THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING STUDY OF THE USE OF NON-SIMULTANEOUS MEASUREMENTS IN TRIANGULATION R odert E. *ese October, 1958 IP-326

iv N. X " easaj

ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to the chairman of his doctoral committee, Professor A. B. Macnee, for his advice and counsel during this study, He is also grateful to the other members of the committee, Professors R. C. F. Bartels, F. Jo Beutler, L. F. Kazda, L. Lo Rauch, J. G. Tarboux, and especially H. H. Goode for their comments and suggestions. The author is indebted to Yo Morita and T, Connors for their constructive criticism during the preparation of the manuscript, to Miss C. Landini who assisted in its preparation, and to the Industry Program of the College of Engineering for the reproduction of the dissertation. The author wishes to express his special gratitude to his wife, Barbara, for her encouragement and patience throughout the study. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.............................................. ii LIST OF TABLES.............................................. LIST OF FIGURES........................................... vi LIST OF SYMBOLS.............................................. ix CHAPTER I DESCRIPTION OF THE PROBLEM.................... 1 CHAPTER II BACKGROUND INFORMATION.................... 4 CHAPTER III DESCRIPTION OF THE ANALYSIS AND ASSUMPTIONS... 12 CHAPTER IV ANALYSIS...................1......8......... 18 Position Error in Terms of Bearing-Reading Error....................................... 18 The Use of Weighting Factors.................. 26 Probability Distribution for Bearing-Reading Error................................ 29 Approximations for Small Bearing-Reading Error.................................... 37 Probability Distribution for Position Error,,.. 47 CHAPTER V CONVENTIONAL THREE-STATION SYSTEM............. 50 Equations for the Position Error.............. 52 Variation in the Position Error............... 54 Probability Distribution for Bearing-Reading Error...................................... 64 Probability Distribution for Position Error... 67 Description of the Computer Program........... 68 Details of the Computer Program............. 71 CHAPTER VI MODIFIED THREE-STATION SYSTEM.................. 90 Equation for the Position Error............... 90 Variation in the Position Error............... 91 Probability Distribution for Bearing-Reading Error....................................... 96 Probability Distribution for Position Error... 102 Description of the Computer Program........... 105 Details of the Computer Program............ 107 Closed Form Solution.......................... 114 CHAPTER VII NUMERICAL RESULTS............................. 9 iii

TABLE OF CONTENTS (CONT'D) Page CHAPTER VI]I APPLICATIONS......................... 138 CHAPTER IX CONCLUSIONS AND POSSIBLE EXTENSIONS,........... 149 APPENDIX A PROGRAM FOR THE CONVENTIONAL SYSTEM............,.* 153 APPENDIX B MODIFIED SYSTEM, PROGRAM I.................... 160 APPENDIX C MODIFIED SYSTEM, PROGRAM II....................... 164 APPENDIX D OTHER VALUE OF O-u.......... 167 APPENDIX E VALUES OFA AND B................................ 168 APPENDIX F CODE WORDS FOR PROGRAM II......................... 169 BIBLIOGRAPHY.O.............................. 170 iv

LIST OF TABLES Table Page 5-1 Expected Value of the Normalized Position Error for a Simple Probability Distribution for Bearing-Reading Error............. 0..........o.. 57 7-1 Geometry Represented by the Twelve Cases in Figure 7-8....................................... 131 8-1 Characteristics of a Fictitious Three-Station Triangulation System.......................... 139 8-2 Effect of Large Changes in Parameter Values on the Magnitude of the Position Error (which is Exceeded with a Probability of O.1) o............I...... 147 A-i Simplified Description of the MIDAC Operations.... 154 V~~~~~~~~5

LIST OF FIGURES Figure Page 3-1 Triangulation System Made Up of Three-Station Units.1......................................... 15 4-1 The Geometry of a Pair of Bearing Measurement Stations............................................. 21 4-2 Error in Bearing Reading due to Target Motion and Non-Simultaneous Measurement.................... 34 4-3 Error in the Approximate Expression for the Position Error..................................... 42 4-4 Probability Density-Function for.............. 47 5-1 A Simple, Discrete Probability Distribution for Bearing-Reading Error............................. 56 5-2 Target Locations Selected for the Examination of the Position Error.............................. 56 5-3 An Arbitrary, Symmetric Distribution for Bearing-Reading Error.............................. 58 5-4 Locus of Points at which & - -/ is Constant...... 60 5-5 Maximum Values of AU on the Loci of Constant Values of -............................ 62 5-6 Region in which Au Does Not Exceed Twice its Value at the Center.......................... 63 5-7 Probability Density-Functions for Bearing-Reading Error and the Components of Bearing-Reading Error.. 65 5-8 Simplified Block Diagram of the Computer Program for the Position Error in the Conventional System.. 70 5-9 Approximation of the Probability Distribution for E................................................ 73 5-10 Approximation of the Probability Distributions for nu aond ot......O............................ 75 5-11 Convolution to Obtain the Probability Distribution for the Variable - r......a..................... 78 5-12 Symmetry with Respect to........................ 80 vi

LIST OF FIGURES (CONTD) Figure Page 5-13 Part One of the Computer Program for the Position Error in the Conventional System......... 83 5-14 Discrete Distribution for Target Direction......... 84 5-15 Part Two of the Computer Program for the Position Error in the Conventional System.......O. 87 6-1 Normalized Geometry for the Two-Bearing-Reading Case............................................... 93 6-2 Variation of the Expected Value of the Position Error for a Simple, Discrete Distribution for Bearing-Reading Error.............................. 95 6-3 The Areas A, and la in the td ) t3 Plane........ 99 6-4 Probability Density-Function for t, the Age of the Bearing Reading which is Used....... 101 6-5 Probability Density-Function for...............01 6-6 Simplified Block Diagram of the Computer Program for the Position Error in the Modified System-..., 106 6-7 Approximation of the Probability Distribution for...108 6-8 Part One of the Computer Program for the Position Error in the Modified System................... 110 6-9 Part Two of the Computer Program for the Position Error in the Modified System.................... 112 6-10 Modified Part Two of the Computer Program for the Position Error in the Modified System........... 113 6-11 Probability Density-Function for ED When 9-B <................. e 115 6-12 Joint Probability Density-Function for ED and -d - When B << t.. 116 7-1 Normalized Position Error for a Three-Station Triangulation System.......................... 121 7-2 Normalized Position Error for a Three-Station Triangulation System............................. 122 vii

LIST OF FIGURES (CONT'D) Figure Page 7-3 Normalized Position Error for a Modified Three-Station Triangulation System.............. 123 7-4 Normalized Position Error for a Modified Three-Station Triangulation System................. 124 7-5 Normalized Position Error for a Modified Three-Station Triangulation System................ 126 7-6 Loci of Constant P (EN > Eo) for Uniformly Distributed Target Direction................. 128 7-7 Loci of Constant P (E, > EO) for Uniformly Distributed Target Direction....................... 129 7-8 Effect of Target Direction on the Normalized Position Error for a Three-Station Triangulation System..................0......................... 130 7-9 Effect of Target Direction on the Normalized Position Error for a Modified Three-Station Triangulation System..................... 0*.. 133 7-10 Comparison of Results Obtained Using Different Probability Distributions........................... 136 7-11 Percent Difference Between the Values of Eo Obtained by Using Two Different Discrete Distributions for Bearing-Measurement Error........ 137 viii

LIST OF SYMBOLS a subscript indicating any station whose bearing reading is old A - closest integer to lT sin (t- ) A1,. a -regions of integration in the t1 t3 plane A u - smallest region within which the measured target position occurs with a specified probability B- closest integer to DT sin (82- k) C- closest integer to I a, ClC, C3 - constants CW4 C~~ Cb )- constants ~'t - polar coordinate (distance) of Station i Z$l - distance from Station i to the intersection of the bearing lines from Stations i and j ct[ - polar coordinate (distance) of the target at the time the bearing measurement at Station i is performed - reference distance in the triangulation system (chosen to be the distance of each station from the center of the equilateral triangle formed by three stations) DN - distance that the target moves in the time,T, normalized with respect toD and O-6B DT - distance that the target moves in the time, T, normalized with respect to D - position error Ec - variable used to represent the square of EN ED - position error normalized with respect to V) EN - position error normalized with respect to D and ~8 EO - an arbitrary value of EN R - computer address corresponding to E ix

LIST OF SYMBOLS (CONT D) ES - transformation of Ec Ex - expected value of ED Eo position error which is exceeded only 10% of the time K - a multiplier which is a function of C K1, KKK3 - constants used to transform EC into ER M - the number of targets under surveillance 7? - the number of bearing measurement stations N - total number of intersections of pairs of bearing lines O - a subscript indicating a particular value of the variable to which it is affixed - a relative storage address p - discrete probability distribution for the variable indicated as a subscript p( ) probability of a particular event, or the cumulative probability distribution for an event indicated within the parentheses.( ) - discrete, conditional probability distribution for the variable and condition listed within the parentheses Q.[Q. lQk - symbols used to represent trigonometric expressions i' M X Q = approximations of 6a,? and QkL' respectively hAd- polar coordinate (distance) of the intersection of the bearing lines from Stations i and j S - subscript indicating the smallest possible value of a variable S - ratio of the standard deviation of U to the standard deviation of EB to - age of the bearing reading from Station i at the time of its use T 7 when all T are the same

LIST OF SYMBOLS (CONT'D) TZ - time between consecutive measurements of the bearing of a particular target at Station i U - discrete variable representing (I v/ - discrete variable representing 6A I/ - discrete variable representing EBy v, - discrete variable representing 6Et V - speed of the target ut - discrete variable representing Et3 a - discrethe triable representing t3 w - discrete variable representing E 3 W~i - weighting factor assigned to the intersection of a pair of bearing lines from Stations i and j X - Cartesian coordinate of the centroid or weighted centroid of the N intersections - Cartesian coordinate of the intersection of the bearing lines from Stations i and j - Cartesian coordinate of the centroid or weighted centroid of the N intersections ~.. - Cartesian coordinate of the intersection of the bearing lines from Stations i and j c; - angles between the bearing lines from Stations 1 and 2, respectively, and the line joining the stations - a combination of other variables A - a small change in the variable which follows it E -a value of E6 EB. - error in the bearing measurement of Station i 6BiM - individual components of 6Bf E - error in the bearing reading at Station i xi

LIST OF SYMBOLS (CONT'D) Eti - component of the error in the bearing reading at Station i due to time delay and target motion - polar coordinate (angle) of Station i - polar coordinate (angle) of the intersection of the bearing lines from Stations i and j - an arbitrary constant 7T - 3.14159. ~ (0 - probability density-function for the variable indicated as a subscript e ( ) - conditional probability density-function for the variable and condition listed within the parentheses T - standard deviation of the distribution for the variable indicated as a subscript _ - a sum performed over all permutations of the values 4~ of L and; from 1 through - - a sum performed over all permutations of the values yibk of o a, and t from 1 through - a sum over all possible values of %o r - arbitrary constant with the dimensions of time - time required to perform an individual bearing measurement at Station i - polar coordinate (angle) of a target traveling a straight path - polar coordinate (angle) of a target at the time the bearing measurement at Station i is performed xii

CHAPTER I DESCRIPTION OF THE PROBLEM A need exists for electronic systems which can determine by triangulation, the position of many moving targets which exist simultaneously. Triangulation is the general method of calculating the position of a target from measurements of the angle of direction (bearing) to the target from a set of at least two known points. When all of the bearing measurements on a particular target are performed simultaneously, the error in the computed position of the target is unaffected by target motion. When the bearing measurements are not performed simultaneously, an error is introduced because of the motion of the target during the time between the measurements used in the calculation of target position. When many targets are to be under surveillance, simultaneous bearing measurement, although desirable, is not always a practical design requiremento In order to obtain simultaneous bearing measurement at several bearing-measurement stations, the selection of a particular target must be made in advance at some central point within the system and then the selected target must be designated to each station. The selection may be made on the basis of an expected bearing angle at each station, a set of electromagnetic parameters of the expected signal, or both. When many targets are under surveillance, expected bearing angle will not uniquely define a target. For each signal received, the use of electromagnetic parameters to define a selected target requires that a measurement of the parameters be performed and the signal discarded if it is not the selected one. The rate at which data is obtained on each target by such a process is substantially less than the rate which would

-2result if all the received information were used. Data rate is an important characteristic of a system which must handle many fast-moving targets. Consequently, triangulation systems which do not require simultaneous bearing measurements are of interest. The use of non-simultaneous bearing measurements requires that the bearing-measurement data, as well as those signal characteristics which are used to distinguish among them, be stored for future use and that the triangulation system possess a capability for selecting the proper data for use in the calculation of target position. A decision to use non-simultaneous bearing measurements in a triangulation system should be based on whether or not the additional error caused by their use can be tolerated. The study herein is aimed primarily toward providing a measure of this error in terms of normalized triangulation-system parameters. The measure is in the form of probability distributions for the magnitude of the error in the calculated position of the target, hereafter called position error. The results of the investigation provide a means for deciding whether or not to use non-simultaneous measurements and, in addition, provide a way of selecting some of the system design parameters. The results provide a way of e-valuating existing and proposed systems as well as modifications to systems. Because the study is concerned with many, simultaneous, moving targets, a situation chiefly associated with radio direction-finding networks, the details and examples which are considered herein are limited to the direction-finding case. This does not preclude the use of the methods employed nor the application of the results to optical, infrared, or any other type of system which uses triangulation.

-3The study described herein is restricted to the two-dimensional case, i.e,, altitude has not been considered. The position error has been investigated in general for a triangulation system made up of an arbitrary number of bearing-measurement stations. Consideration is given to weighting the information from which the target position is calculated according to the expected accuracy of the information. Numerical analysis has been performed only for the case of three bearing-measurement stations, in a special case which is generally useful. The techniques used in the numerical analysis are such that they may be extended easily to other situations.

CHAPTER II BACKGROUND INFORMATION Triangulation is the general method of estimating the location of a target (any desired point) by measuring the angle of direction of the target from a set of known points. Estimation of the location of a target from the measured data may be performed in a variety of ways. The simplest procedure is the use of a plotting board on which the intersection of bearing lines from two known points (bearing-measurement stations) is selected as an estimate of the target location. It is only an estimate because the measurement cannot be performed without error. Both manual and automatic plotting boards have been used, or, of course, direct, trigonometric calculation can be substituted for them. When more than two bearing-measurement stations are used, the bearing lines seldom intersect at a point. Using a plotting board in this case, a human observer may make an estimate of the location of the target on the basis of his experience or with the assistance of a mechanical aid, called a "spider", with which the set of intersections formed by the set of bearing readings is adjusted into a smaller region by adding a correction angle of equal magnitude to each bearing reading. Barfield(2) describes an interesting electrorrmchanical device with which the most probable target location can be obtained. If exactly three bearingmeasurement stations are used, an observer may compare the shape and location of the triangle formed by the bearing lines with the triangles prepared by Stansfield(l9), which show the location of the "most probable point'. Estimation of the location of the target may also be accomplished by use of digital computing apparatus. A computer may be designed to -4 -

-5perform the estimate by making use of any one of a variety of criteria, and can also be used to compensate the measured data for known systematic error. The general problems of radio direction-finding are extensively discussed by Bond(3), Keen(t4), and Ross(l7). Ross conveniently divides the problems into three groups, (1) those dealing with the instrument itself, (2) those dealing with phenomena occurring in the course of propagation of the waves, and (3) those concerned with the interpretation of the bearings once the readings have been obtained. The study described herein is concerned with the interpretation of bearing readings, and, therefore, some attention must be given to the character of the error in the readings. Error in bearing measurement and means for reducing it has received considerable attention.1 However:, reliable measurements of the error due to individual sources of error are difficult to obtain because of the large number of such individual sources and because of their dependence upon many parameters. Ross(18) classifies errors into four groups~ instrumental errors, site errors, propagation errors, and observational errors, and describes a method of estimating "a priori" the probable error of a given bearing. Compensation for systematic error which can be measured in bearing-measurement equipment is accomplished by calibration. For the random error which remains, there is general agreement that, in practice, it is described by a probability distribution which differs very little 1 See, for example, Bowen(4), Horner(ll), and a collection of papers devoted to direction-finding in The Journal of the Institution of Electrical Engineers, Vol. 94, P tI A, London; 197.

-6from the normal or Gaussian distribution, Ross(l8) cites the results of experimental trials which agree with the results predicted on the basis of a normal distribution. In studies of position error, the normal distribution has been used(8,lO,l9)o Position error is studied for two reasons: (1) to evaluate the confidence which can be placed in the estimated target locations obtained from particular triangulation systems as a function of the location of the target with respect to the bearing-measurement stations and as a function of other parameters and (2) to study the procedures by which the probability of error in each bearing reading can be used to obtain the best estimate of a target location. The evaluation will depend, of course, on the estimation procedure which is used. Stansfield(l9) points out that the problem of the determination of the most probable point given by a set of position lines of unequal weight was considered by d'Ocagne (16) in 1893. Stansfield(l9) developed an expression for the conditions which the coordinates of the estimated target position must satisfy in order to be a best estimate of the true target position by the principal of maximum likelihood. He considers the case of two dimensions and an arbitrary number of bearing-measurement stations, assuming a normal distribution for the error in each of the bearing measurements with the absence of systematic error. He also presents a geometric interpretation of his conditions for the case of three bearing-measurement stations. Even in this case, the application of the conditions to the problem of selecting the coordinates of the estimated target position is not simple. Stansfield also examines the probability distribution which describes the position error when his criterion is used. He

-7presents graphs of the fifty percent-probability contours (ellipses)1 as a function of the target location for the special cases of two, three, and four direction-finding stations, when the standard deviation of each bearing reading is two degrees. Harkin(10) has considered the error in the three-dimensional triangulation problem with a normal, circular, bivariate distribution for each of the bearing lines. He does not consider a best estimate of target position but weights each of the bearing lines equally. The studies of the position error which are referred to above consider that either the bearing measurements are performed simultaneously or that any motion of the target is negligible. As pointed out in Chapter I, the use of simultaneous measurement, although desirable, is not always practical. The author(8) has investigated the position error when non-simultaneous measurements are used for the case of two bearingmeasurement stations, with the rather loose assumption that the error in a bearing reading because of its age and target motion can be approximated by a normal distribution with a mean of zero. A more realistic study requires the use of non.-normal probability distributions with non-zero mean values. The use of probability distributions which are not normal, or the combination of independent, random variables by a process other than addition often requires the use of numerical methods. Such is the case in the study described herein. A variety of numerical methods, including methods of sampling which are discussed below, are available for 1 If circular probability contours are desired, they may be obtained by use of a table of "Q.Functions" such as those prepared by Marcum 15),

-8performing convolution, the integration process by which the probability distribution for the sum of independent random variables is obtained.1 When random variables are combined by a process other than addition, more complicated procedures are usually necessary to obtain the resultant distributiono The resultant distribution can be expressed in the form of an integral depending on a parameter, but as Kaplan2 points out, it can easily happen that the integral cannot be expressed in terms of elementary functions, even when the distributions which describe the random variables are expressed in simple equation form. Teichroew(21) points out that even with the use of high speed computers, numerical integration is not always practical. "It is just as impractical to use a high-speed computer for a year to do an integration as it is to do it by hand in 105 years." ~"Distribution Sampling" is another numerical method of obtaining the resultant distribution in which "the basic problem is expressed in probability terms and sampling has been used to solve it; the integral formulation is not necessary for the sampling procedure."3 The value of each of the independent, random variables is sampled at random according to the probability distribution which describes it and the corresponding value of the dependent variable is calculated. The sampling process is repeated many times and the set of values obtained for the dependent variable is ranked in order of magnitude. The probability that the 1 See, for example, Tustin(23) and Truxal(22), 2 Kaplan(13), p. 218. 3 Teichroew(21), p. 3e

