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Continuous forest inventory using multistage unequal probability sampling with partial replacement Huang, Shongming 1988

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C O N T I N U O U S F O R E S T I N V E N T O R Y U S I N G M U L T I S T A G E U N E Q U A L P R O B A B I L I T Y S A M P L I N G W I T H P A R T I A L R E P L A C E M E N T By Shongming Huang B. Sc. F. Nanjing Forestry University, 1985 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L 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 T H E D E G R E E O F M A S T E R O F F O R E S T R Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F F O R E S T R Y We accept this thesis as conforming to the required standard 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 November 1988 © Shongming Huang, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date DE-6 (2/88) A b s t r a c t Continuous forest inventory (CFI) with partial replacement of sampling units (SPR) has been established as an efficient inventory technique. Previous CFI with SPR theory was based on equal probability sampling which might not be as efficient as the unequal probability sampling that is now widely used in forest inventories. In this study, the general theory of multistage sampling combined with CFI using unequal probabilities in a SPR structure on two occasions is discussed. Estimators of the current means and the change in means for both one-stage and two-stage cases are given, along with the estima-tors for the overall variances. The estimators derived for estimating the current means take the same form as those developed by other researchers. However, the estimators proposed for predicting the change in means take simpler forms than those presented in the literature. This might result in greater simplicity when used in data processing. The application of CFI with SPR using multistage unequal probability sampling to a Chinese national forest inventory is presented for illustrative purposes. Some particular points are addressed regarding specific situations in China. Although no precise theoreti-cal demonstration of the gains from using this method is given, the combination of highly efficient sampling with unequal probabilities and the very practical multistage sampling in CFI with SPR inventories can certainly provide an efficient alternative to traditional Chinese inventory systems. 11 Table of Contents Abstract i i List of Tables v Acknowledgements v i 1 Introduction 1 2 Literature Review 8 3 One-stage S P R wi th Unequal Probabil i t ies 18 3.1 Definition of the Sample 18 3.2 Estimator of the Current Mean 24 3.3 Estimator of the Change 28 4 Two-stage S P R wi th Unequal Probabil i t ies 34 4.1 Definition of the sample 34 4.2 Estimator of the Current Mean 38 4.3 Estimator of the Change 42 5 Appl ica t ion to Chinese Forest Inventory 48 5.1 Background on Chinese Forests and Inventory Problems 48 5.2 Design of a Two-stage Sample . 53 5.2.1 General Considerations 53 5.2.2 The Shape and Size of Sample Plots : 55 iii 5.2.3 Sample Size 57 5.2.4 The Layout and Survey of Sample Plots 61 6 Conclusions 63 7 Literature Cited 66 A Derivation of Equation 3.6 73 B Derivation of Equation 3.11 76 C Derivation of Equation 4.19 79 iv L i s t o f T a b l e s 3.1 Various observations and sample plots 19 4.2 Various observations and sample plots 35 5.3 Forest types and their amounts in China 49 5.4 Forest land utilization in China 49 v A c k n o w l e d g e m e n t s I am most grateful to Dr. Peter L. Marshall, my supervisor, for his guidance, encour-agement, enthusiasm, patience, and understanding throughout the period of my gratuate studies at the University of British Columbia. Special thanks to the members of my thesis committee, Dr. A. Kozak, and Dr. J . H. G. Smith for reviewing the thesis draft and making very useful comments. I am very grateful to the State Educational Committee, Government of the People's Republic of China, for financial assistance in form of the National Overseas Graduate Scholarship for two years, and to the Department of Forest Resource Management at the University of British Columbia for providing a teaching assistantship. I thank Nanjing Forestry University and the Ministry of Forestry of the People's Re-public of China for giving me the opportunity to study abroad. I extend my sincere appreciation to the Faculty of Forestry, especially the Department of Forest Resource Management at the University of British Columbia for providing such an excellent learn-ing environment, and to my teachers and my fellow biometrics students at the University of British Columbia. I thank Dr. Val Lemay of the University of British Columbia, and Professor D'eyin Zhou of Nanjing Forestry University for reviewing the thesis draft and for their comments, as well as Mr. Basivi Reddy and Mr. James Thrower for assisting during the early stages of writing this thesis. My appreciation also goes to my fellow Chinese students in Forestry, and to the General Consulate of the People's Republic of China in Vancouver. Finally, I am very grateful to my mother, my grandmother, and my friends in China. vi I shall always feel indebted for their continuing encouragement and moral support. vn To my mother, I dedicate this thesis. C h a p t e r 1 I n t r o d u c t i o n Originally, inventory was a commercial term meaning the preparation of a detailed descriptive list of articles with number, quantity and value of each item (Loetsch and Haller, 1964). In forest resource management, the term inventory has a wider scope. Forest inventories require specialized and often complicated information and consequently expensive techniques. In recent years, the increase in intensive forest management has stimulated devel-opment of integrated or multiresource inventories (Beers, 1978; Lund, 1979; McClure, 1979). Forest inventories have expanded to cover a broader range of topics. A complete forest inventory not only provides information about the estimates of timber quantity and quality, but also provides estimates of growth and drain. When needed, additional information may be collected according to the purpose of the inventory. Many statistical methods have been used in forest inventories to summarize data and estimate resource characteristics. Relying on sampling for collecting important series of data that are published at regular intervals has become common (Cochran, 1977). Forests are dynamic populations and a. survey at infrequent intervals is of limited use. Highly precise information about the characteristics of a population in 1970 and 1980 may not help much in planning that demands a knowledge of the population in 1990. A series of small samples at shorter intervals may be more valuable. A wide variety of sampling techniques and associated statistical procedures is applied to forest inventory designs for both single-occasion and continuous forest inventories. The 1 Chapter 1. Introduction 2 forester is often faced with the question of which sampling technique to use. The answer to this question depends in part on the sampling objectives. Often, the objective is to minimize cost subject to precision limits, or maximize precision subject to cost limits (Fontaine, 1973; Cochran, 1977). Cost is measured by the resources required to conduct the survey. The precision of a sampling procedure is usually measured as the variance of the sample estimator. The best of the several alternative procedures is the one that most nearly meets survey objectives. Stott and Semmens (1962) defined the term "Continuous Forest Inventory" (CFI) as "a precise frequently repeated, and directly comparable measurement of all commer-cial trees in systematically placed sample plots. These plots have fixed radii and are permanently located in the forest. Their treatment, and the treatment of surrounding forest must be analoguous." CFI's use repeated sampling to estimate quantities and characteristics of the forest present at different occasions and the change in the forest during the intervening period. They provide information on forest populations at various points of time. Single-occasion forest inventories can only provide information on the current values of the forest population at a given time. The limitations of single-occasion inventories in terms of estimating resource changes are obvious (Gillis, 1988). There are three different strategies that CFI's can follow. These are: • The sample plots at later occasions are all different from those at previous occasions. A new sample is drawn at each occasion, and the means and totals, as well as standard errors are calculated separately. Estimates of the change or growth would be the difference in the means for the two inventories. Early CFI's usually used this method. It can provide the current forest information in a fast and relatively cheap way. • The same set of sample plots is used on all occasions. This involves the concept Chapter 1. Introduction 3 of permanent sample plots. Estimates of the means, totals, and standard errors for each inventory would again be found as in the case of two separate inventories. Similarly, the differences between the means for each inventory would indicate the change in the forest. • Partial replacement of sample plots from occasion to occasion. This means that a certain portion of the sample plots observed on the previous occasions will be remeasured at a later occasion as well as a new set of sample plots. This technique has been accepted as a valid CFI technique for estimating current forest resource values and changes in these values over time. Detailed accounts of the development and application of this procedure to forest inventory are given by many researchers and many articles have been published on the general statistical theory of sampling with partial replacement. These will be discussed in the literature review. Partial replacement of sample units requires selection of sample fractions on each occasion (i.e., the proportion of permanent and temporary sample plots). These frac-tions can be selected either for the purpose of maxmizing precision or minimizing cost. Cochran (1977) made the following statements about replacement policy for the purpose of maximizing precision: 1. For estimating change, it is best to retain the same sample throughout all occasions; 2. For estimating the average over all occasions, it is best to draw a new sample on each occasion; 3. For current estimates, equal precision is obtained either by keeping the sample or by changing it on every occasion. Replacement of part of the sample on each occasion may be better than these alternatives. Chapter 1. Introduction 4 Since there is nearly always a positive correlation between the measurements on the same unit on successive occasions, the first two replacement policies are obvious. The third replacement policy is less obvious and it can be very flexible. Much of the research has been directed towards this aspect and the partial replacement policy is often decided by some kind of cost function because of the difficulty of optimizing the precision. In a typical CFI there are three quantities which we may wish to estimate. These are: 1 . The change in values of a parameter from one occasion to the next (e.g., the differ-ence between the total volumes on different occasions). One of the main purposes of a CFI is to obtain accurate information about the changes in values from occasion to occasion. 2. The average value of a parameter over all occasions (e.g., the average volume for the whole population at a certain inventory period). This value may be adequate for a population in which changes over time are slow, but this will not often be the case in forest populations so the overall average value may not be as useful as the current average. 