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Development of a digital terrain simulator for short-term forest resource planning Lemkow, Daniel Zwi 1977

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DEVELOPMENT OF A DIGITAL TERRAIN SIMULATOR FOR SHORT-TERM FOREST RESOURCE PLANNING by DANIEL ZWI LEMKOW B.S.F, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1975 A THESIS SUBKITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (The F a c u l t y o f F o r e s t r y ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISIH COLUMBIA J u l y , 1977 (•B) Daniel Zwi Lemkow, 1977 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without ay w r i t t e n p e r m i s s i o n . Department of F o r e s t r y The U n i v e r s i t y of B r i t i s h Columbia 2075 Westbrook Pl a c e Vancouver, Canada V6T 1H5 i i A b s t r a c t T h i s study d e s c r i b e s the development of a desktop computer model ( d i g i t a l t e r r a i n simulator) f o r short-term f o r e s t planning. An overview of the a p p l i c a t i o n s c u r r e n t l y developed f o r the t e r r a i n s i m u l a t o r i s presented: <1) C o l l e c t i o n of the r e q u i r e d t e r r a i n e l e v a t i o n s and f o r e s t i n v e n t o r y data; {2) Determination and production of map o v e r l a y s of the topographic f e a t u r e s : s l o p e , aspect and e l e v a t i o n ; (3) Design of l o g g i n g s e t t i n g s f o r c a b l e systems and placement o f y a r d i n g roads; (4) l o c a t i o n of f o r e s t - a c c e s s roads; |5) D e l i n e a t i o n of viewable areas and pr o d u c t i o n of three dimensional r e p r e s e n t a t i o n s of the t e r r a i n on a two dimensional s u r f a c e ; (6) E s t i m a t i o n of h a r v e s t i n g c o s t s and wood volume pr o d u c t i o n . The study then presents the theory r e q u i r e d to implement each of the above components. Hhere p o s s i b l e , s e v e r a l d i f f e r e n t i i i approaches are developed and compared. The e l e v a t i o n data base, d e s c r i b i n g the study area, i s repr e s e n t e d by a r e g u l a r g r i d of e l e v a t i o n s . The s l o p e , a s p e c t and e l e v a t i o n are computed f o r each of these g r i d u n i t s frost a geometric plane f i t t e d to the ground s u r f a c e using a l e a s t -sguares procedure. An al g o r i t h m f o r producing map o v e r l a y s of these a t t r i b u t e s , i n v a r y i n g combinations, i s gi v e n . The major emphasis i n the d i s c u s s i o n of the s e t t i n g design module i s on the p r e d i c t i o n of loadpaths of c a b l e yarding systems. The road design module concentrates on route p r o j e c t i o n . An a l g o r i t h m i s proposed that a u t o m a t i c a l l y l o c a t e s a t r i a l r o u t e between any two map l o c a t i o n s . For producing t h r e e dimensional r e p r e s e n t a t i o n s both o r t h o g r a p h i c and p e r s p e c t i v e p r o j e c t i o n s are developed along s i t h an a l g o r i t h m to remove the 'hidden areas* i n these three dimensional p l o t s . The c o s t i n g module presents methods f o r in p u t of f o r e s t i n v e n t o r y data, u s i n g the same r e g u l a r g r i d as f o r the t e r r a i n e l e v a t i o n s . , An a l g o r i t h m d e t e r m i n i s t i c a l l y s i m u l a t e s the yarding o f the wood i n each g r i d u n i t c o n t a i n e d by the l o g g i n g s e t t i n g boundaries. F i n a l l y , the l i m i t a t i o n s of the model, due t o computer technology and the q u a l i t y and g u a n t i t y of the data, are examined. Slow ex e c u t i o n speed, i n s e v e r a l i n s t a n c e s , d i c t a t e d the use of the l e a s t a c curate approach; t h i s occurred i n the i n p u t , y a r d i n g design and c o s t i n g modules. T h i s a l s o made i t necessary t o use o r t h o g r a p h i c i n s t e a d of the t r u e p e r s p e c t i v e p r o j e c t i o n f o r producing three dimensional r e p r e s e n t a t i o n s . The determination of earthwork volumes, f o r road c o n s t r u c t i o n , was deemed i n a p p r o p r i a t e , due to the g e n e r a l l y poor p r e c i s i o n of a v a i l a b l e maps. S o i l s i n f o r m a t i o n , although important f o r e s t i m a t i n g c o s t s o f yard i n g and road c o n s t r u c t i o n , was not i n c l u d e d because of i t s l i m i t e d a v a i l a b i l i t y . The map o v e r l a y s of s l o p e , aspect and e l e v a t i o n c o u l d be e a s i l y extended to i n c l u d e s o i l s and land-use data, thereby o f f e r i n g a more complete and u s e f u l r e t r i e v a l system. Although f u l l implementation of the t e r r a i n s i m u l a t o r has not been p o s s i b l e , t h i s study demonstrates the f e a s i b i l i t y of implementing a comprehensive short-term f o r e s t planning model, designed f o r a desktop computer. V TABLE OF CONTESTS PAGE L i s t o f T a b l e s i x L i s t of F i g u r e s • . . • . • » • • • • • » • « • » » « . . . » • » . • . « . . . » . » . . « » • » » x L i s t o f P l a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,/xiv Acknowledgement xv. I n t r o d u e t i o n • . . . . « . » • • • . . . » . . • » . « • • . » » . . . « . » • » » « • • » « • « • • • ^ P a r t I : An O v e r v i e w o f t h e T e r r a i n S i m u l a t o r and I t s A p p l i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CHAPTER 1.0 I n i t i a l P r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.0 I n p u t o f t h e E l e v a t i o n D a t a Base . . . . . . . . . . . . . . . . 10 2.1 T r a n s e c t Method 10 2.2 C o n t o u r Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.0 A n a l y s i s o f T o p o g r a p h i c F e a t u r e s 16 4.0 Y a r d i n g L o c a t i o n and S e t t i n g D e s i g n . . . . . . . . . . . . . 20 5.0 Road L o c a t i o n . . . . . . . . « . . . . . . . . . . . . . . « . . » • « . « • • • . 28 6.0 A e s t h e t i c A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.1 V i e w a b l e A r e a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 T h r e e D i m e n s i o n a l R e p r e s e n t a t i o n s . . . . . . . . . . 33 7.0 H a r v e s t i n g C o s t s and Wood Volume P r o d u c t i o n ..... 40 P a r t I I ; T e r r a i n S i m u l a t o r T h e o r y 45 CHAPTER 1.0 E l e v a t i o n D a t a Base 46 1.1 R e t e n t i o n o f C o n t o u r L o c a t i o n s . . . . . . . . . . . . . 47 1.2 C o n v e r s i o n o f C o n t o u r s i n t o a R e g u l a r G r i d o f E l e v a t i o n s ....................,49 v i CHAPTER PAGE 1.2.1 I n t e r p o l a t i o n Method f o r f i n d i n g E l e v a t i o n s • 52 1.2.2 Surface Eguation Method For F i n d i n g E l e v a t i o n s .................... €1 1.3 Implementation 64 2.0 Input o f the E l e v a t i o n Data Base 67 2.1 T r a n s e c t Method ............................ 67 2.2 Contour Method 70 2.2.1 i n t r y of Contour L i n e s 71 2.2.2 S o r t i n g of Data P o i n t s t o F a c i l i t a t e Searching .................. 71 2.2.3 Generating the G r i d of E l e v a t i o n s ... 76 2.3 Comparison of the Tra n s e c t and Contour Methods 80 2.4 Implementation ...... ....................... 82 2.5 A New Method f o r Data Entry 83 3.0 Topographic Features 84 3.1 Determining the Topographic Features 85 3.1.1 S o l v i n g f o r Average E l e v a t i o n ....... 85 3.1.2 S o l v i n g f o r the Maximum Surface Slope ......................... 85 3. 1.3 S o l v i n g f o r Aspect .................. 87 3.2 Storage of the Slope, Aspect and E l e v a t i o n .............................. 91 3.3 Output of the Topographic Features . 9 1 3.4 Implementation 92 v i i CHAPTER PAGE 4.0 Yarding L o c a t i o n and S e t t i n g Design ............. 95 4. 1 P a r a b o l i c Model f o r Approximating the Loadpath ...... ...... .... ............... 96 4.1.1 Comparison t o the Catenary Model .... 102 4.1.2 Improvement of the I t e r a t i v e Method ...................... 102 4.2 Percent D e f l e c t i o n Rule f o r Approximating the Loadpath 105 4.3 Implementation .............................111 5.0 Road L o c a t i o n ................................... 114 5.1 Manual Road P r o j e c t i o n .....................114 5.2 Automatic Road P r o j e c t i o n .................. 115 5.3 Road L o c a t i o n Output ...................... , 120 5.4 Grade L i n e L o c a t i o n and Earthwork Volumes .......................... 120 5.5 Implementation ............. ................. ,120 6.0 A e s t h e t i c A n a l y s i s . 122 6.1 Viewable Area Assessment ................... 122 6.1.1 Determining the Viewable C e l l s on a S i g h t L i n e ....................... 122 6.1.2 Implementation 126 6.2 Three Dimensional Representations of the E l e v a t i o n Data Base 127 6.2.1 Transforming the E l e v a t i o n Data Base 128 6.2.2 P l o t t i n g the Three Dimensional Repres e n t a t i o n ........................ 136 v i i i CHAPTER PAGE 6.2.3 P l o t t i n g the L o c a t i o n s of Plana m e t r i c D e t a i l s onto the 3D P l o t , 143 6.2.4 Implementation ...................... 146 7.0 H a r v e s t i n g Costs and Wood Volume P r o d u c t i o n ..... 147 7.1 C r e a t i o n of the Wood Volume and S o i l s Data Base 147 7.2 S e t t i n g A n a l y s i s ........................... 149 7.3 Road C o s t i n g .,,..... . . . . . . , 1 5 1 7.4 Implementation ............................. 152 D i s c u s s i o n ...............................................„ 153 CHAPTER 1.0 Economic J u s t i f i c a t i o n of the T e r r a i n Simulator 153 2.0 A d d i t i o n s t o the Current Model 156 2.1 A Hydrology Module 156 2.2 A G r a p h i c a l Information R e t r i e v a l Module 157 2.3 Summary ... .......... ... ... ............. .... 158 C o n c l u s i o n ......... ...................................... 160 L i t e r a t u r e C i t e d ,........ . . . . . . . . . . . 1 6 8 Appendices 170 Appendix A. P o i n t - i n - P o l y g o n Algorithm ...,,.,,,...,171 Appendix B. Menu O p e r a t i o n ......................... 175 Appendix C. Cable Mechanics Theory ................. 178 Appendix D. , P e r s p e c t i v e P l o t t i n g 20 3 i x LIST OF TABLES TABLE PAGE I . System S p e c i f i c a t i o n s of the Hewlett-Packard 9830A Computing System ................................ 6 I I . C o o r d i n a t e s and E l e v a t i o n s of the Four C e l l Corners 52 I I I . Comparison of the E l e v a t i o n E r r o r s from Maps of D i f f e r e n t S i z e s ................................. 59 IV. Comparison of Operation Counts .................... 60 V. R e l a t i v e Execution Times 60 VI. D e f i n i t i o n of Aspect I n d i c e s 90 VII. E r r o r s from the P a r a b o l i c Model with the C a r r i a g e at Midspan f o r a M a d i l l 052 Tension Skidder ..... 103 VI I I . D i f f e r e n c e T a b l e f o r Sag <y) 104 IX. D e f l e c t i o n a t Midspan f o r a M a d i l l 052 Tension Skidder (Chord D e f i n i t i o n ) .............. 107 X. D e f l e c t i o n at Midspan f o r a M a d i l l 052 Tension Skidder (Span D e f i n i t i o n ) ............... 109 XI. Percent Rule E r r o r at Quarter Span f o r a M a d i l l 052 Tension Skidder 112 XII. L i n e Tensions with the C a r r i a g e a t Midspan f o r a M a d i l l 052 Tension Skidder using the P a r a b o l i c Model ....................... 195 X LIST OF FIGURES FIGURE PAGE 1 Hap of the Study Area with the D e f i n i n g Rectangle .... 9 2 E n t e r i n g the Contour L o c a t i o n s f o r a Transect Line ................................... 12 3 Menu f o r the T r a n s e c t Method of E n t e r i n g Contours .... 13 4 Menu f o r the Contour Method o f E n t e r i n g Contours ..... 15 5 Map Overlay f o r the 40 Percent Slope C l a s s e s ......... 18 6 Combination Overlay f o r Areas Between 40 and 79% s l o p e , 500 and 999 Feet E l e v a t i o n and Southeast and Southwest Aspects ............. 19 7 A Cable Yarding System ............................... 21 8 P l o t of Ground P r o f i l e ............................... 23 9 Ground P r o f i l e and Loadpath P l o t s f o r P r o j e c t e d Yarding Roads 25 10 Plan View of P r o j e c t e d Yarding Roads 26 11 P l c t c f Ground P r o f i l e and Loadpath f o r a S i n g l e Yarding Road ............................. 27 12 Plan View P l o t o f Road Network 29 13 Road P r o f i l e s ........................................ 30 14 Road A n a l y s i s Output 31 15 Map Overlay f o r Viewable Areas ....................... 34 16 Determining the I n i t i a l Viewing Angle ................ 35 17 Orthogonal P l o t o f cyp r e s s Bowl P r o v i n c i a l Park ( l o o k i n g north) ................. 37 FIGURE PAGE 18 Planametric Map Showing the Hidden Areas for the Orthogonal Projection of Cypress Bowl Provincial Park ........... ...... ... ............. 39 19 Map of Study Area Showing Covertype Boundaries and Associated Grid Overlay ..................... 41 20 Covertype Map and Associated Species L i s t i n g ......... 42 21 Costing Analyses for Roads and Settings «... 44 22 D e f i n i t i o n of Coordinate System ...................... 51 23 Computing the Elevation for the Point (x,y) Using Three-step Linear Interpolation and Inverse Distance-squared Interpolation .......... 54 24 Accuracy Test Comparing the Three-step Linear and Inverse Distance-squared Techniques for Computing Elevations 57 25 Graph of the Elevation Error as a Function of Map Area .....58 26 Accuracy Test for the Quadratic Interpolation Technique .........................65 27 Double Transect Method for Generating Elevations ..... 69 28 Relationship of Bands, Contour Points and Grid Points 73 29 Using Four Regions to f i n d an Elevation ..............77 30 Comparison of the Accuracy of the Contour and Transect Methods ................................ 81 31 The Surface Plane and i t s Coordinate System .......... 86 32 De f i n i t i o n of the Aspect Classes ..................... 89 ...' x i i FIGURE PAGE 33 Flowchart f o r P l o t t i n g Hap Overlays .................. 93 34 Schematic Drawing of a Running S k y l i n e System ........ 97 35 Loadpath S o l u t i o n f o r a Running S k y l i n e System Using the P a r a b o l i c Model and Newton I t e r a t i o n ............................ 101 36 Using an I n t e r p o l a t i n g Polynomial to P r e d i c t the Loadpath ............................ 106 37 Planametric View of •Automatic' Road L o c a t i o n ........ 117 38 The E f f e c t o f the C o n t r o l l i n g Parameters cn 'Automatic* Road L o c a t i o n .................... 119 39 L o c a t i n g G r i d C e l l s Along the S i g h t L i n e 124 40 Determining the Viewable Area Along a S i g h t L i n e ..... 125 41 R o t a t i o n of One Box of the E l e v a t i o n Matrix .......... 129 42 Flowchart f o r the R o t a t i o n of the E l e v a t i o n Matrix ... 130 43 Viewing C o n t r o l s f o r Orthographic P r o j e c t i o n ......... 133 44 Orthographic P r o j e c t i o n using One Set of P r o f i l e L i n e s and an Angle of R o t a t i o n of 0 degrees ..... 137 45a Orthographic P r o j e c t i o n using Two Sets of P r o f i l e L i n e s and an Angle of R o t a t i o n of 0 degrees ••... 138 45b o r t h o g r a p h i c P r o j e c t i o n using Two Sets of P r o f i l e L i n e s and an Angle of Rotation of 45 degrees .,.,139 45c Orthographic P r o j e c t i o n using Two Sets of P r o f i l e L i n e s and an Angle of R o t a t i o n of 80 degrees .... 140 46 Removing Hidden Areas 142 47 Beternsining Whether a P o i n t i s Within A Polygon ...... 172 x i i i FIGUSE PAGE 48 Determining Whether an I n t e r s e c t i o n Takes Place ...... 173 49 The Henu Coordinate System 176 50 Punning S k y l i n e System 180 51 G r a v i t y S k y l i n e System ............................... 181 52 Highlead System , 182 53 The Three D i f f e r e n t Loading Assumptions f o r the Cable Weight ,184 54 Geometry of a F r e e l y Hanging Cable 185 55 Forces A c t i n g on the Cable Segment OP ................ 187 56 T r a n s l a t i n g the Coordinate System from O ( x * , y ) to A(x,y) ...... 189 57 Geometry of a F i v e L i n e Cable System 191 58 C o n f i g u r a t i o n of the L i n e s at the C a r r i a g e Drum 200 59 Viewing C o n t r o l s f o r the P e r s p e c t i v e P r o j e c t i o n ...... 204 60 P e r s p e c t i v e P r o j e c t i o n . 2 0 6 x i v LIST OF PLATES PLATE PAGE 1. The H e w l e t t - P a c k a r d 9830A Computing System ........... 7 2. / D i g i t i z i n g a T r a n s e c t L i n e - ' . ^ , , w 7 3. Cy p r e s s Bowl P r o v i n c i a l Park ( l o o k i n g n o r th) 38 ACKNOWLEDGE 13 ENT I am g r e a t l y indebted t o Mr. G.G. Young, my s u p e r v i s o r , f o r p r o v i d i n g h i s i n v a l u a b l e a s s i s t a n c e f o r a l l a s p e c t s o f my graduate program. My g r a t i t u d e i s a l s o extended t o the N a t i o n a l Research C o u n c i l of Canada f o r generously p r o v i d i n g f i n a n c i a l support. I would l i k e to thank Dr. P.L. C o t t e l l and Dr. D.H. Wi l l i a m s f o r being on my committee and reviewing the te x t . F i n a l l y , I wish to thank Dr. D. Reimer and Mr. J . Marlow of MacMillan B l o e d e l LTD. f o r the o p p o r t u n i t y t o t e s t the D i g i t a l T e r r a i n Simulator i n an o p e r a t i o n a l environment. I would a l s o l i k e to express my g r a t i t u d e t o the s t a f f , e s p e c i a l l y Mr. A. K e l l y , o f the Company's F r a n k l i n R i v e r Logging D i v i s i o n f o r t h e i r c o o p e r a t i o n and va l u a b l e s u g g e s t i o n s . , 1 I n t r o d u c t i o n Computer models f o r f o r e s t r e s o u r c e p l a n n i n g have been a v a i l a b l e s i n c e t h e e a r l y 1960's, D u r i n g t h e p a s t few y e a r s t h e i r use has a c c e l e r a t e d due, i n p a r t , t o i n c r e a s i n g r e s o u r c e c o n f l i c t s and wood h a r v e s t i n g c o s t s . The nsajor emphasis i n t h e i r development has been towards l o n g - r a n g e p l a n n i n g . Consequently t h e r e has been an absence o f s u i t a b l e models designed f o r o p e r a t i o n a l ( s h o r t - t e r m ) f o r e s t p l a n n i n g . I t i s the o b j e c t i v e o f t h i s t h e s i s to develop a computer model f o r t h i s i m p o r t a n t p l a n n i n g phase. The l o n g range models have commonly been used f o r r e s o u r c e a l l o c a t i o n and i n f o r m a t i o n r e t r i e v a l . The e a r l i e s t example o f the l a t t e r i s Amidon's (1966) g r a p h i c a l i n f o r m a t i o n system. There have been numerous a l l o c a t i o n models de v e l o p e d . Among thes e a r e M a x m i l l i o n (Clutter, 1969), Timber RAM (Resource A l l o c a t i o n Method) (Navcn, 1971) and CARP (Computer A s s i s t e d Resource P l a n n i n g ) by t h e B r i t i s h Columbia F o r e s t S e r v i c e ( W i l l i a m s , e t a l . , 1975). The l a c k o f development work on s h o r t - t e r m p l a n n i n g models was due p a r t l y t o the l i m i t e d computer t e c h n o l o g y a v a i l a b l e . U n t i l r e c e n t l y most computers were l a r g e and r e q u i r e d t i g h t l y c o n t r o l l e d e n v i r o n m e n t s . A l s o , h i g h l y t r a i n e d computer 2 s p e c i a l i s t s were needed to provide an i n t e r f a c e between the user and the computer. These two f a c t o r s precluded the high degree of computer i n t e r a c t i o n t h a t a planner needed to e f f i c i e n t l y d e v ise a f o r e s t r e source development p l a n . The s m a l l desktop computer developed over the l a s t s e v e r a l years has brought the necessary computing power, with the r e q u i r e d i n t e r a c t i v e c a p a b i l i t y , t o the o f t e n remote g e o g r a p h i c a l l o c a t i o n s where the short-term p l a n n i n g i s f r e q u e n t l y done. T h i s has made p o s s i b l e the development of p r a c t i c a l short-term p l a n n i n g models. The United S t a t e s F o r e s t S e r v i c e (OSES), through the P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n i n S e a t t l e , has pioneered development i n t h i s a r e a . Separate programs have been produced t o analyze s k y l i n e design and l o c a t i o n (Carson, 1975), road design and l o c a t i o n (Burke, 1974), and h a r v e s t i n g c o s t s (Burke, 1976). These models are s u i t a b l e f o r the a c t u a l o n - s i t e implementation of h a r v e s t p l a n n i n g . T h e i r use i n the a n a l y s i s of the v a r i o u s a l t e r n a t i v e s i n s h o r t -term plan n i n g , although q u i c k e r than the previous manual approaches, i s s t i l l cumbersome and slow, as they do not use the f u l l p o t e n t i a l of the desktop computer. Map data d e s c r i b i n g the ground c o n d i t i o n s must be entered i n t o the computer f o r each new a l t e r n a t i v e t o be t e s t e d , even i f d i f f e r i n g only s l i g h t l y from the p r e v i o u s t r i a l . T h i s l a c k o f a s i n g l e data base, a v a i l a b l e f o r repeated use, i s i n the author's o p i n i o n , the M g g e s t drawback f o r development p l a n n i n g . An a d d i t i o n a l problem i s that each model i s a separate package, thereby r e d u c i n g t h e i r o v e r a l l c o m p a t i b i l i t y and e f f i c i e n c y . 3 The idea of using a s i n g l e data base, d e f i n e d p r i o r t o a n a l y s i s , f o r o p e r a t i o n a l planning i s not new. C i v i l engineers itave been u t i l i z i n g t h i s concept s i n c e the i n v e n t i o n of the e l e c t r o n i c computer (Meyer, 1969). T h i s form o f planning t o o l i s commonly r e f e r r e d to as a d i g i t a l t e r r a i n model (DTM). Burke (1974a) suggested t h e i r use f o r ge n e r a t i n g s k y l i n e p r o f i l e s . The USES, at the P a c i f i c Southwest F o r e s t and Range Experiment S t a t i o n , has developed a computer package ( f o r a l a r g e computing system) c a l l e d VIEWIT ( T r a v i s e t a l . , 1974) t h a t permits assessment of viewable areas and d e l i n e a t i o n of slope and aspect c l a s s e s . The program can produce o v e r l a y maps showing the vario u s topographic f e a t u r e s . I n d i v i d u a l l y the above planning systems address the a a j o r planning c o n s i d e r a t i o n s . U n f o r t u n a t e l y , because of t h e i r d i f f e r e n t data reguirements and computer hardware, they do not represent an i n t e g r a t e d approach. The computer model d e s c r i b e d i n t h i s t h e s i s i s a TERRAIN SIMULATOR (another name f o r DTM) which was developed at the F a c u l t y o f F o r e s t r y of the U n i v e r s i t y of B r i t i s h Columbia (UBC) (Young and Lemkow, 1976). The o b j e c t i v e of the s i m u l a t o r was to provide a complete short-range planning package f o r a desktop computer. The model uses one data base to permit e f f i c i e n t a n a l y s i s of the v a r i o u s f a c e t s o f the p l a n n i n g p r o c e s s . The f u n c t i o n s of the v a r i o u s planning modules i n the system i n c l u d e : a a) Determination of the topographic f e a t u r e s such as s l o p e , aspect and e l e v a t i o n ; b) Design of s e t t i n g s and placement of y a r d i n g systems; c) L o c a t i o n c f roads; d) D e l i n e a t i o n o f viewable area; e) E s t i m a t i o n of h a r v e s t i n g c o s t s and wood volume production. The t e r r a i n s i m u l a t o r i s d e s c r i b e d i n two p a r t s . , The f i r s t s e c t i o n d e a l s with the g e n e r a l use of the s i m u l a t o r ; i . e . the v a r i o u s f u n c t i o n s i t can perform., No d e t a i l e d attempt i s made to d i s c u s s how t h e s i m u l a t o r would a c t u a l l y be used t o develop a v i a b l e , comprehensive, short-term development p l a n . The second s e c t i o n d e s c r i b e s the t h e o r e t i c a l background f o r each of the modules. 5 Part I - An overview of the T e r r a i n Simulator And I t s A p p l i c a t i o n s The t e r r a i n s i m u l a t o r has been developed on a Hewlett-Packard 9830A desktop computer. System s p e c i f i c a t i o n s and the p h y s i c a l c o n f i g u r a t i o n are given i n Table I and P l a t e 1, r e s p e c t i v e l y . The d i g i t i z i n g u n i t {Plate 2) i s the most important of the v a r i o u s p e r i p h e r a l d e v i c e s . I t permits the r a p i d entry and c o n v e r s i o n of g r a p h i c a l i n f o r m a t i o n (e.g., roads and contour l o c a t i o n s ) i n t o (x,y) c o o r d i n a t e s 1 s u i t a b l e f o r mathematical manipulation by the computer., 1 The u n i t s (Imperial or I n t e r n a t i o n a l System (S.I.)) f o r the c o o r d i n a t e s are dependent on the d i g i t i z e r . , Consequently, when en t e r i n g data from the d i g i t i z e r i n t o the t e r r a i n s i m u l a t o r program the s c a l e must be s p e c i f i e d i n terms of d i g i t i z i n g u n i t s ; e.g., f e e t per d i g i t i z i n g u n i t or meters per d i g i t i z i n g u n i t where the d i g i t i z i n g u n i t s c o u l d be i n c e n t i m e t e r s or i n c h e s . The computations, however, are independent of the u n i t s . For convenience a l l examples employ the I m p e r i a l System. 6 TABLE I SYSTEM SPECIFICATIONS FOB THE HEWLETT-PACKARD 9830A COMPUTING SYSTEM C a l c u l a t o r - Model 9830A - 16K bytes of rea d / w r i t e memory - Matrix ROM (Read Only Memory) - Extended I/O ROM - p l o t t e r C o n t r o l ROM - S t r i n g V a r i a b l e ROM - Advanced Progamming I ROM - Advanced Progamming I I ROM Printer - Model 9866A - 80 characters per l i n e - 250 l i n e s per minute - thermal printing Plotter - Model 9862A - 10" by 15" plo t t i n g surface External Cassette Memory - Model 9865A - 130 feet/minute search speed - maximum of 6 4K bytes storage capacity D i g i t i z e r - Model 986HA - t a b l e s i z e of 36" by 4 8" - t a b l e manufactured by Bendix Plate 2. D i g i t i z i n g a Transect Line 8 1.0 I n i t i a l P r e p a r a t i o n The s i m u l a t o r depends on one data base from which a l l analyses can proceed. For t h i s , an e l e v a t i o n contour map i s r e q u i r e d . A f o r e s t covertype map, with s p e c i e s volumes, i s a l s o needed f o r the C o s t i n g and Wood Volume P r o d u c t i o n Module. The study area i s d e l i n e a t e d on the map by a r e c t a n g l e (Figure 1). At the outset, the user must ensure t h a t the s i z e o f the contained area i s adequate f o r the proper development of h i s pla n , s i n c e no a d d i t i o n s can be made to the data base a t a l a t e r stage. ^ The map p r e c i s i o n w i l l d i c t a t e the l e v e l of a n a l y s i s t h a t can be achieved. G e n e r a l l y , maps of s c a l e 1 i n c h to 400 f e e t , with 25-foot contours, are adequate f o r a l l the plann i n g modules. Maps with the same s c a l e but 100-foot contours should be used with c a u t i o n i n s k y l i n e and road l o c a t i o n s as they possess i n s u f f i c i e n t t o p o g r a p h i c d e t a i l . Maps with s c a l e s o f f e r i n g l e s s d e t a i l (e.g., 20 c h a i n s to the inch) are u s e f u l only f o r topographic f e a t u r e s and v i s u a l assessment, u n l e s s the t e r r a i n i s very uniform. Figure 1. Map of the Study Area with the Defining Rectangle and the 67 Transect Lines 10 2.0 Input o f the E l e v a t i o n Data Base There are a t present two methods f o r o b t a i n i n g the e l e v a t i o n data base: (a) The T r a n s e c t Method; (b) The Contour Method. fhe i n p u t c f the f o r e s t covertype i n f o r m a t i o n i s c o n t r o l l e d by the Harvesting Costs and Hood Volume P r o d u c t i o n Module (Section 7) . 