UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A general theory on common point intersect sampling with special application to downed woody particles Egglestone, Jeffrey N. 1975

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1975_A6_7 E34_5.pdf [ 7.72MB ]
Metadata
JSON: 831-1.0075128.json
JSON-LD: 831-1.0075128-ld.json
RDF/XML (Pretty): 831-1.0075128-rdf.xml
RDF/JSON: 831-1.0075128-rdf.json
Turtle: 831-1.0075128-turtle.txt
N-Triples: 831-1.0075128-rdf-ntriples.txt
Original Record: 831-1.0075128-source.json
Full Text
831-1.0075128-fulltext.txt
Citation
831-1.0075128.ris

Full Text

A GENERAL THEORY ON COMMON POINT INTERSECT SAMPLING WITH SPECIAL APPLICATION TO DOWNED WOODY PARTICLES BY JEFFREY N. EGGLESTONE B. Sc., University of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE" i n the Department of FORESTRY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1975 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 o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t T h e 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 V a n c o u v e r 8, C a n a d a - i i -ABSTRACT A general sampling theory r e f e r r e d to as common point i n t e r s e c t sampling i s developed and assessed. This new technique i s s p e c i f i c a l l y applied to the problem of estimating parameters of populations of downed woody p a r t i c l e s of i n t e r e s t i n f i r e research. The performance of the common point i n t e r s e c t sampling method i s compared to that of the well-established l i n e i n t e r c e p t technique with respect to two l e s s e r (less than 3 inches i n diameter) downed woody p a r t i c l e s populations. Results of these tests i n d i c a t e that proper a p p l i c a t i o n of the new sampling system can y i e l d t o t a l volume estimates of approximately 15 per cent p r e c i s i o n with savings of up to 40 per cent of the t o t a l sampling time required by the l i n e intercept technique. The common point i n t e r s e c t sampling method i s demonstrated to be a useful approach to solv i n g the problem of obtaining estimates f o r numerous a t t r i b u t e s of populations of downed woody p a r t i c l e s . General formulas are also provided which f a c i l i t a t e the a p p l i c a t i o n of common point i n t e r s e c t sampling to the task of obtaining parameters of standing timber such as crown area and average crown diameter from a e r i a l photographs. The common point i n t e r s e c t technique i s shown to be a f a s t and accurate means of sampling f o r e s t material. The new sampling system has been applied rigourously i n only one problem area. The general nature of the common point i n t e r s e c t system suggests, however, that i t has many other applications i n a m u l t i p l i c i t y of s c i e n t i f i c d i s c i p l i n e s . - i i i -TABLE OF CONTENTS Page TITLE i ABSTRACT .N i i TABLE OF CONTENTS i i i LIST OF TABLES v i v LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENTS v i CHAPTER I. INTRODUCTION ,. 1 CHAPTER I I . DEVELOPMENT OF THE COMMON POINT INTERSECT CONCEPT... 7 Brie f review of past general sampling techniques.... 7 Presentation of the Common Point Intersect theory... 10 General discussion 10 Application to downed woody p a r t i c l e s 19 CHAPTER I I I . FIELD TEST TO EVALUATE THE COMMON POINT INTERSECT TECHNIQUE 43 General discussion 43 Fieldwork undergone wi'th respect to the Line Intersect technique 52 Fieldwork undergone with respect to the Common Point Intersect technique 61 CHAPTER IV. ANALYSIS OF THE FIELD TEST DATA 71 CHAPTER V. CONCLUSIONS 77 CHAPTER VI. PRACTICAL APPLICATIONS OF THE COMMON POINT INTERSECT TECHNIQUE 87 BIBLIOGRAPHY 92 APPENDIX I PROOF OF COMMON POINT INTERSECT FORMULA FOR CYLINDERS 102 APPENDIX I I COMPUTERIZED ANALYSIS OF VARIANCE MODEL 106 APPENDIX I I I COMPUTERIZED DOWNED WOODY PARTICLES MODEL I l l - i v -LIST OF TABLES Page 1. Actual and estimated computer-generated lesser downed woody p a r t i c l e volumes 35 2. Intersection angles of lesser downed woody p a r t i c l e s i n the study areas 48 3. Sample means and standard deviations of the orientation random variables for lesser downed woody p a r t i c l e s i n the study areas 50 4. F i e l d p a r t i c l e intercept counts and corresponding volume estimates for the l i n e intersect sampling units 58 5. Relevant s t a t i s t i c s for the l i n e intersect f i e l d sampling units 60 6. F i e l d p a r t i c l e intercept counts for the common point intersect sampling units 64 7. F i e l d p a r t i c l e volume estimates for the common point intersect sampling units 68 8. Sampling time comparison of the l i n e intersect and common point intersect methods 70 - v -LIST OF ILLUSTRATIONS Figures Page 1 Flowchart depicting the development of the general common point intersect sampling concept 17 2 Typical cross-section of sampling unit s^ with special application to downed woody cylinders 25 3 Curves representing actual and estimated computer-^ generated downed woody p a r t i c l e volumes with respect to size class 1 (<_ \ inch) 37 4 Curves representing actual and estimated computer-generated downed woody p a r t i c l e volumes with respect to size class 2 (> % inch <_ 1 inch) 38 5 Curves0 representing actual and estimated computer-generated downed woody p a r t i c l e volumes with respect to size class 3 (> 1 inch, <^  3 inches) 39 6 Map of Study Area 1 depicting placement of both the l i n e intersect and common point intersect sampling units .... 44 7 Map of Study Area 2 depicting placement of both the l i n e intersect and common point intersect sampling units .... 45 8 'Go-no-go' gauge used to determine the diameter size class of each intersecting p a r t i c l e under 3" i n diameter at i n i t i a l point of intersection 56 - v i -ACKNOWLEDGEMENT S The writer wishes to thank Dr. J.H.G. Smith, Dr. J. Damaerschalk and Mr. S. Smith for their suggestions and comments concerning the thesis during i t s review stage. The writer also wishes to thank B.C. Forest Products for permission to sample logging residue i n two of their cutover areas on Vancouver Island, B r i t i s h Columbia. This study was financed by the Canadian Forestry Service as part of a project to improve knowledge of forest fuels and f i r e behaviour i n coastal logging residue. A special note of thanks i s due my parents and family for their continuing moral support during the writing phase of the thesis. - 1 -CHAPTER I Introduction This thesis i s designed with two primary objectives i n mind. The f i r s t i s to present a general technique for non-destructively obtain-ing quantitative estimates for attributes of any community of objects. The second i s to apply t h i s technique to an important s o c i a l problem area. Since the author i s currently employed as a f i r e research o f f i c e r i n the Canadian Forestry Service, the problem area selected i s forest f i r e oriented, namely precise measurement of downed woody p a r t i c l e s populations. A woody p a r t i c l e i s considered 'downed' i f i t has been detached from i t s source and l i e s within s i x feet of the forest f l o o r (Brown, 1974). Attention i s focused primarily on lesser downed woody pa r t i c l e s populations from which two constituent populations are chosen for analysis. Before considering the scope and methods of the thesis, i t seems very appropriate to consider whether i t i s worthwhile or not to develop a new general quantitative sampling technique. H i s t o r i c a l l y , general sampling schemes have been developed from s p e c i f i c sampling methods designed to s a t i s f y very s p e c i f i c needs i n well-defined professional d i s c i p l i n e s . The different physical and mathematical constraints imposed by each separate d i s c i p l i n e have made this established pattern of procedure a necessary rather than optimum one. There currently being no urgent demand for new quantitative sampling techniques i n any of the fire-oriented phytosociological biomes, i t appears that at present a new technique would - 2 -be viewed with interest only on an academic plane. Any new sampling method should be s i g n i f i c a n t l y superior to the most successful e x i s t i n g one when both are applied to a s p e c i f i c problem. Once th i s superiority i s demonstrated the u t i l i t y of the new technique would be established operationally. This thesis demonstrates the superiority of the new sampling technique i n at least one s p e c i f i c problem area. At t h i s time i t i s relevant that a comment be made concerning the problem area chosen for analysis. The significance of populations of downed woody pa r t i c l e s as areas of concern for forest harvesters, land managers and f i r e s c i e n t i s t s as w e l l as numerous other professionals i s not to be l o g i c a l l y disputed (Bailey, 1969; Beaufait and Hardy, i n prep; Deeming, 1972). The amounts, weights, and dis t r i b u t i o n s of larger downed woody materials are of much concern when considering problems such as the assessment of,' logging waste, behaviour of w i l d l i f e , and probability of successful natural or a r t i f i c i a l regeneration (Davis, 1959; Wagener and Offord, 1972). Also the volumes, weights, surface areas and d i s t r i -butions thereof for the smaller downed woody materials play a key role i n rating the f i r e hazard within a pa r t i c u l a r region (Deeming, 1972; Beaufait and Hardy, i n prep; Brown and Roussopoulos, i n prep.). However, i t could be argued that i t seems s i l l y to sample these downed woody pa r t i c l e s d i r e c t l y . A more l o g i c a l approach would be to sample the forest charac-t e r i s t i c s of interest when the woody materials are secured to the standing trees. Such i n i t i a l values would then be combined with mathematical relationships which describe the effects of a given set of disturbances on the forest to arrive at estimates for the desired parameters of the downed woody p a r t i c l e s populations. Although past efforts to apply t h i s approach may have f a i l e d (Beaufait and Hardy, i n prep.), the author - 3 -recognizes this as a viable approach to the problem of quantifying parameters of downed woody p a r t i c l e s populations. Nevertheless, u n t i l a technique i s created which can successfully apply t h i s systems approach to the downed woody materials complex, interim physical methods w i l l have to be used which means at least temporarily that downed woody pa r t i c l e s w i l l have to be sampled. In t h i s thesis, a new general sampling technique i s applied to populations of downed woody materials. Normally when a new concept of sampling i s applied to a problem area i t i s because the old ones are i n some way unsatisfactory. I t i s not immediately apparent that this i s true i n the case of downed woody materials. For example both 0.1 acre plots and long transect l i n e s have been successfully used to measure the larger fuels (Bailey, 1969; Howard and Ward, 1972). Also short transect l i n e s have been e f f e c t i v e l y used to measure the smaller fuels (Beaufait, Marsden and Norum, 1974; Brown, i n prep; Brown and Roussopoules; i n prep.). None of the above methods have disadvantages which seriously impair t h e i r applications. The j u s t i f i c a t i o n of applying a new general sampling technique to downed woody fuels rests upon the insight of the author. He has worked i n populations of downed woody materials for seven years and has t r i e d numerous versions of currently used sampling systems i n many fuel complexes. He believes that the new general sampling concept i s not only feasible when applied to downed woody fuels but also has a good chance of being s i g n i f i c a n t l y more e f f i c i e n t than a l l other previously applied direct sampling systems. The scope and methods of the thesis are relevant to both of i t s major components which are the general theory and the case study. The scope of the general theory i s very broad; i t applies to any group of - 4 -objects whose attributes of interest can be described by functions of suitably well-behaved mathematical expressions. The methods used i n developing the theory are basic theorems and principles of advanced calculus. The case study applies primarily to lesser downed woody p a r t i c l e s . I t i s agreed that a woody p a r t i c l e which intersects a transect l i n e w i l l be described as lesser (greater) only i f i t s width at the i n i t i a l point of intersection i s less (greater) than or equal to 3 inches. Lesser woody p a r t i c l e s are selected for detailed scrutiny because past studies have shown that these components are most l i k e l y to be consumed by the majority of broadcast f i r e s (Steele and Beaufait, 1969; Brown, i n prep). I t should be noted that, although needles s a t i s f y the above agreed d e f i n i t i o n of lesser woody materials, they w i l l not be considered d i r e c t l y i n the case study. Needles play an important role i n the i g n i t i o n process and hence i n the i n i t i a l stages of f i r e growth (Beaufait, 1965). But i n general the extreme d i f f i c u l t y of measuring or counting needles even over r e l a t i v e l y short distances necessitates the use of indirect sampling techniques. Examples of such techniques are regression estimates from lesser downed woody f u e l data (Brown, 1970) and tables of desired needle attributes for specified f u e l types (Fahnestock and Chandler, 1960; Brown, 1970). Regardless of which indirect sampling scheme i s selected, the main objective of the case study can be completely met simply by considering the problem of obtaining estimates for the parameters of interest with respect to a l l subsets of the lesser downed woody fuels population excluding needles. I t should be kept i n mind that the main objective of the case study i s to evaluate the new general sampling technique by comparing i t to the most successful ex i s t i n g one when the - 5 -two are applied to two lesser downed woody fuels populations. There are two basic methods employed i n the development of the case study. The f i r s t i s the generation of mathematical formulae from the new general sampling concept. These formulae w i l l serve to estimate the desired properties of the downed woody fuels populations. Care must be taken that the assumptions made i n a r r i v i n g at the e x p l i c i t estimates r e f l e c t common f i e l d s ituations. Attention must also be paid to the levels of accuracy and precision which the estimates should meet, cost and time constraints under which the new and exi s t i n g techniques may be forced to operate, and the physical and mental tolerance levels of average f i e l d inventory personnel. The l a s t item i n this l i s t i s an especially important one. Quantitative sampling of lesser downed woody fuels i n logging residue for example i s tedious and requires painstaking work. However, at present i t i s highly recommended that this task be undertaken i f r e l i a b l e objective predictions or assessments of f i r e behavior and impact are to be made (Beaufait, Marsden, and Norum, 1974). If the quantitative sampling process used i s incompatible with normal levels of physical and mental tolerance, the non-sampling errors (Husch, M i l l e r , and Beers, 1972; p.201) introduced through improper or careless measurements may w e l l have a major effect upon the precision of the derived formulae. The formulae derived from the general sampling concept w i l l be determined using functional analysis, analysis of variance, events modelling, numerical analysis and parametric s t a t i s t i c a l hypothesis testing. The second method used i n the development of the case study i s the application of the new sampling process to two actual f i e l d situations and a comparison of th i s new technique to the most successful existing one - 6 -with respect to these same two lesser downed woody fuels populations. The f u e l complexes selected for examination are two areas of fresh (less than 1 year old) logging residue (slash). These areas are chosen because during the f i r e season many untreated slash areas become highly flammable. This means that objective quantitative projections of f i r e danger i n slash are needed. Working i n slash then maximizes the relevance and usefulness of the f i e l d exercise. The f i e l d comparison i s made by inspecting pairs of t o t a l sampling times required to obtain pairs of population para-meter estimates;, where the members of each pair of estimates have both common units and a common allowable sampling error (Husch, M i l l e r , and Beers, 1972) for a common percentage of the time. With respect to each subset of the lesser downed woody fuels population, the t o t a l sampling time i s the sum of the t o t a l fuels inspection time plus the t o t a l t r a v e l time between sample; units. Each resultant t o t a l sampling cost i s d i r e c t l y proportional to i t s corresponding t o t a l sampling time. I t i s easy to see then that the t o t a l sampling times provide ranking indices for the two sampling techniques. This procedure i s well established and has already been used i n studies involved with logging residue (Bailey, 1969; Howard and Ward, 1972). Having presented the general layout of t h i s study, interest i s now focussed upon f u l f i l l i n g the two objectives cited at the beginning of the thesis. - 7 -CHAPTER I I Development of the common point intersect concept  Brief review of past general sampling techniques Before attempting to achieve the f i r s t objective of the thesis which i s to present a general non-destructive technique for quantitatively estimating population parameters or attributes for any community of objects, i t seems l o g i c a l to look b r i e f l y at some of the more successful general sampling schemes already i n existence. The two selected on the basis of degree of f l e x i b i l i t y and extent of proven usefulness are transect system sampling (T S S) and l i n e intersect sampling ( L I S ) . T S S designs are simply applications of the solution offered by the 18th century French n a t u r a l i s t Buffon to the needle problem (Bradley, 1972). I t i s important to note that the solution given by Buffon and others (Segebaden, 1964; De Vries, 1973) i s based on the assumption that the objects of interest are randomly distributed throughout the area of concern. The concept involved i n this technique i s quite ingenious and should be b r i e f l y described. Consider an area occupied by objects which are randomly distributed. I f a network of transects i s superimposed on this area those objects which both intersect any transect i n the network and possess the attributes of interest are t a l l i e d . The t o t a l number of relevant intersections i s then inserted into an appropriate formula derived from Buffon's solution to the needle problem. In t h i s way an estimate i s obtained for the desired population parameter. Approximate general formulas for the standard error of the estimate obtained, the transect system spacing required to achieve a specified precision l e v e l , and the number of sample points required to meet a desired degree of precision are a l l given by Bradley (1972). In passing i t should be mentioned that the general - 8 -T S S theory can be extended to permit i t s a p p l i c a t i o n i n any c o l l e c t i o n of objects whose placement and angular o r i e n t a t i o n d i s t r i b u t i o n s can be q u a n t i f i e d . T S S has already been applied s u c c e s s f u l l y to the problem of determining cross-country transport distances f o r r e a l road nets (SegebWen, 1964) , and also to the problem of estimating road lengths (Bradley, 1972). Other applications can be made. For example the t o t a l length and volume of greater logging residue could be estimated with TSS. Also information about stand c h a r a c t e r i s t i c s and lengths of streams could be obtained through the use of TSS. A l l of the above applications of TSS are greatly s i m p l i f i e d by use of a e r i a l photographs. LIS i s c l o s e l y r e l a t e d to TSS and i n f a c t may even be regarded as a s p e c i a l case of TSS, where the network of transects has been reduced to a s i n g l e transect. Several derivations of LIS formulas are a v a i l a b l e (Canfield, 1941; Warren and Olsen, 1964; Van Wagner, 1968; Brown, 1971). The most general discussion of the LIS concept i s that given by De V r i e s (1973, p.4-7). De V r i e s 1 formulation r e s t s 3 p r i m a r i l y upon two assumptions. The f i r s t i s that the objects of i n t e r e s t can be viewed l o g i c a l l y as l i n e segments or shapes of moderate curvature and the second i s that the placement and angular o r i e n t a t i o n d i s t r i b u t i o n s of the population of objects ( i d e n t i f i e d as l i n e segments or shapes of moderate curvature, whichever i s appropriate) can be described q u a n t i t a t i v e l y . I t i s possible to extend De V r i e s ' argument to a more abstract plane making i t independent of the two above assumptions. LIS i s a very popular technique having been applied to range vegetation (Canfield, 1941), greater logging residue (Warren and Olsen, - 9 -1 9 6 4 ; Van Wagner, 1 9 6 8 ; Howard and Ward, 1 9 7 2 ) , and lesser logging residue (Beaufait, Marsden, and Norum 1 9 7 4 ; Brown, i n prep.). Suggestions have also been made regarding i t s possible application i n other forestry-related problems, such as the estimation of standing timber parameters (De Vries, 1 9 7 3 ) . Before proceeding to the new proposed general sampling concept, one point should be stressed. I t i s not 'the intention of the author to imply through the introduction of a new technique that transect system sampling and l i n e intersect sampling are i n any way inadequate. More research conducted over a wide range of sampling problems i s required i n order to make a rigorous, unbiased comparison of the three approaches. - 10 -Presentation of the common point intersect theory General discussion Before developing the common point intersect theory mathematically, a non-mathematical introduction to the general theory w i l l be made. This introduction w i l l help non-mathematicians grasp a basic understanding of the general theory. Suppose there exists a population of objects. To use some forestry examples, these objects could be standing trees, downed stems or branches, borer beetles, f r u c t i f i c a t i o n s of wood destroying fungi, e t c Each population can be described i n terms of variables which w i l l be called parameters. The parameters of concern are l i s t e d by the i n v e s t i -gator, say pj£'. .. ,py. But the investigator may be interested i n more than just estimating pj,...,p^. He may want to estimate some combination of these parameters, say p = u(pj,,.;. ,pjp where uy i s some function. The investigator does not know the value of any of the parameters, but he has defined Pi,.-. .,p^ and so he knows what they mean. He does not know the value of p, but he has defined u and through the meaning attached to each parameter, he knows what p means. So suppose he wants to estimate p. He may do this by introducing a random variable X derived from the meaning of p. A random variable i s simply a function which maps outcomes of some experiment E into r e a l numbers. In this case E i s the process of selecting locations for sampling units of a common size and shape within the population of obj ects. At this point the investigator knows what X means but he does not have a p r a c t i c a l way to evaluate X. To be more s p e c i f i c , suppose p i s the mean number of p a r t i c l e s per unit area. I m p l i c i t i n the d e f i n i t i o n of p i s the random variable Y where Y represents the number of p a r t i c l e s per unit area. Hence set X = Y g^ g 2 where Y g^ g 2 i s a random - 11 -variable mapping each sampling unit ( a l l of common size s^ and shape s 2) into the mean number of p a r t i c l e s per unit area with respect to that sampling unit. p i s a variable consisting of some combination of p^,...,?^ where p^,. . .,p^  are parameters. This means almost by d e f i n i t i o n that they can be easily referenced to random variables since i n effect they are descriptors of the d i s t r i b u t i o n s of random variables. A consideration of the random variables i m p l i c i t within the meanings of p^,...,p^ combined with the d e f i n i t i o n of p leads to the d e f i n i t i o n of X. The investigator knows what X ( S ) means but as yet he does not know how to evaluate X at any & i n a p r a c t i c a l way. The d e f i n i t i o n of X ( S ) combined with the d e f i n i t i o n of p results i n a point estimate (T(X(Sj),...,X(S n))) of p, where i s a set of sampling u n i t s . T i s a function which maps sets of r e a l numbers into r e a l numbers. T i s not unique unless s p e c i f i c properties are required for the point estimator of p. Using standard s t a t i s t i c a l procedures, i t can be assumed that a point estimator of p has been found which s a t i s f i e s a l l required properties. There are two major remaining items to discuss before developing a rigorous mathematical derivation of the general theory. These items involve an arbitrary sampling unit(S) and the random variable X. The sampling unit Sj i s r e s t r i c t e d to being a right c i r c u l a r cylinder because of mathematical considerations regarding X. Without going into s t a t i s t i c a l d e t a i l s , i t suffices to say here that b a s i c a l l y the size and location of S are arbitrary. The l a s t important item i s how to evaluate X at -S i n a p r a c t i c a l way. The concepts involved here are so i n t r i n s i c a l l y linked to mathematical considerations that a meaningful non-mathematical discussion cannot be made. - 12 -Keeping the above introduction i n mind, a mathematical presenta-tion of the general theory i s now made. Consider any community (C) of objects temporarily fixed i n space. Let u be a function which associates to each K-tuple (p^,.. . ,'p^ ) the unique value uCi>]_,... ,p^)=p, where pi Pk are parameters of unknown values describing C. A general technique w i l l now be provided for deter-mining quantitative knowledge about p with respect to C. Having specified u the meaning of p(the image of (p^ p^) under u) i s understood. From the meaning attached to p, i t i s possible to introduce a random variable X defined on the sample space of outcomes of an experiment E which consists of selecting locations for sampling units of a common size and shape within C. The o r i g i n of X i s i n no way mysterious. X i s simply a function consisting of a combination of the attributes con-s t i t u t i n g p; t h i s function i s referenced to the sample space of outcomes of E whose i n f i n i t e union comprises C. For example i f p i s a mean quantity per unit area, X i s introduced as the obvious functional extension of p defined on the sample space of outcomes of E, i . e . X i s a function which maps each sampling unit into a mean quantity per unit area with respect to that sampling unit. Through X and the meaning attached to p, a point estimate for p ( T ( X ( s p ) , . . . , X ( S j p ) can be expressed e x p l i c i t l y i n terms » . . "ii of X ( S . ) and known constants, (ie ,{'1,...,nH', where X ( S J ) i s the value of i % y . x th . i f . 