3. The average value of a parameter for the most recent occasion (e.g., the average volume for the whole population at the time of inventory). This is the most impor-tant item in most CFI's. Most inventories concentrate on the current average and an accurate estimate of this value will ensure not only an accurate estimate of the changes in values, but also an accurate average value over all occasions. In national and other large forest inventories covering broad forest areas, each unit in the population can be divided into a number of smaller units, or subunits. For instance, the forest areas in China can be divided into TV different units according to forest region. These units can be subdivided into subunits according to forest types or ages. Chapter 1. Introduction 5 Suppose a sample of n units has been selected. If subunits within a selected unit give similar results, it seems uneconomical to measure them all. A common practice is to select and measure a sample of the subunits in any chosen unit (Cochran, 1977). This technique is called subsampling or multistage sampling since the unit is not measured completely but is itself sampled. The principal advantage of multistage sampling is that it is more flexible than single stage sampling (Hansen et al. 1953; Freese, 1962). It allows less intensive sampling, and can take the advantage of prior knowledge of a population which can be used to increase the precision or usefulness of the sampling. Multistage sampling is also a cost effective technique (Deming, 1950; Titus, 1979). In many forest sampling situations, locating and travelling to a sampling unit is expensive, while measurement of the unit is relatively cheap (this is partically true in Chinese forest inventories). In these circumstances, it seems logical to make measurements on two or three units at or near each location. It can yield estimators of a given precision at a cost lower than a completely random sample. This eventually reduces to a balance between statistical precision and cost. Substantial work has been done using multistage sampling in CFI's with SPR. Singh (1968), Chakrabarty and Rana (1974), and Rana (1978) used two-stage or multistage successive sampling. Omule (1981) provided an extensive review and derived a complete set of formulas using multistage sampling in CFI's with SPR. One practical problem of multistage sampling in an extensive population is that pri-mary units that vary in size are encountered frequently. If the primary unit sizes do not vary greatly, one method of overcoming this problem is to stratify by the size of the primary unit so that the units within a stratum become equal, or nearly equal (Cochran, 1977). The multistage sample may then be combined with any type of sampling of the primary unit. Results are given for stratified sampling of the primary units in a two-stage Chapter 1. Introduction 6 sample by Hansen et al. (1953) and by Cochran (1977). Often, however, substantial differences in size remain within some strata, and some-times it is advisable to base the stratification on other variables (Cochran, 1977). Concen-trated effort is required to obtain a good working knowledge of multistage sampling when the units vary in size, because the technique is flexible. The units may be chosen either with equal probabilities or with probabilities proportional to size (PPS) or to a prediction of size (3P). Various rules can be devised to determine the sampling and subsampling fractions, and various methods of estimation are available (Kish, 1965; Cochran, 1977; Brewer and Hanif, 1983). The problems of multistage sampling with units of unequal size are of common occurrence. The sampling procedures in CFI's are more complicated. Since the same population is sampled repeatedly, the sampler is in an ideal position to make realistic estimates both of costs and of variances and to apply the techniques that lead to optimum sampling efficiency. One important question is how frequently and in what manner the sample should be changed as time progresses. Many considerations affect the decision. In forest inventories, loss of permanent sample plots and other unexpected occurences may make the inventory very complicated. The question of replacement of part of the sample and the related question of making estimates from a series of repeated samples should be carefully considered. Much of the work in CFI's with SPR has been done using different equal probability sampling techniques. Relatively little research has been done involving unequal prob-ability sampling in CFI's. Sampling with unequal probabilities on single occasions has been studied in detail by Grosenbaugh (1964, 1965, 1971, 1976), Schreuder et al. (1968, 1971), Schreuder (1984). Furnival et al. (1987) and many other researchers. Althrough Van Hooser (1972, 1973) and Yandle and White (1977) presented some general theories about sampling with unequal probabilities in two-stage sampling, and Scott (1977, 1979, Chapter 1. Introduction 7 1984) discussed midcycle updating using PPS and 3P sampling, some questions remain unsolved. This is particularly true for large scale applications, optimum sample designs, sample size calculations and sample allocations. A general theory of multistage sampling with unequal probabilities in CFI's with SPR has not been developed for forestry. Many researchers have shown that sampling with unequal probabilities can be much more efficient than sampling with equal probability, particularly when the sample units have large variations which is often the case in forest inventories. Consideration of cost often dictates the use of multistage sampling. Hence the combination of multistage sampling with unequal probabilities is particularly meaningful in CFI's. The main objectives of this study are: 1. To summarize the use of CFI's with SPR and discuss multistage sampling using unequal probabilities of selecting sampling plots in CFI's with SPR, and combine the general theory of multistage sampling with unequal probabilities and SPR in CFI's. 2. To illustrate the application of multistage sampling with unequal probabilities in CFI's with specific reference to Chinese forest situations. In the following chapters, a literature review is given first, and then, the combination of multistage sampling with unequal probabilities and partial replacement is discussed. The estimators are given for current means and change in means for both one-stage and two-stage SPR with unequal probabilities are given. An application is discussed with special reference to Chinese forest situations and some suggestions and recommendations are given. Finally, the conclusions are presented. C h a p t e r 2 L i t e r a t u r e R e v i e w The concept of CFI was originated by Stott (1947). It involves the concept of perma-nent sample plots (PSP's) which means that the established sample plots are measured more than once. Based on a PSP system, periodic measurements are made to determine primarily tree growth and yield with time or treatment. The original purpose of these CFI plots was to collect volume, growth and mortality information and provide a base for better forest management (Stott and Semmens, 1962; Stott, 1968; Solomon, 1979). In the early stages of CFI development, the plots established at the first occasion were all remeasured at later occasions (Spurr, 1952; Cutter, 1955; Hall, 1959). Through these repetitive measurements, estimates of present volume and change in volume, basal area, ingrowth, and diameter distribution were obtained. Although remeasurement of all plots could provide more precise information, and many researchers argued for its advantages, the general theory of CFI in the early years was not very clear, and the high cost limited its use. With the rapid development of sampling techniques, many more efficient procedures were presented. There is a considerable volume of literature which addressed the statisti-cal theories used in CFI's. Contributions have been made by many researchers. Repeated sampling with partial replacement (SPR) was introduced by Jessen (1942) and its ap-plication resulted in a great gain in sampling precision for estimating the current mean. Patterson (1950), Cochran (1953), Yates (1960) and many other researchers continued to work on the general theory of repetitive sampling. A detailed discussion of the statistical 8 Chapter 2. Literature Review 9 development was given by Omule (1981). Most of the papers reviewed here address the development of CFI with SPR in forestry situations. Bickford (1956) first introduced the concept of SPR into CFI's. In his proposed design for a continuous inventory of the Northeastern United States, he presented the idea of a partial replacement sampling scheme for forest inventories and pointed out some of its advantages. Three years later, Bickford (1959) reported an application of CFI with SPR to the Allegheny National Forest in Pennsylvania. He concluded that inventories need repeated sampling and good designs. He further noted that for an area as large as a national forest, there are usually sizable differences in forest cover and stratified sampling is, therefore, a more efficient procedure. Several sampling strategies deserving consideration were presented including SPR. His proposed method of continuous inventory was almost the same as the procedure later fully developed for CFI using SPR theory by many other researchers. The method consisted of remeasurement of enough of the plots established at the first occasion to obtain the needed accuracy in periodic net growth and establishment and measurement of enough new plots to obtain the desired accuracy in current volume. He then estimated the current volume by combining the independent estimates obtained from the different types of sample plots. A test was made in the Allegheny National Forest. The test made use of: aerial photographs, double sampling, point sampling, remeasured plots, new plots, and adjustment through regression^ To be able to use stratified sampling, area by stratum was obtained independently from aerial photographs, classifying a large number of plots, or by some other adaptions of double sampling. This may be the earliest example of an application of CFI with SPR in forestry. One of the most influential papers on CFI with SPR was by Ware and Cunia (1962). They discussed the sampling and statistical aspects of the use of remeasured permanent plots and partial replacement of the initial sample for forest inventory. A thorough Chapter 2. Literature Review 10 conceptual presentation and theoretical derivation made the efficient sampling design for CFI with SPR initially proposed by Bickford (1956, 1959) appear to be a very attractive scheme. Simple random sampling was combined with partial replacement of sampling units on two successive occasions. The most efficient estimator for current volume at the second occasion was given, along with its variance. Ware and Cunia (1962) concluded that this weighted estimate was the most efficient linear estimate possible given the sample information under their assumptions of known variance. This means in statistical terms, it is the best linear unbiased estimate (BLUE) obtainable. In estimating the change in volume, Ware and Cunia (1962) presented several estima-tors and their ancillary variances according to various remeasurement procedures. The most efficient of these was the difference between the most efficient estimator of current mean volume at the second occasion and the most efficient estimator available at occasion two for the mean volume at occasion one. When using SPR it is necessary to determine how many plots to replace. This involves calculating the optimum sampling fraction. One set of optimum solutions was given for estimating current mean volume, and one for simultaneously estimating both change and current mean volume by Ware and Cunia (1962). They applied a graphical solution which minimized a linear function of the variances. An example of the application of the derived theory was presented for a large scale forest inventory and it showed an increased efficiency and cost saving when using CFI with SPR. Bickford (1963) clarified the application of SPR to CFI by explaining what was in-volved in successive inventories and providing a basis for identifying the most efficient sampling procedure in relation to particular needs. He noted that sampling can be a major problem when facing the various situations, and that, although several theoret-ical studies had been completed, much was still unknown. The formulae given in his Chapter 2. Literature Review 11 paper were similar to those of Ware and Cunia (1962). However, he gave some ratio-nale for choosing among the many types of successive inventories in order to obtain the information desired in useable forms, and at approximately least cost. Bickford et al. (1963) presented an efficient sampling design for CFI which was devel-oped to provide data to meet a pre-set standard of accuracy at least cost for the forest survey of the Northeastern United States as the second survey was made. The essentials of the method were: an initial stratified inventory with some remeasurable plots, remea-surement of some of the initial plots, and establishment of some new plots in a stratified sample upon the second occasion. It was essentially a stratified double sample at the first occasion^ followed by SPR at the second occasion. Formulae were given for estimating total volumes and areas and their sampling error(variances of the estimates), and for cal-culating number of plots to establish and remeasure in order to meet an accuracy goal. The design was readiljr amendable to local intensification, and the principle of SPR was useful even if the first sample was unstratified. They stated that this design should be of particular interest to those concerned with repeated inventories of extensive forested area who would like to obtain the required data at least cost. Cunia (1964) continued work on SPR. He urged the application of CFI with SPR and pursuaded the user to ignore the complicated statistical aspects of the theory. At this stage of development, although CFI with SPR theory had been advocated by many researchers, very few applications of SPR to forest inventory had been reported. The literarure also differed as to which estimators should be used, and many researchers still worked on this aspect, trying to improve the theory. Cunia (1965) extended the theory of sampling populations on two occasions with SPR to multiple regression estimates. He showed that multiple regressions usually are better than simple linear regression estimates. Formulae were derived for the best estimators of current averages and changes in these averages from the first to second occasion and the Chapter 2. Literature Review 12 standard errors of these estimates were also given. An example was given of its application for the Canadian International Paper Company's limits in Quebec. The sampling method used was basically cluster sampling with systematic subsampling. Cunia (1965) also suggested that the theory of SPR on two occasions with multiple regression could be extended to SPR on multiple occasions. Letourneau (1966) reported a forest inventory by continuous forest control. Pleines and Letourneau (1970) further discussed CFI control systems, and later, Letourneau (1979) used simulation techniques to update CFI results. Frayer (1966) conducted a rigorous analysis of available data to test the validity of up-dating timber volume estimates by means of least-squares regression. Since CFI with SPR involves remeasurement of a portion of the initial sample, certain assumptions involving the distribution of regression residuals