2.1 Transect Method For the t r a n s e c t method, the map i s a l i g n e d on the d i g i t i z i n g t a b l e with the r e c t a n g l e d e s c r i b i n g the area to be analyzed p a r a l l e l t o the t a b l e . With the d i g i t i z e r , the user s e t s the o r i g i n of t i e (x,y) c a r t e s i a n c o o r d i n a t e system a t the lower l e f t c o r n e r of the r e c t a n g l e . The upper r i g h t c o rner i s then e n t e r e d 2 thereby d e f i n i n g the r e c t a n g l e . , The user d r a f t s 67 e q u a l l y - s p a c e d t r a n s e c t l i n e s onto the r e c t a n g u l a r map area (Figure 1) . The user then s p e c i f i e s the contour i n t e r v a l and the i n i t i a l contour e l e v a t i o n of the 2 A l l map p o i n t s are entered from the d i g i t i z e r . The c u r s o r i s placed over the point and the a p p r o p r i a t e button pressed (Plate 2) . 11 c u r r e n t t r a n s e c t l i n e . The p o i n t of i n t e r s e c t i o n i s entered f o r each contour that c r o s s e s the l i n e . The user must s p e c i f y whether the next contour to be c r o s s e d i s u p h i l l , d o w n h i l l or l e v e l with r e s p e c t to the l a s t one entered. T h i s i s accomplished by d i g i t i z i n g a p o i n t at l e a s t one i n c h t o the l e f t of the l i n e f o r "up" (Figure 2) and c o n v e r s e l y , d i g i t i z i n g a p o i n t to the r i g h t f o r "down". The l e v e l c o n d i t i o n i s i n d i c a t e d by an "up" and a "down". The e n t i r e process i s c o n t r o l l e d by a 'MENU * (Figure 3 ) . On completion of a l i n e , the computer i n t e r p o l a t e s between the d i g i t i z e d contour c r o s s i n g s to determine the e l e v a t i o n cf 67 e g u a l l y - s p a c e d p o i n t s along the l i n e . The data base w i l l , t h e r e f o r e , always be a r e g u l a r g r i d of 67x67 (4489) p o i n t s . The data base i s of t h i s r e g u l a r form to allow f o r r a p i d p r o c e s s i n g by the planning modules. This t r a n s e c t method should be u t i l i z e d i f the map i s l a r g e and/or complex as l e s s map d e t a i l w i l l be l o s t than i f using the contour method. The work of d i g i t i z i n g i s t e d i o u s and s u b j e c t to human e r r o r s . However, an important advantage of t h i s method i s t h a t upon completion of the l a s t t r a n s e c t l i n e the data base i s ready f o r use. The map shown i n F i g u r e 1 r e q u i r e d approximately t h r e e hours to d i g i t i z e . 12 upper left corner lower left corner 10,000's-1,000's-0 1 2 3 4 5 6 7 8 9 C 0 E H C N M T 0 D 0 0 M V u P 0 E R L F E M I T E E N E N N T D T U R R V Y L -l's -10fs 100's Figure 3. Menu for the Transect Method of Entering Contours. 14 2.2 Contour Method As i n the t r a n s e c t method, the map must he p r o p e r l y a l i g n e d with the t a b l e . The r e c t a n g l e i s s i m i l a r l y d e f i n e d . Data e n t r y i s s i m p l e r than with the t r a n s e c t method: f o r each contour l i n e the e l e v a t i o n i s entered and then the l i n e t r a c e d . I t i s a l s o p o s s i b l e t o e n t e r s i n g l e p o i n t s (e.g., the e l e v a t i o n of a mountain peak) and t r a n s e c t l i n e s . The ground d i r e c t i o n , i . e . , u p h i l l , d o w n h i l l or l e v e l , along the t r a n s e c t l i n e i s i n d i c a t e d using a menu (Figure 4). The method i s extremely quick and s u b j e c t t o f a r l e s s human e r r o r than the t r a n s e c t approach. The sometimes d i f f i c u l t d e c i s i o n of whether subsequent contours are u p h i l l or d o w n h i l l i s e l i m i n a t e d . Also, the process of f o l l o w i n g a l i n e i s f a r more e f f i c i e n t than e n t e r i n g s p e c i f i c p o i n t s . U n f o r t u n a t e l y , due t o l i m i t e d machine c a p a c i t y , t h i s method can only be used f o r e i t h e r s m a l l maps or those with l i m i t e d c o m p l e x i t y . An a d d i t i o n a l drawback i s t h a t upon completion of the e n t r y , which i s f r e q u e n t l y l e s s than 30 minutes, the data base i s not ready f o r use. The computer must then compile the i r r e g u l a r l y spaced p o i n t s d e s c r i b i n g the contour shapes i n t o the same 67x67 r e g u l a r g r i d p o i n t s as used i n the t r a n s e c t method; a process which can take up t o 12 hours. UPPER LEFT CORNER DOWN LEVEL UP CHANGE CONTOUR INTERVAL ENTER A CONTOUR ENTER A D-LINE MOVE MENU DELETE LAST ENTRY RETURN TO KEYBOARD 0 C 0 M P L E T E D 1 2 3 4 5 6 7 8 9 -LOWER LEFT CORNER Figure 4. Menu for the Contour Method of Entering Contours. 16 3.0 A n a l y s i s of Topographic Features T h i s module a c t s as a g r a p h i c a l i n f o r m a t i o n r e t r i e v a l system f o r ground s l o p e , aspect and e l e v a t i o n . Overlay maps showing these t h r e e topographic components, e i t h e r s e p a r a t e l y o r i n combination, can be obtained. T h i s r e p r e s e n t s a s i g n i f i c a n t improvement over the manual approach c u r r e n t l y used. The p o t e n t i a l uses are v a r i e d , and i n c l u d e : a) D e l i n e a t i o n o f areas of steep s l o p e t h a t c o u l d i n d i c a t e s l o p e i n s t a b i l i t y or presence of rock. B) A n a l y s i s of r e g e n e r a t i o n problems and s p e c i e s s e l e c t i o n f o r p l a n t i n g stock. C) Determination of l o g g i n g season: winter versus summer o p e r a t i o n s . D) P r e d i c t i o n of general equipment a l l o c a t i o n ; e.g., s k y l i n e s i n steep t e r r a i n and ground s k i d d i n g i n f l a t areas. In a d d i t i o n to the o v e r l a y map, a t a b u l a r d i s t r i b u t i o n of acreage by c l a s s e s i s produced., A l l c l a s s i n t e r v a l s are d e f i n e d by the user. Slope c l a s s e s are t r u n c a t e d to even m u l t i p l e s of f i v e percent and e l e v a t i o n s to 100 f e e t . Aspect i s c a t e g o r i z e d i n t o nine c l a s s e s : N, NE, E, SE, S, SW, H, NE and f l a t {less 17 than f i v e percent s l o p e ) . An example of a sl o p e map with 40 percent c l a s s e s i s given i n F i g u r e 5. For combination maps the range of each component must he s p e c i f i e d ; f o r example; ASPECT : southeast, southwest SLOPE : 40 - 79% (could be s p e c i f i e d as 75-79$) ELEVATION : 500 - 999 f e e t The r e s u l t s are shown i n Figure 6. Figure 5. Map Overlay for the 40 Percent Slope Classes. Figure 6. Combination Overlay for Areas Between 40 and 79% Slope, 500 and 999 Feet Elevation and Southeast and Southwest Aspects. 20 4.0 Yarding L o c a t i o n and S e t t i n g Design T h i s module a l l o w s the planner to t e s t the p h y s i c a l f e a s i b i l i t y and p o t e n t i a l of c a b l e y a r d i n g systems i n v a r i o u s l o c a t i o n s on the t e r r a i n base. Economic e v a l u a t i o n s (costs and machine production) are done using the Harvesting C o s t s and Hood Volume Pr o d u c t i o n Module. The y a r d i n g L o c a t i o n and S e t t i n g Design J3odule p r o v i d e s the f o l l o w i n g f u n c t i o n s : - i d e n t i f i c a t i o n of u s e f u l l a n d i n g l o c a t i o n s - e v a l u a t i o n of road spacing - f o r e c a s t i n g of machine a l l o c a t i o n - p r e d i c t i o n of p o t e n t i a l areas of d i f f i c u l t y arding The computer r e q u i r e s the maxisium p o s s i b l e e x t e r n a l y a r d i n g d i s t a n c e and the h e i g h t s of the l a n d i n g and back sp a r s . A l s o r e q u i r e d are the weight of the l o a d (logs p l u s c a r r i a g e ) and the weights and o p e r a t i n g c o n f i g u r a t i o n of the c a b l e s ( F i g u r e 7 ) . To s i m p l i f y c a l c u l a t i o n s , these l a t t e r components (l o a d , c a b l e weights, etc.) are accounted f o r by using the percent d e f l e c t i o n yarder snubbing l i n e span chord between yarder and t a i l h o l d midspan 7. A Cable Yarding System backspar y \ ( t a i l h o l d ) guyline stump r u l e 3 . The user must i n p u t the minimum a t t a i n a b l e midspan d e f l e c t i o n . The loadpath (the path the c a r r i a g e takes over the t e r r a i n ) i s then determined by i n t e r p o l a t i o n (Section 4.2). There are thr e e types of a n a l y s i s a v a i l a b l e : (1) The generat i o n of ground p r o f i l e s between any two map p o i n t s l o c a t e d with the d i g i t i z e r ( F i g u r e 8). (2) The d e l i n e a t i o n of area a c c e s s i b l e by a yardi n g system from a c e r t a i n l o c a t i o n ; e.g., a l a n d i n g or t a i l h o l d . The system s p e c i f i c a t i o n s (as s t a t e d above) are needed by the computer along with the l a n d i n g l o c a t i o n (which i s entered from the d i g i t i z e r ) . The computer then p r o j e c t s up t o 12 yarding roads, a r b i t r a r i l y spaced 30 degrees a p a r t . The user can, however, choose a subset o f these 12. I t i s not the purpose of t h i s f u n c t i o n to p r o j e c t the a c t u a l yarding roads t o be used. For each road, the maximum p h y s i c a l l y p o s s i b l e e x t e r n a l y a r d i n g d i s t a n c e i s determined. T h i s i s achieved by p l a c i n g the t a i l h o l d a t the u s e r - s p e c i f i e d maximum yarding d i s t a n c e from the l a n d i n g . 3 The percent d e f l e c t i o n i s d e f i n e d as the r a t i o between the v e r t i c a l d i s t a n c e from tbe chord t o the c a r r i a g e a t midspan and the l e n g t h of the chord (Figure 7). The ' r u l e * i s the minimum a t t a i n a b l e d e f l e c t i o n f o r a s p e c i f i c y a r d i n g system. Elevation (feet) 700 £00 300 | 1 - 1 1 1 - 1—— H— 1 200 M00 E00 B00 1000 1200 1H00 Horizontal Distance (feet) Figure 8 . Plot of Ground P r o f i l e . 24 Load c l e a r a n c e * w i t h t h e g r o u n d i s t h e n c h e c k e d f o r t h e e n t i r e s p a n . I f c l e a r a n c e i s l a c k i n g then the t a i l h o l d i s moved t o w a r d s t h e l a n d i n g and t h e p r o c e s s r e p e a t e d . A p r o f i l e p l o t i s p r o d u c e d o f t h e y a r d i n g r o a d { F i g u r e 9) when t h e b a c k s p a r has been p r o p e r l y l o c a t e d . A p l a n v i e w p l o t ( F i g u r e 10) i s done on c o m p l e t i o n o f a l l t h e y a r d i n g r o a d s . T h i s p l o t c a n be used a s an o v e r l a y t o h e l p d e f i n e t h e s e t t i n g b o u n d a r y on t h e map. <3) O s e r - s p e c i f i e d y a r d i n g r o a d s c a n be e x amined., A g a i n , t h e s y s t e m s p e c i f i c a t i o n s a r e needed a l o n g w i t h t h e l a n d i n g and t a i l h o l d l o c a t i o n s . The c o m p u t e r , a t t h e u s e r ' s d i s c r e t i o n , w i l l e i t h e r move t h e t a i l h o l d t o o b t a i n c l e a r a n c e ( a s d i s c u s s e d above) o r l e a v e i t as i n i t i a l l y l o c a t e d . A p r o f i l e o f t h e l o a d p a t h and t h e g r o u n d l i n e i s shown i n F i g u r e 11. * The minimum a c c e p t a b l e c l e a r a n c e i s d e f i n e d a s b e i n g z e r o f e e t between t h e g r o u n d and t h e c a r r i a g e . The l e n g t h o f t h e c h o k e r s can be a c c o u n t e d f o r by s u b t r a c t i n g t h e i r l e n g t h from b o t h s p a r h e i g h t s . elevation ( f t ) 7S 2BB 2BB H r H 1- -+- •+- H a i a i c sa | 0 - yarding road number 2ZB -I—h 16 H a 20 | j horizontal (stations) l 2 Figure 9 . Ground P r o f i l e and Loadpath Plots for Projected Yarding Roads. ro Figure 11. Plot of Ground P r o f i l e and Loadpath for a Single Yarding Road. 28 5.0 Road L o c a t i o n T h i s nodule i s t y p i c a l l y used i n c o n j u n c t i o n with the yarding module. A common use i s to check whether v a r i o u s l a n d i n g l o c a t i o n s can be connected by a road system. P l a n view (Figure 12) and p r o f i l e ( F i g u r e 13) p l o t s along with s i d e s l o p e s , e l e v a t i o n s and grades (Figure 14) can be e a s i l y generated f o r any road l o c a t i o n . Road l o c a t i o n s are entered i n t o the computer i n two ways. In both methods the maximum a l l o w a b l e adverse and f a v o u r a b l e road grades are r e g u i r e d by the program: (1) Two c o n t r o l p o i n t s are s e l e c t e d (e.g. l a n d i n g s , switchbacks, j u n c t i o n s ) . The computer then l o c a t e s the average g r o u n d l i n e , s u b j e c t to the grade c o n s t r a i n t s . The method employed i s s i m i l a r t o manually p r o j e c t i n g roads from a map with d i v i d e r s . (2) The user determines the l o c a t i o n i n t e r a c t i v e l y . P o i n t s are entered along the proposed route with the d i g i t i z e r ( F i g u r e 12). As each s u c c e s s i v e p o i n t i s entered the grade i s computed. I f the grade exceeds a l l o w a b l e l i m i t s an a u d i b l e warning i s given and a new l o c a t i o n i s expected. Grade s p e c i f i c a t i o n s can be e a s i l y o v e r - r i d d e n i n d i f f i c u l t t e r r a i n . RDRt> 2 2KB « I 553 1*3 213 3E Horizontal (Stations) RDB£> t Horizontal (Stations) Figure 13. Road P r o f i l e s 31 ROAD 1 STATION 0 163 364 536 674 839 1047 1172 1361 1466 1528 1665 1750 1830 1947 ELEVATION (FEET) 1 29 66 98 100 100 101 123 165 186 197 199 200 200 200 SIDESLOPE (PERCENT) 0 30 24 1 0 1 6 31 33 19 3 0 0 5 0 GRADE (PERCENT) 17 18 18 0 0 0 17 22 20 17 1 0 0 0 ROAD 2 STATION 0 201 446 626 672 752 965 1102 1348 1509 1685 1789 1975 2165 2357 2514 2664 2893 3092 3259 ELEVATION (FEET) 168 19 8 200 202 200 207 204 225 238 248 235 236 270 296 300 319 348 395 397 406 SIDESLOPE (PERCENT) 44 3 6 22 8 15 2 48 53 49 43 45 57 17 4 33 76 46 48 14 GRADE (PERCENT) 14 0 1 -5 8 -2 15 5 6 -8 0 18 13 2 12 18 20 1 5 F i g u r e 14. Road A n a l y s i s Output. , (Note: computer output has been re-typed f o r c l a r i t y ) 32 No earthwork volume calculations are done as the terrain data are not sufficiently accurate to warrant such detailed analysis. 33 6.0 a e s t h e t i c A n a l y s i s Two forms of a e s t h e t i c a n a l y s i s are a v a i l a b l e : viewable area ( s i m i l a r to VIEwTI) and p r o j e c t i o n o f three dimensional t e r r a i n r e p r e s e n t a t i o n s onto two dimensional s u r f a c e s 5 . 6.1 Viewable Area The l o c a t i o n and e l e v a t i o n o f the viewing p o i n t , which can be o u t s i d e the data base, are e n t e r e d . The computer then produces an o v e r l a y map showing the viewable area (Figure 15) , and a t a b u l a r summary of the a s s o c i a t e d acreage. S e v e r a l d i f f e r e n t viewpoints can be combined to produce one o v e r l a y . T h i s i s u s e f u l when a n a l y z i n g the viewable area from a road. P l a n a m e t r i c d e t a i l s (roads, s e t t i n g s , etc.) can be p l o t t e d on the o v e r l a y as i n Fi g u r e 15. An i n i t i a l viewing angle can be s p e c i f i e d i n cases where the viewpoint i s p a r t i a l l y screened by a timber edge or cutbank (Figure 16); t h i s i s commonly used when the viewpoint i s o u t s i d e tbe map. 5 Throughout the t e x t the phrase ' t h r e e d i m e n s i o n a l r e p r e s e n t a t i o n * i s used t o d e f i n e a th r e e dimensional r e p r e s e n t a t i o n t h a t i s produced on a two dimensional s u r f a c e . Figure 16. Determining the Initial Viewing Angle ui 36 6.2 Three Dimensional Representations T h i s f e a t u r e produces a three dimensional (3D) r e p r e s e n t a t i o n (pseudo-photograph) of the data base. The viewing d i r e c t i o n s along with the area of the data base being viewed are i n p u t . No p r e c i s e viewpoint i s used as the p r o j e c t i o n i s o r t h o g r a p h i c and not p e r s p e c t i v e . T h i s l i m i t a t i o n i s due to the slow execution time of the computer coupled with the s l i g h t l y more complicated a l g o r i t h m r e q u i r e d to produce p e r s p e c t i v e . P e r s p e c t i v e p l o t t i n g c o u l d be implemented i f deemed necessary and i s d i s c u s s e d i n the t h e o r e t i c a l s e c t i o n . The d i f f e r e n c e s between the two methods are not n o t i c e a b l e when viewing from a long d i s t a n c e as shown i n F i g u r e 17 and P l a t e 3. The area i s c y p r e s s Bowl P r o v i n c i a l Park, north of Vancouver. A planametric map can a l s o be produced showing tbe l o c a t i o n s o f the hidden areas i n the 3D p l o t (Figure 18). Planametric d e t a i l s can be p l o t t e d onto e i t h e r the hidden area map or the 3D p l o t , a i d i n g i n the assessment of the v i s u a l impact of d i s t u r b a n c e s . Plate 3. Cypress Bowl Pro v i n c i a l Park (looking north). Figure 18. Planametric Map Showing the Hidden Areas f o r the Orthogonal P r o j e c t i o n of Cypress Bowl P r o v i n c i a l Park. 7.0 H a r v e s t i n g Costs and Wood Volume Pr o d u c t i o n T h i s module performs an economic assessment of a l o g g i n g plan, u s i n g the l o c a t i o n s d e r i v e d from the yarding and road design s e c t i o n s , c o s t s and p r o d u c t i v i t y can he estimated f o r the various l a y o u t s . The r e s u l t s can be used f o r comparisons of l a y o u t a l t e r n a t i v e s . , An i n v e n t o r y data base must he s e t up before a n a l y s i s can proceed. The v a r i o u s covertype l o c a t i o n s along with t h e i r a s s o c i a t e d s p e c i e s volumes must he entered i n t o the computer. The data base i s s t r u c t u r e d on a g r i d system, the map being d i v i d e d i n t o r e c t a n g u l a r c e l l s . , One p o i n t i s d i g i t i z e d i n each c e l l whose c e n t r o i d l i e s w i t h i n the covertype c u r r e n t l y being entered i n t o the data base ( F i g u r e 19). A s i m p l i f i e d covertype map (Figure 20) i s produced to v e r i f y t h a t the e n t r y i s c o r r e c t ; e r r o r s can e a s i l y be c o r r e c t e d . The r e s u l t i n g data base i s s t o r e d on a data tape. S o i l / l a n d f o r m i n f o r m a t i o n , not p r e s e n t l y used i n the t e r r a i n s i m u l a t o r , c o u l d be s i m i l a r l y handled. A n a l y s i s can now proceed on any s e t t i n g . At present, production data are a v a i l a b l e f o r only t h r e e c a b l e systems (the M a d i l l 052 t e n s i o n s k i d d e r , 90-foot spar h i g h l e a d system and the l l a d i l l g r a p p l e y a r d e r ) . Others c o u l d e a s i l y be added so t h a t v i r t u a l l y any l o g g i n g system c o u l d be s i m u l a t e d . F i r s t , the l a n d i n g l o c a t i o n i s d i g i t i z e d . Then, the s e t t i n g area i s entered using the same g r i d system employed f o r e n t e r i n g the cell 1 6 11 16 21 26 31 subscripts 36 AAAABBBBBBBBBBBBBBBBDHHHHHHHHHHHHHHH COVERTYPE LISTING SPECIES % VOLUME 35 AAABBBBBBBBBBBBBBBBDDDHHHHHHHHHHHHHH 34 AAABBBBBBBBBBBBBBBBDDDHHHKHHHHHHHHHH COVERTYPE A TOTAL VOLUME = 100 33 AAABBBBBBBBBBBBBBBDDDDDHHHHHHHHHHHHH (100's cu.ft.) . HEMLK 50 32 AAB BBBBBBBBBBBBBC CDDDDDHHEHHHHHHHHHH BALSM 30 31 ABBBBBBBBBBBBBBBCCDDDDDDHKHHHHHHHHHH CEDAR 20 30 ABBBBBBBBBBBBBBCCCCDDDDDHHHHHHHHHHHH 29 BBBBBBBBBBBBBBCCCCCDDDDDHHHHHHHHHHHH COVERTYPE B TOTAL VOLUME = 9 0 28 BBBBBBBBBBBBBCCCCCCCDDDDDHHHHHHHHHHH FIR 40 27 BBBBBBBBBBBBCCCCCCCCDDDDDHHHHHHHHHHH HEMLK 40 26 BBBBBBBBBBBCCCCCCCCCDDDDDDHHHHHHHHHH CEDAR 20 25 BBBBBBBBBBBCCCCCCCCCCDDDDDHHHHHHHHHH 24 BBBBBBBBBBCCCCCCCCCCCDDDDDHHHHHHHHHH COVERTYPE C TOTAL VOLUME = 1 2 0 23 BBBBBBBBBBCCCCCCCCCCDDDDDDHHHHHHHHHH HEMLK 50 22 BBBBBBBBBBCCCCCCCCCCDDDDDDHHHHHHHHHH CEDAR 25 21 BBBBBBBBBBCCCCCCCCCCDDDDDDHHHHHHHHHH FIR 25 20 BBBBBBBBBCCCCCCCCCCCDDDDDDHHHHHHHHHH 19 BBBBBBBBBCCCCCCCCCCCDDDDDDHHHHHHHHHH COVERTYPE D TOTAL VOLUME = 1 5 0 18 BBBBBBBBCCCCCCCCCCCCDDDDDDHHHHHHHHHH FIR 60 17 BBBBBBBCCCCCCCCCCCCCDDDDDDHHHHHHHHHH CEDAR 40 16 BBBBBBCCCCCCCCCCCCCCDDDDDDHHHHHHHHHH COVERTYPE E TOTAL VOLUME = 120 15 BBBCCCCCCCCCCCCCCCCCDDEEET.HHHHHHHHHG CEDAR 70 14 BBCCCCCCCCCCCCCCCCCCDEEEEEHHHHHHHGGG HEMLK 30 13 BCCCCCCCCCCCCCCCCCCDDEEEEEHHHHHHGGGG 12 BCCCCCCCCCCCCCCCCCCDDEEEEF.HHHHHGGGGG COVERTYPE F TOTAL VOLUME = 70 11 CCCCCCCCCCCCCCCCCCCDDEEEEEHHHHHGGGGG HEMLK 60 10 CCCCCCCCCCCCCCCCCCDDDEEEEEEHHHGGGGGG CEDAR 40 9 CCCCCCCCCCCCCCCCCCDDDEEEEEEEEEGGGGGG 8 CCCCCCCCCCCCCCCCCDDDDEEEEEEEEEGGGGGG COVERTYPE G TOTAL VOLUME = 0 7 CCCCCCCCCCCAAAACCDDDDEEEEEEEEEEGGGGG 6 CCCCCCCCCAAAAAAAADDDEEEEEEEEEEEGGGGG COVERTYPE H TOTAL VOLUME = 120 5 CCCCCCCCAAAAAAAAADDDEEEEEEEEEEEGGGFF HEMLK 50 4 CCCCCCCCAAAAAAAAAADEEEEEEEEEEEEGGGFF CEDAR 40 3 CCCCCCCAAAAAAAAAAEEEEEEEEEEEEEGGGGFF CYPRS 10 2 CCCCCCCAAAAAAAAAAEEEEEEEEEEEEGGGGGGF 1 CCCCCCCAAAAAAAAAAEEEEEEEEEGGGGGGGGGG Figure 20. Covertype Map and Associated Species Listing. (Note: computer output has been re-typed) 43 covertypes. The computer then s i m u l a t e s the y a r d i n g of the wood i n each s e t t i n g c e l l to the l a n d i n g . The average slope and y a r d i n g d i s t a n c e are computed and used to p r e d i c t the p r o d u c t i v i t y r a t e f o r that c e l l . S o i l / l a n d f o r m i n f o r m a t i o n would prove u s e f u l at t h i s stage as i t w i l l i n d i c a t e g e n e r a l ground c o n d i t i o n s . On completion o f the s e t t i n g , the acreage, volume by s p e c i e s , average y a r d i n g d i s t a n c e , average s l o p e , t o t a l yarding c o s t , number of s h i f t s and the percentage l o s s of the normal p r o d u c t i o n r a t e (due to d i f f i c u l t yarding) are a l l computed and p r i n t e d (Figure 2 1 ) . Another f u n c t i o n of the module i s the e s t i m a t i o n of road c o n s t r u c t i o n c o s t s . The p r o j e c t e d road i s t r a c e d using the d i g i t i z e r . The s i m u l a t o r computes the l e n g t h and a p p l i e s an average cost per m i l e g i v i n g the road c o s t s (Figure 2 1 ) . Once again s o i l i n f o r m a t i o n would enhance the e s t i m a t e s . ACRES = 127. 4 SPECIES FIR 2911.0 56 33.0 267.0 5371.0 259.0 VOLUME (CUNITS) CEDAR CYPfiS HEMLK BALSM TOTAL: 14441.0 AVE VOL/ACRE = 113.3 CUNITS MACHINE: SKIDDER # SHIFTS = 205.3 YARDING COST = $ 265065.5 AVE % PRODUCTION LOSS =8.1 AVE YARDING DISTANCE = 928.2 FEET AVE SLOPE = 22.9 PERCENT 80AD LENGTH =0.4 COST 2>$55,000/MILE = $19924. Fi g u r e 21. Cos t i n g Analyses f o r Roads and S e t t i n g s . (Note: computer output has been re-typed f o r c l a r i t y ) 45 P a r t I I T e r r a i n Simulator Theory Part II of the t h e s i s o u t l i n e s the important theory of each module i n the t e r r a i n s i m u l a t o r . Supporting examples used a r e , of n e c e s s i t y , dependent on the HP9830A computer. The r e f o r e , at the end of each s e c t i o n i s a b r i e f d i s c u s s i o n of the implementation of the theory on t h i s system. A l l a n a l y s e s , i r r e s p e c t i v e of the module, are i n part dependent on e l e v a t i o n s , making the s t r u c t u r e of the e l e v a t i o n data base c r i t i c a l to the development of the modules. ; Consequently i t i s necessary t o d e s c r i b e the nature o f the e l e v a t i o n data base p r i o r to proceeding t o the r e s t of the s i m u l a t o r . 46 1.0 E l e v a t i o n Data Base The e l e v a t i o n data base must provide f o r quick, accurate r e t r i e v a l of e l e v a t i o n s f o r any map l o c a t i o n . In g e n e r a t i n g the e l e v a t i o n s a t r a d e o f f e x i s t s between speed and accuracy, as g e n e r a l l y more computation i s r e q u i r e d to achieve g r e a t e r accuracy. There are two l e v e l s of accuracy r e q u i r e d by the v a r i o u s modules: a) Hiqh l e v e l accuracy: E l e v a t i o n s f o r s p e c i f i c (x,y) l o c a t i o n s on the data base are r e q u i r e d to the p r e c i s i o n of the map and consequently speed w i l l be s a c r i f i c e d . Examples i n c l u d e s k y l i n e ground p r o f i l e s , road p r o f i l e s , c o s t i n g and p r o d u c t i v i t y e s t i m a t i o n and viewable area assessment. b) Low l e v e l accuracy: approximate e l e v a t i o n s , t h a t are r e p r e s e n t a t i v e o f a c e r t a i n v i c i n i t y , are s u f f i c i e n t t o c l a s s i f y the t o p o g r a p h i c f e a t u r e s and produce three dimensional p l o t s . Although the accuracy i s l e s s , r e t r i e v a l time i s more r a p i d . In both cases the accuracy i s measured not only by how w e l l the data base d e s c r i b e s the map but a l s o the a b i l i t y of the map to d e s c r i b e the ground. Even though map-making o r g a n i z a t i o n s are r e g u i r e d t o contour maps t o c e r t a i n standards (Meyer, 1969) the accuracy of the map can be extremely v a r i a b l e and d i f f i c u l t to p r e d i c t . Although c r u c i a l , no comprehensive a n a l y s i s of t h i s 47 p r o b l e m i s p r o v i d e d h e r e . The model t h e r e f o r e c o n s i d e r s t h e map t o be c o r r e c t . Two b a s i c d a t a s t r u c t u r e s can be employed t o s t o r e t h e e l e v a t i o n d a t a f o r s u b s e q u e n t u s e : 1) R e t e n t i o n o f c o n t o u r l o c a t i o n s ; 2) C o n v e r s i o n o f c o n t o u r s i n t o a g r i d o f e l e v a t i o n s . 1.1 R e t e n t i o n o f C o n t o u r L o c a t i o n s . The <x,y) c o o r d i n a t e s d e s c r i b i n g t h e s h a p e o f e a c h map c o n t o u r a r e r e t a i n e d i n t h e d a t a b a s e . C o n s e q u e n t l y , t h e d a t a b a s e s h o u l d p r o v i d e r e s u l t s t h a t a r e a s a c c u r a t e a s t h e map i t s e l f a s t h e r e i s v i r t u a l l y no i n f o r m a t i o n l o s s . U n f o r t u n a t e l y a l a r g e amount o f f a s t - a c c e s s s t o r a g e i s r e q u i r e d . The r e t r i e v a l o f e l e v a t i o n s i s a l s o s l o w and d i f f i c u l t . T h ese two drawbacks make i m p l e m e n t a t i o n on a d e s k t o p s y s t e m i n f e a s i b l e . T h e r e f o r e , t h i s a p p r o a c h i s o n l y o u t l i n e d and no d e t a i l e d d i s c u s s i o n w i l l be g i v e n . F i n d i n g t h e e l e v a t i o n o f an fx,y) p o i n t on t h e map i s d i f f i c u l t b e c a u s e t h e two c o n t o u r l i n e s t h a t b r a c k e t (x,y) must be l o c a t e d . The i n t e r v a l d e f i n e d by t h e two c o n t o u r l i n e s c a n he c o n s i d e r e d a c l o s e d p o l y g o n . To d e c i d e i f a p o i n t i s c o n t a i n e d by t h e p o l y g o n i t i s n e c e s s a r y t o c h e c k i t a g a i n s t e a ch l i n e segment t h a t makes up t h e p o l y g o n b o u n d a r y . T h i s 48 r e q u i r e s a p o i n t - i n - p o l y g o n r o u t i n e . An a l g o r i t h m i s given i n appendix A. The method i s slow because of the exhaustive t e s t made of the boundary. Once the proper i n t e r v a l i s l o c a t e d the e l e v a t i o n i s computed by i n t e r p o l a t i n g between the contour l i m i t s . A desktop computer cannot r e a l i s t i c a l l y handle t h i s t a s k as the slow e x e c u t i o n w i l l s e r i o u s l y d e t r a c t from the i n t e r a c t i v e nature of the model. An a l t e r n a t i v e approach i s t o d i s r e g a r d the contour shapes and simply i n t e r p o l a t e between the c l o s e s t (x,y) contour p o i n t s e l i m i n a t i n g the need t o l o c a t e the c o n t a i n i n g contour i n t e r v a l . S i nce the contour shapes are disregarded the c l o s e s t contour p o i n t s employed i n the i n t e r p o l a t i o n may not l i e i n the proper contour i n t e r v a l , r e s u l t i n g i n decreased accuracy. For example, i f an (x,y) p o i n t i s between the 2900 and 3000-foot contours, some of the c l o s e s t contour p o i n t s used, may have e l e v a t i o n s greater or l e s s than 3000 or 2900 f e e t , r e s p e c t i v e l y . T h i s type of e r r o r w i l l commonly occur i n steep t e r r a i n where the contour l i n e s are c l o s e together. The major time component i n the two approaches i s l o c a t i n g e i t h e r the b r a c k e t i n g contours or the c l o s e s t contour p o i n t s . , T h i s search time can be d r a s t i c a l l y reduced by i n i t i a l l y s u b d i v i d i n g the map i n t o a g r i d of s m a l l c e l l s . I t i s then necessary to examine only the s e c t i o n of the data base that l i e s w i t h i n the a p p r o p r i a t e c e l l (and i t s e i g h t adjacent neighbours) 49 that c o n t a i n s the {x, y) p o i n t f o r which the e l e v a t i o n i s being sought. Even t h i s improvement i s not s u f f i c i e n t f o r repeated use as the process i s s t i l l too slow. fin a d d i t i o n a l problem i s the i n c r e a s e d memory r e q u i r e d because of the more e l a b o r a t e data s t r u c t u r e . 