and solving for x. The point of maximum crown width x i s then; m Ap In (P 2 - P]_) L ( P l " - P 2) J and hence the height of maximum crown width = H - L + x . Figure 10 Crown Shapes Given by Equation (4) Once the f e l l i n g p o s ition has been selected, and the tree f e l l e d , the thinning process proceeds to the extraction phase. For the purposes of t h i s model i t i s necessary to describe the yarding geometry with respect to the residual stand. This geometry i s incorporated i n the thinning simulation and f a c i l i t a t e s a count of interactions during yarding. Skyline Extraction It i s assumed that the f e l l e d trees are delimbed and topped at the merchantable height, but are not bucked. The tree length logs are then yarded at r i g h t angles to the skyline. It i s also assumed that the trees are yarded separately, although i n the f i e l d t h i s may not be the practise, The end of the yarding cable.is attached to the end of the log that i s closest to the skyline. The log i s assumed to pivot on the free end during yarding. The path shown i n Figure 11 as a dotted l i n e i s the assumed path the yarded log w i l l follow. O r i g i n a l l y the f e l l e d tree had stood at point A. The merch-antable tree length log i s depicted on the l i n e AB, and i s winched i n to point C. merchantable log o damaged residual yarding path-^'. o o o o o o c X Figure 11 The Extraction Situation The number of times that the residual trees l i e on yarding paths may be a measure of the p o t e n t i a l damage. Any time that a p a r t i c u l a r tree l i e s on a path i t i s p o t e n t i a l l y scraped. The more times that i t i s scraped, the greater the p o t e n t i a l loss of growth. Once the log has been extracted to the skyline road, the thinning process, i n terms of t h i s study, i s complete. This thinning process can be repeated for each of the marked trees, and thereby simulate thinning, from f e l l i n g to extraction. It i s now pertinent to allude to the computer simulation that has been developed from the geometric model described. The Computer Simulation Program The Fortran IV program prepared for the model i s presented i n Appendix 4. The program has been compiled by a Fortran G/H l e v e l compiler program (plf the I. B. -M. 360/67 system of the University of British.Columbia. An explanation of the o v e r a l l algorithm i s included i n Appendix 4 to a s s i s t the reader i n following the program. The information needed by the program consists of; 1. The- x 1 and y' coordinates for each of the trees i n the thinning area (see Figure 3). 2. The dimensions of the trees; diameters at-breast height> height to the crown, t o t a l height, and merchantable height. 3. Parameters for Equations (3) and (4) to estimate bole diameters and crown widths. 4. The system parameters; 6 . , e . , the constant used •* m i n m m ' to increment ty, (see discussion near Figure 5), and the maximum extension of a f a l l i n g tree into the skyline road. 5. A code to distThg.uish the marked trees from the residual trees. The information generated by the program, and printed i n tabular form, consists of; 1. A description of the f e l l i n g s i t u a t i o n for each tree; the position 0 (see Figure 7), the indentity of the constraint trees, the corridor width, int e r a c t i o n c o e f f i c i e n t s for each constraint tree at each attitude ty, and an i d e n t i f y i n g code for the type of opening selected (> 6 . e t c . ) . = mm: 2. A yarding description; the log length, yarding distance, and the number of residual trees on the yarding path. 3. A summary of the number of potential scrapes for each residual tree. The variables used for an analysis of te s t runs of the compiled program are described i n the following chapter, as i s the sample stand used i n the thinning simulations. AN ILLUSTRATIVE EXAMPLE Objectives 1. To f e l l and extract logs under three d i f f e r e n t thinning prescriptions. 2. To examine t h e o r e t i c a l i n t e r a c t i o n during f e l l i n g . 3. To test the system's s e n s i t i v i t y to changes i n the value of 6 . . mm The Variables Discussed A l l r e s u l t s are discussed.principally i n terms of three variables. These are; 1. The angle v; the angular deviation from the preferred f e l l i n g directions for a f e l l e d tree. It i s the smallest absolute difference of the f e l l i n g d i r e c t i o n 0 * from the y axis. Two examples are shown i n Figure 12. 2. The width of the f e l l i n g c orridor, denoted w. 3. The number of times any residual tree was 'scraped' during the .yarding phase, denoted n. Smooth curves are used i n plots of the synthetic data. These curves are f i t t e d 'freehand', because the numbers of observations are small. A l l frequency d i s t r i b u t i o n s are drawn as smooth curves. The only s t a t i s t i c s given are the expec-ted values, which are often plotted as a function of 6 . . * See Figure 7 of the previous chapter. F i g u r e 12 Examples of Angle v The Stand Information A one-acre sample p l o t of D o u g l a s - f i r (Pseudotsuga * m e n z i e s i i (Mirb.) Franco) was chosen f o r the example. The f o l l o w i n g i n f o r m a t i o n had been measured, or was c a l c u l a t e d ; 1. The' x 1 and y 1 c o o r d i n a t e s and the diameters a t b r e a s t h e i g h t (dbh) had been measured at the stand age of 51 y e a r s . Only those t r e e s with dbh > 6 inches were used i n t h i s study. 2.. The , t o t a l h e i g h t s were estimated by an equation p r o v i d e d by P a i l l e (11)• 3. The h e i g h t to the base of the ' e f f e c t i v e ' crown was-approximated with an e q u a t i o n proposed by Smith e t al_ (1_5) . 4. The merchantable h e i g h t s (assuming a top diameter of 4 inches) were es t i m a t e d by s o l v i n g Equation (3) One of the p l o t s used by P a i l l e (11). for h (see previous chapter). These equations were assumed for want of better information; however, they are considered adequate for the purposes of t h i s example. The d e t a i l s of the stand description, including the equations mentioned above, are given i n Appendix 1. The s p a t i a l d i s t r i b u t i o n s of the trees, i n the x'-y' plane, are shown as stem maps contained in Appendix 2. The Prescriptions . Three thinning prescriptions were made for t h i s example and are described as follows; 1. A Low thinning. This p r e s c r i p t i o n corresponds to * a l i g h t C grade. The residual crop was selected from those trees that were dominant, and provided the most regular espacement . c- v This type of p r e s c r i p t i o n i s commonly applied for thinning Douglas-fir i n the P a c i f i c northwest. 2. A Crown thinning. A heavier cut (grade D) was marked by selecting some harvest trees i n the dominant classes. This type of p r e s c r i p t i o n i s used i n a regime of greater cycle length than one using Low prescriptions. 3. A Graded thinning. The t o t a l weight of t h i s p r e s c r i p t i o n ".was s i m i l a r to that of the Crown B r i t i s h Forestry Commission c l a s s i f i c a t i o n (6). prescription; however, the d i s t r i b u t i o n of the cut was not uniform. Near the skyline, the cut was heavy, and at the back of the area the cut was very l i g h t . This type of prescription has been proposed for use i n ^H^'^a.cfi'f ic^~^or'r^west>,(2_) . To aid i n the marking of the prescriptions, the author used various indices of r e l a t i v e tree s i z e . It i s important to note that t h i s marking was done as i f i t were in the f i e l d , and the trees were selected without the use of any computer programs. Of course, i f the marking were to be done i n the f i e l d there would be more information available.to the marker, and the marking might be a l i t t l e d i f f e r e n t . The d e t a i l s of the prescriptions are given i n Appendix 1, and the residual stem maps in Appendix 2. The frequency d i s t r i b u t i o n s of diameters for the residual stands are compared to the o r i g i n a l d i s t r i b u t i o n i n Figure 13. Note that the variance of the Graded pre s c r i p t i o n i s larger than that of the Crown pr e s c r i p t i o n . The larger variance accrues from the weight gradient which causes a wider range of size classes to be included i n the cut. if req + Uhthinned o Low & Crown i a Graded j dbh 1 Figure 13 Diameter Frequency Distributions The Experimental T r i a l s To meet the objectives stated i n th i s chapter, three sets of t r i a l s were made. In these t r i a l s i t was necessary to assume parameter values for Equations (3) and ( 4 ) . The exponent value used i n Equation (3) was p =•.8, which was selected because the r e s u l t i n g bole shape was very similar to the shape given by the appropriate equation of Kozak et. a l (8) . The parameter values used for Equation (4) were the same as those used to