must be made to justify the use of regression tech-niques. These assumptions include independence, normality, and homogeneous variance of the residuals as well as freedom from error in the independent variables. Data from West Virginia and New Hampshire were tested to determine if the CFI's with SPR model developed by Ware and Cunia (1962) violates any of the assumptions of least-squares re-gression. Frayer (1966) found that all assumptions were justified except homogeneity of the variance. He employed a function of the expected current volume as a weight, and used weighted regression to stabilize the variance. His work suggested that the weighting procedure for estimating variances was more efficient. Frayer and Furnival (1967) presented a method of calculating changes in area from sampling with partial replacement. Previously SPR had been concerned primarily with volume estimates. They showed how the SPR formulae could be used to estimate area and they gave an example based on data taken from the forest survey of Vermont to demonstrate the procedures and efficiency of the estimate. Cunia. and Chevrou (1969) extended theory from two "successive occasions to three or Chapter 2. Literature Review 13 more remeasurements of the same forest population. This was mainly built on the work of Ware and Cunia (1962) and Cunia (1965). Formulae were given for the best estimates of the current values and change in these values from one to any of the subsequent measurements, and the variances of these estimates. However, CFI with SPR on three or more occasions seemed to be very complicated statistically as well as in its practical applications. For consecutive remeasurement on more than two occasions, the correlation structures may be complex and the regression assumptions may not be met. This limits the application of this theory. As we will see later, the CFI with SPR theory has many alternatives, but the applications remained mostly for two occasion cases. Frayer et al. (1971) gave an example of the application of SPR to timber inventories in the Central Rocky Mountains. The results indicated a substantial savings in time and money with CFI using SPR, especially when the correlation between the remeasurement plots is high. See (1974) examined SPR theory on successive occasions to establish the validity of the claims of increased efficiency in comparison with conventional systems. Several examples were discussed and he concluded that for estimation of a mean on more than one occasion, SPR was superior to conventional independent surveys in cost, and for estimation of both current mean and changes in means on successive occasions, SPR was again a superior system. Newton et al. (1974) extended the work on CFI theory from the univariate case to the multivariate case. General formulae were given and used to calculate stand tables for a particular forest area measured on two occasions. Problems related to practical application were discussed. Simple random sampling was the basic sampling design used in gathering the data. Hazard and Promnitz (1974) proposed convex mathematical programming as a method Chapter 2. Literature Review 14 of optimally allocating forest inventory sampling resources under different sampling plans to meet specified precision requirements for several variables.' Prior to this, optimum de-sign was only discussed in a graphical sense by Ware and Cunia (1962). The solutions presented in their paper illustrate that the optimum replacement fraction in CFI with SPR can vary from complete remeasurement to large replacement fractions depending upon the specified precision levels, the population parameters, and the relative costs of obtaining information. Results of a sensitivity analysis were also given, along with an example. Dixon and Howitt (1979) used Kalman linear filters to estimate the current means. General formulas are given and they showed that the partial replacement estimator of Ware and Cunia (1962) was a special case of the Kalman estimator. They also proved that the best variance estimate of Ware and Cunia (1962) was always greater than or equal to the variance of the corresponding Kalman estimator. The theory was developed solely for measurement on two occasions, and only current value estimators were discussed. Peng and Zhu (1985) used Kalman linear filters to estimate the change in means. Two estimated values of the linear filter were used. In addition, the linear filtering formula, which was used in various SPR projects for CFI on two occasions, was given and some associated problems discussed based.on the results of forest inventories made in a forestry bureau of Jilin Province, China. Omule (1981) extended the work of Ware and Cunia (1962) from one-stage SPR to multistage SPR. He noted that theoretical developments in multistage SPR had been restricted to situations in which the sample size and variance are constant over time, and no application of multistage SPR had been reported in forestry. He derived the theory of multistage SPR without the restrictive assumptions of constant sample size or variance on the successive occasions. For practical purposes, Omule (1981) considered only the case in which partial replacement occurs at the primary stage of the multistage design. Chapter 2. Literature Review 15 The multistage design was based on simple random sampling within each stage. For estimators of the current population mean and change in means between two successive occasions, Omule (1981) derived estimators which were B L U E , together with their respective variances for a multistage structure. He also gave the biased estimators of the ratio form of the current population mean and of the change in means between two successive occasions, and made some comparisons among the different estimation methods. An example of two-stage design was given based on data obtained from the Cranbrook Public Sustained Yield Unit in British Columbia. Results shown were con-vincing. He also suggested that this multistage SPR theory maybe particularly useful for inventories of large and diverse forest areas. Omule and Williams (1982) employed dynamic programing to determine replacement fractions. Most of this work was derived from Omule (1981). One example was given and the solutions showed that the optimum results were almost identical to those of other optimization techniques such as the graphical procedure. However, they explained that dynamic programming could be more appropriate since it could handle optimization problems in which the sampling extended over several occasions. Omule and Kozak (1982) discussed different estimators for successive forest sampling with SPR. They assumed that the coefficient of variation of timber volume remains constant over successive occasions and analysed previous estimators for the current mean volume. An efficient estimator of the current mean volume was developed; it was a modification of the estimator by Ware and Cunia (1962). Results showed that the new estimator could provide gains in precision up to 200 percent when used for even-aged and normally distributed forest populations. The estimator was obtained using a simple random design. Omule (1984) presented the multistage SPR theory in a simpler and more under-standable form. A two-stage sampling design on two successive occasions was used as Chapter 2. Literature Review 16 an example. He also noted the possibility of improving the precision by using variable probabilities to select the sampling units at the various stages of the multistage design, especially when the sampling units within each stage are of unequal size. Cunia (1987a) presented some theory on the error of CFI estimates and gave an illustrative numerical example. An approach was shown that introduced the error of volume tables into the total error of the CFI estimates of average volume per tree, average volume per acre, and growth components. Cunia (1987b) extended the approach to CFI systems using SPR. He obtained SPR estimators which were slightly different from the classical estimators. The formulae are derived for the case of SPR on two occasions when (1) volume tables are constructed from linear regressions for which an estimate of the covariance matrix of the regression coefficients is known, and (2) the sample plots or points are selected by simple random sampling independently of the given volume regression function. Recently, Gillis (1988) discussed some problems in estimating change from successive static forest inventories. He noted that the estimated results were frequently masked by artificial differences and CFI should be done with detailed background knowledge. It can be seen from the literature reviewed above that there are many ways that CFI's can be conducted and that many estimation formulae exist. However, the basic sampling design in CFI's with SPR is the same. Equal probability sampling was always used whether the actual design was simple random sampling, stratified random sampling, or multistage sampling. In the following sections, multistage unequal probability sampling with SPR will be described. The SPR estimate procedures used here originated from Ware and Cunia (1962), Cunia (1965), and Omule (1981). The sampling structures for multistage SPR are assumed to be the same as those developed by Omule (1981). The unequal probability sampling theory, which was first developed by Hansen and Hurvitz (1943), is summarized Chapter 2. Literature Review 17 mainly from Cochran (1977). A detailed discussion of unequal probability sampling theory can be found in papers by Hendricks (1956), Kish (1965), and Brewer and Hanif (1983). Chapter 3 One-stage SPR with Unequal Probabilities 3.1 Definition of the Sample Suppose a population consists of N units of unequal size. A random sample of size n1 of the ./V units will be selected with probability proportional to a measure or an estimate of size with replacement (PPSWR) on the first occasion. The probability of selection of the zth population unit is pxi = Ma/Mi, where Mu is the measure of size of that unit and Mi — Y^iLi Mu is the total measure of size of the N units in the population on the first occasion. Further, suppose a random sample (selected by PPSWR) of size np of the n1 sample units is retained for remeasurement on the second occasion. The remaining part of the rii sample unit is ntl, and is measured only on the first occasion. In addition, a random sample of size nt2 of the TV — rii other sample units is selected by ppswr on the second occasion. The probability of selecting the zth unit on the second occasion is p2i — M2i/M2, where M2i is the measure of size of that unit and M2 — S i L i " 1 Mu. Thus, there are rii — np + ntl sample units on the first occasion, and n2 — np + nt2 on the second occasion. Assume the variable of interest on the first occasion is A" and the same variable of interest on the second occasion is Y. The various observations and sample units can be defined in Table 3.1. In CFI's, interest centers on estimating the current population mean fly a n d the change in means A = fiy — fix- Suppose the observations on both permanent sample 18 Chapter 3. One-stage SPR with Unequal Probabilities 19 Table 3.1: Various observations and sample plots name occasion 1 occasion 2 Permanent sample units np np Temporary sample units Tit, Observations on *i vi permanent sample units (i = 1, . .. Tip) (i = i , . . . T l p ) Observations on Vi temporary sample plots nti) (» = !,..., nt2) units and temporary sample units on the first occasion are described by the linear model: Si = Px + eii for i = l,...,N where X{ is the observation on the ith. unit, fix is overall mean of the observations, and en is the difference of the ith observation from the mean. Since the observations are made for different units, all the e^'s are assumed to be inde-pendent random variables with expected value 0 and variance <r\ . On the second occasion, observations on both permanent sample units and temporary sample units are described by a similar linear model: Vi - Mr + e2 i for i = 1,... , N-where yi is the observation on the ith unit, fly is overall mean of the observations, and £2i is the difference of the ith observation from the mean. Again, all the e2i's are independent random variables with expected value 0 and variance Chapter 3. One-stage SPR with Unequal Probabilities 20 <r\ . Further, it will be assumed that Corr(en, e2;) = p. The covariance between X j and yi therefore is Cov(xi,yi) = E{[xi-E(xi)][yi-E(y{)]} = E(eu,ex) = P<rci<rt3. From the sample observations after both occasions, the following four sample estima-tors are obtained: 1 " p x1-nv i = 1 pu Xi nu i=i pu i n j ' n1 np i = 1 p 2 i 1 n t 2 n t 2 i = l V2i where Xp is an unbiased estimate of the population total based on the observations matched on occasion 1 and 2, t , is an unbiased estimate of the population total based on the observations unmatched on occasion 1. Yp is an unbiased estimate of the population total based on the observations matched on occasion 2 and 1, and Yt2 is an unbiased estimate of the population total based on the observations unmatched on occasion 2. The above unbiased estimates of the population total can be converted into unbiased estimates of the population mean by dividing the population size N. Chapter 3. One-stage SPR with Unequal Probabilities 21 1 n * l - = J _ f 1L ^ Nnp fr[ P 2 i -I " t o Nnt2 i = l P2i To derive the variances of these estimators,- let ti be the number of times that the i t h unit appears in a specific sample np. For sampling with replacement, the i t h unit may appear 0, 1, 2,.. . ,np times in the sample. The probability of the zth unit being selected is pu in each draw. This is a multinomial problem. Consequently, the joint frequency distribution of the