-9dependent variable is less than some value is approximated by the ratio of the number of times a smaller value is obtained to the number of times the sampling process is employed. Teichroew(21) points out that this method was introduced by "Student"(20) in 1908. More recently this method has been used to evaluate definite integrals, to solve differential equations, and to invert matrices by analogy, i.e., by approximating the solution of a probability problem which can be formulated in the same way as the non-probability problem at hand. The use of distribution sampling in this application has been given the name "Monte Carlo Methods"1, a name which has carried over to any use of distribution sampling. Although the convergence of the approximate distribution obtained by Monte Carlo methods to the true distribution is relatively rapid in the vicinity of the mean of the distribution, convergence is quite slow at the "tails" of the distribution because only a small fraction of the total number of sets of samples yields values of the dependent variable in this region. Kahn(l2) discusses six techniques that can be used with Monte Carlo methods to improve the accuracy of approximation for a given number of samples. Two of these techniques, systematic sampling and stratified sampling, are applicable to the approximate solution for a complete distribution, a major part of the study described herein. In systematic sampling, the values of one of the independent variables in a multi-variable sampling problem are not determined by chance, but instead each of the values of this one variable is distributed among the total number of samples according to the probability associated 1A collection of papers on Monte Carlo methods has been published in book form in Symposium on Monte Carlo Methods, edited by Meyer, H. A., John Wiley and Sons, New York; 19563

10with each value, This technique does not lead to substantial improvements in accuracy, but as Kahn points out, "'it ordinarily does not cost anything to applby this technitue, so that there is no point in not using ittt1l In stratified sampling.9 the sample space is divided into nonoverlapping sub-sets, for each of w h.tch the conditional probability is calculated, A representative sample o)f the same size is then taken from each sub-set, and the results from each sub-set are combined according to the conditional probability f or each 2 The method used in the stuady herein to calculate an approximate resultant probability distrlibution canrnot properly be called distribution sampling or a Monte Carlo method. However, the method used herein must be compared to Monte Carlo methods because of the current acceptance and interest in them. In the method used herein, rhich might be called "complete systematic sampling", the probability space is divided into a large number of non-overlapping sub-sets by dividing each of the dimensions of the space into a set of non-averlapping interva~lso The possible values of the independent, random variables (dimensions) are grouped and approximated by the value at the center of. the interval, ie,, the probability that the vaLue of the variable is within an interval is assigned to the value at the center of the interval. Each of the large number of sub-sets is sampled once, ioe,,o all of the possible combinations of intervals for all of the i.ndependent, 1 Kahn(l2) p o 154o 2 See Albert(l), p. 44 and Teichroew(21" )p L -20. o,~ ppo i7, 0 ~

-11random variables are considered in a systematic way. Although the number of sub-sets considered in this way may be unusually large,l certain efficiencies are available. The random sampling process is eliminated and, therefore, the generation, storage, and use of random numbers is unnecessary. When a systematic selection procedure is used, the timeconsuming process of computing the value of the dependent variable for each set of randomly selected values of the independent variables may be reduced to a simple calculation based on the value obtained on the previous trial. Such is the case in the study described herein. The procedure described above is an extension of both systematic sampling and stratified sampling to the point where they are identical. It is an extension, applied in grand scale to a digital computer, of a simple technique for combining discrete probability distributions by considering all of the possible outcomes. The computer provides a means for systematically considering all of the possible outcomes as well as for performing the necessary calculations. 1 In the three-dimensional probability space considered herein, the number of sub-sets used was in excess of 125,000.

CHAPTER III DESCRIPTION OF THE ANALYSIS AND THE ASSUMPTIONS The position error in a triangulation system consisting of )Z bearing-measurement stations was investigated in general for any arrangement of the stations with respect to the target. An expression for the magnitude of the position error in a single measurement was obtained as a function of the geometry of the situation and the error in each bearing reading. In developing an expression for position error, it was assumed that the calculated position of the target is the centroidl of the set of all intersections formed by all the bearing readings taken two at a time. The centroid is used because of its suitability for high-speed, automatic computation, a necessity for triangulation systems that are used in determining the positions of many, high-speed targets. The use of a weighted centroid to reduce the expected value of the position error is also considered. The probability distribution for the magnitude of the position error is obtained from the probability distributions for the error in the bearing reading at each of the bearing-measurement stations. Bearingreading error is separated into two components: (1) error in the bearingmeasurement itself, and (2) error due to motion of the target during the time between the measurements used in the calculation of target position. Error in the bearing measurement is assumed to be normally distributed for the reasons described in Chapter IV. The age of the measurement is 1 "Centroid" means the center of mass. In tbhis case each intersection has the same mass. A "weighted centroid" means that unequal weights (or masses) may be assigned to each intersection. -12

assumed to be uniformly distributed over a finite interval because the triangulation system is assumed to operate in the following way. Each bearing-measurement station is operated independently and targets are selected for measurement on the basis of their availability. When many targets are available simultaneously, measurements are performed by sequencing through them in any orderly fashion. At each central point in the system at which target position is calculated, bearing reports and target-identification data from each bearingf measurement station are stored according to the time of arrival of such reports. Each time a new report is received from one station, the most recent bearing information on the same target is selected from the storage associated with each of the other bearing-measurement stations and is used to calculate the position of the target. Therefore, in each calculation, one bearing reading is new; the age of each of the other bearing readings depends upon the rate at which bearing readings are performed at each of the stations. If Tt denotes the time between consecutive measurements of the bearing of a particular target at Station i, then the age of the bearing measurement at the time it is used is described by a probability distribution over the interval from zero to Tr. Because the bearing-measurement stations are operated independently, the distribution for each station is a uniform one. In addition, the age of each of the bearing readings used in the calculation of target position is independent of the others. As explained below, the general method selected for numerical evaluation of probability distributions for position error is not based on the use of such simple probability distributions as the normal

distribution for error in bearing measurement and the uniform distribution for age of the measurement. These distributions were selected because of their general utility and because they are appropriate for the triangulation system in question, and not for reasons of convenience in numerical analysis. The numerical method used is applicable to any theoretical or empirical probability distribution for the components of error, provided that the components are independent. For use in the numerical analysis, the general expressions for the position error in terms of the error in bearing readings were approximated for the case of small bearing-reading error. The approximate expressions are simpler in form and, consequently more convenient to use in numerical analysis. The method used, however, does not require that the approximation be made. The special case of three bearing-measurement stations was selected for numerical analysis because it is an arrangement that is used frequently and because this arrangement is well suited for use in a system designed to cover a large area and made up of three-station units, such as those illustrated in Figure 3-1. The study of the threestation arrangement has been restricted to a study of a symmetrical arrangement with the stations located at the vertexes of an equilateral triangle because this arrangement or those differing only slightly from it are generally used. For this symmetrical arrangement, the variation in the position error is demonstrated to be small for a target located anywhere throughout a region surrounding the center which contains at least half of the area of the triangle, For this reason the numerical analysis has been restricted to this region. A number of probability

-150 0 V v - A Vt V V V t tt Figure 3-1 Triangulation System Made Up of Three-Station Units LEGEND: O bearing measurement station A central station for calculation of target position - information flow

-16distributions for the normalized position error have been obtained in this case for several values of normalized triangulation-system parameters. The results are valid for any choice of the variance of the bearingmeasurement error, provided that it is small, and for any choice of spacing between the bearing-measurement stations. Separate probability distributions were obtained for several values of the error due to target motion and age of bearing readings. This error is normalized with respect to the bearing-measurement error. Separate probability distributions for the normalized position error were obtained for each of 481 directions of target motion with respect to the location of the station whose bearing reading is new, so that the variation in position error with target direction is determined. The separate distributions were combined assuming a uniform distribution for target direction, to provide one probability distribution which describes the position error for an arbitrary target direction. A modification of the triangulation system consisting of three bearing-measurement stations was considered also.. In this modification only two bearing readings, the new reading and the more recent of the other two, are used to calculate the position of the target. The reason for considering this modification is the recognition that in a system in which no attempt is made to compensate for the error due to target motion and age of the measurements, the use of the bearing reading with the greater age may increase rather than decrease the position error. This modified system was analysed in a manner similar to the analysis of the conventional system. The conditions under which this modification Because of symmetry, only 12 different probability distribution curves are necessary for the 48 different target directions.

-17provides a reduction in the position error were determined by comparing the probability distributions which were obtained for both cases.

CHAPTER IV ANALYSIS Position Error in Terms of Bearing-Reading Error A target is assumed to be located at the origin of a twodimensional coordinate system. The locations of nt bearing-measurement stations are specified by the set of polar coordinates, dco and Ax, in which the subscript Z, which denotes the corresponding bearingmeasurement station, takes on all integer values from one to n. Each pair of bearing lines intersects at a point denoted by ho., 1, in which the double subscript indicates the pair of bearing-measurement stations involved and, of course, L. j. The total number of intersections, N, is given by N _ t (-'. (4 1) The location of the target, as calculated from the bearing readings, is the centroid of the set of N intersections. Because the target is located at the origin of the coordinate system, the coordinates of the centroid of the intersections specify the error in the calculated location of the target. The magnitude of this error in terms of the coordinates of.the intersections is obtained as follows. In order to determine the centroid easily, the coordinates of the set of intersections are expressed temporarily in a Cartesian coordinate system in which 9X =t cos 0 and X =. sin e. In this Cartesian coordinate system, the coordinates of the intersections, ( i.,) Ax) -18

-19are X;.. k cia Ca.4 and -= -, + (4-2) The coordinates of the centroid are -~xx ~~~~~~ - -W. (4-3) and i aN I 9*tX ) (4-4) in which denotes a double m performed over all pe tations of in which Z denotes a double sum performed over all permutations of the possible values of L and. The use of permutations of values of L and; requires the factor of 1/2 in the expression for the coordinates of the centroid. The position error, E, i.e., the magnitude of the error in the calculated position of the target, can be expressed in terms of its square by E= X (4-5)

-20The combination of Equations (4-2), (4-3), (4-4), and (4-5) yields: 4N 1-N' ( X~-) = (aN) (N)(4-6) = (Zx..) ~ ooebYck) *- Q + i i i permutations of the subscr ipts, with the exceptions and It in which 2 denotes a quadruple summation performed over all ck l permutations of the subscripts, with the exceptions j 4 X and Rs k An expression for the coordinates of the intersections in terms of the coordinates of the bearing-measurement stations and the error in the bearing readings is obtained by use of Figure 4-1 which. illustrates the geometry of an arbitrary pair o;' bearing-measurement stations. The error in the bearing reading taken at Stations i and j is denoted by 6: and 6; respectively. An expression of the "law of

-21/ Station j / Station i IL Intersection Figure 4-1 The Geometry of a Pair of Bearing-Measurement Stations

-22sines" for the two larger triangles is Lt SC at (4 7) A,4EE At(77-*% —6 ( ) and -ic _ 3 X(4-8) Ai -4;(K +b (r +' or cJ4- a f (',c/ee- (E + -l = A, [( te~. a. -( + (4-9) Equation (4-9) can be rearranged to provide M4i L LckP6. + - Ei) - ___ __i~ e. d- e C (,t - d +6 (4-10) Sin -0 and cos 0> can be obtained by using Equation (4-10) and the trigonometric identity ~Co~~~~ -E53 * ~~ "~ a~ ( (4-11) from which'd., "8:..-codb6c+_ +e. =4 (4-12)' ~,, "d eI

-23The expressions for cos -O.. and sin 8.o for use in Equation (4-6) are obtained from Equations (4-10) and (4-12). The results are:.A;.CM(+.6.). CAi6.( e.4* +.) =, I 6. (4j13) ~~.it [ d j - a. 6.c- - (4, B)- + 1 + os. and d 6.A(trtt im )- A. ( (4 + E) -OR~yL (4-14) An expression for I.. for use in Equation (4-6) is obtained from Equations (4-8), (4-13), and (4-14) as follows: 6. AsV66. ( ( t- -j. s Aift.- a e d. Aim 6. C-IE, + 6 -6.) ]+ _. t F ttje e + (; j 41 5t _i o [d, - d 4t. e in. a 8:. ( + 2 e 2j1/; ain ( ~~~~~~~~~~~~~~~~~~ 8~~~~ B i b C ~ ~ ~ ~ (4~

-24The proper selections of algebraic signs for cos 6, sin, and /,. are determined by consideration of the special cases E. 0 Ct L and 6 = 0 and the geometry illustrated in Figure (4-1). By use of Equations (4-13), (4-14), and (4-15), the summand of Equation (4-6) can be put in the form~ Abt IL. CwJ -4k t Ge A - ~C -e tAA An*? -O('* k k Fd ~i b4B

-25Substituting Equation (4-16) in Equation (4-6) and rearranging the summation into a more useful form yields: N/E'~ Ct E\ ) k (t+ t A(t: + k ) 4N Ea AQ' + (t 6 1 k) L-. 6~..C(~ k (.- ( z)kE Z C (. + A ( i C Aik + k) *C} oe~~~~~R~~~k ~(4-17)

Therefore, the position error can be expressed in terms of the error in the bearing reading and the coordinates of the bearing measurement stations with respect to the target as Nl E = 2 1 cajkl k Ai- ~5 Ak~(n +i -8() a I,;* k (4-18) As would be expected from the geometry of a problem in which the angle coordinate of each bearing-measurement station is measured from an arbitrary reference, only the difference in angle coordinates enters the expression for error. The Use of Weighting Factors In order to obtain the best estimate of target position, each source of information should be weighted according to the possible error produced by it. In this way, the total error tends Ago bI'e reduced by placing a greater emphasis on that information which is most likely to have small error and less emphasis on that information which is most likely -o hI-ve large error.1 Stansfield(l9) properly treats the problem of weighting in triangulation systems in terms of perpendicular distances from the estimated target position to each of the bearing lines. When the error in each of the bearing lines is described by a normal probability 1 ormer'), p. 2 -)4.

-27distribution with a mean of zero, the best estimate of the position of the target, on a maximum likelihood basis, is that position at which the sum of the weighted squares of the perpendicular distances is a minimum. The proper weight for each perpendicular distance is proportional to the reciprocal of the variance of the probability distribution which describes the possible values of each perpendicular distance. For small errors in bearing reading, this variance is approximately equal to the product of the variance of the bearing-reading error with the square of the distance along the corresponding bearing line from the station to the intersection with the perpendicular distance. This weighted, leastsquare estimate is generally used for convenience even when the prob.ability distributions involved are not normal. However, the use of this criterion in the selection of the estimated target location is complicated. It was pointed out in Chapter III that the centroid of the set of all possible intersections of pairs of bearing lines is convenient to use in the calculation of target position. In the use of a weighted centroid, a weight is assigned to each of the intersections. For a best estimate of the target position, the weight should not be assigned according to the possible error in each of the intersections because the error in each intersection is not independent of the error in the others. Even if the weights were to be assigned in this way, the determination of the variance for the distribution which describes the error in an intersection is not simple, as can be seen from Equation (4-15). The weighting factor which is used in this study is obtained in the following intuitive way. Each intersection is formed by two

-28bearing lines, to each of which is assigned the weighting recommended by Stansfield(19) The possible error in each intersection is also proportional to the cosecant of the angle of irn'tersection of the two bearing lineso The weight assigned to each intersection is the product of the weights assigned to each bearing line divided by the square of the cosecant of the angle of intersection, ioe., +.. L D j.;j- (4I-19) in which W. is the weighting factor assigned to the intersection Lc of the pair of bearing lines from Stations i and j, D is some reference distance, a constant, a is the variance of the error in the bearing reading at Station i, and d.. is the distance from Station i to the intersection "' formed by the bearing lines from Stations i and j, as shown in Figure 4-1. The use of a weighted centroid using this weighting factor yields the same estimate of the targe- position as the estimate obtained by Stansfield(1l9) which is more difficult to apply. Whern t~e:a-Erget position is calculsited'by use of a "weighted centroid" the coordinates of the centroid are given by X = 2 W.. x. / X (4-20) Ca- t b~~i b~~~~J-i

-29and Z w.:1.. 1w.. it;>~ L+;~ 6(4-21) which are analagous to Equations (4-3) and (4l4) when an unweighted centroid is used, The magnitude of the position error is given by a W. = W d. a k;it o, sk (i 5 t C- + ) - ti vt.,J - k C k (4-22) Probability Distribution for Bearing-Reading Error In a triangulation system in which bearing measurements are not performed simultaneously, the error, 6., in the bearing reading at the time it is used in the calculation of the target location is made up of an error, 6B, in the bearing measurement itself and an error, E, due to the motion of the target from the time the bearing measurement is performed to the time that the bearing measurement is

-30used in the calculation of the target location.l These errors add dire c tlyr E. = + ( (4-23) L 8< tL CE, the error in the bearing measurement itself, is caused by many factors, such as errors in siting,,iv-ar'iag propagation effects due to refraction of the atmosphere ankid the addi tioL of reflected signals from local terrain features, and calibration errors as well as random errors present in the measurement apparatus. Some of these factors are randomly distributed, i.e., their values are subject to chance. Other factors, called systematic errors, are invariant during a set of repetitive measurements and, therefore, their effect cannot be reduced by averaging the result of repetitive measurements. If the value of a systematic error is known, the result of the measurement can be compensated and the effect of the "error" eliminated. If the value of a systematic error is known and the result is not compensated, the error in the result of the measurement due to the lack of compensation can be determined directly from the equation of the measurement, such as Equation (4-18) for the case of a triangulation system. If the value of a systematic error is unknown but the possible values are described by a probability distribution which expresses the lack of knowledge of the value, this probability distribrution can be used to combine the 1 The position error considered in this study is the difference between the calculated position of the target and the true position of the target at. the rst.9,t -)mm 5.i:ie rmrss rec3>r. measurement used in thie cCalclatelon Was performec.

-31effects of this error with other systematic and random sources of error to determine the probability distribution which describes the total error in the result of a measurement. In such a combination, it is necessary to insure that invariant nature of the systematic error to repetitive measurement is properly treated. In a triangulation system in which one measurement from each bearing-measurement station is used to calculate the location of a moving target, the probability distributions which describe the lack of knowledge of the values of systematic error and the probability distributions which describe the random errors can be combined directly. The error, I, in the measurement of bearing at bearingmeasurement Station i consists of a set of independent, random errors and systematic errors, as described above, denoted by. The total error, equal to the sum of these errors, is (4-24) According to the central limit theoreml, if the independent sources of error are described by probability distributions which have standard deviations which are finite, the probability distribution which describes the result of the summation approaches the normal distribution as the number of such sources of error becomes large. If no one source of error predominates, i.e, if the standard deviations for each of the I Statements of the centra] limit theorem app ar in slightly different form in Goode and Machol 9), p. 112, Cram~r ~7), pp. 114-116, and Woodward(24), p. 16.