1 X yielded by the i — independent re p e t i t i o n of E, i e {/l,. .. ,n<}: for a given • / ? n '' set vSjih. ,. Of course there i s no unique point estimator for p. The ii i ^ i = l form of T(X, X ) (where X. i s a random variable defined on the outcome-1 n i t i l ( ' \'' of the i — independent rep e t i t i o n of E, ie'|l,..-.,n|'f depends upon the choice of u. I t also depends upon any properties which are desirable for - 13 -T(X^,...,Xn) to have. Some examples of desirable properties are unbiased-ness, small mean-square error, closeness and consistency (Ehrenfeld and, Littauer, 1964). I t i s f a i r to assume that a l l properties required by the point estimator have been considered by the:1 investigator and that T(X(sp X(S^)) has been expressed uniquely i n terms of X(Si) and known constants, i e ^ l n| for a fixed function u. Quantitative knowledge about the p r o b a b i l i s t i c location of p w i l l be obtained once the s i z e , shape and location of S^, i e | l , . . . ,n4 - f-' " v has been selected and once X(S£), i e | l , . . . , n | has been expressed i n terms of variables which can be measured i n the f i e l d . This quantitative knowledge about p i s obtained simply by combining the value T(X(S^),...,X(S n)) for a . t. 'v* n given set of | S i | i = l with the d i s t r i b u t i o n of T(X^,...,Xn) acquired by either using existing s t a t i s t i c a l theorems or else applying goodness-of-fit tests to | T ( S ( S ^ ) ,. . • x( sj 1^^))^j° 1 f o r some s u f f i c i e n t l y large No and using standard s t a t i s t i c a l procedures. F i r s t attention i s paid to selecting the s i z e , shape and location th • f' 11 of S^ ,, i = l n. Consider the i t r i a l for any i e ^ l , . . . , n | i n a set of t i l n independent repetitions of E. The i sampling unit (SJ) i s chosen to be a right c i r c u l a r cylinder of radius (r) and height (h/) selected from an i n f i n i t e population of units i n the shape of right c i r c u l a r cylinders, each unit containing a portion of C. Naturally the v a r i a b i l i t y of X i s p a r t i a l l y dependent upon the size and shape of the population elements. Right c i r c u l a r cylinders of common radius are chosen to f a c i l i t a t e the mathematics and r i s chosen so that the v a r i a b i l i t y of X w i l l be small. Logically there should be optimum radius ( r Q ) above which the v a r i a b i l i t y of X does not decrease s i g n i f i c a n t l y . I f preliminary samples cannot be obtained here to help select (r ), the radius used may have to be chosen largely on a basis of personal experience and - 14 -i n t u i t i o n . The height of the i t h sampling unit (S^) i s simply the maximum height of a l l objects of interest .intersecting the i t h open right c i r c u l a r cylinder of radius ( r ) . F i n a l l y the location of i s a function of any important physical, time or s t a t i s t i c a l constraints under which E may be forced to operate. Now attention i s paid to expressing X(S i),i=l,...,n i n terms of variables which can be measured i n the f i e l d . Consider again the i t h sampling unit S. whose basal center point (c - i ), radius (r) and height (h.») are a l l known. Here r can be regarded as either*,.a radius under investiga-tion or an optimum radius selected from either a preliminary sampling analysis or a subjective decision-making process. Project a l i n e segment along the base of from c^ to some fixed perimeter point (p D) on iS^. Let t h i s l i n e segment (L Q) intersecting p Q define a unique zero angle. Then sweeping a l i n e segment (L) of length r around the basal perimeter of Sv^ keeping one of i t s end points fixed at c , i t can be seen that each location of L defines a unique angle 6e [0, 2TTJ , and hence the base of a rectangle R , of width r and height h., V0£[0;,2TT1 (see Figure 2 ) . can be e a s i l y evaluated. From the meaning of X and from known properties of C, F_^  (2ir)=KX(Si) for some constant K. From this understanding of F^, construct a bounded function (f ) defined on Q), 2TT] whose set of discon-t i n u i t i e s (Lang, 1968; p.50) on [o, 2TTJ has Lebesgue outer measure zero, (Taylor, 1965; p.191), such that: The objective i s to express X(S^) i n terms of variables which i t i s possible to attach meaning to a function (1) o - 15 -The right-hand side of (1.) i s well-defined and exists (Speigel, 1963; p.81). Now i f f^(t) can be readily expressed i n terms of measurable variables, the only task l e f t i s to evaluate the right-hand side of (1.). However i f i t i s not possible to express f ^ ( t ) i n terms of measurable variables, i t becomes necessary to approximate g 1 ( t ) = f i ( t ) / k by g ^ t ) , VteJ0,e] V0e[b,2Tr], where g i has both the bounded and 'almost everywhere' continuity properties (Spiegel, 1969; p. 33) of f , and where g^(t) can be expressed i n terms of measurable variables. Then i t i s seen that X(S^) can be expressed as a function of measurable variables by evaluating the right-hand side of (1.) at 6=2TT with.g ±(t) replacing f ± ( t ) , Vte[o,2iT]. The f i n a l remaining problem i n producing X(S^) i s to evaluate the right-hand side of (1.) at 0=2TT with g^(t) replacing f^. Unfortunately th i s may not be a t r i v i a l task. Very often the measurable variables used i n defining g^(t) cannot be ea s i l y expressed i n terms of t mathematically. In other cases the measurable variables of interest are not d i r e c t l y integrable. I f the l a t t e r case arises, g^ i s approximated by a suitable d i r e c t l y integrable function h^ and through integration of IK over [o,0=2Tr], X(S ±) i s evaluated. In the former case however either numerical or Gaussian i n t e -gration methods (Scheid, 1968) must be used. In th i s case, X(S i) i s approximated by: N C<igi(tio) > where N, C^, and to, u=l,...,N 10=1 are determined from the choice of a pa r t i c u l a r integration method applied at a pa r t i c u l a r l e v e l of sampling int e n s i t y . The choice of the method depends upon required precision subject to s p e c i f i c time and cost constraints. From previous comments,quantitative knowledge; about the - 16 -p r o b a b i l i s t i c location of p i s now theoreti c a l l y obtainable. The general sampling technique described above w i l l hereafter be referred to as common point intersect sampling (CPIS), since each sampling rectangle R Q i s generated from a common point, namely the center of the c i r c l e comprising the base of a parti c u l a r c y l i n d r i c a l sampling unit. To c l a r i f y the lo g i c behind the general CPIS theory a flowchart (Figure 1) depicting the major concepts involved i s provided. Now before proceeding to some s p e c i f i c applications of CPIS, a few remarks regarding this new concept should be made. There seems to be two very serious drawbacks to CPIS. One of these i s that knowledge about the d i s t r i b u t i o n of T(X^,...,Xn) i s required. I t i s true that some general behaviour of the d i s t r i b u t i o n of T(X^ X^) should be known i f a parametric s t a t i s t i c a l confidence i n t e r v a l (Ehrenfeld and Littauer, 1974; p.364) i s to be constructed about p. I f th i s information i s not available and furthermore cannot be obtained from p r i o r preliminary sampling or related sampling due to time or cost constraints, d i s t r i b u t i o n -free or non-parametric s t a t i s t i c a l methods can s t i l l be used to construct either a meaningful hypothesis test for p or a rough confidence i n t e r v a l for p. I f the information yielded by the non-parametric investigation i s not s u f f i c i e n t l y precise to be very h e l p f u l , general s t a t i s t i c a l techniques can be used to estimate the standard deviation D(X^,...,X^) of T(X^,...,X^). Then given a set of independent t r i a l s of E, D(X(S 1) X(S£))/T(X(S!) x ( S n ^ gives a measure which expresses the magnitude of the average v a r i a t i o n of T(Xj X-) r e l a t i v e to the size of T(X(S 1),...,X(S^)). This r a t i o can serve as a tentative indicator of how successful T(X(S^),...,X(S^)) can be expected to be and as such can be temporarily used i n place of information regarding the d i s t r i b u t i o n of T(Xj_,. . . ,X^) . In other words an absence of knowledge regarding the d i s t r i b u t i o n of T(Xj X^) has only the effect - 17 -(Pi,..., pk) • ( VALUE OF p| UNKNOWN ) U (DEFINED BY EXPERIMENTER ) (VALUE OF p UNKNOWN ) SIZE , SHAPE AND LOCATION FOR S|, i = l n MEANING OF X,E PHYSICAL MEANING OF Fi(G) ,6 6 (0, 2rr] PHYSICAL MEANING OF Fj ( 2ir)iK9X(S-t ), FOR SOME CONSTANT K CHOSEN TO FACILITATE UNDERSTANDING OF F;. FUNCTIONAL ANALYSIS T KNOWN PROPERTIES OF C T fi = [o,2n]-~IR+s.t. Fi(9) = j fi(t)dt , vee [o,2rr] FUNCTIONAL APPROXIMATION THEORY COST , TIME CONSTRAINTS g" • [0,2 I T ] -HR+s.t. (i) gj « fi/K (ii) c5i HAS NECESSARY ANALYTIC PROPERTIES (iii) r£(t) IS EXPRESSED IN TERMS OF ACCEPTABLE VARIABLES , Vte [O,2TT] INTEGRATION THEORY 21T X(Si)»(hi(t)dt WHERE hi IS DIRECTLY INTEGRABLE, J 0 PROVIDING OF COURSE THAT THE TERMS .COMPRISING £(t) CAN BE EXPRESSED IN TERMS OF KNOWN FUNCTIONS OF t. NUMERICAL ANALYSIS X(Si) = ^  c „ £ u „ ) w = l OTHERWISE PROPERTIES OF POINT ESTIMATORS (X(S,) X(S„)) T(X(S,), X(Sn)) STATISTICAL DISTRIBUTION THEORY CONFIDENCE INTERVAL FOR p Figure 1. Flowchart depicting the development of the general common point intersect sampling concept. - 18 -of reducing the power of CPIS; i t does not prevent CPIS from being a v a l i d sampling method. CPIS seems to have a second serious drawback, namely the problem of obtaining f^ from an understanding of F^. I f the meaning of X i s clear, then with proper selection of there should be no problem i n attaching meaning to F^. A l l that need be remembered when trying to achieve this understanding i s that F^(0) for each 0efo,2ir] simply considers objects i n a portion of the i t h sampling unit <S^ . This means that as 0 approaches 2TT, F_^(0) approaches K«X(S^) i n the most natural way, namely through increasing portions of S^ defined by the sweep of 0 towards 2TT. Note that K i s chosen to convert X(S^) into a variable which i s easier to work with. I t remains then to consider the question of deriving f^ from both an acquired understanding of F^ and a knowledge of what i s meant by a Reimann int e g r a l (Widder, 1947, p.149). I t should be said now that i n general there i s no optimum approach to use i n deriving f^ from F^ through (1). In practice this problem i s usually very easy to solve. Since there i s no preferable procedure to follow when obtaining f^, the investigator must at this point rely largely on his own experience and ingenuity. I t i s the b e l i e f of the author that the best insight into the process of actually getting i s given through example. Hence attention i s now turned to the application of CPIS to downed woody f u e l p a r t i c l e s . - 19 -Application to downed woody p a r t i c l e s Two experiments (E^ and E^) w i l l be conducted upon a community (C) of downed woody f u e l p a r t i c l e s . I t w i l l be shown that the results of these experiments w i l l ultimately y i e l d estimators for a number of s p e c i f i c parameters of .interest. Consider a community (C) of downed woody p a r t i c l e s . Let u(pl,...,pk)=pl, where pl,...,pk are parameters of unknown values describing C and p i i s the mean number of p a r t i c l e s per unit area. Divide C into M subsets such that i f the sampling rectangle (RQ) intersects a f u e l p a r t i c l e (q), q i s said to belong to the i — subset, i e { l M} providing that the width of q taken at i t s i n i t i a l point of intersection l i e s between d^ and D., for some specified d., D. where d. <D., ie{l,...,M}. i i 1 1 i 1 Notice here that q i s i m p l i c i t l y assumed to have a well-defined length and hence a well-defined central axis with respect to which the widths of q are measured. p l ^ i s taken to be the mean number of fuel p a r t i c l e s per unit t i l area with respect to the population of sampling units comprising the i — subset of C. From the meaning attached to p l ^ \ i t i s possible to introduce a random variable (XI ^ ) defined on the sample space of out-comes of an experiment E^"^ where E^"^ consists of selecting a sampling unit with respect to the i subset of C. From the general theory XI i s simply a function which maps each sampling unit into the average number t i l of f u e l p a r t i c l e s per unit area ( i n the i — subset of C) with respect to that sampling unit. Through X I ^ and the meaning of p l ^ a point estimate for p l ^ (Tl ( X I ( l ) ( S l _ ( l * ) ,.. . ,X1 ( l ) ( S l ( l ) ) )) can be expressed - ; 1 n i n terms of XI ^  ( S I . . , j e { l , v ^ . ,n} - 20 -and known constants for a given set {ST.;^ : n . . T l ( X l ( i ) ( S l 1 ( i ) ) , . . . , X l ( i ) ( S l ( i ) ) ) = J_ X l ^ W ^ / n = X l ( i )  1 n j=l J The parent estimator of X l ( l ) ( X P l ^ E £T X l ^ / n ) i s an unbiased estimator ( i ) j = 1 2 of p^ and by applying the Central Limit Theorem (Ehrenfeld and Littauer, 1964; p. 187) i t can be shown that X P l ^ i s also a consistent estimator of p l ^ . The Central Limit Theorem can also be applied together with the d e f i n i t i o n of XI ^  to demonstrate that: ( 2 ) ^ ; ^ , where (3) ( S D l ^ ) 2 -'-^ £ ( X i a ) - X P l ( i > ) 2 SD1 ( 1 ) v£ n~ l j=l 3 has approximately Student's t d i s t r i b u t i o n with (n-1) degrees of freedom (Ehrenfeld and Littauer, 1964; p 189). It i s important to r e a l i z e that (2) has Student's t d i s t r i b u t i o n with (n-1) degrees of freedom i n most cases even when n i s small. This i s true because XI i s an average (i) 2 taken over a large sampling area (fr(r v J) »25) and hence by the Central Limit Theorem has"approximately a normal d i s t r i b u t i o n . I f C contains large continuous areas (»ir(r^) 2) d i f f e r i n g d r a s t i c a l l y i n f u e l p a r t i c l e frequency, not only w i l l the normality of X I ^ be probably violated but C 2 also (SD1 ) w i l l probably take on very high values. To counteract these problems, C should be s t r a t i f i e d wherever feasibly possible into regions of different f u e l p a r t i c l e frequencies with respect to the i t h subset of C and the theory of s t r a t i f i e d random sampling (Freese, 1962; p.28) .applied. I t follows that for a given set of { S l ^ } ? , ( a l l J J—1» ly i n g within one area containing no large continuous sub-areas d i f f e r i n g d r a s t i c a l l y i n f u e l p a r t i c l e frequencies), a confidence i n t e r v a l for p l ( l ) i s X l ( l ) +'t -«/2;ri-l S d l ^1 Jri (Ehrenfeld and Littauer, 1964; p. 271) - 2 1 -where (1-^) = l e v e l of s t a t i s t i c a l inference and t^ oc./2'n-l = v a ^ u e °^ Student's t d i s t r i b u t i o n with (n-1) degrees of freedom at the (l-<*/2.) l e v e l . I t remains to select the s i z e , shape and location of S l ^ ^ and to ( i ) , , , , ( i ) N _ c V 1 U.Jj express X1V"L'' (SI. ^ 1 J) i n terms of measurable variables, Vie-|l,...,MJp-, Vjeil,...,n| . This w i l l now be done. Choose the sampling units i n the i — subset of C to be right c i r c u l a r cylinders of common radius r l ^ ^ . Due to time constraints r l ^ w i l l be determined subjectively. Choose r l ^ s u f f i c i e n t l y large such that the v a r i a b i l i t y of XI ^  i s expected to be small. The height h ' l ^ of the i t h sampling unit S l ^ i s well-defined from the general CPIS theory. A systematic plot sampling design with an equidistant grid pattern (Husch, M i l l e r and Beers, 1972; p.233) i s used to select the location of SI; , j e | ' l , . . . , n j " , due to i t s ease i n application (Husch, M i l l e r and Beers, 1972; p.228). I t should be noted that the use of systematic sampling does introduce a problem i n that now the n repetitions of a r e n 0 longer completely independent. This means that ( S D l ^ ) ^ as defined i n (3.) w i l l not v a l i d l y represent the sample variance (Husch, M i l l e r and Beers, 1972; p. 229). In fact ( S D l ^ ) 2 tends to overestimate the sample variance (Osborne, 1942). A supposedly more representative expression for the sample variance for equidistant grid patterns i s given by using successive difference formulas (Loetsch and H a l l e r , 1964). However, unless the spacing between sampling units becomes coincidental with the pattern of population v a r i a t i o n , the improvement offered by these successive difference formulas becomes ne g l i g i b l e (Husch, M i l l e r , Beers, 1972; p. 229). I t w i l l be assumed that the sampling unit locations have been selected so that no such coincidence occurs, making (3.) v a l i d . In practice this i s almost always done. I f t h i s manipulation process proves awkward or expensive with respect to a pa r t i c u l a r downed woody fuels - 2 2 -population, alternate successive difference formulas (Husch, M i l l e r and Beers, 1971; p.2 3 6 ) may be used to obtain a th e o r e t i c a l l y more r e a l i s t i c ( i ) 2 expression (CD1 ) for the sample variance. The remainder of the argument can then be applied with ( C D l ^ ) 2 replacing ( S D l ^ ) 2 . In order to determine X1^^(S1^^) i n an appropriate form i t i s necessary to select a m u l t i p l i c a t i v e constant k l which w i l l transform X l ^ C S l ^ ) . into a variable which can be related to more e a s i l y . Since p a r t i c l e s are being considered with respect to thei r intersections along transect l i n e s , i t seems natural to select k l as one unit length. Then K l • X l ^ ^ ( S l j ^ ) becomes the average number of p a r t i c l e s per unit length with respect to S l f " ^ provided of course that the units' of length are chosen s u f f i c i e n t l y small. This relates better to the sampling design than does average number of pa r t i c l e s per unit area, a function which associates to each 0e [0,271"]] a non-negative r e a l number (9) representative of the average number of pa r t i c l e s per unit length of transect with respect to the area i n S l ^ ^ defined by a r a d i a l sweep from zero to 0. In construct ing f l ^ , i t seems l o g i c a l as a f i r s t attempt to try where i s a function which associates to each te|o,2Trj a non-negative r e a l number (*") representative of the number of p a r t i c l e s which intersect the transect located at t , 0 _< t <0. This function i s bounded on. £O,2TTJ and i n fact i s a step function by d e f i n i t i o n which ensures i t s 'almost everywhere' continuity on Q),2TTJ. NOW from elementary calculus i t . i s well known that: ' _ • i r ( 4 ) a^Ce) = 0 j a ? 1 ) ( t ) d t , O<0<2TT where (0) i s the average value of the function taken over a l l values of t ranging from 0 to 0>O. I t now becomes obvious for fixed ee(0,2ir],to set f 1 ^ = 0 ^ / ( Q ^ ^ ) since F l * 1 * = / r l ( l ) . Note that for fixed 6E(0,2TT), f l ^ ^ ( t ) i s already expressed i n terms of measurable variables, for each t e(b,8], namely number of relevant p a r t i c l e i n t e r -sections at t . The f i n a l problem then i n producing X l ^ ^ ( S ' l j ^ ) i s to evaluate the right-hand side of (4) at 8=277 with a ^ V r l ^ replacing . This w i l l not be done here for one simple reason. Knowledge about numbers of downed woody p a r t i c l e s i s not currently required as a sig n i f i c a n t direct input for evaluation or prediction of f i r e behaviour and impact. I t should be stated again that throughout t h i s thesis the applications of the CPIS general theory w i l l be focussed on obtaining those parameters which relate most s i g n i f i c a n t l y to fire-oriented a c t i v i t i e s . I t i s true that results of can be combined with results of E^ to y i e l d estimators for a number of downed woody p a r t i c l e parameters. In fact t h i s w i l l be done l a t e r for completeness. But the results of E^ do not relate s i g n i f i c a n t l y to f i r e evaluation, and so an e x p l i c i t expression which estimates the right-hand side of (4.) w i l l not be derived. E^ serves here primarily as a" working example to demonstrate that the process undertaken to get a meaningful "handle" on the general CPIS theory i s not a mysterious one. Each step taken i n working out thi s f i r s t application of the general theory has been l o g i c a l and very straight-forward. I t i s of interest to notice that nowhere i n the argument have any assumptions been made regarding the d i s t r i b u t i o n of the p a r t i c l e s within the CPIS units themselves. Attention i s now turned to E^. The general layout of i s very s i m i l a r to that of E^"^, and reference w i l l be p e r i o d i c a l l y made to E^ throughout the discussion - 24 -of E^' Here u(pl,...,pk) = p2, where p2 i s defined as the mean t o t a l volume of pa r t i c l e s per unit area. Analogous to , E 2 ^ * s a n experiment consisting of selecting a sampling unit within C with respect to the i t h subset of C. Also, analogous to X l ^ , X 2 ^ i s simply a function which maps each sampling unit into the average volume of fu e l p a r t i c l e s per unit area with respect to that sampling unit. The d e f i n i -tions of T 2 ( X l ( i ) ( S 2 j 1 ) ) , . . . , X l ( i ) ( S 2 ^ 1 ) ) ) and ( S D 2 ( i ) ) 2 are both u a * T ^ d ) A I I i • ^ r ( i ) , - p l ( l ) obvious from E . A l l remarks concerning XI and . r  1 SDl ( l )//n" ~XP~2^ - p 2 ^ apply equally to X2 V ' and ... v , respectively. The comments SD2 ( l )//n" regarding both selection and j u s t i f i c a t i o n of the sampling units used i n also apply equally to E ^ . It remains then only to determine X2^^ (S2 ) i n an appropriate form. Since X 2 ^ ( S 2 J i ^ ) i s an average volume per unit area i t makes good sense to select K2 as 7 T ( r 2 ^ ) 2 . Then K2«X2^ ( S 2 ^ ) becomes a t o t a l volume of p a r t i c l e s which i s more eas i l y related to the sampling design. An understanding of F 2 ^ i s now possible and F 2 ^ i s analogous 3 3 i n meaning of F l ^ " ^ with the obvious difference that F2^^ (0) represents a t o t a l volume of pa r t i c l e s with respect to the area i n S2^^ defined by a r a d i a l sweep from 0 to 0>O. The construction of f 2 ^ ^ i s derived from a basic understanding of F2? l ) and the meaning of a Riemann i n t e g r a l . F 2 ^ i s a function which when evaluated at fixed 0 yields a t o t a l volume. I t i s w e l l known that when a cross-sectional area i s swept through an arc, a volume i s generated. (see Figure 2). This volume i s a function of the distance separating the centre of the sampling unit and the cross-sectional region. Thus f2^"^ w i l l involve both the shape of the fuel p a r t i c l e s i n the i t h subset of C and a separation distance factor. More s p e c i f i c a l l y , downed - 25 --26 -woody p a r t i c l e s can be c l a s s i f i e d reasonably well into f i v e geometric d i v i s i o n s : frustums of cylinders, parallelepipeds, cones, parabaloids and n e i l o i d s (Husch, M i l l e r and Beers, .1972; p.120). The presence of p a r t i c l e boundary taper as possessed by cones or p a r t i c l e boundary con-cavity as possessed by parabaloids and neiloids provides a s i g n i f i c a n t complication to both the theoretical and p r a c t i c a l aspects of quantifying downed woody f u e l parameters by working with the fuels themselves. If the i n i t i a l points of intersection are consistently used to compute the fu e l p a r t i c l e widths, and i f the concavity present i s not too severe, i t should be feasible to c l a s s i f y the l a s t three troublesome divisions under frustums of cylinders. I t i s of importance to mention that i n practice the lack of symmetry i f any of a given cross-sectional region i s usually small. But due to time constraints no quantitative analysis was undertaken to support the above c l a s s i f i c a t i o n grouping made by using the above technique of width measurement. Hence further studies are required to define a completely v a l i d set of conditions under which frustums of cones, parabaloids and neiloids may be considered cylinders with respect to geometric cross-sectional form. Continuing then, i t i s seen that i t i s temporarily f a i r to assume that a l l cross-sectional regions are whole e l l i p s e s , truncated e l l i p s e s , or rectangles. Without loss of generality only the f i r s t two of these regions w i l l be .considered. In practice the t o t a l number of parallelepipeds:}• generally comprise a very small proportion of a downed woody fuels population. I f p a r a l l e l -epipeds are of par t i c u l a r interest, an almost i d e n t i c a l and i n fact simpler argument to that offered below can be made by applying the con-cepts below to rectangles as opposed to e l l i p s e s . Now from the understanding - 27 -gained from the above remarks, the values of F2^^ are obtained by summing up the volumes generated by rotating either whole or truncated e l l i p s e s through small arcs. But i f 6^ c\^q» ^2}* w n e r e ~ ^ * S s m a ^ » and i f the f u e l p a r t i c l e cross-section at 6^ i s a whole e l l i p s e , then from elementary calculus, an excellent estimate for the p a r t i c l e volume generated by rotating t h i s cross-section at 9^ through [J9Q, i s : (5) f j ( d ( ± ) > 2 *S(6 x) • ( s e c $ ± ) ) 'CSC($(6 x ) )| • 6 2 - 6 o | where ( d ( i ) ) = an estimate for the quadratic mean p a r t i c l e diameter ( d b a r ^ ) (Brown, 1973) with respect to the i t h subset of C, ieC, M i s ( 9^ ) = the horizontal distance between the centre of the e l l i p t i c a l cross-section at 9^ and the basal center point of the sampling unit, where the horizontal i s defined p a r a l l e l to the orienta-tion of the base of the sampling unit. (sec (tr^)) = an estimate for the mean secant of the p a r t i c l e t i l t Q^) with respect to the i t h subset of C, ief.. ^ CSC($(6L)) = the cosecant of the angle of intersection (<j)(0^)) between the transect at 0^  and the central axis of the fu e l p a r t i c l e , O<$(01)<TT/2, Ve^^.e^. See Appendix I for a complete proof that the volume obtained by rotating an e l l i p t i c a l cross-section of 9^ through J J ) Q , 9 ^ . J i s given by (5). Through summation over relevant p a r t i c l e s , i t follows from the above that a l o g i c a l f i r s t attempt for tlS^ i s v."^ where ,.N , 3 (6) ^ ( i ) ( t ) = ?/4 ( d ( 1 ) ) 2 (sec(^i)) y k=l s[L\ ( t J - C S C ^ 1 ? (t)) ^>3 K»J where m ^ (t) 1 S the number of pa r t i c l e s which intersect the transect at t , and where the kth f u e l p a r t i c l e intersects the transect at t , Vkefl m;"^(t) 3 In E^*^ , a p a r t i c l e intersection i s defined to occur only when both the p a r t i c l e central axis and at least one p a r t i c l e edge intercept the transect. Since esc (<j>£^ (t)) i s bounded on |6, 2TT]by Jl + p^^^where a ^ - 28 -i s the maximum length of a l l p a r t i c l e s i n the i t h subset of C, ie'j(l, . . . , M } , ( i ) r 1 ( i ) ( i ) i t follows that v. i s also bounded on [0,2T TJ. . and <J£ . are J k,j k j j sectionally continuous (Spiegel, 1963; p.26) on Q),2T T]. Also the cosecant function i s continuous on (0,T T/2)J. Hence the composite function CSC°^-^. i s sectionally continuous on [ O , 2 T T J (Lang, 1968; p.51). Thus s a t i s f i e s a l l the required properties stated i n the general CPIS theory. Setting f 2 ^ l ) = v f i ) reveals: 2 2 0 (7) F2< i )(G) = ^ l ) ; ( t ) d t , V6e(0,2Tn] J o 2 I t follows that: 2TT 2TT . X2 ( 1 ) (S2^ l ) ) = F2? l )(27T)/Tr(r2 ( i )) 2 = J" v? 1* (t) d = j g 2 f i ) ( t ) d t , where.; 2 1 o ,.x 0 o 2 r r ( r 2 ( l ) ) 2 (8.)g2 , ( 1 ) ( t ) = y . ( 1 ) ( t)/TT(-r2 ( 1 )f , Vte(o,2Tr] The next step to be taken i n order to express X 2 ^ (S2^"^) e x p l i c i t l y i n terms of measurable variables i s a consideration of the variables comprising (t) as defined i n (8) and (6). An inspection of g 2 ^ (t) reveals that i t i s expressed i n terms of sf H (t) and (t) , K e f l , . . . »mj^^ (t)!