for all N units in the population is the multinomial expression: . , , , • .P11P12 • - -PIN-For a multinomial distribution, the following properties are well known: E(U) = nppu, V(U) = nppu(l - pu),and Cov(ti,tj) = -Uppupxj. It follows that, 1 . Xi A 2 Xif -(ti h t2 h • • • + ijv" Nnp pn P12 PIN 1 N Y N n p " 1 PU The only difference between the formulas X^-Nn~Pti^ and 1 ^ . X i Mnp^l 'pu Chapter 3. One-stage SPR with Unequal Probabilities 22 is that the sum extends over all units in the population in the later formula. In repeated sampling the £t's are the random variables, while the Xi and pu are a set of fixed numbers. Since E(ti) = nppu, hence, E(xp) = £ ( ^ _ f > p P l j ) ^ ) = E(X) = fix • Therefore, the variance of xp is: 1 A Xi V(x-P) = vi—Zu — ) Nnpf^l Pli 1 \£(-)2V(U) + 2J:^Cov(ti,tj)] -jH- E(-)ap«(i - P * ) - 2E - ^ p i ^ i , ] ^ % £i Pi i £2 P J i P!i (E —-*2) -^ 2 np i l l P l i 1 N Y £ P I . ( - - * ) S N 2 T l p i t l P l i where 1 ^ A'-s i = 1H52PK{— - X)2,and i V »=i P i i The unbiased estimate of V(xp) is: Similarly. i N 1 , , X; E(xtl) = E(—— ^2(ntipu)—) — E(X) = px  N n u i=i PIT Chapter 3. One-stage SPR with Unequal Probabilities and E(yP) = E{^-Y,{npp2i)^~) = E(Y) = fiY E(Vt2) = E(-^J2(nt2p2l)^) = E(Y) = liY. The unbiased estimators of these variances are: = A / 2 , . ^ M i - g p f n t 2 1 i V 2 n t l ( n t l - 1 ) 7 \ N2np(np -.1 ^ 2 n t 2 ( n t 2 . - 1 ) and the covariance between mean of the sample plots observed on both occasions Cov(xp,yp) = — N2Tl2p ~[fr[Plr^P2i »2 where 1 n p "r 1 The CFI estimators can now be derived using these terms and formulas. Chapter 3. One-stage SPR with Unequal Probabilities 24 Since, 3.2 Estimator of the Current Mean The most commonly used (Ware and Cunia, 1962; Omule, 1981) estimate of the current mean py on the second occasion is obtained by a linear estimator of the form: yb = axxu + bxxp + cxyp + dxyt2 where ax,bx,cx, and dx are constants. To make this estimator an unbiased estimator of u-y, it follows that: E(yb) = py. E(xtl) = E(xp) = px E(yt2) = E(yp) = fly E(y~b) = a-ipx + bxpx + c i / x y + dxpY = (<*i + bt)fiX + ( c i + dx)pY ax + bx = 0 cx -\- dx = 1. Vb = ax(xtl - xp) + cxyp -f (1 - cx)yt2. (3.1) therefore, then: Hence. The variance of y;, is: V(yb) = al[V(xtl)-r V(xp)) + c\V%) + (1 - cx)2V(yt2) - 2axcxCov(xp,yp) (3.2) = g 2 ( i L + f l ) + C 2 j j + (1 - Cl)2ii- 2 a l C i e P a ^ . • (3.3) ntl np rip . nt2 npl Chapter 3. One-stage SPR with Unequal Probabilities 25 In order to calculate ay and Cy and minimize V(yb), the equation (3.3) is differentiated with respect to ay and Cy and the equations set equal to zero. That is: d[V(yb)] d[V(yb)} d(cy) = 0 = 0. Therefore, 2 f l l ( i l + i l ) - 2 C l ^ ^ = 0 ntl np n p np Tii2 Tip Solving the above two formulas simultaneously, after simplification, yields: _ T)s7n\nix np2s1s2n1n2 - ntlni2r]2 (3.4) d = — L 2 _ L P (3.5) np2s-lS2n1n2 - ntlTii2T)1 where r/ = 8pa£l a£2. The relationship of a a and Cy can be written as: TjTltlCi ay = . SyTlyTlp Substituting the ay and cy into (3.1) yields the best estimate of the current mean: yb = ay(xtl - xp) + cxyp + (1 - cy)yt2 .= {xtl - Xp) + cyyp 4- (1 - cy)yt2 SyTlyTlp = cy{ VTltl (xtl - xp) +yp] + (1 - cy)yt2. synynp Notice that the overall mean of the population on the first occasion, ntlxtl + npXp x — Tly nti - , nP-— xtl H xp. Tly Tly Chapter 3. One-stage SPR with Unequal Probabilities 26 Since np = nx — ntl, the above formulas can be written as: x — x v ni (xtl - xp). Thus, Note that: Vb = c i [ {x - xp) + yp] + (1 - cx)yt2. SlTlp r, _ Cov(xp,yp)n2p _ — Pi sxnp V(xp)npnp where j3i is similar to a regression coefficient. Therefore, Vb = C ! [ ^ i ( i - x p ) +yp] + (1 - ci)yt3 - Ci-Vi + (1 - cx)yt2 where yx = /3x(x - xp) +yp. Substituting the calculated values of (3.4) and (3.5) into (3.3) yields the general formula for calculating the minmized variance of 'V(i/ 0): n p ^ i f i z n x ^ - ntlnt2rj' ntl np np2sis2nxn2 - ntlnt2r]2 np + ^ ^ S 2 7 i p 2 s 1 ^ 2 ^ i " ' 2 — ntlnt?rj2 nt2 Notice np^s1s2nin2 - ntlnt2T]^ npAsxs2nxn2 - ntlnt2T]* nv* sxs2nxnlntn - ntlnt2rj 1 - d - : " •-- • TOp2s152n1Ti2 - ntlnt2rj2 Chapter 3. One-stage SPR with Unequal Probabilities 27 After simplification (See Appendix A), V(yi) can be expressed as: y ( f c ) = — ( 3 . 6 ) If ntl = nt2 — nt and nx — n2 = n (the number of sample plots on both occasions is the same), then: T]s2ntnl n2n2sxs2 — nt2r]2 n2n2s1s2 - n t 2 n 2 ' Substituting these values into ( 3 . 3 ) yields: nzn£s1s2 — nil]1 nt nzn£s-lis2 — nfr2 np i f nsis2n3p 2 s 2 s2 n^n^Si - nfrj2 np nt ns2ntn2p nsxs2nl rj n2n2SiS2 — n\j]2 n2n2psxs2 — n2r}2 n p 2 Notice _ nntn2psis2 - n2r)2 nzn£siS2 — n^rj^ After simphfication, the variance V(y~b) c a n be expressed as: ^ b i ) = - ( l - c n ) . nt Another assumption often made in CFI's states that the variances on both occasions are the same. This means: •Si = s2 = s nt, = n h - nt n1 = n2 — n. Chapter 3. One-stage SPR with Unequal Probabilities 28 The ai and C\ values in this case become: •qsntn2v 0-12 = 9 9 9 2 9 nznps'i — nirj-ns2nl C l 2 = n2nps2 — n2rj2 Substituting these values into (3.3) yields: V(yb2) = ( ^ ^ \ 2 — +(2 2 w f L) 2-+(i-a-n^n^s* — rtfTj^ np nt . Vsntn2p ns2n3p v V 9 9 9 2 9 'V 9 •> 9 2 9 / 9 ' nzripS' — niT]* nzn~sz — nirjz np Noti ce: nntn2s2 — n2n2  1 2 n2n2s2 — nt2rj2 After simplification, the variance V(yb) can be expressed as: V(yb2) = - ( 1 - c 1 2 ) . 3.3 Estimator of the Change The most commonly used (Ware and Cunia, 1962; Omule, 1981) estimation of the change in means between two occasions, A = py — px, is by a linear estimator of the form: A b = a[xtl + b'xxp + c[yp + d\yt7 where a\, b[, c[, d[ are constants. To make this estimator be the unbiased estimator of A , it follows that: E(Ab) = u.Y - px = A . Chapter 3. One-stage SPR with Unequal Probabilities 29 Since, therefore, then. E(xu) = E(xp) = px E(yt2) = E(yp) = pY E(Ab) = a[nx + Kfix + CW + d W = [a\ + b[)px + (c[ + d'^py a[ + b[ = -1 c'i + = 1-Hence, A b = b[xp + c[yp + (1 - cpy t, - (1 + b[)xtl. (3.7) The variance of A(, is: V(Ab) = b^^ + cfVtiJ + il-tfVfa) + (1 + b[)2V{xtl) + 2b'1c'1Cov(xp,yp). Substituting the different variances and covariance into the formula yields: V'(A 6) = b[2fl + cftl + (1 - e p z i l + (1 + 6 p 2 . f i -r 2 6 ' ^ ^ ' ^ . (3.8) n p n p n t 2 n ( l np-1 In order to calculate b\ and c[ and minimize the variance V(Ab), the above formula is differentiated with respect to b\ and c[ and the equations set equal to zero. That is d[V(Ab)} d{b[) d(c[) = 0 = 0. Chapter 3. One-stage SPR with Unequal Probabilities 30 Therefore, 2 6 ' 1 ^ + 2 ( l + & ' 1 ) - ^ + 2 c ; ^ ^ = 0 Tip Tit i Tip 2 ^ - 2 ( 1 - ^ + 2 6 1 ^ ^ - 0. Tip nh np2 Solving the above two formulas simultaneously and simplifying yields: , _ -nls2(Sln2Tip + ntlT)) s1s2n1n2nl - ntlnt2Tj2 , _ nls^s^rip + nt2r])  1 S j s ^ n ^ l - ntlni2ri2' where r/ = 8paeiae2, and the relationship between and c[ can be seen to be: y _ _rip _ ntlrjc\ 1 T i i <SlTiiTip Substituting the calculated values of (3.9) and (3.10) into (3.7) yields: Ab = b'^Xp - xtl) + c[yp + (1 - c'^yt, - xtl = (— ){XP ~xt1) + c-iVv + (i - ci)yt2 - xtl Tli S\TliTlp = c'i{yP -^^-{xp ~ *tj] - — (S P - s t l) + (1 - ci)i/t2 - S t l -5 l T i i T i p T i i Notice that the overall mean on the first occasion is: ntl xtl + r ip ip (3.10) x = ^ x t l + ^ ^ l x p T i i n l = Xp + (xtl - Xp). Til Hence, Chapter 3. One-stage SPR with Unequal Probabilities 31 and Also, 7 1 1 1 - - \ xp - x t l = —(x - xtl). V Cov(xp,yp)nl "•- Pi • Therefore, SjUp V(xp)npnp &b = c[[yp - Pi(xp - x)] - (x - xtl) + (1 - c[)yt2 - xtl = C'IV'I + (i - ci)y<2 - * = y'b-x where: V'l =yP- A ( s P - x), and y'b = c'1y'l + (l-c'1)yt2. This result means that the best estimate of the change on both occasions is the best estimate of the current mean on second occasion minus the overall mean on the first occasion. Now substituting (3.9) and (3.10) into (3.8), we get the general formula for calculating the minimized variance of V (Ab)-- n 2 p s 2 ( S l n 2 n p + n u n ) S l n^i(s 2ni» P + nhv) l 2 s 2 V(AB) = I ; 2 ; ; — s1s2n1n2n^ - ntlnt?T)z np s1s2nin2np - ntlnt2rjl np n l s ^ s ^ r i p + n^ri) s 2 - n ^ 2 ( a i n 2 n p + n t l r j ) S l +ll ~ 2 ; 1- 1 H 2 i — s i s 2 n 1 n 2 n ^ - n t i n t 2 T j z n h s 1 s 2 n 1 n 2 n * - n t i n t 2 r j 2 n t 2 ^ ^ - n 2 p s 2 ( s 1 n 2 n p - \ - n i i T ] ) ^ n ^ 1 ( 5 2 n 1 n p • + n t 2 v ) rj . SiS-iU^nl - n t l n t 2 r } 2 s1s2n1n7npi - ntlnt2r}2 n p 2 This can be simplified (See Appendix B) to give: . V ( A k ) = — ( l - c i ) + - ^ . ( l - ^ c i . ) . (3.11) nt2 TI a Chapter 3. One-stage SPR with Unequal Probabilities 32 The best estimate of variance is a. weighted estimate of the variances divided by the number of temporary sample plots on the second occasion and the total number of sample plots on the first occasion. When n t l = n f 2 = n t and T i j = n2 = n, b\ and c\ become: -s2nl(sxnnp + ntrf) bn = sis2n2n2 - n2rj2 , _ n2psx(s2nnp + ntn) 11 2_9 2 9 ' SiS2nznp — ni7]i Substituting the above values into (3.8) yields: \/(A \ -s2nl(s1nnp + ntri) S l n*ai(* 2wra P + ntrj) 2s2 V{Abl) = — 2 _ 2 — H r~2 2~1~$ ~ SiS^Tlp — TifT/^ Tip SiS^Tl* — n{TIZ Tip _ nlSl(s2nnp + ntV) s2 -s2n2p(sxnnp + ntn) S l '[ 9 9 2 9 J T [ 1 T „ , 2 9 J J S\s2nlnlv - nfri2 nt2 s^n^n* - nfrj2 ntl -s^Ksynrip +n tT/) n2psx(s2nnp + ntv) y I 9 9 2 9 H 9 9 2 9 J 9' SiS2nzn~ — niT)* SiS2nzn^ — niv/ np After simplification, the variance V"(A D l ) can be expressed as: V ( A D l ) = ^ ( l - c ' 1 1 ) + ^ ( l - / 3 1 c i 1 ) . Ti t n In CFI's, the assumption is often made that the population variances are the same on both occasions in addition to equivalent sample size. This means: S1 — S 2 - — s fit, — ntn = n, n-i — n2 = n. The b[ and c[ values become: , _ -5Tip(^TiTip + ntrj)  12 s2n2n2 — nfrj2 n2s(snnp+ntrj) s2n2n2 — nfrj2 Chapter 3. One-stage SPR with Unequal Probabilities 33 Substituting the above values of b'12 and c'12 into (3.8) yields: _ -sn2p(snnp + ntn) s n2ps(snnp + ntn) s ' \*-*b2 ) [ n 5 7 2 9 J ' I 9 9 9 2 9 J slnln^ — nfr)z np s*n*n* — nil}2- np +2[-n^(a7in p + nt7/) 2 j + h + - ^ ( j 5 7 t 7 i p + ^7?) s s2n2n2 — n 2 7/ 2 n t 2 s2n2n2 — 7 i 2 r/ 2 n t sn2(sranp + ntrj) n2s(snnp + nt7/) 77 5 2 7Z 2 n 2 — 7127?2 5 2 7 Z 2 n 2 — 7l2772 7Z2 This variance can again be simplified to: v(A M) = i - ( i - c ' 1 2 ) + r ( 1 - ^ ) -Chapter 4 Two-stage SPR with Unequal Probabilities 4.1 Definition of the sample As in the one-stage case, suppose the population consists of TV primary units (pu). Within each of the TV sample units, there are M secondary units (su). Both the TV pu and the M su within each pu are of unequal size. On the first occasion, a random sample of size rii of the TV pu is selected by ppswr. The probability of selection of the i t h pu at any of these draws is pu — Mu/Mi, where Mu is the measure of size of that unit and M i = Y^iLi Mu is the total measure of the size of the TV units. Within each selected rii of the TV pu, a random sample of size m of the M sample units is assumed to be drawn with probability proportional to a measure or an estimate of size without replacement (ppswor). If a particular su is drawn more than once, the whole sample is replaced, and a new independent drawing of m subunits, again without replacement, is made. Suppose a random sample of size np of the n1 pu is retained for remeasurement on the second occasion together with its respective su drawn from the first occasion. An additional random sample of size nt2 of the TV — rii other sample units is selected with ppswr on the second occasion. Within each selected pu, m su are selected again by ppswor. The probability of selecting the i t h pu on the second occasion is p2i = M2i/M2, where M 2; is the measure of size of that unit and M 2 = Hili" 1 M2i. The various observations and sample units are summarized in Table 4.2. The total number of sample units on the first occasion is riim = (np + ntl)m and 34 Chapter 4. Two-stage SPR with Unequal Probabilities 35 Table 4.2: Various observations and sample plots name occasion 1 occasion 2 Number of matched pu Number of unmatched pu n p  nu np nh Number of matched su Number of unmatched su npm ntlm npm nt2m Observations on permanent sample plots ™' (i = 1,... ,n p ) (j = 1, . . . ,m) V'ij (i = l,...,np) (j = 1, . . . ,m) Observations on temporarjr sample plots (i = 1, ... ,ntl) (j = 1, .. . ,m) Vij (i = 1, . . . ,n i 2 ) (;' = 1,... ,m) the total on the second occasion is n 2 m = {np -f nt2)m. From the observations on both occasions, the following four unbiased estimators of the population means are obtained: 1 ripMi i = l P i i 1 v MuXi nt,Mi Z-i = l pu 1 T i j , M2iy'i 7 l p M 2 P2i 1 Mnyi nt7M2 2- Pu where xp is an unbiased estimate of the population mean based on the observations matched on occasion 1 and 2, xtl is an unbiased estimate of the population mean based on the observations unmatched on occasion 1, yp is an unbiased estimate of the population mean based on the observations matched on occasion 2 and 1, and Chapter 4. Two-stage SPR with Unequal ProbabiHties 36 yt2 is an unbiased estimate of the population mean based on the observations unmatched on occasion 2. To find the variances of these estimators, let ti be the number of times that the zth unit appears in the specific sample np. Since ppswr was assumed in selecting of pu, the ith pu may appear 0,1,2,... ,np times in the sample. The probability of the i t h unit being selected is pu in each draw. This is a multinomial problem as in the one-stage case. The between-unit component can be written as: 1 N Y Its variance, Vj, can be obtained by the same method as in the one-stage situation, that is: 1 N Y npMi ~i Pi* The contribution of the su within a pu to the variance for a unit that is drawn once is 1 Mu(l-f2)Sl where f2 = ^  is the sampling fraction in second stage, and 1 M l '* 1 j=i Its variance, V2, is calculated as: npM\ £ p^m • Adding the variances obtained from pu's and su's to get the total variance of the two-stage sampling yields: ivi\np i = i Pit 1 y Ml(l-f2)Sli V ^ i ^ p2um. = 1+4 nP Chapter 4. Two-stage SPR with Unequal Probabilities 37 where 1 £ ,Xi 1 " M ^ ^ V -and 1 ^Mu{l-h)Sl E The unbiased estimate of V(xp) is obtained as: where / x = jjj is the samphng fraction in first stage, and X{ is the sample total in the ith pu and x is the average of the X{. Similarly, M i n h f r i Pu ntiMifr{ P2U m = - f L l f i VQ \ 1 f n ( Yl Y)2 + 1 f - '2>5* M i n P f^i P2i npM2 fr[ pliiri ' 4- s" 2 ' A 2 _ s2 ^ S2 and the unbiased estimators of these variances are: H"p) = (nP-l)lhM2)2^~*)2 Chapter 4. Two-stage SPR with Unequal Probabilities 38 The co variance between the remeasured sample plots, using the correlation structure given by Omule (1981), is expressed as: Cov(x'p,y'p) = (32/np where (32 = {m-picr^a*, + p2aeia£2)/m. Pi is the true population correlation coefficient between the primary units on the two occasions and p2 is the true population correlation coefficient between the secondary units within the primarj' units on the two occasions. The CFI estimators can now be derived using these terms and formulas. 