-32major sources of error Ks of the same order of magnitude, the convergence is relatively rapid, Cramer points out that it often seems reasonable to assume that experimental errors combine in this way, D Goode and Machol point out that such an assumption may lead to pitfalls 2 However, in a general study in'which measuring equipment as yet unspecified is involved, the central limit theorem suggests that the error in measurement is best approximated by a normal distribution than by any other. Ross(18) cites experimental data in which approximation by a normal distribution was justified, In the quantitative studies of a conventional and a modified three-statlion triangulation system described in Chapters V and VI, the total error in the measurement of bearing is assumed to be normally distributed, This assumption is reasonable for the reasons cited above. The adequacy of the application of the results of the quantitative general study to particular trianguLation systems will depend upon how closely the probability distribution of the bearing-measurement error approximates the normal distribution. It is believed that of the total of all possible bearing-measurement apparatus that would be used in triangulation systems, the distributions for bearing-measurement error can be better approximated by a normal distribution than by any other distribution, especially if measures are taken to suppress the major sources of error in the apparatus. However, the method used to study quantitatively the error in three-station and two-station triangulation systems is not restricted to the use of the normal distribution; any theoretical 1 Cramer(7), ppo 120 and 230. 2 Goode and Machol,(9) p 112

-33or empirical distribution can be used, provided that the error is independent of the other parameters which are considered. The error, HE, in the bearing reading due to the motion of the target from the time the bearing measurement is performed to the time that the bearing measurement is used in the calculation of the target location, is a function of the velocity of the target, the location of the target with respect to the bearing-measurement stations, and t., the age of the bearing reading at the time of its use, Figure 4-2 describes the geometry of the situation. The "law of sines" for the triangle shown in Figure 4-2 is t. c Ad-4 ~(w a i-E ~.(4-25) in which gca 9;.are the polar coordinates of bearing-measurement Station i with respect to the true target position at the time the position is calculated, 9t ). are the polar coordinates of the target location at the time the bearing measurement at Station i is performed, with respect to the same reference, and is the error in the bearing reading due to target motion and age of the bearing measurement at the time the target position is calculated, Equation (4-25), when solved for ~, is 4- (4-26)

Station i t// Target location when / Target path its position Target location_ a when the bearing measurement is performed Figure 4-2 Error in Bearing Feading due to Target Motion ard Womn -nSmnltaneosus Measurement / L 41,a e

-35For a target which is traveling a straight line path, the value of f is the same for each of the stations and may be denoted by q. If in addition, the speed of the target, V, is constant, then I = Vt, (4-27) t~ 1 and -V.. -L Vt ()- (4-28) The triangulation system which is considered in this study is one in which each time a bearing measurement is performed at one bearing-measurement station, this measurement and those measurements already taken at each of the other stations are used to calculate the location of the target, The bearing measurements at each station are performed by cycling through the set of targets in some orderly fashion, independent of the measurements taken at the other stations. The time between consecutive measurements of the bearing of a particular target is denoted by T. T depends on A, the time required to perform an individual measurement. If only one measurement is taken at a time, 7 depends directly upon M, the number of targets under surveillance. In this case, T = (4-29) Because of the independence of measurement among the stations, age, to, of the measurement at Station i, when the measurement is used in a calculation initiated by another station, is described by a

-36probability distribution which is uniformly distributed over the interval from 0 to T. T is a characteristic of the triangulation system and the capacity at which it is operating. Except in unusual systems, T7 will have the same value for each bearing measurement station and can be denoted by T The probability distribution for t can be obtained from the probability distribution for t. by means of Equation (4-28). Although the set of variables denoted by t. are independent, the parameti ters which describe the probability distribution of these variables are not independent; each distribution depends on the parameters V,, and T. If distributions rather than individuaal values of target parameters V and 0 are to be considered, the probability distributions for the error in the calculated location of the tamget for individual values of the target parameters O v.a c.e a ormbie a-orcli t o the proability distributions of those parameters. 2~':: y: c:;:;i!iJ. e shaf t o L ao.-Jv:tao:u1 l -a>i a modified tree-sstation triangulation system described in Chap-ers'd and`L;,- ai uniform distribution for target direction is obtained by combining the results obtained for individual target directions. A z:ifor, distri bution for the age of bearing measurements is used altho:XCg- me-;-c. used in -this study is not restricted to the use of uniform distrib-tions. The method is restricted only to indepenience of the age of measurements at different stations.

-37Approximations for Small Bearing -Reading Error Equation (4-18) for the magnitude of the position error, E, can be expanded into the form N'E~' = Zk ei.kt qkl Q% Zt:,lk (4-30) in which Q1 ( )(E, ) a(. -)AE,- ) ) (4-31) PR = +(2- ek) (- k) + (- k (- k) ) (4 32) and Q = _t(- ~C (5-,) + ~-~~(f-( ( 5-,), (4-33) Equation (4-30) can be simplified by using approximate expressions for Q I Qgk and Q.. which are valid when the bearing-reading error is small. When the bearing-reading error is small, Qj can be approximated by Q, given by q C( )c ( l) a (434)

-38For the terms of the summation described by Equation (4-30) in which ~ =, sin (4- - is zero and the approximation given by Equation (4-34) is exact. For terms of the summation in which the value of t - 9 is in the neighborhood of 0 or TT, the error in the approximation is small be cause lavuje -4>)"n4> - ) << -l (435) For terms of the summation in which the value of'- -0 is in the neighborhood of + 7T/d, the error in the approximation of each such term is large on a percentage basis, but the contribution to the summation on subscript 2 of such terms is small because the corresponding value of Q is comparatively small. For example, when -. - & = - 7T/, 9I QQI= Oa Q4 =o (436) whereas I QE4 = |(N c) J 1 (4-37) for those terms in which j or in which -_. - = 0 or 7T. In general, the percentage error of approximation is large in the terms with small values and small in the terms with large values. Q., and Q.,, which appear in the denominator of the summand of Equation (4-30), can also be approximated when the bearing-reading error is small. These quantities are the same except for the subscript notation, so that a discussion of the approximation of one of them

-39applies to the other. Q9L is approximated by Q9. which is given by 9.. = Ah-j.99)cia(E-6) (4-38) For terms of the summation described by Equation (4-30) in which the value of -a -& is in the neighborhood of + 7T/1, the error in the approximation is small because | mL(,-e. ) (Ai. <' ( -.+19)CO(6. -6.) (4-39) For terms of the summation in which the value of 4 -. is in the neighborhood of O or 7T, the error in the approximation of each such term is large on a percentage basis. Furthermore, the contribution to the summation on subscript I of such terms is large because the corresponding value of Q. is small. These are terms which represent major contributions to the position error. The proper design of a triangulation system will eliminate them by eliminating from the computation those pairs of bearing readings which intersect at angles in the neighborhood of zero or TT or by using the weighting factor given in Equation (419)> The result of these approximations for Q Qk X and Q. when the bearing-reading error is small is N.Ea = L'4 Ek ( (t -)-) Nk=I 6k,( -,) ( -,) - k)4(6- -) /(E- E) j+c' Itk (440o)

-4oAdditional approximations can be made for small bearing-reading error. For the 6. as large as five degrees: 4~E/ E~ 6. and L L (441i) At6k 6k with an error in each which does not exceed 0.13 percent; and (:Se,( 6k ) 1, and (4-42) cab (6. - 6.) 1 I with an error in each which does not exceed 0.6 percent. Using these approximations, Equation (4-40) becomes NaEk = Z Ek C( eeniesk o aps wh - ave4 (4e e43) The entire set of approximations which have been made for small bearing-reading error may be interpreted geometrically as the approximation of the bearing lines from each Station i by lines which

-41are located parallel to the true bearing lines with a displacement of ae. E, in the proper direction.l When the intersection of a single pair of bearing lines is used to estimate the position of a target, the error of approximation due to the use of Equation (4-43) in place of Equation (4-30) can be examined easily by direct comparison. If the bearing-reading error is at most five degrees, the maximum error of approximation occurs when the geometric interpretation of the approximation indicates that the geometry is distorted most, i.e., when = 50 and 6 50 The error of approximation depends upon the angle at which the bearing lines intersect, a -, and the ratio of the distances from the target to the bearing-measurement stations. Figure 4-3 shows this error obtained by direct comparison as a function of - - for the two limiting cases, = d and or /1 = 0. When bearing-reading error is at most five degrees, Figure 4-3 shows that the error of approximation does not exceed twenty percent if 500 ~ I|-11| i 140 O When more than two bearing-measurement stations are used, a centroid or weighted centroid of all of the intersections of pairs of bearing readings is used to estimate target position. The position error in this case is no greater than the maximum of the errors in the individual intersections. Similarly, the error of approximation in calculating the position error by using Equation (4-43) instead of (4-30) is no greater than the maximum of the errors of approximation in the 1 This geometric approximation has beep used as a starting point in other studies such as Frese (8), Stansfieldtl9), and Harkin(10).

-4260 40 20 OR =0 ~2 di cro o0 L. -20 _ =I LLI Q -2 CL i 40:2=-50 -60 o % % % 2%7/3 5%7r/ V Figure 4-3 Error in the Approximate Expression for the Position Error

-43error in each of the individual intersections. If a target is so located with respect to more than two bearing-measurement stations that all of the;. -0 i are within the interval 5~0 0 | ~-. <_ 140~, the error of approximation in the position error does not exceed twenty percent for bearing-reading errors which are at most five degrees. For the special case of a target located at the center of an equilateral triangle formed by exactly three bearing-measurement stations (A-L1 = a 7r/, Figure 4-3 shows that the error of approximation does not exceed six percent for bearing-reading errors which a:re at most five degrees. When a weighting factor is used, approximations for small errors in bearing readings can also be made. The weighting factor is first expressed in terms of known distances, approximating the measured distances from each station to each intersection by the distance from the station to the target, i.e., L a. (4-44) Equation (4-19) then becomes + X_ =.a.d Qi (4-45) This approximation is valid whenever the errors in the intersections are small compared with the distance of the target from the bearingmeasurement stations. It is necessary if the equation for position error is to be a simple one~

-44Using Equation (4-45), the position error as given in Equation (4-22) becomes E; W 1 = DBUtA~ g k W.. 4k 6u (4-46) in which Q, Qkt, Q are the quantities defined in Equations (4-31), (4-32), and (4-33), When the bearing-reading error is small, these quantities can be approximated in the same way as they are approximated when no weighting factors are used. The approximations are given in Equations (4-34) and (4-38). The approximation of I is valid for the same reasons which were presented for the case of no weighting. The approximations of Q;. and c. are better when a weighting factor is used than when it is not, because when Qo. and QkO are in the numerator of the summand, the percentage error is large in the terms with small values and small in the terms with large values. By use of these approximations, Equation (4-46) becomes Z - a A al -o -o ikL cI Ei ek ei k j, _.Lk

-45By use of the approximation for Q. a the weighting factor as given in Equation (4-45) becomes \AI= W.a.ra 9YcQ7 * (4-48) As in the case when no weighting factor is used, the trigonometric functions of the bearing-reading errors in Equation (4-47) can be approximated by the expressions given in Equations (4-41) and (4-42). UTsing these approximations, Equation (4-47) may be written as (4-49)

-46Iwo approximate expressions for the position error have been developed~ Equation (4-43) for the case of no weighting factor and Equation (449) for the case of the particular weighting factor given in Equation (4-19). Chapters V and VI describe the use of these equations to determine the position error for the case of three bearingmeasurement stations. An approximation for small bearing-reading error can also be made for Equation (4-28) which describes the error in a bearing reading due to motion of the target. For the error, t, to be small for all values of -0 -, Equation (4-28) shows that Vt. must be much less than H. With this condition and by replacing tan 6t. by E. Equation (4-28) can be approximated by tt a4 (4-50) For Vt./ l as large as 0.05, the maximum error in this approximation is five percent and it occurs when t.- is close to zero or 7T and t. is very small. As - approaches + i/, the error of approximation is zero. Because t. is uniformly distributed in the interval o - t.' T, as discussed on page 3 6, Equation (4-50) shows that. is uniformly distributed in the interval between zero and V Touw(e-~)t as shown in Figure (4-4),

-47Eti V To~C ( _._-___ 0 VTA,,~ t 0) Figure 4-4 Probability Density-Function for i Probability Distribution for Position Error Expressions for the magnitude of the error in the calculated position of a target have been developed in general, Equation (4-18), for the case of a particular weighting function, Equation (4-22), and for both of these cases with an approximation for small bearing-reading errors, Equations (4-43) and (4-49). These expressions can be denoted by the general expression E = E, e, ), (4-51) which indicates that E is a function of all of the a. S,. S, and S- s. Only the 6. s are described by a probability distribution. Each C. is the sum of an 68. and an, as given in Equation (4-23), Both CB. and et have independent probability distributions. The

-48probability distribution of each %i. is specified directly. Each i is a function [Equation (4-28) or (4-50)] of the variables e.,., V, Y, and U., of which only t. is described by a probability disL tribution. This functional relationship can be denoted by the set of general expressions to = Et(,- o, V At) t) (4-52) The probability distribution for each bt can be obtained directly from Equation (4-52) by using the probability distribution for and the values of the other parameters. The probability distribution for each E, can then be obtained by a convolution of the probability distributions of CZ and Ct using the equation Bi t (4-53) The probability distribution for the position error, E, can then be obtained by a combination of the probability distributions for each of the E. by using Equation (4-51) with values supplied for the parameters other than. The complexity of the combination process requires the use of automatic computing devices. To investigate the error in a particular existing or proposed triangulation system which fits the general model described herein, the values of the parameters used in the analysis can be selected and the necessary combinations of the probability distributions can be performed. To investigate triangulation systems in a more general way, configurations

-49of bearing-measurement stations possessing geometric symmetry can be selected and investigated using system parameters which are conveniently normalized. The probability distributions which describe ES6 J 6ti I and finally the position error, E, can be determined in terms of these normalized parameters. This general type of investigation for a threestation triangulation system is described in the next chapter.

CHAPTER V CONVENTIONAL THREE-STATION SYSTEM A triangulation system which consists of three bearingmeasurement stations is of particular interest because such an arrangement provides an efficient way of providing area coverage when the range of the bearing-measurement apparatus is limited, An example of such an arrangement is shown in Figure 3-1. The position error in a triangulation system consisting of three bearing measurement stations is investigated in the following way. An expression for the magnitude of the error in the computed location of a target in the special case of a triangulation system consisting of three bearing-measurement stations is obtained from the expressions developed in the general analysis (Chapter IV). This special' case is specialized further by considering the three bearing-measurement stations to be located at the vertexes of an equilateral triangle. A discrete and a normal probability distribution for the bearing-reading error are used to investigate the expected value of the position error when the target is located at the center of the equilateral triangle and at several other points. The results of this investigation demonstrate that the variation in the expected value of position error is small as the location of the target is moved within a region around the center of the equilateral triangle at least as large as half the area of the triangle. Because of this small variation, the case of the target located at the center of the equilateral triangle is selected for detailed investigation. The position error in this case is a good estimate of the position error in a large region around this point.

-51The probability distribution of the position error for a target which is located exactly at the center of the equilateral triangle formed by three bearing-measurement stations is obtained from the probability distributions for the error in each of the bearing readings. The complexity of both the individual probability distributions for the bearingreading error and the expression for the square of the position error in terms of these distributions requires that the combination process which yields the resultant distribution be performed with the use of an automatic computing device. The bearing error at each bearing-measurement station is grouped into a set of many small intervals and, thereby, the continuous probability distributions are approximated by discrete distributions. The probability distribution for the square of the position error is constructed by examining all of the possible ways that one interval can be selected from each of the three sets, For the particular digital computer used and for this type of problem, the use of all possible combinations of intervals has many advantages over the "Monte Carlo" method which was first considered for use in the solution of this problem. The result of the digital computer study is a set of cumulative probability distributions for the position error, normalized with respect to both the distance of the bearing-measurement stations from the center of the equilateral triangle formed by them and with respect to the standard deviation of the error in the bearing measurement. The set of distributions consists of separate distributions for the two parameters: (1) target direction and (2) a normalized combination of

-52target velocity and time delay, Probability distributions are also presented for the case of a uniform distribution for target direction. Equations for the Position Error For the special case of only three bearing-measurement stations, Equation (4-43), which expresses the magnitude of the position error when no weighting factor is used and when the error in bearing reading is small, can be expanded into the form' r____ ALO, = 4 (aO!t *,) (k-]_V( / + o + r _ a- (_;_an_ (o-od(a) c~ (O-3) _ct(t4 e,) + + __ 31I(_3 a3L/t(-O) A(t-o3 t-) rC4(t&,) 2(t <(twig) 3 (5-1 3 13

-53For the same case, but with any weighting factor used', Equation (4-22) can be expanded into the form: |v(A-O) Li1 + — 2 - EY3a |z 33:aid~icit-e,)e| -j r21 3 d [ ~ 13 1 Z/3 3i ( ) (5-2) |!-tl ((,,-l t+l&,u,c E(e- c - a, - X which is equivalent to Equation (5-1) when the weighting factors are one. In the numerical analyses to be described, only the weighting factor given in Equation (4-19) is considered.

For coverage of an area (Figure 3-1), triangulation systems composed of units of three bearing-measurement stations use arrangements of stations which differ very little from a symmetric arrangement with each station located at the vertex of an equilateral triangle, because it has been generally recognized that such an arrangement of three stations leads to minimization of the position error. In the following analysis of a three-station system, this symmetric arrangement is used because of its more general utility. The procedures which are used in this analysis are not restricted to this symmetrical arrangement and can be applied equally well to any other particular arrangement of bearingmeasurement stations. Because of the use of a symmetric arrangement of bearings measurement stations, it is convenient to normalize all distances, including the position error, with respect to the distance, D, of each bearing-measurement station from the center of the equilateral triangle. This normalized position error is denoted by ED. This distance, D, is used in the weighting factor also. Variation in the Position Error The variation of the normalized position error with variation in the target location is studied in order to justify the choice of a particular target location for use in a detailed study of the position error in a three-station triangulation system. In this study, both a simple, discrete probability distribution and a normal distribution are used to describe the error in bearing readings. The simple, discrete probability distribution presented in Figure 5-1 shows that bearing.reading error is described by two equally

likely values, +E and -. This discrete probability distribution for bearing-reading error is used to examine the position error at the four specific target locations, designated in Figure 5-2 as the points 0, A, B and C. At each of these points, numerical values of the angles and distances in Equations (5-1) and (5-2) were calculated and substituted in these equations. All distances are normalized with respect to D, the distance of each bearing measurement station from the center of the equilateral triangle. At the point O, Equation (5-1) becomes the simple equation D ~ 9d [~ ~ ~ t Et 6 +6 * (5-3) Equation (5-3) is valid whether or not the weighting factor is used, because at the point O equal weights must be assigned to each intersection. Equal weights are assigned because the variances of the bearing readings are identical and because the distances of the stations to the point 0 are equal. For the other points of interest, the equations which describe the position error are more complicated than Equation (5-3). For each of the points, A, B, C, and 0, and for each of the possible combinations of values of E. given by the probability distributions for 6., numerical values of ED were calculated. The expected values of Eo, Ex, based on these probability distributions, are listed in Table 5-1.1 For the use of the weighting factor, The process for determining EX is described in Chapter VI for a simpler case.

-56a Figure 5-1 A Simple, Discrete Probability Distribution for Bearing-Reading Error /Bearing-Measurement Stations Figure 5-2 Target Locations Selected for the Examination of the Position Error

-57the numerical results presented in Table 5-1 demonstrate that the expected value of the position error for a target located as far from the center of the triangle as points A and B is little different from the expected value for a target located at the center. A maximum difference of eighteen percent occurs when the target is located at point B. TABLE 5-1 EXPECTED VALUE OF THE NORMALIZED POSITION ERROR FOR A SIMPLE PROBABILITY DISTRIBUTION FOR BEARING-READING ERROR Ex (Expected value of the normalized position error, ED) Without Weighting With Weighting Point 0 1.00O 6 1. 00 Point A 1.18E 1.14 E Point B 1.20E 1.186 Point C 1.62E 1 162E i The value without weighting at point C was obtained assuming that the target is on the line joining the two stations when the measured bearing lines to the target do not intersect. Although the expected values of the normalized position error listed in Table 5-1 are not realistic because the probability distribution for bearing-reading error that was used is not realistic, the demonstration that the variation of position error is small in a large region around the center of the triangle can be extended to any more realistic distribution, provided it is symmetric. Consider an arbitrary,

-58symmetric distribution for bearing-reading error such as that shown in Figure 5-3. If this distribution is divided into intervals, the contribution to the position error corresponding to each pair of symmetric intervals can be examined separately in exactly the same way the simple, discrete probability distribution was used. If the variation in each contribution is small, the variation in the total position error is small. 6 EE.E E -6 +6 Figure 5-3 An Arbitrary, Symmetric Distribution for Bearing-Reading Error

-59The variation of the position error in the region around the center of the triangle can be examined using a normal distribution for bearing-reading error, In a previous report, the author has investigated the error in determining the location of a target when two bearingmeasurement stations are used and when the error in bearing reading is described by a normal distribution.l The results of the investigation are presented in the form of an area of uncertainty, Au, defined as the area of the smallest region surrounding the true target position2 within which the measured target position will fall with a specified probability. The results show that A oc Us" 4I'~~(6-')] (5-4) for any specific probability level and for any finite values of the variance of the normal distributions which describe 16 and E.3 This proportionality can be used to demonstrate that the variation in position error is small. The arc of a circle joining two bearing-measurement stations, as shown in Figure 5-4, is the locus of the points at which the bearingmeasurement stations subtend a fixed angle, ioe., at which -LO- 0 is constant. 1 Frese(8), pp. 2-220 2 The smallest region is an ellipse. 3 Equation (5-4) indicates that AU increases without bound as 0a- approaches TT. If the fact that bearing lines are semi-infinite instead of infinite had been taken into account, Au would be bounded. For semi-infinite bearing lines, AU is certainly no larger than the value obtained from Equation (5k4).