generate curve 1 i n Figure 10. This curve looks l i k e a t y p i c a l Douglas-fir forest-grown crown. It may describe a t y p i c a l ' e f f e c t i v e ' crown, and has been used for want of a better description. The t r i a l series are then; Set A. The whole plot was thinned under the three prescrip-tions A. value of 7° was a r b i t r a r i l y chosen for 6 . , as were values for the other system parameters, mm' J * Set B. A single t r i a l was made using the Crown prescrip-t i o n to provide some good examples of f e l l i n g interactions. In t h i s t r i a l a small value was used for incrementing the f a l l i n g attitude , so as to provide much d e t a i l . Set jC. It i s apparent that 6^ -^  I s the most s i g n i f i c a n t parameter. In order to demonstrate the s e n s i t i v i t y of the system to 6 • several t r i a l s were made 2 mm using d i f f e r e n t values of 6 • . These t r i a l s were ^ mm made on a subplot of the stand, for an economy of computer time. The increased bias associated with the boundaries ( r e l a t i v e l y more boundary trees to be constrained) should not detract from the comparison. After a l l , i t i s not the d i s t r i b u t i o n of constraint trees that i s affected by the boundaries, but rather the sizes of the feasible f e l l i n g areas. A detailed description of these t r i a l s i s given i n Appendix 3. Comparing the Three Prescriptions The r e s u l t s of the t r i a l s of Set A are presented as frequency d i s t r i b u t i o n s of variables v, w, and n, and in tabular form. The d i s t r i b u t i o n of v for the three prescriptions (Figure 14) r e f l e c t s the inte r a c t i o n of the stand geometry and the selection system. The differences i n the curves are due to the differences i n the geometry of the residual stands. Note that the curve for the Graded thin i s f l a t t e r than that for the Crown t h i n , which indicates thatJthe low weight of the thinning i n the upper part of the plot i s having a greater e f f e c t than the high weight i n the lower part. Therefore the o v e r a l l constraint upon f e l l i n g i s higher for the Graded th i n and res u l t s i n a greater mean v (less favourable mean d i r e c t i o n ) . However, the d i s t r i b u t i o n s of the corridor width w are si m i l a r for these two prescriptions (see Figure 15), although the means, are d i f f e r e n t . This s i m i l a r i t y r e f l e c t s the si m i l a r o v e r a l l weight. The table that follows the figures shows the weights and the means. 80, |f req. Figure 14 Frequency Distributions of v o Low A crown c Graded Figure 15 Frequency Distributions of w o -Low A Crown • Graded IT^f T T n Figure 16 Frequency Distributions of n  Comparative S t a t i s t i c s for the Three Prescriptions Pres. % t r . rem. % ba rem. Mean dbh A l e f t Expected Values % op. > 6". mm E(v) E (w) E(n) E(Cc) max. 1 2 3 4 5 6 7 8 9 Low 56.5 39. 9 . 96,' 10.8° 12.2' .54 .37 98.1 Crown 77. 3 65. 8 1.0 13.5 17.7 .97 .24 94.0 Grad. 76. 3 65.6 .99 18. 6 23.5 .67 .22 88.4 t r . = trees, rem. = removed, ba = basal area, op. = openings The difference i n the curve shapes for the d i s t r i b u t i o n of v for the Low and the Crown prescriptions may be due to the difference i n average tree' size (see column 4 of ta b l e ) . It was expected that the r e l a t i v e incidence of the small values of v would be less for the Crown pres c r i p t i o n because of greater available f e l l i n g space, but the converse i s true (note the larger E(v)). F e l l i n g of the larger trees i n the Crown prescription may involve the e f f e c t of r e l a t i v e l y more constraint trees. As expected, the Graded t h i n , when compared to the Low t h i n , results i n less potential yarding damage. This difference i s due to the shorter t o t a l yarding distance of the Graded t h i n , and i s evident i n Figure 16 and i n the table. Although there are fewer residual trees for the Grown pre s c r i p t i o n compared to the Low, the yarding damage may be greater. The corresponding larger E(n) i s due to the higher incidence of trees that were scraped several times. It was thought that the trees that would be scraped several times would always be those nearest the skyline. However, the d i s t r i b u t i o n of n i s apparently f a i r l y uniform throughout the residual stands, although trees with n > 3 were always within the half of the thinning area nearest the skyline. Column 8 i n the table was computed by only considering the largest crown-on-crown (Cc) i n t e r a c t i o n values for each f e l l e d tree. No interpretation can be made of these values because i t i s not known what the significance of f e l l i n g damage may be. However, the obvious proportional relationship to E(w) i s evident. The percentage of f e l l i n g openings selected that had a 6 > ^ m ; j _ n c a n be compared for each pre s c r i p t i o n to the expected values of the variables v, w, and n. It should be noted that there were no openings selected where <S> < 6 . , v ^ t f mm' and that the rest of the percentage points are attributed to openings where e 2 £ m ;|_ n- Note that although the average f e l l i n g d i r e c t i o n of the Graded th i n i s the least favourable, (largest E(v)), the incidence of 'free' openings (e > a ^ - ^ ) i s higher, i n d i c a t i n g easier f e l l i n g and yarding. The other s t a t i s t i c s substantiate t h i s advantage. The relationship between the d i s t r i b u t i o n s of opening types and damage should be investigated, as there ma'y.; be an application for t h i s information i n a stochastic prediction model. The information presented i n t h i s section i s not meant to be an exposition of the r e l a t i v e merits of the three prescriptions tested, but i s meant to i l l u s t r a t e the data provided by the model. Such information has hitherto been unavailable. An Example of F e l l i n g Interaction Figure 17 depicts a p a r t i c u l a r f a l l i n g s i t u a t i o n . The x-y-z o r i g i n i s at the base of the marked tree. The direction.of f a l l i s 0 and the view i s shown i n the v e r t i c a l z- (x/cosine.6) plane. The p r o f i l e projections of constraint trees B and C are shown as dashed l i n e s . The corridor width * See Figure 7 of the previous chapter. i n t h i s p a r t i c u l a r case i s 6.4 feet. The other dimensions are indicated on the figure. Figure 18 i s a graph of i n t e r a c t i o n c o e f f i c i e n t against attitude \> for each constraint tree. The points of r e f l e c t i o n indicated by the dotted arrows refer to the change from-> crown-on-crown (Cc) to crown-on-bole (Cb) i n t e r a c t i o n . The ranges of these interactions are • indicated by the numbered l i n e s , where; 1 = the Cc range for tree B. 2 = the Cc range for tree C. 3 = the Cb range for tree B. 4 = the Cb range for tree C. tree ;.104' z 84 48" Figure 17 Three Attitudes During a F a l l Figure 18 Interaction C o e f f i c i e n t Against Attitude of F a l l S e n s i t i v i t y of the System to- 6- . - ••—— —rain The- data discussed i n t h i s section are from the re s u l t s of the t r i a l s of Set C. F e l l i n g Direction The d i s t r i b u t i o n of v for the d i f f e r e n t prescriptions has already been i l l u s t r a t e d for a p a r t i c u l a r value of <5 . . J r mm It would be i n t e r e s t i n g to see how the d i s t r i b u t i o n s depend o n 6 . . Figure 19 i s a plot of the d i s t r i b u t i o n of v for. three mm- ^ JT-values of <$m^n for thinnning under the Low p r e s c r i p t i o n . As expected, the curve becomes f l a t for large values of • <$mj_n. because of the increased constraint upon the selection of the f e l l i n g d i r e c t i o n s . Similar curves were plotted for a wider range of <5 . than that shown i n the figure, and for a l l the mm- z> r . p r e s c r i p t i o n s . An unexpected increase i n the incidence for p a r t i c u l a r classes of v was observed. These p a r t i c u l a r classes were the same for a l l the p r e s c r i p t i o n s , and the increase i n the frequency for the 35°-40° class was the most prominent. The p o s s i b i l i t y of multimodal d i s t r i b u t i o n s had not been expected. Whether t h i s anomally i s due to the system, or tp the geometry of the stand, i s not clear and should be studied. The expected values of v are plotted against &mj_n i n Figure 20, for each p r e s c r i p t i o n . The important attributes of these curves i s that they have a single i n f l e c t i o n point, and that they are asymptotic below 40°. i n comparing the Low and the Crown prescriptions, the curves imply that when &m^n is-'large, the increased space afforded by the l a t t e r permits better positioning of the f e l l e d trees; but when 6 m ^ n i s small, the marginally smaller average tree size (see column 4 i n table of previous section) of the Low p r e s c r i p t i o n i s the c r i t i c a l factor. The expected values for the Graded pr e s c r i p t i o n are consistently.larger than those of the Crown. This difference suggests that the undesirable e f f e c t of the low weight of the Graded thin at the top of the p l o t consistently outweighs the reverse e f f e c t of the high weight i n the lower part of the p l o t . Figure 19 D i s t r i b u t i o n of v for Different Values of 6 . 