1.2 Conversion of Contours i n t o a G r i d of E l e v a t i o n s In t h i s method, the contour i n f o r m a t i o n i s condensed t o form a r e g u l a r g r i d of e l e v a t i o n s . The advantages l i e i n f a s t r e t r i e v a l {for both the *high* and 'low* l e v e l s of accuracy) and r e l a t i v e l y s m a l l memory reguirements making i t s u i t a b l e f o r desktop computers. The b a s i s of the method l i e s i n d i v i d i n g the map area i n t o egual r e c t a n g u l a r c e l l s . Each c e l l s u r f a c e i s d e s c r i b e d mathematically by e i t h e r an i n t e r p o l a t i o n r o u t i n e t h a t uses the c e l l corner e l e v a t i o n s , or, by a s u r f a c e eguation which i s t y p i c a l l y some form of geometric plane. The i n t e r p o l a t i o n approach r e q u i r e s more computational work to generate an e l e v a t i o n but storage requirements are l e s s as only one e l e v a t i o n per c e l l need be r e t a i n e d . The e q u i v a l e n t s u r f a c e equation, {computed only once at the i n p u t stage u s i n g the same g r i d of e l e v a t i o n s as the i n t e r p o l a t i o n method) i s simple to e v a l u a t e , but each of the parameters must be r e t a i n e d thereby i n c r e a s i n g s t o r a g e requirements. 50 The d e f i n i t i o n of the c o o r d i n a t e system i s the same f o r both methods. The r e c t a n g l e , d e f i n i n g the map area, Jsas i t s o r i g i n (0,0) a t i t s lower l e f t corner (Figure 22). The d i a g o n a l l y opposite c o r n e r (upper r i g h t ) i s d e f i n e d by the c o o r d i n a t e p a i r (u,v). The c o o r d i n a t e s are expressed i n d i g i t i z e r u n i t s ( u s u a l l y i n c h e s ) . The r e c t a n g l e c o n t a i n s n 2 c e l l s , each (u/n) (v/n) i n s i z e . The n 2 c e l l s are d e s c r i b e d by a g r i d of (n+1) 2 e l e v a t i o n s with one e l e v a t i o n at each of the four c e l l c o r n e r s . These e l e v a t i o n s are s t o r e d i n a matrix E{j,k) where j i s the x-c o o r d i n a t e and k the y - c o o r d i n a t e . For an (x,y) p o i n t i n the data base the s u b s c r i p t s f o r the c o n t a i n i n g c e l l a r e : j=[xn/u]+1 k=[yn/v]*1 .......... (1.1) where [ J r e p r e s e n t s the n e a r e s t lower i n t e g e r value The s u b s c r i p t s (j,k) a l s o d e f i n e the c e l l ' s lower l e f t e l e v a t i o n , E ( j , k ) . The corresponding l o c a t i o n of (j,k) i n d i g i t i z e r u n i t s i s (s,t) with: s=(j-1)u/n t=(k-1)v/n (1.2) The c o o r d i n a t e s and e l e v a t i o n s f o r each o f the four c e l l c o r n e r s are given i n Table I I . 52 TABLE I I COOHDIN ATES AND ELEVATIONS OF THE FOO.fi CELL C0SNE8S c o r n e r I c o o r d i n a t e s e l e v a t i o n s 1 lower l e f t 2 upper l e f t 3 upper r i g h t 4 lower r i g h t is,t) ( s , t * v / n ) (s+u/n,t+v/n) (s+u/n,t) E<j,k) E*j,k-*1) E ( j + 1,k*1) E(j+1,k) 1.2.1 I n t e r p o l a t i o n Method f o r F i n d i n g E l e v a t i o n s An e l e v a t i o n f o r an (x,y) l o c a t i o n can be d etermined by i n t e r p o l a t i n g between the e l e v a t i o n s f o r t h e c l o s e s t c e l l c o r n e r s i n the d a t a base. The s u b s c r i p t s f o r t h e c e l l t h a t c o n t a i n s (x,y) a r e found u s i n g E g u a t i o n (1.1). The number o f g r i d p o i n t s used w i l l depend on t h e t e c h n i g u e employed. To p r e v e n t b i a s a s y m m e t r i c a l s e t of g r i d p o i n t s s h o u l d be used; e.g., t h e f o u r c e l l c o r n e r e l e v a t i o n s . U s i n g more than f o u r p o i n t s r e q u i r e s u n a c c e p t a b l e amounts of c o m p u t a t i o n time w i t h l i t t l e g a i n i n a c c u r a c y ( S e c t i o n 1.2.2). The two most a p p r o p r i a t e methods developed i n terms o f speed and a c c u r a c y , when u s i n g the f o u r c e l l c o r n e r e l e v a t i o n s , were: 53 1) Three-step l i n e a r i n t e r p o l a t i o n to form a weighted average of the f o u r data p o i n t s 2) Weighted average using ' i n v e r s e d i s t a n c e - s q u a r e d law' Three-Step L i n e a r I n t e r p o l a t i o n f o r Computing E l e v a t i o n s Three s t e p s , each using l i n e a r i n t e r p o l a t i o n , are re q u i r e d t o generate an e l e v a t i o n f o r a po i n t (x,y) (Figure 23) . 1) fin estimate, e^, i s made of the e l e v a t i o n a t (s,y) using E(j,k) and E (j,k*1) : e =E<;j,k)+ (E(j,k + 1)-E{j,k) ) Cy-t)/(v/n) 2) S i m i l a r l y an es t i m a t e , e 2 , i s made of the e l e v a t i o n a t (s + u/n, y) using E (j+1,k*1) and E ( j + 1,k): e 2=E{j* 1 ,k).• (E ( j * 1,k +1) -E ( j * 1 ,k)) (y-t) / (v/n) 3) F i n a l l y , the e l e v a t i o n i s estimated at (x,y) by using the two e l e v a t i o n s , and e 2 # t h a t l i e on the l i n e Y=y (defined by (s,y) and (s+u/n,y)): e_ = e , M e , - e , ) (x-s)/(u/n) Figure 23. Computing the Elevation f or the Point (x,y) Using 3-step Linear I n t e r p o l a t i o n and Inverse Distance-squared I n t e r p o l a t i o n . 55 Inverse Distance-squared I n t e r p o l a t i o n f o r Computing E l e v a t i o n s The e l e v a t i o n f o r a po i n t (x,y) i s found by averaging the four c e l l c o r n e r e l e v a t i o n s by the i n v e r s e of t h e i r distance-sguared t o (x,y) thereby g i v i n g l e s s weight to those c e l l c o r n e r s f u r t h e s t away from (x,y) (Figure 23). Computing the di s t a n c e - s q u a r e d t o the c o r n e r s ; D2= (x-s) 2+ (y-t) 2 (lower l e f t corner) D|= (x-s) 2+ (y-t-v/n ) 2 (upper l e f t ) D*=(x-s-u/n) 2+<y-t-v/n) 2 (upper r i g h t ) Df= (x-s-u/n) 2 + ( y - t ) 2 (lower r i g h t ) 4 The e l e v a t i o n i s then; e =(E{j,k ) / D 2 + E{j,k+1) /D2 + E ( j+1,k+1)/D2 x,y i / o + E(j*1,k) / D 2)/D Where D i s the sum of the r e c i p r o c a l s : D={1/D2*1/D2+1/D2+1/D2) Comparison of the Two I n t e r p o l a t i o n Methods The accuracy was t e s t e d by t r a c i n g a contour l i n e with the d i g i t i z e r and, f o r each p o i n t entered, computing the e l e v a t i o n using both methods. The e r r o r i n e l e v a t i o n a t each p o i n t was determined by s u b t r a c t i n g the a c t u a l contour e l e v a t i o n from the c a l c u l a t e d v a l u e . A sample s i z e o f 500 p o i n t s was taken from f i v e maps of 56 d i f f e r i n g s i z e s b u t o f i d e n t i c a l s c a l e s . , F i g u r e 24 i l l u s t r a t e s a s u b - s a m p l e o f 20 p o i n t s o b t a i n e d f r o m d i g i t i z i n g one c o n t o u r . The e l e v a t i o n e r r o r s f o r e a c h o f t h e 500 p o i n t s were p l o t t e d a g a i n s t map s i z e s ( F i g u r e 2 5 ) . T h e s e r e s u l t s a r e a l s o g i v e n i n T a b l e I I I . T h e s e show t h e o b v i o u s , w i t h a f i x e d number o f e l e v a t i o n g r i d p o i n t s t h e e r r o r and i t s v a r i a n c e w i l l i n c r e a s e w i t h t h e map s i z e . The m agnitude o f t h e e r r o r s w i l l depend on map s c a l e , c o m p l e x i t y o f t h e c o n t o u r s , q u a l i t y o f t h e map d r a f t i n g and t h e a c c u r a c y o f t h e o p e r a t o r » s d i g i t i z i n g . The d i s t a n c e - s q u a r e d and t h r e e - s t e p a p p r o a c h e s y i e l d e d a l m o s t i d e n t i c a l r e s u l t s . . T h i s was c o n f i r m e d s t a t i s t i c a l l y by t e s t i n g t h e n u l l h y p o t h e s i s t h a t t h e d i f f e r e n c e i n t h e mean e r r o r f o r the two methods, u s i n g p a i r e d o b s e r v a t i o n s , was z e r o . I t was assumed t h a t any d i f f e r e n c e s between t h e two methods was n o t d e p e n d e n t on map s i z e . The z - v a l u e was t e s t e d a g a i n s t a f i v e p e r c e n t c r i t i c a l l i m i t : n _ z=((E;(x.-y. ) ) / n ) / ( a / / n ) 1=1 1 1 w i t h x.: e r r o r f o r t h e i o b s e r v a t i o n u s i n g t h e t h r e e - s t e p method t i l y±: e r r o r f o r t h e i o b s e r v a t i o n u s i n g t h e d i s t a n c e - s q u a r e d method a : s t a n d a r d d e v i a t i o n o f t h e d i f f e r e n c e s n ; sample s i z e The v a l u e of z was 1.J9 which i s w e l l below t h e c r i t i c a l l i m i t o f 1.96 hence t h e d i f f e r e n c e i n t h e means i s i n s i g n i f i c a n t . A more p r e c i s e c o m p a r i s o n c a n be made o f t h e c o m p u t a t i o n a l s p e e d o f the two methods. An o p e r a t i o n c o u n t , i . e . , t h e number DISTANCE-SQUARED THREE-STEP LINEAR ERROR (FEET) ERROR 5. 1 5.6 12.0 6.0 2.9 7.2 -11.2 -9. 1 -17.7 -12.8 6.4 5. 1 -10.6 -6.9 9.5 11.5 22.3 23.0 12.3 12.5 5.2 8.9 2. 4 2.4 0.4 0. 1 0.5 4.0 -7.7 -2. 1 -6.4 -7.8 -2.6 -5.7 -2.9 -2.9 6.5 7.2 CONTOUR ELEVATION: 2900.0 NUMBER OF POINTS: 19.0 THE AVERAGE (MEAN) ERROR: 8.0 THE LARGEST ERROR: 22.3 VARIANCE: 92.4 STANDARD DEVIATION: 9.6 CONTOUR ELEVATION: 2900.0 NUMBER OF POINTS: 19.0 THE AVERAGE (MEAN) ERROR: 7.0 THE LARGEST ERROR: 23.0 VARIANCE: 7 7.7 STANDARD DEVIATION: 8.8 F i g u r e 24. Accuracy T e s t Comparing the Three-step L i n e a r and the Inverse Distance-sguared Technigues f o r Computing E l e v a t i o n s . (Note: computer output has been re-typed f o r c l a r i t y ) Elevation Error (feet) S0.00T 4S.B0+ M0.00--3S.00-30.00' 2S.00-20.00-l?J0-10.0B-S.00' 0.0% tS3 eg (500 observations) Map Area (sq.in.) S3 + P4 + + Figure 25.,Graph of the Elevation Error as a Function of Map Area - r i r si B 5*£ tsa m is Ul n OO TABLE I I I COMPARISON OF ELEVATION ERRORS FROM MAPS OF DIFFERENT SIZES Map Distance-squared Method Three-step Method Area Perimeter Variance Mean* Variance Mean (sq.in.) (in.) ( f t 2 ) ( f t ) ( f t 2 ) ( f t ) 315.7 74.2 14.7 14.1 12.2 13.8 261.2 64.8 11.5 9.3 11.1 9.2 150.0 50.0 7.9 6.1 7.3 6.2 80.1 36.0 3.9 2.7 3.8 2.6 30.2 22.4 0.9 0.9 0.9 0.9 * A l l errors are computed as absolute values. VO 60 of a d d i t i o n s , s u b t r a c t i o n s , e t c . , was done f o r each method (Table I V ) . These values can be compared by c o n v e r t i n g them t o , j | TABLE IV | | COMPARISON OF OPERATION COUNTS | | Three-Step Method Distance-squared J i : 1 | 3 a d d i t i o n s 14 J | 6 s u b t r a c t i o n s 8 I | 6 m u l t i p l i c a t i o n s 5 i | 0 d i v i s i o n s 5 I i 1 a measure of time. Using one a d d i t i o n as the b a s i c time u n i t and the r e l a t i v e e x e c u t i o n times f o r the Hewlett-Packard 9830A computer (Table V) the t h r e e - s t e p method r e q u i r e s only 24 time u n i t s as compared to the 70 used by the i n v e r s e d i s t a n c e - s q u a r e d approach. The d e s i r e d method f o r implementation i s t h e r e f o r e | TABLE V 1 | RELATIVE EXECUTION TIMES | 1 1 s u b t r a c t i o n = 1 a d d i t i o n ( | 1 m u l t i p l i c a t i o n =2.5 a d d i t i o n s I | 1 d i v i s i o n = 7 a d d i t i o n s I I 1 L 1 the t h r e e - s t e p method as i t i s t h r e e times f a s t e r while s t i l l being as accurate as the di s t a n c e - s q u a r e d approach. However, t h i s i s on l y t r u e when the data p o i n t s are spaced i n a r e g u l a r 61 g r i d as i s the case f o r the e l e v a t i o n data base. I f t h e p o i n t s are i r r e g u l a r l y spaced then the distance-squared method i s the e f f e c t i v e approach as i t i s independent of the o r i e n t a t i o n of the p o i n t s . 1.2.2 S u r f a c e Eguation Hethod f o r F i n d i n g E l e v a t i o n s In the s u r f a c e equation method, i n t e r p o l a t i n g polynomials are computed f o r each c e l l . They are s u b j e c t t o the c o n s t r a i n t that e l e v a t i o n s generated along the c e l l boundaries are the same no matter which of the adjacent c e l l s u r f a c e equations are used, thereby producing continuous e l e v a t i o n s with no jump-d i s c o n t i n u i t i e s between c e l l s . T h i s property i s c r u c i a l when generating road and s k y l i n e p r o f i l e s , thus, s t a t i s t i c a l l y f i t t e d s u r f a c e s can not be used. A s t a t i s t i c a l approach, however, i s u s e f u l when determining the topographic f e a t u r e s where continuous e l e v a t i o n s are not r e g u i r e d ( S e c t i o n 3). There are many f e a s i b l e i n t e r p o l a t i n g polynomials a v a i l a b l e . The number o f terms i n the eguation w i l l depend on the number of data p o i n t s (nodes) used. There must be the same number o f nodes as parameters being c a l c u l a t e d i n order f o r the polynomial t o be unique (Shampine and A l l e n , 1973). The s i m p l e s t form i s a zero degree system. However, t h i s approach y i e l d s a f l a t s u r f a c e , with no s l o p e , c r e a t i n g a d i s c o n t i n u i t y a t the boundaries. S i m i l a r l y , a f i r s t degree 62 system, that permits s l o p e i n e i t h e r the x or y d i r e c t i o n , has d i s c o n t i n u i t i e s . I t i s t h e r e f o r e necessary to have a t l e a s t a f i r s t degree, t h r e e node system t h a t a l l o w s f o r slope i n both the x and y d i r e c t i o n s . , A f i r s t degree, t h r e e node system i s d e s c r i b e d by mlx+m2y+m3z=1 with z being the e l e v a t i o n and the parameters, m±, the r a t e s of change (slope) with r e s p e c t t o x,y and z. T h i s eguation r e q u i r e s t h r e e nodes. Every c e l l would t h e r e f o r e have to be d i v i d e d i n t o two equal t r i a n g l e s each with i t s own eguation. The t r i a n g l e s i n t r o d u c e the e x t r a computational t a s k of d e c i d i n g which t r i a n g l e the ix,y) po i n t l i e s i n and the reguirement of 6 n 2 elements of s t o r a g e . Besides t h i s l a r g e amount o f st o r a g e and i n c r e a s e d computational e f f o r t the e r r o r s are g r e a t e r than the system to be d e s c r i b e d below. The next p o s s i b l e system uses four nodes and i s of the form EjX+m^y*!^ xy+m^z=1 which i s e q u i v a l e n t to the t h r e e - s t e p i n t e r p o l a t i o n method d e s c r i b e d e a r l i e r . T h i s system best f i t s the g r i d - t y p e data s t r u c t u r e as the fou r r e q u i r e d nodes c o i n c i d e n i c e l y with t he fou r c e l l c o r n e r s . The parameters (m) can be sol v e d f o r by c o n s t r u c t i n g a system of four l i n e a r l y independent eguations, each using one of the four c e l l c o r ner e l e v a t i o n s . The s o l u t i o n , which i s s t r a i g h t f o r w a r d , i s as f o l l o w s : . Am=b t h e r e f o r e m=A - ab 6 3 where A i s the matrix of c o e f f i c i e n t s A= I s t s t E (j,k) ~~ s t+v/n s(t+v/n) E(j,k+1) s+u/n t+v/n (s+u/n) (t+v/n) E(j+1,k+1) s+u/n t (s+u/n) t E(j+1,k) b i s the vecto r of constants b= and m i s the ve c t o r of parameters m= m m. Theref o r e z= { x - m 2y-m 3xy)/m ^ . and z=Ax*By*Cxy+D with A=-m /m 1 4 B=-m /m 2' 4 C=-m /m 3 4 D=1/m, (1.3) T h i s system of eguations i s s o l v e d f o r each g r i d c e l l with parameters A,B,C,D being s t o r e d f o r the c e l l , r e g u i r i n g 4 n 2 64 e l e m e n t s o f s t o r a g e . , E g u a t i o n (1.3) t a k e s o n l y 10.5 t i m e u n i t s to e v a l u a t e . O t h e r s u r f a c e s ( w i t h more nodes) c a n be u t i l i z e d w i t h t h e major p r o b l e m b e i n g how many and w h i c h nodes t o u s e . , S i n c e t h e b a s i c d a t a s t r u c t u r e i s a r e g u l a r g r i d t h e next l o g i c a l f o r m s h o u l d r e g u i r e 9 p o i n t s which y i e l d s a c o m p l e t e q u a d r a t i c ( s e c o n d d e g r e e ) s u r f a c e : m 1x 2y 2+m 2x 2+m 3x*m^xy+m 5y 2+m 6y*m 7y 2x*mgX 2y+m gz=1 The s o l u t i o n i s s i m i l a r t o t h e f o u r node c a s e b u t w i t h n i n e e q u a t i o n s t h a t a r e a l l l i n e a r i n ® ±* E v a l u a t i o n o f t h e e g u a t i o n i s s l o w r e q u i r i n g 50 t i m e u n i t s t o compute one e l e v a t i o n . A n o t h e r s e r i o u s drawback i s t h e i n c r e a s e d f l e x i b i l i t y o f the p o l y n o m i a l g a i n e d t h r o u g h more t e r m s . C o n s e q u e n t l y l i t t l e c a n be g u a r a n t e e d a b o u t t h e a c c u r a c y when e v a l u a t i n g between t h e n o d e s . T h i s was b o r n e o u t i n t e s t s s i m i l a r t o t h e one d e s c r i b e d i n S e c t i o n 1.2.1 ( F i g u r e 2 6 ) . I n many c a s e s t h e r e s u l t s were e x c e l l e n t b u t t h e r e were seme a l a r m i n g e r r o r s (upwards o f 50 f e e t ) i n p l a c e s . 1.3 I m p l e m e n t a t i o n The c r u c i a l f a c t o r i n d e c i d i n g t h e t y p e o f d a t a s t r u c t u r e was t h e l i m i t e d amount o f computer memory and t h e l a c k o f a f a s t - a c c e s s s t o r a g e d e v i c e ( e . g . , d i s c d r i v e ) . C o n s e q u e n t l y , a g r i d - t y p e d a t a b a s e was u s e d e m p l o y i n g a 67x67 a r r a y o f e l e v a t i o n s ; i . e . , n=66. 65 THREE-STEP LINEAR ERROR (FEET) 0.4 2. 3 -3.0 -4.2 1.4 8.2 -0.4 -6.7 -8.4 -4.5 CONTOUR ELEVATION: 2900.0 NUMBER OF POINTS: 10.0 THE AVERAGE (MEAN) ERROR: 4.0 THE LARGEST ERROR: -8.4 VARIANCE: 23.9 STANDARD DEVIATION: .4.9 QUADRATIC ERROR (FEET) 0.0 2.0 -8.0 -0.0 1.0 2.0 -2.0 -1.0 - 1 . 1 -2.0 CONTOUR ELEVATION: 2900.0 NUMBER OF POINTS: 10.0 THE AVERAGE (MEAN) ERROR: 2.0 THE LARGEST ERROR: -8.0 VARIANCE: 8.3 STANDARD DEVIATION: 2.9 THREE-STEP LINEAR QUADRATIC ERROR (FEET) 29.2 5.5 -3.6 -3.8 6.0 0. 1 2. 0 9. 1 6.0 1.1 ERROR 54.7 8.9 0.9 4.8 7.9 -0.0 -0.0 0.0 0.0 1.9 (FEET) CONTOUR ELEVATION: 2500.0 NUMBER GF POINTS: 10.0 THE AVERAGE (MEAN) ERROR: THE LARGEST ERROR: 29.2 VARIANCE: 89.1 STANDARD DEVIATION: 9.4 CONTOUR ELEVATION: 2500.0 NUMBER OF POINTS: 10.0 7.0 TBE AVERAGE (MEAN) ERROR: 8.0 THE LARGEST ERROR: 54.7 VARIANCE: 282.0 STANDARD DEVIATION: 16.8 F i g u r e 26. A c c u r a c y T e s t f o r t h e Q u a d r a t i c I n t e r p o l a t i o n T e c h n i g u e . ( N o t e : computer o u t p u t has been r e - t y p e d f o r c l a r i t y ) 66 Using i n t e g e r storage, one e l e v a t i o n r e g u i r e s one word* of memory thereby using 4489 of the 8000 a v a i l a b l e words. In a d d i t i o n , the c o o r d i n a t e s (u,v), d e f i n i n g the map area r e c t a n g l e , and the map s c a l e were r e t a i n e d as pa r t of the data base., The r e t r i e v a l of the e l e v a t i o n s was accomplished by the th r e e - s t e p l i n e a r i n t e r p o l a t i o n method. 6 One word = 2 bytes = 16 b i t s 67 2.0 I n p u t o f t h e E l e v a t i o n D a t a Base The p u r p o s e c f t h e E l e v a t i o n D a t a Base I n p u t Module i s t o a l l o w t h e u s e r t o e n t e r t h e c o n t o u r l o c a t i o n s , t h a t d e s c r i b e t h e t e r r a i n , i n t o t h e c o m p u t e r . The c o n t o u r i n f o r m a t i o n i s t h e n c o n d e n s e d t o form a r e g u l a r g r i d o f e l e v a t i o n s . The i n p u t and s u b s e q u e n t g r i d c r e a t i o n a r e p e r f o r m e d , a s d i s c u s s e d i n P a r t I , by e i t h e r t h e t r a n s e c t o r c o n t o u r method. B o t h methods p r o d u c e t h e same d a t a base s t r u c t u r e o f an (n + 1 ) 2 g r i d o f e l e v a t i o n s . 2.1 T r a n s e c t Method In t h e t r a n s e c t method t h e i n p u t i s c o n t r o l l e d by t h e menu ( F i g u r e 3) whose o p e r a t i o n i s d e c r i b e d i n a p p e n d i x B. The n+1 t r a n s e c t l i n e s , used t o d e s c r i b e t h e map, c o r r e s p o n d t o t h e columns o f t h e e l e v a t i o n m a t r i x E ( u s i n g t h e same d e f i n i t i o n s g i v e n i n S e c t i o n 1.2). Each t r a n s e c t l i n e has i t s o r i g i n a t t h e bottom o f t h e map ( i . e . , t h e x - a x i s (Y=0)) and t h e e n d p o i n t a t t h e t o p o f t h e map (Y=v) . Upon c o m p l e t i o n o f t h e e n t r y o f t r a n s e c t l i n e j , t h e i r r e g u l a r l y s p a c e d c o n t o u r i n t e r s e c t i o n s a r e c o n v e r t e d t o n+1 e l e v a t i o n p o i n t s s p a c e d a t i n t e r v a l s o f v/n a l o n g t h e l i n e . T h ese p o i n t s c o r r e s p o n d t o t h e i n t e r s e c t i o n s o f t h e rows o f t h e E m a t r i x w i t h t h e column ( t r a n s e c t l i n e j ) b e i n g c o m p i l e d . E a c h 68 o f t h e s e e l e v a t i o n s (E ( j , k ) ,k=1,2,. ., ,n+1) i s computed by l i n e a r l y i n t e r p o l a t i n g between t h e two c l o s e s t , b r a c k e t i n g , c o n t o u r i n t e r s e c t i o n s on t h e t r a n s e c t l i n e . T h e r e f o r e , t h e e l e v a t i o n f o r g r i d p o i n t ( j , k ) , whose map l o c a t i o n i s ( s , t ) ( E q u a t i o n 1.2), w i l l be: E( j , k ) = T ( i , 2 ) * (T (i+ 1 ,2) -T (1,2) ) ( t - T ( i , 1) ) / (T ( i * 1 , 1 )-T ( i , 1) ) w i t h T ( i , r ) t h e m a t r i x o f t h e (y, z) p o i n t s ( c o n t o u r i n t e r s e c t i o n s ) f o r t r a n s e c t l i n e j r=1 : y - c o o r d i n a t e (y) r=2 : e l e v a t i o n (z) i = s u b s c r i p t o f t h e c o n t o u r i n t e r s e c t i o n i n T t h a t i s c l o s e s t t o and s t i l l l e s s t h a n t h e t -c o o r d i n a t e o f t h e g r i d p o i n t ( s , t ) . A f t e r c o m p i l i n g t h e l i n e , the same m a t r i x , T, i s u s e d f o r t h e n e x t l i n e s i n c e c o n t o u r i n t e r s e c t i o n s a r e r e t a i n e d o n l y f o r t h e l i n e b e i n g c o m p i l e d . The major d i s a d v a n t a g e o f t h e t r a n s e c t method i s t h a t t h e e l e v a t i o n s f o r e a c h l i n e a r e c o m p i l e d i n d e p e n d e n t l y . L a r g e e r r o r s , up t o one h a l f t h e c o n t o u r i n t e r v a l , c a n o c c u r i n r e g i o n s where t h e t r a n s e c t l i n e s r u n p a r a l l e l t o t h e c o n t o u r l i n e s . T h e s e e r r o r s c o u l d be r e d u c e d by r u n n i n g a s e c o n d s e t o f l i n e s (T-*-) p e r p e n d i c u l a r t o t h e s e r i e s j u s t d e s c r i b e d ( F i g u r e 2 7 ) . These w i l l c o r r e s p o n d t o t h e rows o f t h e m a t r i x E. In t h i s d o u b l e t r a n s e c t method t h e e l e v a t i o n E ( j , k ) f o r 70 g r i d p o i n t (j,k) would be computed by l o c a t i n g the two c l o s e s t b r a c k e t i n g p o i n t s on T and s i m i l a r l y the two c l o s e s t on TJ- and then weighting these f o u r e l e v a t i o n s by the i n v e r s e d i s t a n c e -sguared method (Se c t i o n 1.2.1) to form the average. The t h r e e -step l i n e a r i n t e r p o l a t i o n method can not be used e f f i c i e n t l y as the e l e v a t i o n p o i n t s would be i r r e g u l a r l y spaced. Although accuracy would be improved, i t i s now necessary to r e t a i n a l l the i n t e r s e c t i o n s f o r the 2(n*1) l i n e s : T : i=1 ,2,3,. . . ,n + 1 and T+ : i=1,2,3,...,n+1 1 X The e r r o r r e d u c t i o n has not been q u a n t i f i e d as memory l i m i t a t i o n s d i d not permit implementation on the HP computer. Also, the amount o f work r e q u i r e d t o d i g i t i z e the contour l o c a t i o n s would be doubled. 2.2 Contour Method The technique used i n the contour method i s s i m i l a r to t h a t d e s c r i b e d i n S e c t i o n (1.1). To form an e l e v a t i o n E ( j , k ) , whose d i g i t i z e r c o o r d i n a t e s are ( s , t ) , i t i s necessary t o s e a r c h through the contour p o i n t s d i g i t i z e d by the user, f o r the c l o s e s t p o i n t s . The i n v e r s e d i s t a n c e - s q u a r e d technique i s then a p p l i e d t o these p o i n t s to o b t a i n the e l e v a t i o n . Searching the e n t i r e i n p u t data i s both time consuming and unnecessary. Thus, to employ a systematic search, the map area i s s u b d i v i d e d i n t o c e l l s and only the r e l e v a n t c e l l s are then searched. The process of computing the e l e v a t i o n g r i d i s done, t h e r e f o r e , i n three phases: 71 1) entry of contour l o c a t i o n s from the d i g i t i z e r ; 2) s o r t i n g of (x #y) contour p o i n t s i n t o s m a l l e r map u n i t s to allow s y s t e m a t i c s e a r c h i n g (data s t r u c t u r i n g ) ; 3} generating the ( n * 1 ) 2 e l e v a t i o n s f o r the matrix E. , 2.2.1 Entry of Contour L i n e s The entry of the contour l i n e s i s c o n t r o l l e d by a menu (Figure 4). The (x,y) p o i n t s are entered i n continuous mode 7 from the d i g i t i z e r as each contour i s t r a c e d . Not a l l the p o i n t s are used. Those c l o s e together are thinned out using a d i s t a n c e t e s t from the l a s t recorded p o i n t (x ,y ): i i r e j e c t (x,y) i f (x-x } 2+(y-y ) 2 < t o l e r a n c e 2 i i The t h i n n i n g , i s a l s o done to ensure that the c o n t o u r s are d e s c r i b e d by an evenly d i s t r i b u t e d s e t of p o i n t s . 2.2.2 S o r t i n g of Data P o i n t s to F a c i l i t a t e Searching The contour p o i n t s can be c l a s s i f i e d i n t o e i t h e r l a r g e g r i d s or bands t o f a c i l i t a t e more e f f i c i e n t manipulation. 7 Two entry modes can be used when d i g i t i z i n g . A s i n g l e point entry allows the user to s e l e c t the p o i n t to be d i g i t i z e d . The continuous mode permits the computer t o take a continuous stream of p o i n t s and i s t h e r e f o r e used when t r a c i n g l i n e s . 72 The Band Data S t r u c t u r e f o r Searching Bands can he c r e a t e d i n e i t h e r the x or y d i r e c t i o n . ; For purposes of d i s c u s s i o n (and implementation) the bands are assumed p a r a l l e l to the x-axis (Figure 28) . The contour p o i n t s are placed i n t o bands, b , each i having a width o f w. ,A t o t a l of u/w bands w i l l be r e q u i r e d to cover the map area. , The point (x,y,z) i s i n b^ i f : (i~1)w<y<iw The p o i n t s are s t o r e d i n band b^ by c o n s t r u c t i n g a f o u r dimensional matrix B ( i , x , y , z ) . I f the computer memory i s l i m i t i n g , an a l t e r n a t e method could be t o s t o r e the p o i n t s a s s o c i a t e d with each band onto separate data tape f i l e s . T h i s would allow i n d i v i d u a l bands to be brought i n t o memory as r e q u i r e d . The search time f o r l o c a t i n g the c l o s e s t contour p o i n t s t o g r i d p o i n t (s,t) can be reduced by s o r t i n g the (x,y,z) contour p o i n t s w i t h i n a band by t h e i r x-c o o r d i n a t e s . The need f o r t h i s w i l l become apparent i n S e c t i o n (2. 2.3) . An important c o n s i d e r a t i o n when using bands i s the presence of an "edge e f f e c t " . When determining the e l e v a t i o n E(j,k) f o r the point (s, t) t h a t i s c l o s e to the band boundary, i t i s h i g h l y probable that the c l o s e s t contour p o i n t s w i l l not a l l l i e i n the same band. T h i s grid point (j,k) T+~-—— digitized contour point — l y l ^ i » • • , —T^f^r-*H* — ? T £ — ft —,*—J.Tfi-C.* ./» -a ^..