-60Target LoCUS I I Station 1 JD Station 2 Figure 5-4 Locus of Points at Which -t - 4 is Constant From the "law of sines" for the triangle shown in Figure 5-4, the product of the distances A/ and Al can be expressed as d J3 = J - D 4(* -K (5-5) in which o< and o are the angles between the line joining the two bearing-measurement stations and the bearing lines from Stations 1 and 2,respectively, and E/D is the spacing between the stations. Equation (5-5) can be simplified by use of trigonometric identities to the form {&.t(- -i) (5-6)

-61On the locus defined by a fixed value of - 0, Equation (5-6) shows that t cki is a maximum when c = o(, i.e., when / equals. Therefore, at any point on the locus defined by a fixed value of -0; - Au is no larger than its value at = d-. Figure 5-5 is a graph of the maximum values of AU as a function of the value of 0- 0 which defines the locus~ The maximum value of A for the locus, -O = 0 /1AO is normalized to unity. When three bearing-measurement stations are used, the area of uncertainty for the three intersections of pairs of bearing lines can be examined. Figure 5-6 shows three stations located at the vertexes of an equilateral triangle. The center of the triangle is the point at which the value of Au was normalized. For each intersection of a pair of bearing lines, Figure 5-5 shows that Au does not exceed twice its value at the center of triangle anywhere in the region between the loci defined by bearing lines which intersect at 710 and at 160~0. The region (Figure 5-6) in which the value of Au for each of the three intersections is no larger than twice its value at the center of the triangle is at least fifty percent of the area of the equilateral triangle. In terms of a distance rather than an area of uncertainty, the position error is no larger than the JT times its value at the center throughout a region around the center which contains at least fifty percent of the area of the triangle. For a value of AU no larger than three times its value at the center, the region contains at least seventyfive percent of the area of the triangle. Thus, in a detailed investigation of the position error, the determination of the position error for a target located at the center

-623,0 2.5 LU0 w.5__ 1.0 w O,5 [r 0 60~ 80~' 100o 120~ 140~ 160~ 180~ 2 Figure 5-5 Maximum Values ofrAu on the Loci of Constant Values of &2-e

-63STATION 3 LOCUS OF &2-. =710 REGION IN WHICH AU 2 STATION I STATION 2 LOCUS OF - = 160~ Figure 5-6 Region in which A Does Not Exceed Twice its Value at the Center

-64of the equilateral triangle is an adequate estimate of the position error for a target located in a region around the center at least as large as half the area of the triangle. Probability Distribution for Bearing-Reading Error At the time the position of the target is calculated in a triangulation system which contains three bearing-measurement stations, the bearing reading from one of the stations is new, and the age of each of the other two bearing readings is described by the uniform probability distribution discussed in Chapter IVo For the purpose of calculation, the new bearing reading is assumed to have been taken at bearingmeasurement Station 1. The assumption is arbitrary because the error in bearing reading at each bearing-measurement station enters into the equation for the normalized position error in the same way. Figure 5-7 is a sketch of the probability distributions for the error and the components of the error in the bearing readings at each of the bearing-measurement stations for a particular target path.'The probability distribution for the error in bearing reading at Station 1, E, is the same as the probability distribution for the error in the bearing measurement itself, BI, because the age of the measurement is zero. The quantity, EBI, and therefore 61, are assumed to be normally distributed for the reasons discussed in Chapter IV. The standard deviations of these distributions are denoted by 0B G The error in the bearing readings at Stations 2 and 3 is given in Equation (4-23) as = ( 5-7) ~B,2

-650 (88 E2 mla t 0 E52 O t2 2 3 Et3E 133 <O 0 VT SIN(Os- #) d, Figure 5-7 Probability Density-Functions for Bearing-Reading Error and the Components of Bearing-Reading Error

-66and 3 83 t3, in which (5-8) the subscripts 2 and 3 refer to the bearing-measurement Stations 2 and 3, respectively, and in which 6. is the error in the bearing reading, 6B is the error in the bearing measurement, and is the error due to motion of the target from the time the bearing measurement is performed to the time that the bearing measurement is used in the calculation of the position of the target. The error in bearing measurement at Stations 2 and 3 is assumed to be normally distributed for the reasons discussed in Chapter IV. If it is assumed also that identical bearing-measurement apparatus is used at each station., and if there are no specific reasons for the characteristics of the error in bearing measurement to differ among the three stationsl, the standard deviations of the probability distributions for E and E3 will also be equal to..B 83 C The probability distributions for tA and 6E, as shown in Figures 4.4 and 5-7, are uniformly distributed between the limits of zero and VTAi- ( - 0)/d, in which C takes on the values 2 and 3, respectively. 1 Differences in the conductivity of the ground and other features of the local environment in addidtion to differences in the propagation paths of the signals received can cause differences in the characteristics of the error even though the measurement apparatus is identical.

-67T For the special case of a target located at the center of the equilateral triangle, d = 3 = X.(5-9) If 1)T is defined as the value of VT, normalized with respect to D, i.eo. VT -D D (5-10) then the limits of the probability distribution for 6t and 3t may be expressed as zero and Drin. (. ) The construction of the probability distributions for E and E3 from the normal and uniform distributions of their components is the first part of a digital computer program for determining the probability distribution for the position error. Probability Distribution for Position Error The probability distribution for the position error for a target located at the center of the equilateral triangle formed by three bearing-measurement stations was obtained by a combination of the probability distributions for bearing-reading error at each station, making use of Equation (5-3). This combination is the second part of a digital computer program for numerically determining the probability distribution for the position error. The use of a digital computer requires that each of the probability distributions be approximated by discrete distributions. The entire probability distribution for position is desired in order that the results be of general use to those who are interested in

-68the value of position error which will not be exceeded except with a very small probability, as well as to those who are interested only in the mean value of position error. If that part of the distribution in the neighborhood of the arithmetic mean of the distribution for position error were of interest only, then the "Monte Carlo" method, which provides reasonably rapid convergence in this neighborhood could be used. Because the probability of values of position error remote from the arithmetic mean of the distribution are of interest, each possible combination, rather than a random sample, of the discrete values of the components must be considered. In addition, systematic consideration of the possible combinations reduces by manyfold the computing time required per combination over that required by the "Monte Carlo" method. The succeeding sections of this chapter contain a brief description and a discussion of some of the details of the computer program used to obtain probability distributions for the position error. Description of the Computer Program The digital computer program which was used to determine the probability distribution for the position error (using the MIDAC computerl) is divided into two parts. In the first part, the probability distributions for bearing-reading error are constructed. In the second part.the probability distribution for the position error is determined. In the first part of the program, the probability distributions which describe the bearing-reading error at each bearing measurement station were approximated by discrete probability distributions 1 MIDAC is the MIehigan Digital Automatic Computer, and is located at the University of Michigan.

69of many ordinates each. The non-normal distributions which describe the bearing-reading error at Stations 2 and 3 were obtained by processing the normal distribution for the bearing-reading error at Station 1I, a distribution which was contained in the computer program. The different distributions which describe the bearing-reading error at Stations 2 and 3 were obtained automatically for each selection of target direction, In the second part of the programs for each possible combination of the ordinates of the three probability distributions, the joint incremental probability, which is the product of the three ordinates, and the corresponding value of the position error were calculated. The value of the position error for each combination of ordinates was used to obtain a storage address in the computer at which the values of the joint incremental probabilities were accumulated. The computer program was designed to cycle through all of the possible combinations of ordinates of the three probability distributions for bearing-reading error in an efficient manner so that the probability distribution for position error was obtained with a minimum of computing time. The result of the computation is a set of numbers which is the desired probability distribution. Depending on the particular normalized parameters used in the computation, the location of a number in the set corresponds to a particular value of normalized position error. Figure 5-8 summarizes, in block diagram form, the brief description of the computer program given above. For convenience in the digital computation, the set of possible values of 6/, E and E were represented by integers and integers plus one half. The variables u, ir, and w are used to denote this representation. Thus, in

Form.the Use the value of Start < probability Compute Cmpute |. Ec(U-o, r~ ) as an distribution P > Ec((,%rc,) address at which for Pu, PI a to accumulate the and P. value of PVV Cycle through all i. combieations of the the P Cycle through \ all values Print out of target I results direction angle, then end, Figure 5-8 Simplified Block Diagram for the Computer Program for the Position Error in the Conventional System

-71Figure 5-8,' tP, and IP, are the three discrete probability distributions for the variables U. Ar, and AJrrespectively. The joint probability of a particular combination of values of u Ar, and w is given by P PPP (5-11 The subscript o, indicates particular values of the variables. EC is used to represent the square of the position error. Details of the Computer Program Each of the probability distributions used in the digital calculation to describe the components of bearing-reading error was approximated by a discrete distribution of many ordinates. For convenience, the elements of the set of possible discrete values of these variables are represented by integers plus one half. The variables U, 1Wr, nr, r and w are used in the description of the computer program to denote these discrete values corresponding to the variables 6E, a. 3,, and E3 respectively. The computer I Ba' ta 83, t3 variables are related to the components of bearing-reading error by a scale factory denoted by s and defined by T& _ (5-12) in which 0u is the standard deviation of the discrete distribution used in the computer program to describe the variable U, and as is the standard deviation of the probability B distribution for error in bearing measurement.

-72The variable V is related to E/, the error in the bearing reading from Station 1, by UL= S & (5-13) If U is considered to be a continuous variable, it is normally distributed with a mean value of zero and a standard deviation, a, which is related to CB by Equation (5-12). For use in the digital computation, EB the normal distribution for J is modified by truncation, i.e., by neglecting both "tails" of the distribution, as illustrated in Figure 5-9. The remainder of the distribution is divided into 50 equal intervals with integer values of U as boundaries and is approximated by discrete ordinates located at the center of the intervals. The discrete ordinates are equal to the area under the normal probability-density curve for the corresponding intervals. The values of the ordinates listed in the computer program were obtained from a table of areas.l In the course of the digital computer study, two standard deviations for the probability distribution for the variable U = SE, were used. These are m = /0 and T = 50/7. In both cases, fifty ordinates were used for the distribution, corresponding to the ranges -2.5 _U L I ~ 2.5C and -3.5T < U- < 3.5(}. The fifty ordinates were listed as constants in the computer program. As previously discussed, the probability distributions for 6BI' CB' %E, and EI are assumed to be identical. Therefore, the probability distribution listed in the computer program for the discrete variable U can be used also as an approximation of the 1 C.arver(6)

-73-. P -5 -4 -3-2-1 0 1 2 3 4 5 U (a) Normal Distribution for EI (b) Normal Distribution for U:eE, Note: This is an illustration only. Actually 50 ordinates were used for the discrete distribution for Pu P U'U -72-7-3/2-~2722 A2 2 /2 (c) Approximate Discrete Distribution for U Figure 5-9 Approximation of the Probability Distribution for Ei

probability distributions for the variables /Mi and W, which correspond to E and 6. Although the distributions are the same, the variables are independent. Equations (5-14) and (5-15) define the variables in and w ~ =sa (5-14) and w = S B3 (5-15) The probability distributions which describe E and 6 ta t3 the components of bearing-reading error due to target motion and age of the bearing measurement, are uniformly distributed between the limits of zero and.DTsin (. - ), as previously discussed. The variables oar and uW are related to 6 and t3 by - s= e65a (5-16) and Ur= Sit3 (5-17) and are uniformly distributed between the limits of zero and SDT sin (a - ) and, S Tsin (4 -,respectively. In order to approximate the continuous uniform distributions by usable discrete distributions, the non-zero limits are approximated by the closest integers, A and B, respectively. The intervals between zero and A and the interval between zero and B are divided into IAI and IBI sub-intervals, respectively. The continuous distributions are then approximated by discrete distributions with ordinates of amplitude I /IAIand /B,1 respectively, centered in the sub-intervals. This approximation is illustrated in Figure 5-10.

-751 PI fir )t~ ta I DTDl(e-) Aaft A DT A a t B = closest integer to sDTn( — ),in which s = C/B Figure 5-10 Approximation of the Probability 3sbtnfr ad B = closest integer to sDT1~ (A- ~in which $ u/a Figure 5-10 Approximation of the Probability Distributions for GC and E

-76/ // If Mr 3, f/zr and IV were considered to be continuous variables with probability density-functions (&r), &tr')jand >/,(/r')respectively, the probability distribution for - = r +Ar can be expressed in terms of the convolution integral 00 CINz (mu IP (Alp) Ar(N r)dNI. (5-18) -00 To describe the method used in the computer program to perform the convolution of the discrete probability distributions of the variables /r and. r, it is convenient to interpret the probability distribution of the variable ir as the sum of the discrete conditional probability distributions of fir for each value of. Let PF ( A r /") be the discrete, conditional probability distribution for the variable Ar for MIr =V l, a particular value of Tr. Then (, lr, "= o" ) -,I (,- =" ). (5.19) The probability distribution for P. is the summation of the products of each of the conditional probabilities and the corresponding probability of the condition, i.e., the probability of the corresponding value of /o,' Therefore, P -2 P P(%") C( Ar I /V -o) -- ZP,,(',r)P,(/,,-l "=) (520o)

-77which is the equivalent of Equation (5-18) for discrete variables. Figure 5-11 illustrates the convolution by summation for the case of A=3. In the program for the computer study, sequential computer storage locations were assigned to sequential values of or over the entire range of values of Ar considered. For each value of A selected, the computer program was designed to have the following operations performed automatically: (1) The probability distribution which was listed in the computer program and used for the computer variables U, /',and rin was multiplied by X/IAI to obtain the set of ordinates for Py (/Nr%") P, (4r- /2") (2) The storage location corresponding to /lsI, the smallest possible value of nr (/5 = -df for A >0 and = - - + /A for A <O), was determined. (3) The set of 50 ordinates for P.(~) Pr (r- <") was added to the contents (original contents were zero) of 50 sequential storage locations starting with the one corresponding to 1Y.. The addition was iterated a total of IAI times with the initial storage location increased by one each time. In this way, the probability distribution for the variable rv was constructed. The probability distribution for the variable wr was constructed in the same way with the simplification, 8 > O, which results from the symmetry described in the following paragraphs. For each value of sDT selected for use in the digital calculation, a set of twelve values of A and B, corresponding to the set of twelve target directions considered, was listed in the computer program~ Values of target direction, ~, over a range of only 7T/A need be considered because of symmetry. To illustrate this symmetry, an expression for the position error in terms of the computer variables is utilized.

-78-3-2 -1 0 2 3 0 1 2 3 I P — ( -) 3 -3-2-1 0 1 2 3 Note: This is an illustration only. Actually 50 ordiV3l,-i) nates were used for the discrete dis tribution -3-2'-1 0 1 2 3 4 5 3 V -3-2 -1 0 I 2 3 4 5 Figure 5-11 Convolution to Obtain the Probability Distribution for the Variable V,

-79Equation (5"3) for the position error can be written in terms of the computer variables as follows: Ha 9Es (5-21) Ec U = u +- - - =er -u/-rur - (uru 4Da in which:,,, u = tr + r j (5-22) Jr = cr + ur (523) and L_ = the variable used in the digital computer program to represent the square of the position error. Ec is a function of the variables U, Ar, Ur, Wr Ur, and also, which determines one of the limits of the range of possible values of ir and or. Therefore, Ec is denoted by EC (, Ur, Ar, w, ). The symmetry of EC with respect to target direction can be examined with the aid of Figure 5-l12 For the particular reference angle arbitrarily selected in Figure 5-12 (a), sin ( I- ) and sin (j - 4), to which A and B are proportional, have been sketched in Figure 5-12 (b). Because (,2> ~ 77)=- i(-e- )(5-24) and A m -in /),a.(4O4)), (525)

STATION 3 y STATION 2 REFERENCE LINE (a) STATION I SINgue -2 e) (b) Figure 5.12 Symmetry with Respect to 4

the following is true. E (u, -r'", w') w:' u; y + 7T) =' EC(uI, ir-)7 - U)'-I UT; -'). (5-26) / / The probability distributions for U, Anr and Wr are even functions of these variables so that EC(U,, N Ir,-<,W,- - E- c C ) - -A,-rV,. (5-27) Equation (5-21) shows that Ec-',-" - -' "~-, ) - EC(u,' C ), (5-28) so that UE.(Uu';w Ur; UT + w) = E(u,,",ww', (5-29) I PII Additional symmetry exists because 2r and wr can be interchanged in Equation (5-21) with no effect on E The fact that the probability distributions for 4r and wr are identical makes this possible. In addition, because.is,., fi- /):= -(s -'7, ~t) (5-3o) and (~-,/,) -.;~,. ( -'- ), (5-31)

-82the following is true: E (u,/ r, Ar, >r-c) = E( U A r N ) Ec(- u,- - -w, -" =t ( Agu) N-r v a, ~r,.' (5-32) Therefore, only the values of ~ in the range 0o < T-/ (5-33) need be considered. The process used in the computer program to calculate Pv and P for each value of 0 considered is summarized in the block diagram for Part One of the computer program which is shown in Figure 5-13. In the digital computer study, target direction is assumed to be uniformly distributed over the interval 0 ~I < < T7. The previous discussion points out that for purposes of calculation, only the range 0 < S < 7T/a need be considered. The uniform distribution over the latter interval was divided into 12 equal intervals, and approximated by 12 ordinates, centered in these intervals as shown in Figure 5-14l A probability distribution for the position error was calculated and printed out for this distribution of target direction. In the process, a separate probability distribution for each value of target position was also calculated and printed out,

Construct P, by adding P,/IAI Start Select values Form to itself IAI times with the of A and Pu/IAI I abscissa augmented by one t each time Construct P, by adding Pu/B To to itself B times'with the Form Part I. abscissa augmented by one P/B Two, I each time (Fig, 5-15) Figure 5-13 Part One of the Computer Program for the Position Error in the Conventional System

Pp la~~~ ~, I I _ tI I' I j I I I I I Figure 5;14 Discrete Distribution for Target Direction For each set of normalized triangulation-system parameters considered, the probability distribution for the position error was constructed by examining every possible combination of values of the variables u., Ar, and Ur. For each combination, the value of E.( wAl) was calculated. Then, the value of PU/T = PU P PI was calculated and added to the contents of one of sixty storage locations determined by the value of Ec(uN) a) r Thus., the probability distribution for position error was obtained as a set of numbers from the set of sixty storage locations. The position of a number in the set corresponded to a particular range of values of E. For the computation, it is convenient to calculate the normalized position error in terms of its square, as given in Equation (5-21). If this form had been used directly to determine the storage. address for Purr, the smaller values of position error would have been compressed into a relatively few storage locations and the larger values of position error would have been spread uneconomically among the majority of the

-85storage locations. Rather than utilize a time-consuming routine to calculate the square root of EC, the transformation K, E= K+ / (5=34) in which K, K,and K3 are constantswas used. K, and Ka were chosen so that o ~ K/(E + K) < 60 (5-35) over the entire range of values of U, A, and U considered, and so that only the value of K,i/(E + K) which corresponds to smallest discrete value of E would be within the interval 59 K, /(Ec K) < 60 (5-36) The constant K was selected to be the computer "Address" of the first of the 60 storage locations used for the accumulation of the probability distribution of position error. Inequality (5-35) can be rearranged to show the range of values of E corresponding to each storage location. For the b-th. storage location, where /I' -'.0 -,I ~< K,/(&C. KD') < A (537) C a ~~~~~(5-37)