}» both of which are d i f f i c u l t to measure d i r e c t l y . In fact both are t o t a l l y impractical to measure d i r e c t l y with respect to lesser downed woody p a r t i c l e s . Hence from this point on, i t w i l l be necessary to consider the lesser downed p a r t i c l e s and the greater downed pa r t i c l e s separately. F i r s t consider ie;{l,.. vJ - ' ,Mi where D. < 3 inches. From (6.), (9) g2^ l )(tr 1 ( d ( l ) ) 2 (sec(^.)) ^> ^ S ^ ( t ) . C S C ( ^ ] ( t ) ) / / T r ( r 2 ( i ) ) 2  4 k=l - 29 -C O - c s c c ^ c t ) ) K» j k=l ( m / i ) ( t ) . 7 r ( r 2 ( i ) ) 2 ) , i f m;^(t) ^ 0 0, i f n i j ^ C t ) = 0 • CD, From the above, the selection of g2j ' Ct) i s obvious. (10) g^ ( 1 )(t) = -CD (i) TT ( d w ) " ( s e c ( T i ) ) - K w V ' ( t ) , i f m u ; ( t ) # 0 .4 J J 0, i f nij^Ct) =0 I ( d ( 1 ) ) 2 ( s e c ^ i ) ) ' K ( l ) - m ; ^ (t) , where K ( l ) i s an appropriate ^ constant g2^^ i s a bounded step function on Q ) , 2T7J and g2^"^(t) i s expressed exclusively i n terms of measurable variables and , Vi el 1,.. . .nf. Hence i t remains to produce Itf . I t may seem somewhat opt i m i s t i c to assume that (i) for each ie|l,...,M|, D| <^  3 inches, there exists a unique constant K ^ ( i ) ( i ) independent of both j and t permitting g2. (t) to be very close to g2. ( t ) . However i f for some j and t, g2^ i s a poor approximation for g2^ ( t ) , i t does not r e a l l y matter providing that the integrals of the-two functions from 0 to 2TT are reasonably close. K (i) was e x p l i c i t l y determined for i = 1,2,3 where (d^Dj] = (0,V] (d 2,D 2] = (VM'3 (d3,D£| = (1",3']] These three subsets were chosen for two reasons. The f i r s t i s that the intervals are s u f f i c i e n t l y small so as to f a c i l i t a t e the effec t i v e use of quadratic mean diameters (as required i n (10.)) and hence accommodate the computation of accurate lesser p a r t i c l e volume estimates. The second reason for t h i s choice i s that the above three subsets correspond - 30 -respectively to 1,10 and 100 hour average moisture time lag divisions for a number of common woody forest materials, (Fosberg, 1970). For each i e{l,2,3} , was determined using simultaneous runs of avone-way c l a s s i f i c a t i o n random components analysis of variance model (AN0VA1) with unequal numbers of observations i n the c e l l s (Ehrenfeld and L i t t a u e r , 1964; p.399) and a two-way c l a s s i f i c a t i o n random components analysis of variance model with unequal numbers of observations i n the c e l l s (Ehrenfeld and Littauer, 1964; p.432). A description and computerized version of the f i r s t of these two models i s offered i n Appendix 2. Since the second of these models i s v i r t u a l l y i d e n t i c a l i n design to the f i r s t model, no computerized version of i t i s required; however, a supplementary description of the two-way model i s included i n Appendix 2. The two-way model was constructed using p a r t i c l e d i s t r i b u t i o n and p a r t i c l e frequency or loading as the two influencing factors. The primary purpose of t h i s model was to determine for each ie{1,2,3^ the ranges of the two above influencing factors under which i t was.possible to assert the existence of a unique constant reasonably independent of both j and t (i) which would permit the i n t e g r a l of §2) to be close to the i n t e g r a l of * ( i ) g%) for each j e { l , . . . , n | , and for each ie{l,2,3|. A thorough description of the processes involved i n the two-way model i s given i n Appendix ,2.. I t suffices to say here that the results of the two-way model indicated that a satisfactory unique constant could exist only when the f u e l p a r t i c l e ttl d i s t r i b u t i o n (with respect to the i — subset of C) was held fixed within the CPIS units being sampled, Vie|l,2,3§. E x p l i c i t determination of for each iej{l,2,3'|' was performed i n the one-way model as described i n Appendix 2'. In order to produce -.31 -i t was necessary to select a suitable sampling radius r 2 ^ f o r each ie{l,2,3}. Due to time constraints, selection of r 2 ^ was made subjectively. More precisely, r 2 ^ = 8.5 feet (sampling area of 1/192 acre) (2) r2 = 16.5 feet (sampling area of 1.51 acre) r 2 v J =28.5 feet (sampling area of 1/12 acre) I t i s noted that some experienced investigators may be more adept than others i n selecting an appropriate r 2 ^ . Hence to eliminate as much guesswork as possible i n the selection of r 2 ^ , an approximate formula for determining r 2 ^ ^ as a function of D_^  i s presented below: (11) Log r 2 ^ ^ r 1.22 + 0.47 log D±)t where D^  i s i n inches and r 2 ^ i s i n feet, V i e { l M} (11.) i s presented here merely as a rule of thumb. I t serves primarily as a guide for the inexperienced investigator and as a reference for the exper-ienced one when no preliminary data to help select r 2 ^ i s available. I t i s of interest to note that the analysis of the one-way model revealed that for randomly distributed f u e l p a r t i c l e s within the CPIS units being sampled the random variable defined by: (12) l . . ( 1 > '£ ka)-CSC(<b ( 1 ) ) (m j f 1 ).TT(r2 ( i)) 2) - K ( 1 ) K^^ -<^ <i>)]/<-r-^ <i>>2, R ( i ) i and j fixed was found to be normally distributed with mean 0. I t was also discovered that i f j i s ordered by loading, the variance of the above r a t i o generally - 32 -decreased with increasing j , Vie){l,2,3*}. This means that the performance * ( i ) A ' ( i ) of g2j as a percentage approximation of g2^ i n the in t e g r a l from 0 to 2TT i s satisfactory for a l l i and j . i and j may, i f desired, be interpreted i n terms of fuel p a r t i c l e s ize and loading, respectively. Note that Rj"^ i s defined only on those values in[b, 2?7]such that (t)-tO. I t i s of interest to note that the one-way model showed that the random variable defined by: mj*^» a n d j fixed was found to have a coeffic i e n t of var i a t i o n reasonably independent of i and i ,Vie{l,2,3j ;Vje£i h}. This means that on a percentage basis the ••'A. (±) r— -1 v a r i a b i l i t y of g2^ over (_0, 2TTJ i s v i r t u a l l y independent on both i and j . Combining the two above points of interest, i t i s seen that i f numerical •y ' A ' ( i ) methods are required to evaluate the integral of g2) , one numerical technique applied at one l e v e l of sampling intensity w i l l probably s u f f i c e for a l l i and j . Numerous runs of the one-way model with randomly distributed fuel p a r t i c l e s yielded values of k ^ within a small neighbourhood of the following: (13) K(2) K (3 ) - 2.83 x 10~ 2 (feet) -1 -2 -1 = 1.49 x 10 (feet) = 8.56 x 10" 3 (feet) -1 ^ (i) g2- (t) has now been e x p l i c i t l y determined i n terms of measurable variables Vie{1,2,3"?}, as previously defined, providing that the lesser p a r t i c l e s considered are randomly distributed within the CPIS units. Attention i s now turned to i e ; { l , . . . ,Mi} where D^ > 3 inches. Suitable formulas for e x p l i c i t l y defined subsets of greater - 33 -downed p a r t i c l e s whose distrib u t i o n s are either random i n the CPIS units or else can be quantified i n the CPIS units can be obtained by a procedure almost i d e n t i c a l to that used above for the lesser downed p a r t i c l e s . Due to time constraints plus the fact that attention i s to be focussed primarily upon the lesser fuels which as has been previously stated are most l i k e l y to be consumed by the majority of broadcast f i r e s , formulas for the larger fuels are not derived. I t i s of interest to note that only a s l i g h t manipulation of (9;,) w i l l y i e l d a formula for g'2ji^ (t) ( i n terms of measurable variables) which applies to greater downed p a r t i c l e s and i s independent of p a r t i c l e d i s t r i b u t i o n . This formula i s presented below: — ( i ) m j ^  (t) -j (14) g 2 ( i ) ( t ) = r ^ 1 - ( t ) ) 2 - ( P ^ 1 - (t)) | » w h e r e j 8 ( r 2 U V \ = l 1 > 3 , 3 (15) P2< i ) (t) = (t) + ( d ( ± ) ) •CSC(41) ( t ) ) ) / 2 (16) P l ( i ) ( t ) = ( 2 S ^ (t) - ( d ( 1 ) ) •CSC(<b^i)(t)))/2 and (sec(YD = 1 k,j *>2 k' J Note that when D^ >3 inches, i t i s f a i r to assume that (sec(Yi)) = 1 (Brown, 1973; p.3). Then from (15.) and (16.), i t i s clear that PI and P2 can be interpreted as distances to the l e f t and right end" points of i n t e r -section, respectively for a whole e l l i p s e , which are not d i f f i c u l t to measure for the greater woody p a r t i c l e s . I f a truncated e l l i p s e i s encountered ( i . e . a p a r t i c l e end) i t i s necessary to extrapolate one edge of the p a r t i c l e to the transect i n order to obtain the proper value providing of course that the p a r t i c l e central axis intercepts the transect. Hence (14.) may be used d i r e c t l y for greater downed fuels of any d i s t r i -bution and loading. - 3 4 -There i s one remaining step to be made so that X 2 ^ ( S 2 j ^ ) w i l l be expressed i n terms of measurable v a r i a b l e s . This step i s the e v a l u a t i o n of the i n t e g r a l of an appropriate f u n c t i o n between 0 and 2T T. Due to time c o n s t r a i n t s and the f a c t the t h e s i s i s designed to focus upon the l e s s e r downed p a r t i c l e s , e v a l u a t i o n of X2^ (S2^) through the i n t e g r a l of an appropriate f u n c t i o n w i l l be done only f o r ie'l 1,2,30:, as p r e v i o u s l y defined. The processes i n v o l v e d here are thoroughly described i n Appendix 3, which incl u d e s a computerized v e r s i o n of a downed woody f u e l s model. A major r e s u l t of t h i s i n v e s t i g a t i o n was that the random v a r i a b l e defined by: (17) V - - h 2 ^ v( i ) [ ^ 7 " ' ] / fe^j 1 ^ ' ' W h 6 r e (18) h2< i>(t) = £ C , . ^ ( t + f ) , ( J 12 ' 4 , i f a) i s odd 2, i f a) i s even w=2 and vf^ = the tru e t o t a l p a r t i c l e s volume i n the j.th sample taken w i t h respect to the i t h subset of C, ie|^ 2 3 } ' . ' was found to be normally d i s t r i b u t e d w i t h mean 0 and very small v a r i a n c e independent of i and j , ^ L e h , 2 , 3 } VjeH,...,n}. The means that V ^ i ) / T r ( r 2 ( i ) ) 2 can be very w e l l approximated by an appropriate a p p l i c a t i o n of Simpson's r u l e (Schied, 1968; p.108) with twelve t r a n s e c t s , V i e | l , 2 , 3 ' } , Vje{l,. .. ,n}. S e l e c t i o n of the placement f o r the f i r s t t r a n s e c t i n a p a r t i c u l a r sampling u n i t does not s i g n i f i c a n t l y a f f e c t the performance of the estimate f o r the average p a r t i c l e volume per u n i t area w i t h respect to that sampling u n i t . A t y p i c a l performance of Simpson's r u l e w i t h twelve t r a n s e c t s as a p p l i e d to subset i i s given i n Table 1 and di s p l a y e d g r a p h i c a l l y i n Figure (±4-2) f o r i e { , l , 2 , 3 | . In summary then X 2 ^ ( S 2 ^ ) can be very w e l l approximated by: - 35 -Table 1. Actual and estimated volumes. computer-generated lesser downed woody p a r t i c l e Actual Volumes Estimated Volumes Class 1 Class 2 Class 3 Class 1 Class 2 Class 3 ( ( f t ) 3 / ( f t ) 2 ) X 10" -2 ( ( f t ) 3 / ( f t ) 2 ) X 10' -2 0.03021 0.2129 1.317 . .0.02853 0.2178 1.413 0.03051 0.2171 1.345 0.02645 0.2087 1.257 0.03015 0.2162 1.322 0.03403 0.2141 1.295 0.02952 0.2070 1.309 0.03196 0.1986 1.204 0.03005 0.2167 1.365 0.02907 0.2004 1.381 0.02918 0.2138 1.305 0.02961 0.1913 1.375 0.0305 0.2133 1.326 0.03060 0.2297 1.680 0.03067 0.2032 1.294 0.02825 0.1977 1.268 0.04528 0.4686 2.665 0.04180 0.4759 2.916 0.04452 0.4681 2.614 0.04369 0.4686 2.905 0.04557 0.4757 2.543 0.04270 0.5299 2.611 0.04462 0.4766 2.658 0.04514 0.4438 2.622 0.04559 0.4709 2.658 0.04405 0.4704 3.023 0.04550 0.4855 2.590 0.03990 0.5042 2.852 0.04489 0.4711 2.653 0.03945 0.4695 2.793 0.04551 0.4688 2.663 0.04225 0.4512 2.889 0.05938 0.7496 3.997 0.06147 0.7815 4.045 0.05922 0.7501 3.897 0.05615 0.7706 3.815 0.06070 0.7336 3.899 0.06012 0.7019 3.810 0.05912 0.7465 3.918 0.05741 0.8017 4.125 0.05912 0.7465 3.918 0.05741 0.8017 4.125 con' t - 36 -Actual Volumes Estimated Volumes Class 1 Class 2 Class 3 Class 1 Class 2 Class 3 ( ( f t ) 3 / ( f t ) 2 ) X 10" •2 ( ( f t ) 3 / ( f t ) 2 X 10 2 0.05993 0.7286 3.878 0.05678 0.7577 3.986 0.05934 0.7294 3.898 0.06048 0.6726 4.147 0.05988 0.7222 3.894 0.06184 0.7065 3.788 0.05967 0.7266 3.978 0.05886 0.7513 4.056 0.07372 0.9804 5.528 0.08106 1.052 5.399 0.07457 0.9973 5.251 0.07258 1.035 5.485 0.07457 1.012 5.166 0.07447 1.038 5.062 0.07359 0.9982 5.173 0.07068 1.045 5.217 0.07470 0.9934 5.152 0.07592 1.012 5.265 0.07550 0.9907 5.231 0.07285 0.9774 5.479 0.07477 1.009 5.358 0.07465 0.9783 5.784 0.07361 1.003 5.232 0.07330 0.9929 5.335 0.09020 1.240 6.539 0.08224 1.310 6.672 0.08955 1.276 6.593 0.08883 1.366 7.117 0.09011 1.266 6.488 0.08386 1.243. 6.533 0.09021 1.257 6.491 0.08278 1.320 6.314 0.08909 1.261 6.487 0.08296 1.290 6.491 0.08826 1.274 6.688 0.09208 1.312 6.913 0.08996 1.263 6.617 0.09009 1.199 6.710 0.08936 1.283 6.387 0.09460 1.405 6.854 1.00 r-0.90 O o O O D ( Q ^" 0.80 ro ^ 0.70 X I < IJJ < 2 3 0.60 0.50 DC £ 0.40 LU _i o > < t-o 0.30 0.20 a—n*— 0 n_l»i SIZE CLASS I (^1/4 INCH) Q — o ACTUAL VOLUMES . . . . . ESTIMATED VOLUMES 0.10 J L ' i I I i J 1 I I J L J L J L J I I I I 40 00 10 12 14 16 18 20 22 UNIT N U M B E R 24 26 28 30 32 34 36 38 Figure 3 . Curves representing actual and estimated computer-generated downed woody p a r t i c l e volumes with respect to size class I (<_h inch) . 1.60 1.00 ^ 1.40 v. to 1.20 eg I O X I < rr < 0.80 £ 0 6 0 a UJ 5 3 0.40 O > < 0.20 O „ \ ' • - N i n SIZE CLASS 2 ( > 1/4 INCH , < I INCH ) * '-'tr-a • ACTUAL VOLUMES • • ESTIMATED VOLUMES • 9-I I I I I I J I I L J 1 I L J 1 I I I I J I I I L J L 10 12 14 16 18 20 22 UNIT NUMBER 24 26 28 30 32 34 36 38 40 Figure 4. Curves representing actual and estimated computer-generated downed woody p a r t i c l e volumes with respect to size class 2 (> \ inch < 1 inch) CO CO 0 . 8 0 £ 0 . 7 0 , — . 0 . 6 0 X < UJ cc < Z> CC UJ 0. UJ 5 _J O > o 0 . 5 0 0 . 4 0 ' 0 . 3 0 0 . 2 0 0 . 1 0 h / O Q D — • — i . — a — o — _ — l a — a 0 D SIZE CLASS 3 .( >l INCH, < 3 INCHES) . a - — o A C T U A L V O L U M E S • • E S T I M A T E D V O L U M E S J L J I I I I I L J I L J I L i I i i i I i I I i I i I I I 10 12 14 16 18 2 0 2 2 2 4 UNIT NUMgER 2 6 2 8 • . 3 0 3 2 . . 3 4 . 3 6 - 3 8 4 0 Figure 5. Curves representing actual and estimated computer-generated downed woody p a r t i c l e volumes with respect to size class 3 •(> 1 inch, < 3 inches) - 40 -12 (19) j ,(d(i)) ( Sec(Yi))-K ( 1 ) .Y~Cw.m(1)(t + ^ ) , Vi<{l,2,3.L V jef t n}, w=l and where t is arbitrary in [O, 2Tf]and 14, if a) is odd C ~ w |2, i f a) is even providing of course that the fuel particles of interest are randomly distributed within 3 2 ^ , Vie|;i,2,3-«}, Vj e | l , . . . , n f . Thus far, two estimators X P l ^ and XP2^ , have been explicitly determined as far as time constraints will permit. X P l ^ is an estimator for P l ( 1 ) , the mean number of fuel particles per unit area with respect to the population of cylinders of common radius r l ^ . XP2^ is an estimator for p2 ( i ), the mean volume of fuel particles per unit area with respect to the population of cylinders of common radius r2^^. Confidence intervals for p l ^ and p 2 ^ have also been produced and they too are as complete as time will permit. If i ef'1,2,30-, not only has the estimate X 2 ^ been completely determined but also a complete ^confidence interval for p 2 ^ has been presented. The term 'complete' i s used here in the sense that a l l terms comprising the confidence interval of interest can be precisely evaluated in any given downed woody particles population whose lesser components are randomly distributed/ within the CPIS units selected for sampling. At the beginning of the section regarding applications to downed woody particles, i t was asserted that the results of experiments and would yield estimators for a number of specific parameters of interest. This assertion will not be verified. Let r l ^ = r 2 ^ , V i e|l M ) . Some common parameters of interest (De Vries, 1973) are: mean total volume per unit area ( p l ^ ) , mean number of particles per unit area ( p 2 ^ ) , mean total weight per unit area ( p 3 ^ ) , mean mid-sectional area per unit area ( p 4 ^ ) , mean total length per unit - 41 -area ( p 5 ^ ) , mean t o t a l volume per p a r t i c l e ( p 6 ^ ) , mean mid-sectional area per p a r t i c l e ( p 7 ^ ) , mean mid-diameter per unit area ( p 8 ^ ) , mean length per p a r t i c l e ( p 9 ^ ) , mean mid-diameter per p a r t i c l e ( p l O ^ ) , mean t o t a l surface area per unit area ( p l l ^ ) , and mean t o t a l surface area per p a r t i c l e ( p l 2 ^ ) . An estimator f o r p k ^ , ^7,10 can be e a s i l y expressed i n terms of X I ^ \ X 2 ^ and s u i t a b l e constants, V i e {1,. . . ,M}. 3 3 ^ Notice that i f k=7 an excellent estimator f o r p k ^ i s Tr fdbar^"* "^ /4, where an estimate f o r d b a r ^ i s obtained from s u i t a b l e tables (Brown, 1973). No confidence i n t e r v a l i s r e a l l y required f o r p k ^ providing the i t h subset i s s u f f i c i e n t l y small. I f a confidence i n t e r v a l i s desired, i t can be obtained through repeated observations of p a r t i c l e diameters i n the i t h subset of C. S i m i l a r l y p l O ^ can be very well estimated by d b a r ^ , where a confidence i n t e r v a l , i f desired, can also be obtained through repeated t i l observations on p a r t i c l e diameters' i n the i subset of C. I t i s l e f t to produce an estimator f o r p k ^ ,k^7,10. « . • . . , ( i ) „(i) _ ( i ) . ( i ) c ( i ) , ( i ) _ ( i ) Estimators f o r p l v , p2 v , p3 , p4 , p5 v ', p6 v ', p8 v , P 9 ( 1 ) , p l l ( i ) and p l 2 ( 1 ) are i n order X P 2 ( i ) , 3cpT(i), S ( i ) • I P 2 ( i ) where S ^ i s the mean s p e c i f i c gravity f o r p a r t i c l e s i n the i t h subset, ^ ( d b a r ^ ) 2 - X P l ^ n X i 2 ( 1 > / | ( d b a r ( i ^ ) , 5 0 P 3 ( 1 ) ( X P 3 ( ± ) E ^ X S ^ / n . where X3^> ^lf>/Xl^>) , d b a r ( 1 ) ^ l ( i J I f f 3 ( i ) / J ( d b a r ( i ) ) 2 , a ( i ) ^ 2 ( i ) , and a % t f 3 ( i ? , where a ( i ) ( i ) 4 / ( i ) i s the average surface area to volume r a t i o f o r the i t h subset ( i e . 0" = /dbar ). Confidence i n t e r v a l s f o r pk-"^,k^6,9,12 are obvious from an inspection of t h e i r estimators and from previous remarks made when discussing experiments E^ and E2> I t should be observed that i t i s possible only to produce a sample c o e f f i c i e n t of v a r i a t i o n f o r p k ^ \ k=6,9,12 unless the d i s t r i b u t i o n of XP3 ^  can be roughly determined. In passing i t i s of i n t e r e s t to note - 42 -that the estimators of p 4 ( l ) , i >_ 3, p 7 ( l ) , i > 3, p 8 ( l ) , i >_ 3 and ' p l O * 1 ^ , i _> 3 can a l l be a l t e r n a t i v e l y obtained ( i e . without using tables f o r estimates of d b a r ^ ) with a CPIS design simply by applying an argument almost i d e n t i c a l to that given i n experiment E^. The only d i f f e r e n c e here would be of course that p a r t i c l e diameters and c r o s s - s e c t i o n a l areas ( f o r i >_ 3) as w e l l as numbers of p a r t i c l e s would be of i n t e r e s t . I t has now been shown that i t i s very possible to attach a meaningful "handle" to the general common point i n t e r s e c t sampling (CPIS) concept. The a p p l i c a t i o n of CPIS techniques to downed woody p a r t i c l e s provides a v e h i c l e whereby the general theory i s mapped into an operational plane. But the value of CPIS procedures on an operational l e v e l has not as yet been demonstrated. I t remains then to give the CPIS system an actual f i e l d test and to compare i t s performance to that of the most suc c e s s f u l l y established technique, l i n e i n t e r s e c t sampling (LIS) (Brown, 1971). For reasons previously stated, the f i e l d t e s t w i l l be concerned only with l e s s e r downed woody fue l s sub-populations. Attention w i l l now. be turned to demonstrating the tentative s u p e r i o r i t y of CPIS over LIS with respect to at le a s t two l e s s e r downed woody p a r t i c l e s populations located i n areas of logging residue. - 43 -CHAPTER I I I F i e l d test to evaluate the common point intersect technique General Discussion In order to f i e l d test the CPIS technique and compare i t s per-formance to that of the LIS technique with respect to lesser downed woody pa r t i c l e s , two areas of fresh (i.e. less than 1 year old) coastal logging residue were selected for reasons previously cit e d . The f i r s t area was a clearcut comprised of 165 acres of tractor logged residue (see Figure 6). I t was situated approximately 8 miles west of Shawnigan Lake which l i e s 30 miles west of V i c t o r i a , B r i t i s h Columbia. The fuel types were predominantly coastal Douglas-fir, western hemlock and western red cedar. The second study area was a 180 acre clearcut of cable logged residue (see Figure 7). I t was located approximately 8 miles southwest of Sooke Lake which l i e s 25 miles southwest of V i c t o r i a , B r i t i s h Columbia. The f u e l types were s i m i l a r to those of the f i r s t area. Both study areas possessed moderate volumes of residue and reasonably continuous terrains of moderate in c l i n a t i o n s (i.e. less than 30° slope). The second study area possessed a noticeable ravine where lesser woody p a r t i c l e s accumulations were evident. Before the two sampling techniques were applied to these areas, each clearcut was examined i n order to determine whether the downed p a r t i c l e s i n the i t h subset Vie|l,2,3|-/were randomly distributed i n the sampling space of cylinders of radius r ^ = r 2 ^ . Recall that the formula offered for X2 (S2j *)> ie|i,2,3-t, je|;l,...,n'} i s v a l i d only when the p a r t i c l e s i n the (i) ' i t h subset are randomly distributed i n S^ , ie{l,2,3}. Hence randomness i n the f i r s t three subsets must be v e r i f i e d i n the sampling units to be sampled at least on a tentative basis, before the CPIS technique can be applied to those subsets. - 44 -Figure 6. Map of Study Area 1 depicting placement of both the l i n e intersect and common point intersect sampling units. Figure 7. Map of Study Area 2 depicting placement of both the l i n e intersect and common point intersect sampling units. - .46 -In order that the p a r t i c l e s may be regarded as randomly d i s t r i -buted i n the sampling units, they should s a t i s f y two properties. The f i r s t i s that the random variable defined by angular orientation of p a r t i c l e projection onto a planer section defined by adjacent ground l e v e l should have approximately a rectangular d i s t r i b u t i o n on £o,2TrJ (Mize and Cox, 1968; p.50) with respect to each sampling unit i n question. The second i s that the random variable defined by distance from the geometric centre of the i — sampling unit (of radius r ^ ) i n question to the contained p a r t i c l e geometric centre-point should have approximately a rectangular d i s t r i b u t i o n on |o, r^J Viel-l,2,3}", for each sampling unit of radius r ^ selected. The f i r s t property was tentatively v e r i f i e d for the i t h subset, Vie|l,2,3"3- on each study area. Verification,.was made by constructing a 2 yi goodness-of-f i t test at the 5 per. .cent significance l e v e l . On each area data for this test were collected by laying out 5 CPIS units (for each subset) systematically over the area to be sampled. Within each unit 20 transects of common length equal to the radius of the sampling unit were positioned systematically i n a unidire c t i o n a l pattern. Transect lengths for the f i r s t , second and t h i r d subsets were 8.5, 16.5 and 28.5 feet respectively. Note that these transects lengths are not c r i t i c a l to the test but that the unidir e c t i o n a l sampling design i s . Note also that the CPIS units were not l a i d out over the entire area but rather only over the area to be sampled. This sampling region i s selected to act as a representative f u e l c e l l for the entire area. Size and location of the fu e l c e l l are largely at the discretion of the investigator. I t need only be said here that a 6 chain by 6 chain square was selected on each clearcut where no portion of either 3.6 acre f u e l c e l l lay within 2.5 chains of any access - 47 -road. Choice of the c e l l s was based on the subjective decision that each 3.6 acre square proportionately reflected the major characteristics of interest on i t s p a r t i c u l a r area. Continuing then i f angular orientation i s t r u l y random i n the j s a m p l i n g unit, i t can be shown that the random variable ( 8 ) ^ defined by the intersection angle between the p a r t i c l e major axis and transect with respect to the j t h unit has probability density function f Q where: (sin u, ue(0,Tr/2]| Vje'tl, . . . ,5} (Van Wagner, 1968) 0, elsewhere Therefore under the assumption of randomness with respect to angular orientation, i t i s expected that the s t a t i s t i c : 2o ,* f D k = observed frequency of angles i n h* i * N 2/* i i u ' k t h p a r t i t i o n <(f D,k - f £ > k ) / f e , k f , where J f e k = expected frequency of angles i n k t h p a r t i t i o n , Vke{l, ..... ,20} would have aX^ d i s t r i b u t i o n with 19 degrees of freedom where fe , k i s evaluated using f g ; Vke;{l,. .. ,20-}. The scheme used here for defining angular class intervals i s based on .05 proportions of area under the sine curve. Hence: 1~ |( fo , k - fe , k) 2/fe , k ] = J- ( fo ,k " fe)2» where f e = f e , k , Vkbjl 20j} k=l L I e k=l , , . J by design. A l l sampling units i n the three subsets constructed for each area passed theX^ goodness-of-f i t test at the .05 significance l e v e l . Observed class p a r t i c l e frequencies for 6 t y p i c a l CPIS units are presented i n tabular form i n Table 2. I t i s d i f f i c u l t to construct a p r a c t i c a l test to either support or contradict the assertion that the random variable defined by distance from the geometric centre of the sampling unit (of radius r ^ ^ ) i n question to the contained p a r t i c l e geometric centre point has a rectangular - 48 -Table 2. Intersection angles of lesser downed woody p a r t i c l e s i n the study areas. Class Intervals Observed Frequency .Size Class 1 Size Class 2 Size Class 3 Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 degrees numbers 0-18.20 18.21-25.83 25.84-31.80 31.81-36.87 36.88-41.40 41.41-45.58 45.59-49.45 49.46-53.13 53.14-56.63 56.64-60.00 60.01-63.27 63.28-66.42 66.43-69.52 69.53- 72.53 72.54- 75.52 75.53-78.47 78.48-81.37 81.38-84.25 84.26-87.13 87.14-90.00 124 156 125 120 122 155 121 156 152 125 146 154 136 144 123 132 151 140 160 122 94 73 96 100 79 101 95 75 101 98 102 103 79 78 80 71 95 98 81 69 31 48 36 34 28 32 30 44 34 47 33 45 48 34 27 47 42 36 50 45 18 29 25 35 19 40 28 33 22 27 34 28 21 23 33 36 35 35 38 24 10 6 18 8 14 9 17 10 19 14 11 15 17 8 13 15 14 8 19 9 18 25 27 29 17 26 14 14 27 13 29 17 30 18 25 18 26 28 19 28 d i s t r i b u t i o n on |^ 0, r ^ J , VieiOl,2,3}', for each sampling unit of radius r ^ selected. Recall that under the assumption