3 ; . m J L n j 40 | E(v) i j 20 0 _ . . , ; 0 10 20 30 6 . mm „ I Figure 2 0 Expectation of v Against 6 . — t d mm F e l l i n g Interaction The* corridor width w i s d i r e c t l y related to the amount of possible i n t e r a c t i o n . The frequency d i s t r i b u t i o n o of w follows the same trends as that of v, including the indi c a t i o n of multimodal d i s t r i b u t i o n . A plot of the expectation of w as a function of 6 m j _ n (Figure 21) shows the same i n t e r a c t i v e e f f e c t of the weight of thinning and i t s d i s t r i b u t i o n as was observed for v. This i n t e r a c t i o n i s evident i n a comparison of the curves for the Crown and Graded prescriptions. When &mj_n exceeds approximately 6 ° , i n t h i s example,. the undesirable e f f e c t of the low weight i n the upper part of the Graded th i n apparently becomes important. o Low A Crown • Graded 30 mm Figure 21 Expectation of w Against <5 ^ n The inverse r e l a t i o n s h i p between the expected l e v e l of f e l l i n g i n t e r a c t i o n , and w, i s demonstrated by a comparison of the curve of Figure 22 with the curve for the Low thinning i n Figure 21. In Figure 22, the ordinate i s the expected maximum crown-on-crown interaction l e v e l that occurs during f e l l i n g . Although t h i s may not be the l e v e l that occurs i n the f i e l d , because of de f l e c t i o n during the f a l l , i t does, however.,- measure the s t a t i c constraint upon f e l l i n g . This information could be used i n conjunction with a description of the dynamics of f e l l i n g so as to predict physical damage. Figure 22 Expected Max. Cc Interaction Against ^ m ^ n for Low Thin In the t r i a l s of Set C there were very few openings selected because e > e . , due to the large value assigned to = mm ^ ^ e . for these t r i a l s . Therefore, i n the sel e c t i o n of the f e l l i min ' d i r e c t i o n , the angle 6 was always considered. The amount of f e l l i n g i n t e r a c t i o n i s determined by the size of the opening and the sizes of the trees. Hence, i t i s useful to record the percentage of selected openings where 5'> ^ m^ n? that i s , the percentage of times the 6 . constraint was not vi o l a t e d . r ^ mm Figure 2 3 i s a graph of t h i s percentage as a function of 6mj_n« Note that the space afforded by the Crown pr e s c r i p t i o n constrained f e l l i n g less than the space of the Graded prescrip-t i o n . The p r o b a b i l i t y of the v i o l a t i o n of the 6 . constraint . J mm could aid i n the prediction of f e l l i n g damage. Figure 23 Percentage of Acceptable 6 Openings Against 6 mm Yarding Damage The e f f e c t of the f e l l i n g directions on the po t e n t i a l yarding damage can be studied as a function of 6m;j_n# by considering the number of scrapes' that the residual trees endure (the variable n). The general 'parabolic' shape of the frequency d i s t r i b u t i o n of n was shown i n Figure 16 and i s , apparently, the generic s h a p e a s s i m i l a r curves were plotted for increasing values of 6 . . However, the expected values for ^ mm c n change markedly as 6 . i s increased (Figure 24). 3 mm E(n) / / / / / / _ n -- o — r o ^ h - o — o — — Low A Crown — • Graded 10 20 30 6 . 1 mm Figure 2 4 Expectation of n Against 6 . 3 — : mm-It i s apparent from the figure that the int e r a c t i o n of the geometry of the pre s c r i p t i o n with, the system i s very s i g n i f i c a n t ; In t h i s regard, note,the difference between the curves for the Crown, and Graded prescriptions. Although the t o t a l weight of thinning was sim i l a r for the two pr e s c r i p t i o n s , the Graded t h i n consistently resulted i n about half the amount of yarding damage. This difference i s due to the weight gradient of the Graded p r e s c r i p t i o n . An important feature of Figure 24 i s the general curve description; namely, a single i n f l e c t i o n point, and an obvious asymptote. These attributes are also those of the curves for the expectation of v (see Figure 2 0). However, the r e l a t i o n s h i p between E(v) and E(n) i s not one of d i r e c t p r o p o r t i o n a l i t y , i n d i c a t i n g the important dependence of damage upon the p r e s c r i p t i o n . The position of the asymptote and the smallest value of <5 ^  close,,t-o i t , may be important parameters for planning. For instance, a <S j _ n value greater than approximately 18° w i l l not r e s u l t i n a s i g n i f i c a n t l y larger E(n) for the Low pre s c r i p t i o n of t h i s example. However, the d i s t r i b u t i o n of n may continue to change. It should now be apparent that the prediction and control of damage i n skyline thinning,, are d i f f i c u l t tasks, and can only be solved by the application of sound physical theory to. the complex b i o l o g i c a l system. This study i s but an i n i t i a l attempt to disclose the relevant aspects of the problem, and to indicate what are the relevant physical par-ameters that should be considered. SUMMARY A three dimensional model of the stand and skyline system has been presented. The model has been used i n a discrete simulation of the thinning process, from f e l l i n g to extraction, as a framework for examining the physical i n t e r a c t i o n . It has been assumed that t h i s i n t e r a c t i o n i s d i r e c t l y proportional to the physical damage that would occur i n the f i e l d . In t h i s study the consideration o% damage has been li m i t e d to that which would occur above the ground i n the residual stand. The simulation program, based on the model, was tested on a sample stand of immature Douglas-fir, t y p i c a l of) the second-growth stands growing i n the coastal region of the P a c i f i c northwest. Despite obvious mensurational d e f i c i e n c i e s in. the example","..the author considers that the assumptions made were s u f f i c i e n t for reasonable hypothetical thinnings of the sample stand. The synthetic data produced by the program, and consisting of an enumeration of interactions during thinning, were evaluated for several t r i a l s . Three d i f f e r e n t prescriptions were examined; Low, Crown,.and Graded. The Low pr e s c r i p t i o n was a conservative approach, that may be t y p i c a l of thinning prescriptions i n the P a c i f i c northwest. The Crown and Graded prescriptions were heavier and of the same o v e r a l l weight (removing 66% of the basal area of trees > 6 inches dbh), but the Graded thin was characterized by a weight gradient, which reduced with increasing distance from the skyline. parameter defined as 6 . was considered the most s i g n i f i c a n t . r min ^ This parameter controls the selection of the f e l l i n g directions for the marked trees. The s e n s i t i v i t y of the system to 6 . m x n was tested for the three prescriptions. The d i s t r i b u t i o n s of the geometric variable were discussed i n terms of the stand description, the prescriptions, and <5 . . The reasons for differences i n the i n t e r a c t i o n l e v e l s mm for varying values of 6 . , that r e s u l t from the i n t e r a c t i o n •J ^ « mm of the s p a t i a l and size d i s t r i b u t i o n s of the trees, were not always i n t u i t i v e l y obvious. In terms of the potential damage the differences between the prescriptions were very apparent and sometimes unexpected. Some of the r e s u l t s of the tests are summarized as follows; 1. Although the Crown prescription providedx'more'space when compared to the Low, the larger trees l e f t were; more constrained during the selection of the f e l l i n g d i rections. 2. On the average the Graded thin was the most constrained i n the selection of f e l l i n g d i r e c t i o n s , although i t had the highest incidence of completely unconstrained C\ openings. Yet/ i t resulted i n far less potential f e l l i n g and yarding damage. The l a t t e r amount was about half that of the others for a l l values of 6 . . mm 3. It was. not • (P,expe<&ted that the d i s t r i b u t i o n of the residual trees that suffered several yarding 'scrapes' would be as uniform as observed. However, the very worst cases were s t i l l near the skyline. 4. The i n t e r a c t i o n of the weight of thinning and i t s d i s t r i b u t i o n (both, s p a t i a l l y and by tree s i z e ) , i s the most important factor for f e l l i n g and yarding i n t e r a c t i o n r d i f f e r e n c e s , that r e s u l t from d i f f e r e n t prescriptions. 