O86K, K and Ec are all greater than zero so that Inequality (5-37) can be rearranged into the form K K 1-K <E <K b.2 C' -a (5-38) Using Equation (5-21), Inequality (5-38) can be further modified to the form 9SA L + K < E' 4 - K L/O J 9 s -/O (5-39) in which S = 6t/cB8 Inequality (5-39) expresses the position error corresponding to each storage location in terms of constants and the two triangulation-system parameters, D and 0B. For each value of the other parameters DT and ~, separate probability distributions were obtained. Part Two of the computer program was designed to obtain the probability distribution for position error by examining every possible combination of U, r}, and Wr in a systematic and efficient manner. A block diagram of Part Two of the program is presented in Figure 5-15, in which the subscript, s, is used to denote the smallest possible value of each variable. In the calculations the possible values were considered according to ascending numerical order. Part Two of the computer program consists of three loops corresponding to three variables whose values must be cycled. In Step 1, Nr5 the initial value of ~r is selected. In Steps 2 and 3, all of the

2. 4. i. 3.~~~~~~ 5. Select Calculate 6 From Select P and Select Select Calculate and P~art One -calculate'Us C (U /zr ur) + K WS SS - _ _ (Fig, 5-13) L4 r and P fix, 7. __ Calculate K3 t 8. Store fic_ U using as an address 12. 11] 10. 9 __ Cycle IAI + 49 Cycle B + 49 times Cycle 50 times Calculate times augmenting Itr augmenting wr augmenting U E by 1 each time by 1 each time by 1 each time andd (us~i)!then then_ then 13. 14. Print out Select next To Part One results value of (Fig. 5-13) Figure 5-15 Part Two of the Computer Program for the Position Error in the Conventional System

-88computation that can be performed after the value of nr has been selected is performed and, war, the initial value of ar is selected. In Steps 4, 5, and 6, all of the additional computation that can be performed after the value of ur has been selected is performed. Then, us, the initial value of u is selected, and duS, and Ec(Us.'rs, Us) are calculated. In the inner loop, which consists of Steps 7, 8, 9, and 10, Pu is calculated and properly stored, making use of Ec(us -4rs Vs), Then, Ec(US+ I,A/rw) is calculated in a simple fashion, making use of Ec (us, nu, r) and other quantities calculated in steps outside of the inner loop, as, indicated in the following equations in which the subscript, O, indicates any particular value of the variable. From Equation (5-21), 0O + 0 U0O1 -4ror - rOUO (5-4o) and EC(uo-,, l 1 0): (Uo+-l) go -, ((o+/)/lrO -.or (Ar.(Il) *- (5-41) Equation (5h41) can be rearranged into the form ~C('Ott i 4 QrO = UO + 4r + Wo - UO4)r -Ar(Jo- - Urw ++I-4-r (5_42)

-89or Ec(u/,;o, w;) = Ec(jo Ir, ) Uo+ / I- -%. (5 43) Equation (5-43) was used to calculate EC in the inner loop of the computer program because this method requires a minimum of computation time on MIDAC, the digital computer which was used. In Steps 10, 11, and 12, the number of cycles through each of the loops is controlled. 50 times for the variable u, (Bt+49) times for the variable U and (IAI + -9) times for the variable r. A total of 50(Bt- +9)( A I t 49) combinations of values of U, fr and w were considered to obtain each probability distribution for position error. After cycling through these combinations, the computer printed out the probability distribution. In Step 13, the probability distribution was read out in the form of a cumulative probability function rather than a probability density function in order that the results be useful directly and to provide a check1 on the results. In Step 14, the number of values of target direction which were considered was controlled. The computer program is listed in Appendix A. The changes necessary in the program to change P. and DT are listed in Appendixes D and E, respectively.?Ec= (Z u)( e )( a known value. F C~~~~~~~~c

CHAPTER VI MODIFIED THREE-STATION SYSTEM A modification of the three-station triangulation system was investigated also. In this modification, only two bearing readings are used to calculate the position of the target. When a bearing report from one of the bearing-measurement stations is received at the central station, the more recent of the reports on the same target from the other two bearing-measurement stations is selected from storage, The''new"l reading and the "more recent" reading are then used to calculate the position of the target. The numerical investigation of this modification proceeds in the same general fashion as the investigation of the unmodified case which was described in Chapter V. Details, of course, are different. The result of the digital computer study is a set of separate probability distributions for the magnitude of the position error for the parameters (1) target direction and (2) a combination of target velocity and time delay. Probability distributions are also presented for the case of a uniform distribution for target direction. The results of the investigation of the modified case and the unmodified case can be compared directly. Equation for the Position Error In this modified case, for the purpose of calculation, the new bearing reading is assumed to have been taken at bearing-measurement Station lo Thus, Equation (5-33), which expresses the magnitude of the position error when the error in bearing reading is small, can be -90

-91i expanded into the form / / + d- d c e t E (e in which the subscript "a" assumes the values 2 or 3, depending upon which bearing-measurement station provides the more recent bearing reading. Of course, no weighting factor can be used because only one intersection is computed. As in the unmodified case, the bearing-measurement stations are assumed to be located at the vertexes of an equilateral triangle. It is convenient to normalize all distances, including the position error, with respect to the distance (D) of each bearing-measurement station from the center of the equilateral triangle. The normalized position error, denoted by ED, is obtained by modifying Equation (6-1) to the form d E ( 6a( D Da *a i(te) (6-2) Variation in the Position Error The variation in the normalized position error as the location of the target is varied throughout the triangle formed by the bearingmeasurement stations is investigated by using the simple, discrete, probability distribution for bearing-reading error (Figure 5-1) that

-92was used in the study of the unmodified case. For this investigation, only two bearing stations need be used. Stations 1 and 2 have been selected arbitrarily. Using this simple, discrete distribution, four equally likely arrangements of bearing-reading error are possible: Case I 6 E E= Case II = -I Case III E e = 6 Case IV / = = The expected value of ED, denoted by X, is given by E 1EI E E (6-3) in which the Roman numeral subscripts indicate the particular arrangements of bearing-reading error. From Equation (6-2) (Cases I and IV and Cases II and III yield identical values for E ), E can be expressed as =X _________ B~~ [aal~~ (~~(6-4) aAf.e-E +[4a (d D 6l.

-93Reference to Figure 6-1, which describes the geometry of this situation, shows by the "cosine law" for a triangle that + 3 (6-5) Equation (6-4) can be simplified, using Equation (6-5), to the form: x At_+ D + (6-6) Station 3 1I D Station 1 eg Station 2 Figure 6-1 Normalized Geometry for the Two-Bearing-Reading Case

-94For numerical computation it is convenient to define the angles oa< and as the angles between the bearing lines to the target and the line joining the bearing-measurement stations. The "law of sines" for the triangle is dt/D 4/D /1 ~(6-7) in which o + <, + e - 7T. Therefore, =Da AA(+ ) - t(6-8) Substitution of Equation (6-8) into Equation (6-6) yields E - LL r- + +'I (6-9) Equation (6=9) was used to calculate values of Ex for a variety of target locations as shown in Figure 6-20 For a target located near the center of the equilateral triangle., the value of Ex is 1.59E. As explained in Chapter V, the values of Ex calculated from Equation (6-9) are not realistic because the probability distributions for bearing reading that were used are not realistic. However, the calculations demonstrated that the variation of Ex is small in a region surrounding

-95STATION 3 2.73E /ID VARIATION IN Ex IS NOT GREATER THAN 10 %/ 2.07.07E / 11.73/ 1.73E 173(' STATION I STATION 2 Figure 6-2 Variation of the Expected Value of the Position Error for a Simple, Discrete Distribution for Bearing-Reading Error

-96the center of the equilateral triangle. The region in which EX does not exceed its value at the center of the triangle by more than ten percent is shaded in Figure 6-2. If the variation in the position error with variation in target location is investigated by using a normal distribution for bearing-reading error, the results obtained in Chapter V apply directly. The region in which the area of uncertainty is no larger than twice its value at the center of the triangle is at least 80 percent of the area of the equilateral triangle. In the numerical investigation to determine the probability distribution for the position error using realistic probability distributions for the error in the bearing readings, only a target located at the center of the equilateral triangle has been considered. This provifdes an adequate estimate of the position error in a region surrounding the center. Probabilit Distribution for Bearing-Readin Error In the modification of the three-station triangulation system in which only two bearing readings are used to calculate the position of the target, the probability distribution which describes the component of the error in the bearing reading from each station due to error in the bearing measurement itself E ~B is the same as in the unmodified case, This component, E8, is described by a normal distribution with a mean of zero and a standard deviation O6' The probability distribution which describes the component due to age of bearing measurement and target motion is obtained in the following way. As already stated. the new bearing reading is assumed to have come from Station 1. The other bearing reading used in the calculation

-97T is from either Station 2 or 3, depending upon which of the stored readings from these stations is the more recent. The ages of the stored bearing readings from Stations 2 and 3 are denoted by t and t. As discussed in Chapter IV, both t and t3 are described by a probability distribution which is uniform over the interval from zero to T, i.e., e = orrt for 0 t~ < T7 Kt elsewhere (6-io) and? I | T for 0 < t3 < T L0 elsewhere (6-11) in which P and t are the probability density-functions for the variables ta and t3, respectively. The joint probability densityfunction for t. and t3 is denoted by P and is given by t t3 e ee= [p = + for O < t< T, O< t <T t*~t rt 13 elsewhere (6-12) If a = and a= 3 denote the use of the stored bearing reading from Station 2 and Station 3,respectively, the probability of a= Z is given by P(=a)= P (tP< t3) fe< (1 A, -Jr a / ~(6-13)

-98in which A, is the area within a region O < t < T and O < t. < T such that t < t. As shown in Figure 6-3, this area is TR ~A' A (6-14) and therefore, P(a, ) = )- (6_15) By similar reasoning, or by use of the fact that P(,=Z3) = I - P(a=a), (6-16) the following can be determined: P(, 3) - _ (6-17) If 7r is an arbitrary constant, the probability that the age of the bearing reading which is used., be it from Station 1 or Station 2, exceeds r is given by P(tor) = P(t>rt>r) fC> = r (6-18) in which Aj is the area within the region O<t < T and O < t3.< T such that t > r and t$ > r. As shown in Figure 6-3, this area is A (T-r), (6-19)

-99t3 T A, t0 T t3 ~3 T 0o 6 hAe nA a tr Figure 6-3 The Areas A, and la in the ta, t3 Plane

-100and therefore, p(t0 > ) for 0 < LT. (6-20) The probability density-function for t_ is obtained by differentiation of Equation (6-20) with respect to 7. 9 aPft7<T) = (T-T) o< T f. (6-20a) ta, Ta This probability density-function is illustrated in Figure 6-4. The relationship between the error in the bearing reading and the age of the bearing measurement when the error is small is provided by Equation (4-50) which is repeated here: ~ — = -......... -(6-21) in which V,Al, L and dcQaare constants. The probability distribution for ~O_ is obtained by use of Equation (6-21) and is shown in Figure 6-5. The error in the bearing reading which is old is given by V'I (6-22) in which E is the error in the bearing measurement itself. The probability distribution for the error in the bearing reading which is old is given by the convolution integral'e JE5.C - Ed) (6-23) c 00

-101Pta T T Figure 6-4 Probability Density-Function for t, the Age of the Bearing Reading which is Used P a~~~da,~, V Ty-Function for to

-102The solution of this convolution integral is the first part of a digital computer program for determining the probability distribution for the position error. Probability; Distribution for Position Error The probability distribution for the position error for a target located at the center of the equilateral triangle formed by the three bearing-measurement stations was obtained by a convolution of the probability distributions for bearing-reading error at each station. This convolution is the second and third parts of a digital computer program for numerically determining the probability distribution for the position error. For a target located at the center of the equilateral triangle (as shown in Figure 5-12), D D I, )(6-24) D D A~e~-e) e (6-25) C-OV- (,-) (= (6-26) and, therefore, Equation (6-2) becomes ED =3 + i j (6-27) in which C depends upon 6B and 6-t according to Equation (6-22).

-103The probability distribution for Et (shown in Figure 6-5) can be described in terms of the quantity, which is defined as =VT D Ai D ( a~4q) = DT (% ~ ~') (6-28) For a particular value of the constant, a, the conditional probability distribution for ED, denoted by (ED I ct), is a function of The probability distribution for ED may then be expressed in terms of the conditional distributions as e = P(~a~)~(~a()E ( +P(=3)e( EIa 3) (6-29) which, by Equations (6-15) and (6-17), can be written as 3 E X (cE ~D} (6-30) D a- b Equation (6-30) suggests that for numerical computation it may be convenient to obtain P for each value of target direction, ~, from ED a pre-calculated set of conditional probability distributions corresponding to a set of values of t Because 1 is an even function of 68, Equation (6-27) shows that only the absolute value of ~ need be considered. Therefore, the second part of the digital computer study of the modified three-station arrangement consisted of tabulating a set of conditional probability distributions for a set of values of | Ja(.

-1o04 In the third part of the digital computer study, selected groups of conditional probability distributions were combined to obtain probability distribution for position error assuming a uniform distribution for target direction, c. This latter distribution, denoted by re, is given by D ED aTr; ED 4) (6-31) o which can be expanded, by using Equation (6-30), into the form UTr ED - =;) I ((I E =3)]d (6-32) o The terms of the integrand depend on 2, as defined in Equation (6-28). Because the integration is performed over a complete cycle of sin (,- ) Equation (6 —32) can be written as E a77IjP(t~la)~, ~ (6-33) Because only the absolute value of {a need be considered and because of the symmetry of the integrand, Equation (6-33) may be rewritten in the form 7r/ 0l:lr

-105In the numerical computation of %, the interval O0' -, was divided into a set of sub-intervals and the integral was evaluated by approximating the value of 0 - within each sub-interval by its value at the center of the sub-interval, Description of the Computer Program The digital computer program which was used to determine the probability distribution for the position error is divided into three parts. In the first part, the probability distributions for bearingreading error are constructed. In the second part, a set of conditional probability distributions for the position error are constructed and stored in the low-speed computer storage. These conditional probability distributions are applicable to both cases, ao =. and a = 3 In the third part of the computer program, groups of conditional probability distributions are combined to obtain a probability distribution for position error which assumes a uniform distribution for the target direction angle. Except for the third part, the computer program differs only slightly in detail from that used in the analysis on the unmodified three-station case. Because only two stations are involved, the required computation time is comparatively small. Figure 6-6 summarizes the computer program in block diagram form. For convenience in the digital computation, the set of possible values of E/ and 6E was represented by integers and integers plus one half. The variables U and 1r are used to denote this representation. Thus, in Figure 6-6, P and P are the probability distributions for the variables u and 4F, respectively The joint probability of a particular combination of

Form the Compute.I Compute Start probability distributions UO VO f or Pu nd P,, Cycle through all Use the value of combinations of the E u,,, Iv) as an values of the adddress for acciuu Store the variables u and ir lating the values Cycle through conditional O Ilf Pu all values of dirdstribution e thenn c ~~~P,(E, ICC itheni Combine the stored con- Repeat for each set ditional distributions N of normalized system to obtain P parameters selected EC end then Figure 6-6 Simplified Block Diagram of the Computer Program for the Position Error in the Modified System

-107values of U and ar is given by sP = P( P (6-35) The subscript, o, indicates particular values of the variables. E is used to represent the square of the position error and. is used to represent K Details of the Digital Computer Program As in the study of the unmodified three-station triangulation system, each of the probability distributions which are used to describe the components of bearing-reading error for the modified system was approximated by a discrete distribution of many ordinates. For convenience, the set of possible discrete values of these variables is repre/!/ sented by integers plus one half. The variables u, IT and /rF are used in the computer program to denote these discrete values and are defined by Equations (5-13), (5-14), and (5-16), in which the scale factor S is defined by Equation (5-12). The same discrete probability distribution for the variables u and 4r that was used in the unmodified case was used in this case also. This distribution is illustrated in Figure 5-90 The discrete probability distribution which describes the variable /r was obtained by approximation of the probability distribution for E, shown in Figure 6-5. The approximation is illustrated in Figure 6-7 in which t is used to describe the distribution for t. If Ur is considered to be a continuous distribution over the interval 0 < /~ <- C in which C is a positive integer, the continuous

-1o8ae~~ ~ta I i a/IT a ~ ~ I 4r"= s~ in which = Figure 6-7 Approximation of the Probability Distribution for Et distribution is divided into C equal intervals, which are replaced by C ordinates, each with a value equal to the area under the continuous distribution in the corresponding interval. The discrete distribution can be expressed in equation form as P,= d (6-36) for values of irw which are integers plus one-half in the interval < < <C The discrete probability distribution for the variable r = Ar/ t vl was obtained by the same method used in the unmodified case described in Chapter Vo For each value of C selected, the computer program was designed to have the following operations performled automatically:

-109(1) The probability distribution which was listed in the computer program and used for the computer variables U and fVr was multiplied by the factor P= (Ar = aI to obtain the set of fifty ordinates for (2) The fifty ordinates were added to the contents (original contents were zero) of fifty sequential storage locations starting with the one corresponding to a, the smallest possible value of Ar, which is = - 2a. (3) Operations (1) and (2) were performed a total of C times, with the value of,r and the address of the first storage location augmented by one each time. In this way, the probability distribution for the variable or was constructed for each value of C selected. This process is summarized in the block diagram shown in Figure 6=8. The probability distributions for the variables U and lr were combined by the process described in Chapter V, with, of course, one less variable described by a probability distribution. The variable, EC, used to represent the square of the position error in this case is defined as EC ='' ED = u -'- U + ~2 UA. (6-37) The probability distribution for position error was obtained as a set of numbers from a set of sixty computer storage locations. The position of a number in the set corresponded to the same particular range of

Form KPu, in which Accumulate KPU Start __(ac___)/_ _ at the proper at the start computer storage addres se s Cycle C times, augmenting the storage addresses by I, and subtracting a/C2 from K each time To Part Two then] (Figs. 6-9 or 6-10) Figure 6-8 Part One of the Computer Program for the Position Error in the Modified System

-111values of EC as described in Equation (5-38). Thus, the corresponding range of values of E is given by + _KP I _D K K-A < E < Ka - ~3s"~ ~ sA L~ $ a (6-38) which is analogous to Equation (5-39) for the convention system. Separate probability distributions were obtained and printed out for each value of C from one through fifty by use of Program I which is listed in Appendix Bo The second part of this program is summarized in the block diagram shown in Figure 6-9.1 The change necessary in this program to change P is listed in Appendix D. In order to obtain a probability distribution for position error which assumes a uniform distribution for the target direction angle, Program II, which is listed in Appendix C, was used to change Program I. With this change, the separate distributions for each value of C were stored in the low-speed computer storage. For each set of 12 values of C corresponding to a particular value of D) and twelve values of target direction angle, 0, the corresponding set of twelve probability distributions was selected from the computer storage and combined. The second part of the program, when changed by Program II, is summarized in the block diagram shown in Figure 6-10. The sets of values of C which were used are listed in Appendix F. The first part of this program is summarized in the block diagram shown in Figure 6-8.