4.2 Est imator of the Current M e a n The most commonly used (Ware and Cuina, 1962; Omule, 1981) estimate of the current mean py on the second occasion is obtained by a. linear estimator of the form: Vb = a2^t, + b2&p + c2yp + d2yi2 where a2, b2,c2,and d2 are constants. To make this estimator an unbiased estimator of uy, it follows that: E{yb)=pY-Since, E(xtl) = E(xp) = px E(yh) = E{yp) = Chapter 4. Two-stage SPR with Unequal Probabilities 39 therefore, E{yb) = a2px + b2px + c2pY + d2pY = (<l2 + & 2 W ' + (c2 + d2)pY-Thus, a 2 + 62 = 0 C 2 + ^ 2 = 1. Hence, Vb = a2ixh ~ XP) + C 2 2 / P + (1 ~ c2 ) y t 2 - (4.12) The variance of yb is f f o ) = al{V(xtl) + V(x.p)} + clV(§p). +(l-c2)2V(y:t2)-2a2c2Cov(x=p,§p) (4.13) 2/4+4' , 4+4'\ , 2 4 + 4' . „ \2S2 + 4' 0 „ „ ^2 .. ., = a2{ 1 J + c 2 h(.l —c2) 2a 2c 2 — -(4.14) 7 l r In order to calculate a2 and c 2 and minimize V(2/f,), equation (4.14) is differentiated with respect to a2 and c 2 and the equations set equal to zero. That is d[V(yb)] d(a2) d[V(yb)] d(c2) = 0 = 0. Therefore, 2 o ! ( f i ± i I + £ i ± i I ) 2 c f t = o ntl np np 4 + 4' , ,4+4' „ 02 2c2 2 2 - 2(1 - c 2) 2 2 - 2a 2— = 0. Tip n i 2 rap Chapter 4. Two-stage SPR with Unequal ProbabUities 40 Solving the above two formulas simultaneously, after simplification, yields: K + s i ) ( s 2 + 4>in2 - ntlnt2^ { ' ; The relationship of a2 and c 2 can be written as: _ f32ntlc2 °2 " (*i + * i > i ' Substituting (4.15) and (4.16) into (4.12) yields the best estimate of the current mean: Vb = a2{xh - i p ) + c2yp + (1 - c 2 )y t 2 / 3 2 n t , c 2 Notice that the overall mean of the population on the first occasion, = — a j t l + — V rii ri! Since n p = Tlx — ntl, the above formulas can be written as: = _ = - ^ i i ^ _ = ^ X Xp — y^t\ *^p)' Hi Thus, Suppose, yb ~ C2L „Ax ~ xP) + yp) + (1 - ci)yt2. \S1 -T S l ) ft ft Chapter 4. Two-stage SPR with Unequal Probabilities 41 where j3r is similar to a regression coefficient. Therefore, Vb = c 2 [/3 r(l - xp) + yp] + (1 - c2)yt2 = c2y2 + (1 - c2)yh, where y2 = f3r(x - xp) + yp. Substituting the calculated values of (4.15) and (4.16) into (4.14) yields the general formula for calculating the minimized variance of V(yb): V(v) - ( /3^S'2 + S'^U^ W-> | -»V 1 | 1 ) (*i + *"){*2 + 4)ninp M + s'i) (s\ + s")(s2 + 4 ' ) n i n2 - ntlnt2f3i np Notice + /j _ (*i + s")(*2 + agViTip , 2 (5 2 + s'2') _ 2 / ^2(4 + w (*1 + * l ) ( * 2 + • S 2 ) n l T i - p jfo (*i + s")(32 + s2)nin2 ~ ntlni3(3% (si + sJ'Xs'rj + s 2).*in 2 - ntlnt20l np' (s[ + s'{(s'2 + s'Drixnt, - ntlnt2/32 1 - c2 -(s[ + s")(s2 + s ' 2 ' ) n i n2 - ntlnt2f3: After simplification, V(yb) can be expressed as: V(yb)={-^^(l-c2): If ntl — nt2 = nt and nx = n2 — n, then: /32(s'2 + s2)npnt a 2 1 C21 n 2 ^ + S'/(5'2 + 4') - V& 2 71(5; + 4)(4 + 5'2')np n 2 W + « I)('2+ 4)-'H% 2' Substituting these values into (4.14), after simplification, the variance V(yb) can be expressed as: m 1 ) = ^ ± ^ ( i - C 2 1 ) . 71, Chapter 4. Two-stage SPR with Unequal Probabilities 42 Another assumption often made in CFI's states that the variances on both occasions are the same. This means: i i a ' i " 711 = 71 2 = n. The a 2 and c 2 values in this case become, (32snpnt &22 c 2 2 n V - n2/32 ns2np n2s2 - n2t(3l' Substituting these values into (4.14) yields: v r x -fasripnt 2 s 02snpnt 2 5 2 l n 2 5 2 - n 2/? 2 n t ™ 2 5 2 - n2/322 n P Notice 2 , ,- ns np 2 s 2 5 + ( n 2 , 2 - n 2 / 3 2 ) ^ + ( 1 - C 2 2 ) ^ (32snpnt ns2np J32 [n2s2-n2t{322 ) [ n 2 s 2 - n2(322 } np nnts2 - n2t/32 1 — c 2 2 = n2s2 — nt2/32 After simplification, the variance V(yb) can be expressed as: V($b2) = - ( l - c 2 2 ) . 711 4.3 Estimator of the Change The most commonly used (Ware and Cunia, 1962; Omule, 1981) estimation of the change in means between two occasions, A = py — px > is estimated by a linear estimator Chapter 4. Two-stage SPR with Unequal Probabihties of the form: A b = a'2xtl + b'2xp + c'2yp + d'2yt2 where a' 2 ,fc 2 ,c 2 ,d 2 are constants. To make this estimator be the unbiased estimator of A, it follows that E(Ab) = ny - px = A. Since, E(xtl) = E(xp) = fix Therefore, Thus, E(Ab) — a2fix + b'2px + c'2pY + d'2pY = (a2 + + (4' + d'2)pY. a2 + K = -1 c'2 + d 2 ~ 1. Hence. A b = b'2xp + c'2yv + (1 - c'2)yt2 - (1 + b'2)xtl. The variance of Ab is V(Ab) •= b'22V(xp) + c'22V(yp) + (l-c'2YV(yt2) +(1 + o ' 2) 2V(I t l) i 262c2COT(x=p,z7p). Chapter 4. Two-stage SPR with Unequal Probabilities 44 Substituting the different variances and the co variance into the above formula yields: {1+b'2f(A±fll+2b'2c'2^. (4.18) In order to calculate b'2 and c 2 and minimize the variance V(Ab), the above formula is differentiated with respect to b2 and c'2 and the equations set equal to zero. That is d[V(Ab)} d{b>2) d(c>2) = 0 = 0. Therefore, 2 6 ;W±i i ) + 2 ( 1 + ^ W ± i i ) + 2 4 f t = „ rip ntl np 24' a ' + ^ - 2 ( l - 4 ) ^ + 5 ' ' + 2 ^ = 0. Tip Tlt2 Up Solving the above two formulas simultaneously and simplifying yields: b' = c, = -np(s'2 + s'2'){{s\ + s'{)n2 + ntlf32) (s[ + s'{)(s2 + 5-2')ni n2 - ntlnt2f3l np(4 +s'l){(s'i + s2')n1 +nt2(32) (4 + 4')(4 + s 2 ) n i n 2 - ntlnt2f3l The relationship of b'2 and c'2 can be seen to be: 2 rii (4+4>i nTJ(3Tc'2 2 Til n : Substituting the calculated values of b'2 and c'2 into (4.17) yields: Afc = b'2(xp - it, ) + c 2 y p + (1 - c 2)j/ t 2 - xtl •)(lp - l t ] ) + c'2yp + (1 - c 2 )y t 2 - i t l Tip ntl(32c2 = ,= n r ( 5 ; + 8'{)nx nuPi ,= • — \ 1 n p, = = .• ,H ,,= Chapter 4. Two-stage SPR with Unequal Probabilities 45 Notice that the overall mean on the first occasion is: ntlxtl + npxp T> , , -•  n t l = , (ri! - ntl) = — ^ t i i I f Tii 7i! — = _L ThL(= - = ^ — Xp i v""'i XV)' Hence, and Xp - x t l = — - ( i - x f J . Also, Substituting the above results into Ab yields: A f c = c'2[yp - 0T{xv - x)\ - (1 - l t l ) + (1 - c'2)y t 2 - a £ l = c2y'2 + (l-c'2)yt2-x = y'b-x where y ' 2 = yp - f3T(xp - x). Now substituting the calculated values of b'2 and c'2 into (4.18), we get the general formula for calculating the minmized variance of: = ^-Ms'z-r + s")n2 +ntl/32)M + s'{) ( s i - f s")(s 2 + s 2 ) r i i 7 i 2 - ntlnt2(32 np nP(s[ + s';)((S'2 + s'Dm + nla/32) l 2 f Y 2 + s2') + (si + s'{)(s'2 + s 2 ) 7 i i 7 i 2 -ntxni2Bl np Chapter 4. Two-stage SPR with Unequal Probabilities 46 + [1 _ % ( - s i + + 4 > i + ntM M + s'2') (s[ + s'{)(s'2 + a2/)ra1n2 - n t l n t 2 /3 2 2 ntj -n p(s' 2 + s'2')((si + s'{)n2 + ntl/32).2{s[ + s'() fl ( 5i + • s i ' ) ( 5 2 + •s'2')n1n2 - n t l n t 2 /? 2 2 n t ] [ 2 , - n p ( 5 2 + s2')((sj + s j > 2 + 7i t l / 3 2 ) 7i p (s j + sj')((s2 + s 2 > ! + n t afo) (s[ + s'{)(s'2 + s'i)nin2 - ntlnt2/3^ (s[ + s'^s!, + s 2 ' ) n i 7 i 2 - ntlnt2f3l np This can be simplified (See Appendix C) to give: \ / (A f c ) = ^ ± ^ ( l - c 2 ) + M+iil(i-^Ca). (4.19) The best estimate of the change in means can be seen to be a weighted linear com-bination of a regression estimate, the mean obtained from the temporary sample units on the second occasion, and the overall mean on the first occasion. The best estimate of variance is a weighted estimate of the variances divided by the number of sample units on the different occasions. When ntl = nt2 = nt and nx — n2 — n, b'2 and c'2 become: _ ~(s'2 + 4 ) n p ( ( * l + S2)n + ntP2) 2 1 ~ {s[ + s'{)(s'2 + s 2 > 2 - 7i 2 / 3 2 c np(s'i + + s'2')n + nt(32) (s>1+s1)(s2 + s'i)n*-n?ft ' Substituting the above values into (4.18), after simphfication, the variance V(Abi) can be expressed as: F ( A b ! ) = ^ ± ^ ( l - c 2 1 ) + ^ ± i l ( l - / 5 r c 2 1 ) . (4.20) Tif 71 In CFI's, the assumption is often made that the population variances are the same on both occasions in addition to equivalent sample size. This means: (si + s") = (s'2 -f s 2) = s nt, = • nt2 = nt Tii — n2 — n. Chapter 4. Two-stage SPR with Unequal Probabilities The b'2 and c'2 values become: _ -snp(sn + nt(32)  2 2 ~ s2n2 - n2f32 2 2 ~ s2n2-n2(32 ' Substituting the above values of b'22 and c'22 into (4.18) yields: , . -snp(sn + ntf32) 2 s nps{sn +n t/3 2) 2 s ^ b 2 > L S 2 n 2 _ n 2 / 3 2 J U p + i g 2 n 2 _ n2p% i n p _ nps(sn +n t/3 2) 2 a -ara p(aTi 4- n ^ ) ^ - , + l 52 n 2 - n2/32 J n t 2 + [ + s 2 n 2 - n2(32 1 ' - , s n p(5n 4- nt/32) nps(sn + nt(32) (32  [ s2n2-n2/32 l [ s2n2-n2(32 lnp This variance can be simplified to: y ( A f c 2 ) = - ( l - c 2 2 ) 4 - - ( l - & c 2 2 ) . nt n Chapter 5 Appl ica t ion to Chinese Forest Inventory 5.1 Background on Chinese Forests and Inventory Problems China is the world's third largest country covering an area of 9.6 million square km. Its land extends 5,500 km from north to south, crossing the tropics, subtropics, warm temperate zone and cold temperate zone, and 5200 kilometers from east to west, stretch-ing upwards from the Pacific all the way to top of the world-Mount Everest situated in the hinterland of the Asian continent. China's major landscapes are mountains, high plateaus, deserts and extensive river plains. Its varied geographical and climatic condi-. tions give it perhaps the greast variety of plants in the world. The forests in China are highly variable. In the south (Hainandao, Guangdong, Guangxi, Yunnan and Taiwan), there are large areas of tropical monsoon rain forest and small stretches of tropical rain forest. Going progressively northwards, one finds subtropical humid evergreen broadleaf forests, warm temperate zone deciduous broad leaf forests, temperate zone mixed broadleaf-conifer forests and finally, temperate and frigid zone conifer forests. In addition, there are large mountain conifer forests on the Tibet-Qinghai Plateau and the Inner Mongolia-Xinjing Plateau and large diversiform-leaved poplar forests in the Talimu River Valley in Xinjing. There are also desert forests on the fringes of deserts and dryland areas. According to the latest forest inventory results released by the Ministry of Forestry in 1984 (Ministry of Forestry, 1984), China's stocked forests cover 115,277,400 ha which 48 Chapter 5. Application to Chinese Forest Inventory 49 is 12 percent of the total land area. Thirteen provinces and autonomous regions are over 20 percent forested. Table 5.3 illustrates the forest types and their amounts. Table 5.3: Forest types and their amounts in China Type of forest Area (10,000 ha) Percent Protective forests 1000.24 9.1 Timber forests 8062.84 73.2 Fuel forests 369.09 3.4 Special forests 129.90 1.2 Economic forests 1128.04 10.2 Bamboo forests 319.96 2.9 Total 11527.74 100.0 It is estimated that China has about 250 million ha (26 percent of its total land area) suitable for forests (Hsiung, 1980), but one-third of that still remains unforested. Table 5.4 illustrates the problems. Table 5.4: Forest land utilization in China Utilization Million ha Percent Nurseries 0.12 0.08 New plantations 4.15 1.8 Thinly-stocked forests 15.63 6.2 Shrub lands 29.75 11.9 Nonforested lands 85.82 34.2 Stocked forests 115.27 45.9 Total 251.01 100.0 There are six forest regions in China. These forest regions are divided according to the administrative authorities and forest types. A brief description of each follows. Chapter 5. Application to Chinese Forest Inventory 50 1. The northern coniferous forest region. Most of China's remaining forest is in the north. Located in the extreme north of Manchuria, this region includes the northern part of Daxinganling. Adminis-tratively, the area covers some 200,000 square kilometers, and embraces part of Heilungjian province and the Inner Mongolian autonomous region. The vegetation type in this region is strongly reminiscent of the northern coniferous forests of Eu-rope and North America, with many common or closely related species. The region is almost entirely forested. The forest is mainly overmature with corresponding deficiencies in young-age classes. The distribution of these forests is irregular both in volume and area. 2. The mixed coniferous and deciduous broadleaved forest region. This type forms a transition between the coniferous forest in the north of Manchuria and the deciduous broadleaved forest in the south. It covers much of the Xiaoxd-nanling and the Changba mountains. China's richest timber resources are located here. Vast forests remain which provide some 30 percent of China's present timber requirements. Like the northern coniferous forest, the forests in this area are mainly overmature with corresponding deficiencies in younger age classes. However, the distribution of these forests is more regular than that of the northern coniferous forest region. 