of randomly distributed p a r t i c l e s i n the CPIS units the random variable defined by (12.) was found to have a normal d i s t r i b u t i o n with mean 0. Now i t . has been shown that i t i s tentatively v a l i d to assume that the random variable defined by angular orientation (as previously specified) has a rectangular d i s t r i -bution on £o, 2T T J. Therefore i t can be seen that i f the above assertion regarding the random variable defined by distance (as previously specified)„ i s true, the random variable defined by (12.) should have approximately a normal d i s t r i b u t i o n with mean 0. Using t h i s l o g i c , a parametric s t a t i s t i c a l two-sided hypothesis test comparing a mean against 0 (where the population variance i s assumed to be unknown) was set up at the .05 significance l e v e l for each study area. Five widely spaced CPIS units were selected systematically on each area 1,2,3)'. Note that the comments made e a r l i e r j u s t i f y i n g the use of systematic sampling designs apply equally here. Now within each unit, 24 evenly spaced transects were l a i d ( i ) Tri 2 4 out and the image of t under R. (see (12.)) was found for each t e-({ - r y l , 3 ' 1 i=l where t=0 was chosen randomly. This process led to the computation of f i v e sample means and f i v e sample variances for each ie;{l,2 ,3} with respect to each study area. This data i s presented i n Table 3 i n which 'orientation random variable' i s to be i d e n t i f i e d with R^  ^  . For each j e f l , ... ,5'} and for each iefCl,2,3 ;} the j t h sampling unit yielded a sample value which lay outside the rejection region constructed under the n u l l hypothesis which i s i d e n t i f i e d with the assumption that the random variable defined by distance from the geometric centre of the j t h sampling unit (of radius r ^ ) to the contained p a r t i c l e geometric centre point has a rectangular - 50 -Table 3. Sample means and standard deviations of the orientation random variable for lesser downed woody p a r t i c l e s i n the study areas. Unit Sample Mean Sample Standard Deviation Number c l a s s ± class 2 Class 3 Class 1 Class 2 Class 3 Area 1, 2 Area 1, 2 Area 1, 2 Area 1,.2 Area 1, 2 Area 1, 2 1 .072 .027 .096 .092 .0978 .108 .183 .169 .230 .228 .283 .302 2 .023 .005 .064 .021 .110 .124 .137 .133 .170 .159 .264 .296 3 .013 .049 .099 .068 .138 .116 .157 .195 .237 .221 .331 .285 4 .044 .039 .092 .080 .113 .118 .178 .106 .220 .194 .269 .282 5 .015 .059 .077 .025 .135 .137 .201 .206 .193 .206 .324 .325 - 51 -di s t r i b u t i o n on [ o , r ^ j , Vietl.2,3}, Vjefl 5|. Combining the results from the two above s t a t i s t i c a l t e s t s , i t was considered tentatively v a l i d to apply the CPIS technique to the above CPIS units on each of the two study areas with respect to the f i r s t three downed woody p a r t i c l e subsets. - 52 -F i e l d work undergone with respect to the l i n e intersect technique Within each fuel c e l l 25 l i n e segments of common length 8.5 f t were systematically placed uniformly over the entire f u e l c e l l using an equidistant grid pattern (see Figures 6 and 7). The orientation of each transect was determined randomly. Random transect orientation i s not r e a l l y necessary here. Note that i t requires no more time to implement than does un i d i r e c t i o n a l transect orientation. J u s t i f i c a t i o n of the use of a systematic sampling design with equidistant grid pattern has been previously made. I t remains tip; j u s t i f y both the number and common length of the l i n e segments used. The use of 25 transects w i l l be explained f i r s t . Consider the well-known formula for an i n f i n i t e population of sampling units where the random variable of interest i s normally distributed: (21) N - ( C V ) 2 ^ - cc/2; N - l ) 2 / Z 2 where N = number of sampling units CV = c o e f f i c i e n t of sample v a r i a t i o n l-(oc= l e v e l of s t a t i s t i c a l inference t ^ j 2 ; N-1 = Student's t value at the (l-*/2) l e v e l with (N-1) degrees of freedom Z = degree of precision (expressed as a proportion of the sample mean) An inspection of tKe behavior of the t d i s t r i b u t i o n and application of the Central Limit Theorem reveal almost immediately that the minimum number of samples (N m^ n) required to meet precision l e v e l Z can be 2 2 expressed as: (22) ^ m^ n = 3.84(CV) /Z , providing N m i n i s s u f f i c i e n t l y large ( i . e . N- in-_> 25) regardless of the d i s t r i b u t i o n of the random variable of i n t e r e s t . I t follows that i f CV can be estimated from a preliminary sample, then N m i n can be expressed roughly as a function of Z. With respect to downed woody p a r t i c l e s , i t usually i s of interest to determine N m^ n u~ 53 -for Z within a small neighborhood of 0.15 (Howard and Ward, 1972; Brown, 1973). Hence for 15 percent precision, the following formula holds: (23) N m i h ^ 170.67 £(CV)pJ2, providing N m i r i; i s s u f f i c i e n t l y large ( i . e . N^"_> 25) = A- ( C V ) p s 2 where (CV) p g = c o e f f i c i e n t of preliminary sample v a r i a t i o n The v a l i d i t y of (23.) i s c r u c i a l to the comparison of LIS and CPIS since (23.) indicates the minimum number of samples required to meet 15 percent precision at the 95% confidence l e v e l . I t may be argued that (23.) leads to erroneous results unless the c o e f f i c i e n t of sample v a r i a t i o n s t a b i l i z e s for sample sizes greater than the preliminary sample s i z e . Unless the lesser downed woody pa r t i c l e s within the LIS units are pathologically dis t r i b u t e d , i t i s i n t u i t i v e l y l o g i c a l that the r a t i o s of the sample means to the sample variances should s t a b i l i z e beyond some minimum sample size usually quoted as 25 (Freund and Williams, 1958; Ehrenfeld and L i t t a u e r , 1964; Husch, M i l l e r and Beers, 1972). No rigorous proof of t h i s conjecture i s offered i n the l i t e r a t u r e because (23.) i s t r a d i t i o n a l l y regarded as only an approximation of N ^ » not as a precise replacement. Unfortunately a thorough investigation of j u s t how precise (23.) i s when applied to downed woody fuels was not feasible. Therefore the v a l i d i t y of (23.) w i l l have to rest temporarily upon s t a t i s t i c a l i n t u i t i o n . Note that a more precise knowledge of the adequacy of (23.) i s not v i t a l to the comparison between LIS and CPIS. For example i f (CV) p g i s smaller than CV with respect to a p a r t i c u l a r subset, N m^ n should be larger than the estimate given by (23.). Hence for comparison purposes (23.) works i n favour of LIS i n t h i s case. Alte r n a t i v e l y i f (CV) i s larger than CV with respect to a p a r t i c u l a r ps subset, the use of the oversized (CV) p g w i l l mostly l i k e l y be offset by the - 54 -fact that (A)in (23.) i s at least 4% larger than i t should be. Even i f (23.) overestimates N . for a p a r t i c u l a r subset, this would have to be mm ' very large to have a serious effect upon the sampling time required because the average measurement time per LIS point i s r e l a t i v e l y small. Thus (23.) i s s u f f i c i e n t l y credible for comparing LIS with CPIS. A preliminary sample size of 25 was chosen i n order than (23.) be v a l i d when applied to the comparison of LIS and CPIS. An explanation of the common transect length of 8.5 f t w i l l now be given. In previous studies, optimum lengths for LIS transects applied to lesser downed p a r t i c l e s have been suggested as: 6.8 f t for material less than 3 inches i n diameter at intersection (Brown, i n prep), 6.56 f t for p a r t i c l e s less than 1 inch i n diameter at intersection and 9.84 f t for p a r t i c l e s from 1 inch to about 4 inches i n diameter at intersection (Brown, i n prep). Although no s t a t i s t i c a l j u s t i f i c a t i o n was given to support the selection of these lengths, they have proven satisfactory for previous investigators. The choice of minimum adequate transect length was d i f f i c u l t to make without a thorough analysis to evaluate the performance of the LIS technique when applied with different transect lengths. I t was again necessary to resort to i n t u i t i o n and experience supported with a l l relevant information available i n the l i t e r a t u r e . The transect length chosen was 8.5 f t which agrees very w e l l with Brown, i n prep. I f the transect length was selected greater than or equal to 25 f t and the random variable of interest was an average per unit area, (23.) may be regarded as reasonably v a l i d i n many cases without the r e s t r i c t i o n that N . >25. mm-Having completely specified and j u s t i f i e d the lengths, number and locations of the LIS units on each area, attention i s now turned to the data collected at each LIS unit and the calculations performed upon that data. At each LIS unit, every lesser downed woody p a r t i c l e whose central axis (and at least one of whose edges) intersected the 8.5 f t transect segment was c l a s s i f i e d into one of the three previously defined subsets using a go-no-go gauge (Brown, i n prep.; p. 5)(see Figure 80. Also the times taken to locate each LIS point and to perform necessary p a r t i c l e measurements at each point were recorded. The device used to define each LIS transect segment consisted of two wooden stakes approximately 3 f t long, each of which was sharpened at one end f o r the purpose of easy in s e r t i o n into the ground. Attached to each stake was a small metal ring with a wingnut to which :;was fastened one end of the 8.5 f t cord representing the transect. These wingnut devices permitted the LIS transect segment to be adjusted for f u e l depth and slope. At each LIS point, the transect segment was oriented p a r a l l e l to the adjacent ground l e v e l . This procedure i s well-established for l i g h t to moderate slash (Brown, i n prep). Two sets of formulas were used to "Convert the preliminary LIS raw data into values required for the comparison of the LIS and CPIS techniques. The f i r s t set of formulas which converted p a r t i c l e intercept counts into average t o t a l volumes per unit area i s presented below: (i ) = / 2 ^ ( 1 ) ^ -(24) VOLT' = TT ( d w ) (secCVi))/68 K J 1 ^ , (Brown, 1973),! Vie{l,2,3}; Vje{l,...,25}';, Where K-^^number of p a r t i c l e intersections (as previously defined) i n the j t h LIS unit, je|l,...,25} with respect to the i t h subset, i e l l , 2 , 3 } . ' - 56 -Figure 8. 'Go-no-go' gauge used to determine the diameter size class of each intersecting p a r t i c l e under 3" i n diameter $i i n i t i a l point of intersection. V O L ^ = average t o t a l volume per unit area i n the j t h LIS unit, j e { l 25} v 3 2 with respect to the i t h subset, i e { l , 2 , 3 } ' , ( ( f t ) / ( f t ) ). and IjvOL^^J^J^where'^45 = K ^ , V j e f l 25}; (4) and V 0 L j i s s i m i l a r l y defined) obtained for each study area are presented i n tabular form i n Table 4 . Values computed from ( 2 4 . ) were then used to ' 4 for each study area. The entries i n this sequence i=l calculate ) ( C V ) ( l ) PS were then inserted into ( 2 3 . ) to arrive at the values offered i n Table 5 . 2 Note i n Table 5 that 384 -(CV) i s simply the right-hand side of ( 2 3 . ) , evaluated at Z=.10. I t i s now evident that ( 2 4 . ) combined with ( 2 2 . ) (where CV may be i d e n t i f i e d with (CV) p g) w i l l y i e l d the minimum number of LIS units required to obtain a population mean estimate with a specified degree of precision. The second set of formulas combines the results of ( 2 4 . ) and ( 2 2 . ) with the raw LIS sampling time data to obtain the minimum time required to obtain a population mean estimate with a specified degree of precision. This second set of formulas whose derivation i s obvious i s presented below: r i , ie{< 1 , 2 , 3 , 4 < } ( 2 5 ) T 2 ( ± ) = D ( 1 ) ( C ( I ) ) + ' T J ( ± ) [ P + ( M ( 1 ) ; where T 2 ^ = minimum t o t a l sampling time required (hours) using the LIS technique for the i t h subset, i e { l , 2 , 3 , 4 ' } . D ^ = t o t a l distance walked to obtain p a r t i c l e measurements (chains) = 6 ( U ( 1 ) - l ) / v T J ( i ) + 6 ( 1 ) + 1, where (i ) U = the minimum number of LIS units required for the i t h subset, . i e 1 ; l , 2 , 3 , 4 ^ 6 ( i )= min Ze]t+: /\S^ + Z^ £ whereji+ represents the set of a l l positive integers. Table 4. F i e l d p a r t i c l e intercept counts and corresponding volume estimates for the l i n e intersect sampling units. Unit Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fuel l Component Count Fuel Volume Estimate Class 1 Class 2 Class 3 Class 4 (Total) Class 1 Class 2 Class 3 Class 4 (Total) Area 1, 2 Area 1 , 2 Area 1, 2 Area 1 , 2 Area 1 , 2 Area 1 , 2 Area 1 , 2 Area 1, 2 ( f e e t ) 3 / ( f e e t ) 2 X 10 3 44 195 2 15 1 3 47 213 0.73 3.24 0.63 4:74- 3.13 9.39 4.49 17.37 33 126 6 6 1 1 40 133 0.55 2.09 1.90 1.9.0 3.13 3.13 5.58 7.12 187 87 26 16 5 4 218 107 3.10 1.44 8.22 5.06 15.65 12.52 26.97 19.02 175 187 16 12 3 3 194 202 2.91 3.10 5.06 3.79 9.39 9.39 17.36 16.28 151 75 14 8 2 2 167 85 2.51 1.25 4.42 2.53 6.26 6.26 13.19 10.04 105 210 10 11 1 7 116 228 1.73 3.49 3.16 3.48 3.13 21.91 8.02 28.88 227 102 27 13 4 5 258 120 3.77 1.69 8.53 4.11 12.52 15.65 24.82 21.45 157 153 13 14 2 9 172 176 2.61 2.54 4.11 4.42 6.26 28.17 12.98 35.13 77 65 8 11 1 11 186 87 1.28 1.08 2.53 3.48 3.13 34.43 6.94 38.99 168 192 12 17 3 4 183 213 2.79 3.19 3.76 5.37 9.39 12.52 15.94 21.08 186 87 19 13 3 1 208 101 3.09 1.44 5.95 4.11 9.39 3.13 18.43 8.68 78 42 14 4 2 3 94 49 1.29 .70 4.38 1.26 6.26 9.39 11.93 11.35 66 59 5 20 1 4 72 83 1.10 .98 1.57 6.32 3.13 12.52 5.80 19.82 28 60 22 9 5 1 55 70 .46 1.00 6.89 2.84 15.65 3.13 23.00 6.97 Cont'd. Table 4 cont'd. Unit Fuel Components Count Fuel Volume Estimate Number Class 1 Class 2 Class 3 Class 4 (Total) Class 1 Class 2 Class 3 Class 4 (Total) Area 1, 2 Area 1 , 2 Area 1 , 2 Area 1 , 2 Area 1. , 2 Area 1, 2 Area 1, 2 Area 1, , 2 (feet) 3 2 -3 J / ( f e e t ) X 10 15 18 230 5 16 3 3 26 249 .30 3.82 1.57 5.06 9.39 9.39 11.26 18.27 16 51 57 9 16 2 1 62 74 .85 .95 2.82 5.06 6.26 3.13 9.93 9.14 17 68 122 14 29 6 13 88 164 1.13 2.03 4.38 9.16 18.78 40.69 24.29 51.88 18 193 93 40 21 9 4 242 118 3.20 1.54 12.52 6.64 28.17 12.52 43.89 20.70 19 237 115 21 20 4 4 262 139 3.93 1.91 6.57 6.32 12.52 12.52 23.02 20.75 20 218 90 23 20 4 10 245 120 3.62 1.49 7.20 6.32 12.52 31.30 23.34 39.11 21 130 98 19 22 3 10 152 130 2.16 1.63 5.95 6.95 9.39 31.30 17.50 39.88 22 75 130 4 15 0 6 79 151 1.25 2.16 1.25 4.74 0 18.78 2.50 25.68 23 7 92 4 21 0 5 11 118 .12 1.53 1.25 6.64 0 15.65 1.37 23.82 24 119 69 9 8 2 7 130 84 1.98 1.15 2.82 2.53 6.26 21.91 11.06 25.59 25 61 254 12 32 7 7 80 293 1.01 4.22 3.76 10.11 21.91 21.91 26.68 36.24 I VO I - 60 -Table 5. Relevant s t a t i s t i c s for the l i n e intersect f i e l d sampling units. Area Class Mean Standard Coefficient of ,.,_,2 - , o / / ™ 7 \ 2 Volume Deviation Variation (CV.) 1 7 0- 6 7< C V) 3 8 ^ C V > ( f t ) 3 / ( f t ) 2 ( f t ) 3 / ( f t ) 2  X 1Q~3 X 10~ 3 1 1 1.90 1.24 0.65 72 162 1 2 4.45 2.84 0.64 70 157 1 3 9.26 6.84 0.74 93 210 1 4 15.61 9.84 0.63 68 152 2 1 1.99 0.99 0.50 43 96 2 2 4.92 2.10 0.43 32 71 2 3 16.03 10.30 0.64 70 157 2 4 22.94 11.83 0.52 46 104 - 61 -(This subformula i s clear from the fact that a square grid pattern super-imposed uniformly on a 3.6 acre fuel c e l l i s used for each LIS design). = average chain walking time per unit distance (hrs./chain) with respect to the i t h subset, ie'{l,2,3,4} p = average pin placement time per sampling unit (hrs./LIS unit) ( M ^ ) = average p a r t i c l e measurement time per LIS unit (hrs./LIS unit) with respect to the i t h subset, ie{l,2,3,4}. Examples of minimum t o t a l LIS times required to obtain population mean estimates of specified precision levels are given i n Table 8. A detailed discussion of the LIS f i e l d work which was undergone has been presented. Also formulas converting the LIS f i e l d data into values which can be used to compare the LIS and CPIS techniques have been derived and discussed. Hence i t i s now appropriate to discuss the next topic of interest, namely the CPIS f i e l d work which was undertaken. F i e l d work undergone with respect to the Common Point Intersect Sampling Technique Five sets of concentric c i r c l e s were systematically placed within Note that the centre CPIS unit i s located at the center of the fuel c e l l . Each set of c i r c l e s consisted of three c i r c l e s of r a d i i 8.5, 16.5 and 28.5 f t . , i n which f u e l measurements were made for the f i r s t , second and t h i r d subsets respectively. Inscribed within each of the above sets of c i r c l e s were 12 radius transect segments, one every 30°, with the location of the f i r s t transect randomly selected. These radius segments were easily located using a compass and yardstick. Five CPIS units were selected here only on a tentative basis. I f computations using data from these preliminary CPIS units revealed that 5 samples were not s u f f i c i e n t to obtain a population mean estimate within a small neighbourhood of 15 percent precision, then more CPIS units would be collected using a different systematic CPIS design. The selection of r a d i i for the CPIS units with respect to each subset has been previously j u s t i f i e d . The use of 12 transects i s required i n order that the image of t under h2? l ) ( see (18.)) can be computed for some randomly selected te:fo,2Tf], j e f l 5?;, ie.{l,2 ,3}. I t i s h 2 ^ ( t ) for the chosen t which w i l l serve as the very good estimate of / T r ( r 2 ^ ) 2 (see (17.)). Having both specified and j u s t i f i e d the siz e , shape and tentative number and locations of the CPIS units on each study area, attention w i l l now be focussed upon the data collected at each CPIS unit and the calculations performed on that data. Consider any of the 12 transects radiating from the center of the j th CPIS unit i n which p a r t i c l e measurements are to be performed with respect to the i t h subset. Every downed woody p a r t i c l e belonging to the i t h subset whose central axis (and at least one of whose edges) intersected the transect was counted. Determination of whether or not a p a r t i c u l a r intersecting p a r t i c l e was a member of the i t h subset was made using the x go-no-go gauge previously described. This process was repeated for a l l 12 transects V j e t l 5;), Vied,2 , 3 ) . - 63 -The t a l l i e s obtained are expressed i n tabular form i n Table 6. Also the times taken to locate each CPIS unit and to perform necessary p a r t i c l e measurements at each unit were recorded. Each of the 12 transects inscribed within every CPIS unit was defined by a device almost i d e n t i c a l to that used i n the section on l i n e intersect f i e l d sampling. The only difference here i s the cord which now i s 28.5 f t long and i s marked with flagging tape at both 8.5 f t and 16.5 f t . As i n LIS, two sets of formulas were used;:to convert the pre-liminary CPIS raw data into values required for the comparison of the LIS and CPIS techniques. Analogous to the f i r s t set of formulas used i n LIS, the f i r s t set of formulas used i n CPIS converted p a r t i c l e intercept counts into average t o t a l volumes per unit area: 12 (26) V ^ 1 ) / T r ( r 2 i ) 2 = h 2 ? i ) ( t ) = ^ C ^ ^ t + ^ ) , t arbitrary i n [o, 2TT] and where / T r ( r 2 ^ ) 2 = average t o t a l volume per unit area i n the j t h CPIS unit j e t l , . . . , 5 ) , with respect to the i t h subset, ie{1,2,3} ( ( f t ) 3 / ( f t ) 2 ) y C(JJ = (4» i f (1) i s odd 2, i f 0) i s even and gH^Ct)' = TT . ( d ( 1 ) ) 2 • (se c ( ^ i ) ) - K ( i ) - m ^ i ) ( t ) , Vte[b,2Tr] Note here that: ( d ( 1 ) ) = 0.0091 f t . ( d ( 2 ) ) = 0.0439 f t . } ( d ( 3 ) ) = 0.1400 f t . (Brown, 1973) -64-Table 6 . F i e l d p a r t i c l e i n t e r c e p t counts f o r the common p o i n t i n t e r s e c t sampling u n i t s . U n i t Transect _ ., n , _ , Number Number F u e l C o m P o n e n * S i z e Class 1 S i z e Class 2 S i z e Class 3 S i z e C l a s s 4 ( T o t a l ) Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 1 1 41 158 26 32 11 21 78 211 1 2 34 110 25 20 12 16 71 146 1 3 47 69 17 24 4 12 68 1.105 1 4 67 81 27 31 16 13 110 125 1 5 75 49 87 23 21 20 183 92 1 6 9 1 57 24 39 14 15 129 111 1 7 75 97 13 17 8 20 96 134 1 8 106 92 22 34 13 21 141 147 1 9 79 94 34 27 a 14 121 135 1 10 151 83 28 17 12 18' 191 118 1 11 135 147 32 18 . 12 19 179 184 1 12 68 128 27 32 13 20 108 180 2 11 I89 93 28 41 7 19 224 153 2 2 78 38 54 27 a 12 140 77 2 3 61 39 32 11 7 20 100 70 2 68 26 12 20 11 15 91 61 2 5 144 - 49 17 ! 9 7 22 168 90 2 6 101 28 21 .29 11 21 133 78 2 7 79 47 23 33 11 19 113 99 2 8 137 62 !6 25 12 x 3 165 100 2 9 343 81 56 28 24 20 423 129 2 10 118 64 50 34 9 18 177 116 con't. -65-Table 6 con't Unit Transect „ , „ , . Fuel Component Count Number Number Size Class 1 Size Class 2 Size Class 3 Size Class 4 (Total) Area 1 Area 2 /Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 "2 M i 212 51 81 26 5' 24 298 101 2 122 148 80 39 29 9 25 196 134 3 1 61 71 38 27 8 33 107 131 3 2 112 75 29 19 17 17 158 111 3 3 78 29 41 14 ' 16 12 135 105 3 4 72 51 35 14 11 12 118 77 3 5 134 44 32 17 16 11 182 72 3 6 95 43 48 17 9 10 152 70 3 7 54 48 22 21 3 13 79 82 3 8 41 54 20 19 10 18 71 91 3 9 44 124 22 38 9 15 75 177 3 10 51 30 37 19 13 14 101 63 3 11 82 31 18 21 12 23 112 75 3 12 156 33 24 24 13 32 193 89 4 1 88 117 16 28 5 31 109 176 4 2 90 151 47 24 11 24 148 199 4 3 128 70 27 13 11 22 166 105 4 4 89 118 22 27 10 23 121 168 4 5 50 112 15 • 20 10 17 75 149 4 6 149 177 26 31 10 28 185 236 4 7 112 97 17 32 •4 15 133 144 4 8 97 81 17 25 7 22 1 121 128 con't. Table 6 con't. Unit Number Transect Number Fuel Component Count Size Class 1 Size Class 2 Size Class 3 Size Class 4 (Total) Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 4 9 42 31 21 26 11 14 74 71 4 10 554 122 26 26 8 12 88 160 4 11 131 59 49 29 5 20 185 108 4 12 138 134 22 43 6 19 166 196 5 l 177 109 30 28 18 9 225 146 5 2 130 81 21 21 7 14 158 116 5 3 208 28 45 10 11 12 264 50 5 4 225 23 38 20 7 17 270 60 5 5 255 29 32 21 10 20 297 70 5 6 442 41 36 21 10 23 488 85 5 7 310 45 36 18 19 16 365 79 5 8 185 24 39 7 11 5 235 36 5 9 135 54 22 37 13 31 170 122 5 10 135 61 23 34 14 22 172 117 5 11 138 38 20 22 8 24 166 84 5 12 95 91 24 26 10 25 129 142 and (secCf-j)) = 1.40 (sec(Y 2)) = 1.13 J (Brown, 1973) (sec(T 3)) =. 1.10 Also note that Is evaluated i n ( 1 3 . ) , V i e d , 2 , 3 } and that ro^P (t) i s defined i n (6 . ) V i e { l , 2 , 3 } , Vj ejCl , . . . ,5} and Vte[o ,2Trj. [ [ v j ( i ) / i f ( t 2 < 1 > ) 2 ] j ' . 1 ] ^ a n d g v W / i r C t W ] ' ± were obtained for each study area and are presented i n Table 7. (26.) was then combined with a special case of (21. ) (N > 3) i n order to obtain the minimum number of CPIS units corresponding to at least one sample mean of precision l e v e l as close to,15% as possible. Note that a sample size of two was not .'/considered s u f f i c i e n t . This i s done to reduce the probability that the spacing between sampling units coincides with any pattern of pa r t i c l e s population va r i a t i o n not immediately apparent. Note also that (21. ) i s meaningful with respect to CPIS because the random variable of interest here i s an average taken over a s u f f i c i e n t l y large sampling area (see page . (20' . ) ) . Now once a satisfactory number (N) of CPIS units has been obtained for some Z close to 15%, the second set of formulas combines the value obtained for N with the raw CPIS time data to obtain the minimum time required to obtain a population mean estimate with degree of precision (2). This second set of formulas (25i) i s almost completely analogous to the second set of formulas (see (25 . ) ) used i n L I S . The only exception here i s that now: I = 5 . 5 0 chains i f three samples are used (27) D.