From Part One Select Select Sele ct Calculate Calculate (Fig. 6-8) P dU t I Art~u+r~ nd~ ~~ n (~~~ig~~~6-8) ~~~US. and. P. and. ScCU~"-'s+ K K + K,/(Ec* Ka) Store P using Us'vj K3' ~ Ks') as an address Cycle C + Pr9 times Cycle 50 times Calculate augmenting Por augmentingsiti(usrorith MoifiISy and by 1each time t2 U + t Ar E(U+1,4t K by 2 each tilme theniI thenj Print out Augment C To Part One results 6 b Fg -8) Figure 6-9 Part Two of the Computer Program f or the Position Error in the Modif ied System

From Select Select Select Calculate Calculate Part One > P2 a Ul I nd P and (Fig. 6-8) an. U S an Store P. using K, K/(Ec tK,) as an address Cycle C+49 times Cycle 50 times Calculate augmenting Ir augmenting (U *I + IiJ and by 1 each time'a +IU.l 2 I +' rs and by 2 each time (_ then then Store results on the. Cycle 50 times Form I magnetic drum and augmenting C P / | 71check that storage by 1 each time ERI is correct Read in 12 then[ code words Print out I Combine the corresponding results ~ 112 distributions Figure 6-10 Modified Part Two of the Computer Program for the Position Error in the Modified System

-114Closed Form Solution For the special case of 0eB very small in comparison with, an expression for the position error can be obtained in closed form, even for the case of uniformly distributed target direction. For a target located at the center of an equilateral triangle formed by three bearing-measurement stations, Equation (6-27) for the modified system is gK -tE * (6-39) D 3 39) When EB (6-40) Equation (6-39) can be approximated by E' (6-41) 3 and., therefore, E = F t for (6-42) The probability distribution for ED is obtained from the probability distribution for t and Equation (6-42). As explained previously, [Equations (6-32) and (6-33)] only the case a-=d need be considered. Therefore, the probability distribution for ED as obtained from Figure 6-7 and Equation (6-42) is as shown in Figure 6-11. The quantity, is defined in Equation (6~28) as 4=D T (<g ) * (6_43)

115E F3J Figure 6-11 Probability Density-Function for ED When "' << D6t. As also explained previously, only values of - - - within the interval 0 < 0 -'Jr- need be considered. For uniformly distributed target direction, the joint probability distribution for ED and 4.- is as shown in Figure 6-12. This joint distribution has a triangular crosssection normal to the 4- axis and, of course, it satisfies the condition that Tr oDT (Ain,) eE,#- gEZ d (t- ) = I. (6-44) o o0

-ul6ED~ =, TSIN(%2-0) 40 ED =XDTSIN# Figure 6-12 Joint Probability Density-Function for ED and 2- when Cj~<< CE /~~~~ E, c(

-117Within the region in which it is non-zero, E can be expressed as T _ IA|,JT)' 3 ED,D a La- T2 D ) Db ( a IaJ (6-45) The probability that ED exceeds some arbitrary value, X, is P(E, 2) ___ __ _t - D.t4( _ )] d (t-4) i7T -., /,_A\ 7T( (6-47)

-118Performing this integral yields: ~P(ED A) = I - n (d -D ) + sA 3XZ. r X I + I-DTa(6 (6-48) Values of P(ED > ) were calculated and are plotted in Figure 7-5 using notation developed in Chapter VII. This graph is a limiting case of a set of graphs obtained by using the digital computer.

CHAPTER VII NUMERICAL RESULTS The position error in a triangulation system consisting of three bearing-measurement stations located at the vertexes of an equilateral triangle was calculated and is presented in the form of cumnulative probability distributions. For the presentation of the results, normalized system parameters are used which are independent of the choice of T in the computer program. These parameters are: EN = (7-1) rB and ~ VT T _ N D r (7-2) in which EN is the normalized magnitude of the position error, E is the magnitude of the position error, D is the distance from the center of the equilateral triangle to each of the bearing-measurement stations, 97 is the standard deviation of the error in the bearing measurements at each station, DN is the normalized distance that the target moves in the time between consecutive bearing measurements of a target at each bearing-measurement station, V is the speed of a target traveling a straight-line path, -~ll9g

-120T is the time between consecutive bearing measurements of a target at each bearing-measurement station, and DT is the distance the target moves in the time T. Figure 7-1 shows the cumulative probability distribution for the normalized position error for the three-station triangulation system in which all three bearing measurements are used to calculate the position of the target. Target direction is uniformly distributed. Figure 7-2 is an expanded view of the same graph for the higher values of position error. In both figures, the ordinate is P ( EN> Eo), the probability that E, will exceed the corresponding value of the abscissa. Four graphs are presented for four values of D. The value D = 0 corresponds to the use of simultaneous measurements. Figures 7-1 and 7-2 describe the magnitude of the position error if the system parameters are knowno These figures clearly show the additional position error which results when non-simultaneous measurements are used. These figures also provide a way of specifying system parameters to meet particular restrictions on the position error. Examples of these uses are presented in Chapter VIII. Figure 7-3 shows the cumulative probability distribution for the normalized position error for a modified three-station triangulation system in which only two bearing readings, the new reading and the more recent of the other two, are used to calculate the position of the target. Figure 7-4 is an expanded view of the same graphs for the higher values of position error. Eight graphs are presented for eight values of DN These graphs can be used in the same way as the graphs of Figures 7-1 and 7T2.

10 0.8 UNIFORMLY DISTRIBUTED TARGET DIRECTION 0.6 DN 3.5 bJ A 0.4 oN 3 LL2 DN = 2.5 N= 0.2 0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Eo Figure 7-1 Norm-alized Position Error for a Three-Station Triangulation System

0.4 UNIFORMLY DISTRIBUTED 0.3 L \ \ \ \ TARGET DIRECTION DN =3.5 DN = 3.0 0.2 DN= 2.5 rI 0.1 1.0 2.0 3.0 4D 5.0 6D Eo Fi-gure 7-2 N8ornalJzed Position Error for a 7hree-Station Triang-lation System

1.0 0.8 UNIFORMLY DISTRIBUTED TARGET DIRECTION DN =7.0 N as C.6Y ~ DN = 6.0 DN:5.0 N A D 0~~~~~~~ DN=4.0 0-' DN:3.0 4 0.4 _ ~~ ~ DN.: A z I \\` C\K(\/- ~~~~D#:0 z ~LI. 0.2 0~~~~~~~~~~~~ 0 1.0 2.0 3D 4.0 5.0 6.0 7.0 Eo ]~igu 7-3 ~raie oiinko o ~dfe he-tto Pinuain Sse

o.4 UNIFORMLY DISTRIBUTED TARGE'T DIRECTION DN =6.0 DN z.0 DN =4.0 0.2 DN=1. D =O. JJ aLJ 0. 1 4.0~~~~~~~~1 0 3.0 LO2.0 Eo,Moafia TIhreeStationTi positioon Error foraMofe Figure 7-4 N~ormalz

-125In Figure 7-5, the abscissas of the curves presented in Figures 7-3 and 7-4 have been divided by DN. The probability distributions which describe the position in the modified three-station system are given in terms of P([En/D~> [EO/Dl)~j or P( E/VT] > [EO/DNJ); which are identical according to Equations (7-1) and (7-2). By presenting the distributions in this form, they can be compared with a graph of Equation (6-48), which is a solution in closed form for the probability distribution when the value of DN approaches infinity.l The graph for this limiting case has been added to Figure 7-5. When compared with the set of calculated probability distributions, this graph of the limiting distribution demonstrates that little would be gained in using a digital computer to calculate many distributions for values of DN greater than seven. A comparison of the graphs for DN = 0 (e.g., no target motion) shows, as would be expected, that for any selected value of Eo P(EN > EO) is greater for the modified three-station system than for the conventional system. As the value of DN is increased in both cases, P(EN >EO) increases less rapidly for the modified system because the information which is most likely to be in large error is omitted in the calculation of target position in the modified system. In the range of values of DN from 3.5 to 4.o0, P(EN > E) is approximately the same for both systems. For values of D1 greater than 4.0, P(EN > Eo) is less for the modified system than for the conventional system. Thus, The approach of DN to infinity corresponds to the case of position error due to error in bearing measurement being negligible in comparison with the error due to time delay and target motion.

1.0 0.8 0~6 w~ 0 0-6 0.4 ~ 0 Iz~~ I A 0.2 UNIFORMLY DISTRIBUTED TARGET DIRECTION 0 0 0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 Eo/DN Figure 7-5 Normalized Position Error for a Modified Three-Station Triangulation System

-127from the standpoint of minimizing the position error, these graphs indicate that the modified type of system should be employed if a threestation triangulation system is to operate in an environment in which DN is greater than 4.0o The comparison of the conventional and modified systems is illustrated further in Figures 7-6 and 7-7. The same data used in the previous figures have been used to construct a family of curves for both systems for several values of constant P(EN > Eo). As DN increases for both systems, a greater range of values of position error is included at each probability level. For a selected probability, the value of DN at which the range of values of position error is the same for both systems is quite evident. Minor extrapolation is required because the range of values of DN investigated for the conventional system was somewhat restricted. For this reason, the calculated points on these graphs have been indicated, The data previously presented are for the case of target direction which is uniformly distributed. Of interest also, are the probability distributions which describe the position error for particular target directions. For the conventional three-station system, a set of such distributions has been plotted in Figure 7-8 for the case of )N = 3 and for several values of | - I-, the absolute value of the angle between the target path and the line joining the center of the equilateral triangle with the station whose bearing reading is new. Table 7-1 lists the values of e - - | for each of the cases listed in Figure 7-8. The curves in Figure 7-8 indicate the variation in position error that can be expected in a sequence of measurements as each of the stations becomes

-128Q28 P(EN> E ) =0.2 - 0.3 Q24 0.20 0.16 0.12 0'I 0.08 LEGEND: MODIFIED SYSTEM 0~04 F~ —O- CONVENTIONAL SYSTEM 0 2.0 4.0 6.0 80 DN Figure 7-6 Loci of Constant P(EN> E0) for Uniformly Distributed Target Direction

-1296.0 LEGEND: —- MODIFIED SYSTEM - 0 CONVENTIONAL SYSTEM 5.0 4.0 P(EN > EO) =0.05 N /.0 0 6.0 Figure 7-7 Loci o ConatantP(E N> Eo) 0.20 or 1.0 0 200. 04.0 60 80 Uniformly Distributed Target Direction

1.0 0.8 CASEC 7:8 CASE: 6TABLE 7-I 0.4 4,l 020 0.5 1.0 1.5 2.0 2.5 3.0 35 Eo Figure 7-8 Effect of Target Direction on the Normlized Position Error for a Three-Station friangulation System

-131TABLE 7-1 GEOMETRY REPRESENTED BY THE TWELVE CASES IN FIGURE 7-8 3 71.250 108.750 251.250 288.75 4 6375~ 116.,250 243,750 296.250 5 56.256 123.75 236.250 303 75~ 6 48.750 131.250 228.750 311.25 7 41.25" 138.750 221 250 3188750 8 33.750 146.25" 213.750 326.25~ 9 26.25 ~ 153 75 ~ 206.25 ~ 333 75 ~ 10 18,75" 161.250 198.750 341.25" 11 11.25 ~ 168.75 ~ 191.25 348.75" 12 3.750 176.250 183.750 356.25 When Station 1 provides the more recent bearing reading

-132the one to provide the new reading. The observed, extreme variation with target direction corresponds in magnitude to the variation observed in the probability distributions for uniformly distributed target direction when DN is varied between zero and 3.5. However, if target direction with respect to the system rather than with respect to the station providing the new bearing readings is considered, each station is equally likely to be the one providing the new reading. In this case, the position error is described by a combination of the three conditional probability distributions for the absolute values of the angles between the target path and the lines joining the center of the eqcuilateral triangle with each of the three bearing-measurement stations. The variation with target direction among the resultant distributions is negligible. The limiting curves of the set of probability distributions which describe the position error differ by at most 0.02 in the P(EN> Eo) coordinate and they can be approximated by the corresponding curve for uniformly distributed target direction. For the modified three-station triangulation system, a set of probability distributions which describes the variation in position error with target direction has been plotted in Figure 7-9 for the case of DN = 3. For the modified system, target direction affects the position error by way of Io- j, the absolute value of the angle formed by the intersection of the target path and the line joining the center of the equilateral triangle with the station which provides the more recent of the two bearing readings which are old. The entire range of variation of the probability distributions is covered by the four curves plotted for four values of this angle. Comparison of Figures 7-8 and 7-9

DN=3 Q8 0.6 0.4.,.0 Z 02 0 a5 1.0 1.5 2D 2.5 3.0 3.5 Eo Figure 7-9 Effect of Target Direction on the Normalized Position Error for a Modified Three-Station Triangulation System

.134shows that when DiN = 3 the variation due to target direction is approximately twice as great for the conventional system. Thus, the modified system has some advantage if the probability of large error is to be minimized, even at values of DN as low as 3. If target direction with respect to the system rather than with respect to the station providing the old bearing reading is investigated, considering that each station is equally likely to provide the old reading which is used, the variation in the probability distributions which describe the position error is negligible, as in the case of the unmodified system, The limiting curves of the set of probability distributions which describe the position error in this case differ by at most 0.04 in the P(EN> E) coordinate, and they can be approximated by the corresponding curve for uniformly distributed target direction. The results of this study are based on a normal distribution for the error in bearing measurementso The numerical results of this study are in error because the normal dristribution was truncated and then approximated by a discrete probability distributional In order to estimate this error, the probability distribution for the position error for DN = 2~5 in the conventional three-station system was calculated using two different discrete probability distributions for the error in bearing measurements. The distributions differ in the points of truncationo One of the distributions results from truncation at U = 5 35.U the other at U ~= ~.5 2 Both of the discrete distributions consist 1 The points of truncation determine the range of values of bearingmeasurement error considered. 2The variable u is used in the digital computer program to represent the error in the bearing measurements. The quantity %i, is the standard deviation of the continuous distribution from which the approximate discrete distribution for the variable u is obtained.

-135of a set of probability values for the same fifty values of U. Because of the difference in the points of truncation, the sets of probability values and the relationships of the variable u to the error in bearing measurement are different for the two distributions. The discrete distribution which results from truncation at u = 3.rSs was used to obtain the results presented in the previous figures. The particular distribution for position error for DlN = o. for the conventional three-station system (Figure 7-2) is repeated in Figure 7-10, using an expanded abscissa. For the same case, the probability distribution obtained by use of the discrete distribution which results from truncation at.J = z- c.6, is presented for comparison in the same figure. The difference in the two probability distributions for position error is small. As expected, the discrete distribution which was truncated at U = + -.5 d; produces a higher probability that large position error will occur. It is believed that most of the difference between the two probability distributions for position error is due to the difference in the points of truncation rather than to the use of discrete distributions. The percent difference in the values of Fo for the cumulative distributions shown in Figure 7-11 was calculated and plotted as a function of the probability level. In the range of probability values in which these calculations can be performed accurately (0.05 to 0.9), the difference does not exceed three percent.

0.4 DN =2.5 Q3 0.2 0: 50/7 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 l~LJ A Z ~~~~~~cr:10 LIi U 0.1 0 10 1.5 2.0 2.5 3D0 35 E0 Figure 7-10 Comparison of Results Obtained.Using Different Probability Distributions

3.0 u 2.0 z r LIJ Li IL 1.0 Ld o I I I 0 0.2 0.4 0.6 0.8 ID P (EN> Eo) Figure 7-11 Percent Difference Between the Values of Eo Obtained by Using Two Different Discrete Distributions for Bearing-Measurement Error

CHAPTER VIII APPLICATIONS The numerical results presented herein can be used (1) to estimate the position error in the evaluation of existing or proposed threestation triangulation systems, (2) to study the effect on position error of proposed modifications to the system, and (3) to specify system design parameters when requirements on position error have been established. The following fictitious example illustrates these uses. A triangulation system consists of three bearing-measurement stations located at the vertexes of an equilateral triangle. The distance, D, from the center of the triangle to each of the bearing-measurement stations is fifty mileso The standard deviation, HB, of the error in the bearing measurement at each station is 0.057 radians. The time required at each station to perform one measurement is 5.0 seconds. Targets with speeds of 0.2 miles per second are expected. The characteristics of this fictitious triangulation system are summarized in Table 8-1. It is assumed that this system may operate in either the conventional or the modified mode. For a target located near the center of the triangle, it is desired to determine the magnitude of the position error that will be exceeded only ten percent of the time when ten targets are under surveillance. From Equation (4-29), the time between consecutive bearing measurements of a target at each bearing-measurement station is T - MY= 50 seconds, From Equation (7-2), the normalized distance that the target moves in the time T is VT/D B = 3.5 From the graph for P(EN > Eo) 0 ol in Figure 7-7, the vealue f Eo which corresponds to A = 3.5 is 2086. -1380~~~

-139TABLE 8-1 CHARACTERISTICS OF A FICTITIOUS THREE-STATION TRIANGULATION SYSTEM D = 50 miles B = 0.057 radians r = 5.0 seconds M = 10 T = 50 seconds V = 0.2 miles per second = VT/.D B 3.5 E = 2.86 (from Figure 448) E = ETD 0 =8.16 miles From Equation (7-1), the magnitude of the position for this value of E, is given by E = EoDcB = 8.16 miles. Thus, the magnitude of the position error will exceed 8.16 miles with a probability of 0.1. The results of this analysis have been included in Table 8-1. Study of the effect of minor modifications to the system is most easily accomplished by use of an equation for the position error and perhaps a Taylor series expansion of the equation about the values of the parameters which represent the present system. Again, for the purpose of illustration, it is assumed that the position error which will be exceeded only ten percent of the time is of interest. Therefore, the graphs in Figure 7-7 of the locus of P(EN E) = 0.1 for both the conventional and modified three-station system are used. The locus for

-140othe modified three-station system is observed to have zero slope at DN = 0 and has, for large values of DN, an asymptote which is a straight line with a slope of 0.58. This latter observation is made from the graph of P([EN/D~] > [Eo/D~J) for.DN = ~ in Figure 7-5o This asymptote, expressed in equation form is Eo 0 C + C DN (8-1) in which Ca equals 0.58 and C, estimated from Figure 7-7, is 1.0. In spite of some similarity, the locus is not a hyperbola. The entire locus could be approximated by a quotient of higher degree polynomials; a quotient of low degree polynomials is not a satisfactory approximation. However, for the purpose to be served herein, a simple second degree polynomial is sufficient. For both the conventional and modified three-station systems, that part of each locus represented by the graphs plotted in Figure 7-7 can be approximated reasonably well1 by the parabola E C3 + CD + CSDN (8-2) in which all of the constants are positive. C = 1.75, C = 0.0965, and C = 0.0624 for the conventional three-station system, and C3 = 2.52, C = 0.0400, and C = 0.0376 for the modified three-station system. From Equations (7-1) and (7-2), the position error which is exceeded 1 The deviation of the approximate equation from the points plotted in Figure 7-7 does not exceed 0.03 (1.2 percent) in the E coordinate.