3. The deciduous broadleaved forest region. This zone skirts the southwestern end of the Manchurian plain to take in the penin-sulas of Shantung and Liaoning, the northern China plain, the northern slopes of the Tsingling mountains and the western half of the Shansi highlands. It covers almost 1,000,000 square km in nine provinces. Most of the accessible forest in the deciduous broadleaved region has been logged and the present potential for timber Chapter 5. Application to Chinese Forest Inventory 51 production is not high. This area has a long history of drought and flood. There is an obvious need for afforestation of protection forests in conjunction with a water conservancy scheme, and considerable scope for the development of farm forestry and shelter planting on the plains. At the present time, China has an ambitious reforestation program in the region; 4. The mixed deciduous and evergreen broadleaf forest region. This region is a transition between the deciduous and the evergreen broadleaved forest zone and forms a belt of 300,000 square kilometers along the Hanjing river in the west and the lower reaches of the Yangtze in the east. It runs through parts of eight provinces. The existing forest, as in most accessible parts of China, has been cut over a long period and most of the natural forest has disappeared. Only the protected areas around the secluded temples in the mountains and hills give an indication of the natural forest. This area is densely populated and forms part of the traditional rice bowl of China. 5. The evergreen broadleaved forest region. This forest region in China extends in a broad belt some 1,750,000 square km in area across the southern provinces and include parts of Zejing, Anhuai, Jiangsu, and Guangxi autonomous regions. Much of the natural forest of this area has been cleared for crop cultivation. Most of the evergreen species are utilizable timber trees and, where readily accessible, they have been extensively logged. Excessive tree cutting and poor methods of cultivation and reclaimation have made the forests in this region into lean secondary forest types. 6. The tropical monsoon rain forest region. The tropical monsoon rain forest is restricted to southern China and forms a narrow belt 250,000 square km in area along the sourthern edge of Fujian and Guangdong Chapter 5. Application to Chinese Forest Inventory 52 provinces. The tropical regions in general, and the rain forest in particular, have been a great reservoir of plant stocks. Although the tropical forests are different in many respects from those of the temperate forests, they have important similarities. While irreversibilities are more common in the tropics, in the vast majority of cases the tropical forest has the resiliency to restore itself. However, often the new forest will be different than the original in the composition of species, etc. This is particularly true when the forest areas harvested or destroyed are large. Existing natural forest in the tropical zone in China is limited to the hills of some provinces. In the southern part of this region, there is still a large amount of unused forest. However, it is difficult to access these forests. There was no planned inventory of China's forests for a long time. The first inventory was conducted between 1953 and 1961. This survey was actually conducted in different forest regions, using point sampling and visual estimation. No permanent sample plots were established and the data obtained at this period were only used to provide basic information such as the approximate, total forest areas. From 1973 to 1977, a more detailed forest inventory was conducted by provinces. An average volume of 78.26 m 3 /ha (Ministry of Forestry, 1980) was found. In several forest provinces such as Guanxi and Heilongjiang, a basic CFI system was established in this period. Some detailed checks and remeasurements have been made on a small scale in recent years. However, forest inventory is still a weak link in managing Chinese forests. Due to a lack of adequately trained personnel and equipment, the data obtained through the early forest inventories have seen very limited use and are not precise (Huei, 1980; Zhou, 1980). Many problems still exist. Lack of suitable inventory information has made forest management decisions very difficult to make. Chapter 5. Application to Chinese Forest Inventory 53 Forestry is playing an increasingly important role in recent years in China's social and economic development. Demands for timely and accurate forest inventory data are becoming more urgent. Forest inventory systems are now being established on a national basis to provide the required data. New and improved sampling methods are being tried in different forest types and regions. The following section will outline the steps necessary to apply the methodology developed in the previous chapters to China and discuss aspects of the design of a national forest inventory for China based on CFI principles. The purpose is to illustrate points which should be addressed, rather than providing a design which should be implemented. 5.2 Design of a Two-stage Sample 5.2.1 General Considerations It is very important to define clearly the various uses of CFI's in forests. In order to design and implement an inventory which best solves the problems, a clear idea, of the objectives must be decided before starting the work. In China, objectives center on current average volume, forest areas, stand structures, and growth patterns. These objectives should not be defined by an inventory specialist alone; they must be defined jointly by the people who will make use of inventory results and the inventory specialist. The inventory specialist should design an inventory which will provide the users with the information they need in a suitable form and with the required precision (Fontaine, 1981). Because CFI's can produce information on a number of variables besides volume, the priority of CFI objectives to be met has to be clearly assessed before designing an inventory. Additional requirements also have to be known before starting the inventory. Fontaine (1973, 1981), and Loetsch et al. (1973) gave a detailed discussion of these Chapter 5. Application to Chinese Forest Inventory 54 aspects. Some major points are summarized here to illustrate how CFI's should be applied. 1. Acquire general information. For example, determine the authority responsible for the inventory, available information and data on the area to be inventoried from past surveys, reports, maps, general descriptions of the forest, and conditions of terrain, transportation facilities, accessibility, and other necessary information. 2. Determine the exact purpose of the inventory. Detailed specifications of the objec-tives should be listed, for example, nature of the information required, exact limits and size of the area to be inventoried, and precision required. A clearly defined purpose provides a focus for the inventory foresters in designing and executing the details of the inventory. 3. Decide on which type of design to use. Many sampling designs can be used with CFI's. The selection of a sampling design is not limited only to the theoretical optimum. It is the combination of many theoretical and practical considerations. Some design possibilities include systematic or random locations, clustered or single locations, points or plots, one-stage or multistage. 4. Establish correct measurement procedures. For example, knowing the instruments that will be used in the inventory, provide detailed instructions on all techniques. Field test the instruments and the measurement procedures that will be used and provide forms for recording of observations. Outline a procedure for locating and establishing sample plots, etc. 5. Review compilation procedures and production of the final report. For example, provide detailed formulae for estimating means, totals, and their sampling errors, skeleton tables, and detailed descriptions of all phases of calculation from raw data Chapter 5. Application to Chinese Forest Inventory 55 on original forms to final results. It is also important to consider the analyses which will be made on the results in the final report. 5.2.2 T h e S h a p e a n d S i z e o f S a m p l e P l o t s It appears that no single kind of sample plot shape is the best for Chinese forest inventories. The establishment of CFI sample plots should differ according to the different natural and economic situations in the various regions in China. It is recommended that in the eastern part of China (the deciduous broadleaved forest region and the mixed deciduous and evergreen broadleaf forest region), which largely consists of plains, the use of clusters of circular sample plots is better. Cluster sample plots are popular in northern Europe, especially in Sweden, where they are used in national forest inventories. This approach can profit from the advantages of circular plots while having at the same time sufficiently large sampling units. Hagberg (1957) discussed this method in detail. Janz (1975) made some improvements on Hagberg's method. Although cluster sample plots have many advantages, they also have some disadvan-tages, especially in mountainous regions. Also, for the same size of plot, the cluster of circular plots may have a longer total perimeter and consequently more borderline trees than equivalent rectangular sampling plots. However, these deficiencies will not influence the adoption of cluster sample plots in the eastern part of China which largely consists of plains. A cluster of 2 to 5 circular plots can be used. In southern China (the evergreen broadleaved forest region and the tropical monsoon rain forest region) which largely consist of mountains and rain forests, while the trans-portation systems are very limited, variable area single plots should be used. Variable area plots are mostly used in North America based on point sampling theory. In the southern provinces of China where the rain forests are common, these sample plots are used widely at the present time. It has been proven that this type of plot is very efficient. Chapter 5. Application to Chinese Forest Inventory 56 It can gain the required precision at lower cost because in each sample plot, fewer trees are measured. Plot centers can be relocated very easily when the sample plots become permanent plots. On the negative side, this kind of sample plot can cause some trees in the sample location to be missed unless caution is taken. Small mistakes can cause a big difference in the estimate of the current volume and especially in estimates of the change in volume. It should also be mentioned that the larger the sample plot, the more precise the results. However, the cost of the plots will also increase. It is therefore necessary to stipulate basal area factors (BAF's) for different regions. In Guangxi and Guangdong where the tropical forests are more intensively managed and need more accurete inventory data, smaller BAF's (more trees per plot) are desirable. In Yunnan, Hainandao, and part of Gueizhou where the forests are mainly unexplored and the accessability is not good, larger BAF's (less trees per plot) can better serve inventory purposes. In the northern part of China (the northern coniferous forest region and the mixed coniferous and deciduous broadleaved forest region) where forests are regulated well and the topography is simple, the use of rectangular sample plots would appear best. Rect-angular sample plots are used extensively in most Chinese provinces. They usually cover an area of 0.06 to 1.0 ha. Rectangular plots have some advantages. They have clear boundaries and each tree has a smaller probability of being ignored or double measured than with other types of plots. The disadvantage of this kind of plot is that sometimes there are too many trees in a sample plot which may increase survey costs. Since the forests in China have a great diversity even in a very small area (especially in the southern and eastern parts), any small sample plot will not provide representative data. It is recommended that a relatively large plot size with a relatively small number of total sample plots should be used for Chinese forest inventories. A large plot size may increase the measurement cost,.but the relatively small number of total sample plots will Chapter 5. Application to Chinese Forest Inventory 57 greatly reduce the cost of transportation, a major cost in Chinese forest inventories. The plot sizes suggested for different regions are as follow. They should be jointly considered when used in designing sample plots. 1. In the evergreen broadleaved forest region and the tropical monsoon rain forest region, a large plot size of 1 ha to 1.5 ha should be used. Such a large plot may increase surveying cost by a large amount in many other countries. However, the low labor cost in China (approximately 2 to 3 Canadian dollars per day) will keep the costs reasonable. The large plot size will reduce the total number of sample plots in these regions. This will greatly reduce the cost of transportation and minimize the total inventory costs. 2. In the northern coniferous forest region and the mixed coniferous and deciduous broadleaved forest region, a sample size of 0.8 ha to 1 ha should meet the inventory purposes. Northern forests are well regulated and the topography is simple. In-ventories are much easer than in the southern tropical regions. The transportation facilities in this area are also readily accessble. 3. In the deciduous broadleaved forest region and the mixed deciduous and evergreen broadleaved forest region, a sample size of 0.2 ha to 1 ha should be used. The better economic situation and transportation capacities in these areas make an inventory relatively easy to conduct. Past inventory results showed that a relatively small sample plot in this region not only met the inventory purposes, but also were very practical. 