„,^™™—4 Figure 28. Relationship of Bands, Contour Points and Grid Points. —i 74 makes i t n e c e s s a r y t h e r e f o r e , t o a l s o c h e c k t h e c o n t o u r p o i n t s i n t h e c l o s e s t , a d j a c e n t band. The s e l e c t i o n o f t h e band w i d t h i s i m p o r t a n t a s i t a f f e c t s t h e number o f c o n t o u r p o i n t s t h a t l i e w i t h i n a band. I f narr o w bands w i t h few p o i n t s a r e u s e d t h e n i t w i l l n o t be s u f f i c i e n t t o m e r e l y c h e c k the two bands a s d i s c u s s e d a b o v e . C o n v e r s e l y , a wide band w i t h many p o i n t s w i l l d e c r e a s e t h e l i k e l i h o o d o f m i s s i n g a r e l e v a n t d a t a p o i n t . T h i s d o e s however i n c r e a s e t h e s e a r c h t i m e w i t h i n a band t h e r e b y c r e a t i n g an t r a d e o f f i n e f f i c i e n c y . The o p t i m a l band w i d t h w i l l o c c u r when ( s , t ) i s b r a c k e t e d by two c o n t o u r l i n e s . S i n c e two bands a r e a l w a y s b e i n g u s e d , i . e . , t h e band c o n t a i n i n g ( s , t ) and t h e c l o s e s t b a nd t o ( s , t ) , o n l y one c o n t o u r p e r band i s needed. C o r r e s p o n d i n g l y , t h e band w i d t h would be e g u a l t o t h e a v e r a g e d i s t a n c e ( i n d i g i t i z i n g u n i t s ) between c o n t o u r l i n e s . A f a c t o r o f s a f e t y s h o u l d , however, be i n t r o d u c e d as t h e a v e r a g e d i s t a n c e between c o n t o u r s c o u l d v a r y s u b s t a n t i a l l y w i t h i n t h e s t u d y a r e a , e s p e c i a l l y i n e x t r e m e l y v a r i e d t e r r a i n . I t i s d i f f i c u l t t o e s t i m a t e ho* many c o n t o u r l i n e s s h o u l d be i n c l u d e d p e r band t o remove t h i s u n c e r t a i n t y . A l t h o u g h n o t t h o r o u g h l y i n v e s t i g a t e d due t o l i m i t e d c o m p u t e r c a p a c i t y , two c o n t o u r l i n e s a r e p r o b a b l y s u f f i c i e n t making t h e band w i d t h t w i c e t h e a v e r a g e d i s t a n c e between c o n t o u r s . , 75 The other .important - parameter used t o c o n t r o l the number of p o i n t s i n a band i s the t h i n n i n g t o l e r a n c e . Using t h e above r a t i o n a l e of having two contours per band, a t h i n n i n g t o l e r a n c e equal t o one h a l f the band width should be used. For a contour running p e r p e n d i c u l a r t o the o r i e n t a t i o n o f the bands, t h i s t h i n n i n g t o l e r a n c e w i l l ensure t h a t no more than two contour p o i n t s on the contour l i n e w i l l be contained w i t h i n any one band. The band method i s most u s e f u l when memory l i m i t s the complexity of the data s t r u c t u r e or t h e r e i s no f a s t - a c c e s s storage device, l i k e a d i s c , t h a t permits quick swapping of search u n i t s i n and out of memory. The G r i d Data S t r u c t u r e f o r Searching The g r i d data s t r u c t u r e i s analogous t o banding simultaneously i n two d i r e c t i o n s . The g r i d s used are not to be confused with the e l e v a t i o n g r i d f ( j , k ) and can be considered as "macro-grids". Using a c e l l s i z e of w, (u/w) <v/w) c e l l s w i l l be needed. A contour p o i n t (x,y,z) i s i n c e l l , G ( h , i ) , i f : (h-1)w<x<hw and (i-1)w<y<iw The p o i n t s can be s t o r e d i n t h e i r a p p r o p r i a t e c e l l s , G {h,i), by extending G t o f i v e dimensions: G ( h , i , x ,y,z). The s e l e c t i o n of the c e l l width, w, and the t h i n n i n g t o l e r a n c e i s made on the same b a s i s as used f o r the bands. 76 The "edge e f f e c t " i s more pronounced i n the g r i d approach as the c e l l s have twice the number of edges. Consequently, i t i s necessary to search the t h r e e c l o s e s t , adjacent c e l l s to the p o i n t (s,t) i n a d d i t i o n to the c e l l t h a t c o n t a i n s ( s , t ) . . The g r i d data s t r u c t u r e r e g u i r e s a f a s t - a c c e s s s t o r a g e device as there are more data s t r u c t u r e u n i t s to handle than i n the band approach., 2.2«3 Generating the G r i d of E l e v a t i o n s The e l e v a t i o n E(j,k) f o r the p o i n t (s,t) i s computed by d i v i d i n g the area around (s,t) i n t o p egual r e g i o n s (e. g. , f o u r t h s (p=4) , s i x t h s (p=6)) and f i n d i n g the c l o s e s t p o i n t i n each r e g i o n (Figure 29). I t i s not s u f f i c i e n t to merely l o c a t e the p c l o s e s t p o i n t s as t h i s would c r e a t e a b i a s i f the p o i n t s are concentrated t o one s i d e of ( s , t ) . I n c r e a s i n g the number of r e g i o n s w i l l r e g u i r e more computational work but the l e v e l of accuracy w i l l a l s o i n c r e a s e . The d e c i s i o n as to which p o i n t s t o check w i l l depend on the type of data format used t o r e t a i n the contour p o i n t s ; i . e . , bands or g r i d s . I f bands are used then i t i s necessary t o determine which band c o n t a i n s ( s , t ) . T h i s band, b , i s indexed by: i = f t / w j * 1 . i The p o s i t i o n of (s,t) w i t h i n b., with r e s p e c t t o the x-d i g i t i z e d contour point hi — REGION 3 closes t x-ccordinate to (j,k) i n band b.;' subscript band b i+1 REGION 1 closest x-coordinate to (j ,k )^ i n band subscript ^---^••^closest point i n regicjn 1 band b. , i - l Figure 29. Using Four Regions to f i n d an Elevation. 78 c o o r d i n a t e , i s e a s i l y f o u n d u s i n g a s t a n d a r d b i n a r y s e a r c h ( C o n t e and d e B o o r , 1973) a s t h e p o i n t s i n t h e bands have b e e n p r e v i o u s l y s o r t e d ( S e c t i o n 2.2.2). The p o i n t w i t h i n band b , i h a v i n g t h e c l o s e s t x - c o o r d i n a t e t o s , w i l l o n l y l i e i n one o f t h e p r e g i o n s a r o u n d {s,t).„. l e t t i n g t h e s u b s c r i p t o f t h i s p o i n t w i t h i n b . be c, i t i s n e c e s s a r y t o examine s e v e r a l o f t h e o r d e r e d c o n t o u r p o i n t s w i t h i n b t h a t l i e t o e i t h e r s i d e o f p o i n t c t o e n s u r e t h a t a l l t h e r e g i o n s a r o u n d ( s , t ) a r e r e p r e s e n t e d . , I n o r d e r t o m a i n t a i n an u n b i a s e d s e a r c h , t h e s e l e c t i o n o f t h e number o f p o i n t s t o be examined s h o u l d c o r r e s p o n d t o t h e band w i d t h and t h i n n i n g t o l e r a n c e u s e d . Because t h e c r i t e r i o n u s e d i n d e f i n i n g t h e s e two p a r a m e t e r s was t o have one band w i d t h c o n t a i n i n g e i t h e r two c o n t o u r l i n e s o r two c o n t o u r p o i n t s ( d e p e n d i n g on whether t h e c o n t o u r l i n e s a r e p a r a l l e l o r p e r p e n d i c u l a r t o t h e bands) i t i s n e c e s s a r y t o c h e c k f o u r c o n t o u r p o i n t s on b o t h s i d e s o f p o i n t c . , To e l i m i n a t e t h e "edge e f f e c t " i t i s a l s o n e c e s s a r y t o p e r f o r m t h e same s e a r c h a s d i s c u s s e d above on t h e c l o s e s t , a d j a c e n t band. T h i s band i s i d e n t i f i e d by c o m p a r i n g t h e y-c o o r d i n a t e o f ( s , t ) t o t h e m i d p o i n t ( w i t h r e s p e c t t o y) o f t h e band b.. I f t<w(i-1/2) t h e n t h e c l o s e s t band i s b . , ; o t h e r w i s e X 1-1 i t i s b , . I n t o t a l t h e n , 18 c o n t o u r p o i n t s must be c h e c k e d . i+1 I f a g r i d s y s t e m i s u s e d t o r e t a i n t h e ( x , y , z ) c o n t o u r p o i n t s t h e n s e a r c h i n g f o r t h e c l o s e s t p o i n t s becomes s i g n i f i c a n t l y e a s i e r . The s u b s c r i p t s o f t h e c e l l , t h a t c o n t a i n s ( s , t ) , a r e found f r o m : 79 h=£s/w]+1 and i=[t/w] + 1 A l l the p o i n t s i n c e l l G(h,i) are searched along with the three c l o s e s t , adjacent c e l l s t o ( s , t ) . These three c e l l s are found i y comparing {s,t) to the c e n t r o i d <w(h-1/2),w (i-1/2)) of the c e l l ( h , i ) . For example, i f s<w(h-1/2) and t<(w(i-1/2) then the three c l o s e s t c e l l s would he G ( h - 1 , i - 1 ) , G<h-1,i) and G (h, i-1) . The checking of the (x,y,z) contour p o i n t s f o r the c l o s e s t p o i n t i n i t s r e s p e c t i v e r e g i o n around (s,t) i s the same r e g a r d l e s s o f whether the g r i d or band search s t r u c t u r e i s used. Each contour p o i n t examined (18 i n t o t a l i f the band method i s used) must f i r s t be p l a c e d i n i t s a p p r o p r i a t e r e g i o n about ( s , t ) . The r e g i o n i s determined from the angle, e , between (s,t) and (x,y,z) (with z ignored) where: 9= arctangent ( ( y ~ t ) / ( x - s ) ) The angle 6 must be c o n v e r t e d to a f u l l 360 degree range as most computers use arctangents between p l u s and minus 90 d e g r e e s 8 . The r e g i o n number, r (1<r<p), i s found from: r=[ 9/(360/p) ]+1 The d i s t a n c e i s computed between (x,y,z) and (s,t) from d 2=(x-s) 2 + (y-t) 2 T h i s d i s t a n c e , d 2 , i s compared to d 2 (the d i s t a n c e between the s Converting an angle o b t a i n e d from an arctangent f u n c t i o n to a f u l l 360 degree range i s accomplished by 6 = |e| and i f x>s, y>t then 6= 9 i f x<s, y>t then 9= 180-9 i f x<s, y<t then 9= 180+9 i f x>s, y<t then 9= 360-e 80 c l o s e s t p o i n t yet found and (s,t) i n r e g i o n r ) . I f d 2>d 2 then r (x,y,z) i s r e j e c t e d . Conversely i f d 2<d 2 then d 2=d 2.•„ The e l e v a t i o n of the r e g i o n i s a l s o updated: e r=z. When a l l the a p p r o p r i a t e p o i n t s have been checked the e l e v a t i o n i s computed using the d i s t a n c e - s q u a r e d technique and the c l o s e s t p o i n t i n each of the p r e g i o n s . 3 (j,k)={e l/d 2+e 2/d 2*...+e p/d 2)/(1/d 2+... + 1/d2) 2.3 Comparison o f the T r a n s e c t and Contour Methods The t r a n s e c t and contour methods were compared i n a manner s i m i l a r to t h a t d e s c r i b e d i n S e c t i o n 1.2.1. Contours were t r a c e d and the average d i f f e r e n c e i n e l e v a t i o n between the a c t u a l and the computed contour e l e v a t i o n was determined f o r each method (Figure 30). For the contour method only the band technique of computing the E(j,k) was used as the g r i d s t r u c t u r e exceeded the c a p a b i l i t y of the computer. In the example shown i n F i g u r e 30 the data base was c r e a t e d by d i g i t i z i n g the 100-f o o t contours only. The e r r o r i n the t r a n s e c t method was g e n e r a l l y g u i t e uniform i r r e s p e c t i v e of whether the contour b e i n g t e s t e d had been used t o form the data base. The o p p o s i t e was t r u e f o r the contour method. I t was s u p e r i o r to the t r a n s e c t method only when generating e l e v a t i o n s along contour l i n e s t h a t were used to c r e a t e the data base. As seen from Fig u r e 30, t h e e r r o r i n the 2825-foot contour i s s i g n i f i c a n t l y higher than f o r the 2800-foot contour. T h i s e r r o r c o u l d be 81 CONTOUR METHOD ELEVATION ERROR (FEET) 3.00 6.91 20. 34 10. 50 5.89 6.50 -0.96 -1.00 5. 46 0.98 CONTOUR ELEVATION: 2800.00 NUMBER OF PGINTS: 10.00 THE AVERAGE (MEAN) ERROR: 6.15 THE LARGEST ERROR: 20.34 VARIANCE: 39.82 STANDARD DEVIATION: 6.31 TRANSECT METHOD ELEVATION ERROR (FEET) 4.68 10.29 7.47 -10.68 -4.01 0. 12 4.79 -10.78 12.50 3.50 CONTOUR ELEVATION: 2800.00 NUMBER OF POINTS: 10.00 THE AVERAGE (MEAN) ERROR: 6.88 THE LARGEST ERROR: 12.50 VARIANCE: 65.49 STANDARD DEVIATION: 8.0 9 CONTOUR METHOD TRANSECT METHOD ELEVATION ERROR (FEET) -12.31 -13.32 10.32 12.26 2 1. 39 -17.70 17.44 5.20 -13.58 -5.83 ELEVATION ERROR (FEET) -1.25 -1.23 0.54 -3.89 -7.50 -6.59 10.29 2.75 -4.33 -8.02 CONTOUR ELEVATION: 2825.00 NUMBER OF POINTS: 10.00 THE AVERAGE (MEAN) ERROR: 12.93 THE LARGEST ERROR: 21.39 VARIANCE: 211.62 STANDARD DEVIATION: 14.55 CONTOUR ELEVATION: 2825.00 NUMBER OF POINTS: 10.00 THE AVERAGE (MEAN) ERROR: 4.64 THE LARGEST ERROR: 10.29 VARIANCE: 30.86 STANDARD DEVIATION: 5.56 F i g u r e 30. Comparison of the Accuracy of the Contour and Transect Methods. (Note: computer output has be re-typed f o r c l a r i t y ) 82 reduced i f the number o f search r e g i o n s was i n c r e a s e d and/or by r e t a i n i n g a higber percentage of the contour c o o r d i n a t e s obtained when d i g i t i z i n g the data base; i . e . , t h i n n i n g fewer p o i n t s . The other obvious p o s s i b i l i t y i s to d i g i t i z e more contours. The p o t e n t i a l improvement was not t e s t e d as machine c a p a c i t y was l i m i t i n g . 2.4 Implementation Both the t r a n s e c t and contour methods have been implemented on the HP 9830A computer., Although the contour method i s s u p e r i o r t h e o r e t i c a l l y , i t can only be used f o r s m a l l maps or those with l i m i t e d d e t a i l due to l i m i t a t i o n s of the HP9830A computer; i . e . slow execution speed, l a c k of a f a s t a c c e s s storage d evice and a s m a l l memory. Using a t h i n n i n g t o l e r a n c e of 0.20 inches and a complexity s i m i l a r to that shown i n F i g u r e 1, the l a r g e s t map s i z e t h a t can be e f f e c t i v e l y accommodated by the contour method i s approximately 12 inches sguare. The t r a n s e c t method i s r e s t r i c t e d to one s e t of 67 l i n e s ( i . e . , n=66) each with a c a p a c i t y of 200 contour i n t e r s e c t i o n s . There was i n s u f f i c i e n t memory to allow f o r implementation of the double t r a n s e c t method. In the contour method only 2300 (x,y,z) contour p o i n t s can be r e t a i n e d . To improve search time f o r l o c a t i n g the c l o s e s t p o i n t s when g e n e r a t i n g the e l e v a t i o n s , h o r i z o n t a l l y o r i e n t e d 83 bands a r e u s e d . To make t h e i m p l e m e n t a t i o n p o s s i b l e a f i x e d number o f bands (20) i s a l w a y s u s e d making t h e band w i d t h , w, v a r i a b l e . The bands a r e s t o r e d i n d i v i d u a l l y on d a t a t a p e f i l e s and have a c a p a c i t y o f 200 p o i n t s e a c h . The e l e v a t i o n s were d e t e r m i n e d u s i n g f o u r r e g i o n s . 2.5 a New Method f o r Data E n t r y The major drawback t o t h e e n t i r e s i m u l a t o r i s e n t e r i n g t h e d a t a . I t i s a t i m e c o n s u m i n g , t e d i o u s p r o c e s s which w i l l t e n d t o d i s c o u r a g e many p o t e n t i a l u s e r s . One r e l a t i v e l y new, p r o m i s i n g a p p r o a c h i s i n t h e use o f o r t h o p h o t o g r a p h s . O r t h o p h o t c g r a p h s a r e c r e a t e d by r e m o v i n g t h e d i s t o r t i o n i n o r d i n a r y a e r i a l p h o t o g r a p h s due t o t h e e l e v a t i o n d i f f e r e n c e s o f the l a n d s c a p e (Meyer, 1969) ; t h e p r o c e s s c a n be e i t h e r manual o r automated. The d i s t o r t i o n i s removed by s y s t e m a t i c a l l y a s s e m b l i n g a p h o t o g r a p h i c m o s a i c o f t h e o r i g i n a l a e r i a l p h o t o g r a p h . E a c h s m a l l u n i t making up t h e m o s a i c i s o p t i c a l l y a d j u s t e d t o a s i n g l e , common e l e v a t i o n . I n h e r e n t i n t h e p r o c e s s t h e r e f o r e i s t h e d e t e r m i n a t i o n o f t h e l a n d s c a p e e l e v a t i o n s . T h i s i s done on a r e g u l a r p a t t e r n , t y p i c a l l y u s i n g a g r i d o r t r a n s e c t a p p r o a c h , w i t h t h e e l e v a t i o n s and t h e i r r e s p e c t i v e (x,y) l o c a t i o n s b e i n g r e c o r d e d on a d a t a t a p e . T h i s d a t a c a n t h e n be r e l a t i v e l y e a s i l y c o n v e r t e d i n t o t h e r e g u l a r g r i d o f e l e v a t i o n s u s e d by t h e t e r r a i n s i m u l a t o r by e i t h e r t h e t r a n s e c t c r c o n t o u r methods t h e r e b y e l i m i n a t i n g t h e d i g i t i z i n g p h a s e . 84 3.0 T o p o g r a p h i c F e a t u r e s The T o p o g r a p h i c F e a t u r e s Module p r o v i d e s a f a c i l i t y f o r o b t a i n i n g map o v e r l a y s and a c r e a g e d i s t r i b u t i o n s f o r g r o u n d s l o p e , a s p e c t and e l e v a t i o n . The s l o p e , a s p e c t and e l e v a t i o n a r e d e t e r m i n e d f o r e a c h g r i d c e l l i n t h e e l e v a t i o n d a t a b a s e . T h i s t i m e c o n s u m i n g p r o c e s s i s done o n l y once f o r any d a t a s e t w i t h t h e r e s u l t s b e i n g s t o r e d on d a t a t a p e . 3.1 D e t e r m i n i n g t h e T o p o g r a p h i c F e a t u r e s The s l o p e , a s p e c t and e l e v a t i o n f o r e a c h g r i d a r e computed u s i n g t h e s u r f a c e e g u a t i o n f o r a l i n e a r p l a n e : m lx*m 2y*a 3z=1. O n l y t h r e e nodes a r e r e q u i r e d f o r t h i s e q u a t i o n ( S e c t i o n 1.2.2). I n c l u s i o n o f a f o u r t h node, t h e r e b y u s i n g a l l t h e f o u r c e l l c o r n e r e l e v a t i o n s , r e q u i r e s a l e a s t - s q u a r e s s t a t i s t i c a l f i t . With f o u r n o d e s , f o u r l i n e a r l y i n d e p e n d e n t e g u a t i o n s c an be d e v e l o p e d . T h e s e c a n be e x p r e s s e d i n m a t r i x n o t a t i o n a s Am=b wi t h A,m,b d e f i n e d i n S e c t i o n 1.2.2. The t h r e e p a r a m e t e r s , m, a r e s o l v e d u s i n q l e a s t - s q u a r e s by T T m (A A) ^ A b .*..,.,*-...,.,.,,,,....*..... :,....{3. 1) The s o l u t i o n o f t h e s y s t e m g i v e s : z — c^ " 3. x^ljy •*.• • * «• • • • * • • • • • ^ 3 • 2 ^ w i t h c = 1/m 85 a = b = which i s t h e e q u a t i o n f e a t u r e s . 3.1.1 S o l v i n g f o r A v e r a g e E l e v a t i o n The a v e r a g e e l e v a t i o n , z, i s e v a l u a t e d a t t h e c e n t r o i d o f the c e l l ( j , k ) . The c o o r d i n a t e s o f t h e c e n t r o i d a r e : x=(s+u/n/2) and y=(t+v/n/2) S u b s t i t u t i n g t h e s e v a l u e s f o r x and y i n t o E q u a t i o n (3.2) y i e l d s : z = c*a(s+u/n/2)+b(t+v/n/2) which i s t h e a v e r a g e e l e v a t i o n f o r t h e c e l l ( j , k ) . 3.1.2 S o l v i n g f o r t h e Maximum S u r f a c e S l o p e The p l a n e , d e f i n e d by E g u a t i o n (3.2) ( F i g u r e 3 1 ) , has t h r e e i n t e r c e p t s : z - i n t e r c e p t i s c (x,y=0) ( p o i n t C) x - i n t e r c e p t i s - c / a (y,z=0) ( p o i n t A) y - i n t e r c e p t i s - c / b (x,z=0) ( p o i n t B) The t a n g e n t o f t h e a n g l e , 6^, wh i c h i s s u b t e n d e d by t h e p l a n e and t h e x - a x i s , r e p r e s e n t s t h e r a t e o f change ( s u r f a c e s l o p e ) o f z w i t h r e s p e c t t o x. S i m i l a r l y , t h e t a n g e n t o f t h e a n g l e , e y , i s t h e s u r f a c e s l o p e o f z w i t h r e s p e c t t o y. The maximum s u r f a c e s l o p e o f z w i t h r e s p e c t t o bot h x and y i s t h e t a n g e n t o f 8 . The a n g l e 6 (^CPO) i s s u b t e n d e d by t h e p l a n e and t h e P P . . -m /m 1 3 -m2/m3 f o r d e t e r m i n a t i o n o f a l l t o p o g r a p h i c 86 z (elevation) Equation of the surface plane: z = c + ax + by l i n e of maximum slope (CP) Figure 31. The Surface Plane and i t s Coordinate System. 87 l i n e segment PO with P l y i n g on A~B. , I f t h e l e n g t h of PG i s i then the tangent ofZ.CPO w i l l be ( c / i ) . T h i s tangent w i l l be a t a maximum when i i s a minimum. T h i s occurs only when PC i s the a l t i t u d e of t r i a n g l e ABC and PO i s the a l t i t u d e of t r i a n g l e OAB. noting that OP i s the a l t i t u d e of AOAB, the value of i can be determined from the c o s i n e o f ZBOP m u l t i p l i e d by the l e n g t h of OB which i s the y - i n t e r c e p t (-c/b). L e t t i n g Z.EOP be a then: cos a = i / (-c/b) Rearranging: 1/i=-b/c(i/cos a ) , . , , , . . . . < 3 . 3) Using the r e l a t i o n s h i p 1/cos a =/1 + t a n 2 a i n Eguation (3.3): 1/i=(-b/c) /1+tan 2 a The tangent a. can be r e p l a c e d by (a/b) s i n c e Z.BOP=^EAO and tan a = (-c/b)/(-c/a)=a/b Therefore 1/i={-b/c) /1+a 2/b 2 and i=-c//a 2+b 2 Hence, tan e - c / i = /a 2+b 2 ( s i g n can be ignored) P ' The maximum s l o p e , i n percent, i s t h e r e f o r e {/a 2+b 2 )100/map s c a l e .........................(3.4) 3.1.3 S o l v i n g f o r Aspect The aspect of the s u r f a c e plane i s determined by s e l e c t i n g any p o i n t on the ray OP (Figure 31) and then determining the compass guadrant i n which i t l i e s . A convenient p o i n t i s 88 (a,b,0) which d e f i n e s the angle, 0, of the ray OP to be the arctangent of (b/a). The point (a,b,0) can be e a s i l y shown to l i e cn OP by c o n s i d e r i n g the t r i a n g l e O&P. a+3=90 and s i n c e a = arctangent o f (a/b), 3 must be the arctangent of (b/a). Dsing Eguation (3.2), i f a>0 then dz/dx>0; i . e . , when proceeding east (x i n c r e a s i n g ) a c r o s s the s u r f a c e plane the e l e v a t i o n w i l l a l s o i n c r e a s e . Consequently, by d e f i n i t i o n , the aspect w i l l be w e s t e r l y . , Extending t h i s l o g i c to the other three quadrants y i e l d s a set of r e l a t i o n s h i p s o p p o s i t e to those employed i n S e c t i o n 2.2.3 t h a t can be used to c o n v e r t 3 to the f u l l 360 degree aspect range (Figure 32). Namely: 3=101 i f a<0, b<0 then 3=3 i f a>0, b<0 then 3= 180-3 i f a>0, b>0 then 3= 180+3 i f a<0, b>0 then 3= 360-3 The angle 3 can be converted i n t o one o f the e i g h t main aspects (N, NE, E, SE, S, SW, W, N » ) . For convenience the angle 3 i s r o t a t e d by 22.5 degrees thereby s h i f t i n g the boundaries o f the aspect c l a s s e s onto even U5-degree m u l t i p l e s (Figure 32). 89 Figure 32. D e f i n i t i o n of the Aspect Classes. 90 The map o r i e n t a t i o n 9 t h a t was use d when d i g i t i z i n g t h e c o n t o u r s must be a c c o u n t e d f o r . Assuming t h e a n g l e of t h e o r i e n t a t i o n i s p o s i t i v e f o r c o u n t e r - c l o c k w i s e r o t a t i o n s o f n o r t h t h e n t h e e q u a t i o n f o r g i s ; B = B + 2 2 . 5 - r o t a t i o n * 3 6 0 {360 i s added t o e l i m i n a t e n e g a t i v e a n g l e s ) To e n s u r e t h a t B i s n o t g r e a t e r t h a n 360: B = 8 -£ B /360]360 The a n g l e B can now be c o n v e r t e d t o t h e a p p r o p r i a t e a s p e c t i n d e x { T a b l e VI) by I=[ B/451+1 The v a l u e o f I i s s t o r e d as t h e a s p e c t i n d e x f o r c e l l | j , k ) . "3 i I I TABLE VI DEFINITION OF ASPECT INDICES A s p e c t e a s t n o r t h e a s t n o r t h n o r t h w e s t west s o u t h w e s t s o u t h s o u t h e a s t 9 When e n t e r i n g t h e e l e v a t i o n d a t a b a s e w i t h t h e t r a n s e c t method i t i s more a c c u r a t e t o have t h e map a l i g n e d s o t h a t t h e t r a n s e c t l i n e s a r e p e r p e n d i c u l a r t o t h e m a j o r i t y o f t h e c o n t o u r s . C o n s e q u e n t l y t h e a s p e c t s computed f o r t h e " t a b l e n o r t h " w i l l n o t be t h e same as f o r t h e t r u e n o r t h . The map o r i e n t a t i o n between t h e t r u e n o r t h and t h e t a b l e n o r t h must t h e r e f o r e be added t o t h e d e g r e e o f a s p e c t computed f o r t h e s u r f a c e p l a n e . 91 3.2 Storage of the Slope, aspect and E l e v a t i o n The t o p o g r a p h i c f e a t u r e s i n d i c e s are c a l c u l a t e d only once and then s t o r e d i n a t h r e e - d i m e n s i o n a l matrix F ( i , j , k ) , with i=1 s l o p e i=2 aspect i=3 e l e v a t i o n and (j,k) the c e l l s u b s c r i p t s . 3.3 Output of the Topographic Features For topographic c l a s s maps the c l a s s i n t e r v a l , c, d e f i n i n g the v a r i o u s c l a s s ranges (Figure 5 ) , must be d e c l a r e d by the user. In the case of an aspect map, c i s a u t o m a t i c a l l y set to one i n order to produce each of the e i g h t a s p e c t s . : Tbe c l a s s index i s determined f o r each c e l l and s t o r e d i n a (n+1) 2 matrix I? ( f o r subsequent p l o t t i n g o f the map overlay) by P ( j , k ) = f F ( i , j,k)/c]+1 For example, i f the map o v e r l a y was d e s i r e d i n 30-percent s l o p e c l a s s e s then c=30. I f f o r c e l l (j,k) the s l o p e i s 48 percent, then the c e l l i s i n the 30 to 59 percent s l o p e range, which i s the second c l a s s . Hence P(j,k)=2. a t a l l y can be kept of the frequency of the v a r i o u s c l a s s e s i n a v e c t o r T with the c l a s s number, P ( j , k ) , b e i n g the s u b s c r i p t . C o n t i n u i n g with the above example, i t i s necessary to increment the frequency counter f o r c l a s s 2; i . e . , increment 92 T ( P ( j , k ) ) by one where P(j,k)=2. To produce a combination map the topographic f e a t u r e s f o r each c e l l are checked a g a i n s t the a l l o w a b l e ranges s e t by the user ( F i g u r e 6). The corresponding elements of P (the same p l o t t i n g matrix used f o r the c l a s s maps) w i l l be e i t h e r t r u e (1) or f a l s e (0) depending whether the c e l l has the s p e c i f i e d s l o p e , aspect and/or e l e v a t i o n . The combination o v e r l a y mapping could e a s i l y be extended to i n c l u d e other forms of data; f o r example, s o i l s , land-use, w i l d l i f e and wood i n v e n t o r y . The p l o t t i n g of the o v e r l a y map i s done i n two phases. F i r s t , the boundaries of the v a r i o u s c l a s s e s are p l o t t e d ; then, area l a b e l s are added. The o v e r l a y i s most e f f i c i e n t l y produced by u s i n g a g r i d map ( F i g u r e 6) with the s m a l l e s t p l o t t i n g u n i t corresponding t o one g r i d c e l l o f the e l e v a t i o n data base. The C l a s s boundary l i n e s w i l l t h e r e f o r e be e i t h e r v e r t i c a l {"north-south") or h o r i z o n t a l ("east-west"). The boundary l i n e s w i l l only be needed t o separate c e l l s of d i f f e r e n t c l a s s e s . The p l o t t i n g a l g o r i t h m i s g i v e n i n the f l o w c h a r t shown i n F i g u r e 33. 3.4 Implementation Memory r e s t r i c t i o n s made i t impossible t o r e t a i n both the e l e v a t i o n data base and the complete s e t of topographic f e a t u r e s i n computer memory a t the same time. . The topographic f e a t u r e s matrix was conseguently reduced i n s i z e t o a g r i d of 34x34 93 note: plot(x,y,d) moves the plotting pen to (x,y) with pen down (d). no plot (x-l,y,d) penup true Do east/west lines fir s t ; do lines in pairs; 1st east y=l z -plot east line fi r s t ; do west line on return trip penup -^true x<-4 x<66 X-f-X+l initialize plot pen status to up yes no action required; cells are the same. cell above the same? lower pen at start of cell. plot(x,y,up) penupfalse complete rest of line; plot(66,y,d) yes finished pair. y4-y+l do return trip for) the west line; method same as fori east direction i x-s-66; y-t-y+1 finished east line. do north/south lines. Figure 33. Flowchart for Plotting Map Overlays. 94 (1156) c e l l s . E a c h o f t h e s e t o p o g r a p h i c c e l l s was t h e r e f o r e f o u r t i m e s t h e s i z e o f t h e e l e v a t i o n d a t a b a s e c e l l a nd c o n t a i n e d n i n e e l e v a t i o n s . . The l e a s t - s q u a r e s f i t , E q u a t i o n ( 3 . 1 ) , used t o g e n e r a t e t h e t h r e e p a r a m e t e r s o f t h e s u r f a c e p l a n e , was done u s i n g n i n e e g u a t i o n s i n s t e a d o f t h e f o u r as d e s c r i b e d i n S e c t i o n 3.1.1. T h i s r e s u l t e d i n g e n e r a l l y l e s s p r e c i s i o n as t h e v a r i a n c e i n c r e a s e d b e c a u s e t h e t h r e e - p a r a m e t e r , f l a t p l a n e was b e i n g f i t t e d t o f i v e more e l e v a t i o n p o i n t s . To c o n s e r v e s t o r a g e , t h e t o p o g r a p h i c f e a t u r e s f o r e a c h c e l l were combined i n t o one 1 6 - b i t word. T h i s was a c h i e v e d by l i m i t i n g t h e e l e v a t i o n t o 1 0 0 - f o o t c l a s s e s w i t h a maximum e l e v a t i o n o f 12,700 f e e t . , T h i s r e s u l t e d i n 128 c l a s s e s u s i n g s e v e n o f t h e 16 b i t s . S i m i l a r l y , s l o p e was r e s t r i c t e d t o f i v e -p e r c e n t i n t e r v a l s t o a maximum o f 165 p e r c e n t , r e q u i r i n g f i v e b i t s t o r e p r e s e n t t h e 32 c l a s s e s . A s p e c t r e q u i r e s n i n e c l a s s e s and f o u r b i t s w i t h t h e n i n t h c l a s s f o r f l a t a r e a s ( d e f i n e d a s t h o s e u n d e r f i v e p e r c e n t s l o p e ) . T he o v e r l a y naps were done u s i n q t h e a l g o r i t h m i n F i g u r e 33. 