-141only ten percent of the time can be expressed in the form: C VlaT; EO C D + C VT D (8-3) 10.3 ES 4 D-(8-3) The partial derivatives of EIo with respect to D, O. B, and T, as obtained from the approximate Equation (8-3), assuming these variables to be independent, are: aE E,, C VT~ LN 8 = I'C CD D:o (8-4) aD a D 07 C3Z)3 5 B' = Elo C V 1E,0 _ _ C_ and ds C3D CDN, and (8-5) E c2,CSVaT V- T [C4-+CsDN]V. aT s C4V +; = C4 + CSdw V (8-6) i/a For values of DN less than (C3/CS), each of the partial derivatives is greater than zero, which indicates that a small increase inD, 6D or T causes an increase in E For the parameters D and B, the /o Ea effect on E is a maximum at D = 0 and is smaller for larger values of DNO For the parameter 7, the effect on E/o is a minimum atD = 0 and is larger at larger values of Due Equations (8-4) and (8-5) indicate that the partial derivatives of Eo with respect to D and TB are zero at DN (C3/C C /o Er a /V ES

-142and negative for DN in excess of this value. Obviously the approximation is not valid in this region; DN = (C,/63) // corresponds to DN = 5D4 for the conventional, and DN = 8.2 for the modified threestation system, values which are beyond the range of points plotted in Figure 7-7 from which the approximation was made. For the modified three-station system, Equation (8-1) can be used to examine the partial derivatives of Elo at large values of DN Assuming the variables D, B' and T to be independent, these partial derivatives are: a E1 = -C cE (8-7) E/B C Ke oE [0 j J.o 2D)i(8-9) all of which are positive. If 4D/D, aTB/5B Z and aT/T denote small fractional changes in the parameters D a';B and Ttrespectively, the corresponding small change in Eo is approximately E/0 [3 C. o 3 [DN L L[+ J eB T ~(8-l)

-143for values of oi. which are small, and 0 ClmBID'E + (8-11) for the modified system when the value of iD is large. For the modified system, the coefficients C and C-DL in Equation (8-11) are the limiting values for large values of D. of the coefficients C- CD and (Ci tCD )c N, respectively, in Equation (8-10). The relative effect on Eg of the same fractional change in I, %,and T can be compared by comparing the coefficients in Equation (8-10). The relative effects are the same when C3 - xC r e (8-12) or when %D 3 C + - + (8-13) N I / For the conventional three-station system, Equation (8-13) yields lN " 4.2, which indicates that throughout the range of values of DN in which the conventional system should be used in preference to the modified system, greater reduction in position error will be accomplished by a reduction in D or XT than by the same fractional reduction in T.1 1 When X, OB, and T are independent variables, the equality of the relative effects of D and B on Eo is a consequence of the definition of E and Dv

For the modified three-station system, Equation (8-13) yields DN ~ 7.2, which indicates that in the range O < DN < 7, E/O is more sensitive to a fractional change in D or 7B than to the same fractional change in T. Depending on the nature of the triangulation system, the parameters D, and T may or may not be independent. Variation in the parameter D corresponds to variation in the area of surveillance. If T is directly proportional to the number of targets under surveillance, and if the expected number of targets under surveillance is directly proportional to the area of surveillance, then T =C6D, (8-14) in which Cs is a proportionality constant. T must be considered dependent upon any variation in D but must also be considered to be independent when D is constant because T may be changed in other wayso For this case, the partial derivative of E. with respect to D, as ob/0 tained from the approximate Equation (8-3), is: -"D C83 + 6D 6 EB 3 Al D)8(8-15)

-145= For this case, the partial derivative of o with respect to jD as obtained from Equation (8-1), which is the limit for large values of DN, is aEa D d/o- + C tV J D ( C dC DN)B. (8-16) The partial derivatives of Elo with respect to 0B and T are the same as those obtained for the case of all independent parameters. In terms of small fractional changes in the parameters, the change in EUo is "EEo ~ (q'+ c.D~ + 3C5DN ) ~B D D + (c3, CDA - (C3 DN) G;B D for values of D which are small, and E (C} + 4DN D D + C, D + CO DN% D AT (8-18) e~D~`D T

for only the modified system when the values of DN are large. The coefficients of Equation (8-17) and (8-18) demonstrate that AEo is most sensitive to fractional changes in D at all values of DN e Dependence among parameters may exist in other ways, depending upon the exact nature of the triangulation system. For example, the standard deviation of the bearing- measurement error may be related to the time T, the distance D, or both, If functional relationships which describe the dependence are obtained, the effect of small changes in the parameters can be examined in'the same way as those considered herein. Study of the effect of modifications to the system in which large changes in parameter values are involved requires direct use of the graphs presented in Chapter VII. To illustrate the effect of large changes, the following modifications to the system described in Table 8-1 have been considered~ Modification A D = 25 miles, Modification BB P = 0.285 radians, and Modification C T = 25 seconds. In each case, the parameter value has been decreased by half. Each modification and combination of the parameters was analysed in the same way as the original system was analysed; the results are tabulated in Table 8-2. Dependence of T upon D, as described in Equation (8-14), has been considered also in a modification designated by the symbol A'. In all of the cases, it has been assumed that the system operates as a conventional three-station system for DN < 3.5 and as a modified three-station system for DN > 3.5.

-147TABLE 8-2 EFFECT OF LARGE CHANGES IN PARAMETER VALUES ON THE MAGNITUDE OF THE POSITION ERROR (WHICH IS EXCEEDED WITH A PROBABILITY OF 01) -D, N E (miles) Unmodified System 3.5 2.86 8.16 A 7.0 4.64 6,61 7.0 4.64 6.61 c j 1.75 2~0 5.71 AB 14.0 8.1* 5 76* System with the listed AC 35 2 modifications BC 3.5 2.86 4.0o8 and combinations of ABC 7.0 4o64 3 30 modifications: A' 1.75 2.0 2.86 A'B 35 2.86 2.04 A'C i 00875 1.86 2~65 A'BC 1.75 2.0 143 * estimated

l148Knowledge of the position error as a function of the system parameters and knowledge of the importance of changes in each parameter provide a foundation upon -which values of these parameters can be specified in design when requirements on position error have been establishedo The state of the art determines the minimum value of TB that can be used and partially limits the minimum value of T for a fixed number of targets under surveillance. A value of D can then be selected to satisfy the position error requirements~ If the value of D so selected is impractical, ife., does not provide a sufficient area of surveillance, then advances in the state of the art are required. The relative importance of the parameters suggests which of the parameters should receive the most attention in advancing the state of the art in order to satisfy the system requirements.

CHAPTER IX CONCLUSIONS AND POSSIBLE EXTENSIONS This study provides a measure of the additional error in locating a target by triangulation when non-simultaneous, rather than simultaneous, bearing measurements are used. It extends the previous work on the analysis of the error in triangulation systems and investigates the effect of a geometric approximation used in previous studies. The results suggest areas in which future study would be profitable. This study extends the work of the authorl, who first considered the error in triangulation systems which use non-simultaneous bearing measurements by loosely approximating the effect of the nonsimultaneous measurements. The approximation consisted of increasing the variance of a normal distribution used to describe the error in the bearing measurements. This present study uses a separate, realistic probability distribution for the components of bearing-reading error due to non-simultaneous measurements. In previous studies(8,10,19), the geometry in the vicinity of the target has been approximated in order to simplify the mathematical expressions used in the studies. This study determines the error in this approximation by deriving the equations for position error without approximating the geometry. This study demonstrates the advisability of using a weighted centroid of the intersections of all possible pairs of bearing lines, whereas previous studies(l9) have considered a weighting procedure which is more complicated.. 1 Frese(8). -149

-150For a triangulation system consisting of three bearingmeasurement stations, this study determines the conditions under which greater accuracy is obtained when only two, rather than three, bearing readings are used to calculate the location of the target. These conditions had not been determined previously. The methods used in this study are applicable to future studies of the error in triangulation systemso These future studies include numerically investigating the position error when more than three stations are used and extending the range and scope of numerical investigations of the three-station system. For triangulation systems which employ four or more bearingmeasurement stations, the general expression for the position error which was developed in Chapter IV, and the numerical procedures presented herein can be used to calculate probability distributions for the position error. Each additional station considered adds an additional dimension to the probability space. In such a study, if the number of intervals used in the approximations of the probability distributions for the components of bearing-reading error is the same as the number of intervals used in the study described herein, the computing time required would be increased by a factor of approximately sixty for each additional station consideredol Reduction of the number of intervals used will decrease the accuracy of the calculations. However, the 1 The computer (MIDAC),which was used to obtain the numerical results presented herein, performs multiplication in approximately three milliseconds and, when most efficiently programmed, performs addition and logical operations in 0~43 milliseconds. The computing time required to calculate each of the probability distributions presented in Figure 7-1 (with the exception of DA = 0) was from 8-1/2 to 9 hours.

-151results of this study as presented in Figures 7-10 and 7-11 indicate that the number of intervals may be reduced a small amount with little effect on the overall accuracy because the error due to truncation appears to predominate For the conventional three-station system, probability distributions for the magnitude of the position error can be obtained for values of DN greater than those considered herein in order to obtain a quantitative measure of how much reduction in position error is produced by the use of the modified three-station system. The calculations of each such probability distribution would require computing time only slightly in excess of the time required to calculate the distributions presented herein. However, the calculation of such distributions does require the use of a digital computer with a high-speed storage capacity of greater than 512 words in order to accommodate component distributions over a larger range of arguments if the size of the approximation intervals is not to be increased. For any number of bearing-measurement stations, the equations and numerical methods presented herein can be used to describe the magnitude of the position error in any combination of the following situations: 1o a target located at a point other than at the center of the system, 2. a non-symmetrical'arrangement of the bearing-measurement stations, 1 The computer program, listed in Appendix A, utilizes 511 of the 512 high-speed storage cells of the MIDAC computer. Only a small saving in the use of these cells could be accomplished by changes in the program.

-1523. a non-normal probability distribution for bearingmeasurement error, 4o a non-uniform distribution for the age of the bearing measurements, and 50 large rather than small bearing-reading error. Four times as much computing time would be required to obtain each probability distribution for uniformly distributed target direction in the first and second situations than in the special case numerically treated herein because of differences in symmetry. The third situation requires only a change of the numerical distribution which is stored in the program and does not require additional computation time. The fourth situation may not require additional computing time nor additional computer storage space if the distribution to be used for age can be described by a simple equation. Changes in the computer program are necessary, of course. If an empirical distribution is to be used in the fourth situation, additional storage space in the computer is requiredo The fifth situation would substantially increase the required computing time because of the additional arithmetic operations to be performed.

APPENDIX A PROGRAM FOR THE CONVENTIONAL SYSTEM The program listed below was used with the MIDAC computer to calculate a set of cumulative probability distributions for the normalized position error for the case of uniformly distributed target direction. Intermediate results for each of twelve target directions were obtained also. This particular program applies to the cases A-u = 10 and sDT = 25. A modification to this program for Oru = 50/7 is presented in Appendix D. Modifications for SDT = 125/7 and 150/7 are presented in Appendix E, MIDAC, the digital computer which was used to calculate the probability distributions for position error, is a three-address, serial, general purpose computer.l The programs presented herein make use of the Magic I system of MIDAC, an automatic programming system which translates a program which is written in floating address form into the correct computer language for computation.2 A MIDAC instruction word is made up of four parts; o<,,, and an operation symbol, which are written in the order listed. Table A-1 contains a simplified description of the logical and arithmetic operations performed by MIDAC in terms of these four parts of each MIDAC is described in detail by Carr and Scott(5) Appendix IX.1 2 The Magic I system, which includes other features such as automatic error diagnoses and a library of frequently used sub-routines, is described by Brown, J. H., "Programming for the Magic I System," Section II3 of a book edited by Carr and Scott(5). -155

1a54TABLE A-1 SIMPLIFIED DESCRIPTION OF THIE MIDAC OPERATIONS Operation Symbol Description ba Augment C. by -o< If 3 > Cs, B. If ~ C8 a set C, = O and continue ex L1 = F<1? [(3] ad [61 [ ~ [(+ L ri Read in o( words from input station or drum address ( to c< cells starting with address, su [s Y [J ] - [] cn If [3] > [o],- O If [P] E [<],continue, s n I[ j< = [x I ( X~ [ cm If > o If [!- EIJ),continue. fi Store C, + I in ~ of < and -* Y dY x ] r / mr [r~], (rounded) ro Read out rX words from o< cells starting with address P to output station or drum address W. bd [rJ is a binary coded decimal number equal to [.] LEGEND:~ C is the base counter, a counter used in cycling. C. is the instruction counter, which is set to the address of the instruction being performed. I- means "set C. to". [ means "the contents of"t means a logical "and" rather than arithmetic addition.

-155instruction word. The symbols o', P, and ( are used to designate either numbers or the addresses of storage cells, depending on the nature of the operation. By affixing a minus sign to an address, the address is automatically augmented by either of two counters, the base counter if the operation symbol is prefixed by a minus sign, or the instruction counter if not. A minus sign affixed to a number in a "read in" or "read out" instruction specifies that the contents of the cells are coded typewriter symbols and not numerical quantities. The address which precedes the instruction words is the floating address which is not assigned to a computer address until translation takes place.

- 156 - PROGRAM cd faaO4 000 000 000 ba clear the base counter faaO5 leOl e02 eOO ex set T#, the 0 tally, to its initial value (usually zero) faa06 eOO bOO a07 ad put unmodified instruction in a07 faaO7 000 000 000 ri put selected value of A x 2-3 and B x 2-36 -001 002 a07 baJ in cOO and lcOO, respectively cOO e02 2c00 ex IAI x 2-3 in 2c00 faa08 -dOl -dOl -dOl -su clear 155 cells from address dOl -001 155 a08 ba through address 73d02 lbOO e02 a12 ex put unmodified instructions 2b00 e02 b05 ex 1 in al2 and bO5 2eO1 3e01 cl0 su brs = -24 x 2-36 in cOl fOO cOO a09 cn if A > O * a09 cOO le02 3c00 sn A x 2-24 in 3c00 cOO a12 a12 ad add A to the addresses in 3c00 a12 a12 ad Y and 9 of instruction a12 cOO cl0 cl0 ad (%+ A) x 2-36 in cOl faaO9 2e01 2c00 alO cm if IAI 1 + a10O e02 e02 4c00 ex 1 - 2-44 I/IA I in 4c00 512 001 all fi + all faal0 2c00 2e01 4c00 dv l/IAl in 4c00 faall 4c00 -dOO -d03 -mr P fi (/ ( r"= A o") = Pnr(o")>P, (,- -I. ") -001 050 all baJ \stored in 50 cells starting with address d03 leOO leOO leOO su clear TA, the'A~' tally a12 e26 b05 ex use instruction a12 (already modified) to modify instruction bO5 so that the use of P1, starts at Pr(A) faal2 000 000 000 ri accumulate P, at the proper -001 050 a12 ba address determined by instruction a12 2e01 le00 le00 ad augment T., by 1 4e01 a12 al2 ad augment g and r of instruction a12 by 1, i.e., augment N4r" by 1 le00 2c00 al2 cm if IAI > TA -- a12 faal3 3b00 e02 a16 ex put unmodified instruction in a16 faal4 lcOO 2e01 5c00 dv 1/B in 5c00 faal5 -dOO 5c00 -d03 -mr>f PF(cw wt =wO ) = p,,, ( Pw,) P(U - -") -001 050 al5 baJ in 50 cells starting with address d03 2e00 2e00 2e00 su clear T., the B tally faal6 000 000 000 ri\ faccumulate P, at the proper address -001 050 al6 baJ determined by instruction al6 2e01 2e00. 2e00 ad augment T, by 1 4e01 a16 a16 ad augment 3 and I of instruction a16 by 1, i.e., augment'i by 1 2e00 lcOO a16 cm if B > Ts - al6 faal7 e04 -dOO -dO3 -mr P /12 in 50 cells starting -001 050 a17 baJ kwith address d03 faal8 2c00 5e01 c02 ad (49 + JAJ ) x 2-36 in c02 lc00 5eOl lc02 ad (49 + 8 ) x 2-36 in lc02 3eO0 3eO0 3e00 su clear T_, the Nr tally fab05 000 000 000 ri P (r.)n in 3c01 cOl cl0 2c01 mB %.AX 2-28 in 2c01 4b00 e02 b06 ex put unmodified instruction in b06 2e01 3e01 lcl0 su rs = -24 x 2-36 in lcOl 4e00 4e00 4e00 su clear T,, the "r tally fab06 000 000 000 ri P, (%.) P, (w) in 2e27 lcl0 cl0 4c01 su ( u,-r. ) x 2-36 in 4c01 4col lc01 5c01 m~ or, ( er.- r- ) x 2-28 in 5c01 5cO1 2c01 6c01 ad [. +.(r. (-,r)]6 x 2-28 in 6c01 6e01 3e01 c04 su us = -24.5 x 2-36 in co4 lcl0 cl0 lc04 ad (ro +.o) x 2-36 in 1c04 faal9 c04 lc04 2c04 su (u - w. -o.) x 2-36 in 2c04 2c04 c04 3c04 ml us (u,-. - r) x 2-28 in 3c04 3c04 6c01 4co4 ad Ec ( u,, r,.) x 2-28 in 4c04

-157faa20 4c04 e01l c05 sn E x 2-13 in c05 2e01 lc04 5c04 su ( I - vJ - w-r)x 2-36 in 5c04 e05 c05 e25 ad (Ec + K.) x 2-13 in e25 5c04 3e02 5e25 sn (I -, -.r) x 2-13 in 5e25 c04 6e02 4e25 sn u, x 2-12 in 4e25 faa22 a27 001 a25 fi - a25, the start of the inner loop subroutine 2e01 4e00 4e00 ad augment T, by 1 2e01 lc1 lc1Ol ad 7e01 bO6 b06 adJ augment U by 1 4e00 lc02 b06 cm if 49 + B > T, r b06 faa23 2e01 3e00 3e00 ad augment TV by 1 2e01 cl0 cl0 ad 7e01 b05 b05 ad augment ir. by 1 3e00 c02 b05 cm if 49 + IAJ > T,r + b05 faa24 c06 co6 c06 su clear cell c06, the cell at which the cumulative distribution is accumulated 001 leOl 001 ro /print out code words which 001 eOO 001 ro J identify the results faa28 -lc4 c06 c06 -ad construct the cumulative distribution c06 001 lc06 bd convert the result to a decimal number 001 1c06 001 ro print out the result -001 o60 a28 ba - a28 until all 60 results are printed out faa29 4eO2 eOO eOO ad augment T$ by 2 eOO 5e02 aO6 cm if 24 > T - a06 000 000 000 ri HALT faa25 e25 e29 le29 dv EA (uO,vr, wr) x 2-24 = K, x Z-3/(E+K) x.' in le29 -dO3 2e27 le27 -mr PUr, (u,, nr, cr,) in le27 le29 e26 le25 ex 3A' i.e., 3 digits of E x 2-24 in le25 le25 le26 3e25 ad (452 + )x 2-24 in 3e25 3e25 2e26 e27 sn (452 + 13 ) x 2-36 in e27 e27 4326 a26 ex: /modify instruction a26 so that the 3e25 3e26 a26 exJ results are accumulated properly 5e25 4e25 2e25 ad (2 Uo + 1 -,. - wr ) x 2-13 in 2e25 2e25 e25 e25 ad [Ec(uo+,) - K;]- x 2-13 = [E(u0) + iK] x 2-13 + (2 uJ + 1 - nv. - w ) x 2-13 in e25 4e25 e28 4e25 ad 2( UO + 1) x 2-13 = 2 U. x 2-13 + 2 x 2-13 in 4e25 faa26 le27 000 000 ad accumulate PvUWin the proper cell -001 050 a25 ba -* a25, i.e., cycle through all values of U in order to accumulate Pc (ER jI r= ~r., ar t o) in 60 cells starting with cell number 452 faa27 512 001 000 fi -* la22, i.e., return to the main program fae25 \ fassign six cells for ac6e25 j \temporary storage fae28 daOO1 001 2 x 2-13 fae26 daO05 Offf 0001c4 (constants entered as -0000000000 c hexadecimal numbers Offf 00000 fff fae27 fassign three cells for ac3e27 J temporary storage fae29 daOO1 OOOOOOd6f2c K, x 2-37 ac2e29 assign one cell for temporary storage cd fab00 fOO -e03 -cOO -ad -dO3 -21d01 -21d01 -ad fO 000 3c01 ad -dO3 -dO2 -dO2 -ad fixed instructions which are modified in 3cOl dO2 2e27 mr) use elsewhere in the program

-158faccOO assign six cells facOl ac7cOl assign seven cells facO2 ac2cO2 J assign two cells for temporary storage facO4 ac6cO4 assign six cells facO5 assign one cell fac06 ac2c06 assign two cells def fadOO.1988 -2d Ob.2526 -2d Ob.3179 -2d Ob.3961 -2d Ob.4886 -2d Ob.5967 -2d Ob.7213 -2d Ob.8635 -2d Ob.10234 -ld Ob.12008 -ld Ob.13950 -ld Ob.16043 -id Ob.18270 -ld Ob.20596 -Id Ob.22989 -ld Ob.25405 -ld Ob.27795 -ld Ob.30109 -id Ob.32289 -ld Ob.34285 -Id Ob.36040 -ld Ob.37511 -ld Ob.38651 -ld Ob the set of constants, P (u), entered as decimal fractions.39432 -ld Ob.39828 -ld Ob.39828 -ld Ob.39432 -ld Ob.38651 -ld Ob.37511 -ld Ob.36040 -ld Ob.34285 -ld Ob.32289 -ld Ob.30109 -ld Ob.27795 -ld Ob.25405 -ld Ob.22989 -ld Ob.20596 -ld Ob.18270 -ld Ob.16043 -ld Ob.13950 -ld Ob.12008 -ld Ob.10234 -ld Ob.8635 -2d Ob.7213 -2d Ob.5967 -2d Ob.4886 -2d Ob.3961 -2d Ob.3179 -2d Ob.2526 -2d Ob.1988 -2d Ob