5.2.3 Sample Size Althrough the recommended plot size varied from region to region, an individual CFI sample plot of 1 ha will be assumed for sake of simplicity in subsequent discussions. Chapter 5. Application to Chinese Forest Inventory 58 As seen previously, the total forest area is approximately 115,277,400 ha. Suppose a sampling intensity of 0.02 percent is used on a national level (previous sampling intensities have ranged from 0.005 percent in the south to 0.1 percent in the north for large scale inventories). Then, on the national level, a total area of approximately 23,600 ha should be sampled, requiring 23,600 sample plots. The average cost for establishing and measuring a sample plot in China varies from region to region. In the tropical south where the forest conditions are very complex, a sample plot usually costs 500 Yuan ( 3 Yuan= 1 Canadian dollar). The average cost in the northern regions is 400 Yuan while in the eastern regions it is 350 Yuan. Suppose an average cost of 400 Yuan per plot is used for the national inventory system. For a total sample size of 23,600, a total cost of 9.44 million Yuan is needed for the first stage of the national inventory system. This cost will include: 1. Transportation and locating sample plots. 2. Measurement costs and compensation for the workers. 3. Auxiliary materials (maps, recording papers and tables, pens, etc.) and equipment (raincoats, bags, etc.). The cost for measurement equipment is not included. Of the 23,600 sample plots, a certain portion will remain as permanent sample plots and will be remeasured on a second occasion. This portion is not only decided by the inventory results from the first occasion, but also by the total budget available at the second occasion. For the sampling design presented in Chapter 3 and 4, the number of sample plots that should be remeasured on the second occasion is also decided by the individual values of each sample plot and the total values of all sample plots, as well as additional requirements such as the required precision, expected population variance, co-efficient of variation, correlation coefficient, etc. Some rough data for these requirements Chapter 5. Application to Chinese Forest Inventory 59 in China are provided by Guang et al. (1983) from past inventory results. The coeffi-cient of variation usually ranges from 20 percent to 50 percent, the correlation coefficient ranges from 0.6 to 0.9, and the required precision for a large scale inventory is 5 to 10 percent. This aspect will not be discussed in detail and only the basic sample size calculation formula used by many researchers in a simple random design will be given as a rough guide for calculating the portion that should be left as permanent sample plots: where np is the number of permanent sample plots, na is the total number of sample plots, p is the correlation coefficient between the sample plots measured on the first occasion permanent sample plot, and Ct is the cost for establishing and measuring a temporary sample plot. plots is 450 Yuan. The average cost of establishing and measuring the new plots on the second occasion is the same as on the first occasion, 400 Yuan. A correlation coefficient between the sample plots measured on the first occasion and remeasured on the second occasion is approximately 0.8. Hence, Ct = 400, Cp = 450, p = 0.8, and na = 23, 600. Substituting the values into the formula yields np=7585, which means that within the 23,600 established sample plots, a total number of 7,585 sample plots should remain as permanent sample plots and be remeasured on the second occasion. It is therefore recommended that for a national CFI system in China, a total number of 8,000 to 10,000 permanent sample plots should meet all the purposes of the inventories. In order to retain a total number of 23,600 sample plots on the second occasion, 13,600 to 15,600 temporary sample plots would need to be established. and remeasured on the second occasion, Cp is the cost for establishing and measuring For example, suppose the cost of relocating and remeasuring the established sample Chapter 5. Application to Chinese Forest Inventory 60 It must be emphasized that the above results can only serve as a rough guide for a national CFI system in China. Determining the "best" sample size involves optimizing the sample design. This means finding the design that for a specific cost gives the smallest error for the parameter to be estimated or for a specified error yields the least cost. This is very complicated for a two-stage design using unequal probabilities with SPR on more than one occasion, since good information on the sampling and measurement errors, as well as biases is never simple to obtain. In many textbooks and manuals the calculation of the optimum is restricted to the estimation of the optimum number of sample units at each stage based on a simple cost formulation. Loetsch and Haller (1964) gave an example of optimization of a two-stage sample based on equal probability for the one occasion case using inventory data collected in Thailand. Other researchers have used different cost functions in calculating the optimum sample size in a two-stage sample design. These include: 1. C = C\n + C2nm. where C is the total cost for a inventory, Cj i i is the total cost for the number of primar3r units in the sample, and C2nm is the total cost for the total number of second-stage units. This cost function is discussed in detail by Cochran (1977) and is often used when the travel costs between units are unimportant. 2. C = C-^n + CtyJn + C2nm. This cost function is presented by Hansen et al. (1953) and used when the cost of travel (Ct) between primary units is substantial. where Cw is the cost of walking a unit distance, Ce is the cost of establishing a plot or point, Cm is the cost of measuring the characteristic of interest, yJnA is the average minimum distance among randomly drawn plots in the area A, and 3 Chapter 5. Application to Chinese Forest Inventory 61 T is the average number of elements observed and measured at a location. This function was presented by O'Regan and Arvanitis (1966). A general discussion concerning the optimum sample size was given by Fontaine (1973). He assumed that the standard error of the estimate (SE) of a given sample, and the total cost (C) are functions of various design characteristics (xx, x2, ... , Xk). For a given cost of C 0 , to minimize the standard error SE, it must follow: SE{xi, x2,Xk) = minimum C(x1,x2, ...,xk) = C 0 Optimum sample size is obtained by taking the derivative with regards to every charac-teristic involved. 5 .2 .4 T h e L a y o u t a n d S u r v e y o f S a m p l e P l o t s The layout of a two-stage sample design should be carefulljr planned. There are man3r layout forms that could be used in a Chinese national CFI system. In order to provide more precise estimates, it seems reasonable that different forest regions should be treated as distinct populations and different layouts should be employed in each region. However, the forests and topography in China are so complicated that this approach is seldom possible in practice. Given the present situation in China, systematic layout of plots has the most promise for a national CFI system. The vast majority of previous inventories have used some form of systematic selection. Systematic sampling provides good coverage and good representation. It is also good for mapping. The following steps are necessary in designing the layout of the sample plots: 1. Divide the whole nation into uniform portions. The total forest area in China is 115,277,400 ha. This is comprised of forest production farms of approximately Chapter 5. Application to Chinese Forest Inventory 62 50,000 ha in size. Assuming a uniform size of by 50,000 ha means that there will be 115,277,400/50,000=2306 farm size regions in the country. 2. From the 2,306 production farms, systematically select 200 from which to be sam-pled further. 3. Classify each of the selected farms according to forest types and ages. Then divide the farm into different units according to the classification. These units will be used as primary sampling units and they usually cover an area of 20 ha to 1,000 ha. 4. Choose the primary units according to probability proportional to the area of the units. Fifteen primary units should be selected in each farm using this method. 5. Select eight plots (secondary units) wTithin each primary units by probability pro-portional to predicted volume. The above procedures will ensure: 1. A total sample size of 8x15x200=24,000 will be selected. This number is approxi-mately equal to the required total number of sample plots which is 23,600. 2. A two-stage unequal probability sampling estimate procedure presented in Chapter 4 can be used in each of the selected 200 farms. 3. A final national inventory result can be obtained by using the results from the 200 production farms. Chapter 6 Conclusions The previous chapters discussed the general theory of multistage sampling combined with CFI using unequal probabilities in a SPR structure on two occasions. Only one-stage and two-stage SPR with unequal probabilities were discussed. Estimators of the current means and the change in means for both one-stage and two-stage cases were given. Although no precise theoretical demonstration of the gains from using the presented method was given, the combination of highly efficient sampling with unequal probabilities and the very practical multistage sampling in CFI with SPR in Chinese forest inventories can certainly provide an efficient alternative to traditional systems. General theories of CFI with SPR, unequal probability sampling, and multistage sampling have been developed by manj' researchers. Their applications in forestry have been examined in depth. CFI with SPR estimation structure has existed for a long time and the main difference among the various CFI methods is the way to obtain the least variance or to obtain the least combination of the variances from each component. In this study, multistage unequal probability sampling was the method used to obtain these variances. The results in Chapter 3 showed that for one-stage SPR with unequal probabilities, the estimator of the current mean takes the same form as presented by many researchers. However, after using unequal probability sampling, the variances obtained and the con-stants involved in estimating the current means differ from those reported for equal probability sampling. The estimator of the current mean in .two-stage SPR with unequal 63 Chapter 6. Conclusions 64 probabilities also showed a similar form as the one-stage case, with the variances and constants differing from the one-stage case. This resulted from a similar linear form and estimation structure. It is difficult to tell whether the estimators derived in this study are better or worse than the estimators obtained using other methods. Further analysis and comparison might be done in this area. In estimating the change in means, the conventional linear estimate form is used again, and the results from one-stage SPR with unequal probabilities showed that the best estimator of the changes on both occasions is the difference between the best estimate of the current mean on second occasion and the overall mean on the first occasion. A similar result is also obtained for two-stage SPR with unequal probabilities. The overall variances in estimating the change in means for both one-stage and two-stage SPR with unequal probabilities on two occasions are obtained by substituting the different variances from independently observed sample plots (temporary sample plots) and from dependently observed sample plots (permanent sample plots). Results showed that in the one-stage case, the best estimate of the total variance is a weighted estimate of the different variances. For two-stage sampling, a similar result is obtained with the variances and constants differing from the one-stage case. The estimators of overall variances developed take simpler forms than those presented in the literature which might result in greater simplicity when used in data processing. The application of CFI with SPR using multistage unequal probability sampling in Chinese national forest inventories is discussed for illustrative purposes. Inventory sys-tems are a weak link in Chinese forest resource management and the nation is now undergoing an ambitious program trying to establish a good inventory system. Some researchers have suggested that large scale remote sensing images combined with ground sampling design is a better way to achieve accurate results. Others have suggested that current social and economic situations might not allow the wide use of remote sensing Chapter 6. Conclusions 65 images in forestry, and thus, more emphasis should put on developing and using more efficient and practical sampling designs. It will be benefical if a CFI with SPR using multistage unequal probability sampling could be combined with the use of available re-mote sensing results, especially in forest region classifications, forest type identifications, and area calculations. The establishment of a national CFI system in China should use all kinds of useful auxiliary facilities and information that might reduce the complexity of the inventories. China's inventory problems can never be solved by a uniform sampling design. Some suggestions made in Chapter 5 may be used as a rough guide in actually designing the inventories. More research is needed regarding how the design can be implemented more efficiently and how the optimum sample size could be calculated. With the improving understanding of current inventory methods and with the practice of new inventory designs, a better inventory system will eventually be established in China, and forest resource management decision making will benefit from improved information on the current status of the resource. Chapter 7 Literature Cited Beers, T. W. 1978. Developing efficient estimation techniques for integrated inventories. USDA For. Serv. Gen. Tech. Rep. RM-55. pp. 270-275. Bickford, C. A. 1956. Proposed design for continuous inventory: a system of perpetual forest survey for Northeast. USDA For. Sev. Eastern Technical Meeting, Forest Survey. Cumberland Falls, K Y . 