95 '; 4.0 Yarding L o c a t i o n and s e t t i n g Design The prime f u n c t i o n o f the Yarding L o c a t i o n and S e t t i n g Design Module i s to a s s e s s the p h y s i c a l f e a s i b i l i t y of v a r i o u s cable systems i n d i f f e r e n t l o c a t i o n s . T h i s i s achieved by comparing the loadpath (the path the l o g s f o l l o w when yarded t o the landing) to the ground p r o f i l e . I f the two i n t e r s e c t such t h a t the f r o n t end of the l o g s can not be c l e a r of the ground then the backspar l o c a t i o n i s probably i n f e a s i b l e . The major problem, then, i s to determine the l o a d p a t h . Theory of Cable Mechanics Cable theory i s w e l l documented i n the l i t e r a t u r e . The USFS has done e x t e n s i v e r e s e a r c h i n t h i s area (Carson 1970,1971,1975). Recently c a b l e theory has been examined a t UBC {Appendix C). A d i f f e r e n t mathematical f o r m u l a t i o n i s r e q u i r e d f o r each cable system depending on i t s geometry {number and c o n f i g u r a t i o n of the l i n e s ) and method of o p e r a t i o n . For a p a r t i c u l a r system, s e v e r a l d i f f e r e n t models can be developed depending on the governing p h y s i c a l assumptions. The most t h e o r e t i c a l l y c o r r e c t model i s c a l l e d a catenary model. T h i s approach assumes t h a t the c a b l e weight i s d i s t r i b u t e d u niformly along i t s l e n g t h . The 96 s o l u t i o n i s complex an d s l o w t o e v a l u a t e and i s t h e r e f o r e n o t s u i t e d f o r f r e g u e n t use on a r e l a t i v e l y slow c o m p u t e r . I t d o e s however s e r v e as an e f f e c t i v e benchmark f o r o t h e r l o a d p a t h m o d e l s . T h e r e a r e two models t h a t a r e u s e f u l f o r t h e t e r r a i n s i m u l a t o r l e v e l o f a n a l y s i s . , {1) P a r a b o l i c a p p r o x i m a t i o n . (2) P e r c e n t D e f l e c t i o n R u l e . 4.1 P a r a b o l i c Model f o r a p p r o x i m a t i n g t h e L o a d p a t h The b a s i c t h e o r y f o r t h e p a r a b o l i c model i s d e v e l o p e d i n a p p e n d i x C. Tbe p a r a b o l i c model d i f f e r s f r o m t h e c a t e n a r y a p p r o a c h b e c a u s e t h e c a b l e w e i g h t i s assumed t o be d i s t r i b u t e d u n i f o r m l y on t h e c h o r d and n o t on t h e c a b l e i t s e l f a s f o r t h e c a t e n a r y a p p r o a c h . The c a b l e s y s t e m d i s c u s s e d h e r e i s a r u n n i n g s k y l i n e { F i g u r e 3 4 ) . The l o a d p a t h i s d e f i n e d by E g u a t i o n 6 f r o m a p p e n d i x C: y=x (L-x) (2R+ x <w |*w3) • (L-X) 2w 2) ) / {4HL) -Ex/L . . . .. . . . . . {4. 1) w i t h y = y - c o o r d i n a t e o f t h e l o a d p o s i t i o n x = x - c o o r d i n a t e o f t h e l o a d p o s i t i o n L = span l e n g t h { h o r i z o n t a l ) R = w e i g h t o f l o a d p l u s c a r r i a g e H = h o r i z o n t a l t e n s i o n i n t h e h a u l b a c k c a b l e segment Figure 34. Schematic Drawing of a Running Skyline System. VO 98 between the c a r r i a g e and the backspar E = e l e v a t i o n d i f f e r e n c e of the spars w = e f f e c t i v e l i n e weight on the subchord of the 1 haulback segment between the l a n d i n g and the c a r r i a g e w2 = e f f e c t i v e l i n e weight on the subchord of the haulback segment between the c a r r i a g e and the backspar w^  = e f f e c t i v e l i n e weight on the subchord of the mainline plos s l a c k p u l l e r The e f f e c t i v e l i n e weights, w , w , w^  are found using w.=w?/cose. where w. i s the a c t u a l c a b l e weight and 6. i s tbe 1 1 1 1 i angle o f the subchord of the r e s p e c t i v e c a b l e segment. The h o r i z o n t a l t e n s i o n , H, i s the governing parameter of the c a b l e system and i s d e f i n e d by Eguation 7 of Appendix C: H=Tcos(arctan ((E+y)/(L-x) +w2 (L-x)/{2H) )) (4.2) The s o l u t i o n f o r y i s n o n - t r i v i a l as y i s c o n t a i n e d i n both s i d e s of Eguation (4.1). Because a c l o s e d form s o l u t i o n i s not p o s s i b l e an i t e r a t i v e approach i s needed. Newton's method f o r s o l v i n g n o n - l i n e a r e g u a t i o n s i s employed because i t i s easy to implement and converges q u i c k l y and dependably. F o l l o w i n g i s a d e s c r i p t i o n of the i t e r a t i v e s o l u t i o n of y i n Equation ( 4 . 1 ) : The b a s i c Newton i t e r a t i v e r e l a t i o n s h i p i s : y =y.-f ( y j / f (y.) 99 To s i m p l i f y development, some common values are e s t a b l i s h e d . Let A=x(L-x) <2H+x(wi+W3)*2w2 (L-x) ) / (4L) B=Ex/L 0={E+y)/(L-x) +w 2(L-x)/{2H) M=Tcos (arctan(0) ) Therefore Eguation (4.1) can he r e w r i t t e n as: y=A/M—B T h i s can be rearranged t o the form f (y) = 0: f (y) =Y+B-A/M=O The problem i s now to f i n d f * ( y ) , the d e r i v a t i v e of f ( y ) . T h i s i s d i f f i c u l t s i n c e the w± are f u n c t i o n s of the sag y through the r e l a t i o n s h i p w =w»/cose with 9 a f u n c t i o n of y. i i i i The h o r i z o n t a l t e n s i o n , H, a l s o i s a f u n c t i o n of sag, y. I n t u i t i v e l y , changes i n » . and H with r e s p e c t t o y are r e l a t i v e l y s m a l l compared to the remaining components of the d e r i v a t i v e which concern the geometry of the system. The approach taken, t h e r e f o r e , was t o s i m p l i f y the d e r i v a t i v e of f (y) by assuming «, and H are independent o f y. T h i s r e s u l t s i n an approximation t o the t r u e f*(y) which a t worst w i l l r e t a r d s l i g h t l y the r a t e of convergence of Newton's i t e r a t i v e r e l a t i o n s h i p . The c o r r e c t i o n f a c t o r f ( y ) / f M y ) w i l l s t i l l converge to zero as y 1 converges to y, p r o v i d i n g f ( y ) i s c o r r e c t l y formulated. I t i s important to note t h a t although w i and H are assumed independent o f y they must be r e c a l c u l a t e d at the beginning of each i t e r a t i o n i n order f o r f ( y ) t o be c o r r e c t . 100 The v a l u e o f H c a n be o b t a i n e d f r o m E g u a t i o n ( 4 . 2 ) . , U s i n g t h i s s i m p l i f i c a t i o n t h e d e r i v a t i v e o f f ( y ) becomes f ' ( y ) = d ( y + B -A/M)/dy T h u s : f * (y) = 1-Adm-t/<3y=1-A/m2 (dm/dy) Now dm/dy=-Tsin ( a r c t a n (U)) d ( a r c t a n (0)) /dy and d ( a r c t a n (U) )/dy=1/J (1+U 2) (L-x)} T h e r e f o r e M»=-Tsin ( a r c t a n (U) )/{ (1+U 2) (L-x)) and f ' (y) = 1-AHVfl2 The i t e r a t i o n e g u a t i o n becomes, y =y *iy *B-A/M)/{1+AW*/M 2) i + l " i i w h i c h i s t e r m i n a t e d when IY ~ y i n t o l e r a n c e i + l i I n i t i a l v a l u e s a r e r e q u i r e d f o r b o t h y and H. S u i t a b l e v a l u e s a r e H = T c o s ( a r c t a n (E/L) and y=0 The s l o p e E/L a c t s as an e s t i m a t e o f t h e c a b l e s l o p e o f t h e h a u l b a c k a t t h e t a i l h o l d . The i n i t i a l g u e s s o f y=0 i s s u f f i c i e n t as Newton*s method i s e x t r e m e l y i n s e n s i t i v e t o s t a r t i n g v a l u e s f o r t h i s a p p l i c a t i o n . . The c o n v e r g e n c e p r o p e r t i e s a r e shown i n F i g u r e 35 w h i c h a r e t y p i c a l o f most s i t u a t i o n s . N o r m a l l y o n l y two t o t h r e e i t e r a t i o n s a r e r e q u i r e d t o c o n v e r g e w i t h i n a t o l e r a n c e o f 0.1 f o o t . POSITION FROM SAG CONVERGENCE TOLERANCE = 0.1 FOOT LANDING (FT) (FT) X= 100 Y= 0 20. 15827618 20.17601203 *** FINAL ESTIMATE X= 200 Y= 0 33.99335120 34.07230361 *** FINAL ESTIMATE X= 300 Y= 0 41.42970191 41.60286993 41.6056442 *** FINAL ESTIMATE X= 400 Y= 0 42.38437297 42.64587 68 42.65G26602 *** FINAL ESTIMATE X= 500 Y= 0 36.75436297 37.03962774 37.04484928 *** FINAL ESTIMATE X= 600 Y= 0 24.38456978 24.57728104 24.5815718 *** FINAL ESTIMATE X= 700 Y= 0 4.973321633 4.999950411 *** FINAL ESTIMATE X= 800 Y= 0 -22.26108228 22.0122756 •22.0163119 *** FINAL ESTIMATE X= 900 Y= 0 -59.74894408 •56.86285058 •56.85955859 *** FINAL ESTIMATE F i g u r e 35. Loadpath S o l u t i o n f o r a Running S k y l i n e System Using the P a r a b o l i c Model and Newton I t e r a t i o n . (Note: computer output has been re-typed f o r c l a r i t y ) 10 2 4.1.1 Comparison to the Catenary Method The p a r a b o l i c model provides r e s u l t s t h a t are extremely c l o s e to those obtained from a catenary s o l u t i o n . An example i l l u s t r a t i n g t h i s i s given i n Table VII which shows the e r r o r s i n t e n s i o n s and l o a d p o s i t i o n s f o r v a r y i n g v a l u e s of E and L a t midspan. T h i s example i s f o r a running s k y l i n e system { M a d i l l 052 t e n s i o n skidder) with a 35000 pound l o a d and a maximum al l o w a b l e haulback t e n s i o n of 56500 pounds {the approximate l i n e p u l l c a p a c i t y of the yarder i n t e r l o c k ) . The a b i l i t y of the p a r a b o l i c t o c l o s e l y approximate the catenary has a l s o been v e r i f i e d i n f i e l d s t u d i e s {Guimier, 1977). The d i f f e r e n c e s between the two models w i l l always be s m a l l p r o v i d i n g the subchords approximate the c a b l e s between tbe l a n d i n g and the c a r r i a g e and the c a r r i a g e and the t a i l h o l d . T h i s c o n d i t i o n w i l l minimize the e f f e c t of the u n d e r l y i n g assumption i n the p a r a b o l i c model that the cable weight i s d i s t r i b u t e d on the subchords and not on the c a b l e i t s e l f . In most c a b l e y a r d i n g setups the subchords do approximate the c a b l e s unless the system i s g r o s s l y underloaded. 4.1.2 Improvement of the I t e r a t i v e Method Although no other method converges more q u i c k l y than Newton's method, the s o l u t i o n i s s t i l l f a i r l y slow e s p e c i a l l y i f the i t e r a t i o n must be invoked at s e v e r a l p o i n t s along the span. One p o s s i b l e improvement, when computing at s e v e r a l p o i n t s i n 103 TABLE V I I ERRORS FROM THE PABABOLIC MODEL WITH THE CARRIAGE AT MIDSPAN FOR A MADILL 052 TENSION SKIDDER LOAD INCLUDING CARRIAGE = 35000 L B . TENSION (HAULBACK) AT INTERLOCK = 56500 LB. WEIGHT OF HAULBACK/FOOT = 2.34 LB. WEIGHT OF MAINLINE+SLACKPULLER = 4.13 LB. SPAN = 1400 F T . ERRORS IN TENSIONS AT THE CARRIAGE CHORD SLOPE% 100 ERROR IN LOAD POSITION (FT.) 0.0686 HORIZONTAL COMPONENT VERTICAL « AXIAL « 50 0.0053 HORIZONTAL VERTICAL AXIAL -0.0132 -0.0249 HORIZONTAL VERTICAL AXIAL -50 HORIZONTAL VERTICAL AXIAL -100 -0.0542 HORIZONTAL VERTICAL AXIAL HAULBACK LNDG-CARR (LB.) 0.34 0.79 - 0 . 17 -0.20 -0.74 -0.05 0.08 •0. 33 0.04 •0.63 1.23 0.09 •2.41 2. 20 0. 13 HAULBACK CARR-TAIL (LB.), -5.63 4.23 -0.25 -1.82 2.55 0,04 0.02 -0.30 0.20 -0.90 -2.82 -0.53 -3.46 -4.59 -0.05 MAINLINE + SLACKPULLER (LB.) -11.60 -8.79 - 3 . 84 - 3 . 44 -4. 18 - 1 . 81 -0.03 1.02 0.41 -1.13 4.42 0.63 -4.54 7. 28 2.56 104 the span, i s to use s u p e r i o r i n i t i a l guesses. T h i s can be achieved by use of a d i f f e r e n c e t a b l e . A d i f f e r e n c e t a b l e i s a t a b u l a r e x p r e s s i o n of an i n t e r p o l a t i n g polynomial (Conte and deBoor, 1973). Using the loadpath r e s u l t s shown i n F i g u r e 35 a d i f f e r e n c e t a b l e can be c o n s t r u c t e d and i s shown i n Table V I I I . The f i r s t column of TABLE VI I I DIFFERENCE TABLE FOR SAG (Y) 100 200 300 400 500 20. 18 34.07 41. 60 42.65 37.04 I D i D* 13.89 7.53 1.05 -5.61 1 -6.36 J -6.48 | -6.66 d i f f e r e n c e s approximate the f i r s t d e r i v a t i v e ; second column of d i f f e r e n c e s approximate the second d e r i v a t i v e , e t c . As seen from Table V I I I , the second d i f f e r e n c e i s almost constant i n d i c a t i n g t hat the f u n c t i o n can be approximated by a parabola. Values f o r the f u n c t i o n i n the d i f f e r e n c e t a b l e can be e x t r a p o l a t e d by u s i n g the approximate d e r i v a t i v e v a l u e s : y =y *D»+D2 <D:«delta n) i + l i i i where Df=y.-y. . i i i - l D 2=Di-D» i i i - l - (y.~y. . ) - (y. -y. 0 ) i i - l i - l i-2 The above eguation f o r y can be s i m p l i f i e d t o y i e l d : i+l 105 y =3y -3y +y 1 i + l Ji i - l *l-2 For example, an estimate for y can be made at x=500 by using the second to l a s t diagonal of the table: y = 42.65 • 1.05 - 6.48 = 37.22 This represents an error of only 0.18 f o o t . Employing the difference table to compute i n i t i a l values of y w i l l v i r t u a l l y guarantee convergence i n one i t e r a t i o n thereby almost halving the computational time required. 4.2 Percent Deflection Rule for Approximating the Loadpath The percent rule method i s applicable to any cable system. The percent deflection (for a system) i s defined here as the minimum attainable midspan load displacement from the chord divided by the length of the chord (Figure 36) and i s usually known through l o c a l experience. The application of the percent rule, as currently used i n the forest industry f o r predicting loadpaths, has two serious drawbacks: (a) One universal deflection value i s used f o r one system regardless of the yarding conditions. (b) The percent deflection defines the load position only at midspan providing l i t t l e information about the rest of the span. The effect of applying one deflection value for a l l yarding conditions i s c l e a r l y i l l u s t r a t e d (for a running skyline system) in Table IX. The deflections given are computed using a L loadpath defined by the i n t e r p o l a t i n g polynomial y=x((4Ym-E)/L+2x(E-2Ym)/L ) node 3; (L,E) (backspar) Figure 36 . Using an Interpolating Polynomial to Predict the Loadpath. o Ov I TABLE IX DEFLECTION AT MIDSPAN FOB MADILL 052 TENSION SKIDDER (CHORD DEFINITION) LOAD INCLUDING CARRIAGE = 350 00 LB., TENSION (HAULBACK) AT INTERLOCK = 56500 LB. HEIGHT OF HAULBACK/FOOT = 2.'34 LB. HEIGHT OF H8INLINE+SLACKPULLER = 4.13 LB. CHORD SPAN - FEET SLOPE5S 800 1000 1200 1400 1600 1800 2000 (PERCENT DEFLECTIONS AT MIDSPAN) 100 9. 4 9. 6 9. 7 9. 9 10.0 10. 2 10. 4 90 9. 3 9. 5 9. 6 9. 8 9.9 10. 1 10. 3 80 9. 2 9. 4 9. 5 9. 7 9.8 10. 0 10. 1 70 9. 2 9. 3 9. 4 9. 6 9.7 9. 9 10. 0 60 9. 1 9. 2 9. 3 9. 5 9.6 9. 8 9. 9 50 9. 0 9. 1 9. 2 9. 4 9.5 9. 7 9. 8 40 8. 9 9. 0 9. 1 9. 3 9.4 9. 5 9. 7 30 8. 7 8. 9 9. 0 9. 1 9.3 9. 4 9. 5 20 8. 6 8. 8 8. 9 9. 0 9.2 9. 3 9. 4 10 8. 5 8. 6 8. 8 8. 9 9.0 9. 2 9. 3 0 8. 4 8. 5 8. 6 8. 8 8.9 9. 0 9. 2 -10 8. 3 8. 4 8. 5 8. 7 8.8 8. 9 9. 1 -20 8. 2 8. 3 8. 4 8. 6 8.7 8. 9 9. 0 -30 8. 1 8. 2 8. 3 8. 5 8.6 8. 8 8. 9 -40 8. 0 8. 1 8. 3 8. 4 8.6 8. 7 8. 9 -50 7. 9 8. 1 8. 2 8. 4 8.5 8. 7 8. 9 -60 7. 9 8. 0 8. 2 8. 4 8. 5 8. 7 8. 9 -70 7. 8 8. 0 8. 2 8. 3 8.5 8. 7 8. 9 -80 7. 8 8. 0 8. 2 8. 4 8.5 8. 7 8. 9 -90 7. 8 8. 0 8. 2 8. 4 8.6 8. 8 9. 0 -100 7. 8 8. 0 8. 2 8. 4 8. 6 8. 8 9. 0 -90 9. 2 9. 5 9. 7 10. 0 10.2 10. 5 1 0. 8 -100 9. 7 10. 0 10. 3 10. 5 10.8 1 1. 1 11. 4 108 catenary model. The average midspan deflection i n the Table IX i s approximately nine percent. The r e s u l t of using a sing l e deflection value for a l l conditions i s c l e a r l y indicated since actual deflections range from 7 .8 to 10. fJ percent causing a maximum error of 33 .6 feet. These errors are of s u f f i c i e n t magnitude to cause concern. This problem can be a l l e v i a t e d by using deflection tables s i m i l a r to the one in Table IX (Lysons and Mann, 1967 ) . Often the percent deflection i s defined not by the chord length but by the span length. The application of a universal deflection value defined i n t h i s manner yields even greater errors as the e f f e c t of the elevation difference of the spars i s t o t a l l y ignored. The large variations of the deflections calculated i n t h i s fashion are shown i n Table X. The other drawback of the percent rule i s i t s i n a b i l i t y to predict the loadpath for the entire span. This can be overcome by using an i n t e r p o l a t i n g polynomial. There are three known points on the loadpath: landing spar, midspan load position (Ym) and t a i l h o l d (Figure 36). These can be used to generate a second degree polynomial. Starting with the general expression of a second degree polynomial; p(x)=a 0+a 1(x-x 0)+a 2 (x-x Q) {x-x^ The c o e f f i c i e n t s a , a , a can be found by recursively using 0 1 2 this expression and the three nodes (Shampine and A l l e n , 1973 ) . TABLE X DEFLECTION AT MIDSPAN FOR MADILL 052 TENSION SKIDDER (SPAN DEFINTION) LOAD INCLUDING CARRIAGE = 35000 LB. TENSION (HAULBACK) AT INTERLOCK = 56500 LB. WEIGHT OF HSUL8ACK/FOOT = 2.34 LB. WEIGHT OF MAINLINE*SLACKPDLLER = 4.13 LB. CHORD SPAN ~ FEET SLOPES? 800 1000 1200 1400 1600 1800 2000 (PERCENT DEFLECTIONS AT MIDSPAN) 100 13.3 13.5 13.8 14. 0 14.2 14. 4 14.7 90 12.5 12.8 13.0 13.2 13. 4 13.6 13.8 80 11.8 12.0 12. 2 12.4 12.6 12. 8 13.0 70 11.2 11.4 11.5 11.7 11.9 12. 1 12.2 60 10.6 10.7 10.9 11. 1 11.2 11.4 11.6 50 10.0 10.2 10.3 10.5 10.6 10. 8 11.0 40 9.5 9.7 9.8 10.0 10. 1 10.3 10.4 30 9. 1 9.3 9.4 9.5 9.7 9.8 10.0 20 8. 8 8.9 •9.1 9.2 9.3 9.5 9.6 10 8.5 8.7 8.8 8.9 9. 1 9.2 9.3 0 8.4 8.5 8.6 8. 8 8.9 9.0 9.2 -10 8.3 8.4 8.6 8.7 8. 8 9.0 9. 1 -20 8.3 8.5 8.6 8.7 8.9 9.0 9.2 -30 8.4 8.6 8.7 8. 9 9.0 9.2 9.3 -40 8.6 8.8 8.9 9.1 9.2 9.4 9.6 -50 8.9 9.0 9.2 9.4 9.6 9.7 9.9 -60 9.2 9.4 9.6 9.7 9.9 10. 1 10.3 -70 9.6 9.8 10.0 10.2 10.4 10.6 10.8 -80 10.0 10.2 10. 5 10.7 10.9 11.2 11.4 -90 10.5 10.7 11.0 11.3 11.5 11.8 12.0 -100 11.0 11.3 11.6 11.9 12.2 12.5 12.8 110 Using the f i r s t node (p 0 = Q,xQ = 0) (landing) P<x 0)=a 0 Therefore a o = p o = 0 Adding the second node (p =Ym#.x =1/2) (midspan) p (x )=a +a (x -x ) * 1 0 1 1 0 Therefore a = (p -a ) / (x -x ) 1 **1 0 1 0 a =2Ym/L F i n a l l y , using the t h i r d node (p 2 = E,x 2=L) (backspar) P(x 2)=a Q+a 1(x 2-x ( )) • . a 2 ( x 2 - x 0 ) ( x ^ x ^ Therefore • a 2 = ( P 2 - a 0 - a i { X 2 - X 0 ) ) / ( X 2 - X 0 ) ( X 2 - X 1 > a 2 = 2 (E-2Ym) / L 2 S u b s t i t u t i n g the v a l u e s o f , a^, a^ i n t o the gene r a l polynomial and s i m p l i f y i n g y i e l d s y=x({4Ym-E}/L+2x(E-2Ym)/L 2) (4.3) Ev a l u a t i o n o f t h i s e quation i s easy but can be made s i m p l e r by co n v e r t i n g i t to a d i f f e r e n c e t a b l e form. Assuming the l o a d p o s i t i o n s are t o be determined at r e g u l a r i n t e r v a l s of I , the r e c u r s i v e r e l a t i o n s h i p i s : D*=D* , *D 2 X x-1 y ,=y . , i i - i i The i n i t i a l D* i s s e t to y1 which i s c a l c u l a t e d from Eguation (4.3). The D 2 i s the second d e r i v a t i v e which i s const a n t . I t i s found by d i f f e r e n t i a t i n g Eguation (4.3) twice to y i e l d : D 2=I 24 (E-2Ym) / L 2 111 The r e s u l t s from t h i s i n t e r p o l a t i n g polynomial compare f a v o u r a b l y t o those o b t a i n e d from catenary model (Table XIJ p r o v i d i n g the midspan d e f l e c t i o n i s known. These e r r o r s are r e l a t i v e l y i n s i g n i f i c a n t c o n s i d e r i n g the accuracy o f the 25-foot contour maps t h a t are g e n e r a l l y used. The e r r o r s shown i n tbe t a b l e were determined at qu a r t e r span by f i r s t computing the midspan d e f l e c t i o n u s i n g the catenary model. Then, a p p l y i n g Eguation (4.3), the l o a d p o s i t i o n a t qu a r t e r span was determined and compared t o the corres p o n d i n g l o a d p o s i t i o n d e r i v e d from the catenary model. I t should be noted that the maximum e r r o r w i l l not occur e x a c t l y at quarter or t h r e e - q u a r t e r span (the order o f magnitude of the e r r o r a t qu a r t e r and t h r e e - q u a r t e r span i s v i r t u a l l y the same) but w i l l vary s l i g h t l y depending on the op e r a t i n g c o n d i t i o n s . 6.3 Implementation The f r e q u e n t use of the loadpath c a l c u l a t i o n s d i c t a t e d tbe use o f the percent r u l e because the slow e x e c u t i o n speed of the computer made computation times f o r catenary and p a r a b o l i c methods e x c e s s i v e . The ground p r o f i l e i s determined a t 31 p o i n t s along the i n i t i a l l y d e f i n e d span. The l o a d p o s i t i o n s are then computed at these p o i n t s using Equation (4.3) and a d i f f e r e n c e t a b l e . The loadpaths generated are reasonably accurate (Table XI) as l o n g as tbe user employs d e f l e c t i o n s from a t a b l e l i k e t h a t given i n T a b l e IX. 112 TABLE X1 PERCENT RULE ERROR AT QUARTER SPAN FOR A MADILL 052 TENSION SKIDDER LOAD INCLUDING CARRIAGE = 35000 LB. TENSION <(HAOI.-BACK) AT INTERLOCK = 56500 LB. WEIGHT OF HAULB ACK/FOOT » 2.34 LB. WEIGHT OF MAINLINE*SLACKPULLER = 4.13 LB. CHORD SPAN - FEET SLOPE% 800 1000 1200 1400 1600 180 0 2000 (ERRORS IN FEET) 100 -5.6 -7.3 - 9 . 1 - 11. 1 -13.2 -15.4 -17.8 90 -5.0 -6.6 -8.2 -10. 0 -1 1.9 -13.9 -16.0 80 -4. 5 -5.9 -7.3 -8.9 -10.6 -12.4 -14.3 70 -4.0 -5.2 - 6 . 5 -7.9 -9.5 -11. 1 -12.8 60 -3.5 -4.6 -5.8 -7.0 -8.3 -9.8 - 1 1 . 3 50 -3.0 -4.0 -5.0 -6. 1 -7.3 -8.5 -9.9 40 -2.6 -3.4 -4.3 -5.2 -6.3 -7.3 -8.5 30 -2.1 -2.8 -3.6 -4.4 -5.2 -6.2 -7.2 20 -1.7 -2.2 -2.8 -3.5 -4.2 -5.0 -5.9 10 -1.2 -1.7 -2. 1 -2.7 -3.3 -3.9 -4.6 0 -0.8 - 1 . 1 -1.5 - 1 . 9 -2.3 -2.9 -3.4 -10 - 0 . 4 -0.6 -0.8 - 1 . 1 -1.5 -1.9 -2.3 -20 0.0 -0.1 -0.2 - 0 . 4 -0.6 -0.9 -1.2 -30 0.4 0.4 0.4 0. 4 0.3 0. 1 -0.0 -40 0.7 0.8 0.8 0.9 0.9 0.8 0.7 -50 1 .1 1.3 1.4 1.6 1.7 1. 8 1.8 -60 1.4 1.7 2.0 2.2 2.4 2.6 2.7 -70 1.7 2.1 2.5 2.8 3. 1 3.4 3.7 -80 2. 1 2.5 3.0 3.4 3.8 4.2 4.6 -90 2.4 3.0 3.5 4. 0 4.6 5. 1 5.6 -100 2.7 3.4 4.0 4.6 5.3 5.9 6.5 113 Actual yarding roads to be used in harvesting cannot be projected as there i s too much uncertainty about the ground shape due to the generally low accuracy of the available contour maps. The effectiveness of this module i s therefore limited to general development planning.. 114 5.0 Road L o c a t i o n The b a s i c component o f t h e Road L o c a t i o n Module i s t h e p r o j e c t i o n of r o a d s l o c a t i o n s . T h i s i s a c c o m p l i s h e d i n e i t h e r 'manual* o r ' a u t o m a t i c ' mode. 5.1 Manual Road P r o j e c t i o n The r o a d l o c a t i o n i s e n t e r e d v i a t h e d i g i t i z e r . The o n l y c o m p u t a t i o n r e q u i r e d i s t o t e s t w h ether t b e g r a d e between s u c c e s s i v e t r i a l p o i n t s i s w i t h i n a l l o w a b l e a d v e r s e and f a v o u r a b l e l i m i t s . L e t (x.,y .) be t h e c o o r d i n a t e s a n d z. t b e e l e v a t i o n o f t h e X X X l a s t r e c o r d e d p o i n t o f t h e l o c a t i o n . Then a new p o i n t ( x , y # z ) i s a c c e p t a b l e i f : - | a i < 1 0 0 { z - z j / ( m a p s c a l e / ( x ^ x ) 2 + ( y ^ y ) 2 < I f I where I a] and | f j a r e t h e maximum a l l o w a b l e a d v e r s e and f a v o u r a b l e r o a d g r a d e s . X f t h e above i s t r u e t h e n t h e newest p o i n t on t h e r o a d l o c a t i o n i s : x = x and y =y i+l - i+l 7 I f t b e g r a d e l i m i t s have been v i o l a t e d t h e n a s u i t a b l e 115 warning i s given and a replacement p o i n t ( x r , y r ) i s expected. An o v e r r i d e ( i . e . accept (x,y) i r r e s p e c t i v e o f the slope) can be i n c o r p o r a t e d by t e s t i n g i f the two p o i n t s are w i t h i n a c e r t a i n d i s t a n c e of each other: use (x,y) i f (x r~x) 2+ (y r-y) 2 < t o l e r a n c e 2 The c o o r d i n a t e s and a s s o c i a t e d e l e v a t i o n s can be s t o r e d i n three v e c t o r s X,Y and Z. To permit s e v e r a l roads i n memory a t one time a f o u r t h v e c t o r , H, can be added which c o n t a i n s the road numbers f o r the corresponding elements i n X,Y and Z. 5,2 Automatic fioad P r o j e c t i o n The technique l o c a t e s , i n the h o r i z o n t a l plane, the average grade between the s t a r t and end point s by s e a r c h i n g i n s u c c e s s i v e i n t e r v a l s towards the f i n i s h . The method prov i d e s a h e u r i s t i c s o l u t i o n and i s not guaranteed t o be o p t i m a l . Let the s t a r t i n g and f i n i s h i n g p o i n t s be (x ,y 1# z ±) with (i=0) and ( x f , y f , z f ) , r e s p e c t i v e l y . The average grade (q) between the two i s given by: g=1Q0{z f-z i )/(mapscale /{x f-x ±) 2+ ( y f - y ± ) 2 ) An immediate check on f e a s i b i l t y can be made. I f g i s exceeds the grade l i m i t s (a,f) then the process i s terminated. 1-he s e a r c h i n g f o r the grade g w i l l be done at a d i s t a n c e o f 1 1 6 L from ( x ± r y ± ) , along the l i n e AB (Figure 37) . The l i n e AB i s constructed perpendicular to the l i n e between ^ x±'^±^ a n d (x , y ). The intersection of the two l i n e s i s at (x ,y ); f f s s x =x.+Lcos0 and y = y.+Lsin8 S 1 S ' 1 with e =arctan ((y. -y ) / (x. -x )) i f i f Searching proceeds along A l for the point (x,y), such that the grade between (x,y) and ( x i»Y 1) 1 S e < J i a l to the average grade g. Points along AB can be expressed as a distance (d) from <* s,y s): x=xs+dcos ( 6+90) and y=ys + dsin ( 6*90) ........ (7. 1) The grades between tbese points and (x 1,y i) can therefore be expessed as a function of d. When the correct value of d i s found the resultant grade s w i l l egual g. Hence: f (d) =s-g=0 .... , . ,,, . .. ... (7. 2) This i s a non-linear eguation i n d and can be solved by using secant i t e r a t i o n . Let d Q and d^ be the f i r s t two, a r b i t r a r i l y chosen, guesses and f Q and f 1 the respective function values of Eguation ( 7 . 2 ) . Subseguent estimates of d can be made by d = d - f (d -d )/(f - f ) J+l 3 3 3 J-l 3 j - l The i t e r a t i o n can be stopped when: |d^ + 1~d^1<tolerance. The value of f # 1 1 should be near zero. The method works well when J+l the ground p r o f i l e along A~B i s r e l a t i v e l y uniform; i n i r r e g u l a r Figure 37. Planametric View of 'Automatic' Road Location. 118 t e r r a i n the r e s u l t s are h i g h l y u n s t a b l e . An i t e r a t i o n counter provides a safeguard a g a i n s t non-convergence* The c o o r d i n a t e l o c a t i o n of (x,y) f o r the f i n a l i t e r a t i o n value of d i s found using Eguation {7.1) and becomes the new s t a r t i n g p o i n t of the road: x. , =x and y , =y i+l 7 i + l The process i s repeated u n t i l the endpoint of the road i s reached. There are t h r e e parameters t h a t c o n t r o l the l o c a t i o n process (Figure 38): (1) The d i s t a n c e , L, t h a t AB i s from ( x . , y . ) . T h i s value i i c o n t r o l s the number o f s t e p s r e q u i r e d to l o c a t e the road and should be chosen t o ensure t h a t no major breaks i n the topoqraphy are missed. A s u i t a b l e value i s the average h o r i z o n t a l d i s t a n c e between conto u r s . T h i s can be d i f f i c u l t to estimate as t h e r e i s o f t e n s u b s t a n t i a l v a r i a t i o n w i t h i n a map. (2) The d i s t a n c e , d, along AB. T h i s c o n t r o l s the f l e x i b i l i t y of the r o u t e ; a l a r q e value w i l l allow the road to wander. Through experimentation i t was concluded t h a t d should not be g r e a t e r than twice the value of I. (3) The s t o p p i n g t o l e r a n c e . Tolerance should be f i n e enough to assure that the proper d (f (d)=s-g=0) has been l o c a t e d but too s m a l l a value w i l l r e g u i r e too many i t e r a t i o n s . A 119 Figure 38. The Effect of the Controlling Parameters on 'Automatic' Road Location 120 s u i t a b l e t o l e r a n c e value was found to be f i v e percent of the maximum a l l o w a b l e d, as s p e c i f i e d i n (2) above. 5.3 l o a d l o c a t i o n Output Three forms of output are a v a i l a b l e : p r o f i l e and plan-view p l o t s and, ground i n f o r m a t i o n along the road l o c a t i o n . The ground i n f o r m a t i o n ( e l e v a t i o n , s i d e s l o p e s , grade and d i s t a n c e from the s t a r t o f the road) i s computed f o r each c o o r d i n a t e p a i r d e s c r i b i n g the road l o c a t i o n . The s i d e s l o p e i s determined by computing an e l e v a t i o n o f f to the s i d e o f the road and using i t to c a l c u l a t e the s l o p e to the road. 5.4 Earthwork C a l c u l a t i o n s and G r a d e l i n e L o c a t i o n The procedures f o r a l l o w i n g g r a d e l i n e l o c a t i o n and subsequent c a l c u l a t i o n s of volumes are w e l l documented i n the l i t e r a t u r e (Burke, 1974). T h i s f e a t u r e has not been i n c o r p o r a t e d i n the Road L o c a t i o n Module because the contour maps a v a i l a b l e i n B r i t i s h Columbia are not s u f f i c i e n t l y a c c u r a t e to p r o v i d e the necessary d e t a i l r e q u i r e d i n earthwork volume c a l c u l a t i o n s . 