-159fadOl ) assign eighty-one cells ac81dOl fad02 ac74dO2 p assign seventy-four cells fadO3 for temporary storage ac5OdO3 J assign fifty cells fae00 assign five cells ac5eO 01 fae O1 daO08 0000000000f used to multiply by 215 by shifting 0 initial value of T 000000001 1 x 2-36 000000019 25 x 2-36 000001001 used to modify an instruction 000000031 49 x 2-36 ooooooooo8 0.5 x 2-36 000001 used to modify an instruction fae02 da007 fffffffffff extractor OOOOOOOOOOc used to multiply by 212 by shifting 00000000009 used to multiply by 29 by shifting 00000000017 used to multiply by 223 by shifting 000002 2 x 2-24 000018 24 x 2-24 00000000018 used to multiply by 224 by shifting fae03 da024 -000000015 000000016 -000000013 000000018 -000000010 000000019 -00000000e 000000019 -00000000b 000000019 -000000008 000000019 -000000005 values of A and B 000000018 -000000001 000000016 000000001 000000015 000000005 000000013 00o000008 000000010 OO000000b OOOOOOOOe fae04 def.83333333333 -ld Ob 1/12 fae05 daOO1 03a08 K, x 2-13 bca04 computation begins with the instruction at address aO4

APPENDIX B MODIFIED SYSTEM, PROGRAM I The program listed below, in floating address form, was used with the MIDAC computerl to calculate and print out a set of fifty cumulative probability distributions for the magnitude of the position error for the modified three-station system for values of C from one through fifty. In this program, cu = 10. Appendix D contains a modification to this program for - = 50/7. 1 See Appendix A for a brief description of MIDAC operations and programming. -160

-161PROGRAM cd faa04 000 000 000 ba clear the base counter faaO8 -dOl -dOl -dO1 -su f clear 99 cells starting -001 099 a08 ba with address dOl bOO e02 a12 ex /put unmodified instruction lbOO e02 b05 exJ in cells a12 and b05 eOl leOl cOl su. = -24 x 2-36 in c01 e0l eOO alO cm if C > 1 b alO faaO9 -dOO e02 -d01 -ex f P = P, (I v ) in 50 cells -001 050 a09 baJ starting with dO1 512 001 a12 fi 4 a18 faalO eOO leO2 cOO sn AC x 2-28 in cOO eOO eOO lcOO ma Ca x 2-28 in lcOO cOO 2e01 2c00 su (AC - I) x 2-28 in 2c00 lc00 2c00 3c00 dv ( C -I )/C = Pr,,(I/,) in 3c00 lcoo 3e01 4c00 dv Z/C~' in 4c00 leOO leOO leOO su clear Tc x 2-26, the C tally faall -dOO 3c00 d03 -mr ( n (( r) P) (" -) faal2 000 000 000 ri r accumulated at the proper address -001 050 all ba Kdetermined by instruction al2 3c00 4c00 3c00 su P, (/UO + I) P( in 3c00 eOl leOO leOO ad augment TC by 1 4eOl a12 al2 ad augment /iv by 1 leOO eOO all cm if C >C - - all faal8 eOO 5e01 c02 ad (49 + C ) x 2-36 in c02 2eO0 2eO0 2e00 su clear T x 2-36, the ur tally fabO5 000 000 000 ri Pt (i2) 2n 2e27 cOl cOl lcOl mb,r x 2-2 in lcO0 faal9 6eO1 leOl c04 su uS = - 24.5 x 2-36 in c04 cOl c04 lc04 ad ( Vs + N ) x 2-36 iR lc04 lc04~ c04 2c04 ml s (us + v.) x 2'2 in 2c04 lcol 2c04 3c04 ad rCa + u,(u$ +;.)] x 2-28 = Ec(u5, /tr) in 3c04 3c04 2e02 cO5 sn E, (u, r, ) x 2-13 in c05 e05 co5 e25 ad (C + K ) x 2-13 in e25 faa20 co4 le02 lc05 sn 81 Us x 2-28 in lc05 eOl cOl 2c05 ad (1 +,r ) x 2-36 in 2c05 2c05 3eO2 3c05 sn (1 +,r ) x 2-28 in 3c05 3cO5 lc05 4c05 ad (2 Us +1 + 14r ) x 2-28 in 4c05 4co5 2e02 2e25 sn (2 us + 1 +. ) x 2-13 in 2e25 faa22 a27 001 a25 fi -, a25, the start of the inner loop subroutine eOl 2eO0 2eOO ad augment T., by 1 eOl cl0 cOl ad 7e01 b05 b05 ad augment ir by 1 2e00 c02 b05 cm if 49 + C > T, - b05 faa24 c06 c06 c06 su clear cell c06, the cell at which the cumulative distribution is accumulated 001 eOO 001 ro read out C -001 eO6 001 ro carriage return identifies results faa28 -elO c06 c06 -ad construct the cumulative distribution c06 001 lc06 bd convert the result to a decimal number 001 lc06 001 ro print out the result -001 o60 a28 ba -4 a28 until all 60 results are printed out faa29 eOl eOO eOO ad augment C by 1 faa30 -elO -elO -elO -su f clear 60 cells starting with -001 o60 a30 ba ~address elO -004 e06 001 ro read out four carriage return instructions eOO 8e01 a04 cm if 50 > C -4 aO4 000 000 000 ri HALT

-162faa25 e25 e29 le29 dv E, (u,, It,) x 2-24 = K, x 2-37/(Ec + K. ) x 2-13 le29 e26 le25 ex #8Rr, i.e., ~ digits of E, x 2-2 in le25 le25 le26 3e25 ad (452 + IR ) x 2-24 in 3e25 3e25 2e26 e27 sn (452 + Ot ) x 2-36 in e27 e27 4e26 a26 ex modify instruction a26 so that the 3e25 3e26 a26 ex \results are accumulated properly 2e25 e25 e25 ad [Ec(u+* I) + KA] x 2-13 = [ E (u) + K ] x 2-13 + (2 + 1 + 1 + ) x 2-13 in e25 2e25 e28 2e25 ad (2 [u.+ 1I + 1 + ir; ) x 2-13 = (2 U, + 1 + r ) x 2-13 + 2-12 in 2e25 -dOO 2e27 le27 -mr P (U.) Pr (Av) in le27 faa26 le27 000 000 ad accumulate PU P,. in proper cell -001 050 a25 ba - a25 and cycle through all values of U in order to accumulate Pc ( ER 4r = r.) in 60 cells starting with cell number 452 faa27 512 001 000 fi -* la22, i.e., return to main program fae25 fae25 ) assign six cells for temporary storage fae28 daOOl 001 J a constant, 2-12 fae26 daO05 000fff OOOlc4 -0000000000c constants entered as hexadecimal numbers 00Offf 00000 fff fae27 ac3e27, assign three cells for temporary storage ac3e27 fae29 daOl0 000000d6f2c K, x 2-37 ac2e29 assign one cell for temporary storage cd fab00 dO3 -dOl -dOl -ad / fixed instructions used at fOO dOl 2e27 ad addresses a12 and bO5 facOO d ac5cOO J assign five cells facOl ac2cOl assign two cells facO2 assign one cell for temporary storage fac04 ac4cO4 J assign four cells fac05 ac5cO5 J assigt five cells fac06 ac2cO6 J assign two cells def fadOO.1988 -2d Ob.2526 -2d Ob.3179 -2d Ob.3961 -2d Ob.4886 -2d Ob.5967 -2d Ob.7213 -2d Ob.8635 -2d Ob.10234 -ld Ob the set of constants, P,(u), entered as decimal fractions.12008 -ld Ob.13950 -ld Ob (Continued on the following page).16043 -ld Ob.18270 -ld Ob.20596 -ld Ob.22989 -ld Ob.25405 -ld Ob.27795 -ld Ob

-163-.30109 -ld Ob.32289 -ld Ob.34285 -ld Ob.36040 -ld Ob.37511 -ld Ob.38651 -ld Ob.39432 -ld Ob the set of constants, Pu(u), entered as decimal fractions.39828 -ld Ob.39828 -ld Ob.39432 -ld Ob.38651 -ld Ob.37511 -ld Ob.36040 -ld Ob.34285 -ld Ob.32289 -ld Ob.30109 -ld Ob.27795 -ld Ob.25405 -ld Ob.22989 -ld Ob.20596 -ld Ob.18270 -ld Ob.16043 -ld Ob.13950 -ld Ob.12008 -ld Ob.10234 -ld Ob.8635 -2d Ob.7213 -2d Ob.5967 -2d Ob.4886 -2d Ob.3961 -2d Ob.3179 -2d Ob.2526 -2d Ob.1988 -2d Ob fadOl } {assign ninety-nine cells acqq99dO1 J for temporary storage fadO3 assign one cell for temporary storage fae03 daOOl 000000001 J C x 2-36, initially set at C = 1 ac3eOO assign two cells for temporary storage faeOl daOO9 000000001 2-36 000000019 25 x 2-36 0000001 2-28 0000002 2 x 2-28 000001001 used to modify an instruction 000000031 49 x 2-36 0000000008 0.5 x 2-36 000001 used to modify an instruction 000000032 50 x 2-36 faeO2 daOO4 fffffffffff extractor 00000000009 used to multiply by 29 by shifting 000000000 f used to multiply by 215 by shifting 00000000008 used to multiply by 28 by shifting fae05 daOO1 03a08 Ka x 2-13 fae06 daOO4 ec ec code for carriage return, used for the format of the results ec ec ac452 fael0 } assign one cell for temporary storage bca04 computation begins with the instruction at address a04

APPENDIX C MODIFIED SYSTEM, PROGRAM II The program listed below provides a change in Program I by which the set of fifty cumulative probability distributions for the position error is stored on the magnetic drum of the computer instead of being printed. Each of these cumulative distributions may be interpreted as a conditional distribution for the corresponding value of C. The program then provides that any sub-set of twelve conditional distributions may be selected (with replacement) and combined to obtain a resultant distribution assuming that each of the selected values of C is equally likely. This combinatorial procedure is used to obtain cumulative probability distributions for the position error for an arbitrary target direction. The selection of the twelve distributions is accomplished by listing the twelve corresponding values of C on an auxiliary tape which is processed by the computer. These values are listed in Appendix D for the several values of sDT which were used. n164

-165PROGRAM change aca24 cd 512 001 a34 fi -* to drum storage subroutine ac4eO6 cd faa34 ell ell ell su clear cell used as temporary storage for distribution sum eOO le13 le12 sn 8C x 2-36 in lel2 a37 lel2 2e12 ad (1064m + 8C ) x 2-36, drum address for distribution storage in 2e12 2e12 2e13 3e12 sn (1064m + 8 C) x 2-24 in 3e12 2e12 4e26 a32 ex add proper drum address to instruction a32 3e12 e26 la32 ex add proper drum address to instruction 1a32 faa31 -elO clO -lell -mr multiply distribution by 1/12 ell -lell ell -Su f obtain the distribution sum -001 o60 a31 bal and place in cell ell a36 001 a35 fi -* distribution sum check subroutine fOO cll a34 cm -, a34 if distribution sum check is not correct faa32 061 ell 000 ro store the distribution plus the distribution sum on the drum starting with drum address (1064m + 8C) 061 000 451 ri read the distribution plus the distribution sum from the drum starting with address 451 b36 001 b35 fi -* distribution sum check subroutine fOO cll a32 cm - a32 if distribution sum check is not correct eli 001 e12 bd /print out distribution sum 001 e12 001 rok ito indicate progress of computation faa33 -elO -elO -elO -su clear cells used for storage -001 o6o a33 ba of the distribution eOl eOO eOO ad augment C by 1 eOO 3e13 a04 cm if 51 > C -+ a04 fab20 013 001 e14 ri read in a code word followed by twelve values of C x 2-12 from an auxiliary tape fOO e28 e12 ad set T., the distribution tally, to 1 x 2-12 fab21 -iell -lell -lell -su\f clear 60 cells starting -001 060 b21 baJ kwith address lell fab24 a38 e12 b25 ad add TD to - of instruction a38 and put the result at address b25 fab25 000 000 000 ri select the ( 7T - 1)'th value of C x 2-12 which was read in from the auxiliary tape and put it at address le12 with a scale change to C x 2-21 a37 le12 2e12 ad add 8 C to, of the instruction at address a37 and put the result at address 2e12 2e12 e26 b22 ex put the drum address (1064m + 8 C ) in f of the instruction at address b22 fab22 061 000 451 ri read the distribution plus the distribution sum corresponding to the ( T - 1)'th value of C from the drum to 61 cells starting with address 451 b36 001 b35 fi -+ distribution sum check subroutine fOO cll b22 cm -* b22 if distribution sum check is not correct fab23 -452 -lell -lell -ad j accumulate the sum of the twelve distributions -001 060 b23 ba ~in 60 cells starting with address lell e28 e12 e12 ad augment TD by 1 e12 4e13 b24 cm if 13 > T-, b24 fab26 001 e14 001 ro print out a code word which identifies the results -001 e06 001 ro read out a carriage return instruction ell ell ell su clear cell ell

fab27 -lell ell ell -ad construct the cumulative distribution ell 001 cll bd convert result to a decimal number 001 cll 001 ro print out the decimal result -001 o6o b27 ba -* b27 until all 60 results are printed out -004 eo6 001 ro read out four carriage return instructions 512 001 b20 fi f- b20 and repeat the distribution combination process for the next set of twelve values of C. (end of main program) faa35 cll cll cll su clear cell used for temporary storage -ell cll ll -adl fobtain the distribution sum -001 061 la35 baj and place it in cell cll faa36 512 001 000 fi return to the main program fab35 cll cll cll su clear cell used for temporary storage -451 cll cll -ad /obtain the distribution sum and -001 061 lb35 baj place it in cell cll fab36 512 001 000 fi return to the main program faa37 000 1064m 1064m ri dummy instruction used only for the address it contains faa38 e14 e13 le12 sn fixed instruction used at address b25 fael2 assign four cells for ac4el2 k temporary storage fael3 daO05 -00000000009 used to multiply by 2-9 by shifting 00000000003 used to multiply by 23 by shifting 0000000OOO c used to multiply by 21 by shifting 000000033 51 x 2-36 OO 13 x 2-12 faclO def.83333333333 -1 Ob 1/12 facll assign one cell for temporary storage fae14 acl3el4 assign thirteen cells for temporary storage faell assign one cell for temporary storage bca04 computation begins with the instruction at address a04

-167APPENDIX D O'TER VALUE OF 0u The program listed below was used to change the computer programs listed in Appendixes A and B for C = 50/7. PROGRAM change acd00 def.157 -3d Ob.251 -3d Ob.394 -3d Ob.6o6 -3d Ob.914 -3d Ob.1352 -2d Ob.1961 -2d Ob.2788 -2d Ob.3889 -2d Ob.5319 -2d Ob.7134 -2d Ob.9381 -2d Ob.121 -Id Ob.15301 -Id Ob.18977 -ld Ob.23078 -id Ob.27522 -1d Ob.32186 -1d Ob.36911 -ld Ob.4151 -id Ob.45776 -Id Ob.49503 -ld Ob.52496 -id Ob.54591 -id Ob.5567 -id Ob.5567 -ld Ob.54591 -Id Ob.52496 -ld Ob.49503 -ld Ob.45776 -ld Ob.4151 -ld Ob.36911 -ld Ob.32186 -ld Ob.27522 -Id Ob.23078 -ld Ob.18977 -ld Ob.15301 -id Ob.121 -Id Ob.9381 -2d Ob.7134 -2d Ob.5319 -2d Ob.3889 -2d Ob.2788 -2d Ob.1961 -2d Ob.1352 -2d Ob.914 -3d Ob.6O6 -3d Ob.394 -3d Ob.251 -3d Ob.157 -3d Ob bca04

APPENDIX E OTHER VALUES OF A AND B The following sets of values of. the constants A and B were used in the program listed in Appendix A at address fae03~ when the distribution for the variable u is described by 0, = 50/7. The value of sDT for each set of values of A and B is listed at the head of each column. The set of values listed in Appendix A are used for sDTr = 3.5. sDT= 2.5 5DT= 3.0 oooooooo8 000000009 00000000a OOOOOO0Oc oooooooo6 000000007 OQOOOOOc OOOOOOOOe 000000003 000000004 000000003 ooooooo4 00000000d 000000010 000000001 000000001 000000OOOOOOOOf 000000012 -000000001 -000000001 000000010 000000014 -000000003 -000000004 000000011 000000014 -oooooooo6 -000000007 000000012 000000015 -000000008 -000000009 000000012 000000015 -00000000 -OOOOOOOOc 000000012 000000015 -00000000 -00000000e 000000012 000000015 -00000000d -000000010 000000011 000000014 -00000000f -000000012 000000010 ooooooo14 -168~

APPENDIX F CODE WORDS FOR PROGRAM II Each group of twelve code words (values of C ) listed below was used in Program II to select and combine twelve probability distributions for position error in order to obtain a distribution for the case of uniformly distributed target direction. The code words apply to the case of u = 50/7. The values of s5T are listed at the head of each column. 0.5 1.0 1.5 2.0 2,5 3.0 3.5 001 001 001 001 001 001 002 001 001 002 003 003 oo004 005 001 002 003 005 oo6 007 008 002 003 005 oo6 oo8 009 0Ob 002 oo4 oo6 008 OOa OOc OOe 002 005 007 009 00c OOe 011 003 005 oo008 OOb OOd 010 013 003 oo6 009 OOc OOf 012 015 003 oo6 OOa OOd 010 013 016 003 007 OOa OOe 011 014 018 oo4 007 OOa OOe 012 015 019 004 007 00b OOe 012 015 019 4,0 4.5 5s0 5.5 6~o 6.5 7T0 002 002 002 003 003 003 003 oo6 007 007 008 008 009 OOa 009 OOa OOb OOd OOe OOf 010 OOd OOe 010 011 013 015 016 010 012 014 016 018 Ola Olc 013 015 018 Ola Ol Olf 021 015 018 Olb Ole 020 023 026 018 Olb Ole 021 024 027 02a Ola Old 020 023 026 02a 02d Olb Ole 022 025 029 02c 02f Olc 020 023 027 02a 02e 031 Olc 020 024 027 02b 02e 032 =169

BIBLIOGRAPHY 1. Albert, G. E., "A General Theory of Stochastic Estimates of the Neuman Series for the Solutions of Certain Fredholm Integral Equations and Related Series,," Symposium on Monte Carlo Methods, (edited by Meyer, H. A. ), John Wiley and. Sons, pp. 37-46, New York; 1956. 2. Barfield,- R. H., "Statistical Plotting Methods for Radio DirectionFinding," The Journal of the Institution of Electrical Engineers, Vol. 94, Part III A, No. 15, pp. 673-675y, London; 1947. 36 Bond. D. S., Radio Direction Finders, McGraw-Hill Book Company, New York; 194.' 4. Bowen, K. C., "Sources of Error in U-Adcock High-Frequency DirectionFinding," The Journal of the Institution of Electrical Engineers, Vol. 102, Part B, pp. 529-532, London; July, 1955. 5. Carr, J. W,, and Scott, N. R., Notes on Digital Computers and Data Processors, College of Engineering, University of Michigan, Ann Arbor, Michigan; 1956. 6. Carver, H. C., Mathematical Statistical Tables, Edwards Brothers, Ann Arbor, Michigan; 1950. 7. Cramer, Ho, The Elements of Probability Theory, John Wiley and Sons, New York; 1955. 8. Frese, R. E., An Evaluation of the Accuracy of the AN/TSQ-9 System, Engineering Research Institute Report No. 2359-2-S, University of Michigan, pp. 2,-22 (unclassified portion of a classified report); June 1956. 9. Goode, H. Ho, and Machol, R. E.,o System Engineerin, McGraw-HiUll Book Company, New York; 1957o 10. Harkin, B., "The Expected Error of a Least-Squares Solution of Location from Direction-Finding Equipment," Australian Journal of Applied Science, Vol. 7, No. 4, pp. 263-272; December, 1956. 11. Horner, F., "The Accuracy of the Location of Sources of Atmospherics by Radio Direction-Finding," The Journal of the Institution of Electrical Engineers, Vol. 101, Part III,pp 383- 3 Nember, 1954. 12. Kahn, H,, "The Use of Different Monte Carlo Sampling Techniques,` Symposium on Monte Carlo Methods, (edited by Meyer, H. A.),, John Wiley and Sons,* pp. 146-190, New York; 1956. 13. Kaplany W., Advanced Calcul.ust Addison-Wesley Publishing Company, Cambridge, Massachusetts; 1953. -170

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