5. Some, of the generic d i s t r i b u t i o n s of the geometric variables may become multimodal for large values of 6 . . . J ^ min This study has been e s s e n t i a l l y exploratory, and much more work i s required. Some suggestions are made af t e r t h i s summary. Physical damage to the trees i s a very important factor i n reducing future growth. In skyline thinning i t may be the most c r i t i c a l damage component. It i s hoped that t h i s study has shown that the prediction and control of t h i s damage are complex s i l v i c u l t u r a l and engineering tasks, and worthy of future i n v e s t i g a t i o n . It i s also hoped that the model presented here may; serve as a basic framework for such deliberations. It i s not known whether 6 • can be used as a mm parameter for accurate simulation of the decision process that selects the d i r e c t i o n of f a l l . Because of i t s attributes it.has been assumed i n th i s study that i t can. Perhaps the constraint should be expressed as a function of the p a r t i c u l a r ,6\ and other tree and stand variables. It i s also not known how 6 . could be translated into simple f i e l d guidelines mm ^ ^ comprehendible to the average workman. It may not be possible to express s a t i s f a c t o r y guidelines, and therefore the control over f e l l i n g may have to be exercised through t r a i n i n g programs^, for f i e l d personnel. These problems can only be solved empirically. The enumeration of int e r a c t i o n can be used for estimating the actual physical damage, and i t s economic consequence in:, terms of the thinning regime. The physical damage must f i r s t be related to the enumeration produced by the simulation. This could be achieved i n several ways; 1. F i e l d experimentation. 2 . F u l l - s c a l e simulation with entire trees. 3. Application of t h e o r e t i c a l mechanics (statics and dynamics). The loss of po t e n t i a l growth due to t h i s damage could be estimated through b i o l o g i c a l growth models, and the r e s u l t i n g wood y i e l d loss equated with a conversion loss to be discounted i n an economic analysis. 1. Adamovich, L. 1969. Present and planned research and development of logging techniques i n thinnings i n North America. Proc. of Int. Union of For. Res. Organizations (IUFRO) meeting on 'Thinning and mechanization^, Royal C o l l . of For., Stockholm, Sweden, Sept. 1969. p 236-241. 2. Adamovich, L. 1968. Problems i n mechanizing commercial thinnings. Fac. of For., Univ. of Br. Columbia, mimeo. 24 pp. Ci3. Ager, B. H. 1969. Thinning and mechanization. Some views on the research problem and an analysis of harvesting costs. IUFRO i b i d , p 2-16. 4. Browne, J. E. 1962. Standard cubic foot volume tables for the commercial tree species of B r i t i s h Columbia. For. Surveys and Invent. Div, B.C. For. Service, V i c t o r i a , B.C.. 107 pp. 5. Day, W. R. 1966. B i o l o g i c a l aspects of thinning conifer plantations. Forestry 39(2). p 191-212. 6. Hummel, F. C., G. M. L. Locke, J. N. R. J e f f e r s , and J. M. C h r i s t i e . 1959. Code of sample plot procedure. B r i t i s h For. Comm. B u l l . No. 31. 113 pp. 7. Kurkkuinen, M. 1969. A study on tree i n j u r i e s caused by mechanized timber.transportation. IUFRO, i b i d . p 136-140. 8. Kozak, A., D. D. Munro, and J. H. G. Smith. 1969. Taper functions and t h e i r applications i n forest inventory. Reprint from For. Chron. 45(4). 6 pp. 9. Myrhman, D. 1970. Intagning av staende trad. (Handling of trees i n a v e r t i c a l position} 5. Forskningsstiftelsen Skogsarbeten, Sweden, Rep. No. 13. 25 pp. 10. Newnham, R. M., .and S. Sjunnesson. 1969. A Fortran program to simulate harvesting machines for mechanized thinning. Can. For. Service, Dept. of Fish, and For., For. Mng. Inst. Info. Rep. FMR-X-2 3. 48 pp. and app. 11. P a i l l e , G. 1970. Description and prediction of mortality of some coastal Douglas-fir stands. Ph. D. thesisj Fac. of For., Univ. of B. C., 300 pp. 12. Pienaar, L. V. 1965. Quantitative theory of forest growth. PhD thesis, C o l l . of For., Univ. of Wash. 176 pp. 13. Savina, A. V. 1956. (The physiological j u s t i f i c a t i o n for the thinning of f o r e s t s ) . Trans, from the Russian by B. Toker, Israel program for - s c i e n t i f i c trans., 1960. Office of Tech. Services/ U. S. Dept. of Comm., Wash. 91 pp. 14. S i l v e r s i d e s , C. R. 19 64. Developments i n logging mechanization i n eastern Canada. H. R. McMillan lecture, Univ. of B. C., Feb. 1964. 29 pp. 15. Smith, J. H. G., J . W. Ker, and J. Csizmazia. 1961. Economics of reforestation of Douglas-fir, Western Hemlock, and Western Red Cedar, i n the Vancouver Forest D i s t r i c t , Fac. of For., Univ. of B. C , For. B u l l . No.,3. 144 pp. 16. Tamm, C. 0. 1969. Site damages on thinnings due to. removal of organic matter and plant nutrients. IUFRO, i b i d . p 175-179. . APPENDICES Appendix 1 Stand and Pr e s c r i p t i o n D e t a i l A pure, even-aged, and n a t u r a l l y generated stand of Douglas-fir (Pseudotsuga  menziesii (Mirb.) Franco), enumerated at an age of fifty-one years. A one-acre square p l o t , with only those trees with diameter at breast height larger than s i x inches included i n t h i s summary. Equations 1. Total height; H = -19.3101 + 9.1109dbh - . 1769dbh2+ . 13602ba/ac , (Faille" (11)) 2. Height to crown; he = -7.32 + .343dbh + .288H + .413A , (Smith et a l (15)) 3. Gross volume; logV .= -2.658 + 1.7401ogdbh + 1.1331ogH , (Browne (4)) where; H =* t o t a l height i n feet he = height to the crown base i n feet V = gross volume i n cubic feet dbh = diameter outside bark at breast height in inches ba/ac = stand basal area per acre i n square feet A = stand age i n years S t a t i s t i c s ..'Stand No. of trees Average dbh' Average H Total V Total ba Total 283 9.7 ins.. . 76.2 f t . 5180 cu. f t . 15 7.8 sq. f t Pres. Cut Leave Cut Leave Cut Leave Cut Leave Cut Leave Low 160 123 8.3 11.5 .68.9 85.6 1862 3318 63.0 94.8 Crown 219 64 9.0 12.0 72.7 88.0 3259 1921 103. 9 53.9 Grad. 216 67 9.1 11.7 73.1 86.1 3253 1927 103.5 54.3 U1 Unthinhed Stand • • • 4- Yarding Direction 4-Low Thinning Crown Thinning Graded Thinning • • • Plot Bounds * 'Parameters Set Prescr. x 1 dr. y' dr. 6 . mm mm r inc 1^ top Y e x t A Low, Cr. and Gd. 0 to 208.71' 0 to 208.71' 7° 30° 5° 5' 15' B Crown 100 to 208.71' 0 ,to 208.71' 7 0,2,4 30 2 5 15 10,15 80 5 5 15 20 Low 0 to 0 to 12.5, r> 100' 208.71* 17.5, 22.5, 25,30 35 80 80 5 15 Crown & Graded 0 to 100' 0 to 208.71' 0,6, 12,18 24 ,30 80 80 5 15 * Corresponding Fortran variables; 6 . - DEMIN or DEM (defined i n the text) mm e . - EMIN or EM (defined i n the text) mm ip. - the complement of 9 0 i s decremented i n the i n c program by PHIDCR (defined i n the text) 1^ - TOPL (defined i n the text) top y e - EXT, the maximum distance that a f e l l e d tree x may extend into the skyline road Appendix 4 The Fortran IV Program The main program i s used to read i n the stand and pr e s c r i p t i o n information. The thinning process for each marked tree.proceeds by c a l l i n g the applicable subprograms. As the trees are thinned, the res u l t s are stored, and printed out at the end by the subprogram OUTPUT. The programs that do the thinning are; 1. DECIDE; selects the f e l l i n g d i r e c t i o n . 2. FELL; simulates f e l l i n g between the two constraint trees. 3. LATYRD; simulates extraction of the log. The functions that are discussed i n the text are; 1. DELTA; calculates the angle 6 . 2. SEP; calculates corridor half-width s.. 3. BOLE; taper function,(Equation ( 3 ) ) . 4. SHAPE; crown shape (Equation ( 4 ) ) . 5. XWIDE; height of maximum crown width (Equation ( 5 ) ) . 6. CRUSH and BASH; for in t e r a c t i o n c o e f f i c i e n t s Cc and Cb, (Equations (1) and (2)). The> other programs used by the major components are; 1. SORT and KUT; sorting algorithm used by DECIDE and LATYRD. 2. BISRCH; a searching algorithm that finds -;tvhe bounding elements of a range i n a sorted array. Used by DECIDE and LATYRD. 