5.5 Implementation The data s t r u c t u r e (200x3 array) allows f o r 200 road c o o r d i n a t e s . Column one i s the road number; columns two and three h o l d the (x,y) c o o r d i n a t e s . 121 The parameters c o n t r o l l i n g the automatic road p r o j e c t i o n were chosen through t r i a l and e r r o r such that the road generated did not wander e x c e s s i v e l y or d i s r e g a r d important t e r r a i n f e a t u r e s . S u i t a b l e parameters are: L = 0.125 i n c h e s * map s c a l e 1ABJ = 0.5 inches * map s c a l e t o l e r a n c e f o r convergence = 0.01 inches., To conserve memory, the e l e v a t i o n s were not s t o r e d f o r each p a i r of Jx,y) road c o o r d i n a t e s . These c o u l d be generated whenever needed by using the (x,y) c o o r d i n a t e s t o compute the e l e v a t i o n from the e l e v a t i o n data base. 122 6.0 A e s t h e t i c A n a l y s i s The A e s t h e t i c A n a l y s i s Module c o n s i s t s o f two main p a r t s : the viewable area assessment and the three dimensional r e p r e s e n t a t i o n s . 6 .1 Viewable Area Assessment The viewable area assessment f u n c t i o n a l l o w s the user t o determine the area o f the data base t h a t i s viewable from a c e r t a i n l o c a t i o n . , A person's l i n e o f s i g h t i s s i m u l a t e d i n a f u l l 360 degree spectrum from the viewpoint l o c a t i o n (x*,y*,z*) which i s entered from the d i g i t i z e r . The map area i s s u b d i v i d e d i n t o a g r i d of n 2 c e l l s t h a t correspond to the e l e v a t i o n matrix, E. The c e l l s are checked by p r o j e c t i n g s i g h t l i n e s from the viewpoint to each of the 4 (n-1) border c e l l s . Each c e l l , upon completion of the a n a l y s i s , w i l l be e i t h e r viewable (true) or hidden ( f a l s e ) . 6 . 1 . 1 Determining the Viewable C e l l s on a S i g h t L i n e The d i g i t i z i n g c o o r d i n a t e system i s s c a l e d t o correspond to the s u b s c r i p t s of the e l e v a t i o n matrix. x* = nx*/u*1 and y* = ny*/v*1 123 The sight lines are intersected with each c e l l encountered between the viewpoint and the border c e l l . , To simplify computations a l l c e l l s are examined at the same r e l a t i v e position as the viewpoint has within i t s c e l l . The border subscripts (x b,y b) are s h i f t e d to the same r e l a t i v e position as the viewpoint (Figure 39) by x = x* - f"x*] +xu and y^  = y* - IY*] • Y, b b b b The length of the sight l i n e i s found using b •= /(y b-Y*) 2 * (x b-x*) a The planametric angle, 0, of the sight l i n e i s computed from 0= arctan{ (y*-Y,)/(x*-x,)) D b Commencing at the viewpoint with (x i,y i) (i=0) and x±=x* and y =y* the next point on the l i n e to be examined i s : x i + 1 = x± * Dcos 0 and Y±+i ~ J± * Dsin0 The value D i s the increment required to advance between two c e l l s along tbe sight l i n e . By setting D to be the smaller of L/(y*-y ) and L/(x*-x ) every c e l l i n the grid w i l l have been b b examined at least once upon completion of a l l the sight l i n e s (Figure 45) . The c e l l containing (x ,y ) i s viewable i f the slope, i+l i+l s , from (x ,y , ) to the viewpoint i s greater than the i+l i+l ' i+l largest slope yet encountered for the sight l i n e (Figure 40). The slope i s calculated using s = (z -z*)/D . A t a l l y f o r " i+l i + l the c e l l i s kept in d i c a t i n g the number of times the c e l l has been checked and the number of occasions i t was viewable. Upon completion of a l l the sight l i n e s , each c e l l w i l l be defined as (x t ?,y b) bore \ \ er c e l l \ \ \ L A \ L / ( y , - y * \ \ VAX y b -y* / e • i \ \ ^ viewpo: nt (x*,y*) L / ( x . - x * f '-'"'point i n j r i d c e l l to 1 je checked ; c e l l e \ x,-x* b Figure 39. Locating Grid C e l l s Along the Sight Line. (S3 4> Figure 40. Determining the Viewable Area Along a Sight Line. 126 viewable (true) i f at l e a s t 50 percent of the checks made were t r u e . Tbe o v e r l a y map showing the viewable areas i s produced using the a l g o r i t h m i n Fig u r e 33. fin allowance must be made f o r viewpoints l o c a t e d o u t s i d e of the map area. The only adjustment r e q u i r e d t o the procedure already o u t l i n e d i s t o s k i p over f i c t i t i o u s c e l l s ; i . e . , those that l i e o u t s i d e the map area. An allowance must a l s o be made f o r cases when o b s t r u c t i o n s , not represented by the e l e v a t i o n data base, scr e e n the viewpoint l o c a t i o n (Figure 16). T h i s i s accomplished by g i v i n g the user the o p p o r t u n i t y to o v e r r i d e the i n i t i a l viewing angle s e t by the computer f o r each s i g h t l i n e . 6.1.2 Implementation Memory l i m i t a t i o n s d i c t a t e d the use of a g r i d s i z e of o n l y 29x29 (741) c e l l s . In a d d i t i o n , c e l l s were deemed viewable i f at l e a s t one s i g h t l i n e t e s t showed the c e l l to be viewable. T h i s e l i m i n a t e d the need to keep two t a l l i e s f o r each c e l l but in t r o d u c e d a b i a s i n favour o f viewable a r e a s . 127 6.2 Three Dimensional R e p r e s e n t a t i o n s o f the E l e v a t i o n Data Base There are two b a s i c types of three dimensional p r o j e c t i o n s ; o r t h o g r a p h i c and p e r s p e c t i v e . Producing t h r e e dimensional (3D) r e p r e s e n t a t i o n s of the data base i s a three s t e p process r e g a r d l e s s of the p r o j e c t i o n : (a) t r a n s f o r m i n g the e l e v a t i o n data base to the c o r r e c t o r i e n t a t i o n f o r p l o t t i n g ; (b) p l o t t i n g t h e 3D r e p r e s e n t a t i o n of the e l e v a t i o n data base and c o n c u r r e n t l y removing the hidden areas from the p l o t ; (c) p l o t t i n g the l o c a t i o n s of the planametric d e t a i l s , l i k e roads and s e t t i n g boundaries, on the 3D r e p r e s e n t a t i o n o f the e l e v a t i o n data base. S e v e r a l parameters must be d e f i n e d ; (a) area of the data base to be viewed; (b) the c a r d i n a l viewing d i r e c t i o n (N,S,E,W); (c) the viewing angle w i t h i n the c a r d i n a l d i r e c t i o n ; (d) the viewing angle above the h o r i z o n ; (e) the viewpoint l o c a t i o n ( p e r s p e c t i v e ) ; (f) the viewing a p e r t u r e ( p e r s p e c t i v e ) . The area to be viewed i s d e f i n e d by a r e c t a n g l e that i s p a r a l l e l with the r e c t a n g l e o u t l i n i n g the map area of the data base. T h i s r e c t a n g l e i s l o c a t e d by d i g i t i z i n g i t s lower l e f t (x , y ) and upper r i g h t (x ,y ) c o r n e r s . These c o o r d i n a t e s must L L R R be converted to s u b s c r i p t s using Eguation (1.1): 128 j =[x n/u]+1 and k =[y n/v ]+1 Li Li LI Lt j =[x n/u]+1 and k =[y n/v]+1 K. K K K 6.2.1 Transforming the E l e v a t i o n Data Base The elements w i t h i n the e l e v a t i o n matrix must be r o t a t e d to c o i n c i d e with the c a r d i n a l viewing d i r e c t i o n . T h i s i s done because, although the t r a n s f o r m a t i o n to be d e s c r i b e d can handle any r o t a t i o n between 0-360 degrees, the hidden area a l g o r i t h m must work from the foreground to the background of the e l e v a t i o n matrix. Consequently, i f the foreground e l e v a t i o n s i n the matrix have t h e i r c o o r d i n a t e s r o t a t e d to be i n the background {rotation=180) the e l e v a t i o n s would s t i l l be p l o t t e d f i r s t but should i n f a c t be done l a s t . T h e r e f o r e the matrix of e l e v a t i o n s must be " p h y s i c a l l y " r o t a t e d to the a p p r o p r i a t e c a r d i n a l d i r e c t i o n . The f o l l o w i n g r o t a t i o n s are r e g u i r e d : - i f l o o k i n g NOBTH then no r o t a t i o n i s needed - i f l o o k i n g SOUTH then r o t a t e 180 degrees c l o c k w i s e - i f l o o k i n g EAST then r o t a t e 270 degrees cl o c k w i s e - i f l o o k i n g WEST then r o t a t e 90 degrees c l o c k w i s e To conserve memory the matrix i s r o t a t e d w i t h i n i t s e l f . The e l e v a t i o n matrix can be considered a s e r i e s of (n+1)/2 c o n c e n t r i c boxes. The boxes are i n d i v i d u a l l y r o t a t e d by s y s t e m a t i c a l l y r o t a t i n g each symmetric group of f o u r elements; i . e . , one element per s i d e ( f i g u r e 41). The r o t a t i o n a l g o r i t h m i s presented i n F i g u r e 42. 129 130 j=n+l ^4 J-J-I j<<l+(n+l)/2)* yes no k=k-l k<n+3-j yes no t = E(n+2-j,k) E(n+2-j,k) = E(k,j) E(k,j) = E(j,n+2-k) E(j,n+2-k) = E(n+2-k,n+2-j) E(n+2-k,n+2-j) = t Figure 42. Flowchart for the Rotation of the Elevation Matrix * The symbol pair Q refers to the nearest lowest integer. 131 The s u b s c r i p t s d e f i n i n g the viewing area of the matrix must a l s o be r o t a t e d : l o o k i n g west: l o c k i n g South: y=n«-2-kR x=n+2-j R k R = n + 2 - * L lo o k i n g East: j =n+2-k L R k = L n + 2-x R k = L y The a c t u a l viewing t r a n s f o r m a t i o n can now be a p p l i e d . The angle , 9 , i s the viewing r o t a t i o n w i t h i n the c a r d i n a l d i r e c t i o n and must be a p p l i e d f i r s t . The angle, 3, i s the r o t a t i o n above 132 the h o r i z o n and i s a p p l i e d second (Figure 43). The angle 9 i s a r o t a t i o n about the z - a x i s ( e l e v a t i o n ) and the angle 8 i s a r o t a t i o n about the x - a x i s . A r o t a t i o n about the y - a x i s Idepth) would t i l t the data base which was deemed u n d e s i r a b l e and was t h e r e f o r e not i n c l u d e d . , The r o t a t i o n f o r a p a i r of axes i s simple and i s found i n most g e n e r a l mathematic books; x*=x.cose +y.sin6 y•=y.cos8-x.sin6 Th i s r o t a t i o n can be a p p l i e d t o the three dimensional case (Newman and S p r o u l l , 1973): Ro t a t i o n about the z - a x i s (6) i s (x',y»,z')= (x,y,z) cos e - s i n e 0~ s i n 0- cose 0 0 0 1 R o t a t i o n about the x - a x i s (8) i s (x* ,y» ,z*)= (x,y,z) 0 cos s i n 0 - s i n I cos I These two t r a n s f o r m a t i o n s can be combined; (x*,y<,z*)= (x,y,z) cos s i n 0 - s i n e c o s i cose cos sin8 sm6 sxn I -cose s i n I cos8 r o t a t i o n about (6) r o t a t i o n above (3) viewing d i r e c t i o n (x',y',z') coordinate system for the object (x,y,z) coordinate system of the projection Figure 43. Viewing Controls for Orthographic P r o j e c t i o n . 134 T h e r e f o r e , x*=x.cos8 + . y.sin9 y*=-x.sin9cosg + y.cosecosg + z . s i n g z*=x.sin0sin3 - y.cos0sin3 +z.cos 3 At t h i s stage i n the development the o r t h o g r a p h i c p r o j e c t i o n i s complete. P e r s p e c t i v e r e g u i r e s an a d d i t i o n a l two t r a n s f o r m a t i o n s which are d e s c r i b e d i n Appendix D. I t i s assumed t h a t the a d d i t i o n a l t r a n s f o r m a t i o n s c r e a t i n g the p e r s p e c t i v e p r o j e c t i o n have now been a p p l i e d making the remainder of the d i s c u s s i o n a p p l i c a b l e to both p r o j e c t i o n s . The t r a n s f o r m a t i o n ( p e r s p e c t i v e or orthographic) must be a p p l i e d to each (x,y,z) p o i n t of the e l e v a t i o n data base. The y* i s not r e q u i r e d i n the o r t h o g r a p h i c p r o j e c t i o n (but i s i n the p e r s p e c t i v e as i s shown i n Appendix D) as there i s no adjustmemt made f o r depth. Tbe 3D e f f e c t i s c r e a t e d by p r o j e c t i n g the transformed data onto a two-dimensional screen placed i n f r o n t of tbe data base. T h i s i s accomplished by using the {x*,z») c o o r d i n a t e s from each p o i n t . The z* v a l u e s are r e t a i n e d i n the o r i g i n a l e l e v a t i o n matrix. The x* v a l u e s need not be s t o r e d as they can be generated as r e q u i r e d . To s i m p l i f y c a l c u l a t i o n s the o r i g i n a l (x,y) p l a n a m e t r i c l o c a t i o n s f o r the e l e v a t i o n g r i d , E, are s t o r e d i n two v e c t o r s X and Y. The l o c a t i o n f o r E ( j , k ) i s t h e r e f o r e (X (j),Y ( k ) ) . Tbe t r a n s f o r m a t i o n to d e r i v e x' f o r p o i n t (j,k) can be o b t a i n e d from 135 x* = X(j) • Y(k) with X=Xcos9 and Y=Ysin6 To s c a l e the 3D p l o t f o r output i t i s necessary to determine the maximum and minimum v a l u e s f o r z*. The corresponding range i n x* i s found from using the s u b s c r i p t s d e f i n i n g the viewing area: x 1. =X(j )*Y(k ) and x» =X(j )+Y(k ) mm L L max R R The x* v a l u e s should he s c a l e d such t h a t t h e i r range i s egual t o the number of s u b s c r i p t s i n the h o r i z o n v e c t o r , H, to be used i n the removal of hidden areas. T h i s w i l l allow " d i r e c t " a d d r e s s i n g between the x* values and the h o r i z o n v e c t o r ; i . e . , the e l e v a t i o n at p o i n t x* i n the h o r i z o n v e c t o r w i l l be H(x') . T h e r e f o r e ; Xrange = x* - x* . and; max mxn X = (X-X ( j )).(number o f S u b s c r i p t s i n h o r i z o n vector) /Xrange Y = (Y-Y ( j L ) ). (number of s u b s c r i p t s i n h o r i z o n vector) /Xrange The p l o t t i n g i s now s c a l e d : - i n the x - d i r e c t i o n : 0 t o the number of s u b s c r i p t s i n h o r i z o n v e c t o r - i n the z - d i r e c t i o n (where z i s now the y - d i r e c t i o n of the two dimensional s c r e e n ) ; z* t o z* . max mm 136 6.2.2 P l o t t i n g the Three Dimensional Representation The most complicated aspect o f p l o t t i n g i s the removal of those s e c t i o n s of the data base t h a t are hidden from view. The 3D r e p r e s e n t a t i o n can be produced by p l o t t i n g two s e t s of ground p r o f i l e s ; one s e t corresponding to the rows and the other t o the columns of the data base. F i g u r e 44 shows an example of using only row p r o f i l e s . , F i g u r e s 45a,b,c produced using the UBC computing c e n t e r l i b r a r y program UBC:PERSP on the IBM 370/168 computer system, shows the i n c l u s i o n of both s e t s of p r o f i l e s f o r three d i f f e r e n t viewing d i r e c t i o n s . I t i s apparent t h a t with the r o t a t i o n being zero the column l i n e s become almost v e r t i c a l and c o n t r i b u t e l i t t l e to the p e r c e p t i o n of the area. Conversely when the r o t a t i o n approaches 90 degrees the s i t u a t i o n i s r e v e r s e d . T h e r e f o r e the row p r o f i l e s are used when the r o t a t i o n i s l e s s than 45 degrees; column p r o f i l e s are used when the r o t a t i o n exceeds 45 degrees. The p l o t t i n g method i s t h e same f o r the p e r s p e c t i v e and o r t h o g r a p h i c p r o j e c t i o n s . The l i n e s are p l o t t e d s e g u e n t i a l l y commencing at the foreground of the r o t a t e d data base. Each p o i n t along the p r o f i l e l i n e i s checked a g a i n s t the viewing h o r i z o n . The p o i n t i s p l o t t e d only i f i t i s above the h o r i z o n ; i . e . , viewable. I n i t i a l l y the h o r i z o n v e c t o r , H, i s s e t to z e r o . The k p r o f i l e l i n e (corresponding to one row of the e l e v a t i o n matrix) i s p l o t t e d from r i g h t to l e f t proceeding from g r i d p o i n t z*. ure 44. Orthographic Projection using One Set of P r o f i l e Lines and an Angle of Rotation of 0 degress. Figure 45a. Orthographic Projection using Two Sets of Profile Lines and an Angle of Rotation of 0 degrees* *The plot is produced using a UBC Computing Centre library program called UBC:PERSP. 00 Figure 45b. Orthographic Projection using Two Sets of P r o f i l e Lines and an Angle of Rotation of 45 degrees.* *The plot i s produced using a UBC Computing Centre l i b r a r y program c a l l e d UBCrPERSP. p—• CO Figure 45c. Orthographic Projection and an Angle of Rotation of 80 *The plot i s produced using a UBC Computing using Two Sets of P r o f i l e Lines degrees.* Centre l i b r a r y program called UBC:PERSP. to grid point z». where {j=n*1,n ,n-1,.. . ,2). To create better resolution, points along the l i n e projected between the two grid points are also checked. These points, P i, are determined frcm using the eguation of the l i n e between z'. and z* . Each point i s checked against i t s corresponding j-J-»im-position in the horizon array. The points are chosen at in t e r v a l s of one unit in the x-direction thereby occurring at the same freguency as the elements in the horizon array. This i s made possible by the scaling described e a r l i e r . The more points along the l i n e the better the plot guality (Figure 46). The points P (x ,y ± r% ±) are generated by z_, =z_, * m. where z_ =z *. , p i \ - i J . % i , k x =x + 1 where x p =x» p i p i - l p0 J ' k m. = ( « 3 , k - * , j _ i f k)/<* , j > k-x , j _ i > k ) < sl°Pe of-the line) where x ] j k = I ( j ) + Y(k) and k=current row (profile) j=current grid point on row k. The elevation of each point, P i, i s checked with i t s corresponding horizon elevation: If z >H (x ) then P. i s viewable and the pl o t t i n g pen i i 1 i s moved to (x p , z p ). Upon reaching (x p , z p ) tbe pen i s i i i i lowered. Consequently i f tae pen was previously down then P i (the previous point on the profile) was viewable and a li n e i s drawn. Otherwise, i f the pen was previously up then P. was hidden and no l i n e i s drawn. The horizon i vector i s updated with the higher elevation: H (Xp )=z p . grid point E(j,k) on p r o f i l e k intermediate elevation generated between E(j+1,k) and p r o f i l e k-1 Intermediate elevations are computed and checked against each element i n the horizon array. ' 1 [ ' 1 j 1 ' | 1 ' j 1 ' [ i ' I ! i I l i [ . ' I I I I [ I l | I I | l I | i i I l I | I i j 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Subscripts of the horizon array Figure 46. Removing Hidden Areas. 143 I f z p <H (Xp ) then P_ i s hidden. The p l o t t i n g pen i s l i f t e d but remains at i t s c u r r e n t p o s i t i o n (the l a s t viewable point) , 6.2.3 P l o t t i n g the L o c a t i o n s of P l a n a m e t r i c D e t a i l s on the Three Dimensional R e p r e s e n t a t i o n The l o c a t i o n s of the p l a n a m e t r i c d e t a i l s , l i k e roads and s e t t i n g s , are p l o t t e d onto the three dimensional r e p r e s e n t a t i o n by a p p l y i n g the same t r a n s f o r m a t i o n used to generate the 3D p l o t . The only d i f f i c u l t y i s d e c i d i n g whether a planametric d e t a i l i s hidden from view. A p o s s i b l e s o l u t i o n would be a t each (x,y) p o i n t , d e s c r i b i n g the l o c a t i o n , to search back on a l i n e p a r a l l e l to the viewing d i r e c t i o n u n t i l the foreground i s reachedio., I f t h i s s i g h t l i n e does not i n t e r s e c t with the ground then the (x,y) p o i n t f o r the l o c a t i o n of the planametric d e t a i l i s viewable. T h i s would be a very slow process. A good approximation i s a t t a i n a b l e by " f l a g g i n g " the e l e v a t i o n s i n the data base array t h a t are hidden, by, s e t t i n g the s i g n s of the e l e v a t i o n s t o negative as the 3D p l o t i s being produced. P o i n t s d e s c r i b i n g the l o c a t i o n could then be checked by merely t e s t i n g 1 0 In the case o f p e r s p e c t i v e p r o j e c t i o n , i t i s necessary to search back on a l i n e t h a t ends at the viewpoint as the s i g h t l i n e s are not p a r a l l e l . 144 whether t h e e l e v a t i o n g r i d c e l l t h a t i t l i e s i n , i s v i e w a b l e {plus) o r n o t ( m i n u s ) . The e r r o r i n t h i s a p p r o a c h o c c u r s b e c a u s e t h e e n t i r e c e l l a r e a w i l l be e i t h e r h i d d e n o r v i e w a b l e . The p l o t t i n g o f t h e l o c a t i o n s i s g u i t e s i m p l e . The (x,y) c o o r d i n a t e s f o r the l o c a t i o n o f t h e p l a n a m e t r i c d e t a i l s a r e d i g i t i z e d and t h e n s c a l e d t o c o r r e s p o n d t o the s u b s c r i p t s o f t h e e l e v a t i o n m a t r i x : x s=nx/u and y s=ny/v The p o i n t s ( x s , y s ) must be r o t a t e d t o t h e c a r d i n a l d i r e c t i o n o f view: i f N o r t h no r o t a t i o n i s r e q u i r e d i f West - t=y s ys=n~xs i f E a s t - t = x s x s=n-ys y s = t i f S o u t h - x s = n - x s y =n-y J s -' s The c o o r d i n a t e s (j#k) o f t h e g r i d c e l l c o n t a i n i n g ( x s , y s ) a r e f o u n d u s i n g j = [ * s ] and k=Iys] The x* c o o r d i n a t e o f t h e 3D r e p r e s e n t a t i o n c a n be f o u n d p r e c i s e l y by u s i n g t h e o r i g i n a l t r a n s f o r m a t i o n x , = x . c o s e + y . s i n e . T h i s i s cumbersome however a s t h e s c a l i n g used must a l s o be a p p l i e d . A s i m p l e r method t h a t i s a s a c c u r a t e u s e s 145 i n t e r p o l a t i o n : x*=x (j) * (X { j+D -x (j)) * (x s- j ) *y (k) (Y (k+1 )-Y (k)) * (y s -k) S i m i l a r l y t h e e l e v a t i o n , z*, can be found u s i n g t h r e e - s t e p i n t e r p o l a t i o n ( S e c t i o n 1.2.1): 22 = 2,j+l,k *<2'j+l,k+l "2 ,j+l,k M Y s - k ) z« = z 1 + ( z 2 - z 1 ) ( x s - j ) - where a l l z* v a l u e s a r e ta k e n as a b s o l u t e v a l u e s t o a v o i d t h e +/- f l a g s . The p l o t t i n g pen can now be moved t o ( x * , z f ) . The d e c i s i o n as t o whether t h e p o i n t i s hidden c r not depends on t h e s i g n o f z' i n t h e v i c i n i t y of ( x s , y g ) . , A s i m p l e but e f f e c t i v e c r i t e r i o n i s t o t e s t t h e s i g n o f z' ( x s , y s ) where the v a l u e s ( x s , y s ) a r e rounded t o the n e a r e s t i n t e g e r v a l u e and not t r u n c a t e d as done above, i . e . , j = [ x s + .5] and k = [ y s + . 5 ] . A p l a n a m e t r i c map showing t h e hidden a r e a s of t h e 3D p l o t can be ge n e r a t e d by u s i n g t h e plus/minus d e s i g n a t i o n o f t h e e l e v a t i o n v a l u e s as t r u e and f a l s e i n d i c a t o r s . The map can then he produced u s i n g the p l o t t i n g a l g o r i t h m i n F i g u r e 33. 146 6.2.4 Implementation The slow e x e c u t i o n speed of the HP9830A computer made the use of t h e p e r s p e c t i v e p r o j e c t i o n i n f e a s i b l e . The o r t h o g r a p h i c p r o j e c t i o n r e g u i r e d approximately one hour depending cn the viewing c o n t r o l s . The h o r i z o n v e c t o r was l i m i t e d to o n l y 200 elements. 147 7.0 Harves t i n g Costs and Wood Volume Production The Harvesting C o s t s and Wood Volume Pr o d u c t i o n Module i s p o t e n t i a l l y the most important of a l l analyses i n a s s e s s i n g whether a h a r v e s t i n g p l a n d e r i v e d from the other s i m u l a t o r modules i s f e a s i b l e . Tbe e l e v a t i o n data base must be augmented by wood volume and s o i l data. , The wood volume data are r e a d i l y a v a i l a b l e , u s u a l l y i n the form of f o r e s t i n v e n t o r y maps. However, the s o i l / l a n d f o r m data tend to be l a c k i n g , l i m i t i n g the e f f e c t i v e n e s s of the p r o d u c t i v i t y e s t i m a t e s . The module c o n s i s t s of two p a r t s : c r e a t i o n of the wood volume and s o i l s data base; and,, the a n a l y s i s o f the v a r i o u s s e t t i n g a l t e r n a t i v e s (volumes and c o s t s ) . 9.1 C r e a t i o n of the Wood Volume and S o i l s Data Base There are two p o s s i b l e data base s t r u c t u r e s : r e t a i n i n g the shape of the covertype boundaries; or, using a g r i d system with each c e l l being designated as a c e r t a i n covertype and/or s o i l type. The former i s d i f f i c u l t t o implement as attempting to ov e r l a y one polygon ( s e t t i n g boundary) onto s e v e r a l polygons (covertypes) i s a problem t h a t f a r exceeds the c a p a b i l i t y o f the desktop computer., The g r i d system, however, i s simple, e f f i c i e n t and d i r e c t l y compatible to the e l e v a t i o n data base 148 s t r u c t u r e . The method of entry of the i n v e n t o r y (wood volumes and s o i l s data) i n t o the data base can a l s o be accomplished i n two ways. The e a s i e s t from a user's s t a n d p o i n t i s to t r a c e the polygon boundary of each covertype and then have the computer determine which c e l l s are l o c a t e d w i t h i n the polygon. The range and domain of the boundary p o i n t s are used t o i n d i c a t e which of the n 2 c e l l s i n the g r i d can f e a s i b l y be c o n t a i n e d by the polygon. The c e n t r o i d of each of these c e l l s i s used i n a p o i n t - i n - p o l y g o n a l g o r i t h m (Appendix A) t o determine i f the c e l l i s a c t u a l l y w i t h i n the polygon boundary. A s i m p l e r method to implement i s having the user d i g i t i z e one p o i n t i n each c e l l c o n t a i n e d by the covertype polygon (Figure 19). In t h i s way t h e user performs the p o i n t - i n - p o l y g o n task. The c e l l s u b s c r i p t s f o r an (x,y) p o i n t from the d i g i t i z e r are determined u s i n g Eguation (1.1)., J=[xn/u]+1 and k=[ yn/v 1+ 1 An e d i t f a c i l i t y t o c o r r e c t e r r o r s ( o f t e n o c c u r r i n g at the boundary r e g i o n s of the polygons) must be provided. A simple g r i d map (Figure 20) w i l l a i d i n l o c a t i n g these e r r o r s . C e l l s can then be p r o p e r l y a l l o c a t e d using the entry method d e s c r i b e d above. 149 9.2 S e t t i n g A n a l y s i s I f the data s t r u c t u r e i s a g r i d then the l o g g i n g s e t t i n g area oust he s i m i l a r l y d e f i n e d . The c e l l s that are con t a i n e d by the s e t t i n g boundary are entered as before; i . e , e i t h e r by t r a c i n g the boundary and then using a p o i n t - i n - p o l y g o n r o u t i n e or manually d i g i t i z i n g p o i n t s i n each c e l l . In a d d i t i o n t o the s e t t i n g boundary, the l a n d i n g l o c a t i o n (x ,y ) and the type o f yarding system t o be used are r e q u i r e d . The p r o d u c t i v i t y and wood volume estimates are made by d e t e r m i n i s t i c a l l y s i m u l a t i n g the yarding of the wood t h a t l i e s on each c e l l . P r o d u c t i v i t y i s c a l c u l a t e d by using a r e g r e s s i o n eguation o r a l o s s - f a c t o r approach. The l o s s - f a c t o r approach commences with a normal production r a t e f o r the c e l l . Then, c e r t a i n deductions are made from t h i s value a c c o r d i n g to v a r i o u s y a r d i n g c o n d i t i o n s . The i n i t i a l p roduction r a t e and the deductions are machine dependent. An example of l o s s - f a c t o r s ( d e r i v e d from l o c a l e xperience or production s t u d i e s ) f o r a 90-foot spar h i g h l e a d system a re; - yarding d i s t a n c e ; l o s e f i v e percent production f o r each 100 f e e t beyond 600 f e e t ; - ground s l o p e ; l o s e f i v e percent p r o d u c t i o n f o r each 20 percent of s l o p e beyond 60 percent; - f o r d o w n h i l l yarding l o s e f i v e percent p r o d u c t i o n . In t h e above example the l o s s e s are a d d i t i v e which i n seme circumstances w i l l not be c o r r e c t . The yardi n g time f o r each 1 5 0 c e l l i s computed from time •= volume / ad j u s t e d production r a t e There are many important v a r i a b l e s a s s o c i a t e d with e s t i m a t i n g the p r o d u c t i v i t y of any y a r d i n g system: - type of y a r d i n g system - machine c h a r a c t e r i s t i c s - mechanical a v a i l a b i l i t y and u t i l i z a t i o n - yarding d i s t a n c e and d i r e c t i o n - ground s l o p e - crew experience - p i e c e volume - turn volume - s p e c i e s - weather - type of t e r r a i n - g u a l i t y o f l a n d i n g - amount o f d e f l e c t i o n . A l l of these components c o u l d be i n c o r p o r a t e d i n t o the module p r o v i d i n g t h e i r e f f e c t on yarding i s q u a n t i t a t i v e l y known. Some of these, however, such as weather and crew experience, are probably too s p e c i f i c f o r the development planning l e v e l . S pecies and p i e c e volume s i z e can be estimated from the i n v e n t o r y data while the s o i l / l a n d f o r m data can provide i n f o r m a t i o n on the t e r r a i n t y p e . The crew e x p e r i e n c e , weather and l a n d i n g q u a l i t y are a l l h i g h l y v a r i a b l e and i f used would have to be provided by tbe user f o r each s e t t i n g examined. 