^"^ / = 7.24 chains i f four samples are used = 11.12 chains i f f i v e samples are used - 68 -Table 7. F i e l d p a r t i c l e volume estimates for the common point intersect sampling units. Unit Fuel Volumes Estimate Number Class 1 Class 2 Class 3 Class 4 (Total) Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 Area 1 Area 2 ( f e e t ) 3 / ( f e e t ) 2 X 10 3 1 1.32 1.52 4.58 4.33 11.34 15.79 17.24 21.64 2 2.06 0.85 5.53 4.33 9.16 16.65 16.75 21.83 3 1.34 0.86 4.98 3.22 10.63 15.84 16.95 19.92 4 1.44 1.82 4.15 4.45 7.59 18.98 13.18 25.25 5 3.19 0.84 4.87 3.51 9.97 16.40 18.03 20.75 - 49: -Examples of minimum t o t a l CPIS times required to obtain popula-tion mean estimates of specified precision levels are given i n Table 8. Before proceeding to the analysis of the f i e l d test data, i t i s appropriate to b r i e f l y summarize the process by which Table 8 was obtained. Consider any iell,2,3,4<} with respect to each study area. F i r s t formula (26.) was used to derive jjV^^/T\(T2^)2jv_^, i i 4. This sequence or a suitable subsequence or a suitable summation thereof was then used i n (21.) with to derive an N^>3 such that at least one of i t s corresponding sample means had precision ( Z ^ ) ( s e e column 5 of Table 8) within some small neighborhood of 15%. The N ^ so obtained was inserted together with raw CPIS time data into (25f) (see (25.) and (27.)) to obtain T l ^ (see column 4 of T.able 8) , where T l ^ refers to the minimum t o t a l CPIS time required to obtain a population mean 'estimate of precision . Next formula (24.) was used to derive ^VOLj^J j = l * -"P1*8 sequence was used to obtain (CV) ^ w h i c h was combined with Z ^ to obtain N^^ through ps min & the use of (22.) where C V ^ may be i d e n t i f i e d with ( C V ) ^ . F i n a l l y the ps N ^ so obtained was inserted with raw LIS time data into (25.) to obtain mm T 2 ^ which i s defined analagous to T l ^ (see column 3 of Table 8). I t i s important to r e a l i z e that t h i s process cannot be s i m p l i f i e d because there i s no formula for the CPIS technique analagous to (22.) (where CV i s i d e n t i f i e d with ( c v ) p s ) used for LIS. This i s due primarily to the fact that the t o t a l sampling time for the minimum number of CPIS units required to obtain a population mean estimate with a specified degree of precision i s far more sensitive to the coe f f i c i e n t of sample v a r i a t i o n than i s the sampling time for the corresponding minimum number of LIS u n i t i . - 70 -Table 8. Sampling time comparison of the l i n e intersect and common point intersect methods. Area Class Line Intersect Common Point Time hours Intersect Time Precision Total Samplingl'Time Gain hours % of l i n e intersect Sampling time 1 1 1 1 2 2 2 2 2 2 2 2 3 4 2,3 2 2 3 4 4 2,3 2^3 5.54 2.46 13.21 6.28 1.47 2.24 3.01 5.64 7.63 3.70 4.12 3.34 13.11 2.04 18.41 9.02 13.64 5.09 13.11, 18.41 2.64 17.56 2.01 14.50 1.84 14.14 6.10 15.57 4.58 10.28 4.30 17.56, 14.14 3.35 14.50, 14.14 +39.71 +17.07 +31.72 +18.95 -79.59 +10.27 +38.87 - 8.16 +39.98 -16.22 +18.69 - 71 -CHAPTER IV Analysis of the f i e l d test data Before interpreting the results l i s t e d i n Tables 4 through 8, there are two points concerning the f i e l d test that warrant discussion. These are considered below: The f i r s t of these two points relates t c t h e fact that two s t a t i s t i c a l hypothesis tests are conducted on each area p r i o r to applica-t i o n of the CPIS technique. These tests were performed i n order to either v e r i f y or reject the necessary assumption that the downed p a r t i c l e s i n the i t h subset were randomly distributed i n the cylinders of radius (i ) ' ^ r to be sampled, Vi.ej(l,2,3$. As has been noted before, the data for these two hypothesis tests (see Tables 2 and 3) consistently give good c r e d i b i l i t y to the claim of p a r t i c l e placement and orientation randomness within the CPIS units sampled i n each study area. Now this required condition of p a r t i c l e placement and orientation randomness within the sampling cylinders may seem l i k e quite a severe r e s t r i c t i o n on the CPIS technique when applied to lesser downed fuels. It should be realized that randomness i s required only with respect to the CPIS units being sampled and not with respect to either the entire f u e l c e l l or a l l CPIS (i) units within the f u e l c e l l . This means that the CPIS formulas with K as evaluated i n (13.) may be applied to any population of lesser downed fuels with the one provision that randomness i s present within the CPIS units being sampled. I t should be mentioned here that the general theory on l i n e intersect sampling (De Vries, 1973) u t i l i z e s randomness but this time i t i s the p a r t i c l e s population that i s considered to be randomly distributed. With respect to lesser downed fuels, LIS i s some-what more th e o r e t i c a l l y f l e x i b l e than CPIS i n that the general LIS theory u t i l i z i n g randomness can be modified to v i r t u a l l y overcome bias due to nonrandom patterns of angular orientations of p a r t i c l e major axes (Van Wagner, 1968; De.Vries, 1973). Unfortunately the price of this f l e x i b i l i t y i s that three times as many LIS units are required (De Vries, 1973). The second point that warrants discussion i s the fact that p a r t i c l e intercept counts were made on each LIS'and each CPIS transect with respect to a l l three subsets. An alternative approach would have been to use Grosenbaugh's (1967) 3P subsampling procedures (Beaufait, Marsden and Norum, 1974). These 3P procedures were not applied to the second and t h i r d subsets because i t was f e l t that here the reduction i n sampling time offered by the 3P system was not s u f f i c i e n t to offset the s t a t i s t i c a l errors which these procedures introduced. Grosenbaugh1s 3P subsampling techniques were not applied to the f i r s t subset because i t was ^considered too d i f f i c u l t to make ocular estimates (required by the 3P system) of the numbers of twigs ((0-V'3) intercepting most transects. This intersection count estimation process required by 3P subsampling was deemed too d i f f i c u l t for twigs because i n many cases these f i n e r p a r t i c l e s were uniformly layered, making the number of twig interceptions not only impossible to estimate but also next to impossible to count. This d i f f i c u l t y i n counting twig interceptions was a point of much concern. When layers of p a r t i c l e s were encountered, i t becomes necessary to disturb s l i g h t l y the l o c a l p a r t i c l e s i n order to obtain a v a l i d intersection count. In the fieldwork undergone i n the study areas, t h i s process was not found awkward unless the p a r t i c l e s of concern were twigs i n layers. Then the process became mentally exasperating. Through much painstaking e f f o r t , twig interception counts were obtained for both study areas (see Table 4 for LIS twig counts and Table 6 for CPIS twig counts). I t was decided that i f the f i v e smallest preliminary CPIS units did not y i e l d a twig population mean volume estimate with approximately 15% precision, no more CPIS units would be sampled, thus preventing a comparison of LIS and CPIS to be made for twigs. This decision was reached because the frequency of occurrence of twig layers was so great that i t made the task of counting twig intersections next to f u t i l e . Since the twig data analysis (which u t i l i z e d data from the f i v e smallest CPIS units only) yielded mean volume estimates of about 40% precision for both study areas, no comparison was made of LIS and CPIS with respect to twigs. This fact i s reflected i n the absence of figures for the f i r s t s i z e class i n Table 8. I t i s suggested that i f mean twig volume estimates are required i n future investigations, regression equations expressing mean twig volumes per unit areai.as functions of mean p a r t i c l e volumes per unit area for at least the second and t h i r d subsets should be developed for the important f u e l types. These regression relations could then be used i n place of actual physical twig data. I t i s important to observe that when sampling i n slash, the u t i l i z a t i o n standard implemented may have a s i g n i f i c a n t effect on either the general form of regression model selected or the estimates for the regression c o e f f i c i e n t s used. Due to time constraints, suitable regression equations w i l l not be developed here. Before analyzing the data i n Tables 4 through 8, i t should be stressed that the CPIS technique has not f a i l e d i n i t s application with respect to twigs. The analysis with the available data has revealed that sampling twigs i n the two study areas i s not p r a c t i c a l . In areas where i t i s p r a c t i c a l to sample twigs and no regression equations for twig volumes are available, the CPIS technique can be applied providing that either a bigger sampling unit radius i s chosen or al t e r n a t i v e l y a larger number of CPIS units are considered. Tables 2 through 8 l i s t both raw f i e l d data and important values computed from that f i e l d data. The values l i s t e d i n Tables 2 and 3 have already been discussed. The data given i n Tables 4, 6 and 7 are straight forward and require no further comment. In Table 5, i t suffices to point out that the number of LIS units (see column 7) required to obtain a population mean volume estimate with 10% precision i s more than double that required to obtain a population mean volume estimate with 15% precision (see column 6). I t should be observed that this i s a very high price to pay for an increase of 5% s t a t i s t i c a l accuracy. I t remains only to consider Table 8 which deserves the most attention since i t summarizes the perfor-mances of both the LIS and CPIS techniques when applied to the lesser downed fuels of the study areas. An inspection of Table 8 reveals immediately that on the f i r s t study area the performance of CPIS was consistently superior to LIS. As previously mentioned the sampling times offered i n Table 8 are represen-tative of t o t a l sampling times required to obtain population mean volume estimates with specified degrees of precision ( i n a small neighbourhood of 15%) at the 95% confidence l e v e l . Notice that on area 1, the minimum time gain offered by CPIS was about 17% on 2.5 hours of LIS time. Also notice that on area 1 the maximum time gain offered by CPIS was about 40% on 5.5 hours of LIS time. These figures c l e a r l y reveal that on study area 1 the t o t a l sampling times required by LIS to obtain lesser f u e l volume estimates of approximately 15% precision at the 95% confidence l e v e l can be s i g n i f i c a n t l y reduced through proper application of the CPIS concept. However the information presented i n Table 8 for study area 2 i s not so - 7,5 -straight-forward. Recall that i n the general discussion of the application of CPIS to downed woody p a r t i c l e s , i t was mentioned that s t a t i s t i c a l problems arose when the population of f u e l p a r t i c l e s contained large continuous areas d i f f e r i n g d r a s t i c a l l y i n f u e l p a r t i c l e frequency. Recall also i t was pointed out there that i f such d i s t i n c t areas occurred, i t would be advisable to s t r a t i f y the fuels population i n a meaningful way and apply the theory of s t r a t i f i e d sampling. Now i t was remarked e a r l i e r i n the general discussion of the f i e l d test that there was a noticeable ravine on study area 2 where lesser woody p a r t i c l e accumulations were evident. A closer inspection of t h i s ravine revealed that the f u e l p a r t i c l e frequencies of only the f i r s t and second diameter size classes (subsets) were dramatically high. Hence i t became necessary to s t r a t i f y the fuels populations of the f i r s t and second diameter size classes only. I t i s suggested here that a bias i n f u e l p a r t i c l e frequency of occurrence for the fuels of the f i r s t two size classes was present because i t i s probable,.that the d i s t r i b u t i o n of f u e l placement for the smallest p a r t i c l e s becomes skewed when the d i r e c t i o n of log p u l l interacts with unusual features such as s i g n i f i c a n t continuous undulations or i r r e g u l a r i t i e s i n the t e r r a i n . Going back to Table 8, notice that on study area 2 size classes 2, 4 and (2, 3) each have two sets of data associated with them. Notice also that one set i s underlined and one i s not. The underlined set refers to results of computing appropriate mean volume estimates with proportional s t r a t i f i c a t i o n (Ehrenfeld and Littauer, 1964; p. 388); the non-underlined set refers to results without proportional s t r a t i f i c a t i o n . The differences are highly s i g n i f i c a n t , indicating that f u e l s t r a t i f i c a t i o n by f u e l loading has a dramatic influence here especially upon the performance of CPIS. I t should be remarked that the increases i n the t o t a l LIS times after s t r a t i f i c a t i o n are consistent with the corresponding decreases i n precision l e v e l s . Fuel s t r a t i f i c a t i o n has a favourable effect upon both LIS and CPIS. This effect i s not exposed so obviously for LIS because LIS i s sensitive to a decrease i n precision l e v e l within a neighbourhood of 15%. Note that size class (2, 3) refers to the pair of size classes not the i r j o i n t grouping into the si z e class, ( V , 3"). The inclusion of size class (2, 3) i s j u s t i f i e d by remarks made e a r l i e r to the effect t h a t . i t was theor e t i c a l l y feasible to obtain volume estimates for size class 1 from volume estimates for both size classes 2 and 3 using regression analysis. Proper application of CPIS techniques to the problem of obtaining mean downed woody volume estimates of approximately 15% precision at the 95% confidence l e v e l has consistently resulted i n s i g n i f i c a n t t o t a l LIS sampling time reductions with respect to both study areas considered. The time constraints imposed upon this investigation prevent a rigorous v e r i f i c a -t i o n of the claim that CPIS i s s i g n i f i c a n t l y superior to LIS with respect to a l l downed woody f u e l applications. On the other hand the findings presented i n Table 8 cannot be dismissed as simply interesting. These findings offer concrete confirmation that common point intersect sampling (CPIS) i s a f l e x i b l e and viable sampling technique which deserves much attention. At th i s point the data for the f i e l d test has been presented and analyzed. I t i s appropriate now to consider the o v e r a l l significance of the thesis, and the inferences which can be made from the information the thesis has provided. - 77 -CHAPTER V Conclusions A general sampling technique referred to as 'Common Point Intersect Sampling' (CPIS) has been developed and discussed and tested operationally with encouraging results. Appropriate CPIS formulas were derived with respect to downed woody fu e l s . Using these formulas the performance of CPIS was compared to that of l i n e intersect sampling (LIS) i n two fresh cutover areas. Proper application of the CPIS technique yielded lesser fuel volume estimates of about 15% precision with savings of up to 40% of the t o t a l sampling time required by the LIS technique. a The general theory of CPIS as presented i s extremely f l e x i b l e . I t can be applied to the problem of obtaining quantitative estimates for attributes of any community of objects temporarily fixed i n space. I t i s imperative for the reader to r e a l i z e that the basic concepts involved are seated i n sound l o g i c . To emphasize t h i s point a consideration of a l l the seemingly negative aspects of CPIS w i l l now be made. The f i r s t apparent negative property of CPIS i s that there i s no cut and dried general formula for obtaining the common radius of the c y l i n d r i c a l sampling units used. This property i s closely linked to the problems of selecting the appropriate number of transects to lay out i n each CPIS unit and of specifying a s u f f i c i e n t number of CPIS units which w i l l result i n a satisfactory estimate. Although preliminary sampling holds some promise for solving these problems there i s l i t t l e doubt i n the mind of the author that a t r u l y satisfactory answer l i e s i n the art of simulation modeling. Using a systems analysis approach, the system i n question can be subjected to a multitude of sampling designs with - 78 -s t a t i s t i c a l and cost c r i t e r i a used as the basis for selecting the best set of outcomes. Time constraints forbid the construction of such a general systems model which would be interfaced with different general sampling techniques. In essence then what i s being said i s that a t r u l y s a t i s f a c -tory answer to the general problem of selecting both the number of CPIS transects and the number and size of the CPIS units l i e s outside the scope of the thesis. U n t i l such time as a general systems model i s b u i l t , each investigator w i l l be forced to either construct and experiment with events models sim i l a r to those presented i n this thesis, conduct preliminary sampling, or else rely on his own experience and i n t u i t i o n i n order to quantify precisely the CPIS system which w i l l best serve him. The second apparent negative property of CPIS i s that i t i s not immediately clear how to get the function whose int e g r a l i s to be used i n obtaining an estimate for the parameter of inter e s t . Of course i t can be argued that the multitude of situations which can arise here i s so great that i t prevents the s p e c i f i c a t i o n of a precise technique. But there i s a related procedure embedded i n the general theory and reinforced i n the CPIS applications. More s p e c i f i c a l l y , i f the attribute i s a one-dimensional average, try to relate i t through the m u l t i p l i c i t i v e constant k to the image of a function whose component terms can be easily t a l l i e d either i n the f i e l d or on a photograph. I f successful, t h i s image then leads d i r e c t l y to a form of the desired integrand using for example the concept of the average value of a function. The argument used for i s a perfect example of t h i s . I f the attribute i s two-dimensional, the integrand w i l l probably be either one or two dimensional depending upon what character-i s t i c s can be measured. A good example here i s that surface area i s - 79 -generated by rotating c u r v i l i n e a r length. F i n a l l y i f the attribute i s three dimensional, the integrand w i l l probably be either one or two dimensional, again depending upon the inputs involved and any simplifying assumptions which are v a l i d to use. A good example here i s that volume i s generated by rotating surface area. In practice fihe selection of the integrand i s almost always si m p l i f i e d by the fact that only certain f i e l d variables can be measured to satisfactory precision. See Chapter VI for a further discussion of this problem. The general common point intersect sampling concept has been applied to the problem of obtaining estimators for important parameters of downed woody fuels populations. A general approach has been presented which permits the complete s p e c i f i c a t i o n of CPIS designs and formulas for computing t o t a l volume estimates of randomly distributed downed woody par-t i c l e s . This approach was followed i n d e t a i l with respect to the lesser downed fuels, and was followed i n general with respect to the greater downed fuels. A general formula which expressed CPIS unit radius as a function of maximum p a r t i c l e diameter of interest was presented for -randomly d i s t r i -buted p a r t i c l e s . Also a general CPIS formula independent of p a r t i c l e d i s t r i b u t i o n was offered for computing t o t a l volumes of greater downed fuels. These results w i l l now be summarized by the following suggested step-by-step f i e l d procedure to be followed when t o t a l volume estimates of downed woody pa r t i c l e s are of interest: 1) specify the set of diameter size classes of interest, |(di» D i ] j i = 1 . d i > V . 2) select a representative f u e l c e l l i n the area of concern. Avoid access roads and unusual topographic features such as rock - 30 -outcrops. I f the f u e l c e l l has no large continuous areas of unusually high or low fu e l frequencies, apply steps 3 through 8 to the entire fuel c e l l . Otherwise s t r a t i f y the f u e l c e l l by loading and i f desired apply steps 3 through 8 to each stratum. I f this amount of d e t a i l i s not required for each stratum, simply do steps 3 through 8 applying the theory of proportional s t a t i f i e d sampling (Ehrenfeld and Littauer, 1964; p. 388). (Note that i f this l a s t process i s conducted, the N sampling units are selected on a basis of appropriate represen-tation i n the fuel strata rather than on an arbitrary basis as implied by (5) and (6). 3) Lay out a systematic sampling design consisting of about seven common point intersect sampling units, each of common radius (M) r where: T r(M)-» , o o i n / - 7 i ^ DM i n inches r ^ 1 ' rn feet Within each of these 7 large c i r c l e s there are (M-l) concentric c i r c l e s , where the radius of the i t h c i r c l e i s defined by: T - r ( i ) ^ T\ DJ i n inches Log r 1.22 + 0.47 l o g ^ , i i U i V r < x ) i n feet, V i e{l,... ,M-1} Make sure that the 7 c i r c l e s are spread such that they cover the entire f u e l c e l l uniformly. 4) Set i = 1. 5) Set N = 3, j = 1, and k = 0. Select N widely spaced large CPIS units from the 7 units l a i d out. 6) Consider the i t h c i r c l e i n the j t h large CPIS unit. I f the fu e l p a r t i c l e s i n the i t h diameter size class i n the i t h c i r c l e - 81 -are randomly distributed (assume this unless i t i s very obviously f a l s e ) , choose a radius transect (t) randomly using some set of random numbers. Each radius transect i s defined by a sub-(M) length of cord of length r j o i n i n g two or more wooden stakes, each sharpened at one end to f a c i l i t a t e easy i n s e r t i o n into the ground. Transects are l a i d out geodetically ( i . e . p a r a l l e l to the adjacent topographic surface) not horizontally. The cord should be attached to each wooden stake with a s l i d i n g ring equipped with a wingnut to permit the cord height to be raised or lowered according to f u e l depth. Count the number of p a r t i c l e s i n the i t h diameter size class (d-pD^/J which intersect this transect. Note that an intersection occurs only i f both the p a r t i c l e central axis and at least one of the p a r t i c l e edges intersect the transect. Then rotate the radius transect through a 30° arc and repeat the above process. This procedure i s carried out a t o t a l of 12 times to obtain the sequence • . .12 {'m?±) (t + ^ ) 1 , where m ^ C t + ~ ) , wefl,...,12i i s the number of p a r t i c l e s i n the i t h diameter size class which intersect the transect at (-^  + t) inscribed within the i t h b concentric c i r c l e i n the j th large CPIS unit . This sequence i s then inserted into (19.) to obtain the desired volume estimate for the i t h c i r c l e i n the j t h CPIS unit. Note however that i f the p a r t i c l e s i n the i t h c i r c l e (where D£<3") are not randomly distributed, no legitimate volume estimate can be obtained using (19.) with as.evaluated i n (13.), Vie][ 1,2,3.}. Now suppose that the fuel p a r t i c l e s within the i t h c i r c l e - 82 -(where D^ >3") are not randomly distributed. Then repeat the sampling procedure described above for randomly distributed p a r t i c l e s , with the exception that for each relevant i n t e r -section the distances to the l e f t and right most points of the intersection (along the p a r t i c l e edge) are t a l l i e d and not the-number of relevant p a r t i c l e intersections. As indicated previously i f a truncated e l l i p s e i s encountered ( i . e . a p a r t i c l e end), i t i s necessary to extrapolate one edge of the p a r t i c l e to the transect i n order to obtain the proper value assuming of course that the p a r t i c l e central axis intersects the transect. The data so obtained are inserted into (14.) which i s then used to obtain the desired volume estimate for the i t h concentric c i r c l e (3<i<M) i n the j t h large CPIS unit.- This estimate i s described below: 12 10=1 Coog2 ( t + iiE) j e f 1 N ) ie$4 K], V 3 " ' 4, i f a) i s odd 2, i f to i s even 7) Increment j by 1 and repeat (6.) u n t i l j=N+l+k. 8) Take the N volume estimates obtained above and compute: 1^ 0.975; N-l)'^CV^i/Cj? E ( T S ) ^ where (CV) ± = coefficient:of'sample ' ' v a r i a t i o n for the i t h diameter size class. t0.975; N-1 = value from Student's t d i s t r i b u t i o n with (N-1) degrees of freedom. If (TS)^ 1^ < 0.15, the desired 95% confidence i n t e r v a l with 15% precision N has been obtained. Hence increment i by 1 and repeat steps (5) through (8) - 83 -u n t i l i=M+l. However i f ( T S ) ^ > 0.15, increment K by 1 repeat steps N (6), (7) and (8) with j=N+K u n t i l either (TS) ( l )< 0.15 or j=8. Then N — increment i by 1 and repeat steps (5) through (8) u n t i l i=M+l. Note that (6), (7) and (8) are repeated with only one new added sampling unit. This means that i t i s necessary to select only one new CPIS unit each time (6), (7) and (8) are repeated for a p a r t i c u l a r ie{-l Mj. I t should be observed that the above guidelines are not applicable to p a r t i c l e s which l i e i n the (0"-V'3 diameter size class. Upon investigation of these smaller p a r t i c l e s i t was discovered that i n many downed woody fu e l complexes, f i e l d problems ( i . e . clumps or p i l e s pf>. p a r t i c l e s i n t e r -spersed with needle mats) were encountered making impractical the twig counting process or even the estimation of twig counts. I t i s recommended that for • the (o"-V3 size class, an interim suitable regression model be developed. This model would express p a r t i c l e volume i n the (o"-V'J size class as a function of p a r t i c l e volumes i n the (%"-l"] and (l"-3"3 size classes and any other factors (e.g. type of disturbing influence and u t i l i z a t i o n standard for a harvesting operation) deemed important. Due to time constraints the above step-by-step procedure necessarily includes s l i g h t abuses of that portion of the general CPIS theory previously developed for greater downed p a r t i c l e s . The above procedure can s t i l l v a l i d l y serve as a sound tentative setiof guidelines for a l l downed woody par t i c l e s pending further studies of the performance of CPIS techniques with respect to greater downed p a r t i c l e s . I t can v a l i d l y be argued that there are many sampling problems of more economic importance than those involved with downed woody materials. In fact i n Chapter VI suggestions are given as to how the - 84-common point intersect technique may be applied i n other forestry-related areas. Perhaps the strength of CPIS should have been!;jtested i n some other forestry problem, such as the task of estimating important standing timber parameters. The fact that the CPIS concept was not applied there has been amply j u s t i f i e d . I t i s the claim of the author that the problem of obtaining lesser downed fu e l volume estimates i s an extremely d i f f i c u l t one, and as such serves as an excellent test for evaluating any general sampling scheme. The common point intersect technique did more than pass t h i s test on two study areas; when properly applied i t proved i t s e l f to be s i g n i f i c a n t l y more time economical than l i n e intersect sampling, which to date i s the only economically r e a l i s t i c means of obtaining quantitative estimates for physical parameters with respect to lesser downed woody material. I t could also be argued quite v a l i d l y that the new sampling technique as presented i s unappealing to the average potential user. This disenchantment stems almost e n t i r e l y from the use of mathematics beyond the scope of the average potential user. More than l i k e l y he w i l l be hesitant about using a sampling technique which he does not f u l l y understand. Note that there i s nothing complicated about picking locations for some points and imagining c i r c l e s of a common specified diameter about those points. There i s nothing complex about insc r i b i n g a few spokes at specified angles within these c i r c l e s and counting p a r t i c l e intersections along each spoke and/or making a few simple measurements at each intersecting p a r t i c l e . I t should not be d i f f i c u l t to see that these p a r t i c l e counts and/or intersections within a pa r t i c u l a r c i r c l e result i n an estimate of some attribute of concern with respect to that c i r c l e . The only thing which the average potential user may f i n d hard to follow i s the higher mathematics used i n converting the p a r t i c l e intercept counts and/or measurements to estimates of lengths, surface areas, volumes or whatever i s of interest. The use of advanced mathematics here could be frowned upon because i t complicates the theory. But i t i s because of the higher mathematics that the f i e l d work can be reduced to a minimal l e v e l . The average potential user i s most interested i n how much work he has to do and how much he has to spend to get what he wants. The common point intersect technique has been shown to make substantial reductions i n both of these areas at least with respect to lesser downed woody materials. There i s good promise that with the use of CPIS e f f i c i e n t solutions for other sampling problems can be found. In the step-by-step f i e l d procedure offered on p.74-78 there are some formulas whose evaluation i s p e r i o d i c a l l y required i n the f i e l d . The author does not expect the average potential user to take these guidelines verbatim into the f i e l d with him. But rather these guidelines are intended to form a basis for developing a f i e l d guidebook which would permit the user to look up i n a table the volumes, etc. f o r his p a r t i c u l a r data. The l e v e l of resolution of these tables i s primarily a function of the accuracy desired by most users. The average potential user i s never expected to understand mathematical formulas. At t h i s stage a l l the potential user need be aware of i s that the common point intersect technique was tested on two d i s s i m i l a r slash areas and was shown to require s i g n i -f i c a n t l y shorter sampling times than the most successful sampling technique currently available, namely l i n e intersect sampling. F i e l d packages w i l l come l a t e r . The r e a l power of common point intersect sampling l i e s i n the - 86 -fact that i t i s extremely v e r s a t i l e . I t can be likened to cluster sampling and as such i s most b e n e f i c i a l i n those cases where the cost of selecting and locating a population element fa r exceeds the cost of determining the contribution which that element makes to the estimate for the attribute of interest (Freese, 1962; p.64). A great deal of e f f o r t has been made to convince the reader that common point intersect sampling i s a sound and viable non-destructive technique which deserves the attention of i n v e s t i -gators from almost every s c i e n t i f i c d i s c i p l i n e . I t i s almost redundant to say that more studies i n many more sampling problem areas are required i n order to thoroughly evaluate the worth of t h i s new sampling concept. But t h i s i s s t i l l not a v a l i d reason to ignore common point intersect sampling u n t i l i t s value has been conclusively determined. Through numerous arguments i t has been indicated that applications of CPIS techniques are feasible now at least i n the forestry-related areas of downed woody material and standing timber (see Chapter VI). The two main objectives of th i s thesis were stated as the pre-sentation of a new general sampling theory and the application of this theory to an important s o c i a l problem area. But these objectives are only a means to an end, which i s the f u l l operational use of common point intersect sampling. The achievement of this end i s what th i s thesis i s a l l about. - 87 -CHAPTER VI P r a c t i c a l applications of the common point intersect technique Thus f a r the common point intersect sampling technique (CPIS) has been applied only to the problem of estimating parameters for populations of downed woody materials. But t h i s i s only one small problem for which CPIS offers a promising solution as t h i s chapter w i l l demonstrate. Before discussing alternative applications of CPIS consider f i r s t a s i m p l i f i e d version of the sampling concept upon which CPIS i s based. CPIS selects c y l i n d r i c a l sampling units enclosing elements of the population of interest. CPIS hopes to choose these units s u f f i c i e n t l y large so that the v a r i a b i l i t y among them with respect to the attribute of interest i s small. Within each unit a number of l i n e transects are inscribed. Regarding each sampling unit i n two dimensions as a c i r c l e , each transect corresponds to a r a d i a l segment. Many measurements are made i n each c i r c l e at places defined by the intersection points of the transects and relevant population elements. For a given c i r c l e these measurements are inserted into appropriate formulas which given an estimate of the attribute of interest with respect to that c i r c l e . When applying CPIS to the problem of estimating some physical property of interest, i t i s usually of benefit to ask the question 'What geometric figure when rotated about the centre of a c i r c l e w i l l y i e l d that physical property?'