3. POLAR; calculates polar coordinates for DECIDE. HITONE; determines which of the constraint trees i s h i t f i r s t during f e l l i n g . Used i n FELL. LOGIC; a control program used i n int e r a c t i o n calculations of FELL. The Fortran IV source l i s t i n g follows. l 2 3 * , 5 6 7 8 9 l O 11 1 2 I 3 C c c c c c c c c c c c c c c c c • c c c c c c c c c c c c c c c c c c c c c c c c c c c c* 2) 3 ) S O U R C E P R O G R A M OF A G E O M E T R I C , D E T E R M I N I S T I C M O D E L , F O R S I M U L A T I N G T H E F E L L I N G A N D E X T R A C T I O N C F U N B O O K E D L O S S . THE S Y S T E M B E I N G M O D E L L E D I S A S K Y L I N E E X T R A C T I O N M O D E L U N D E R THE- F O L L O W I N G A S S U M P T I O N S 1) T H E F E L L I N G A N G L E D E C I S I O N C A N B E S I M U L A T E D B Y S P E C I F Y I N G T H A T T R E E S A R E F E L L E D I N T O THE M Q S T O P E N P O S I T I O N C O N S I S T E N T - W I T H T H E O B J E C T I V E O F M I N I H I Z I M G L A T E R U Y A R D I N G D I S T A N C E T H E P A R A M E T E R S S P E C I F I E D T O 'MEET T H l S - O B J E C T J V C ' D E M I N ' A N D ' E M I N * T H E E F F E C T I V E R A D I U S F O R F E L L I N G C O N S I D E R A T I O N S I S D E F I N E D B Y T H E H E I G H T O F T H E T R E E L E S S A C O R R E C T I O N F A C T O R • T C P L ' F E E T T H I S R A D I U S C A N N O T E X T E N D I N T O T H E S K Y L I N E K O R E T H E ' E X T ' F E E T T R E E S A R E F E L L E D W I T H I N T H E P L O T , A N D O N L Y T W O C O N S T R A I N T T R E E S A R E R E C O G N I S E O WHICH D E F I N E T H E F E L L I N G A N G L E ' D E L T A ' T H E U N B U C K E O MERCHANTABLE L O G S A R E Y A R D E D A T R I G H T ANGLES T O T H E S K Y L I N E , W H I C H I S R E G A R D E D A S I M O V A B L E T H E L O G S A R E C H ' J K E R E D A T T H E L O W E R E X T R E M I T Y A N D A S S U M E D T O T E N D TO P I V O T O N T H E O T H E R E N D . A N Y R E S I D U \ L T R E E S L Y I N G U N T H E W I D E S T P O L Y G O N I N T H E Y A R O I N G T R I A N G L E A R E C O N S I D E R E D P O T E N T I ALLY.. S T E M . S C A R R E D t T . G J i G R I N G T H E E F F E C T OF THE O T H E R C H O K E R E D L O G S O N T H E P O S I T I O N O F T H E Y A R D I N G TR I A N G L E T H E E F F E C T OF S L O P E C A N C E I G N O R E D I M P O R T A N T P A R A M E T E R S A N D V A R I A B L E S 1) ' D E M I N 1 I S T H E F E L L I N G A N G L E S U B T E N D E D B Y T H E TWO C O N S T R A I N T T R E E S , ' E H I N • I S A F R E E U N O B S T R U C T E D S P A C E A N G L E . B O T H G I V E N I N D E G R E E S • P , P I . P E , A " A R E T H E P O W E R S A N D C O E F F I C I E N T U S E D TO D E S C R I B E T H E S H A P E O F T H E B O L E A N D T H E C R O W N ( L O N G I T U D I N A L L Y ) T H E P H Y S I C A L I N T E R A C T I O N S B E T W E E N T H E C U T T R E E A M D T H E TWO C O N S T R A I N T T R E E S A R E C A L C U L A T E D F O R P O S I T I O N S O F T H E F A L L A N G L E D E C R E M E N T E D BY ' P H I D C R ' D E G R E E S • 1 S T A I S R E S I D U A ! . S = C , » c M O V A i . S = l ' K I N O ' , T H E T Y P E O F F E L L I N G P O S I T I O N . G V E R C G N . - . T R A * \ i i : D - l » U M C 3 S T f t U C T £ J ' = 2 , C O N S T R A I N T - i , , G E . ' D E M i IM' =<V Z) 3 ) •V) . L T . ' 0 E M I N ' I N P U T / O U T P U T L O G . U N I T 5 5 5 5 5 3 6 6 6 a R O U T I N E T I T L E M / P R Q S P L O T B O U N D S N / P R O G P A R A M E T E R S E T C . K / P R O G F O R M A T F O R OAT A K / P R O G F O R M A T F O R R E S I D U A L T R E E N O S . M / P R Q G R E S I D U A L T R E E N U M O E P . S M / - P R Q G B A S I C D A T A - M / P ' R O S P A R A M E T E R L I S T I N G ' M / P P . O G T A B L E H E A D I N G S i ' V P P . O G R E S U L T T A B L E O U T P U T R E S U L T S ( W I T H O U T I N T E R A C T I O N C O E F F S . ) O U T P U T C O M M O N X I 5 0 0 ) , 7 ! S C O ! , D ! 5 0 0 ) , H (15 0 0 ) - H L C I 5 0 0 } $ I S T 4 T I 5 0 0 3 / A R E A / X M i N , X H A X i Y M I N t Y A \ IRANKX/ X R K . l 5 0 0 ) ,N>:R i '300 i ,<'"•! TOT / P O W E R S / P , P I . P E , A '•<?. r F E L G L , S E P , :" - i --a ! 5 0 0 ) C R S . H ! 4 D ,2 ;• : ^SH{ -^0 >,: ) •• BR S i f t 9 0 , ?. C O M M O N C O M M O N C O M M O N C O M M O N C O M M O N NT .1 , NT 2, R1, THSTEP! <90) / D E C P U T / / F E L P U T / l i N 0 E X ( 9 0 , 2 ) C O M M O N / Y RO P U T / C i S T . N I T S , ! -i H ; T S (500 ) D I M E N S I O N F M T 1 2 0 ! , F C H M » 2 0 ! D I M E M S I O N TI TL E ! 2 0 ) , I S T A R i ! ? 0 ) D I M E N S I O N H T M C i S O C ! D A T A R A O / 5 7 . 2 V 5 7 5 / O A T A 1 S I « R / 12 C * ' * ' / R E A D ( 5 , - 3 0 ! ( T I T L E ! ! > , . ' = '. , 2 0 ) 14 50 F0RMATt20A4) 15 PO 11 1=1»500 C INITIALIZE REMOVAL SVATUS ARRAY 16 11 C GET ISTAT(I)=1 THINNING PLCT BOUNDS 17 REAOt 5, SCO] XMiM , XsMAX» YMINt YKAX i a 500 F ORMATI4F10.2) 19 WRITE16,61) (T TTLE I I ) , 1 = 1 ,20 i , ( I STAR ! J i , J = 1 » 1 2 0 ) 20 61 FORMAT(•1 ' ,25X, 20A4/12OA I//) C GET FELLING CONSTRAINT PARAMETERS, FUNCTION POWERS, ETC. 21 READ I 5 ,501) D E W , E M , E X T , P H I O C R , P , P I , » £ , A , TOPL, ISOST 22 DEMIN=DEK/RAD 23 EMIN=EH/RAD 24 H RI T E (6, 60 ) 25 60 FORMAT(* PARAMETERS U S E D 1 / ' DEM IN EM IN l » PI PE A TGPL 1 ,2.3X, ' I SORT'/• 2 1 FEET DEG.S * , 2 5 X , ' F E E T * ) EXT PHIDCR DEGREES ' , 26 HR1TF16,501} OEM,EM,EXT,PHIDCR , P , P I , ?E ,A ,TOPL,ISORT 27 501 C GET rORMAT(9F6.2,25X, I 1) THE -VARIABLE INPUT FORMATS 2 8 READ!5,504) t F M T( I ), 1=1,20 ) 29 READ(5.504) (FORMfl) ,1=1,20) YJ 50-'i FORMAT!20A4) n X S I O £ = X M A X - X M ! N 32 YS ID£*YM.* .X -YK IN 3 3 DQ 9 1=1,500 O READ IN THE RE 31 DUAL TREES 34 RE AD! 3 , FORM j£UC=10 ) ID 35 I STATiIQ>=0 36 9 CONTINUE 3 7 10 DQ 1 1=1,SCO 33 NTOT=I- l 39 IRHITSi!1=0 P S 3 X , C* READ IN BASIC TREE DATA 40 RSAOCA»Fi-!T,END=22) XI I ) ,.Y ( i ) , D i.I ) , H i I) , Hl.C i I ) s SiTr-X (I) 41 XR!Ui)=xn> 42 NXR( I i= l 4 3 1 CONTINUE C* SORT TREES - AND FIND BOUNDING ELEMENTS OF THiK.Ni.NG PLOT 44 2 2 i F I iSORT.NE.1)GO TO 2 45 V.RITES6,6'2) 46 62. FORMAT'.' DATA PREVIOUSLY SORTED 3Y X-COORDINATE' ) 47 GO TO 3 43 2 CALL 50RTiXRK,NXR,NTOT ) 49 3 CALL 3 I SRCH J X*K, NTCT ,XM!N ,Xi-<AX , N'L,NM ,KENTS J 50 W R i r e ( 6 , 6 0 0 ) X SIDfc,Y SIOE,HE N TS 51 600 FORMAT{/' THE AREA TO BE THINNED ! S ' , P 8 . 2 , ' 3 Y ' , F 8 . 2 , » FEET} TOTAL 1 NUMBER OF TREES = ' , I6//) 52 WRITE 16,601) 53 601 FORMAT t 1 MARKED FELLING CONSTRAINT CORRIDOR EFFECTIVE « , '.'ANGLE FELLING INTERACTION COEFFICIENTS YASDT'S RHS! DUAL S * / 2' TREE DIRECTION TREES WIDTH LOG IGTH. PHI 3 ' CROWN ON 8GI.E ON CROWN ON DISTANCE POSSIBLY ' / 1 I X » T-EG 4S ' ,1CX,*FEET FEET DECS CROKfi CROWM •, TX, 5'ROLE FEET H IT ' ) 5'. NGOT--0 55 DC 5 I = .NL,KH 56 I F ! I STAT INXRt I ) ) . E C O ) GO TO 5 57 NG0T=N3OT * i 58 ir=.NXR(I) 59 CALL DEC IDE U 5 , C 7 , I T , D E M I N , E M I N , H T M C ( I T ) , C .7, TOPL ) 60 CALL F E L L I N S T E ? S,R1 ,R2 , S E P R , I T , N T 1 ,M T2 » PHIDCR ) 61 GO TO 12 62 6 NSTEPS=0 63 12 ELGG=H7MC(IT) 64 - I F I F E L G L . L E . 130. )GC TO 13 65 I f ! F E L G L - 2 7 0 . > 14 ,15 ,16 66 14 « L P H = ( r E L G L - l 8 0 . J / R A O 67 17 Y3=EL0G*SINlALPH) 63 I F I Y 3 . L E . Y 1 I T ) ) G O TO 13 69 ELQG=Y< I T S / S I N I A L P H ) 70 GC TO 13 71 15 I F ( Y ( I T ! . L T . E L C G ) E L C G = Y I I T ) 72 GO TO 13 73 16 A L P H = ( 3 6 0 . - F £ L G L )/RAO 74 GO TO 17 75 13 CALL L A T Y R D ! I T , F L O G , F F L G L ) 76 CALL O U T P U T ! I T , N S T EPS) 77 GO TO 5 73 7 NG0T=N30T-1 79 5 CONTINUE SO WRITE (6, 604) NGtJT 81 604 F O R M A T ( / / ' ACTUAL NUMBER OF STEMS REMOVED:»»16 > 82 WRITE 16,605) 0 3 605 FORMAT(' iPOTONTIAL HITS TO R E S I D U A L S ' / / 7 X , ' T R E E ' , 6 X t « H I T S ' ) 34 DO 8 1 = 1 ,.NTOT 85 I F ! I f t H I T S m . G T . 0 ) K P . ! T E ( 6 » 6 0 6 ) ' I . I R H ITS( I) 06 606 FORMAT 121.10) 8 7 8 CCNTIHtic 6 3 STOP S9 EN: 3 90 SUOROU T I ME D E C I D E ! * . * , U , D E M ! M.E.MIN,HMCH,EXT,TOPL) C * * * DECIDES VJHEP.E TO PUT TREE UNDER THE STRATEGY THAT IT SHOULD 3E WITHIN THE C SETTING BOUNDARIES, TOWARD THE. SKYLINE ; IF ALLOWED UNDER THE EXT CONSTRAINT C AND AS PERPENDICULAR TO'THE SKYLINE AS P O S S I B L E . 91 -i.MP'l.K. iT LOGICAL (0) 92 COMMON X 1 5 0 0 ! , Y ! 5 0 0 ) , 0 ! 5 0 0 J , H { 5 0 0 ) , H L C < 5 0 0 ) « I S T A T J 5 0 0 ) 93 COMMON / A R E A / XHIM,XHAX,YMIN,YMAX 94 COMMON /.RANX.X/ XRK(5C0),NXR 1500) ,NTOT 95 COMMON / O E C P U T / NT 1 ,NT2 ,R1 , R 2 , F-ELGL, SS?R, ELQG,X'INQ ('500 ) 96 DIMENSION 8L14) ,BUi 4) ,ASC1!70) ,NSi' I 70 ) , ASQ2 ( 70 ! , N«2 (70 ) , AS03 170), LNQ3C70 5 , ASQ4! 70 5 ,MQ4( 70) ,MVi 4) ,M2 (4) ,DEL (4) , F A L ! 4/ , O A X ! 4 ) , ! .QK! 4! , 2NK !•'; ) , J L ! 4 ! , JU (4) 97 DIMENSION ASQl 70,4) , .N0(70,4) ,MF.NTS!4) J 98 EQ'J I VALENCE ! ASQ i 1 , 1) , A SO 1! 1) ) , t ASOi 1 ,2 ) , ASQ2 I 1) ) , ! ASQl 1 ,3 ) , AS 03 i 1 j 1 ! ) , U S Q ( l v4 ) ,ASS4 ( 1 ) ) , INC! 1 , 1! ,N01 ! l ! ) , 1 NO ( 1, 2) t N 0 2 ! 1) ) , t NQ! 1 , 3 ) , 2HQ 3 ( I I i -, ( NO ( I , 4 ) , NC4 ( 1 ) ) j 99 DATA P I / 3 . 1 4 1 5 9 3 / , S A D / 5 7 . 2 9 5 7 6 / 5 C* GET FELLING C i R C L E RADIUS | 100 R/; Y - H! [ V) - T0PL i 101 I ? 1 HMCH.GT.RAY) R A Y=H!<CH C* GET BOUNDS ON X AND Y, AND QUADRAT BOUND ARRAYS Bl. AND BU 5 02 NUMX-NT01 ' !.03 00 ', I 3. ; 4 10 4 i !,L( : ) = 0. 105 91KI> = 1 . £ 1 0 106 !.Ci< 115=0 107 MENTS? i 3 = 0 103 1 CONTINUE . 109 XLO=X( m - r u « Y 110 X H I » X U T ) * R A Y 111 Y L G - Y ! I T l - R A Y 112 YHI=Y( IT ) + RAY 113 i F I X L O . G E . X M . i N i C O TO 2 C INTERSECTS L E F T BOUNDARY 114 XLO=XHIN 115 X!.