1 5 1 The yarding d i s t a n c e i s computed by: d = /(x L-x)2 + <yL-y)2 with (x,y) the c o o r d i n a t e s o f the c e l l being "yarded".,, The ground s l o p e i n the c e l l w i l l be e g u a l t o the slope computed from the Topographic Features Module. I f the s l o p e data cannot be kept i n memory then the sl o p e can be estimated by computing the e l e v a t i o n f o r a p o i n t near the c e n t r o i d t h a t l i e s on a l i n e t h a t extends from tbe c e n t r o i d to tbe l a n d i n g . . T h i s a r b i t r a r y point { x V y * } i s found from: x* =x+d.cos6 and y'=y*d.sin0 with d = the d i s t a n c e t o (x,y) from (x* ,y*) e = arctangent ( (y L-y) / (x L~x)} A summary of the o p e r a t i n g s t a t i s t i c s f o r the s e t t i n g can be e a s i l y produced (Figure 2 1 ) . 9 . 3 Boad Co s t i n g Approximate road c o s t i n g can be determined by t r a c i n g the route l o c a t i o n with the d i g i t i z e r and computing the l e n g t h : L i + l = L i * / ( X i + l - X i ) 2 + < * i + i "V* with (x,y) being the road l o c a t i o n c o o r d i n a t e s . 152 S o i l / l a n d f o r m data are paramount f o r p r o p e r l y e s t i m a t i n g road c o s t s . The c e l l s c o n t a i n i n g the s o i l s data t h a t are i n t e r s e c t e d by the road can be determined from Eguation (1.1). Using the s o i l data f o r each c e l l a proper cost c o u l d then be made e i t h e r through a r e g r e s s i o n or a l o s s - f a c t o r approach. 9.3 Implementation The slow e x e c u t i o n speed of the computer d i c t a t e d the use of the manual e n t r y method f o r c o n s t r u c t i n g the wood volume d a t a base. The p o i n t - i n - p o l y g o n approach was t r i e d but proved t o be too slow. A l i m i t o f 30 c o v e r t y p e s , each having up t o s i x s p e c i e s , was imposed. The stand formula c o n s i s t e d of the combined s p e c i e s volume per acre along with the percent composition o f each s p e c i e s . A l o s s - f a c t o r approach (derived from l o c a l experience) was employed with data a v a i l a b l e f o r three machines: a H a d i l l 052 t e n s i o n s k i d d e r , a 90-foot h i g h l e a d spar and a M a d i l l grapple yarder. 153 D i s c u s s i o n 1.0 Economic J u s t i f i c a t i o n of t h e T e r r a i n Simulator The general d e f i c i e n c e s of the modules i n the t e r r a i n s i m u l a t o r , as c u r r e n t l y implemented, have been p r e v i o u s l y o u t l i n e d i n the t h e s i s . For the most p a r t , the drawbacks were due to t h r e e main f a c t o r s : the inadequacies of the Hewlett-Packard 9830A computer; the low p r e c i s i o n of the e l e v a t i o n data; and, the l a c k of s o i l s , b y d r o l o g i c a l , land-use, p r o d u c t i v i t y and c o s t i n g data. These l i m i t a t i o n s g e n e r a l l y d i c t a t e d the use of the l e a s t p r e c i s e approaches. Given these drawbacks the important g u e s t i c n to be answered i s whether or not the t e r r a i n s i m u l a t o r i n i t s present form c o n s t i t u t e s a u s e f u l t o o l f o r short-term f o r e s t p l a n n i n g . Does i t i n f a c t j u s t i f y the expenditure of some $40,000 to $50,000 on a computer l i k e the HP9830A? T h i s guestion was p a r t i a l l y answered during the summer of 1976 when the model was t e s t e d i n an o p e r a t i o n a l environment a t MacMillan B l o e d e l ' s F r a n k l i n River Logging D i v i s i o n . For the a n a l y s i s , three study areas were s e l e c t e d , each being at a d i f f e r e n t stage of development. The areas were approximately 1000 acres i n s i z e with the only a v a i l a b l e data 154 being wood i n v e n t o r y and e l e v a t i o n contour maps. The e v a l u a t i o n of the t e r r a i n s i m u l a t o r was p r i m a r i l y based on i t s a p p l i c a b i l i t y f o r the planning of h a r v e s t i n g o p e r a t i o n s . The f i r s t area to be examined had r e c e n t l y been harvested, p r i m a r i l y with a l o n g - l i n e , M a d i l l 052 t e n s i o n s k i d d e r c a b l e system with a maximum yarding d i s t a n c e of 2 00C f e e t . O r i g i n a l l y the area was t o be logged using o n l y the c o n v e n t i o n a l h i g h l e a d system which has a maximum y a r d i n g d i s t a n c e of l e s s than 800 f e e t . This approach r e q u i r e d the b u i l d i n g of approximately one mile of road. Upon completion of t h i s access road, i t was d i s c o v e r e d that a t e n s i o n s k i d d e r , s t r a t e g i c a l l y l o c a t e d a t s e v e r a l p o i n t s along e x i s t i n g roads could reach the m a j o r i t y of the wood i n the area to be harvested, making the c o n s t r u c t i o n of the just-completed road unnecessary. T h i s same a f t e r - t h e - f a c t c o n c l u s i o n was reached by using the t e r r a i n s i m u l a t o r . With the model i t was p o s s i b l e t o p r e d i c t , w i t h i n an accuracy of 50 f e e t , the l o c a t i o n s of the s e t t i n g boundaries used i n the t e n s i o n s k i d d e r o p e r a t i o n s . T h i s unneeded road, c o s t the company an amount e q u i v a l e n t to the p r i c e of the computing system capable o f o p e r a t i n g the t e r r a i n s i m u l a t o r . There i s , of c o u r s e , no guarantee t h a t with the s i m u l a t o r t h i s mistake would have been avoided as i t i s s t i l l the r e s p o n s i b i l i t y of the planner to d e v i s e the a l t e r n a t i v e s . The other two study areas were i n v a r i o u s p r e l i m i n a r y stages of p l a n n i n g development. Although e x t e n s i v e t e s t i n g was 155 not done on these two areas, the s i m u l a t o r d i d prove u s e f u l i n h e l p i n g t o decide which y a r d i n g systems would be needed along with t h e i r r e g u i r e d deployments. I t a l s o helped to assess what f i e l d work was necessary to e s t a b l i s h the o p e r a t i o n a l h a r v e s t i n g plan. The g e n e r a l c o n c l u s i o n d e r i v e d from the t e s t s was t h a t the s i m u l a t o r c o n s t i t u t e s a powerful t o o l f o r p r e l i m i n a r y h a r v e s t planning. I t s main advantage i s that i t g i v e s the planner the o p p o r t u n i t y t o examine a l a r g e number of a l t e r n a t i v e s not normally p o s s i b l e using the e x i s t i n g manual technigues. With the o f t e n complex c o s t t r a d e o f f s between y a r d i n g systems and the high .cost of road c o n s t r u c t i o n , i t i s imperative that a l l p o s s i b l e o p t i o n s are i n v e s t i g a t e d . R e a l i s t i c a l l y , t h i s can o n l y be achieved with tbe a i d of a computer model l i k e the one developed i n t h i s t h e s i s . , For the other p l a n n i n g c o n s i d e r a t i o n s , such as a e s t h e t i c s , the improvements or savings gained from using the s i m u l a t o r are d i f f i c u l t to q u a n t i f y . They are nonetheless important c o n s i d e r a t i o n s f o r j u s t i f y i n g the a c q u i s i t i o n of a desktop computer. The desktop computer has other v a l u a b l e uses i n a l o g g i n g d i v i s i o n . These i n c l u d e the c o m p i l a t i o n of f i e l d notes f o r roads (Burke, 1974) and y a r d i n g p r o f i l e s (Carson, 1975) and v a r i o u s types of accounting procedures. 156 2.0 a d d i t i o n s to the Current Model The modules presented i n t h i s t h e s i s , c o l l e c t i v e l y , do not r e p r e s e n t a l l the components needed i n f o r e s t development planning. Two important components t h a t c o u l d be i n c o r p o r a t e d , with l i t t l e d i f f i c u l t y , would be a hydrology module and a g r a p h i c a l i n f o r m a t i o n r e t r i e v a l module. The f o l l o w i n g d e s c r i p t i o n s of these two modules are meant onl y to o u t l i n e what might be done. 2.1 A Hydrology Module a hydrology module could be designed to analyze water flow through the study area. The b a s i c i n p u t c o u l d c o n s i s t of the l o c a t i o n s and c l a s s i f i c a t i o n s o f the streams and the water flow p r o p e r t i e s of the s o i l types i n the study area. The stream l o c a t i o n s c o u l d be entered i n an analogous way to that used f o r the read network; i . e . , by d i g i t i z i n g the l o c a t i o n s . The water flow p r o p e r t i e s would best be i n c o r p o r a t e d as part of the c l a s s i f i c a t i o n system used to d e s c r i b e the s o i l types. water flow could then be simulated by s y s t e m a t i c a l l y a n a l y z i n g each c e l l i n the data base - assuming t h a t a g r i d type data s t r u c t u r e i s being used. One approach t o s i m u l a t i n g water flow c o u l d commence with the c e l l of the highest e l e v a t i o n i n the study area. The water 157 i t holds (e.g., from spring-runoff or from a major storm which could both be inputs) could be "passed" onto i t s adjacent neighbours according to i t s aspect and ground slope as calculated from the Topographic Features module. The other ouputs, l i k e t r a n s p i r a t i o n and evaporation, should also be considered. The amount of water transferred would depend on the c e l l ' s ground slope and s o i l type. This process could then be repeated f o r each c e l l i n the study area by examining them i n descending order of elevation. Water accummalated i n a c e l l intersected by a stream could be "moved" along the stream. Using t h i s type of water flow simulation, various types of analyses could be performed: - estimating possible sedimentation problems and the resultant water quality - estimating required culvert and bridge sizes - predicting possible erosion control problems - estimating t o t a l water guantity and supply. 2.2 A Graphical Information Retrieval Module A graphical information r e t r i e v a l module could be designed as an extension of the Topographic Features module already developed. In addition to slope, aspect and elevation the map overlays could include s o i l s , land-use (like the Canada land Inventory) and wood inventory data. This function would have widespread use i n general resource a l l o c a t i o n . For example, i f 158 a g e n e r a l resource development plan was being devised f o r the study a r e a , i t would be f e a s i b l e with t h i s type of module to generate map o v e r l a y s d e p i c t i n g the v a r i o u s c o n f l i c t areas between res o u r c e s and/or uses. As i n the combination mapping i n the Topographic Features module, an acreage summary would a l s o be produced. For i n s t a n c e , i t may be d e s i r a b l e t o know which regions i n the study area had both good h a r v e s t i n g p o t e n t i a l and s e n s i t i v e w i l d l i f e h a b i t a t . The module c o u l d s e a r c h f o r a l l the c e l l s i n the data base t h a t contained these two resource uses and then show the l o c a t i o n s of these areas on a map o v e r l a y along with the a s s o c i a t e d acreage. The e f e c t on the resource base of land-removals could a l s o be determined. By examining a l l the data base c e l l s i n the removal area i t would be p o s s i b l e t o determine how much of a c e r t a i n resource, such as wood volume, had been removed. The c e l l s i n the r e g i o n c o u l d be i d e n t i f i e d to the computer by d i g i t i z i n g the r e g i o n boundary and then using a p o i n t - i n - p o l y g o n r o u t i n e (Appendix A). To implement such a module i t would be necessary t o b u i l d a l a r g e data base which w i l l r e g u i r e c o n s i d e r a b l e amounts of computer memory and/or f a s t - a c c e s s s t o r a g e such as t h a t obtained with a d i s c d r i v e . Each m u l t i - d i m e n s i o n a l c e l l i n the data base w i l l have to hold i n f o r m a t i o n oa each of the resources and la n d -uses being c o n s i d e r e d i n the study a r e a . The data base could be creat e d i n a manner s i m i l a r t o t h a t used t o enter the wood i n v e n t o r y i n t o the Harvesting Costs and Wood Volume Pr o d u c t i o n 1 5 9 module. 2.3 Summary Two modules have been suggested as p o s s i b l e a d d i t i o n s t o the t e r r a i n s i m u l a t o r model. These are a hydrology module f o r e s t i m a t i n g water flow p a t t e r n s through the study area; and, a g r a p h i c a l i n f o r m a t i o n r e t r i e v a l module t o l o c a t e and summarize the v a r i o u s resource and land-use t r a d e o f f s . Other p o s s i b i l i t i e s f o r extending the a p p l i c a t i o n s of the t e r r a i n s i m u l a t o r model undoubtedly e x i s t . 160 C o n c l u s i o n The management of f o r e s t r e s o u r c e s has become an extremely complex problem., Tbe e x i s t i n g procedures used i n the planning process have not, i n many cases, been a b l e to provide the necessary s o l u t i o n s . The computer model presented i n t h i s t h e s i s was developed t o i e l p meet t h i s d i f f i c u l t c h a l l e n g e . Pa r t of the task i n c r e a t i n g the model was the i n t e g r a t i o n of s e v e r a l developments, such as VIEwIT ( T r a v i s e t a l . , 1975), i n t o one planning package. , I t was designed to deal with the f o l l o w i n g a s p e c t s of short-term resource planning; - c o l l e c t i o n of the r e g u i r e d t e r r a i n e l e v a t i o n s and f o r e s t i n v e n t o r y data; - d e t e r m i n a t i o n and p r o d u c t i o n o f map o v e r l a y s of the t o p o g r a p h i c f e a t u r e s ; s l o p e , aspect and e l e v a t i o n ; - design of l o g g i n g s e t t i n g s f o r c a b l e systems and placement o f y a r d i n g roads; - l o c a t i o n of f o r e s t - a c c e s s roads; - d e l i n e a t i o n o f viewable areas and production of three dimensional r e p r e s e n t a t i o n s of the t e r r a i n ; - e s t i m a t i o n of h a r v e s t i n g c o s t s and wood volume production., 161 A Review o f - t h e M o d u l e s D e v e l o p e d i n t h e T e r r a i n S i m u l a t o r The f o r m a t used f o r t h e e l e v a t i o n d a t a b a s e was a r e g u l a r g r i d o f e l e v a t i o n s . The most a c c u r a t e f o r m a t , however, would r e t a i n t h e l o c a t i o n s of t h e o r i g i n a l e l e v a t i o n c o n t o u r s . I n t h i s method t h e r e would be v i r t u a l l y no i n f o r m a t i o n l o s s . U n f o r t u n a t e l y , t h e a l g o r i t h m r e g u i r e d t o g e n e r a t e an e l e v a t i o n f o r a map l o c a t i o n f a r e x c e e d s t h e c a p a b i l i t y o f t h e HP9830A comput e r . F o r t h e r e g u l a r g r i d a p p r o a c h , e a c h c e l l s u r f a c e was d e s c r i b e d m a t h e m a t i c a l l y by e i t h e r an i n t e r p o l a t i o n r o u t i n e t h a t used t h e c e l l c o r n e r e l e v a t i o n s , o r , by a s u r f a c e e g u a t i o n method w h i c h was t y p i c a l l y some f o r m o f g e o m e t r i c p l a n e . The i n t e r p o l a t i o n a p p r o a c h r e g u i r e s more c o m p u t a t i o n a l work t o g e n e r a t e an e l e v a t i o n b u t s t o r a g e r e g u i r e m e n t s a r e l e s s a s o n l y one e l e v a t i o n per c e l l n e e d be r e t a i n e d . The e q u i v a l e n t s u r f a c e e g u a t i o n method, {computed o n l y once a t t h e i n p u t s t a g e u s i n g t h e same g r i d o f e l e v a t i o n s as t h e i n t e r p o l a t i o n method) i s s i m p l e t o e v a l u a t e , but e ach o f t h e p a r a m e t e r s must be r e t a i n e d t h e r e b y i n c r e a s i n g r e q u i r e d s t o r a g e c a p a c i t y . Two methods were p r o p o s e d f o r f o r m i n g t h e e l e v a t i o n d a t a b a s e . The s i m p l e s t method u s e d e q u a l l y s p a c e d t r a n s e c t l i n e s t o c o v e r t h e map o f t h e s t u d y a r e a . The c o n t o u r i n t e r s e c t i o n s on e a c h l i n e were d i g i t i z e d and t h e n u s e d t o compute t h e e l e v a t i o n s a t e g u a l l y s p a c e d p o i n t s cn t h e t r a n s e c t l i n e . T h e r e f o r e , t h e d a t a b a s e was a l w a y s i n t h e f o r m o f a r e g u l a r g r i d . The p r o b l e m 162 with the method was t h a t each l i n e was compiled independently which c o u l d i n t r o d u c e l a r g e e r r o r s when the contours run p a r a l l e l t o the t r a n s e c t l i n e s . To avoid t h i s problem, a second set of t r a n s e c t l i n e s c ould be run p e r p e n d i c u l a r to the f i r s t group., The g r i d of e l e v a t i o n s c o u l d then be computed by merging the two s e t s o f l i n e s . T h i s approach would i n c r e a s e d i g i t i z i n g time as w e l l as the amount of storage r e g u i r e d and was t h e r e f o r e not c o n s i d e r e d . The second approach developed f o r c r e a t i n g the e l e v a t i o n data base was the contour method. Each contour l i n e was d i g i t i z e d and s t o r e d i n memory. The e l e v a t i o n was computed f o r each g r i d point by using the e l e v a t i o n s from the c l o s e s t d i g i t i z e d p o i n t i n each of the f o u r quadrants surrounding the g r i d p o i n t . To l e s s e n the search time r e g u i r e d to l o c a t e the c l o s e s t p o i n t s , the d i g i t i z e d p o i n t s were i n i t i a l l y c l a s s i f i e d i n t o s m a l l e r map u n i t s . T h i s made i t necessary t o examine o n l y those p o i n t s t h a t were contained i n the map u n i t t hat h e l d the g r i d p o i n t . The accuracy of the contour method was s u p e r i o r t o tha t of the t r a n s e c t method; but, due to memory r e s t r i c t i o n s , the HP9830A system c o u l d only handle s m a l l maps or those of l i m i t e d complexity when using the contour method. , The topographic f e a t u r e s ( s l o p e , aspect and e l e v a t i o n ) were computed f o r each c e l l of the e l e v a t i o n data base from a geometric plane f i t t e d to tbe ground s u r f a c e using a l e a s t -sguares procedure., The parameters of the r e s u l t a n t plane eguation were then used t o determine the s l o p e , aspect and 163 e l e v a t i o n . Hap o v e r l a y s showing these t h r e e topographic f e a t u r e s , e i t h e r s i n g l y or i n v a r y i n g combinations, were produced u s i n g the area of one data base c e l l as the s m a l l e s t p l o t t i n g u n i t . The Yarding L o c a t i o n and S e t t i n g Design module conc e n t r a t e d on developing the c a b l e mechanics theory needed t o p r e d i c t the loadpaths of s k y l i n e c a b l e systems. The most t h e o r e t i c a l l y c o r r e c t approach (the c a t e n a r y model) was not used because s i m i l a r accuracy could be achieved using the s i m p l e r , p a r a b o l i c model., f i n the p a r a b o l i c f o r m u l a t i o n , c a b l e weight was assumed to be d i s t r i b u t e d on the chords of the c a b l e s , whereas i n the catenary model, c a b l e weight was d i s t r i b u t e d on the c a b l e i t s e l f ) . Newton's method f o r s o l v i n g n o n - l i n e a r eguations was used to s o l v e the p a r a b o l i c model. The i t e r a t i o n eguation was easy to formulate and converged g u i c k l y and dependably t o the s o l u t i o n f o r the l o a d p o s i t i o n . Even with f a s t convergence, however, the e x e c u t i o n time u s i n g the HP computer was e x c e s s i v e . The percent d e f l e c t i o n r u l e was t h e r e f o r e needed t o p r e d i c t the loadpath. I t expressed the minimum a t t a i n a b l e midspan l o a d d e f l e c t i o n as a percentage of the chord d i s t a n c e between the tops of the l a n d i n g and back spars. To determine the loadpath f o r the r e s t of the span, an i n t e r p o l a t i n g polynomial was f i t t e d to the t h r e e nodes; i . e . , the midspan l o a d p o s i t i o n and the tops cf the l a n d i n g and back spars. 164 R e s u l t s from the percent d e f l e c t i o n approach compared fa v o u r a b l y to the catenary model, provided the midspan d e f l e c t i o n was known., These d e f l e c t i o n s are r e a d i l y a v a i l a b l e i n t a b l e s that are produced using the catenary model. The i n a c c u r a c y of the a v a i l a b l e contour maps l i m i t s the module to development pl a n n i n g ; i t cannot be used f o r the a c t u a l o p e r a t i o n a l planning phase., In the Road l o c a t i o n module, two methods f o r the p r o j e c t i o n of roads are pro v i d e d : "manual" and "automatic". In the "manual" method, p o i n t s along the road l o c a t i o n were d i g i t i z e d by the user. The only computation r e g u i r e d was to t e s t whether the grade between s u c c e s s i v e t r i a l p o i n t s was between the s p e c i f i e d a l l o w a b l e adverse and f a v o u r a b l e grade l i m i t s . The "automatic" road p r o j e c t i o n r o u t i n e used an h e u r i s t i c a l g o r i t h m to l o c a t e i n the h o r i z o n t a l plane, the average grade between the d i g i t i z e d s t a r t and f i n i s h p o i n t s of the r o u t e . A secant s e a r c h , employed at s u c c e s s i v e i n t e r v a l s along the r o u t e , l o c a t e d the average grade i n each of these i n t e r v a l s . The determination of earthwork volumes f o r road c o n s t r u c t i o n was deemed i n a p p r o p r i a t e due to the in a c c u r a c y of the maps. To determine t h e viewable area, a viewpoint was s p e c i f i e d from which s i g h t l i n e s were p r o j e c t e d t o each border c e l l i n the e l e v a t i o n g r i d . The c e l l s t h a t i n t e r s e c t e d the s i g h t l i n e were judged viewable i f the sl o p e f o r the l i n e of s i g h t between i t and tbe viewpoint was g r e a t e r than any s i m i l a r l y c a l c u l a t e d slope on the s i g h t l i n e . The o v e r l a y was produced u s i n g the 165 same p l o t t i n g a l g o r i t h m put f o r t h i n the Topographic Features module. There were two methods developed f o r producing three dimensional r e p r e s e n t a t i o n s of the study area: o r t h o g r a p h i c and p e r s p e c t i v e p r o j e c t i o n . The p e r s p e c t i v e p r o j e c t i o n was not implemented due to the added t r a n s f o r m a t i o n s r e q u i r e d . The three dimensional e f f e c t was produced by p l o t t i n g l i n e s t h a t were p e r p e n d i c u l a r t o the l i n e of s i g h t onto a two dimensional s u r f a c e . The hidden areas were removed by e l i m i n a t i n g p o i n t s on the p r o f i l e l i n e s t h a t were at a lower e l e v a t i o n than the corresponding p o i n t s i n the pre v i o u s p r o f i l e . The l o c a t i o n s o f planametric d e t a i l s (roads and s e t t i n g boundaries) were p l o t t e d on the t h r e e dimensional r e p r e s e n t a t i o n s by a p p l y i n g the same t r a n s f o r m a t i o n used t o generate the r e p r e s e n t a t i o n . Planametric d e t a i l s t h a t were i n a p o s i t i o n t h a t was hidden from view were not p l o t t e d . In the Ha r v e s t i n g Costs and Hood Volume P r o d u c t i o n module, two methods were developed to enter the wood i n v e n t o r y and s o i l s data i n t o the g r i d - t y p e data base. These were: (1) d i g i t i z i n g a p o i n t i n each c e l l c o n t a i n e d by the covertype; or, (2) by d i g i t i z i n g the covertype boundary and then using a p o i n t - i n -polygon r o u t i n e to determine which c e l l were contained i n the covertype. An a l g o r i t h m d e t e r m i n i s t i c a l l y simulated y a r d i n g the wood from each g r i d c e l l i n the l o g g i n g s e t t i n g . P r o d u c t i v i t y estimates c o u l d use e i t h e r a l o s s - f a c t o r or r e g r e s s i o n eguation approach; o n l y the l o s s - f a c t o r approach was used. Road 166 c o n s t r u c t i o n c o s t s were estimated by d i g i t i z i n g the road l o c a t i o n , computing i t s l e n g t h and then determining the c o s t using e i t h e r a l o s s - f a c t o r or r e g r e s s i o n eguation approach. In the implementation of the model, however, a constant c o s t per mile was assumed due to lack of data., The p r i n c i p a l l i m i t a t i o n s o f the c u r r e n t v e r s i o n o f the s i m u l a t o r , are due t o t h r e e main reasons: - the ina d e g u a c i e s of the Hewlett-Packard 9830A computer; - the l a c k of accurate t e r r a i n data; - the l a c k of data on s o i l s , hydrology and l a n - u s e . The l i m i t e d computer technology, i . e . * slow execution speed, small memory and l a c k o f a f a s t - a c c e s s storage d e v i c e , was the most c r i t i c a l of these drawbacks. T h i s meant t h a t i n the ma j o r i t y of the cases the l e a s t p r e c i s e a l g o r i t h m bad to be used. Future development of the t e r r a i n s i m u l a t o r should c o n c e n t r a t e on s e v e r a l main areas; <1) As computer technology improves along with tbe a v a i l a b i l i t y and g u a l i t y o f data, the v a r i o u s r e s t r i c t i o n s due t o these drawbacks should be removed from the model. The HP9830A computer i s p r e s e n t l y being r e p l a c e d by a new generation of desktop computers which appear t o provide the necessary execution speed and f a s t - a c c e s s storage c a p a b i l i t y needed to remove many of the c u r r e n t l i m i t a t i o n s of the model. The use of orthophotographs f o r o b t a i n i n g 1 6 7 the elevation data base should also be thoroughly investigated. (2) New planning components should developed such as a hydrology module to analyze water flow through the study area and a graphical information r e t r i e v a l module that would be able to produce map overlays for various resource and land-use combinations. although f u l l implementation has not been possible, t h i s study has demonstrated that the t e r r a i n simulator, even in i t s present form, i s a useful planning t o o l that can aid i n the management of forest resources. 168 L i t e r a t u r e C i t e d Amidon, E.L., 1966. MIADS2: An alphanumeric Map Info r m a t i o n Assembly and D i s p l a y System f o r a Large Computer. , USDA, Fo r e s t S e r v i c e , Research Paper, PSW-38. 12 pp. Burke, D. 1974., Automated A n a l y s i s of Timber Access Road A l t e r n a t i v e s . P a c i f i c Northwest F o r e s t and Range Experimental S t a t i o n , USDA, F o r e s t S e r v i c e , PNW-110. 40 pp. 1974a. S k y l i n e Logging P r o f i l e s from D i g i t a l T e r r a i n Models. Proceedings, 1974 S k y l i n e Logging Symposium, U n i v e r s i t y of Washington, S e a t t l e . pp52-55. , 1976. Automated Yarding Cost E s t i m a t i o n . Proceedings, 1976 S k y l i n e Logging Symposium, Vancouver. pp63-68. Carson, W.W., 1975. Programs f o r S k y l i n e P l a n n i n g . P a c i f i c Northwest F o r e s t and Range Experimental S t a t i o n , USDA, For e s t S e r v i c e , PNW-31. 40 pp. — and C N . Mann., 1970. A Technique f o r the S o l u t i o n of S k y l i n e Catenary Eguations. P a c i f i c Northwest F o r e s t and Range Experimental S t a t i o n , USDA, F o r e s t S e r v i c e , PNW-110. 18 pp. - 1971. A n a l y s i s o f Running S k y l i n e Loadpath. P a c i f i c Northwest F o r e s t and Range Experimental S t a t i o n , USDA, F o r e s t S e r v i c e , PNW-120. 