. Consider f i r s t the problem of estimating the length of a stream or road network. The physical property of interest i s length. So what geometric figure when rotated about the centre of a c i r c l e y i e l d s length? The answer i s simply a point. So for a given CPIS unit l e t X^^(0) be the distance along the base of that unit from the centre of the unit to the j—r.point formed by the intersection of the transect at 0 with the i — stream or road intersecting the transect at 0. Let in. (0) be the number of - 88 -til such points formed by the intersection of the transect at 0 with the i — stream or road intersecting the transect at.0. Also l e t n(0) be the number of stream or road intersections intersecting the transect at 0. Then 2ir ' . n(0) m±(6) ^ y - j=X~ X V ^ ( 0 ) d 0 gives a measure of the t o t a l length of streams or 0 roads within the CPIS unit of interest. Notice that the result i s inde-pendent of stream or road d i s t r i b u t i o n or frequency. Consider next the problem of estimating large numbers of f r u c t i f i -cations of wood-destroying fungi, large numbers of migrating w i l d l i f e or large numbers of host-attached plant parasites. The insight gained from the discussion of experiment should provide the key to these problems. For a fixed CPIS unit of radius r l e t n(0) be the number of population 2TT elements intersecting the transect at 0. Then 1 ( n(0)d0 gives a 2fTr J 0 measure of the average number of elements per unit area within that CPIS unit providing of course that the units of r are s u f f i c i e n t l y small. This result i s also independent of d i s t r i b u t i o n or frequency of the elements of concern. The problem of estimating crown area (Husch, M i l l e r and Beers, 1972; p.106) and mean crown diameter (Husch, M i l l e r , and Beers, 1972; p.49) with CPIS techniques w i l l be considered next. Time constraints permit only a general investigation here and t h i s i s limited as above to the generationfpf the integrand i n (1.) with respect to each of the above two stand parameters. Focus upon the problem of estimating crown area using the CPIS concept. Suppose that a e r i a l photographs are available which permit crown boundaries to be delineated. Let p i be a parameter representing the average crown area per unit area. Similar to experiment E 2 (see 'Application 2 to downed woody f u e l s ' ) , l e t k be TTr . The F.. (2TT) represents a t o t a l crown - ,89 -area with respect to the area i n the j t h CPIS unit. Using notation as i n the application of CPIS to downed woody p a r t i c l e s , a function f^ i s desired such that: F j (2TT) = ^ f j ( t ) d t , where FJ(2TT) = T r r 2X(S^), and 0 X(S.) = value taken on by X at S., je{l,...,N}, where X again refers to a J j random variable defined on the sample space of outcomes of experiment Now i n order to produce fjCt ) 1 1 1 terms of variables which can be e a s i l y measured, i t i s necessary to make some assumptions regarding the regions enclosed by the crown boundaries. I t i s v a l i d to assume that each region defined by a crown i s a c o l l e c t i o n of connected sub-regions (Taylor, 1965; p.76). I f desired, this can be taken to mean that each sub-region can be traced without ever havingi>to l i f t the pencil. The attribute of concern here i s a surface area which i s two-dimensional. So what geometric figure when rotated about the centre of a c i r c l e yields surface area? The answer i s a l i n e segment. A basic understanding of calculus reveals that i f a l i n e segment of small length AX i s located approximately a distance X away from the o r i g i n , a surface area of (Ax) ((X«A'<9) i s generated by sweeping the l i n e segment through A8 radians. I t follows t r i v i a l l y that: * j ( t ) p i > j fj(t) = h YZ. X_ [ e ^ ' j ( t ) " bk.i.j(t)] • V t £ f ° > 2 7 T J > w h e r e i = l k=l nu(t) = number of crowns intersecting the CPIS transect located at the t within the j t h CPIS unit. P i . j = number of l i n e segments formed by the i n t e r s e c t i o n of the transect at t with the i t h crown intersecting the transect at t, (t) i e {1,...,m.(t)}, within the j t h CPIS unit. = furthermost end-point of the k J i l i n e segment formed by the intersection of the transect at t i n the j t h CPIS unit with the i t h crown intersecting the transect at t, ke{l > • • • » (t) i e { l m j ( f c ) } i d e n t i c a l to e k > i > j ( t ) with the exception that b k y i , j ( t ) refers to the innermost end-point. f j i s sectipnally continuous on [0,2TrJand thus s a t i s f i e s a l l required a n a l y t i c a l properties. Note that the only inputs required by f.. are l i n e a r distances which are most often e a s i l y obtained from a e r i a l photographs. I t i s important to r e a l i z e that the CPIS formulas produced to obtain .. estimates of crown area are independent of tree d i s t r i b u t i o n and also v i r t u a l l y independent of crown shape. The problem of estimating mean crown diameter with CPIS techniques i s completely analagous to experiment (See 'Application to'downed woody p a r t i c l e s ' ) . The only assumptions required are those used d i r e c t l y above, namely that the a e r i a l photographs used permit a meaningful delineation of crown boundaries, and that each crown may be regarded as a c o l l e c t i o n of connected sub-regions. Since crown diameters are usually well-defined except for some hardwoods (Sayn-Wittgenstein, 1960), they can be measured d i r e c t l y and used with an analogue to the average function value formula presented i n experiment E^ (See (4.)) t o ; y i e l d the desired estimate. for other stand parameters such as number of trees per unit area by species (where species can be i d e n t i f i e d ) , mean crown diameter per tree by species, mean height per tree by species (where tree height can be either estimated or else regressed from other measurable tree c h a r a c t e r i s t i c s , mean tree With CPIS techniques i t i s eas i l y possible to obtain estimates volume per tree by species, etc. Note that these estimates can be inserted into suitable regression equations to produce estimates f o r component oven-dry weights of t o t a l - t r e e , bole-wood, bole-bark, total-crown and branches of different sizes (Kurucz, 1969). Now consider the problem of estimating the surface areas of lakes, watersheds, large r i v e r systems, or areas occupied by different herbaceous and ligneous species. The application of CPIS to these problems i s si m i l a r to the application of CPIS i n estimating crown area for standing timber. So l e t a^(0) and b^(0) be the distances along the base of some CPIS unit from the centre of the base of the unit to the l e f t and right hand end-points respectively of the j-£il segment formed by the intersection of the transect at 8 with the i — population element intersecting ?the transect at 6. Let n^(0) be the number of l i n e segments formed by the intersection of the transect at 6 with the i J ^ l population element intersecting the transect at 9. Also l e t m(9) be the number of population elements intersecting the transect at 9. Then gives a measure of the t o t a l surface area occupied by the populat ion elements within that CPIS unit. Note that this result i s independent of the d i s t r i b u t i o n or frequency of the elements of concern. This r e s u l t can also be used to measure rate of seasonal change or annual growth by comparing figures computed from photo-graphs or direct measurements taken at appropriate times. BIBLIOGRAPHY Acton, F.S. 1970. Numerical methods that work. Harper and Row Publ., New York, 541 p. Anderson, H.E. 1964. Mechanisms of f i r e spread, research progress report No. 1. U.S.D.A. For. Serv. Res. Pap. INT-8, 20 p. Int. For. and-Range Exp. Stn., Ogden, Utah. Anderson, H.E., A.P. Brackebusch, R.W. Mutch, and R.C. Rothermel. 1966. Mechanisms of f i r e spread, research progress report No. 2. U.S.D.A. For. Serv. Res. Pap. INT-28, 29 p. Int. For. and Range Exp. Stn., Ogden, Utah. Anderson, H.E. 1969. Heat transfer and f i r e spread. U.S.D.A. For. Serv. Res. Pap. INT-69, 20 p. Int. For. and Range Exp. Stn., Ogden, Utah. Bailey, G.R. 1969. An evaluation of the l i n e - i n t e r s e c t method of assessing logging residue. Can. Dept. Fish. For., For. Prod. Lab. Inf. Rep. VP-X-23. 41 p. Bailey, G.R. 1970. A si m p l i f i e d method of sampling logging residue. For. Chron. 46(4): 288-294. Beaufait, W.R. 1965. Characteristics of back f i r e s and headfires i n a pine needle fuel bed. U.S.D.A. For. Serv. Res. Note INT-39. Int. For. and Range Exp. Stn., Ogden, Utah. Beaufait, W.R., M.A. Marsden, and R.A. Norum. 1974. Inventory of slash fuels with 3P subsampling. Gen. Tech. Rep. INT-13, 17 p. Int. For. and Range Exp. Stn., Ogden, Utah. Beaufait, W.R. and C.E. Hardy. The M i l l e r Creek-Newman Ridge prescribed f i r e study: f i r e research summary. Manuscript i n preparation, U.S.D.A. For. Serv. Int. For. and Range Exp. Stn., Ogden, Utah. 10. Blanchard, R.K. 1938. Progress report on fu e l studies. Unpublished report, U.S. Forest Serv., C a l i f . Forest and Range Exp. Stn., 20 p. 11. Bormann, F.H. 1953. The s t a t i s t i c a l e f f i c i e n c y of sample plot size and shape i n forest ecology. Ecol. 34: 474-487. 12. Bradley, D.P. 1972. Improve forest inventory with access data -measure transport distance and cost to market. (U.S.D.A. For.. Serv. Res. Pap. NC-82). North Cent. For. Exp. Stn., St. Paul, Minn. 21 -p., i l l u s . 13. Brown, J.K. 1966. Forest f l o o r fuels i n red and jack;;pine stands. U.S.D.A. For. Serv. Res. Note NC-9. North Cent. For. Exp. Stn., St. Paul, Minn. 14. Brown, J.K. 1970. Ratios of surface area to volume for common fine fuels. For. S c i . 16(1): 101-105. 15. Brown, J.K. 1970. V e r t i c a l d i s t r i b u t i o n of f u e l i n spruce-fir logging slash. U.S.D.A. For. Serv. Res. Pap. INT-81, 9 p. Int. For. and Range Exp. Stn., Ogden, Utah. 16. Brown, J.K. 1971. A planar intersect method for sampling f u e l volume and surface area. For. S c i . 17(1): 96-102. 17. Brown, J.K. Calculation of downed woody fuel quantities using the planar intersect method. Manuscript i n preparation, U.S.D.A. For. Serv. Int. For. and Range Exp. Stn., Ogden, Utah. 18. Brown, J.K. Inventory of downed woody material and duff. Manuscript i n preparation, U.S.D.A. For. Serv. Int. For. and Range Exp. Stn., Ogden, Utah. -94 -19. Brown, J.K. and P.J. Roussopoulos. Sampling small f u e l volumes using the planar intersect method. Manuscript i n preparation, U.S.D.A. For. Serv. Int. For. and Range Exp. Stn., Ogden, Utah. 20. Byram, G.M., H.B. Clements, E.R. E l l i o t t and P.M. George. 1964. An experimental study of model f i r e s . U.S.D.A. For. Serv. Tech. Rep. No. 3, 36 p. Southeastern For. Exp. Stn. 21. Canadian Forestry Service. 1970. Canadian forest f i r e weather index. Can. For. Serv., Dept. Fisheries and For., Ottawa. 22. Canadian Forestry Service. 1970. Forest f i r e behaviour system. P a c i f i c Forest Research Centre, V i c t o r i a , B.C. 23. Canfield, R.H. 1941. Application of the l i n e interception method i n sampling range vegetation. J . Forest. 39: 388-394. 24. Cartan, H. 1963. Elementary theory of analytic functions of one, or several complex variables. Addison-Wesley Publishing Co., Inc., Reading, Mass. 226 p. 25. Chandler, C.C. 1955. The c l a s s i f i c a t i o n of forest fuels for wildland f i r e control purposes. Unpublished Master's thesis, Univ. of C a l i f . Berkeley, 91 p., i l l u s . 26. Curry, J.R. and W.L. Fons. 1940. Forest f i r e behaviour studies. Mech. Eng. 62(3): 219-225. 27. Davis, K.P. 1959. Forest f i r e control and use. McGraw-Hill Book Co., Inc., New York,. 584 p., i l l u s . 28. Deeming, J.E., J.W. Lancaster, M.A. Fosberg, R.W. Furman and M.J. Schroeder. 1972. The National Fire-Danger Rating System. U.S.D.A. For. Serv. Res. Pap. RM-84, 165 p. Rocky Mt. For. and Range Exp. Stn., Fort C o l l i n s , Colo. -95 -29. De Vries, P.G. 1973a. A general theory on l i n e intersect sampling with application to logging residue inventory. Mededelingen Landbouwhogeschool Wageningen 73-11: 23 p. Nederland. 30. De Vries, P.G. 1973b. Line intersect sampling over populations of a r b i t r a r i l y shaped elements. MededelingenLandbouwhegeschool Wageningen 73-19: 21 p. Nederland. 31. Draper, N.R. and H. Smith. 1966. Applied regression analysis. John Wiley and Sons, Inc., New York. 407 p. 32. Ehrenfeld, S. and S.B. Littauer. 1964. Introduction to s t a t i s t i c a l method. McGraw-Hill Book Co., Inc., New York. 533 p. 33. Emmons, H.W. 1965. Fundamental problems of the free burning f i r e . 10th Symp. (Int.) on Combustion, The Combustion Inst., Pittsburgh, pp. 951-964. 34. Fahnestock, G.R. 1960. Logging slash flammability. U.S.D.A. For. 0 Serv. Res. Pap. No. 58, 67 p. Int. For. and Range Exp. Stn., Ogden, Utah. 35. Fahnestock, G.R. and J.H. D i e t r i c h . 1962. Logging slash flammability after f i v e years. U.S.D.A. For. Serv. Res. Pap. 70, 15 p. Intermt. For. and Range Exp. Stn., Ogden, Utah. 36. Fons, W.L. 1946. Analysis of f i r e spread i n l i g h t forest fuels. J . Agr. Res. 72: 93-121, i l l u s . 37. Fons, W.L. 1950. Heating and i g n i t i o n of small wood cylinders. Ind. and Eng. Chem. 42: 2130-2133, i l l u s . 38. Frandsen, W.H. 1973. Effect i v e heating of fu e l ahead of spreading f i r e . U.S.D.A. For. Res. Pap. INT-140, 16 p., i l l u s . Intermt. Forest and Range Exp. Stn., Ogden, Utah. 39. Freese, F. 1962. Elementary forest sampling. U.S.D.A. Agr. Handbook No. 232. 91 p. 40. Freund, J.E. and F.J. Williams. 1958. Modern business s t a t i s t i c s . Prentice-Hall, Inc., Englewood C l i f f s , N.J. 539 p. 41. Gibbons, J.D. 1971. Nonparametric s t a t i s t i c a l inference. McGraw-H i l l Book Co., New York. 306 p. 42. Glass, B. 1972. An elementary introduction to dynamic programming -a state equation approach. A l l y n and Bacon, Inc., Boston. 402 p. 43. Goodall, D.W. 1952. Quantitative aspects of plant d i s t r i b u t i o n . B i o l . Rev. 27: 194-245. 44. Hansen, M.H., W.N. Hurwitz, and W.G. Madow. 1953. Sample survey methods and theory. Vol. 1. Methods and applications. John Wiley and Sons, New York. 638 p., i l l u s . 45. Hawley, L.F. 1926. Theoretical considerations regarding factors which influence forest f i r e s . J. Forest. 24: 756-763. 46. Howard, J.O. and F.R. Ward. 1972. Measurement of logging residue -alternative applications of the l i n e intersect method. U.S.D.A. For. Serv. Res. Note PNW-183, 8 p. Pac. Northwest For. and Range Exp. Stn., Portland, Oreg. 47. Husch, B., C.I. M i l l e r and T.W. Beers. 1972. Forest mensuration. Ed i t i o n 2. The Ronald Press Co., New York. 410 p. 48. Klor, G.J. 1969. An approach to general systems theory. Van Nostrand Reinhold Co., New York. 323 p. 49. Kurucz, J. 1969. Component weights of Douglas-fir, western hemlock, and western red cedar biomass for simulation of amount and d i s t r i b u t i o n of forest fuels. Univ. of B.C., Fac. of For., M.F. thesis (mimeo). 116 p. - 97 -50. Lang, S. 1968. Analysis I. Addison-Wesley Publ. Co., Inc., Reading, Mass. 460 p. 51. Loetsch, F. and K.A. Haller. 1964. Forest Inventory. Vol. 1. Munich - BLV Verlagsgesellschaft. 52. Matthews, D.N. 1937. Small-plot method of rating forest fuels. J. Forest. 35: 929-931. 53. Maxwell, A.E. 1961. Analysing q u a l i t a t i v e data. Methuen and Co. Ltd., London. 163 /p. 54. McMillan, C. and R.F. Gonzalez. 1965. Systems Analysis. A computer approach to decision models. Richard D. Irwin Inc., Homewood, 111. 336 tP« 55. Mize,J.H. and J.G. Cox. 1968. Essentials of simulation. Prentice-H a l l , Inc., Englewood C l i f f s , N.J. 234 :/p. 56. Morris, M.G. 1973. Estimating understory plant cover with rated microplots. USDA For. Serv. Res. Pap. RM-104, 12 p. Rocky Mt. For. and Range Exp. Stn., Fort C o l l i n s , Colo. 57. Osborne, J.G. 1942. Sampling errors of systematic and random surveys of cover type areas. Jour. Amer. Stat. Assn. 37: 256-76. 58. Rothermel, R.C. and H.E. Anderson. 1966. Fir e spread characteristics determined i n the laboratory. U.S.D.A. For. Serv. Res. Pap. INT-30, 34 ;'p. Int. For. and Range Exp. Stn., Ogden, Utah. 59. Rothermel, R.C. 1972. A mathematical model for predicting f i r e spread i n wildland fuels. U.S.D.A. For. Serv. Res. Pap. INT-115, 40 ;.;p. Int. For. and Range Exp. Stn., Ogden, Utah. - 9 8 -60. Roussopoulos, P.J. and Von J. Johnson. 1973. Estimating slash f u e l loading for several Lake States tree species. (U.S.D.A. For. Serv. Res. Pap. NC-88). North Cent. For. Exp. Stn., St. Paul, Minn. 8p., i l l u s . 61. Sayn-Wittgenstein, L. 1960. Recognition of tree species on a i r photographs by crown characteristics. Can. Dept. of For., For. Res. Div., Technical Note No. 5. 62. Scheid, F. 1968. Schauta's outline of theory and problems of numerical analysis. McGraw-Hill Book Co., Inc., New York. 422 ;;ip. 63. Segebaden, G. von. 1964. Studies of cross-country transport distances and road net extension. Studia Forestalia Suecica 18: 70 -p. i l l u s . 64. Segebaden, G. von. 1969. Studies on the a c c e s s i b i l i t y of forest and forest land i n Sweden. Studia Forestalia Suecica'76: 76 ,fp. i l l u s . 65. Smith, J. 1968. Computer simulation models. Hafner Publ. Co., New York. 112 vp. 66. Smith. J.H.G. 1970. Weight, size and persistance of needles of Douglas-^fir, western hemlock, western red cedar, and other B.C. conifers. U.B.C. Fac. of For. (mimeo) 6 ;<>p. and tables. 67. Smith, J.H.G. 1970. Development of sampling and simulation methods for description of forest fuels. F i n a l rep. on Can. Dept. Fish, and For., Can. For. Serv. EMR Proj. F-53, Univ. of B.C. Fac. of For., Vancouver, B.C. 68. Snedecor, G.W. 1956. S t a t i s t i c a l methods applied to experiments i n agriculture and biology. Edition 5. Iowa State C o l l . Press, Ames. 534 /p., i l l u s . 69. Spiegel, M.R. 1963. Schaum's.outline of theory and problems of^ advanced calculus. Schaum Publ-Co., New York. 384 ^p. 70. Spiegel, M.R. 1969. Schaum's outline of theory and problems of re a l variables - Lebesgue measure and integration with applications to Fourier series. McGraw-Hill Book Co., New York. 194 p. 71. Steel, R.G.D. and J.H. Torrie. 1960. P r i n c i p l e s and procedures of s t a t i s t i c s with special reference to the b i o l o g i c a l sciences. McGraw-Hill Book Co., Inc., New York. 481 t»p.» i l l u s . 72. Steele, R.W. and W.R. Beaufait. 1969. Spring and autumn broadcast burning of i n t e r i o r Douglas-fir slash. Mon. Forestand Conserv. Exp. Sta. B u l l . No. 36, 12 pp. 73. Steward, F.R. 1971. A mechanistic f i r e spread model. Science and Technol. 4(4): 177-86. 74. Stockstad, D.S. 1967. Ignition properties of fine forest f u e l s : A problem analysis. Unpublished report, U.S. Forest Serv., Int. For. and Range Exp. Stn. Northern Forest F i r e Laboratory, Missoula, Mont. 17 !p. 75. Storey, T.G., W.L. Fons and F.M. Sauer. 1955. Crown characteristics of several coniferous tree species. U.S. Forest Serv., Div. Fire Res., Interim Tech. Rep. AFSWP-416, 95 p., i l l u s . 76. Tate, M.W. 1957. Nonparametric and shortcut s t a t i s t i c s . Interstate Printers and Publishers, Inc., Danville, 111. 171 ,p. 77. Taylor, A.E. 1959. Calculus with analytic geometry. Prentice-Hall, Inc., Englewood C l i f f s , N.J. 762 /'p. 78. Taylor, A.E. 1965. General theory of functions and integration. B l a i s d e l l Publ. Co., New York. 437 ^p. - LOOT -79. Thomas, G.B. 1951. Calculus and analytic geometry. Edition.3. Addison-Wesley Publ. Co., Inc., Reading, Mass. 1010 ;p. 80. Van Wagner, CE. 1968. The l i n e intersect method i n forest f u e l sampling. For. S c i . 14(1): 20-26. 81. Van Wagner, CE. 1970. New developments i n forest f i r e danger rating. Can. Dept. Fish, and For. Can. For. Serv., Infor. Rep. PS-X-19. 82. Van Wagner, CE. 1972. Heat of combustion, heat y i e l d , and f i r e behaviour. Can. Dept. Fish, and For., Can. For. Serv. Info. Rep. PS-X035. 83. Van Wagner, CE. 1973. Rough prediction of f i r e spread rates by fuel type. Can. Dept. Fish, and For., Can. For. Serv. Info. Rep. PS-X-42. 84. Wagener, W.W. and H.R. Offord. 1972. Logging slash: i t s breakdown and decay of two forests i n northern C a l i f o r n i a . U.S.D.A. For. Serv. Res. Pap. PSW-83, 11 j'p. Pac. Southwest For. and Range Exp. Stn., Berkeley, C a l i f . 85. Wald, A. 1947. Sequential analysis. John Wiley and Sons, Inc., New York. 212 ,/p. 86. Walker, J.D. 1971. A simple subjective f u e l c l a s s i f i c a t i o n system for Ontario. Can. Dept. Fish and For., Can. For. Serv. Info. Rep. O-X-155. 87. Warren, W.G. and P.F. Olsen. 1964. A l i n e intersect technique for assessing logging waste. For. S c i . 10(3): 267-276. 88. Widder, D.V. 1957. Advanced calculus-second edition. Vol. 1. Edition 2. Prentice-Hall, Inc., Englewood C l i f f s , N.J. 520 j p . - 101 -89. Wilson, W.J. 1959. Analysis of the s p a t i a l d i s t r i b u t i o n of foliage by two-dimensional point quadrats. New Phytol. 58: 92-101, i l l u s . - 1.02"*-APPENDIX I Problem: To prove that the volume obtained by rotating an e l l i p s e at 9j r R R l through a small arc |8o , J , O^f©]^.^* i S 8 i v e n D y (5) • Proof: Fix 6 1 i n je^ jQ^J* L e t t n e e l l i p s e be denoted by E l . Let the center-point of E l by (S, Yo) and l e t the l e f t and right hand end-points of E l be vXl and X2 respectively. F i n a l l y l e t d, j , and $• be defined as i n (5) and assume that ?f>0. E l l i e s on a rectangle R (cross-section of some r i g h t - c i r c u l a r cylinder) inscribed within the population of concern. Let the base of R define an X + axis and the other-edge of R intersecting the centre of the c i r c u l a r base of the sampling unit define a Y + axis. i^ R R R Let the major axis of E l be incl i n e d at an angle -V, "-^< ^<-^ , from the X+; axis. From simple geometric considerations i t can be seen that the e l l i p s e (E2) defined by: .2 2 (X-S) (Y-Yor _ , W2J2 jTiigp ^ 1 intersects E l at (S, Yo4 ). This gives one point on E l . Now 2 consider the e l l i p s e (E3) defined by: .2 „ .2 ( X - S ) " , (Y - Yo)^ . , -r • . - . . ( d/ 2)2 + idcBfS,\7 = 1 a n d the l i n e defined by m Y = (X-S) cot <j> + Yo where <f> = s i n x(X2-Xl) , 0<.<jr/2 From elementary calculus the length of the l i n e Y = X cot 4> which l i e s to the right of (0,0) and which i s enclosed by E3 i s Now l e t (.(Xo+X^ ) = X^,Yo), X_.>6, be a point on E l . Then from geometric considerations, i t i s seen that a second point on E l i s : Xo + dcsc({> > Y o 2ih.+cotz*sin2Y These two points lead to the equation of E l i n the (X,Y) system. To see •R t h i s , rotate the X and Y axes through ^ i n either a clockwise or counter-clockwise d i r e c t i o n , whichever i s appropriate to obtain a new (X',Y') system. The equation of E l i n the (X',Y') system then becomes: (X'-S') 2 + (Y'-Yo') 2 = 1, where ( a / 2 ) 2 ( d/2> 2 (*) X' = Y/sin.cc+ Xcos a and Y' = Y.c°s q c _ X s i n * Inserting the two points on E l into (*) y i e l d s : a = dcsc^secY (**) ) >0 i f rotation i s counterclockwise and cos 00 = cote}) , where sin«c "^csc2())sec^Y.-l <0 i f rotation i s clockwise Note that these l a s t two equations also hold i f Y= 0. Consider a small v e r t i c a l s t r i p of the area enclosed by E l . Let this s t r i p have width AX and height t(X). The area of t h i s s t r i p i s approximately t(X)"AX. When this s t r i p R R R i s rotated about the X axis through a small arc of length X(0„ -0„ ) = X'A0 , the resultant volume generated i s approximately t(X).AX«X.A0 . Using the continuity of the height difference curve (maximum height less minimum height for each X) defined by E l , i t follows that the volume generated by rotating E l about the X axis through a small arc 0 R 0 R _0 '2 _ R of measure A0 ::is: - 104 -X2 A0 J X-t(X)dX XI From (*) and the quadratic formula, i t can be shown that: X2 X2 J X-t(X)dX = ^ j xV XI XI B 2 - 4AC dX, where A = 4 csc2(j> 2 2 B = 8(X-S) (1 - esc (fi-sec y) cos a • sin 1* 2 2 2 2 2 2 9 C = 4(X-S) (1 + cot (jisin y) sec y - d esc <J> sec y S i m p l i f i c a t i o n using (**) results i n +/2 +/~2 2 2 /B -4AC = 4 csctf)secY / d esc (j) - 4(X-S) •Therefore X2 X2 \ X . t ( x )dx--2££I f x y _/ csctp J XI XI X2-S d 2csc 2 c|)-4(X-S) 2 dX Jj (u + s)^d 2csc 2<J) - 4u 2 du, where u = X-s, s=S CSCfJ) Xl-S X2-S - 4 secy \ (u + s) + / l - H!i2_EL_L d u Xl-S (d/2) 2 ~o - d secy j {^osl + s } | s i n g [ { - d c s c | s i n g J d B > where 3 = cos" 1 f^ H-fiSL.