37pp. Bickford, C. A. 1959. A test of continuous inventory for national forest management based upon aerial photographs, double sampling and remeasured plots. SAF pro-ceedings, 1959:143-148. 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Sci. 11:480-502. Cunia, T. 1987a. On the error of continuous forest inventory estimates. Can. J . For. Res. 17:436-441. Cunia, T. 1987b. On the error of estimates from successive forest inventories. Can. J . For. Res. 17:446-447. Cunia, T. and R. B. Chevrou 1969. Sampling with partial replacement on three or more occasions. For. Sci. 15:204-224. Cutter, D. D. 1955. A permanent plot system of survey for the continuous inventory on ponderosa pine stands in the southwest. J. For. 53: 186-189. Deming, W. E. 1950. Some theory of sampling. John Wiley and Sons, Inc. New York. 602pp. Dixon, B. L., and R. E. Howitt. 1979. Continuous forest inventory using a linear filter. For. Sci. 25:675-689. Fontaine, R. G. 1973. Manual of forest inventory with special reference to mixed tropical forests. FAO manual, Rome. 200pp. Fontaine, R. G. 1981. Manual of forest inventory with special reference to mixed tropical forests. FAO Forestry paper, Rome. 200pp. Frayer, W. E. 1966. Weighted regression in successive forest inventories. For. Sci. 12:464-472. Frayer, W. E. 1967. A systematic bias in the interpretation of CFI results. USDA For. Serv. Res. Note NE-60. 4pp. Frayer, W. E. , and G. M . Furnival. 1967. Area change estimates from sampling with partial replacement. For. Sci. 13:72-77. Frayer, W. E. , R. C. Van Aken and R. D. Sullivan. 1971. Application of sampling with Chapter 7. Literature Cited 68 partial replacement to timber inventories, central Rocky Mountains. For. Sci. 17: 160-162. Freese, F. 1962. Elementary forest sampling. USDA For. Serv. Southern For. Exp. Stat. 91pp. Furnival, G. M . , T. G. Gregoire, L. R. Grosenbaugh. 1987. Adjusted inclusion probabili-ties with 3P sampling. For. Sci. Vol. 33, No. 3. pp:617-631. Gillis, M . D. 1988. Estimating change from successive static forest inventories. For. Chron. Vol. 64:352-354. Grosenbaugh, L. R. 1964. STX-Fortran-4 program for estimates of tree populations from 3P sample-tree measurements. 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Promnitz 1974. Design of successive forest inventories: optimiza-tion by convex mathematical programing. For. Sci. 20:117-127. Hendricks, W. A. 1956. The mathematical theory of sampling. Scarecrow Press. 364pp. Hsiung, W. Y . 1980. Forests and forestry in China. The H. R. MacMillan lectureship in forestry. Faculty of Forestry, The University of British Columbia. Vancouver, B.C. 18pp. Huei, Z. 1980. Analysis of the data obtained through the national forest inventories in the past years in China. J . of Forest Management and Surveying (Chinese). 1980(2):32-38. Janz, K. 1975. The Swedish national forest survey. Second FAO/SIDA training course on forest inventory. Jessen, R. J . 1942. Statistical investigation of a sample survey for obtaining farm facts. Iowa Agric. Exp. Stn. Res. Bull 304, 104p. Kish, L.T965. Survey sampling. John Wiley and Sons., Inc. New York. 643pp. Letourneau, L. 1966. A forest inventory by continuous forest control. For. Chron. 42: 414-419. Letourneau, L. 1979. Updating continuous forest inventory by simulation. Forest Re-sources Inventories: Workshop Proceedings. Colorado State Univ., Fort Collins. Vol. I. pp:327-332. Loetsch, F. and K. E. Haller. 1964. Forest inventory. Vol. I. Munchen. B E R N WEEN, 436pp. Loetsch, F., F. Zohrer, and K. E. Haller. 1973. Forest inventory. Vol. II. Munchen. B E R N WIEN. 469pp. Lund, H. G. 1979. Linking inventories. Forest Resources Inventories: Workshop Proceed-ings. Colorado State Univ., Fort Collins. Vol. I. pp:2-9. McClure, J. P. 1979. Multi-resource inventories-a new concept for forest survey in the Chapter 7. Literature Cited 70 southeast. Forest Resources Inventories: Workshop Proceedings. Colorado State Univ., Fort Collins. Vol. I. pp:23-28. Ministry of Forestry (China). 1980. China's forestry and its role in social development. Forestry Press. 43pp. Ministry of Forestry (China). 1984. A brief account of China's forestry. Forestry Press. 20pp. Newton, C. W., T. Cunia, C. A. Bickford. 1974. 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Kalman's linear filtering and its application to the estimation of forest resource dynamics. Scientia Silvae Sinicae. Vol. 21, No. 2:122-131. Pleines, W. and L. Letourneau. 1970. The continuous forest control inventory system. For. Chron. 46: 39-43. Chapter 7. Literature Cited 71 Rana, D. S. 1978. Ratio method of estimation in multi-stage successive sampling on two occasions. Amer. Stat. Assoc., Proc. Social Stat. Sec.:289-291. Schreuder, H. T., J . Sedransk, and K. D. Ware. 1968. 3-p sampling and some alternatives, I. For. Sci. 14:429-453. Schreuder, H. T., J . Sedransk, K. D. Ware, D. A. Hamilton. 1971. 3-p sampling and some alternatives, II. For. Sci. 17:103-118. Schreuder, H. T. 1984. Alternative estimators for point-Poisson sampling. For. Sci. 30:803-812. Scott, C. T. 1977. Unequal probability sampling for updating inventory estimates. M.S. thesis. Univ. of Ga., Athens. 62pp. Scott, C. T. 1979. Midcycle updating: some practical suggestions. Forest Resources Inven-tories: Workshop Proceedings. Colorado State Univ., Fort Collins. Vol. I. pp:327-332. Scott, C. T. 1984. A new look at sampling with partial replacement. For. Sci. 30:157-166. See, T. 1974. Forest sampling on two occasions with partical replacement of sample units. M.Sc. thesis. Faculty of Forestry, The Univ. of B.C. , Vancouver, B.C. 63pp. Singh, D. 1968. Estimates in successive sampling using a multi-stage design. American Stat. Assoc. J . 63(321):99-112. Solomon, D. S. 1979. Permanent plots in forest research. Forest Resources Invento-ries:Workshop Proceedings. Colorado State Univ. Fort Collins. Vol. I. pp:327-332. Spurr, S. H. 1952. Forest inventory. The Ronald Press Co., New York. 476pp. Stott, C. B. 1947. Permanent growth and mortality plots in half the time. J . For. 37:669-673. Stott, C. B. 1968. A short history of continuous forest inventory east of the Mississippi. J . For. 66:834-837. Stott, C. B., and G. Semmens. 1962. 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A p p e n d i x A D e r i v a t i o n o f E q u a t i o n 3.6 The variance of yb is V(yb) = a 2 [ y ( x f J + y(x p )] + c?y(y p) + ( l - C l ) 2 1 / ( y t 2 ) -2aic1Cov(xp,yp) (A.22) = + f l ) + c > f i + (1 - C l f i l - 2 a l C l ^ ^ (A.23) ntl np np nt2 npl and, „ = . ( A . 2 4 ) npzsis2nin2 - n^n^rj2 C l = — -. (A.25 nplsis2nin2 - ntlnt2rj/ where rj = 8paeia£2. The relationship of a\ and c\ can be written as: Vnu c i a i = • sininp Substituting the calculated values of (A.24) and (A.25) into (A.23) yields the general formula for calculating the minimized variance of V (yb): t„ > . r\s2n\nti , , . 1 1 . V{Vb) = — ' — 2 + — npis1s2n1n2 — ntlnt?rjz ntj np 7 i p 2 s 1 s 2 " - i n 2 - ntlnt2r]2 np n , -ie. c n - T . . . _ T>. n. •p*s1s2n1n2 - ntlnt2r)' nh np2s1s2n1n2 - ntlnt2r]2 np2sls2n1n2 - nunt2rj2 np2 73 Appendix A. Derivation of Equation 3.6 74 Therefore, v r - ^ y2s22nytiSl 1 , 1 N V{yb) = -j—2 ^ — + —) {np2sls2nln2-ntlnt2T}2)2 nti np s2s\n2ni _| 1 2 1 p  ( n p 2 S i S 2 T i i n 2 - n t l n t 2 r / 2 ) 2 + / x aig2wm; 2 s 2 n p 2 s 1 s 2 r a 1 n 2 ntlnhr]2 nt2  2r]2slnpintln1s1 (np2s1s2n1n2 - ntlnt2v2)2' So, T / / - N 7/ 527Z 7l t l 5l 1 1 ( n p 2 s 1 s 2 7 i 1 n 2 - ntlnt27)*y ntl np ( n p 2 s i s 2 n 1 r i 2 - n f l n f 2 r / 2 ) 2 + ( 1 - 2 j ) ' ^ n p , ! 5 i 5 2 7 i 1 n 2 - ntlnt2T}* nt2 T)2 slnlrit^nx ( n p 2 s 1 s 2 n 1 n 2 - ntlnt2n2)2 ( n p 2 s 1 s 2 7 i 1 7 i 2 - n t l r a t 2 7 / 2 ) 2 + ( 1 - 2 S l S 2 7 l i n > 2 " ) -n p 2 s 1 5 2 r i 1 7 i 2 - ntlnt2nz nh S22nlniSl I 1 2 N r i p 2 s 1 5 2 n 1 n 2 - nhnt27]'1 nt2  s2 r • s 1 s 2 n 1 n ^ — [7—2 2 r 2 - ( n i « i s 2 n n t - 7/ n t l n t 2 ) + (1 - c j Notice, 1 - c a = •SiS27ian 27i t 2 - ntlnt2T}2 rap2s1s2n1n2 - ntlnt2T}2 Appendix A. Derivation of Equation 3.6 75 Thus, V(yb) = — — i ~ j ( l - c i ) + ( 1 - c i ) = i l ^ - d X d + l - d ) A p p e n d i x B D e r i v a t i o n o f E q u a t i o n 3 .11 The general formula for calculating the minmized variance of V(Ab): - n 2 5 2 ( 5 1 7 i 2 7 i p + ntlr]) S l s^n^nl - ntlnt2rj2 np  ! ^ nlsl(s2n1np + nt2rj) 2s2 s^sin^nl - ntlnt2j]2 np + [ i _ nlsAs2ninP + n t 2 v ) ,2 s2 s-is^^nl - ntlnt2T)2 nt2 —nps2(s\n2np + ntlTj) 2 s1 5 1 5 2 n 1 n 2 7 i 2 - ntlnt2r]2 nt2 ,2 , + [1 + ( ^-%s2(sin2np + ntlr]) n^As^rip + nt2v) v s1s2n1n2n2 - nunt2rj2 s-^s^n^n2 - ntlnt2rj2 np Notice this variance can be split into two parts: V(Ab) = VA + VB. where and , nlsi(s2ninp + nt2Tj) 2s2, 5 1 s 2 n 1 r a 2 n | - n t l n t 2 v 2 np  + ^ _ nlsAs^rip + ni2ri) ^2 s2 s 1 s 2 n 1 n 2 n 2 - ntlnt2rj2 nt2 ^-n2ps2(sAn2np + n t l77) 71^1(^2^1% + nt2rj) '7? S-LS^I^TI2 - ntlnt2rj2 s1s2n1n2n2 - ntlnT2T}2 n2 5 i 5 2 7 i 1 n 2 n 2 - ntlnt2v2 np 76 Appendix B. Derivation of Equation 3.11 + j 1 + -s2nl(s1n2np + ntlT]) ]2 S l + [ 5i52nin2n^ - ntlnt2rf ntl  -n2s2(sin2np + ntlr)) 7 i ^ i ( a 2 T t 1 T O p + r t^T/) T/ 5 i 5 2 7 i a T i 2 n 2 - ntlnt2rj2 s^s^^n2 - ntlnt2Tj2 n2v Hence T . 5 ! S 2 7 l 2 1 ^ = _F>2 iy2 ( 7 l t 2 W p 5 i S 2 T l i + U t 2 SiUpT) -51n2np7i t2r/ - n t ln^772)l + (1 - c'i)2~ A ' l'52 T l p 1 r , 2 2 2\1 i f i I \2 = Q2 ( " t 2 T t 5 ^ 2 7 1 ! - T i t 2 5 i 7 l 7/ - 72.^71^7? ) + (1 - C j ) where Q = s152n1n27ip — ntlnt2T)2, and $2 = • 5 2 7 i i n p + T i t , 7 / . Note that: 1 , _ nt2nls1s2n1 - n^s^y - ntlnt2y2 So the variance V 4 can be written as: V ™ t 2 ™t 2 = ^-(l-cj^-.+a-cj] = — ( 1 - 4 ) . T i t j Similarly, the variance VB can be written as: T, -sassn2 1 'B = n 2 —[yiKV1*2™2 + n t i S 2 n p v -s2nxnpntlri - ntint,y2)] + (1 + &i ) 2 — = ^ 7 ^ —[<?iK1n251527i2 - ntls2n2r] - ntlnt2T}2)} + (1 + fc'J2 where Q•= 5 1 5 2 n 1 7 i 2 7 i 2 — ntlnuy2, and Appendix B. Derivation of Equation 3.11 Qi = Sin2np + ntlT]. Note that: _ ntlnp2sls2n2 - n t l5 2n^T/ - ntlnt2ri2 Q So the variance VB can be written as: vB = ^ [ < ? i ( i + K)] + (i + K ) 2 -Qntl ntl _ i l _ i l ! ^ _ & 5 * c i 7 l f l T l t l T i l T i l - i l _ (i _ 'HR\-^liifi 7l t l T i j Tlx - _£I!^L _ ^ i 5 i C i T l t l T i i T i i = - ( l - M ) . T i i Hence, the variance V(Ab) can be expressed as: ^0 = ^ ( 1 - 0 + ^ ( 1 - ^ ) . T l t 2 T i j A p p e n d i x C D e r i v a t i o n o f E q u a t i o n 4 . 1 9 From the general formula for calculating the minmized variance: (-np(s'2 + 4X(4 + s'{)n2 + ntMM + s'l) (4 + 4 X 4 + 4) rM™2 - ntlnt2f3l np  | rx p(4 + 4)((4 + 4 ) ^ + ntM 2{s'2 + s'l) (4 + s'{)(s'2 + 4 ' ) n i r a2 - ntlnt2Pl np M 4 + 4X(4 + 4> i + ™t2&) l 2(4 + 4) + [ 1 -+ [1 + (4 + 4 X 4 + 4 ) 7 l i n2 - ntlnt2f32 ntt  -n p(4 + 4X(4 + 4 > 2 + ritMM + 4) (4 + 4 X 4 + 4 ' ) n i n2 - ntxnt2fil ntl , 2r - n p(4 + 4X(4 + 4 > 2 + wtl/32) np(4 + 4)((4 + s'^n, + ntM p2 ^ (-si + 4 X 4 + 4 ) n i n 2 - ntlnl2(32 i[(s[ + s'j(s2 -f 4 ' ) n i n2 - ntlntMnp Notice this variance can be split into two parts: V{Ab) = V1 + y 2 (C.30) where V1 = and ^ ( 4 + 4) 2((4 + 4> i + " t , j9 2 ) 2 (4 + 4) ((4 + 4X 4 + 4 ' K™2 - ntlnt20i)2 np +[i _ V 4 + 4X(4 + 4 'K + ntM M + 4') (•si + 4 X 4 + 4')rMn2 - ntlnt2Pl nt2  -np(4 + 4X 4 + 4)&((4 + 4 > 2 + ntM(i4 + 4> i + nt,/32) ((4 + 4')(4+4>m2-nt lnt a/?I) 2 n2(4 + 4) 2((4 + 4 'K + %,/3 2) 2 ( , i + s'l) ((4 + 4X4 + 4> i ^ . -ntlnl2f32y 79 Appendix C. Derivation of Equation 4.19 80 , [2 , ~K + * 2>P( ( *I + *1>2 + ntlP2),M + -si) ( 5 i + *i ) ( *2 + a 2 ' ) n i n 2 - ratlratj/3f ntl  { -rh^'i + *"){*2 + ^ M ^ ' i + a?)ra 2 + ntl/32)((s'2 + s'Dm + ntaf32) {{s[ + s'{)(s2 + s'l)nin2 - nt^Pt)2 Hence, \r (si + si)(s2 + "SD71? 1 ri u i i " W ' i ii\ \ a \f i , "\ a a'i Vi = jr2 QsK^ i + f l i ) ( ( s 2 + fl2)rei +«-t 2P2j(-s 1 + sJn 2 /3 2 - ri t l /3 2 V "-42 , / i „ / \2 ( 5 2 + 5 2 ) + 11 ~ C2> nt2 {s[+s'{)(s'2 + s'i)np 1 <?2 rat, + ( l _ c ' i ) 2 ( 5 2 + 4) [QaKK + s'l){s'2 + s'^n, - nti{s\ + s'{)np(32 - ntlnt2(S22)} nt2 where Q = (s[ + s")(s'2 + 5 2 ' )n. 1n 2 — ntint2fil, and g 3 = (a2 + s 2 > 1 + n < 2/3 2. Note that: , _ rat2(^ + Q(4 + 4')nx - 71^ (^ 1 + 4)np/32 - ntlnt2{32 ° 2 ~ : — Q ~ ~ -So the variance Vi can be written as: • _ (s[+s';)(s'2 + s'^np j , M + *Z) »'i = n Qs(l - c2)J + (1 - c2) V™<2 T l f 2 - —— ( ! - c 2 ) 7j r- (1 — c2)J = W±i2(i -4) . Similarly, the variance V 2 can be written as: T/ ( ^ l + )('S2 + S2)nP ^ n Unr, ( J A ' n"\( J A o"V~ 1 „ 2/ / , ll\Q V2 = — U4[(ntl(s1 + s1)(s2 + s2)n2 + nu (s2 + s2)/32 -(s>2 + s';)nint A - n t ln t 2/3 2 2)] + (1 + ^ i f l l i S <?2 nt, + ( 1 4 - O 2 ^ ^ •' [<?4(ra t l (si + «i)(s2 + • s 2) n 2 - ratl(s'2 + s2)np02 - ratlnt2/?|)] Appendix C. Derivation of Equation 4.19 81 where Q - (s[ + 4 X 4 + s2)n-in2n2 - ntlnt2f32, and Q4 = (4 + 4 > 2 + n t l /3 2 . Note that: 1 + 6; = n t l(4 + 4 X 4 + 4 ) w 2 - n t l ( f l 2 + s'l)npp2 - ntint2P2 So the variance V2 can be written as: (4 + 4)(4 + 4 > p [g 4 Q(l+6 2 ) ] + ( l + 6 ' 2 ) 2 ( s ' 1 + S ' 1 ' ) li+ iI) ( 1 + t, ) |(4±i|K :Q 1 + 1 + i, ] 4+4')/n n P ntiA-4 ) 4+4') (4 + 4 K #.(4 + 4)4 4 + 4) _ ( 1 _ np _ #(4+4)4 4 + 4 W A-(4 + 4)4 ' i + < ) ( i - # 4 ) . Hence, the variance V(A{,) can be expressed as: ™t2 (C.31) 

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