9 pp. C l u t t e r , J.L. and J.E. , Bethune, 1969. „• Maximum P r o d u c t i v i t y from Industry-owned F o r e s t Lands. Amer. Astr o n a u t S o c , Nat. Meet 15 and Oper. Res. Soc. 35, Pap. I l l B4. Denver, C o l o . pp6 9-54. Conte, S.D. and C. DeBoor, 1972., Elementary Numerical A n a l y s i s ^ 2nd e d i t i o n . McGraw-Hill Book Company, New York. 396 pp. Guimier, D.Y., 1977. Experimental Study of Logging Cable Systems. Masters T h e s i s (unpublished), U n i v e r s i t y o f B r i t i s h Columbia, Vancouver. 175 pp. 169 Lysons, H.H. and C.N. Mann, 1967. S k y l i n e Tension and D e f l e c t i o n Handbook. P a c i f i c Northwest F o r e s t and Range Experimental S t a t i o n , USDA, Fo r e s t s e r v i c e , Research Paper PNW-39. 41 pp., Meyer, C.F., 1969. Route Surveying and Design, 4th e d i t i o n . I n t e r n a t i o n a l Textbook Company, Scranton, Pen n s y l v a n i a . 635 pp. Navon, D.I., 1971. Timber RAM - A Long Range Planning Method f o r Commercial Timber Lands under M u l t i p l e Use Management. USDA, Fo r e s t S e r v i c e , Research Paper PSW-70. 22 pp. Newman, B.M. and R.F. S p r o u l l , 1973. P r i n c i p l e s o f I n t e r a c t i v e G r a p h i c s , McGraw-Hill Computer Science S e r i e s , New York. 607 pp. Nordbeck, S. and R. Bengt, 1967. Computer Cartography - Poi n t i n Polygon Programs, The Royal U n i v e r s i t y o f Lund, Sweden. , 34 pp. Shampine, L.F. and R.C. A l l e n , 1973. Numerical Computing : an I n t r o d u c t i o n . W.B. Saunders Company, P h i l a d e l p h i a . 258 PP. S t u d i e r , D.D. and V. W. B i n k l e y , 1974. Cable Logging Systems, USDA, Fo r e s t S e r v i c e , D i v i s i o n of Timber Management. 211 PP. T r a v i s , H.R., G.H. E i s n e r , S.D. Iverson and C.G. Johnson, 1975. VIEWIT: Computation of Seen Areas, Slope and Aspect f o r Land-use Planning. P a c i f i c Southwest F o r e s t and Range Experimental S t a t i o n , USDA, F o r e s t S e r v i c e PSW-11. 41 pp. Wi l l i a m s , D.H., J.C. McPhalen., S.H. Smith, M.M. Yamada and G.G. Young, 1975. An Operations Research Approach t o Management Un i t P l a n n i n g , (unpublished), F a c u l t y of F o r e s t r y , U n i v e r s i t y of B r i t i s h Columbia, Vancouver. 21 PP. Young, G.G. and D.Z. , Lemkow, 1976. D i g i t a l T e r r a i n S i m u l a t o r s and T h e i r A p p l i c a t i o n t o F o r e s t Development P l a n n i n g , Proceedings, 1976 S k y l i n e Logging Symposium, Vancouver. pp81-99. 170 APPENDICES 171 APPENDIX A Point-in-Polygon Algorithm There are several possible approaches for deciding i f an (x,y) point i s subtended by a polygon. A l l the methods require an exhaustive search of the polygon boundary except i n the case of a s t r i c t l y convex shape (Nordbeck and Bengt, 1967)./ Polygons cannot, however, be r e s t r i c t e d to only convex shapes. A simple and effective method i s to project a ray from the (x,y) point being examined. I f i t crosses the polygon boundary an odd number of times i t must be subtended by the polygon (Figure 47)» The polygon boundary i s represented by a series of p l i n e segments that are defined by p+1 (x,y) points. The point to be checked has the coordinates (x*,y*). The computations are s i m p l i f i e d by using a ray that has the defining equation x=x*. The k*"*1 l i n e segment, defined by the points ( x k,y k) and ( x k + 1 >y k + 1) > intersect the ray CA (Figure 48) at ( x ^ V ^ with x.=x*. l To f i n d the point of intersection (x_^,y^) i t i s necessary to compute the parameters b and m for the equation of the k*"*1 l i n e segment (y=mx+b). The parameter b (the y-intercept) can be solved by substituting (x^,y^) into the equation of the l i n e : y, =mx, + b k k Therefore b = y -mx K. IC The slope of the l i n e , m , i s found from is. \ = ( yk+i - y k ) / ( x k + i - V 172 OUTSIDE no crossing; no crossings; OUTSIDE 2 crossings; OUTSIDE Figure 47. Determining i f a Point is Within a Polygon 173 ( xk+r yk+l } (x^,y^); intersection r e a l ( xk+l , yk+l ) ^ V y k } (x. y.); intersection point i s f i c t i t i o u s -1-» i J, C(x*,y*) ^ k + l ' W (x^,,y_^); intersection point i s f i c t i t i o u s Figure 48. Determining i f an Intersection Takes Place. 174 The equation for the k , line segment is therefore tn y = m^x - xfc) + y k . , . (1) The y-coordinate of the point of intersection (x_^,yj is computed by evaluating Equation (1) with x=x*. Therefore y i = "^(x* ~ x] c) + yk If y* then the intersection is fictitious as i t takes below C on the ray CA. If y^ >y* then the intersection occurs on CA but i t is necessary to test i f (x^,y^) is on the line segment. This will be the case i f l x k + l " X J = l X * " X k l + l X * " X k + l l If the line segment doesnot cross CA then (x^,y^) is not on the line segment and l X k + l " \ I * > X* - X k > + l X * " X k + l l Each of the p line segments must be examined in this fashion and a record kept of the number of intersections. On completion, i f the number of intersections (n) is odd then (x*,y*) is inside the polygon. A suitable test is Odd i f n - 2(n/2) =1 where () is the nearest lower integer. 175 APPENDIX B Menu Operation The two menus used f o r entering e i t h e r the transect or contour l i n e s represent a graphical approach for allowing the user to c o n t r o l the program which i s normally done from keyboard. This allows the operator to concentrate on the d i g i t i z i n g rather than continuously moving between the d i g i t i z e r and the keyboard. For the purposes of discussion, the transect menu w i l l be presented. The menu i s - d i v i d e d into regions which are a l l referenced to the top l e f t corner of the menui"Each region has associated with i t a c e r t a i n program c o n t r o l . As seen from Figure 49, a p o t e n t i a l problem a r i s e s when the menu i s not properly aligned. To a l l e v i a t e t h i s , the top ( x t> v t) a n c * t n e bottom (x^,y^) l e f t corners of the menu are d i g i t i z e d . The menu coordinates can be rotated through an angle of 0 so that the l i n e between ( x ^ y ^ ) and (x^,y^) i s p a r a l l e l to the y-axis. The angle 6 i s defined by 6 = a r c t a n g e n t ( ( x b - x t ) / ( y t - y b ) ) Rotating the menu coordinate system through the angle c f 0 i s given by x'=x,cos9 + y.sin9 .(1) y'=y.cos9 - x.sin8 . .. (2) Using equations (1) and (2), each point entered i n the menu must be rotated to the 'new' menu coordinates. This must be done to the:reference point of the menu: x' = x cos0 + y sin0 t t Jt y' = y cosQ - x sin0 ' t ; t t The menu i n Figure 49 i s based on quarter-inch squares; therefore, each d i g i t i z e d (x,y) point must be converted to the subcripts of the squares. F i r s t i t i s rotated to (x',y') using Equations (1) and (2). the subscripts are found from: Figure 49. The Menu Coordinate System. are found from: 4(x' -xp + 1 + 1 where I I i s the lowest nearest integer. The command selected by the user i s determined by examining ( x ' , y r ) : i f x' = 6 then the command chosen i s 'COMPLETED' i f x' = 7 then the command chosen i s 'END OF ENTRY' i f x' = 8 then the command chosen i s 'MOVE MENU' i f x' = 9 then the command chosen i s "'CHANGE CONTOUR INTERVAL' i f x' <6 then a number i s being entered. The base ten exponent w i l l be equal to 5-x'. The number chosen w i l l be y ' (5-x'), A f u l l f i v e - d i g i t number can be found by accumalating each entry u n t i l a 'COMPLETED' command i s chosen. APPENDIX C The cable mechanics theory developed i n t h i s appendix i s p a r t i a l l y derived from the p a r t i c i p a t i o n of the author i n a graduate-level university course e n t i t l e d Cableways with fellow graduate students, Mr. D.G. Guimier and MR. D.I. Anderson and chaired by Professor G.G. Young. 179 There are numerous cable logging systems i n use today and i t i s possible to model any of these. Generally, each w i l l require a di f f e r e n t formulation p r i n c i p a l l y depending on the number of l i n e s and t h e i r operational configuration. Some of the most common cable systems are shown i n Figures 50, 51 and 52. For any cable system the model, at a distance x from thei' landing spar, " locates the load position at a v e r t i c a l displacement of y such that a l l forces i n the yarding system w i l l be i n equilibrium. Only those forces encountered i n the s t a t i c condition are considered. The dynamic forces are extremely d i f f i c u l t to compute and are generally not predictable. This problem i s c l e a r l y i l l u s t r a t e d when attempting to model the highlead cable method of logging. The precise type of models to be discussed are not suitable as loads are u s u a l l y dragged over the t e r r a i n to the landing spar. This practice r e s u l t s i n large, unpredictable forces. Various models can be devised for any one system. The differences are due to the assumptions used for the cable weight d i s t r i b u t i o n . These assumptions y i e l d different equations for describing the shape of a freel y hanging cable. Consequently the balancing of the forces i n the system, although si m i l a r i n nature, w i l l be mathematically d i f f e r e n t . There are'three possible ways to di s t r i b u t e the cable weight: 1) The cable weight i s uniformly distributed over the horizontal span of the system. GUYLINE Figure 50. Running Skyline System, (from Studier and Binkley, 1974) oo o Figure 51. Gravity Skyline System, (from Studier and Binkley, 1974) i—• oo BLOCK STRAP Figure 52. Highlead System. (from:Studier and Binkley, 1974) 183 2) The cable weight i s uniformly d i s t r i b u t e d over the chord of the system, 3) The cable weight i s uniformly d i s t r i b u t e d over the length of the cable system i t s e l f . These d i f f e r e n t approaches are shown i n Figure 53. The greatest accuracy i s achieved using assumption 3. This model i s c a l l e d a catenary. Although the catenary i s t h e o r e t i c a l l y most cor r e c t , assumptions 1 and 2 provide an easier formulation and l i t t l e s a c r i f i c e i n accuracy (Guimier, 1977) . The chord assumption (2) i s the more accurate of the two and consequently development i n t h i s t h e s i s i s based on i t . Providing the cables are t i g h t , which they normally are i n logging systems, the r e s u l t s are extremely good. Deri v a t i o n of the Equation f o r the Shape of a Freely Hanging Cable  Geometry The basic geometry i s i l l u s t r a t e d i n Figure 54 with the following d e f i n i t i o n s i n use: A : l e f t hand support B : r i g h t hand support L : span; h o r i z o n t a l distances between supports E : d i f f e r e n c e i n e l e v a t i o n between supports AB : chord between supports G : angle between the chord and the h o r i z o n t a l with 6=arctan(E/L) P : any point on the cable 184 Case 3. The cable weight is uniformly distributed over the length of the cable system itself. Figure 53. The Three Different Loading Assumptions for the Cable Weight. 185 Figure 54. Geometry of a Freely Hanging Cable. 0 : point of maximum sag of the cable; acts as the o r i g i n f o r the (x',y') coordinate system Forces The tension at any point on the cable acts along the tangent to the cable at that point. T : tension i n the cable at P P 11^  : h o r i z o n t a l component of the tension at P Vp : v e r t i c a l component of the tension at P T^ : tension i n the cable at support A T_ : tension i n the cable at support B cu* : weight of the cable per unit length Employing the assumption that the cable weight acts on the chord: to = u)'/cos6 As shown i n Figure 55 the forces acting on the system (the section of the cable between 0 and P) are: H h o r i z o n t a l tension at 0 H + V = T tension at P P P P tu = x fw weight of cable between C and T The equations of equ i l i b r i u m are: ; -> Sum of the v e r t i c a l forces : V - tux' = 0 P -> -*• Sum of the Horizontal forces : -H + H =0 p Sum of the moments about P=0 : Hy' - u)x'(x'/2) = 0 Rearranging the moment equation y i e l d s : 2 2H • u ; Equation (1) represents the equation of the cable shape within the (x',y') coordinate system whose o r i g i n i s at 0 (the point of maximum sag). By inspection the equation describes a parabola, hence the name 187 Figure 55. Forces Acting on the Cable Segment OP. 188 'Parabolic Approximation'. Equation (1) i s not convenient i n i t s present ' form since the o r i g i n of i t s coordinate system moves depending on the geometry and the tensions. The next step, i s to translate Equation (1) to a fixed point which i s normally at one of the supports; support A i s used. Equation of the Cable Shape i n the (x,y) Coordinate System, with Origin  at Support A The tr a n s l a t i o n i s defined by: x' = x - a y" = y _ bj with a,b shown i n Figure 56 Substituting for x' and y' i n Equation (1) y i e l d s : y + b = a)(x-a) 2 (2) 2H The values a and b are found by employing the l i m i t conditions (x=0, y=0) and (x=L, y=E). The f i r s t gives: 0 + b = o)(0-a) 2 2H Therefore: b = ma2 2H S i m i l a r i l y the second y i e l d s : E + b = a)( L - a ) 2 2H Simplifying: E = o ) ( L 2 - 2La + a 2) - b 2H Substituting for b gives: E = M(L 2 -2La) 2H Rearranging: a = Therefore: b = oi 2H L HE 2 ~ oiL 'L _ HE 2 wL 189 Figure 56. Translating the Coordinate System from 0(x',y') to A(x,y). 190 Substituting the values of a and b into Equation (2) gives: y + uifT. Ktfiz = /,Jir _ TT UTT!12 2H r L _ 231 2 = CO r x -f L _ HE_" 2 coL J 2H 2 coL > Simplifying: y = cox2 _co 2H " H HE coL x And finally: 2H E _ coL L 2H x (3) This is the parabolic equation describing the shape of a freely hanging cable with the coordinate system origin at the left hand support (A). Tangents to the Cable The tension acts along the tangent of the cable. The tangent of the cable at any point P is given by the first derivative of Equation (3) with respect to x. -2 (E COL! cox' 2H dy _ cox dx ~ H L 2H — .- HILL L 2H tangenta = slope of the cable at x Therefore: tana = cox , X H E _ coL L 2H (4) Derivation of the Model for a Five-Line Cable System The five-line cable system is common in logging and is often referred to as the running skyline (Figure 50)'. i The five cable segments (Figure 57) are: 191 Figure 57. Geometry of a Five Line Cable System. 1 : haulback segment between support A and carriage 2,2' : haulback segment between carriage and support B 3 : mainline segment 3' : slackpuller segment. Additional d e f i n i t i o n s are: coj : weight per unit length of haulback o>2 '• weight per unit length of mainline 0)^ : weight per unit length of slackpuller R : weight of load (carriage and logs) C : carriage Q^yd^' angles of subchords with: 8j = arctan (y/x) 8 2 = arctan ((y+E)/(L-x)) a C l ' aC2' aC3' aC2'' aC3' ! a n S i e s °^ cables at the carriage with the horizontal. The general load position equation can be derived by using the equation,? that describe the equilibrium of the carriage. Sum of horizontal forces: H^ +H^ +H^ , = H^ +B^ , Sum of v e r t i c a l forces: V 1 + V 2 + V 3 + V 2 ' + V 3 ' = R The v e r t i c a l components, V\, can be replaced by: = H^tanct^, , i Therefore: H l t a n a C l + H 2 t a n a C 2 + H 2 ' t a n a C 2 ' + H 3 t a m C 3 + H 3,tana c 3, = R (5) Tana can be replaced by Equation (A) evaluated at x=0: 1 t a n a a x 2H2^ ; L replaced by x as the span of segment 1 i s x uni t s . 193 t a n a C 2 = E + y "gO-x^ L-x 2H„ ; span i s L-x; difference i n elevation i s E+y tana. C2' t 3 n a C 3 = 2H 9, , L-x tana. C3' t o o X _ _L_ x 2H_J x 2H 3,J With a>1 - to'/cosej^ a)'/cos6. U3 = t 0 3 / ' c o s e i "2* = w j / c o s S 2 co^  1 = to^,/cos6^ Substituting these values of the tangents into Equation (5) yi e l d s ; H, C 0 , X ' £ _ _!_ x 2Hj + H, J t i * (°2 ( L- X ) L-x ~ 2H„ + H 2, ( mv (L-x) + H, w.xi /• to0 ,x x _ JL . + Hv y. _ 3 x 2H3J (x 2H3,J L-x = R 2H_, J At t h i s point several s i m p l i f i c a t i o n s can be made. Assuming the haulback segments 2 and 2' are p a r a l l e l and there i s no f r i c t i o n at the t a i l h o l d sheave (support B) then: V = V . C2 C2« 2 2' H2 = H 2 ' T = T C2 C2' Therefore the v e r t i c a l force balance equation becomes: ^(H^Hg+Hg.) - to l X/2 - co3x/2 - to3,x/2 + 2H 2 (y+E) / (L-x) x - co2(L-x) = R Using the horizontal equilibrium of forces Hj+H3+R"3, = 2H 2 and gathering terms yields: 194 2H2y_ - x(w1+a)3+w3,) + 2H2' 2+1 -a>2(L-x) = R 2H2 x L-x L-x = 2R + x (u^ +a^ +u^ ,) + 2o)2(L-x) Ly - yx + yx + Ex = 2 R + * ( " i + " 3 + 'V> + 2 ^ (L-x) x(L-x) 4H„ Therefore: _ x(L-x)(2R + x(to.+u)0+w-,) + 2u)„(L-x)) y — x J J / — i^ x 4H2L L To simplify slightly let u^u^u^i (the combined weight of the mainline and the slackpuller). The equation for the load displacement y, given x, is therefore: • m x(L-x)(2R + x(co1+to3) + 2co2(L-x)) _ E x ^ ( g ) 4H2L L~ Two problems exist in solving for y using Equation (6). First, the values of to^ , k>2, w^' depend on the angles of the subchords AC and CB. These are computable only i f the y value for the carriage location is known. A direct solution for y is therefore not possible anil an iterative solution is required. The second problem is in determining H9. This value constrains the solution and is dependent on either the maximum allowable tension in the haulback (breaking strength adjusted by a factor of safety), the maximum line pull that the yarder can exert, or, in extreme conditions, the maximum allowable tension in the mainline. The latter occurs in uphill yarding when the chord (AB) slope is excessive (e.g. 100 percent). This is illustrated in Table XII. 195 TABLE XII LINE TENSIONS HITH THE CARRIAGE AT MIDSPAN FOR A MADILL 052 TENSION SKIDDER USING THE PARABOLIC MODEL LOAD INCLUDING CARRIAGE = 35000 LB. TENSION (HAULBACK) AT INTERLOCK = 56500 LB. HEIGHT OF HAULBACK/FOOT = 2.34 LB. HEIGHT OF HAINLINE+SLACKPULLER = 4.13 LB. SPAN = 1400 FEET CHORD PERCENT SLOPE% MIDSPAN DEFLECTION 100 9.9 HORIZONTAL COMPONENT VERTICAL " AXIAL » HAULBACK LNDG-CARR (LB. ) 46319.18 -34 372.72 57679.75 HAULBACK CARS-TAIL (LB.) 36151 .88 44944.28 57679.76 MAINLINE * SLACKPULLES (LB.X 2598 4.55 -20515.96 33107.59 50 HORIZONTAL VERTICAL AXIAL 9.4 54482. 28 -16670.04 56975.50 46930.98 32306.17 56975.4 2 39379.65 -12942.34 41451.78 0 HORIZONTAL VERTICAL AXIAL 8.8 55502.11 8908.62 56212.55 55502.10 8908.68 56212.90 55 502.07 8 274. 02 56116. 13 -50 HORIZONTAL VERTICAL AXIAL 8.4 46087.54 30695. 34 55373.86 526 03.00 •17296.91 55373.64 59118.45 38 89 8.45 70767.44 -100 HORIZONTAL VERTICAL AXIAL 8.4 34869. 15 41850.57 54473.23 42810.80 -33683.19 54473.03 50752.39 60515.73 78980.56 196 Examining the first two cases, the maximum will be expressed in terms of axial tension. If the maximum is due to machine capacity then T will occur In the haulback at the yarder. Otherwise i t will occur max J in the haulback at either support A (yarder) or B depending upon which is higher in elevation. In either case T can always be expressed at the max support B. If i t occurs at B no adjustment is required. On the other hand i f T is at A then the corresponding tension at B can be found by max employing the catenary relationship which states that the difference in tension between points on a cable is equal to the cable weight multiplied by the elevation difference between points (Carson, 1971). This is possible providing there is no loss of tension at the carriage in the haulback; i.e., '^QI='^Q2' This is a valid assumption since the carriage rides on sheaves. Therefore: if T occurs at B then T=T max max if T occurs at A then T=T -Ecu' max max 1 The horizontal tension, U^* c a n n o w ^ e expressed by using the tangent of the cable at B: H„ = Tcos(arctan(tana__)) Using Equation (4) for the cable slope: (7) H2 = Tcos(arctanfy+E + ( L x)iJj2^ (L-x " 2H„ 2 } Depending on the iterative technique used for the solution of Equation (6) this expression (7) may not be suitable as c a n n o t be solved directly. Equation (7) can be rearranged by employing the relation: 1/cosa = / l + tanza The expression for H_ becomes: H„/l + tan2an„ = T 197 Therefore: E ^ / l + J+E (L-x) a) L-x 2H 2 2 = T Squaring both sides: H.2 '1 + V+E + (L-X)OJ 2' 2" = T2 ^ X 2 H 2 ' > Expanding: H| 1 + y+E L-x 2 + 2 y+E] fL-x)q)?+ f (L-x) ui ] L-xj l 2H 2J 2H, = T Gathering terms and simplifying: H|fl + 1 l l -y+E 2] + H2(y+E)o)2 + f(L-x)o) ?l 2-T2 =0 ... (8) H 2 can be solved by using the formula for a quadratic equation! H 2 = -b + A2 - 4ac 2a with a = 1 + y+E L-x b = to2(y+E) c = f(L-x)o) 1 2 _ T2 I f the c r i t i c a l tension occurs i n the mainline then H 2 must be solved i n a more complicated manner. must f i r s t be expressed i n terms of the horizontal tension i n the mainline and the slackpuller (H^) as the constraint i s now i n the mainline and the slackpuller combined. Using the force balance at the carriage: H 1 + H 3 = 2 H 2 H l = 2 H 2 " H 3 And; H 2 = (^ + i y / 2 Also, since there i s continuity of tension at the carriage i n the haulback then: T = T Cl C2 V I + tan2a c l - T C L 198 And H„ 1^ + tan 2a = T 2 C2 C2 Using the same expansion in the derivation of Equation (8) a 2 H 2 + b 2 H 2 + And; a ^  + b ^ + (L-x)t 2 = T 2 C2 XO), 2 - T 2 C l (9) (10) with: a^ = 1 + a 2 = 1 + X x y+E L-x b 2 = -(y+E)o)2 Substituting the alternate forms of and H2: Hx - 2H 2 - H3 H2 = (H1 + H 3)/2 into Equations (9) and (10) gives: a l H l + b l H l + And; a2K| + b ^ + xco. 2= a i ( 2 H 2 - H 3)2 + b 1(2H 2 - H3) + Xo), f(L-x)q)?) 2 = a^Hj + H 3)2 + b 2(H 1 + H^ ) + (L-x)c Simplifying; a'jHj + bjHj = a i ( 2 H 2 - H 3)2 + b 1(2H 2 - H3) (11) a 2 H 2 + b 2 H 2 = a 2 ( ( H l + V / 2> 2 + b 2 ( ( H l + H 3> / 2 ) •••• <12> Using Equation (11) can be solved by using: H l = ^ i 4 ^ ! " 4 a i c i '> c i = a i ( 2 H 2 ~ H 3 ) 2 + b l ( 2 H 2 " V This expression for can now be used in Equation (12) which will yield an expression for H2 in terms of H^only: H3 = Tcos(arctan X _ ™ o x 2 ^ ) H 3 can be found by rearranging and using the same approach used to obtain Equation (8). Therefore: fi • w 2' -H 3(y. 3) + f^3 • 12 J 2= T 2 199 And; = -b^ + /b^ - 4 a 3 c 3 ~ 3 with = 1 + b 3 = -yo,3 C3 = XU>_ 2 J 2 _ T2 and; T i s the maximum mainline+slackpuller l i n e tension at the carriage. If the carriage i s above the support A then T=T . Conversely, i f r r max J' the carriage i s below A then T=T -ytu'. This value of tension, T , max 3 max i s for both the mainline and slackpuller combined. Separating the two requires knowledge of their working configuration i n the carriage. As seen from Figure 58 the carriage drum i s i n equilibrium when the tension i n the mainline (T m) i s equal to the tension i n the slackpuller (T g) plus weight of the load (not including the carriage). Therefore: T m = T g + R' (R' i s the weight of the logs) The combined tension, T , i s : max T = T + T max m s = 2T + R' s So, i f the mainline tension i s the l i m i t i n g value then rearranging the above i n terms of T only y i e l d s : m J J T = 2T - R« (13) max m Solution of using Equation (12) can now be achieved. Iteration i s required as a closed form i s not possible. carriage drum tong l i n e (supporting the load R') Figure 58. Configuration of the Lines at the Carriage Drum 201 Deciding Where C r i t i c a l Tension Occurs i n the Cable System The c r i t i c a l tension can be due to one of the following three reasons: 1) l i n e capacity of the haulback i s exceeded (T^) 2) l i n e p u l l capacity of the yarder i s exceeded (T ) 3) l i n e capacity of the mainline i s exceeded '(T ) The load position, y, i s found f i r s t using either or T , whichever Is l e s s , for T . Then H_ i s computed by finding H 0 and H,. max 3 r J & 2 1 i s evaluated using Equation (10) and i s found when the value of y i s determined i n the i t e r a t i v e technique. Therefore using the equilibrium of horizontal forces at the carriage = 2H^ - H^. The tension i n the mainline+slackpuller at the carriage, i s found from: TC3 " V1 + t a n Z a C 3 with tan 2a^2 7. _ 2£w-x 2H 3j This i s converted to the maximum tension (T ) i n the mainline + max slackpuller: i f the carriage i s above support A then X i f the carriage i s below support A then T =T„_+ytoI ° r r max C3 J 3 The maximum allowable mainline tension (T ) i s exceeded i f : m T < T + R1 from Equation (13) m max  2 If the c r i t i c a l mainline tension has not been exceeded then the solution for y i s acceptable. I f not y must be resolved using as defined by Equation (12). 202 Derivation of the Model for a Three Line Cable System The three line cable configuration is commonly referred to as a shotgun or gravity system (Figure 51). The formulation is identical to that for a five line system. The resultant equation for the load displacement, y, i s : y = x(L-x)(2R + u>2(L-x) + x(co ], + o> ) ) _ Ex _ _ _ _ L The expression for tensions is identical to those for the five line system except the horizontal force balance equation at the carriage is + = H2. The appropriate changes are needed in Equations (11) and (12). The solution when the critical tension is in the snubbing line (line segment 3) is simpler as there is only one line and no separation of tensions is required as was done for the slackpuller and mainline (Equation (13)). 203 APPENDIX D PERSPECTIVE PLOTTING Three dimensional representations using perspective are created i n a s i m i l a r mannerj to orthogonal projections (Newman and Sp r o u l l , 1973) In a d d i t i o n to s p e c i f y i n g the viewing angles, about (6) and above ( s ) , the l o c a t i o n of the viewpoint (x ,y ,z ) and the viewing aperature (a) are required. These viewing controls are depicted i n Figure 59. The r o t a t i o n angles 6 and 8 are applied to each (x,y,z) data point i n the same way,1 as f o r the orthogonal p r o j e c t i o n . (x'.y'.z',!) = (x,y,z) cose sine 0 0 (-sin6cosg) (cos6cosg) sing 0 (sinesinB) 0} (-cosOsing) 0 cosg 0 0 1 The coordinate system (x'.y'.z') for the object i s then translated such that the viewpoint ( x ^ y ^ z ^ ) becomes the o r i g i n . Therefore: (x",y",z",l) = ( x ' , y \ z ' , l ) 1 0 0 0 0 1 0 0 0 0 1 0 -x -y -z 1 v v v ' These two transformations ( r o t a t i o n and t r a n s l a t i o n ) can be combined: (x'.y'.-.z'.l) = (x,y,z,l) cos6 (-sinOcosg) sin6 (cos6cos6) 0 sing -x -y v v (sin6sing) (-cos6sin8) cos 8 -z 205 The object can now be; represented i n three dimensions by projecting the (x',y',z') points, that describe the object, back towards the viewpoint onto a screen placed between the object and the viewpoint as shown i n Figure 60. Assume the screen i s square i n shape with width 2b and i s placed at a distance/ "a" from the viewpoint. Using si m i l a r triangles (AEBP = AEAQ) the point P(x',y',z') i s projected onto the screen as P(x ,y ,z ) with: s' Js' s x x' and z z' s = —, s = —, y' y' a J a 1 Therefore: x = ax 1 and z = az s I a t y y These points can be scaled to the screen size by dividing by b: t \ x = s and z = r i s a_ bjy« iy The values of a and b are never required as t h e i r r a t i o (a/b) represents the cotangent of the viewing aperature angle Then, i x ~ x' s y'tan(a/2) zj ° y'tan(a/2) The object i s now plotted on the two-dimensional screen using the coordinates (x ,z ). The o r i g i n a l y' coordinate i s not required as no s s depth i s used. The point (x ,z ) becomes (x,y) of the screen coordinate s s system. Depending upon the r e l a t i v e sizes of the; aperture angle and the object certain regions of the projection could l i e outside the viewing pyramid. Points (x ,z ) w i l l l i e inside the viewing pyramid s s i f : 207 -y' < x' < y' tan(a/2) -y' < z' < y' tan(a/2) If a point l i e s o u tside^it i s not plotted as part of the projection. To obtain a clean and eye-pleasing p l o t , l i n e s that cross the viewing pyramid boundary should be truncated at the boundary and not merely stopped at the closest viewable point. This clipp i n g problem i s discussed by Newman and Sproull (1973); a suitable alogrithm i s also provided. The removal of hidden areas can be done i n the same manner as for orthographic projections. ; Perspective p l o t t i n g , although more r e a l i s t i c , i s far more d i f f i c u l t to implement than orthographic projection. The increased execution time becomes excessive especially when attempting to " c l i p " areas outside the viewing area (unless i t can be done automatically by the pl o t t i n g device being used). For t h i s reason i t was not used i n the Terrain Simulator package developed for the Hewlett-Pacieard 9830A desktop computer system. 

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