=Xf I T J - X H I N 116 B L ( 2 ) = S Q R T f R A Y * * 2 - X L * ' 2 ) / X L 117 3 L ( 3 ) = B L ( 2 ) 118 2 I F ( X H ! . L E , X H A X ) C - 0 TO 3 C INTERSECTS RIGHT BOUNDARY 119 XHi=XMAX 120 XR =X MAX - X ( IT) 121 B L ' d >=S0RT(RAY**2 -XR**2) /XR 1'22 B L ( 4 ) = B L f l ) 123 3 I F I Y L 0 . G E . Y M I N 1 G 0 TO 4 C INTERSECTS BOTTOM BOUNDARY 124 YLO=YMIN 125 YB=YtITJ-YMIM 126 I f ( R A Y . G T . { Y B + E X T ) ) G O TU 300 12 7 GC TO 4 128 300 2L0G-- (Y6-RAY) /1EXT + YB) 129 BUI I ! =YB/SCRTI £ L 0 G * * 2 - Y B * « 2 ) 130 8U!2 3-B'JIl ) 131 4 I F ( Y H I . L E • YMAX)G0 TO 5 C INTERSECTS TO? BOUNDARY 132 YHI-Y MAX 133 YT=YMAX-Y(IT) 13'. BUJ3j = YT/SCiRT!i lAY**2-Yr**2J 135 B i J m ^ B U O ! C * GET SUBSET BOUNDED BY XLO AND X H ! , .VJD THEN QUADRAT SETS OF ABSOLUTE SLOPES ' 136 5 CALL SISRCHiXRKtNUMXtXLQtXVil , J L C f JH . l .MTS» .137 ! U = 1 138 K2=l 139 K3=l 140 K4=l l ' f . l RAYSQ=RAY**2 142 00 6 I=JLO,JHI 143 IF I N.XP.! I i » E Q . I 7 i GO TG 6 144 I F ! Y t M X f c m > . G C - . Y H I . A N D . Y( NXRI 1)5 . L E . Y L 0 3 G 0 TO 6 145 I F ! Y i.NXK i I) ) , G E . Y { I T i') GO "0 66 i<>6 ' IF ! I STATJMXRI I ) i . E C . t i GO TO 6 147 66 0X-X.RK 1 I )-X i 1 T) 148 0Y=Y5NXfH U i - Y ! ! T ) 149 R0=DX**2*DV**2 150 I F J R Q . G E . R.S Y.SG GO iO 6 151 0 1 = D X . 5 T . O . 152 C-2=CX. LT „ 0 . 153 0 3 = D X . E 0 „ 0 . 154 0 4 - - D Y . G T . 0 . 155 05--OY-. L T . O . 156 C 6 = D Y . E Q . O . 157 IF ( ,NQT.03-!G0 TO 7C 153 OX=1.6-.lO C NOW CY CANNOT- HE lf.H0 l * '-'•'< IS 159 IF (05) GO ii) 160 IF!04 )G3 TO 74 161 70 ! F ! . N O T . 0 6 3 G C TO 7 5 162 0Y=1 .E-10 C NOW DX CAMMOT BE ZERO IF OY IS 163 I F ( G l ) G O TO 71 164 1FIQ2JGO TO 73 165 75 I F I O D G O TC 76 166 IF(G4)GO TO 73 167 GO TO 72 168 76 I F ( 0 4 ! G 0 TC 74 169 GO TO 71 170 71 A S Q l t K l ) = A E S ( O Y / O X ) 171 NQK K l ) = NXR( I ) 172 K l = K i+l 173 GC TO 6 174 72 ASQ2(K2)=A3S(OY/DX) 175 NC2(K2 )=NXR( I) 176 K2=K2*1 177 GO TO 6 178 73 A S G 3 1 K 3 ) = A 3 S ( D Y / O X ) 179 NG3 1K3 J=NXRU 1 180 K3=K3*1 181 GO TO 6 182 74 ASQ4(K4!=ABS!CY/DX) 183 NG4IK4)=NXR I I ) 184 X 4 = K 4 U 185 6 CONTINUE 186 NK(1)=X1-1 137 N K i 2 ) = K 2-l 188 N K ( 3 i = K 3-l 139 NK[4)=K4-1 C * RANK ARRAYS BY SLOPE IF MORE THAN ONE ELEMENT AND FIND BOUNDING ELEMENTS C CHECK FOR NULL QUADRATS OR UMUSEASLE CUAORATS 190 DO 10 1=1,4 191 IFifJKI I! . L E , 0 ) L O K ( 192 I FIBL I I 1 .GE.TJUII ) JLCXI I J=l 193 10 CONTINUE 194 I F U O K U ) . N E . O J G O TO 12 195 I F t MK 11) . GT . 1.) CALL SOR T t ASQ1 , NQ1 ,MK ! 1 ! > 196 CALL B I SRCH! ASQ1 ,NK( i i , 3Li 1 ) , BU 11) , . ; L i 1) , J U t 1 J.MTS) 19 7 KENTS(1 )'-MTS 198 1F1MTS.GE.1 )G0 TO 12 199 L0K!1)=-1 200 12 1F!L0:<(2 ) .NE.0 ;G0 TO 14 201 I F I NK 12 ) . GT . 1 ! CALL SORT f ASC2 , N Q 2 , ' « ( 2 ) > 202 CALL BI SRCH i ASQ2 ,NK { 2 5 ,31(2 ! ,BU(2> , J L U ) , JU(2 ! :• M T S ) 203 NENTSI 2 i IS 204 I F t M T S . G E . l ! G 0 TO 14 205 LCK(2)=-1 206 14 I F ! L 0 K ( 3 ) . N E . C ) G 0 TO 16 207 I F I N K l 3 I . G T . I J C A L L SOR T(A SQ 3» NQ3 ,MX13 J ) 2 08 CALL BISRCHi A5C3 , NK i 3 ) , BL ( 3 i , BU i 3 ) , j L ( 3 ! - J ij( "3 ! , M T S ) 209 MENTS13!=MTS 21C IF IMTS.GE,1 )GO TO 16 211 LCK{3)=-1 212 16 ! F ! L C K ( 4 ) . N E . C J G O TQ 18 213 IF (NK( 4) . G T . 11CALL S!;R '! i A S04, MQ4 , -if.! 4.! ! 2 14 CALL BISRCHi AS C4 ,"•;;< i ; , :': L i 4 i , 3 0 ( 4 i > J L • 4 ! , J! J ( 4.' , M T S i 215 ME NTS(4)= MTS 216 I F I M T S . G E . 1 )G0 TO 13 217 L 0 K ( 4 ) = - l C* FIND BEST PCS IT ION IN EACH FEASIBLE QUADRAT 218 13 DO 19 J = l , 4 219 D £ M A X = - 1 . E 1 0 220 I F C L Q K I J ) . E G . 1 ) G O TO 19 221 IFILGKI J K M E . - U G C TO 20 j 222 21 D A X I J ) = 0 . •1 22 3 DA L = ATANt3U I J ) ) - A T AN(3L(J ) ) 1 221 22 I r ( E r i I N . G T . O A L ) G C l TO 19 j 225 L 0 K I J ) = 2 C A. 'FREE ' SPACE CHANCE IN QUADRAT J ; DAX 3E1NG LARGEST .C-T. EMI N ANGLE [ .226 C E L ( J ) = 0 A L 1 227 0AX(J)=0AL 228 GO TO 19 229 20 I F ( M E N T S ( J ) . G T . 1 ) G O TO 23 230 0 A L = A T A N ( B U ( J ) J - A T A N 1 A S C I J U ! J ) , J ) ) 2 31 GO TO 22 232 23 LV=JU;J) 233 I N=-LV-1 234 24 D E = D E L T A ( I T . N Q I L V , J ) , N U I N , J ) ) 235 I F ( O e , L T . O E M I N ) G O TO 25 236 LCKtJ)=4 r A . G E . DEMIN CHANCE IN CUADRAT J ; FAL BEING HIGHEST ABSOLUTE SLOPE 237 0EL(J)=DE 238 M l ( J ) = N Q ( L V » J J 2 39 !•'2 ( J) = N G ( I N , J) 240 F A L { J ) - A S Q ! L V , J ) 241 GO TO 19 2 42 25 I F t O E . L T . D E H A X ) G O TO 26 243 DEMAX=DE -244 MXLV=LV 245 •MXIN=IN | 2 46 26 ! F ( I M . E O - J L I J ) ) G 0 TO 2 7 247 LV'=IH 24J !M=LV-1 2 49 GO TO. 2 4 250 27 LCXI.J5=3 C A , L T . DEM IN CHANCE IN QUADRAT J ; DEHAX BEING LARGEST DELTA AVAILABLE 251 DGLt JJ--=DEMAX 252 ft 1 ( J1 = NQT MXLV , J ) . 253 M2iJ)=NQ'.MXIN, J) 2 54 19 C*« C * CONTINUE PICK THE FINAL FELLING POSITION CALCULATE' DELTA U!*.<>• FREE ' SPACE BETWEEN 1,2 AND .3,4 255 1=1 256 D E L T X = - 1 . E 1 0 257 DETAX=-1 .E10 ! 2 53 DEMX=-1.E10 259 3 3 • I F ! LGKI I ) „EQ , 1 . O R . L O M I* 1 ) . EG . 1 ) GO TO 39 260 l F I B U m . N E . l.C-10 . C R . B U ! I * 1 ) . N E . 1 .E10JG0 TO 39 26 1 IFIMEMFSII 1 . G T . O .AND. MENTSI I + 1 ) . G T . 0 ) G O TO 43 2 62 I F ! NENTSI I ) . E Q . O . A N D . CENTS { I + I ) . £ 3 .0 J'GO TO 51 263 ! F ! M E N T S I I ) . E Q . 0 ! G C TO 5C 26 4 DEM = PI - A T AM!A SQIJ U M ) , I ) ) - A T AN 1 ft L ( I • 1 ) ) 2 6 5 K A 3 E = I 2 6 6 GC TO 52 26 7 50 D E M - P I - A T A N IA S C I J UI ! * 1 ) , I + I ) ) - A T A N ( B L,( I) ) 26 S GO TO 52 5 1 DEM = PI-ATAMI OL i I i )-ATAN{ DL ( H i l l 2 7 '3 52 I F I D E M . L T . E I ' I N I G C TO .3 9 , 271 I F ! 0 E M . L E . D E M X ) G 0 TO 3 9 2 72 CEMX--0EM 2 73 GC TO 46 274 43 NTl=NQtNXII) ,1> 275 I!T2=NQ{?!Kl 1*1) ,1*1) 276 53 DELT=DELTA( I T S N T ' . , N T 2 ) 277 I F I D E L T . L T . C E M I N I G G TO 54 273 I F t D E T A X . G t . O E L D G O TO 39 279 C£TAX=CELT 280 KD1=NT1 231 KD2=NT2 232 GO TO 39 283 54 I F ( D E L T . L E . D E L T X J G O TO 39 284 DELTX=DELT 2 35 NT1X=NT1 236 NT2X=NT2 287 39 I F i I . E Q . 3 J G 0 TO 46 286 1=1+2 . 239 GO TO 33 290 ^6 I F I D E M X . E Q . - i . C l O G G TO •'•.4 291 I F I K A S E . E O . l ) G O TO 55 292 ALGL=AT ANi BL ( I ! 1 • D E M X / 2 . 293 GO TO 56 294 55 ALGL=A7ANl 8L ( 1 *• 1 ) H-DEMX/2 . 295 56 I f ( A L G L . G T . P I / 2 . 1ALGL=PI-ALGL 296 OX=DEMX 297 I F ! I . E 0 . 3 5 G O TC 45 293 F E L G L - R A D * ( P I » P I - A L G L > 299 GO TO 35 300 45 Fi:LCL = RAD* t P I - A L G L ) 301 GO TO 35 30 2 4 4 OX=C. 30 3 JZ=0 304 DO 30 J = l ,4 30 5 I F ( L O X i J ) , . N E . 2 ) G O TO 30 306 I F ' D A X U ) . L S . D X ) G 0 TO 30 307 CX-DA.'UJ) 303 JZ=J 309 30 CONTINUE C FOUND A ' F R E E ' CHANCE IN JZ IF JH .NE .0 310 I F I J Z . G T . O J G G TO 31 311 KIND!IT)=4 C FIRST CHECK . G 6 . OEM IN CHANCES BETWEEN 1 ,2-Oft 3 ,4 312 I F - ( C G T A X . E 0 . - 1 E 1 0 ! G 0 TO 47 313 N71=KDl 314 NT2=K02 315 GC TO 41 316 47 FALX=0. 317 00 23 J - - \ , 4 318 l?a0K(J).NE..4.)G0 TO 28 319 I F f F A L ! J) . L E ' . F A L X I G O TO 23 320 FAL X =F A L ( J ) 321 JZ=J C FOUND A . G E > DEM I.N CHANCE :N J I F Jl . N E . 0 322 28 CONTINUE 323 I F i J Z . G T . O i G O TO 29 324 K IN D f I T ! = 3 C TAKE LARGEST . LT » OEM IN CHANCE A / A I L A1M. E 325 O E L X = . - l . S i O 3 26 DO 22 .1 = 1,4 32 7 I F f L O X I J ) . N E . 3 J G 0 TO 32 328 IF (DEL ! J ) . L E . O S L T X K : C fO 32 329 I F i D E L C J ) . L E , D E L X ) G C TO 22 3 3 0 C K L X = D E L ( J > 331 JZ = J C . FOUND A . L T . DE^IN CHANCE IN JZ IF JZ . N t . 0 3 3 2 32 CONTINUE 3 3 3 I f f J Z . t O . O J G O TO 40 3 3 4 I F I D E L X . - G T . D E L T X J G C TO 29 335 40 I F I D E L T X . E Q . - l . E l O J G O TO 42 3 3 6 NTl=NTiX 3 3 7 NT2=NT2X 3 3 3 GO TO 41 3 3 9 42 K I N O t I T ) = l C* COULDN'T FIND A FEASIBLE POSITION; THEE NOT TO BE 340 WRITE(6,600) IT 341 600 F O R M A T ! I 6 » 3 X , * TREE NOT CUT - - OVER-CONSTPA I NED') 342 RETURN 2 343 31 A N G L = A T A N ! S L ( J Z 1 l + D X / 2 . 344 I F 1 J Z • NE • 1 ) GO. TO 3 3 345 FELGL= I2 .«PI-A:-JGL )*RAD 346 GC TO 35 347 3 3 I F ! J Z . N E . 2 ) G 0 IG 3 4 348 FELGL=(PI+ANGLS*RAO 349 GO TO 35 350 34 i F ! J Z . N E . 3 ) G Q TO 36 351 FtLGL = !P1-ANGL J*RAO 352 GO TO 35 3 5 3 36 F E L G L-= AM G L * 3 A D 3 54 35 SE?R=RAY*SIN(OX/2 . ) 355 ! U N D l i T ) = 2 356 RETURN I 357 29 NT1"Ms I J Z ! 353 NT2=M2!J;) 359 41 CALL POLAR'.! T . N T l j R l , TH1 ) 360 CALL P O L A R C I T , N T 2 , R 2 , T H 2 ) 361 I F J T H l . G T . O . .AND. T H 2 . G T . 0 . > G 0 TO 5 7 36 2 I F ( T H 1 . E 0 . 0 . . A N D . TH2 . G T . P I ) T H 1 = 2 , * PI 363 I F J T H 2 . E Q . 0 . . A N D . T H 1 . G T . P I > T H 2 » 2 . » f M 364 57 GA=ABS!TH2-THI) '36 5 S E P R - S E P ! R 1 » R 2 S G A ) 3 6 6 I ? ( T H 2 , G T . T H 1 ) C 0 TO 43 367 FELGL = RAD*UH2*ARS INISEPR/R2) ) 363 GO TO 4 9 369 48 FELGL = RAO* I TH t >• ARS IN ( SEPR/R IJ ) 370 4 9 I F I S 2 . G T . R 1 ) G 0 TO 37 371 TR-R1 372 TN=sm 3 7 3 R1=R2 3 74 NT1=NT2 375 R 2 = TR 376 MT2=TN 377 37 RETURN 378 END 379 - SUBROUT!ME 3 1 S R C H i A . N . X L , X H , J L , J H , « E N T S ) C * PROVIDES (HENTSJ DEFINING ELECENTS ! J L » J H J FOR THE C I X L . X H ) IN THE SORTEC L I S ! A i N ) 330 DIMENSION AIM) C* ChECiv TO SEE I F P.CUNOS- OUT OF ARRAY R A 'J G t 381 I F ( X H . G £ . A ! 