}, 0 <j3 < rr u d = d esc -fr secy j {dcsc|cosB + s } ^ 2 ^ f 0 . d csc^secy j d e s c i j + g j s i n 2 3 d 6 2 o o 105. -TT 2 v 2 = d •csc<j),secY'S \ sin 8df3, since 1st int e g r a l i s 0. = d «s«sec J. »csc(j) 4 ^8 - sin'2gj 8=.TT B-0 TT 9 = — d •s,secY,cscd> 4 Therefore the volume generated by rotating E l about the X axis through R R~ a small arc Jol v2 R R R of measure (0„ - 0 ) = A8 i s : 2 o "2pd 2 'S'secY' csc<j>.A0R. r R R~I Since 0^ ^ was arbitrary i n [ Q * ^ » (5.) i s established. QED. - 106 -APPENDIX I I The computerized analysis of variance model (ANOVA 1) offered i n this thesis i s designed for a PDP-11 computer. It i s primarily a simultaneous runs one-way c l a s s i f i c a t i o n random components model (Ehrenfeld and L i t t a u e r , 1964; p.391-399) with unequal numbers of observations i n the c e l l s . This program both generates the values of the response variable of interest and also does the standard analysis of variance computations. It also examines both the assumption that the terms comprising the T matrix (see ANOVA 1) are normally distributed with a common 0 mean and common variance and the assumption that the terms comprising the E matrix (see ANOVA 1) are normally distributed with a common 0 mean and common variance. Both these assumptions are required to be v a l i d i n order to correctly apply the standard parametric hypothesis test for homogeneity of 0 variance (Ehrenfeld and Littauer, 1964; p.396) with respect to the terms comprising the above mentioned T matrix. This program was devised f i r s t to determine the conditions under which i t was feasible to replace: (t) m J 1 . J 2 (t) * 0, where: t refers to transect location i n 0, 2TT • j ^ refers to p a r t i c l e d i s t r i b u t i o n j 9 refers to p a r t i c l e loading i refers to diameter size class (see (5.) and (6.)) by a constant K; independent of j , , j 0 and t such - 107 -that not only the i n t e g r a l but also the shape of g2^ (see (10.)) over p 0 , 2TV J would be approximately the same as the i n t e g r a l and shape respectively of g2^ (see (9.)) over [O,2'TVJ. Having tent a t i v e l y established these conditions, the program was then adjusted to evaluate <K , for i {1,25,3} (see 0-0.)). ANOVA 1 was necessary because for many downed woody p a r t i c l e s i t i s not feasible to measure either distance to transect/particle intersection or angle of transect/particle intersection as required by (9.). Determination of the conditions under which i t was possible to replace S ( i , j ^ , j 2 , t ) by was made by setting up ANOVA 1 for a two-way c l a s s i f i c a t i o n run (Ehrenfeld and'Littauer, 1964; p.432-434). The two influencing factors were f u e l p a r t i c l e loading and d i s t r i b u t i o n . A wide range of lesser f u e l p a r t i c l e loadings (0.75 particles/sq.ft-50.00 p a r t i c l e s / sq.ft.) and distrib u t i o n s was tested here under different f u e l p a r t i c l e length d i s t r i b u t i o n s . Let S ( i , jj^,J2) be the mean value of S taken over a l l te [O,2TT], i , j ^ , and j 2 fixed. Then an interesting result of this analysis was that S(i,j^,J2) was found to be reasonably independent ( i e . within s t a t i s t i c a l l i m i t s ) of J2« No such independence could be found for j i . This was found true V±e3[l,2,3^, Vj 1»j2 t e s t e d » and for every fuel p a r t i c l e length d i s t r i b u t i o n tested. Hence due to time constraints i t was decided to determine K\^\ ie-,{l,2,3'} for only the most common lesser fu e l p a r t i c l e s d i s t r i b u t i o n , namely randomly distributed f u e l p a r t i c l e s with respect to the CPIS units. Before proceeding to the one-way c l a s s i f i c a t i o n ANOVA 1 set-up, three relevant points should be b r i e f l y discussed. The f i r s t one relates to the fu e l p a r t i c l e d i s t r i b u t i o n s which were used as levels of the second influencing factor i n the two-way c l a s s i f i c a t i o n ANOVA set-up. A m u l t i p l i c i t y -108 -of angular orientations and p a r t i c l e placements was used here. For example, par t i c l e s were randomly oriented and randomly placed, randomly oriented and placed i n clumps or st r a t a , u n i d i r e c t i o n a l l y oriented and randomly placed, and u n i d i r e c t i o n a l l y oriented and placed i n clumps of i n t e n s i f i e d f u e l accumulations. The second point which merits consideration i s the choice of f u e l p a r t i c l e length d i s t r i b u t i o n s . Three dis t r i b u t i o n s were tested here, each applied using a variety of length ranges. The f i r s t d i s t r i b u t i o n was generated using a random number generator based on the Lehmer M u l t i p l i c a t i v e Congruential Method (Mize:-. and Cox, 1968; p.68) where the general formula used i s : _ n , . This formula i s embedded i n a l i b r a r y function subprogram (RAN) on the 16 32 PDP-11. RAN sets x equal to 2 +3 and m equal to 2 ; _,• i s an a r b i -t r a r i l y chosen odd number. RAN has the unfortunate feature that the actual order of the cycle i t generates i s not ea s i l y determined. However RAN has 20 been demonstrated on the PDP-11 to have actual orders of at least 2 for .A several choices of U q . Only these choiges were used i n generating the random number cycles for both ANOVA models. The maximum ranges used here for the f i r s t , second and t h i r d subsets were (V'-2'), ( l " - 3 ' ) , and (6"-8') respectively. Both the second and t h i r d length d i s t r i b u t i o n s were generated using skewed versions of RAN. The second length d i s t r i b u t i o n simply a l l o t t e d one-half of the p a r t i c l e s to the f i r s t one-third of each range and the second length d i s t r i b u t i o n a l l o t t e d one-half of the p a r t i c l e s to the f i r s t two-thirds of each range. These length d i s t r i b u t i o n s were subjec-t i v e l y chosen to provide dramatically different p a r t i c l e length di s t r i b u t i o n s i n the modelling procedures. The t h i r d point which should be mentioned here - .109 -i s that there i s a general purpose analysis of variance program designed for the PDP-11 cal l e d ANOVA which can accommodate up to 5 influencing factors, one hundred response variables, 50 transformation cards and 400 observation c e l l s . Use of ANOVA would eliminate the need for a portion of the ANOVA 1 model. ANOVA i s d e f i n i t e l y a viable approach here but was not used because the author had written a large portion of ANOVA 1 prior to commencement of the thesis. The one-way c l a s s i f i c a t i o n ANOVA 1 set-up derived , Vie{l,2,3} for randomly distributed p a r t i c l e s . Numerous runs were made here to test the v a l i d i t y of the two underlying assumptions (stated at the beginning of APPENDIX Tl) required for application of s t a t i s t i c a l parametric hypothesis 2 tests. Using X goodness-of-fit tests at the .05 significance l e v e l , both assumptions were found to be correct. The, analysis performed by ANOVA 1 for the one-way c l a s s i f i c a t i o n revealed that the random variable defined by ^ , i and j fixed (see (12.)) was normally distributed with mean 0 (using a X"2 goodness-of-f i t test at the .05 significance l e v e l ) , V i , j tested. A secondary result of the analysis was that i f j i s ordered by loading for fixed i , the variance of generally decreased with increasing j for a l l i . These results may be interpreted to mean that with respect (i ) to randomly distributed p a r t i c l e s , not only the int e g r a l of g2. (see (10.)) * (i ) r 1 but also the shape of g2. over J_0, 2TTJ i s approximately the same as that of the in t e g r a l and shape respectively of ^ 2 j ^ ^ (see (9.)) over (jO,27TJ Vj t es t ed and V i e {1,2,3j. where: K ( 1 ) =2.83 X10"" 2(feet) - 1 K ( 2 ) = 1.49 .X10~ 2(feet) - 1 K ( 3 ) = 8.56 .^X10" 3(feet)" 1 A HO' -A further interesting result of ANOVA 1 for the one-way c l a s s i f i c a t i o n case i s that the random variable defined by m ^ » 1 a n <l j fixed (see (6.)) has a coeffic i e n t of var i a t i o n reasonably independent of i and j , V ie{l,2,3}, Vj tested. This can be interpreted to mean that on a percentage basis the *> ( i ) r I v a r i a b i l i t y of g2^ ' over [0,2TrJis reasonably independent of i and j . Combining this observation with previous remarks, i t follows that one numerical integration technique applied at one l e v e l of sampling intensity w i l l probably produce the same degree of accuracy for the i n t e g r a l of g 2 . j o v e r [o,2rr], V Le!{'l,2,3J, Vj as before, F i n a l l y i t i s observed that the one-way c l a s s i f i c a t i o n ANOVA 1 set-up computes the non-parametric Spearman's rank-difference c o e f f i c i e n t (Tate and Clelland, 1957; p.13) with respect to distance to p a r t i c l e central axis intersection and cosecant of p a r t i c l e central axis intersection. Results of this investigation showed that i n 90% of a l l CPIS units with randomly distributed p a r t i c l e s , the random variable represented by distance to p a r t i c l e central axis intersection and the random variable represented by the cosecant of p a r t i c l e central axis intersection were uncorrelated at the .05 significance l e v e l . The one-way c l a s s i f i c a t i o n ANOVA 1 set-up presented below i s very simple and straight-forward. I t i s supplemented with many comment statements explaining important procedures and definint important matrices to assist i n the reader's understanding of what has been done. r - I l l -APPENDIX I I I A ( i ) From APPENDIX I I i t i s now possible to express g2j (t) as follows: gM^Ct) = A C ± )m ( 1> (t) , : V i e t l , . . . , * * ! VjeU n.}, Vte[o,2Trj where A ^ can be e x p l i c i t l y determined V i e f l MvL 2 T T " ,. Since X2 ( i )(S2< i )/£ C g ^ (t) d t , Vie{l,...,M} J •* 3 0 u • Vjetl n} (see (8.)) i t i s clear that: 2TT (28.) X 2 ( i ) ( S 2 a ) ) - A ( i ) j m i ( i ) ( t ) d t 3 J where i s referred to as the p a r t i c l e intercept counting function. Recall that X 2 ^ ( S 2 ^ ) refers to a t o t a l p a r t i c l e volume per unit area. Since the images of the p a r t i c l e intercept counting function cannot be eas i l y expressed i n terms of t mathematically, numerical techniques must be used to estimate the right-hand side of (28.). The main purpose of the downed woody f u e l model i s then to determine constants C ^ , P ^ and such to ' to x that: 2Tf Q ( i ) ( 2 9 . ) A ^ ] m ^ C O d t * A ( 1 ) Z C ^ . m ^ V 1 * ) Due to time constraints , P ^ and are determined only for randomly distributed f u e l p a r t i c l e s with respect to ie{l,2,3'} as previously defined. In order to f i n d G ( l ), P ^ 1 * and o/1^, V ie{l,2,3f, three different integration techniques were proposed. Each was examined at f i v e different levels of sampling i n t e n s i t y , namely 6, 8, 10, 12 and 14 transects. This range was chosen because preliminary investigations with randomly distributed matchsticks suggested that the desired l e v e l of sampling intensity lay somewhere between 6 and 14 transects. The three techniques tested were: Gaussian-Legendre • (Scheid, 1968; p.126), Simpson's rule (Scheid, 1968; p.108) and the average function value method (Thomas, 1951; p.257). Thus i n a l l 15 techniques were examined for each p a r t i c l e diameter size class. Two c r i t e r i a were used as a basis for selection of the optimum integration formulas. The f i r s t c r i t e r i o n stipulated that the integration formula of interest was required to y i e l d a volume estimate within 10% of the corresponding true volume for each of at least 90% of the c i r c u l a r sampling units simulated. The second c r i t e r i o n stated that the integra-tion formula of interest must y i e l d a volume estimate within 15% of the corresponding true volume for each of at least 97.5%, of the c i r c u l a r sampling units simulated. Obviously i f more than one of these integration formulas s a t i s f i e d these c r i t e r i a for a par t i c u l a r diameter size class, selection was based on the le v e l of sampling intensity required and ease i n application and understanding. After conducting several runs of the downed woody fuels model for each diameter size class, the technique chosen using the above c r i t e r i a was Simpson's rule with 12 transects. I t i s important to r e a l i z e that t h i s formula worked best for a l l p a r t i c l e diameter size classes tested. Hence i t was found that: 4, i f w i s odd j 2, i f w i s even (i) _ t + TTW, for arbitrary t a) 6 , Vie {1,2,3} Q ( i ) = 12 Hence i t may now be asserted that: 12 (30) X 2 ( l ) ( S 2 ^ x ) ) ^ A ( i ) Y c w m - 1 ^ + ___)» t arbitrary i n [ 0 , 2 7 7 ] ^ Vie [ 1 , 2 . 3 } Vje j l , . . . , n ^ It- i s important to r e a l i z e that t i s arbitrary i n jo,27r]. The reason for t being allowed to be arbitrary i n [ 0 , 2TT] w i l l now be examined i n more d e t a i l . For fixed i and j as above, l e t h'2^ be a random 3 variable defined by the following 12 (D ,^ _ ^ n Tod) h2 J (t) = y C u ) g 2 j ( t + Vte [0,2^, where = 01=1 4, i f w i s odd 2, i f w i s even Furthermore for fixed i and j6as previously specified l e t V^ *^  be the true t o t a l p a r t i c l e s volume i n the j t h sampling unit taken with respect to the i t h p a r t i c l e diameter size class. Then the random variable defined by: ( V J ^ M r ^ ) 2 - h 2 ^ l ) ) / ( v j l ) / T r ( r ( l ) ) 2 ) , i and j fi x e d , was found to be normally distributed with mean 0 and small standard deviation (usually under 0.05) reasonably independent of i and j , V i and j , as previously specified. This means that (30.) holds independent of the choice of t . Although this result may seem obvious, i t i s nevertheless an important one to confirm, since i t s i g n i f i c a n t l y simplified the application of the common point intersect concept. A t y p i c a l run of the downed woody fuels model designed for a PDP-11 computer i s presented below. This i s only one run with one choice of three tested p a r t i c l e length d i s t r i b u t i o n s and one choice of many tested p a r t i c l e frequency ranges (see Appendix ' I I). As previously indicated, t h i s run i s set up for randomly distributed p a r t i c l e s . However i t can easily be altered to accommodate non-randomly distributed p a r t i c l e s . The downed woody f u e l model i s almost self-explanatory. Numerous comment statement! have been inserted to assist the reader i n his understanding of the compu-tations and procedures involved. $JOB FUEL [131,1] ) 1 ••' DATE >-20-MAR»75 TIMES-13I02UB SDEL *,* PIP V10»82 #*,»/DE SEXECUTE 5 ~1 FORTRAN V06.13 »SY>FUEL i L P KBII/QN/CK/CO '10. 9 FORTRAN VBt., 13 1 1 3 1 0 3 1 0 3 20..MAR-7J5 R A G E — : 1 I H I S _ P R06R.AJM_I.5_P_ R.I.M.A R.ILY A.. ONE-W.A_Y_C.L A_S.SJF_I.CA.T_I.ON-_ AN 0 1 A J A N B.0J-C C O M P O N E N T S MODEL S E T UP FOR S I M U L T A N E O U S RUNS, T H I S MEANS T H A T C WITH QNLY._S_-lG_lt-J_LX^^^ BE U S E D _ A S _ A TWO-WAY. I C C L A S S I F I C A T I O N ANOVA MODEL. !f_ 0 0 0 1 8.YJ.E. _T_I.tlEJU 8) D Y M E N S I O N T L 0 C T S ( 2 4 ) , N U M ( 5 ) , T L Q C T C 2 4 ) , C 0 S L C 2 4 ) , S I N L ( 2 4 ) , T A N L ( 2 4 ) , 0 0 0 2 J 1CQNLC24._,.CQN2-(2M,.RDC.QEF_X5,A).^ 2 4 ) , Y I J D 0 T C 5 , 8 ) , Y I D D 0 T C 5 ) , Y D J D 0 T ( 8 ) , T ( 5 , 8 ) , E ( 5 , 8 , 2 4 ) , S V 1 1 J 1 ( 5 , ' 8 ) , S O 3.1.1 J X t . 5 _ , . 6 J _ . _ N . S A ^ ^ ( 8 ) , S S 4 B 1 W J C 8 ) , F M B 1 W J C 8 ) , S S W 1 W J C 8 ) , F M W 1 W J C 8 ) , T S 1 W J 1 C 8 ) , V A L I J K C 5 r 8 , 2 4 ) , V A L _ 5 S T A C S , 8 ) ~ 0 0 0 3 E Q U I V A L E N C E ("SDCOL J , S C O L J , C 0 N 2 ) , ( S S B 1 W J , F M B 1 W J , C 0 N 1 ) , ( S S W 1 W J , F M W 1 W J J . J . L O i I L , J _ S J / J J ^ 0 0 0 4 KA«0 0 0 0 5 _L_& CV a 'COMMON R A D I U S OF C I R C U L A R S A M P L I N G U N I T S , JtZ$h RE_A0(5.,.6.)_CJL 0 0 0 7 b FORMAT C F 5 , 2 ) _£ K 1 L • J R A N 5 E C T_N U MJ3EJL 0008 DO 402 Kll»l,24 0009 Tjjr.T.?tK^n-fKll-n«6.283l86/24. 0010 402 C O N T I N U E C NJJM » NUMBE1 _ O F _ P J J _ _ U J _ ! l l ^ ^ I N THE S A M P L I N G U N I T C O N S I D E R E D . _ 0 0 J 1 R E A D ( 5 , 2 0 0 ) N U M 0 0 1 2 2 0 0 F O R M A T ( 5 1 5 ) • C T L M I N » MINIMU]1_PJLRJ_ICJ-.E_L_J£1 H T L M A X » M A X I M U M P A R T I C L E L E N G T H p S E C B 8 MEAN S E C A N T FOR P A R T I C L E T I L T •= — ffl Pi . 1 V C DbAR » Q U A D R A T I C MEAN D I A M E T E R B K A n f S . P B l . T L M I N . T L M A X i S E C B i D B A R J i YJ Vi I J 0 0 1 4 2 0 1 FORMAT (4F8..6) n c P T N F FTI F 1 fP00,2,U, J V A R ) — — -\ lulu 1 J 001b fflfi. 1 7 D E F I N E F I L E 2 ( 2 0 0 , 2 , U , K V A R ) nPF -TNF FTI E 1 f 960. 1. U. I VAR) — — j WWW 0 0 1 8 I V A R H 111 • F U E L L O A D I N G NUMBER • — — —-D 01 01 1 Q C LOOP OVER F U E L L O A D I N G DO 114 11 l • l . 5 — WU I 7 0 0 2 0 c N F U E L P « N U M ( I 1 1 ) LOOP OVER F U E L D I S T R I B U T I O N — _ 0 0 2 1 ffl fJi 3 3 DO l i b Jll»l,8 ( "A l l T T M F ("TIMERS ' • — D wwcc 0 0 2 3 777 W R I T E ( 6 , 7 7 7 ) T I M E R FORMAT ( 2 5 X , ' T H E T I M E I S ' , 8 A 1 ) — — -\ W W c ** 0 0 2 5 ffl _ t. t i ' DO 6 0 7 K H » i , 2 4 R I I M f K l l l l P I . " *j WW CD 0 0 2 7 010.3 A 607 N ( K U ) * 0 C O N T I N U E — " W icJ c O C T L O C T 0 « L O C A T I O N OF I N I T I A L T R A N S E C T T L O C T 0 « b . 2 8 3 1 6 b » R A N ( K A , K B ) : — n W W c ~ 0 0 3 0 0I01 "< 1 JVARsl K V A R B i ~~ _> in 9 V *i J 4. 0032 IC O U N » 0 T H I S S E T OF N E S T E D D O » S (UP TO 1 1 5 ) F I N O S T H O S E P A R T I C L E S ( I N A — — 8 D 7 h — • • 5 D A _ — F O R T R A N V06.13 13»03'»03 20-MAR«75 PAGE 2 C r C ( 1 c c 1 \ T c c 1 1 0033 0034 0035 0036 0037 0038 0039 0040 0041 004 2 0043 0044 0045 0046 0047 004B 0049 0050 0051 0052 0053 0056 0057 0053 0062 0063 0064 C O S E C A N T C C S C ) O F T H E A N G L E 0 £ _ I J 1 T X R ^ J J ^ J J _ R _ E A C H P A R T I C L E . " E T D T ^ A I R I S _ T H E N - M U L T I P L I E D T O G E T H E R A N O T H E P R O D U C T S A D D E D Ul 0065 0066 C _C_ c 0069 0070 FOR EACH TRANSECT IN THE GIVEN SAMPLING UNlT) IS DIVIDED BY THE NUMBER (N) OF PARTICLES INTERSECTING ITS ClRRESPO^DlJ^A JJL*NSEC_T_, EACH AVERAGE (YIJK) SO OBTAINED C O M P R T S E S 0N E 6 BS E R V A f IQ N FOR THE CELL (SA_MPLING ^UNIT)„BEXNG_C0NSIDEREQ. EACH CELL HAS NSAMP AVERAGES NFUELP • NUM~BER O F FUEL PARTICLES IN CURRENT SAMPLING UNIT, LOOP OVER FUEL PARTICLES___ DO 115 M»1,NFUEL> 11=0 0071 0072 0073 LOOP OVER TRANSECTS DO 115 Kll«l,24_ I F C K l i . E Q . l ) GO TO 130 GO TO 141 131 151 T L O C T C K i n = T L O C T S ( K l l ) * T L O C T 0 C O S L ( K 1 1 ) _ " C P S C T L O C T C K 1_1)) " S I N L C K 1 I.) " S I N C T L O C T (K 11)I ) I . I » I I * 1 I'FCM.NE.l) GO TO 180 CON 1 C11) « C^/^_*CpSLJK_UJ. 180 CON2 (11)o CCV/2,)*SINL £K11) IF CCOSL_0< U )_.EQ. 0,.J_GOJTO_550_ T A N i CK 11 ) s S I N L ( K l 1)/COSL CK11) IF(COST.EQ.0.3 GO Tp_555_ ^ T T f A N T , E O , T A N L t K l D ) G O T O l l S X I N B Y I N T / C T A N L C K I D ' T A N T ) T F 7 7 I N ~ . G E . X M I . N . A N D . X I N . L E , X M A X ) G O T O 565 GO T O 115 550 IF'CCOST . E O . 0 . J GO TO 115 I F C 0 . . G E . X M I N . A N D . 0 . . L E . X M A X ) GO TO 580 115 0054 GO TO _ . 0055 555 YIN » T A N L C K i n*XMIN 580 IFCYIN.GE.YMIN.ANO.YIN.LE.YMAX) GO TO 585 - T r C Y T N ^ E . A M I N l C B . , C 0 N a ( I I ) ) . A N D . V l N 1 . L t . A M A X l C a . i C 0 N 2 t X I ) ) J GO 1TO 906 ; 115 llll .65 I? UM IN|GE.AMINl(0.,CONl(Iin.AND.XMlN.LE.AMAXlC0.,CONl ( i nn ~ ' 1TO 9 0 7 " 0061 GO TO 115 GO 565 rF(XIN.GE.AHINlC0,,CONl(II)).AND.XIN.LE.AMAXlC0..CONiaT))) GO TO 1605 J GO TO l l S 605 YIN»TANT*XIN*YINT_ 906 GO TO 606 YINsYINT 0067 907 XINoXMlN 0068 606 N ( K 1 U »N(K11)*1 952 F l N T ' S Q R T ( X I N * * 2 * Y I N « * 2 ) I C O U N » I C O U N * r „_ T r a W U N T G T . 2 0 0 . O R . J i l , G T , « ) G O T O 952 WRITECl'JVAR3 D I N T  A N G I N T * A B S C T L P C f T K l 1 ) - T H E T A ) I CO 12-- M O . 9 . 0 0 7 4 0075 0076 0077 0078 00J_i-0082 0063 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0097 0098 0099 0104 0105 0106 0107 0 1 1 0 0 1 1 1 0112 0113 0114 0115 _X_X »_A B 5 C 51N (MG.I.N-t)-) AN INTERSECTION IS SAID C c „„ „ TO OCCUR ONLY AXjJ3_JUNJ)_JU__^ ^^  IT IS ASSUMED THAT THE INVERSE SINE OF BOTH THE PARTICLE ISECT _. O F FUEL PARTICLE C E N T R A L L I N E . IT  Assuntu in»i i n t i m t i w . a * i . t « r THE RATIO 01 A M1ER_U_J^ JJ1E_J1HJLJF-U.E.I_^^ EQUAL TO .00451 RADIANS C15.5 MINUTES), _I£ C X X . L T . .m4_5.5L_X.XXJl 0_45_5 CSC»i,/XX .J_U__PJJll_^ 6° TQ 953 WRITE(2 fKVAR) CSC OR 953 SUM (K 1 U°SUMJK1,1)»P!NT_«CSC. 0080 115 .CONTINUE 0081 ICOU NT.» 0_ S I J K O M 0 , D O 9 0 0 K11 B | _ , _ 2 4 I F C N ( K l l ) , E Q , 0 ) GO TO 608 GO TO 609 608 I C P U N T a.ICOUNT*1 V i J K c n i i J i i # K i n « 0 , _feB9__SIJK0K»SIJK0K + YIJK ( 1 1 1 , J l l . K l l ) _ _ 900 CONTINUE NS AMP C 1 1 1 , J 1 1 ) »24-IC0UNT re, i c ~r YTJOOTT"MXTTfX -WHICH CONTAINS THE SAMPLE MEANS FOR THE CELLS. V yjLOJTtll.!» J l D ' S I J K O K / N S A M P C I l l , J i l l GO TO 998 IF C J 1 1 . G T . 4 ) IDsJVAR-1 0095 " WRITE C6, 1002) 0096 1002 FORMAT C' H 1 » J 11» I D •iE NUMBER O F C C S A M P L E _ V A L U E S T A K E N I N T O A C C O U N T I N T H E T C W U T A T I O N O F R D C O E F •/' « , " W I T H R E S P E C T T O L O A O I N G N U M B E R ' , I 5 , ' A N 2 D D I S T R I B U T I O N N U M B E R ' , 1 5 , ! ± ' _ j l l L — - ^ ^ m p - - :  ~ R D C " O E " r ^ ~ S P £ A R M A N ' S R A N K - D I F F E R E N C E C O E F F I C I E N T F R O M H E R E T O S T A T E M E N T N U M B E R 0 1 4 5 . T H E P R O G R A M S I M P L Y C O M P U T E S R D C O E F , J V A R = 1 DO 908 KKK»1,ID READ (l'JVAR) RANKDICKKK) 0100 908 CONTINUE 0101 INDIC°0 ; 0102 626 ITALLY»1 0103 ISCORE°ID  957 INOt"0 I I N a l IFCITALLY.EO.l) ITALES«ITALLY-1 GO TO 1 4 1 7 12108 : \W\i CONTINUE 0 1 0 J D O 9 5 5 ITALsl., ITALES 955 T F T R A N K D I C I " I N ) , E Q , I T A L ) G O T O 1 4 1 1 C O N T I N U E 1411  1417 G O T O 1417 I I N B I I N * ! G O T O 1416 C O N T I N U E GO TO 1418 DO 1 4 1 2 I T A L * 1 » I T A L E S i3;03:03 20-MAR-.75 PAGE I 0119 0120 J_J_21 0122 _0_123_ 0124 0125 0126 0127 0128 0129 0130 JSJJLL 0140 0141 0142 0143 0144 0145 0146 '0147 0152 0153 0154 0155 0T56 0157 "0T5T" 0159 0160 0161 0162 0163 "0T6T" 0165 0~f6~6 0167 1412 1418 CONTINUE _ C O N J J i i y E 610 610 IF(RANKDI(IIN),LE,RANKDI(KKK)) GO TO 610 _IIN = KKK__ __ CONTINUE SVDI NT BR_AN KOI(IIN) RANKOItilN)»ITALLV JP_?_s^_KJ<J<»J.t_IO IF(RANKDI(KKK).NE.SVDINT) GO TO 621 IF (KKK ,EO 11 IN) JS.0 TO 621 RANKDI(KKK)»lfALLY IN O J J L I N 0 T i l 6 ~ 9 L ~~ 8 6 ~~ _' rr -A 0132 621 CONTINUE 0133 IF(INOT.EQ.0) GO TO 956 0134 WRITE(6,951) INOT,SVDINT,INDIC,ITALLY _ j 5 951 FORMAT C '__li__i_UMBE»._PF_tI.E.$._H EM_-__-__-l._e.l A. A, 2J_LL 0136 956 IT ALLY "IT ALL Y + l 0137 IS C 0 R E »ISX OR E_C_LNPT +1) T . 3 8 IFCISCORE.GT .0) GO TO 957 0139 IJilJJlDJC.EQ.l) 6 0 T 0 6 2 3 IND.C»lNDIC*l JVAR»1 KVARel DO 909 KKKo_l.jp  "WRlTECi'JVAR) RANKDICKKK) READ (2 *KVAR) RANKDI (KKK) 909 CONTINUE GO TO 626 0148 623 S D R A N 2 8 0, 0149 J _ V A R " J _ "BTS0 DO 910 KKMlTTD 0151 READ CI' JVAR) RANKCS (KKK) "9T- CWTTWE"'"' 911 998 901 TT6~ DO 911 KKK»1,ID S~D~R AN2aSDRTNT+TfTANK CS ("KW) -R AWDI (KKK) )~**2 CONTINUE , RO'C 0 E"F"CI11T3 ll)»l 1"6,» StfRTN^TTCriW^ DO 901 Kll»ti24 "WITE"(nTATT) N ( K l l ) : CONTINUE ~C"0'NTI'« U'E SIJDOJ«0. HBcTB-C DO 628 Jll»l,8 SIJD0J»SIJDOJ»YIJDOTCIlliJll) "TTJTTTTNW YIDDOT « MATRIX WHICH CONTAINS THE OVERALL SAMPLE MEAN FOR EACH -RW, Y I D D 0 T ( I H ) » S I J D 0 J / 6 , NSSAMPS0 12. . n 9 ~_T6"S" 0169 0170 0171 DO 630 J 1 W , -SYDJDB0, T>~CT~629 1 1 1 * 1 , 5 . SYDJDaSYDJD + Y I J D O T C I U . J U ) _ ^SS'AHP « THE TOTAX~^MB'ER^T^B'STir^TIWSnrFr iO.L CULLS ruK A"UT FORTRAN V 0 6 . 1 3 1.3J.03.JB3 2 0 - M A R i l 5 PJU5.E 5. C T U M I T A M F n i l Q DIIKIR — j, 1 0 1 7 2 C a Jl nUL.l-nTitUU.w—nUW.i-i.* . — N S S A M P « N S S A M P + N S A M P ( 1 1 1 , J l 1 ) rnNTTNUE 5 _ 9 0 1 7 3 6 2 9 C Y D J D O T • MA T R I X WHICH C O N T A I N S T H E O V E R A L L S A M P L E MEAN T A K E N OVER r n n R F S P f i M n T N f i r n I S IN A L L PUNSi — I — — — 8 - 4 - i — O CVI 0 1 7 4 C Y D J 0 0 T C J l l ) « S Y D J D / 5 . r* Aki v r l i ne — ~ e — - 01 1 0 1 7 5 0 1 7 6 6 3 0 l- U N T J !N.V t — — — — SYD D D = 0 , Fi — — " Zl 0 1 7 7 0 1 7 8 00 9 ii l.l 1 » 11 a • • S Y D D D s S Y D D D + Y I D D O T C I 1 1 ) 0 1 7 9 6 3 1 C (*0N"fTNUE ™ ~ ™~ YDDDOT » O V E R A L L S A M P L E MEAN T A K E N OVER A L L C E L L S AND A L L R U N S . w n n n n T B < i v n n n / S — — 0 1 8 0 0 1 8 1 Y U U U U ' " O T U u u / j i __ — — — S O V E R P » 0 , n n L i s i 1 1 s 1 . S — •—• 0 1 8 2 0 1 8 3 U U o pO i.JLi_r_t_i_5 — • — DO 6 3 6 J 1 1 M , 6 c w T i l l f T t l . T M l o P i ~ - ~ 0 1 8 4 0 1 8 5 O V i l J l V * i i l-V-L*-/ _-*Li — ' 99..^ *^ - ^ ^ - . . . ^ i ^ i . P i k . m rr c v t k i r v A D n n C w T A T T D W T A f c f P K J flVFR A L L C E L L S C S O V E R P B O V E R A L L S A M P L f e _ _ S JA N I ^ K I L U J ^ X ^ L A J J J J 1 — L * J \ t j N — W J L * - Z * —n ± ± — y _ * » » i » — — c AND A L L RUNS, c n w F D P o « i f t W F R P * f Y T J k n i i . J l l f K i l 5 - Y DODOT) **2 — 0 1 8 6 c f-S V I 1 J 1 » MATRIX WHICH C O N T A I N S T HE S A M P L E S T A N D A R D D E V I A T I O N FOR EA C H C E L L , 0 1 8 7 S V I l J l C I l U J i n - S V I l J K I l l . J i n ^ C Y I ^ C I l l r J l l i K i n - Y I J D O T C I l l . J l t ) 1 ) * * 2 — •— c k = M A T R I X WHICH C O N T A I N S D I F F E R E N C E S W I T H I N C E L L S FOR A L L R U N S , e n n . H i . K l H s V T . I K f I 1 1 . J U . K i n - Y I J D O T CI 1 1 1 J U J - = 0 1 8 8 0 1 8 9 6 3 5 T ° = T M A T R I X WHICH C O N T A I N S P T P F F B F N C F S B E T W E E N C E L L S FOR A L L R U N S . 0 1 9 0 T C I l l , J l l ) a V I J D O T t I l l , J l l ) - Y D J D O T ( J i l ) < s n y F n P B < . n v F R P - r f f ? ' ) - N S A M P f I l l » J l l ' ) ! l * Y D D D 0 T * * 2 ) — ! 0 1 9 1 0 1 9 2 S V I l J r ( I l l , J l l ) « S V I l J l ( I l l . J i l ) - C C 2 4 - N S A M P C I l l . J l l ) ) * Y I J D 0 T ( I l l , i J l l ) * * 2 ) . „ — . —-0 1 9 3 t u i S D I 1 J 1 C I 1 1 , J 1 1 ) « S Q R T C S V I 1 J 1 C I i l , J i i ) / C N S A M P ( I l i , J l l ) - l » • nnklTTKIIIC —— • 0 1 9 4 0 1 9 5 0 50 L U IN | X IN UC . — . — S S D F E = 0, ^ ^ n F T s P I \ 'I 0 1 9 6 0 1 9 7 OOUr 1 *• V m . — —— DO 8 111 = 1 r 5 VMFFToCI _ - • i i — 0 1 9 8 0 1 9 9 S D F T = 0 . n n . t k 1 i >1 . ? U • 0 2 0 0 0 2 0 1 LfU 1 J ^1 i " i / t ^ _ -—- —— -X M E F E » 0 . cntrc-era . — — — 0 2 0 2 ... 0 2 0 3 p U ' u " y f —— • I C O = 0 DO 10 J 1 1 • 1, 8 • —- — — 0 2 0 4 0 2 0 5 I F C Y I J K C H 1 , J 1 1 , K U ) , E Q , 0 , ) GO TO 22 V M C C C » V M F F F * F f T l 1 . . I l 1 . K l l 1 — 0 2 0 6 0 2 0 7 3 3 An u r c * " n u r w " ^ » » \ * »• . § *» * * * * * . — — — — GO TO 10 T r n = T r n * i s f 0 2 0 8 0 2 0 9 C C 10 i U U f c A w W ~ 4 — — — — C O N T I N U E y M F F F s X M E F E / f S - I C O i " - • ~ V 12 n . 0 2 1 0 0 2 1 1 (12 I 3 DO 11 J l l « 1 . 8 T F f Y T ,IK f T 1 1 • J 11 • K 1 1 ) n E Q . 0 .) GO TO U : 10 9 B e 1 c 0 2 1 3 m _ . a S D F £ B S D F E * C E f I U » J l l # K i n - X M E F E ) * * 2 r n w T T W i i F — 8 7 0 - 1 * 0 2 1 5 * * S D F E » S 0 F E / ( 7 - I C 0 ) 6 5 4 — 1 cn CO CO o cz s IT rn z ru i v i n i n w nj n C ; o I - >ci < _ .IS o iv! M Cl CO r - JC- s: 3D| *5i cn - c m CO o CD C_ —l<U 3 "*ll X ru'CED' » »| cn cn cn Z i CD cn] »— J> s : o 3 CO S IS ru ru CD <S <9| ru ru s s i ru ru •* Z i 2 cn cn cn co; cn Z j CD CO, i — j» o ; c, »-« ^ n I cj. t - . <3 e> a ru ru ru » • cn o o C3j o r a a i s H I I — m V »-• • x z Cn JT > X 3 0 ~ I •-> t-» n »— o <3 S> ru ru is -c! cn rs c o o z < -* m t-i -D Z n cr co m • .3 S C9 K ru ru ft* ru.ru ru W W W W , W Irf co ^ i , a - i x - c : iw -4 O •-. z | | T . <9 (9 ru ru ru m o o >» -~ o o . o>j 3t c-t — ( Z t -s rn • :* " S i «« —I rjl O -«1 in Co -4 33 - j -< *• *-! - - ! w ~ | -12H (S <3 ru ru ua ru 19 J3 ru ru ru ru IS IS ru ru ru ru cr un <s s ru ru ru ru t 04 (S s ru ru ru ru ru »— ts s> ru ru ru >-is >D S (9 ru ru P s cr O CO o o CO c l •— D >- -H U ^ .