1 ) . A N D . X L . L E . A t N U G O TO 2 382 : MEN fS=C 383 1 RETURN C* FIND JL FIRST 384 2 K l = l 335 KH=N 336 3 M=KL+tKH-KL)/2 337 D=XL-A(M) 383 I F [ { M - K L ) . E Q . C ) G Q TO 7 339 ! F ( D ) 4 r 5 , 6 390 4 KH=;4 391 GC TO 3 392 5 JL=M 393 GC TO 11 394 6 KL = M 395 GO TO 3 396 7 J L = H * i C* MAKE SURE JL IS AT BOTTOM OF LIST OF EQUAL VALUES 39 7 II I F ( [ J L - 1 ) . E Q . O J G O TC 12 398 IF [ X C G T . A! J L - 1 ) )GC TO 12 399 ' J L = J L - 1 400 GO TO 11 C * NOW FIND JH 401 12 KL=JL 402 Kh = N 403 13 M=KL+ i K H - K L ) / 2 404 D=XH-AIM) 405 I F ! ( M - K D . E Q . O J G O TO 17 | 406 IF(O) 14 ,15 ,16 I 407 14 KH=M ! 403 GO TO 13 j 409 15 Jh=M 410 GO TO 21 4 11 16 KL=M 412 GO TO 13 413 17 JH=M-l C* MAKE SURE JH IS AT TO? OF LIST OF EQUAL VALUES 414 21 I F ! ! JH + 1 ) , £ Q . ( N+ 1! ! GO TO 22 415 I F I X H . L T . A I JH-J-1) ) GO . TO 22 416 JH=JH*1 417 GO TO 21 C * MOW SEE HOW MANY ELEMENTS SPANNED BY ( J L . J H ) 413 22 HENTS = J H - . J L * l C* IF MENTS o L E . 0 THEN BOUNDS DO NOT DEFINE A SET 419 GO TO I 420 END 4;U SUBROUTINE SORTJA,NUN,N> 422 DIMENSION A ( N! ,NUM{"!) 42 3 DIMENSION MINI 20) »M A X i 20! C M! , \ AND MAX ADEQUATE FOR N UP TO 1 ,000 ,000 424 L = 0 42 5 MN = 1 426 MX=N 427 10 ! F ( MX. LE . MMGC TO 49 428 11 MD =K'J 7 [ A , Nlif i M.N , MX ) C NOW IDENTIFY LONGER AND SHORTER S1J3LI ST S 429 TFjMX-HD.GT.MD—MN)GO TO 25 4 30 I F I M X - M 0 . G E . 2 I G C TC 13 4 31 MX==MD-1 432 GC TC 10 4 2 3 13 L - L • 1 C PUT AWAY RECORD OF LONGi:R LIST 434 K1 N f L) = M N 43 5 I iAX(L)=HD-l 4 36 M:«=KD*l 437 GO TO 11 438 25 I F ( M 0 - K N . G e . 2 ) G O TC 26 439 MN=HD+1 440 GO TO 10 441 26 L=L + 1 . C PUT AWAY RECORD . Gr LONGER LIST 442 M IN { L > --MD-s- -443 MAX(Li=MX . 444 MX=HD~1 445 GC TO U 446 .49 I F ! L . S Q . O ) R E T U R N C SORTING COMPLETE IF NO DEFERRED TASKS C OTHERSiISE SELECT A DEFERRED TASK 44 7 MN'=MIN(L ) 443 MX=HAX(L) 449 L=L-1 450 GC TO 11 451 END 452 FUNCTION K'JTI A,NUM»L ,H) .. 453 DIMENSION A(M),NUM(M) C THIS FUNCTION WORKS GN SUB LIST RUNNING FROM ACL) TO AIM) 4 54 X=A(L) 455 NIT--NUMIL) 456 i - L 457 J=H*1 C NEXT 3 LINES 1.0OX BACKWARDS FROM END FOR SMALL ENTRIES 453 I J = . ) - l 459 I F ! ! . E O . J ) G O TC 99 460 I F ( A I J , ) . G £ . X ) G G TO 1 C MOVE' .A! J') INTO THE GAP 461 A ! I i = A ! J ) 462 .NUI1 ( I) =NUM 1 J ) C NO it SEARCH FORWARD FOR. LARGE ENTRIES 463 5 I--=1 + 1 464 I F ( I . E Q . J ) G O TC 99 465 ( F I A ( 1 ) • L E . X ) G O TO 5 466 A ! J ! = A ! I ) 467 MUM!J) = HUM 1 I ! C MOVE A i l ' . INTO THE GAP AND GC TQ 1 463 GO TO 1 469 99 KUT--I 4 70 A!KUT)=--X 471 NUM{KU T;— NIT 472 RETURN 4 73 END 474 FUNCTION D E L T A ! I F , I S l , IS?) C*« GIVES FELLING ANGLE DELTA FOR TREE ' I F ' BETWEEN I S 1 , ! S 2 47 3 COMMON X 1 5C 0 ! i Y I 500 ) » 0 I 509 ; • H( 300! , HLC-! 500 ),.1 S T AT.( >500 ) 476 CAVA P I i i . 1 4 1 5 9 3 / 477 DXl=X! I S i)-X 11F) 478 C X 2 - X !.I S2 ) -X ! I F- 1 4 79 OYl=YiIS I )-Y 1 IF ) 480 0Y2 = YI 1S2 )-Y 11F) 401 SEPSC=(0X2-DXl.) *-«2+ (DY2-CY1 1**2 482 SEP=SCRTISEPSOl-433 484 4 35 436 437 433 489 490 5 491 I 492 493 2 49 4 495 3 496 4 497 498 499 5C0 501 C 5C2 503 504 505 5C6 507 508 5 09 510 511 512 513 514 515 516 517 513 519 520 521 52 2 523 5 24 i 52 5 52 6 C c 52 7 528 5 29 5 i 0 531 5 32 533 5 34 53 5 5 36 53 7 R1SQ=IDXI**2*0Y1**2) R2SQ=IDX2**2+0Y2**2) Rl=SCR'f IRISQJ R2=SQRTIR2SQ) I F l ! SEP-A3SIR2-X J.I ) . C T . l . c - 4 130 TO 5 GA=0. GC TO 4 1FIK1SQ+R2SG-SEPSC) 1 ,2 ,3 GA=Pl-ARCOSl(DX1*DX2 + 0Y1*DY2i/!R1 *R2)) GO TO 4 GA=PI /2 . GO TO 4 GA=AHCOS!10X1*0X2 + 0 Y 1 * C Y2 ) / t R 1 *R 2) ) •PI = R1 *D I IF) / !C(1S1 ) * D ! IF !) . P2 = R 2 * D I I F ! / ! D ( I S 2 ) * D < IF)) OELTA = G A - A R S I N ( . 5 * D ( I F ) / P I 1 - A R S 1 N I . 5 * 0 1 I F ) / P 2 ) RETURN END SUBROUTINE POLAR! I F , I S ; ROS,THE) = GIVES POLAR CCORDS. FOR ' I S ' , ORIGIN"AT ' I F ' IMPLICIT LOGICAi.IL.) COMMON X ! S C O ) , Y I 500) ,D(5CO) ,H(500) , H L C ( 5 0 0 ! , I ST AT!500) DATA P I / 3 . 1 4 1 5 9 3 / X S I D E = X ( I S ) - X ( I F ) Y S I D E = Y { I S l - Y I I F ) R0S=SQRT(XSIDS**2 > YSI0E-*2 ) L l = X S I D E . £ Q . O . . L 2 = Y S I G E , E Q . C . 1.3-XSIDE . G T . 0. L 4 = Y S I D E . G T . O . L 5 = X S I D E . L T . C . L 6 = Y S I 0 E . L T . C . 7HS=-1. IF 1 L 2 . A N 0 . L 3 ) T H E = C . I F i H . AN0.L4>THE = P I / 2 . I F ! L 2 . . A N 0 . L 5 ! T H C = PI I F ! L l . A N D . L 6 > T H E = 3 . * P 1 / 2 . I F I T H E . G E . 0 . ) G C TO 1 THE=ARSIN!ABS!YSIOE)/ROS) I F ! L 4 . A N D . L 5 ! T H E = P I - T H £ I F ! L 6 . A N D . 1 5 ) T H E = T H E f P I I F ! L f t . A N D . L 3 ) T H E=2.* P 1 - T H E RETURN END FUNCTION S E P I A , B , G A ) > CALCULATES F E L L I N G CORRIDOR H AL F-WI DT F" ANGLE GA MUST BE < PI RADIANS DATA P I / 3 . 1 4 1 5 9 3 / IF i A . G £ , G ) G Q TC 1 T - a a = a D = T U=B*SIN(GA> Vi-- SQRT (8**2-u*'»2) i F ( C A . G T . P I / 2 . ) ~ ! -• W ) Z=A-W 3 E = A T A N ( C / Z ) C = SCR T (/:>:•*2 +U* •> 2 ) / 2 . 533 tt=SCRT<C**2*A**2-2.*C*A*C0St8E)) 539 DE = ARSIM!C*SIN(6f:) /R) 540 TH = BE -s-DE 541 SEP=C*SIN(TH) 542 RETURN 543 END 544 SUBROUTINE TELL (NSTEPS,R1 ,R2 , S EP , iPL -. IS1, I S2, THDCR) C * * * SIMULATES FELLING ' I F L ' BETWEEN • I S1' , * I S 2 * 545 COMMON XI S C O ) , Y ( 5 C C ) , D I 5 C O ) , H ( 5 0 0 ) , H L C I 5 0 0 ) , I S T A T ! 5 0 0 ) 546 COMMON /POWERS/ P , P I , P E , A 547 COMMON / F E L P U T / T H S T E P ( 9 0 J , C R S K (90 , 2 ! , 3 S H t 9 0 , 2 ) , B R SHI 9 0 , 2 ) , 1INDEXI90,2) 548 DIMENSION R A 0 ( 2 ) , I S D I 2 ) 549 . DATA RDN/57 .29576/ C<"» IF TREE CAN F A L L UN Cb S T H UC TE 0 RETURN 5 50 NSTEPS=0 55J ZF=H! I F L ) - H L C ! I FL ) 552 ZS1=H! IS). I -HLC ! I SI ) 553 2S2 = H( I S2 1-H1.C I I S2) 554 HCWF=XWIDE!ZF) 555 HCWS1=XWIDEIZS1 ) 556 KC WS2 = XWiD£IZS2) 557 CwF=2.*SHAPE(ZF,HCWF) 550 C R S l = S H A P E ! Z S l . H C W S l ) 5 59 CRS2- SHAPE (ZS2.IICWS2 ) 560 IF'. !CWF*CRSl*C*S2> . LE . ! 2 . * S £ P ) ) RETURN C « * FALL OBSTRUCTED - CONTINUE ViITH ALGORITHM 561 DC i l l N=l,2 562 00 l l t l M=1,9Q 56 3 I F IN ..EQ , >.) THSTEP I M ) = 0. 564 INDEX(M,N)-0 565 CKSH(M,N)=0. 566 &KSH(M,N)=0. 567 3SHIH,N)=0. 568 1111 CONTINUE 569 111 CONTINUE 57 0 RADII)=R1 571 RAD(23=R2 j 572 I S D ( i ) = 5 S l 57 3 !SDI2!= iS2 C * FIND ANGLE THAT FALL ING TREE MAY BEGIN TO BE OBSTRUCTED AT 574 CALL ! l i TONE I H { I Fl. ) , H U S.i 3 I S 2 ) ,R I , R 2 ,1 T , PHI ) j 575 1=1 ' i 576 1 I 577 TKSTEPtI)=PHI 573 THETA=PH1/RCN 579 2 F7AC = RA0 ! J ) / f : O S ! T H F . T A ) 530 STAG = SG3TfFTAC**2-P.AD( J > * * 2 » 581 INDEX! i , J ) = L O G I C ( IFL , 1 S O U ) , F TAG ,ST4C) 582 IFI INDEX I I , J ) . E C . I ) GO TO 100 523 I r { I N D E X ! I , J ) . E C . 2 ICO TO 2C0 584 IF ( INDEX! I , .1) . r.Q . 3 ) GO TO 3C0 505 IF < INCE-X I ! , J.) . E C 4 iGO TO 400 586 I F ! -J. EQ..2 iGO TC 3 •: a r 4 j - 2 539 GO TO 2 539 '.00 I F ! J . E C . l i GO TO 4 590 I F ! INDEX'. I , 1 > . L T. 4 ) GO TO 3 59 1 5 NSTE?S=! 592 RETURN 593 100 S-IT 1 I SO i J) l - H L C I ISO! J l ) 594 X I = S T A C - H L C ! 1 S 0 ( J ) ) 595 C E X S = S H A P E ( S , X I ) 596 S = H( I F U - H L C I I F L ) 597 X i = F T A C - H L C i I F l ) 598 C E X F = S H A P E ; S . X I > 599 CRSKl I , J ) = C R U S H i C E X S , C E X F , S E P ) 600 I F U . E Q . l ) G C TO 4 601 GO TO 3 602 200 S=HI 1 F D - H L C ! I F L ) 603 X I = F T A C - H L C I I F L ) 604 C E X F = S H A P E I S , X I ) 605 LiEXS = 30LE (HI I SO! J) ) ,01 ISO! J) ) , STAC 606 ORSHI I , J ) = 8 A S H ( C E X F , B E X S , S E P ) 607 I F U . E Q . l ) GC TO 4 603 GO TO 3 609 300 S = H l I S O ! J ) l - H L C I ISO(J) ) 61 0 Xl=STAC-Hl .Ct I S C I J ) ) 611 C E X 3 = S H A P £ 1 S , X I ) 612 BEXF = 8 0 L 6 ( H ( I F L ) , 0 ( T F L ) ,FTAC ) 61 3 3 S H ( I , J ) = B A S H ( C E X S , B E X F , S E P ) 614 I F ( J . E G . 1 ) GO TO 4 615 3 P H I = P H I - T H D C R 616 IFIPHI . L E . O . )G0 TO 5 61 7 1 = 1+1 618 GO TO 1 619 E N D SU BRO'JT IHZ HIT ONE ( HTF , HT 1, HT 2 , R • , R 2 , I , THET A ) DATA RDN/57 .29576 / H YP 1= SOR T IR 1 * 2*HT 1**2 ) UYPZ-sSQIlT CR2**2*HT2**2) AL 1 = A T A N I Ii 11/R I ) A!. 2-AT AM ( HT 2/R 2 ) I F t H T F . L t . HY PI J A L1 -- ARC OS ! R1 / HT F ) I F ( H T F . L T. H YP 2 ) A 1.2 = A RC 0 S! R2 / H T F ) I :'• I A l 2. GT . AL 1)GO TO 1 1=1 f H E T A = A!. I * R 0 N RE TURN 1=2 THETA=AL2*RQN RETURN END 636 FUNCTION L O G I C ! I F L , I S 0 , r P , S P ) 637 IMPLICIT I.GG1CALIK) 638 COMMON XI5C0) , Y ( 5CC) , 0 (; 5 C 0 ) »H 1 5 0 0 ) , H L C I 5 0 0 ) » I 5 T A T ( 5 0 0 ) 6 39 K1 = H L C ( 1 F L 1 . L E . F P . A N D . F P . L t . H i IFL) 640 !<2 = FP« LT , HLC ! I F L ) 641 K.3 = F P . G T . H I If-"L) 64 2 K4 = HLC i i SO) . I E . SP. AND. SP . ! _E . i l l I SD ) 643 K5 = S P . L T . HL C ! I S 0 ! 644 K6 = S P - G T . H ! i S O 645 I F ( K 3 . OR . K -!:.'! L OG I C •'- 0 64 & I F ! K l . A N D . K 4 ) L G G I C = 'l 647 - I F ( K I . A N D . X 5 ) L C S I C = 2 648 I F t K 2 . AND .'K4 ) L CGIC = 3 64 9 I F I K 2 . A HO - K 5 ! L CO IC - •'• 621 622 !623 624 62 5 62 6 627 6 28 62 9 6 30 631 632 I 633 634 6 35 650 RETURN 651 END 652 FUNCTION B A S H l C E X , B O L E , S E P ) C*s= 3CLE ON CROWNi OR CROWN ON BOLE» INTERACTION 653 B A S H = 1 . - ( S E P + B C L E ) / ( C E X + 2 . * 2 C L E ) 654 I F ( 3 A S H A T . 0 . ) 8 A S H = C.-655 RETURN 656 END 657 FUNCTION CRUSH[CEXS,CEXF,SEP> C * * CROWN ON CROWN INTERACTION 653 CRUSH = 1 . - S E P / I CEXS-i-CEXF ) 659 I F f C R U S H . L T . C . J C R U S H - C . 660 RETURN 661 END 662 • • FUNCTION B O L E tH T , D B H , X ) C » * GIVES BOLE RADIUS AT ANY HEIGHT X 66 3 COMMON /POWERS/ P , P I , ? E , A 664 B C L E = I D 3 H / 2 . ) * ! I H T - X)/ ( H T - 4 . 5 ) ) * * P 665 RETURN 666 END 667 FUNCTION S H A P E « X , X I ) C * * GIVES CROHN RADIUS AT POINT ' X I ' 668 COMMON /POWERS/ P , P I , P E , A 669 S H A P E = A * ! X I * * P I - X I S - P E / X * ~ ( P E - P I )) 6 70 RETURN •• I 671 END 672 FUNCTION XHIDE(X) C*'» GIVES POINT IN CROWN WHERE THERE IS GREATEST SPREAD C FIRST DERIVATIVE OF 'SHAPE* SET TO 0, SOLVED FOR X 673 COMMON / P O K E R S / P , P I , P E , A 67'. C1=PI*A j 675 C 2 = { P E * A ) / X * * J P E - P I ) . ! 676 X W ! D c = E X P i A I . 0 G ( C 2 / C l ) / ( P I - P E ) ) 677 RETURN 673 END 679 SUBROUTINE L AT YR 01 I TR, F.HCH, ATT ) C * * COMPUTES POSSIBLE HITS TO RESIDUALS WJ'ILE YARDING I TR LATERALLY C AMD PUTS HITS INTO IRHITSI! ) C BASICALLY ROUTINE CALCULATES WIDEST POLYGON SEGMENT LYING IN C THE YARDING TRIANGLE ABC; CHOKER SET TO B j 680 COMMON X ( 5 0 0 ) , Y i 5 C 0 ) , 0 1 5 0 0 ) , H I 5 C C 3 » H L C ( S C O ) » 1 S T A T i 500) i 681 CCHKON / ARE hi X . u I N , X M AX , V M IN, Y M AX j 6S2 COMMON /RANXX/ XRK(500) ,NXR(500) ,NTOT I 63 3 COMMON / YRDP UT / DIST, ' ! ITS, IA HI T S ( 5 00 S j 684 CINENS ION YTRL 190 )-NTRL190) j 635 DIMENSION ASLI90) i 686 DATA R A O / 5 7 . 29 5 76/ C»* FIRST GET TRIANGLE SUBSET Of- RESIDUAL TREES C GET COORDINATES OF TRIANGLE 687 ;*!TS = 0 688 Y O Y M I N 689 I F ' A T T . G T . 0 , ) GO TO 1 690 XB=X!ITR) 691 Y3=Y1ITR) j i ( I ( I i 692 XA-XS+HMCH 693 YA=Y3 694 GO TO 11 695 1 A A - 1 8 0 . - A T T 696 IFIAA) 2 , 3 , 4 697 2 YA=Y(ITR) 693 XA = X(ITR ) 699 A C - 2 7 0 . - A T T 700 IF(AC) 5 , 6 , 7 701 5 ANG=360.-ATT 702 D Y = H !•! C ri * SI N I a ?•  G / R A D) 703 I F I D Y . L T . I Y A ~ Y . M I N ) ) GO TO 21 704 YB-YHIN 7C3 XB = XA * ( X B - X fIN)*AT AN(ANG/RAD) 706 GO TO 12 707 21 YS-YA-OY 7C0 XB = XA«-SCRT(HfCH«*2-OY»*2 ) 709 GO TO 12 710 6 C I S T - Y I ITR) -HMCH-Y111N 711 RETURN 712 7 ANG--AA 713 OY-HMCH«SlN i ANG/RAO> 714 I F ( C Y . L T . ! Y A-Y , M I N ) ) GO TO 22 715 YB-YHIN 716 X 3 = X A - ( X A - X l J I H) * A T A N ( A N G R A 0 ! 717 GO TO 11 718 22 YB=YA-DV 719 X3-XA-SQRT(H.MCK**2-DY**2 ) 720 GO TO 11 721 3 X S - X { I T R ) 722 Y3=Y!ITR) 723 XA«X3-HMCH 7 24 YA = YB 72 5 GO TO 12 726 4 Y3=Y(ITR) 72 7 XB-=X;ITR) 7 23 A 3 - 9 0 . - A T T 729 I F i A B ) 8 , 9 , 1 0 730 8 ANG=AA 7 31 OY-HHCH*S iH{ANG/RAO! 7 32 YA=YD*DY 7 33 X A =X0-SQRT i HMCH**2~DY**2) 734 GC TO 12 7 35 9 DiST=Yi. i TM1-YPIN 736 RE TURN 73 7 10 ANC---6TT 7 38 0 Y *HMCH* 31 N I ANG /RAD) 7 39 YA-Y3-JQY 740 XA-XB+SQRT1HMCH**2-OY**2) 741 1 I XLG-X3 74 2 XMi-XA 743 GO TO 13 7 44 i 2 XLO=XA 74 5 :•;;•! t =xs 746 13 D! ST--YS3-YMIN 747 XC = XB C GET SUBSET IN RANGE ( X L O t W l ) 74 8 CALL Ri SRCH i XRK,NTCT , xCO, ".HI , 749 C* I F ' . H T S . L c . DRETUKN NOW GET TRIANGLE SUBSET C GET ABSOLUTE SLOPES FROM A TO 6 AND TO C 750 NT = 0 751 AHSL=1.E10 752 X8DXA=XB-XA 753 I F I X B D X A . E Q . O . ) R E T U R N 754 23 A S L A 3 = A 8 S ! ' Y O - V A ! / X B D X A ) 755 25 A S L A C = A 8 S l ( Y C - Y A J / X B D X A ) 756 26 GG 14 1=JLC,JH1 757 NUM=NXRtI) 758 I F I N U H . E Q . I T R J G O TO 14 759 I F ! I ST AT(NUH ) . E C . 1 )GO TO 14 760 C I F ( Y ( N U M ) . G E . Y A ) G O TO 14 THE TREE IS IN THE RECTANGLE C GET ABS. SLOPES TO TREES IN RECTANGLE AND SEE IF ARE If 1 TRIANGLE ABC 761 X D X A = X R K i ! ! - X A 762 I F ( X D X A . E O . 0 . I G O TC 14 76 3 27 ASLA=ASSI1Y<NUM)-YA) /XDXA) 7 64 23 I F I A S L A . L T . A S L A B . O R . A S L A . G T . A S L A C ) G O TO 14 C PUT Y-COORD. AND TREE NUMBER INTO TRIANGLE ARRAYS YTRL AND NTRL 765 NT=NT*1 766 YT RL(NT)=Y1NUH ) 76 7 NTRL1 NT) =NUfJ C FIND TREE OP.IGUAL NUMBER WITH I'INIMUH ABS. SLOPE FROM A - VIZ? MTKL 768 I F I A S L A . G E . A M S D G O TO 14 769 AMSL=ASLA 7 70 M7RL=NTRL(NT) 771 14 C* CONTINUE IF TRIANGLE SUBSET . G T . 1, CONTINUE WITH ALGORITHM 7 72 I F ( N T . G T , 1 ) G 0 T O 15 77? I F I N T . E Q . O J G O TG 141 7 74 IRHITS INTRL t 1! ) = IP.HITS! NTRL I 1) ) + 1 775 N I T S = N J T S + 1 7 76 141 RETURN C=> SORT SUBSET BY YTRL AND KEEP TRACK OF ORIGINAL TREE NUMBERS IN NTRL 7 77 15 CALL SORT I Y T R L , N T R L , N T ) C * FIND SCfUEO ARRAY NUMBER XI THAT HAS HI N. ABS. SLOPE FROM A C THE ORIGINAL TREE CORRESPONDING TO Kt IS KTRL 778 OC 16 1=1,NT 779 IF (NTRL 11 ) . N E . f ' T R D G G TO 16 780 KI=I I 781 16 CONTINUE | C* GO THROUGH REMAINING SUBSET AND SCORE HITS FOR THE POLYGON POINTS, WHEN XI 702 17 IRH i TS(NT R L ( K I ) ) = IRH IT 5(NTRLIKI ! )* I | 733 N!TS=NITS+1 C SUBSET COUNTER NO FOR REMAINDER GF TRIANGLE 734 N0--KI-1 785 I F ( N D . E C O ) RETURN 786 Af!SL=l .E10 787 ACXC = A B S ! X C - X ( N T R L i K I i ) )• 783 I r t A D X C . G T . C . ) G 0 TC 30 C SCORE HITS FOR TREES 1. Y I '•: G O N VERTICAL KI T O C SLOPE LINE ! 739 00 20 K=l ,NC i 790 I F (X f NTRL !!• ) ) . N E . X C ; GO T C 20 : 791 IRHITS! N TR L ! K) ) = I R H i ; S I 1 <>.{ '.<) ! > 1 J 792 NlTS-NITS+1 j 793 20 CONTINUE ] 794 RETURN j 795 30 A S LK IC = ABS ! ! Y i RL ( K I ! - Y C ! /'Ai) <C ) ! C FINC NEW KI IN SET NO j 796 KJ=KI 797 793 799 SCO 801 802 803 804 805 806 19 807 C S 808 8C9 810 811 812 . 29 313 014 815 316 817 313 8 19 820 821 822 823 824 601 025 826 2 827 3 828 GOO 029 600 630 8 3!, 3 32 83 3 8 34 33 5 336 8 37 8 38 339 340 8 41 a ' i i j " c 3 4 3 0 4 4 8 4 5 6000 u -* O 8 4 7 iOOO 848 6 001 8 49 2 0C0 850 3 51-DO 19 1=1,NO A 0 X - A 8 S ! X C - X ! N T R L U > ) ) r F i A O X . O E . A D X C l G O TO 19 XI .XKI-X(NTRL{I.l J - X t N T R L I KI ) ) A S L ! I J = A3S.( ! YTRL f I ) - Y T R L ! KIS J / X I X K I ) IF I A S L( I ) . G T . A S L K I C ) G O TO 19 I F l A S L m . G t . A P S U G O TO 19 AMSL = A S L ( I ) K J = 1 C O N T I N U E I F I K J . E Q . K I ) R E TURN ;QRE H I T S FOR T R E E S ON XI TO "C S L O P E L I N E . CO 29 J=KJ , N 0 I F ! A S L ( J) . N E . A S L K I O G Q TO 29 IRHI TSINTRL U> J--IRHI TSINTRL! J ) ) n N i T S - N I T S U CONTINUE KI= K J GO TO 17 END SUBROUTINE OUTPUT! IT,HARK) IMPLICIT L O G I C A L t O ! COMMON / D E C P U T / NT 1 .MT2.R1 , R 2 , f-ELGL , SE PR > E LOG »K IND ( 500) COMMON / F E L P U T / T H S T E P ( 9 0 ) , C R S H ( 9 0 , 2 ) , O S H !90, 2 ) , S R S H t 9 0 , 2 ), 1INDEX(90 ,2) COMMCN / Y R O P U T / 01 S T , N I T S , 1RHITS(500 ) C 0 R R - 2 . * S E P R IF! i - iARK.NE ,G)G0 TO 2 WRITEI6.601) I f » F E L GL »£ L OG,D i ST > N I T S • F O R M A T ( I 6 . F 9 . 1 , 4 X , ' N O T A P P L I C A B L E " , 4 X , F 9 . 1 , 6 X , ' F E L L I N G COMPLETED ' , ! • WITHOUT OBSTRUCTION' , 6 X , F 7 . 1 ,17) GO TO 3 WRITEI 6, '600) IT , F E L G L , N T 1,NT 2 . C O R R , E L O G , D I S T , N I TS WRITE18,C00> I T , F E L G L , C O R R , E L O G . D I S T , N I T S , K I N D ! I T ) FORMAT ( 1 5 , 4 F 9 , 1,216), FORMAT t I 6 , F 9 . 1 , 1 3 , • , 5 , 1 4 , 2 F 9 . 1 , 4 9 X , F 7 .1 ,217) I F f M A R K . E O . O J G G T C 1 WRITE OUT ANY INTERACTION COEFFICIENTS KOUNT-1 CO 6 N = l , .MARK MK1-C-MK2=0 MK3=0 01= C R S H ( N , l ) . E C O . G 2=CRSH ( M , 2).EQ.O. 0 3-QSH'M,1 ! , E Q.O. G 4 = B S H(N , 2 ) . E Q.O. G 5=BRSH ( N . I ).EQ.O. C 6 = B R S H ! N , 2 ) . E C . O . ' i F I C i . AND.02 ..AND.03.. AND .04 . AND.05 . AM 0..06 > GO TO 6 I F i. KOUNT.G T. 1) GO TC 1 0 0 0 WRITE(6 ,6000) THSTEP!N> FORMAT l « + ' ,50X , F 5 - 1 ) GO TO 2000 WRIT E ! b , 6 0 0 1 ) THSTEP!N) FORMAT(51X ,F5 . 1 ) KOUNT=KOUNT+1 DO 7 K = l , 2 IF I INDEX ( N,K ) . . E C . C . O R . 1NCEXIN.K) . E Q . 4 1 G 0 TO 7 852 I F I M K I . E Q . 1 ) G O TO 3 353 I r ( C R S ! H N , K ) . E Q , 0. IGO TO 3 3 5't MK1 = 1 355 VJRITE!6,6Q2 ) ICRSHIN.M) , K = l , 2 ) 356 6C2 FORMA T{« + • , 5 5 X , E 7 . 3 , * , * , F 5 . 3 ) 057 8 I F(MK 2 . E Q . 1 ) G O TO 10 353 IF( BSHi-N.K ) , E Q . O . )GG TO 10 859 i « 2 = l 860 - WRITEI6.604) (B S H(N,M) ,M = 1,2) 861 604 FORMAT C * * ' , 6 6 X , F 7 . 3 , f , • , F 5 . 3 ) 362 10 I F ( K K 3 . E C . 1 ) G 0 TO 7 863 i ' F ( B R S H ( N , K ) . E G . C . ) GO TO 7 864 HK3 = 1 St. 5 wRITE ' 6 ,606 ) ( B K S H(N , M i , >M 1', 2 ) 066 606 FORMA T1 • , 8 1 X , F 7 . 3 , ' , ' , F5 .3 i 867 .' 7 CONTINUE 36S 6 CONTINUE 86 9 1 RETURN 870 END 

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