•- CO cn cn a o - i i • i jm -a cn co o cn x o - l n cn r » co cn o o Co o z o - i H i O CO a o o : CO z CO c jo m l II CO CO o cn O -HI " I PJ -"111 O O l c n j o O CO z o i|<o -» ~n t—« PI Z a cr cn PI CO o CO • IS liw «9 < 3; 3> •X) I »» re * - f o c n ro r - v j D m i J.3J 031.03 2 _ J L M A R » 7 5 P-A.GE 7. 0_6J5 8.01 C O N T I N U E . . . 0 2 6 4 GO TO 6 4 3 C S T A T E M E N T NUMBER 0 2 4 - TO 0 2 6 7 I S AN E X T E N D E D RANGE OF T H E 1 1 S D Q  J:, C LOOP', T H I S P O R T I O N OF T H E PROGRAM P O S I T I O N S T H E G E O M E T R I C C E N T E R i i i C P01NJ_J3JL_£ACH.J_AJLLI__I_E^ T I O N C T H E T A ) 7i C AND L E N G T H C T L J OF T H E C E N T R A L A X I S OF E A C H P A R T I C L E . ' C A L L _ O J _ E . R _ T . E i - M , S _ ^ FROM P R E V I O U S I C ' COMMENTS AND THE FORMAT S T A T E M E N T S BELOW, H P . 5 141 T H E T A B 6 . ? B 3 I 8 6 * R A N ( K A . K B ) ;  0 2 6 8 • X C B C V * R A N ( K A , K B ) - C V / 2 , 0 2 6 7 Y C ° C V - R A N C K A , K B ) » C V / 2 .  0 2 6 8 C O S T = C Q S ( T H E T A ) 0 2 6 9 S . I M j L S . I . N i l t i E T A ) : 0 2 7 0 T L » ( T L M A X - T L M I N ) « R A N ( K A , K B ) * T L M I N 0 2 7 1 ILH_.IL/.S.E.CJ3  0 2 7 2 C Q S T = T L H * C O S T / 2 , 0_2_3 S.IJiTJLl..H_SJ.N_Z_2_, 0 2 7 4 X l ' X C + C O S T 0.2_7_5 Y.iJLY.C * S I.N..T 0 2 7 6 X 2 " X C - C 0 S T 0 2 7 7 Y_2=JT_C_SJM 0 2 7 8 I F ( C O S T , E r _ , 0 . ) GO TO .556 0 2 7 9 IAJ__s S I N T / C O ST , :  0 2 8 0 Y I N T » Y C - T A N T * X C • ' 0 2 8 1 5 5 6 XM I N _A_M I N 1_C XJ_,_X 2 ) 0 2 8 2 X M A X ' A M A X l C X 1 , X 2 ) 0283 YM IN» AM INI CY1_Y2J , _ _ 0284 YMAX»AMAX1 (Y1,Y2) 0285 130 IFCM.EQ.l) GO TO 131 : 0266 GO TO 151 0287 643 IVARsl 0288 DO 902 111 •> 1 (i S 0289 DO 902 Jll»l,6 _ — 0290 SFQRMO0, 0291 DO 903 K l l " l i 2 4 — 0292 READ(3«IVAR) N ( K l l ) ! 0293 T F f Y . J K ' I l l , J l l f K l l ) , E Q , 0 , ) GO TO 878 ....... : _ _ _ 0294 V A L I J K C I l l , J l l . K U ) » C Y I J K ( l i i , J l l , K U ) - Y D D D O T ) / Y I J K ( X l l , J l l , K U ) 0295 GO TO 879 — 0296 878 V A L I J K C I H , J 1 1 , K U ) " 0 . 0297 879 SFORMoSFORM + V A L I J K C I U , J U . K U ) 0298 903 CONTINUE 0299 SFORM=SFORM/NSAMP(Ill,Jll) 0300 SDIJKS0, 0301 DO 904 K l l " l , 2 4 . 0302 SDIJKoSDIJK+CVALIJKCIll,J 11,K11)-SFORM)**2 0303 904 CONTINUE 0304 _DIJKaSDIJK-Ct24-NSAMPCIl1,J11))*SFORM**2) 0305 SDIJKsSQRTCSDIJK/CNSAMPCIil, J l l . - D ) 0306 WRITEC6, 1003) 111, JU,SFORM fSDUK 0307 1003 FORMAT C' ','THE SAMPLE MEAN AND STANDARD DEVIATION FOR THE ERROR T l'ERMS WITH RESPECT TO LOADING NUMBER',15/, ' ','AND DISTRIBUTION NUM 1 2BER',15,'ARE RESPECTIVELY',E16,6,'AND',E16,6) . 0 3 0 8 D O 9 0 5 K H » 1 , 2 4 0 3 0 9 V A L I J K C I U , J U , K 1 1 ) B V A L I J K ( I 1 1 , J 1 1 . K U ) / S O I J K 0 3 1 0 9 0 5 C O N T I N U E 1 FORTRAN V06.13 13:03!03 20-MAR-75 PAGE 8 . . . . . 0311 VALSTACI11, Jli)BABSCSFORM)/(SOlJK/SQRTCl.*NSAnPCIll, J i m ) 0312 • 0313 688 WRITE(6,688) 111 ,J11,(NCK11),K11«1,24) . FORMAT(' •LOADING"',13,IBXi'DISTRIBUTIONS',13/' ',2415) 1 CM 0314 0315 902 CONTINUE WRITE(6,644) 7 0316 0317 644 WRITE(6,645) ( ( N S A M P C I U . J l l ) , J l l " l , 8 ) , I U » 1 , 5 ) FORMAT(' ','THE C I , J ) ENTRY IN THE FOLLOWING 5*8 MATRIX REPRESENTS 1 THE NUMBER OF SAMPLES'/' ','IN CELL CI»J)> WHERE I STANDS FOR FUEL 2L0ADING AND J FOR FUEL DISTRIBUTION.') 0318 0319 645 FORMAT C' ',8C5X,I2n WRITE (6 f 646) 0320 0321 646 WRITE(6,647) CCRDCOEFC111,J11),J11»1,4),111 a 1,5) FORMAT C' ','THE C I . J ) ENTRY IN THE FOLLOWING 5*4 MATRIX REPRESENTS 1SPEARMANS RANK DIFFERENCE COEFFICIENT'/' ','FOR CELLCI.J) WITH 2RESPECT TO THE FOLLOWING TWO VARIABLES'/' ' , ' ( . . ) DISTANCE TO 3POINT OF INTERSECTION BETWEEN TNE TRANSECT AND FUEL PARTICLE 4CENTRAL AXIS'/' ','(2.) COSECANT OF T H E ANGLE OF INTERSECTION 0322 647 5IMPLICATED IN ' (1,) ,') FORMAT C" «,4C5X,E16,4)) 0323 0324 WRITE(6,648) WRITEC6,649) (CYIJDQT(111,J11),J11»1,8),111 a 1,5) 0325 648 FORMAT C ' «,'THE ( I . J ) ENTRY IN THE FOLLOWING 5*8 MATRIX REPRESENTS 1 THE SAMPLE MEAN IN CELL C I . J ) . * ) 0326 0327 649 FORMAT C ' ',6E16.4) WRITEC6.650) 0328 0329 650 WRITEC6,651) CYIDD0TCI11),Ili»l,5) FORMATC ', 'THE ITH ENTRY IN THE FOLLOWING COLUMN VECTOR 0330 651 1 REPRESENTS THE SAMPLE MEAN'/' ','TAKEN OVER THE ITH ROW OF CELLS') FORMAT (' ',30X,E16.4) 0331 0332 WRITEC6.652) WRITEC6.649) ( Y O J D O T ( J l l ) , J l l " l , 8 ) i 0333 652 FORMATC ', 'THE JTH ENTRY IN THE FOLLOWING ROW VECTOR REPRESENTS 1 THE SAMPLE MEAN'/* ','TAKEN OVER THE JTH COLUMN OF CELLS.') i 0334 0335 654 WRITE(6,654) YDDDOT FORMATC ', 'THE OVERALL SAMPLE MEAN »»,E16,4) 0336 0337 657 WRITEC6.657) NSSAMP FORMAT (' ',"TOTAL NUMBER OF COMPARTMENTS OR SAMPLES »',I5) • 0338 0339 WRITEC6,6bl) WRITE(6,649) (FMB1WJ(J11),J11 a 1,8) 0340 661 FORMATC ','THE JTH ENTRY IN THE FOLLOWING ROW VECTOR REPRESENTS M 1SB'/' ','WITH RESPECT TO THE JTH COLUMN,') 0341 0342 WRITE(6,662) WRITE (6, 649) (FMWIWJ(JH) , J11" 1, 8) 0343 662 FORMATC ','THE JTH ENTRY IN THE FOLLOWING ROW VECTOR REPRESENTS M 1SW'/' «,«WITH RESPECT TO THE JTH COLUMN,') 0344 0345 WRITE(6,663) WRITE(6,649) ( T S 1 W J 1 ( J l l ) , J11 a 1, 8) 0346 663 FORMATC ','THE JTH ENTRY IN THE FOLLOWING ROW VECTOR REPRESENTS T 1 HE RATIO OF'/' ','MSB TO MSW WITH RESPECT TO THE JTH COLUMN,') T i? 0347 0348 WRIT£(6 664) WRITE(6j&49) ((S0I1J1 C I l l , J l l ) , J l t " i , 6 ) , I i l « t , 5 ) n 10 0349 664 FORMATC 'THE" C I V J ) " ENTRY IN THE FOLLOWING S*8 MATRIX REPRESENTS 1 THE SAMPLE STANDARD DEVIATION IN CELL (I,J) , ') 9 a 0350 0351 WRITE(6,666) WRITE(6,649) (SDCOLJ(J11),J11»1,8) 7 6 0352 666 FORMATC ' , ' T H E J T H E N T R Y I N T H E FOLLOWING'ROW VECTOR REPRESENTS T 5 4 FORTRAN V-fe.15 L3J-03.I_0J5 20-MAR-75 PAGE 9 ' 0 _ J 1 HE SAMPLE STANDARD DEVIATION*/' ', 'TAKEN OVER THE JTH COLUMN OF CE E 2LLS.') * ' ' 0353 WRITEC6.667) SDQVEP 5 c \ J ! 0354 667 FORMATC ',10X,»THE OVERALL SAMPLE STANDARD DEVIATION o«,Elb,4) 9 7 • 0355 WR_I.T_E_C_6|_6_66„) '. : L 0356 WRITEC6.669) (C(VALIJK (111, J11,K11),K11s1,24),JJlB1,8),111a1,5) 8 0357 668 FORMAT C ','IN THE FOLLOWING ARRAY EACH PAIR OF ROWS CONTAINS 24 S 6 1 AMPLE VALUESC12 VALUES PER ROW)'/' ','COMPUTED FOR A PARTICULAR SA ° 2MPLING UNIT, THE ARRAY CONSISTS OF FIVE GROUPS OF VALUES.EACH GROU 1 3P'/,' ' . 'CONTAINING EIGHT SUBGROUPS. WHERE EACH SUBGROUP CONSISTS z 40F A PAIR OF R O W S C H GROUP'/'•'.'REFERS TO A FUEL LOADING CLASS 5, AND EACH SUBGROUP REFERS TO A FUEL DISTRIBUTION IN THAT CLASS,') 0358 669 FORMAT (' ', 6 E16.4) 0359 WRITEC6.670) 0360 WJ_n_L (_/A7JL)-_(i__AJ^ 0361 670 FORMATC ','THE Q , J ) ENTRY IN THE FOLLOWING MATRIX REFERS TO A SA 1MPLE STATISTIC COMPUTEP FOR C E L L ( I . J ) . ' / ' '.'THIS ENTRY WILL BE CO 2MPARED TO DIFFERENT VALUES TAKEN ON BY THE T DISTRIBUTION WITH'/' 3','(NSAMP(I11, J LI ) - l ) DEGREES OF FREEDOM.') ; 0362 671 FORMATC *,8E16,4~) 0363 WRITEC6.672) SDFT :  0364 672 FORMATC ','IF A COMMON STANDARD DEVIATION DOES EXIST FOR THE T TE 1RMS.AN ESTIMATE FOR IT IS',E16,4) 0365 WRITEC6.673) SDFE 0366 673 FORMAT (' ','IF A COMMON STANDARD DEVIATION DOES EXIST FOR THE E TE I RMS.AN ESTIMATE FOR IT IS',E16,4) 0367 WRITEC6.674) • 0368 WRITEC6,675) (CT1111,J11),J11B1,8),111B1,5) 0369 674 FORMAT (' ', 'IF THE T TERMS IN THE ANOVA MODEL ARE NORMALLY DISTRI IBUTED WITH 0 MEAN AND COMMON VARIANCE'/' ',"'WE WOULD EXPECT 95 PER 2CENT OF THE VALUES IN THE FOLLOWING MATRIX TO LIE BETWEEN -2.36 AN 3D 2.36'/' ', 'SINCE THE VALUES BELOW ARE EXPECTED TO LIE ROUGHLY ON 3 A T DISTRIBUTION WITH 7 DEGREES OF FREEOOM.') • 0370 675 FORMATC '.8E16.4) 0371 W_R_I_T_E (6,_676) , 0372 WRITEC6,677) (C (E (111, J11,K11),J11«1,8),K11B1,24),111 a 1,5) 0373 676 F0 RM A T C * , 'SIMILAR TO TH E T MATRIX ABOVE WE EXPECT 9S PERCENT OF 1 THE VALUES IN THE FOLLOWING E MATRIX'/' '.'-FOR FIXED 111 AND J l l -2T0 LIE BETWEEN T.025 AND T.97S WHERE T HAS ( N S A M P ( I l l . J l l ) - l ) DEGR 3EES OF FREEDOM,') 0374 677 FORMATC '.8E16.4) 0375 CALL EXIT 0376 ENj) SEOD ROUTINES CALLEDt TIME , RAN , COS ' , SIN , AMlNl , AMAX1 , SORT ABS , EXIT L _ 1 2 OPTIONS a/QN./CK./OPIl./GO n 'i_ BLOCK. LENGTH 9 MAIN, 11522 (055004). 8 [ 7 **COMPILER CORE** 6 PHASE USEO FREE 5 DECLARATIVES 00622 14422 4 ; EJLECU T.ABL &S 0196 9_J 3 0 7 5 •3 ASSEMBLY 03399 16285 .13!«a:.?9 02-AUG-74 PAGE.. . i. C T h i s PROGRAM I S A DOWNED WOODY P A R T I C L E S M O D E L . IT I S C U R R E N T L Y 1 C . C SET UP KOK RANDOMLY D I S T R I B U T E D WOOOY C O M P O N E N T S . HOWEVER I T CAN E A S I L Y tiE A L T E R E D TO ACCOMMODATE N D N - R A N D O M L Y D I S T R I B U T E D CM 1 c c • c c P A R T I C L E S . T H I S PROGRAM P O S I T I O N S A L L F U E L P A R T I C L E S W I T H I N C I R C U L A R S A M P L I N G U N I T S , IT THEN E V A L U A T E S THREE T E C H N I Q U E S E A C H A P P L I E P AT HIVE: D I F F E K E N T L E V E L S OF S A M P L I N G I N T E N S I T Y IN ORDER TO ' D E T E R M I N E THE OPTIMUM S A M P L I N G T E C H N I Q U E S TO USE FOR O B T A I N I N G THE -. - • - - :-c c TOTAL WOOOY R E S I D U E VOLUMES FOR THE P A R T I C L E D I A M E T E R S I Z E C L A S S E S OF I N T E R E S T , 0 2 0 1 . 0 0 0 5 B Y T E I I M E R ( 8 ) D I M E N S I O N C 0 E F F ( 1 4 , 5 , 3 ) , T L O C T N (14 , 5 ) , T L 0 C T 3 £ 14 , 5 , 3 ) » NUM ( 5 ) , TLOCT C1 •._PI003_ 1 4 , 5 , / J ) , C 0 S L ( 1 4 , 5 , 3 ) , S I N L ( 1 4 , 5 , 3 ) , T A N L ( 1 4 , 5 , 3 ) , C O N K 1 5 0 ) , C 0 N 2 ( 1 5 0 ) .. D I M E N S I O N . Y I S 1 D ( 5 ) , . .XIS1SD.(5). , .XI.Y.E .S.D . . (5,.3., .51 • .. 00(54 R E A L I S V M A P , I S V V A P , I V M A P , I V V A P , I S O , I D ( 4 0 ) , I D S , I S D g , I S V E M , I V E M , I S V E I V , I V E V , I V E S T ( 5 , 3 , 4 0 ) , I S I P U A ( 8 , 5 ) c c T L M I N n MINIMUM P A R T I C L E L E N G T H . TLMAX B MAY I MUM P A R T I C L E L E N G T H c c S E C 8 » MEAN S E C A N T FOR P A R T I C L E T I L T r 1 i\ 1 ,1 OtiAR » Q U A D R A T I C MEAN D I A M E T E R LsZWllA fue> MW 0 00 5 R E A L T L M I N , . T L M A X , S E C 8 , D 8 A R D E F I N E F I L E . 2 ( ? , 4 , U , K V A R ) - ' 0 0 0 B • KVAl< = l • K A » 0 0 0 H 9 '001 0 K 3 « 0 I C R « 1 . 0 0 1 1 001.?. Krt.CR«l R E A D ( 5 , 2 5 , E N D « 1 0 0 ) ICODE 0 0 1 3 00.1 4 as FORMA f ( I ? ) P f c A D ( 2 ' K V A P ) K A , K B 0 0 1 5 00 ) 6 R £ A D ( 2 ' K V A R ) I C R , K K C R W R I T E ( 6 , 1 0 1 ) I C R , K K C R . 0 0 1 7 '101' FORMAT ( 1 0 X , ' S T A R T - U P L O A D I N G A T ' , 1 3 , 5 X , ' S T A R T - U P D I S T R I B U T I O N A T * , 1 1 3 ) 00.1 8 c 100 I B s S A M P L I N G I N T E N S I T Y 0 0 ' 3 5 0 I B = 1 , 5 . . 0 0 1 9 c I 8 F a a * I 6 + 4 C U E F F » MATRIX WHICH C O N T A I N S THE C O E F F I C I E N T S FOR A L L F I F T E E N 00?0 "c T E S T E D S A M P L I N G F O R M U L A S . " « E A D . ( 5 , 3 0 0 ) ( C 0 £ F F ( I C , I B , 2 ) , I C 1 , I B F ) 0 0 3 1 0 0 2 2 3 5 0 . 3 0 0 C O N T I N U E • • • FORMAT C 1 0 F 8 . 6 ) 3 0 2 3 ' 0 0 2 4 DO 3 5 1 1 9 s 1 , 5 I b E a 16 + 2 c c T L O C T N » M A T R I X WHICH C O N T A I N S THE N O R M A L I Z E D L O C A T I O N S ( B A S E D ON . . A F I X E D 0 A N G L E ) FOR A L L T R A N S E C T S USED TO O B T A I N c c P A R T I C L E S DATA FUR F I V E S A M P L I N G FORMULAS OF I N T E R E S T ( I E . G A U S S I A N - L E G E N D R E I N T E G R A T I O N F O R M U L A S ) 0 0 2 5 ' 05526 R E A D ( 5 , 3 0 1 ) ( T L O C T N C I C I B ) , I C B 1 , I 8 E ) R E A D ( 5 , 3 0 0 ) ' ( C O E F F ( I C , I B , 1 ) , I C « 1 , I 8 E ) 0027 0 0 2 8 3 5 1 3 0 1 C O N T I N U E . FORMAT ( 7 F 9 . 6 ) 0 0 3 9 0 0 3 0 D E F I N E F I L E 3 ( 6 0 0 , 2, U , I V AR), D E F I N E F I L E 1 ( 4 0 , 2 , U , J V A R ) 0 0 3 1 0 0 3 ? I V A R a ( ( ( I C P - 1 ) * 8 + K K C R ) * 1 5 ) - 1 4 J V A R s ( I C R - 1 ) * 8 + KKCR FORTRAN. ..Vttb.13 M'3 3 '. .2034 ._13t.42.1 3.9 _ . 02*AUG-.7.4 RAGE-1^ CV) DO 4*4 IP a 1 , 5 _.: I r i F a p n I B + 4 _I.C_s_Jl.U HB'r£Jl_O.F__lfiA NSECTS Q 0035 DO 4«4 IC a 1 , I8F ... 0036 _.• . CQEF c(IC,IB,3).»6.28318b/IBF_ 00 3 7 4 0 4 CONTINUE __. 0038 . DO 401 IBal ,5 -. 0039 . IBFs?* ID + 4 e.0AH_ 1 RD..=J.b.F_/.2 0 0 4 1 I U S * J . l 5 F / 2 + l . . 0 0 4 2 _;. DO 4 3 1 IC«IBS, IBF 0043 TlOCTNCIC,IB)«A8SCTLOCTN(IND-,IB)) . 0 0 a 4 _ C O E F F C I C I B , 1 ) "COEFF CIND, IB ,1.) -0 0 4 5 . I M O = » I N O - 1 0H4fi 401 CONTINUE : ; ;  C IA a INTEGRATION TECHNIQUE(IE, GENERAL SAMPLING FORMULA) . 004 7._.._ ... DU 4Q2 ..IA«1,2 _ : 0048 DO 402 IB»1,5 .0049 ...—ISFs_* In + 4 . . . : 0050 DO 402 1C«!,IBF -0051 • • IF CIA...EQ.U-G-Q TO 4J3.JL .0052. 0053 C TLOCTS a MATRIX WHICH CONTAINS THE STANDARD" LOCATIONS ( I E , BASED ON C _ FIXED V ANGLE) FOR ALL TRANSECTS USEO TO OBTAIN PARTICLES C DATA FOR ALL SAMPLING FORMULAS OF INTEREST, TLOCTS (IC, 16,1 A).a (I C-.l)*6. 21)3 1 86/IBK _ _ _ _ _ TLOCTS tIC, IB, I'A + 1) "TLOCTS (IC,IB, IA) GQ TO 4 0? ; :  0055 403 TLOCT3CIC,ia,IA)«3,141593*CTLOCTN(IC,IB)+l,) .2.0.58 _4.0_____ CONTINUE :  N U M = NUMBER OF POPULATION ELEMENTS EMBEDDED IN THE SAMPLING UNIT 0057 J_0_?J_ • C ..C . _1ML ... CONSIDERED... RE A D . 5 , 2 0 0 ) N U M FORMAT C5IS) DBAR 0059 READ C5,20l) TLMIN, TLMAX, SECB, .0.0 6.0 ?tU F.O »_. M A T .C.4 F. .ft .6) : •_ • 0061 ICC=0 . C AVCQN_CORRESPON[)S TO YODDOT IN THE ANOVA MODEL CONTAINED IN . C THE PRECEDING ANOVA MODEL, 0062 READf5i6) AVCCN ; ; ;  C?bi 6 FORMATCF5.2) .,. .... .... .. C CV a COMMON RADIUS OF CIRCULAR SAMPLING UNITS. 0084 R E A D (5, 1) CV 0085 7 F O R M A T c F s . ? ) • _ . ; 1__ C CF a •COEFFICIENT WHICH PRECEDES THE INTEGRAL OF THE PARTICLE C I N.TFRCE PT C PUNTING FUNCTION. CF IS A VEHICLE WHICH PERMITS fHE INTEGRAL DF A SIMPLE STEP-FUNCTION TO BE MAPPED INTO A TOTAL PARTJC.LE VOLUME ESTIMATE. : i 2 3 4 5 G 12. I I I U . 9 006b ._«0fOL 0066 ,0089 CFaOBAR**2*SECB*AVCQN/CCV**2) _P.O,.2?g_KKl«J,«0_ 00 222 IA«l,3 00 2?? IH».l,5 C 7 6 — 5 4 — i 0070 007 1 0072 0H73_ 0074 IVEST(I3,IA.,KKI)«0, 222 CONTINUE __ _____ DO 223 I a 1,-5 _.00 223 KKal,8 I S 1 P U A U«, I) »0. MRT.R_A__.VJ3.6 , 1 3_ O 0075 2 2 S _ CONTINUE _• C THIS S'"T OF NESTED DO'S CUP TO l i b ) FIRST FINDS THOSE PARTICLES ' • C CIM .A PARTICULAR SAMPLING UNIT) WHICH INTERSECT F.ACH TRANSECT CM CONSIDERED FOR EACH OF THE FIFTEEN SAMPLING FORMULAS OK INTEREST. WITH THIS INFORMATION,._THE_ ;SET__0F OO'S COMPUTES..FIF.TEEN . TDT.AI VOLUME ESTIMATES FOR EACH OF THE FORTY SAMPLING UNITS SIMULATED WITH RESP.CT TO.A GIVEN PARTICLE DIAMETER SIZE .CLASS. ..THESE...VALUES-ARE STORED IN IVEST. THEN THIS 5ET OF DO'S CONSIOERS EACH PARTICLE Si MULAl i__ANn. DETERMINES THAT PORTION OF IT WHICH INTERSECTS THE C SAMPLING UNIT OF INTEREST. THIS PORTION IS THEN CONVERTED TO A C PAKTJAL PARTICLE VOLUME. THROUGH SUMMATION .OVER ...RELEVANT. _ C PARTICLES, THIS LAST PROCESS RESULTS IN THE TRUE TOTAL PARTICLES C.... VQLUME_..PEi" UNIT AKEA_CSTORED._I._t_I.S 1 P.U.A)__W.1IHIN._E..ACH...OF .THE_f.OR.TJL C SAMPl INT, UNITS SIMULATED, _C LOOP OVER FUr.L LOADING ; ; O 0 . 7 b 0 0 7 S 0 0 7 9 ' enpsn DO l i b I»ICR,_ NH«NUM(I) LOOP OVER FUEL DISTRIBUTION DO U l KXsKKCR,d CALL TIME(TI MER) WRITE (fe, 7 77 ) TIMER 0 0 8 1 0 0 8 2 0 0 8 3 0(164 777 FORMAT (25X,.'THE TIME IS',8A1) _ICC».ICC + t •_ . _ ' _ __ C TLOCT. - LOCATION OF 0 ANGLE FOR THE CURRENT SAMPLING UNIT O F C _ •__ INTEREST. ; K V A R = 1 ' W R I T E C 2 ' K V A R ) K A . K B . ' ' __ 0.85 0 0 8b "00.8 7" 0„8fi _0089_ 009(_ 009 l " "009 2" 0093 WRITE(2'KVAR) I,KK _WHITE(b,2fa) I,KK " _ • 2fe-" FORMAT (IPX., 'LOAOI NG s ', 1 5 , 10X, 'DISTRIBUTI ON o ' , 1 5 ) TLOCT0 Bf-.28318b*RAN(KA,KB) _ C LOOP OVER FUEL PARTICLES • " " " " " " . DO U l M=1,NP I I « 0 LOOP OVER I N T E G R A J 1 0 N_ _T E C H N I Q U E S ( I Ej_5E_l E R A L S A M P L I N G F O R M U L A S ) _ 00 115 I A - 1 , 3 LOOP OVER SAMP_LING__INTENSITJES , oo u_ I H B I , 5 '"""' ~~ : I B F i 2 * I B + « 7.094 0295 039b 0,97 0098 131 LOOP OVER T R A N S E C T S : ' DO 115 IC=1,IbF _ • ' _ IFCIC.EO.l.AND.IB.EQ.l) GO TO 140 GO TO 130 • "T L 0 C T CIC , "18 ,1 A')"- T L OCT S" CIC, 18,I A) +TL 0 CT 0 COSL(IC,Ifl,IA)sCOS (_TLC_C T (IC , IB, I A_) •12. j u 10. ') 8 -7 h-o 0 0 9 9 S " l N L(iC,I.riA")*SlN ( T L O C T r i C,IB.,IA)) _ 0 1 0 0 _ 1 5 1 1 1 * 1 1 + 1 _:  0 1 0 J . "~ 'IF ( M.NF.jfGO "TO" " l ' 8 „ " " ' " ~ _ 0 1 0 2 ____C0 N 1_(II)«CV / 3,*C0SL (IC, IB, IA) 0 1 0 3 CilN'a ("II)VCV/2,*SINL('I'C", 1 8 , l A ) " " 0 1 0 4 1 8 0 IFtCOSL ( I C , I H , I A ) . S Q . 0 . ) GO TO 5 5 0 T AN L CIC", 1671 A"") Bg frCTfC , I"&7l AT/COSL ( I C , IB , I A) IF(C03T,E-.~.) GO TO 5 5 5 0105 010b "0 10 7" 0106 '0109' IF"(TANT,F.ij , T ANL ( I C ,18 ,1 A)~) GO TO 560 X I N « Y I N J / ( T A N U I C > _ I B , I _ A _ ) - T A N T _ _ _ _ _ _ _ I F ( x IW ,GE, X M I N , AN0, X I N , L E ,'X'M'AX)' GO TO 5 6 5 " o 4^ C5 o c \z.z <- -12& ° S S 8 S S | S S p S S S 3 -a -3 3 3 s s ; s s s s s •3 a ru ru ru ru Co X u. Ul L >_r. •c. OE a 3 Icn 3 ! I 3 5! 9 S 3 5 K j — e > ui 'JI ui ui c 01 t w w 'J]:c C c fc s. :-x> cr cr ( C- I i C- 42: C kjJ _*J o- >- UJ UJ-W I j l Xi iw rv »— S Ct cr tjr. * . 1 -T| i-« t-i j-r, ^ 2 Z • -< ro cc ; < ' H H "O 2 1 <><'—<< ;-< o ro • • t~» ._C -i cr* 2 — i n ro r • ; c o » —I i» • ru 1 M X :x i-i X >. I—• •—I 2 !2 ru ^ a u rr, jr. x x 212 i ru ru > * * 1 -«—1 • • • Z O - < - < c r > H Co 2 :Z O ,0 2 iZ ru — cr • z o • 2 z —* —« 1 Cn cn TO TO r r o o . rn ;m »-c >-* > • cn cn x x n o 3 ] ~ —• >^ t> t~ ' X X ^ ^ » O O Iru cr !-« s 3 4= cr i icr-w I CD CD o o ru ru • * a a • -•— > c » o X -r 2 •-« —i -I _i m • d u« m rn « • w 2 (VI Xl o - cn *-• a o O —I >*—<"»—»»-* ; X P-I -Ti < — X 2 . m - » w <r> • *-« o. x _> • O —I X 1 2 • *—* • cr. —c fcD frt OJ —c cn > . - . >-l '_. ' • 2 cn ^ t~ ;:. — . - C5_ 5> -C 1 o n _ ^ r O n cn j i c i ci cn »-* o H " " 7 —I I D , « —I ^ o cr. *^ 1-" 1 j > " . o v n ; 0?' * !' o —4} *-> 1 O O : CC T> -t •— T) CD jo o -n o o -n —c cr x —4 —i x | O S O 3 2 p •_r 2 )> • >D -7- X cr. tx •_: • m ^1 _p CD j — CD ] ^4 W ' U4 ru — 1 s ru ru — t— CL 4T p- -^1 cr u". «^ cw o 2 P. a 2 rs a> CD o 0 ^ 0 0;~r. 3 2 0 0 ^ 3 »—: 2 a - 2 - c • CS. ?S. • £~ CD Ul CD '-I m, .— 2 ; _ I o 2, 5 2' O ! O o (2 cn U1 0 ai rt ' -* 1> — >. 2 '—» 2 • —I • -c r" -c 1 2 0 2 1— O >— I 2 - . 2 fr. > T. *» 2 c 2 h > n > ; < x —4 >: , ~ri t-t o -^v 2 2 2 ; rt; o i 2 ; ru : 2 X ^ —i ^ o —< r- I .2 O 2 }J3 —( o « Ix • . — I 2 r~C .T -un : C3 , L- ° ° CD C/< !,-n - i ^> r~ r~ p-t cn ?-* c"» rr. o t r p m — rr: e •-< 13 H _ • cc: • rr • CD t~ CD O " O cr cr s ' s a LT - cn f r> -o- CD , o >-' o : ID —C CT> —< E. O P . O a- 7~t a- < 3 • s < t> c_n - J ui cx-iTt cr 1 2 : -> i x 1 O i O 1 rt! 0 1 o i :c» '^ 3 •:z o o in CD S IS 1 u • : • cn or Ml > 2 O <s 01 * 2 o -r, . • w 2 -. -f J—1 z tc -C X -« 2 2 J3 I Co • • Cr , _^ >C m -4 |2 C m ; -< 1 2 ! CD i O i • ui r cc : ui j 1 ° ; __> ( 2 cn 1 & i (|CC • p. to E ru CD 6 CD <-O © o S -3 S t3 S _ *Sf_ rufvn_r_f\j|r\jr\jf\; S» s • cr un — i -< x • _B.-r.rurv' _ —. n H —< r. -c x • a o o o o| cr. 01 i w - t cr> n o>| __ - M o •"-«! — i ni z -s. o - i —-; o • 3 S 9 _ -s _ X •_/> o -t J) -O <fl l_T £ : -< .< - I i _ r - r* »— cr o X —1' rj M : H v- n »—J cr zjco -1 n — — . -4 r, : M -4 -4 n K •< ' - i r r c_ o » ;r- _: H ° A •x cr t» _ CO j> o o o in rs a —s—< • a> — i x; X : r~ rn; rr. 2_ • o a _• - m o o -< < t> r-* II x cr z -r\i — i - o I ^ UJ 3 2 =55 _» S -3 S o n - -^ o X * •—• — X i >• o o z ~n *v - . —t ro i - I • - m C> CO c o -< o 2 TJ —4'-4 —1'« .T> O o ' —4 rn *— T J x _r _j — I - i s rr] cn H --• - "t> c X | 0 3T • -c;-4 rr. *> M X, » o I -4 - .Z s 03 C~ CP C~ C-] >j tr - ^ _ s s s> <s 3 S 3 g 3 SI CE CD On —I —J — J ~4 " ^ i " 4 5 1 ru _ o a> cr ai *; iw ru —- _ _ « cr cr [-T . - 1 2 9 -_ cr - - i o •-I r> oj - r o c m -j c o s rr, • 3 o n > >-• m _ •Z O ! —I ix a rr _ • I X Oj m •_ ru x »-i o u» crI x M o ! co m 1-1 x! > Z l 3 2 : Z O co m: o z . X: cn CT" *-• m m| n \ o I 3. D! Z X -4 .: tr mjr- o - i x o 3c o _: i-» *-* ^ o x —; O 2 1-4 —• —» X -« -4 _ "C _ w m . c z: 2 *> c: m — r» » • ^ <— z. x. -4 o -c x -c n r o " H « c Z Z 2 ! H -| ^ |\J —« ! o . n B a o —4 -< X —. O »-* i-i J> ^l -4 -< CD I— l-i C —4 71 r- o i II r_ —t n —i CO « O C- o e —t 3j t- ~ i X X OV—I <2 2 2 •Z. j> LJI \J \J —^ a > w M ODj X " " C H O i-l »-4 cn Co —I »— O -D C C — l> , J » • - » * »— ^ : x o !-4 -4 cr —. * X x —i 1—1 2 2 »— ro i + Z 2 " —. cr —>. ru • • Z j i w -< _r> i "n •—• o • • —> 2T ><»——•' H II O ru > cr i • •n — i .x; i : IT. * • • x cr> tr J - cr c _ J ru T i j —< r_i -< —i *-«*- . ~> i o c r- o r - ~i p i o |x x x . — i ' i i — i u — i n X j a ; » a > o »• «— ' a ac a; * 2 : H -4 cr. >ioi >J yi * jiXi - i 1^  •J1 J1 Lfl >^ a ~4 ro z + -- I - l 2 2 O —4 * i • 3 ' -4 D \t°\ — ru • o tCr F O R T R A N . V 0 6 . . 1 3'_. 0 2 0 8 . 0 2 0 9 J ? . c U J i . 5 5 6 0 2 1 1 0 2 1 2 . H 2 1 3 0 2 1« 0 2 1 5 0 . 2 . L b -0 2 1 7 0 2 1 8 . . 0 2 1 9 0 2 2 0 -0 2 2 1 J 2 2 £ 2 -130 Y _ N T = Y r : - TANT * x c : ~ _ _ X M I N ' A I U N I ( X I ,X2). JLtl.kX _Arl AJ( : — Y M I N » A M I N 1 ( Y 1 , Y 2 J YhAX»AMAXl ( Y l ,Y2). _ -n M A x ? = A h A X l ( X l * * 2 + Y l « * 2 , X 2 * « 2 * Y 2 * * 2 ) O M A x ' S Q R T (DMAX2) I F ( M . t Q . l ) GO TO 131 _G.Q __Q__bj ; 7 0 0 J V A H s 1 I V A rt = 1 i c c » 0 _ I S V H A K s f l , I i i V V A P a 0 , 0 2 2 3 P 2 2 « _ 0 2 2 5 0 2 2 6 . 0 2 2 7 0 ? ? 8 .210.-. fe«fl0 DO 2 1 0 KKsl,8 READ (l.'.J VAR) . 1S1PUA..(KK,.I) : ISVrlAprlSVMAP + ISlPUA (KK , I) CONTINUE - -WRITE ( 6 , 6 0 0 0 ) ((IS1PUA(KK,I),KK«1,8),I«1,5) FORMAT ( ' ' i B E 1 6 . 4 ) 0 2 2 9 .0230.. 0 2 3 1 0 . 2 3 2 . 0 2 3 3 0 2 3 4 P.2.3JL 0 2 3 6 0 2 3 / . . . 0 2 36 0 2 3 9 _ 3 2 0 0 - 0 2 . 1 1 -0 2 4 2 0 2 « 3 IVMAP B HE AN.TRUE TOTAL PARTICLES VOLUME PER UNIT AREA TAKEN OVER . __ THE FORTY CIRCULAR SAMPLING UNITS SIMULATED, I V M A P s I £ V r A P / , 0 . D O .. 215 _ . i e i , 5 : : : 00 215 KK» 1 ,8 . .TSVVAP«ISVVAP+(IS1PUA{KK_ D-IVMAP) *«2  215 CONTINUE C _ IVVAP B SAMPLE VARIANCE CF THE ...TRUE T0T.4L PARTICLES VOLUME.PER C UNIT AREA TAKEN OVER ALL FORTY CIRCULAR SAMPLING UNITS C SIMULATED. : I V V A P B I 3 V V A P / 3 9 . n u i i » i , 5 ; ; ; DO 1 KK=1,6 ICC?I.CC*1 : : DO 1 I A a l , 3 DO 1 1 Ho 1 ,5 _ ._ RE AO (3'1 VAR) 1 VEST (18,I A,ICC) , CONTINUE 600J C W R I T E ( 6 , 6 0 0 1 ) (((IV£STCIB,lA,lCC),IBBl,5),IAnl,3),ICCBl,40) F u P M A T ( ' . . . ' , . 5 ( 2 X r E 1 6 1 4 ) ) _ _ 02«:<L 0 g 4 5 L O O P OVER INTEGRATION TECHNIQUES ( I E , GENERAL SAMPLING FORMULAS) DO o 5 I A » 1 , 3 . ' . , • ' •  LOOP OVER SAMPLING INTENSITIES no 90 , I B B I , 5 - ; 0246 _B.21.7„ _2J.8_ I S O « 0 , I T B V ; •' ' __. . .. _ LOOP OVER FUEL LOADING _00 9 5 _ I s l _ 5 _ _ 0 2 4 9 L O O P O V E R F U E L D I S T R I B U T I O N D O 95 K K » 1 , B  0 2 5 0 I T = I T f l 0251 I D ( I T ) B I S 1 P U A _ ( K K , I)_-1 VEST (IB, I A, IT) .__ C "." ID B MATRIX WHICH CONTAINS THE ERRORS INVOLVED IN USING THE _ t CURRENT VOLUME ESTIMATING TECHNIQUE (ONE OF FIFTEEN TESTED) •"" C •• "WITH RESPECT TO THE FORTY CIRCULAR SAMPLING UNITS SIMULATED, EACH FORTPAN . V_6, 13 13:<I2:39 02-AUG-74 PAGE 7 C ERROR "HERE IS EXPRESSED AS A FRACTION OF THE CORRESPONDING TRUE 025? c VOLUME. I D ( I T ) s I D ( I T ) / I S i P U A ( K K , I ) 1 PI?5a _ 95 ISOsISO+IDCIT) CONTINUE ^ — _ 0 0255 C ' IDS « MEAN OF THE 10 MATRIX UNOER CONSIDERATION, IDBBisO/flU, 0c?56 025 7 ISO2*0. I T = -i 0256 0259 00 105 1=1,5 DO 105 * K s l , 8 0260 0261 IT«IT+1 ISD,JaJ[Sl)2+(IO(IT)-IDB)**2 0262 105 '•' C CONTINUE TSD » STANDARD DEVIATION OF THE 10 MATRIX UNDER CONSIDERATION 0263 C TSDsSuRT (1,*ISD2)/SQRT(3 9.) CVAR « COEFFICIENT OF VARIATION DF THE ID.MATRIX UNDER 026U c CONSIDERATION. CVARrTSU/IDb c c CkWM is A SAMPLE STATISTIC TO BE USED IN THE STANDARD PARAMETRIC P4THE0 COMPARISON TEST FOR MEANS. 0265 0266 CRVVMnAHStJ. Dc/(T5D/SQRT («0.))) WHITE(6,b002) IDB,TS0,CRVVM,CVAR,IB,IA 0267 "60'02 FORMATC ','IUB=',E16,-/' ', ' T SO* ', E1647 ' ', ' CR V VM » ' , E16, H / ' ','C l V A R s ' . E l f e . a / ' ' f ' I S » ' , I 3 / ' ' ,* IAn*,13) 0266 C DO 2« ITT = 1,<J0 UNDER THE HYPOTHESIS THAT THE RANDOM VARIABLE 0EFINE0 BY THE ERROR C c INVOLVED IN USING THE CURRENT VOLUME ESTIMATING TECHNIQUE IS NORMALLY DISTRIBUTED WITH MEAN 0, THE NEWLY DEFINED ID MATRIX c-._. C0NT4INS VALUES OF A RANDOM VARIABLE HAVING A T DISTRIBUTION WITH 39 DEGREES. OF FREEDOM, """0269 0270 20 ID(ITT)"ID(ITT)/TSD. -CONTINUE. 027 1 0272 WRITE 16,21 ) 10,I A wRHE(6,19) .(ID(ITT),ITT»1,<I0) 0273 21 FORMATC ','ThE FOLLOWING MATRIX CONTAINS VALUES OF A RANDOM VARIA 1BLE I/'HICH IS HYPOTHESIZED TO HAVE'/' ','A T DISTRIBUTION WITH 39 D 2EGREES OF. FREEDOM, HENCE 95 PERCENT OF.THESE VALUES SHOULD L I E ' / ' 3','BETWEEN T.025,39 AND T,975,39. HERE IB"',15,'AND IA»',I5) 027« 0275 19 90 FORMAT (' '•.8E16.6) . CONTINUE 027b ' 0277 86 DO 12. IB»1,5 ISVEM«0, 0278 0279 ICC»0 00 1=1,5 0280 028 1 OU 230 KK» 1,8 I C C * I C C t l 0282 0283 230 ISVEMsISVtM+IVESTCIB,IA,ICC) CONTINUE ' • C C IVEM » MEAN ESTIMATED TOTAL PARTICLES VOLUME PER UNIT AREA WITH RESPECT TO THE CURRENT ESTIMATING TECHNIQUE. THIS MEAN IS 1 028a c TAKEN OVER ALL FORTY SIMULATED CIRCULAR SAMPLING UNITS, IVEIHIS.VEM/40, 1 0285 . 0286 ISVEVr0. ICC*?. 0287 . DO 135 I» i , 5 .FORTRAN ..V-_6.13. 0286 .. .. 0289 1 3 I «a.«.39.—:__32»AUG-.7« P-AGE-PU 135 KK«1 (6..__ - _. ICC=ICC+1 _t_S-V-L--a^-VtV.t..CXVEST (IE , IA , ICC!) r l VEH)-«-ft-2 1 CM tor 0291 0292.,, 1 35 C C - C. c C O N T I N U E I V E V "...SAMPLE V A R I A N C E . O F T H E . . E S T . I M A T E Q . T O T A L - P A R T I C L E S . . „ V O L U M . E . . . P E R _ . UNIT A R E A K I T H R E S P E C T T O T H E C U R R E N T E S T I M A T I N G T E C H N I Q U E , _ T H I S S A M P L E V A R I A N C E I S T A K E N O V E R A L L F O R T Y - - S I M U L A T E D C I R C U L A R S A M P L I N G U N I T S , T V E v « J S V E V / 3 9 T : C K V V V a T N E R A T I O O K E S T I M A T E D V O L U M E S A M P L E V A R I A N C E ( F O R T H E C .. C : CU."rENT ESTIMATING TECHNIQUE) TC THE TRUE VOLUME SAMPLE C VARIANCE, UNDER THE HYPOTHESIS THAT THE POPULATION __.C VARIANCES-ARE..EQUAL j-XRV.V.V-.HAS...AN-F..DISTRIEUT10N..-WI.T.H._(39r..3-9) : C DEGREES OF FREEDOM, CRWV IS A SAMPLE STATISTIC TO BE USED IN THE r STA.MTUPH PAPAMFTRTP TEST FOR rOHPARTNT, TWO VARTANf.FS WITH A Twn-0293 _H23JL C SIDED ALTERNATIVE. NOTE THAT WITH RESPECT TO ANY PARTICULAR C _. . ESTIMATING..TECHNIQUE , TMI3.TE S.T_M U.S.L.B E_P_A SSF. O.IN OR DE 8_F_0R_T.H E _ C CORRESPONDING PAIRED COMPARISON TEST FOR MEANS (SEE ABOVE) TO BE ,C CORRECTLY APPLIED. . _ CRVVV=IvEV/( l,*IVVAP) WRITE (6,6005) IV E V , IV V A P i C R V V V , 16... .1 A. 0295 6003 FORMATC ' , ' I VEVs',E16.4/' ' , ' IV V AP » ' , E 1 6 . 4/' ' , ' CRVVV»',E 1 6 , 4/" ' 1 , ' I B » ' , J 3/' ' , ' I A r ' , J 3 ) ; , __1 . C TT IS IMPORTANT TO REALIZE THAT GENERATING A POPULATION WITH C _ .SIMILAR M E A N Ai\D _ V Afi J ANCE. .TO THE. POPULATION .OF. .1 NT F. R. E ST ...I S NOT C SUFFICIENT, SINCE, IN PRACTICE, SAMPLE SIZES ARE SMALL, IT IS _ C HIGHLY DESIRABLE THAT THE ERROR TERM INVOLVED WITH THE SELECTED  C ESTIMATING .0256 1 20 .CONTINUE 0297 85 CONTINUE .0293 CALL. EXIT 0299 END 'TECHNIQUE BE SMALL FOR EACH SAMPLING UNIT. $EOD ROUTINES CALLED! AbS__ , T IME , _R AN_ SORT , EXIT COS _» 3_I_N __AMINl , AMAX1 OPTIONS VON,/CK,/0PU bLOLK '" " LENGTh 1' AIN , ...„8 3 8 9 (04 ? 5.8.2)..*. »*CQMpi|.ER CORE** XJ-.PHASE USED FREE DECLARATIVES 00622 14418 EXECUTAKLES 018P3 13217 ASSEMBLY 0 3.K25_i_6.b 0.5_ 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0075128/manifest

Comment

Related Items