1、0 AMSC NIA MIL-STD-600002 MC m 7777722 008B412 i m pTiiq MI L-STD-600001 26 FEBRUARY 1990 MILITARY STANDARD MAPPING, CHARTING distribution is unlimited. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-STD-600001 MC 77777LL 0088L112 6 MI L-STD-600
2、00 1 FOREWORD 1. This military standard is approved for use by all Departments and Agencies of the Department of Defense. 2. Beneficial comments (recommendations, additions, deletions) and any pertinent data which may be of use in improving this document should be addressed to: Director, Defense Map
3、pingAgency,AlTN: PR, 861 3 Lee Highway, Fairfax, VA 22031 -21 37 by using the self-addressed Standardization Document Improvement Proposal (DD Form 1 426) appearing at the end of this document or by letter; II Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
4、IHS-,-,-CONTENTS PARAG RAPH PAGE 1 . 1 . 1 1.2 1.3 2 . 2.1 2.1.1 2.1.2 2.2 2.3 3 . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4 . 4.1 4.2 4.3 4.4 4.4.1 4.4.2 4.5 4.6 5 . 5.1 5.2 SCOPE . Scope . Purpose Applicability APPLICABLE DOCUMENTS Government documents . Specifications. standards. and handbooks . Other Go
5、vernment documents. drawings. and publications Non-Government publications Order of precedence DEFINITIONS . Absolute horizontal accuracy . Absolute vertical accuracy . Accu racy Datum (geodesy) Random error . Relative horizontal accuracy (point-to-point) Relative vertical accuracy (point-to-point)
6、Systematic error . GEN E RAL REQU i RE ME NTS . Intended use of accuracy . Accu racy requi remen ts . Accu racy requirement de fi ni ti on . Formulas (sim pl if ied) Circular error Li near error . Selection of normal distribution Accuracy note DETAILED REQUIREMENTS . General A bso I u t e accu racy
7、2 2 2 2 2 2 7 7 7 iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PARAGRAPH 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 6 . 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.2.7 MIL-STD-600001 CONTENTS Relative accu racy . Point positio
8、ns Variance-covariance matrix Error propagation relating to triangulation Application of triangulation output . Error propagation from sample statistics Sample statistics when the diagnostic and product errors are independent . Sample statistics when the diagnostic and product errors are dependent S
9、ummary of sample statistics methodology . Absolute accu racy com put at ions . Point-to-point relative accuracy computations . Alternate error propagation from sample statistics . Accuracy influenced by bias . NOTES Intended use . International standardization agreements I n t e r n at ion al St and
10、ardizat io n Ag ree m e nts (STAN AGs) Quadripartite Standardization Agreements (QSTAGs) . Air Standardization Coordinating Committee International MC and to all levels involved in the preparation, maintenance of MC = sample standard deviation (meters) 1.645 = normal deviate for 90% confidence level
11、 4.5 Selection of normal distribution. The normal distribution function was selected since it closely fits the actual observed frequency distributions of many physical measure- ments and natural phenomena. In addition it makes error analysis a more tractable prob le m. 4.6 Accuracv Note. MC but is s
12、till stated in terms of a horizontal component and a vertical component. As in the case with absolute accuracy, the horizon- tal uncertainty is stated as a CE and the vertical error is stated as a LE. 5.4 Point pos itions. Point positions derived from measurements of photographic images are usually
13、referenced to an earth fixed Cartesion coordinate system. Avariance- covariance matrix defining the uncertainty of this computed position relative to this coordi- nate system is determined by standard error propagation techniques utilizing apriori esti- mates of errors associated with the computatio
14、nal parameters. The apriori estimates of the errors associated with these computational parameters are usually in the form of avariance- covariance matrix and includes all of the covariances resulting from the correlation of the parameters. The parameter variance-covariance matrices used to assess p
15、roduct accura- cies result from (1) statistics accumulated from redundant observations of the parameters, or (2) statistics propagated through computations required to determine the parameters from redundant indirect observations. An example of such computations are those required to accomplish leas
16、t squares triangulation to update exposure station positions and camera attitudes. 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-STD-600001 5.5 Variance-covariance matrix. A primary goal of any evaluation scheme should be the construction of
17、the variance-covariance matrix associated with any position depicted in the product. The generation of such matrices will likely utilize standard error propagation techniques andor sample statistics resulting from the comparison of positions extracted from the product to their known positions. Such
18、points are referred to as diagnostic points. Ultimately the success of any evaluation method depends on its ability to approximate these variance-covariance matrices. The variance-covariance matrix relating the errors of two geographic positions will be defined. This is followed by a summary of meth
19、ods used in the determination of this matrix in various circumstances. Finally, the computation of the absolute CE and LE and the relative point-to-point CE and LE is presented. To define acovariance matrixconsidertwovectors, denoted by U and V, whosecomponents are random variables. The cross-covari
20、ance of the two vectors is defined by where E is the expectation of the random variable and is defined as the sum of all values the random variable may take, each weighted by the probability of its occurrence. lhe covariance of U is when U = V. Suppose that the geographic position of two points, and
21、 their cross-covariance matrix has been determined. Let the two positions be denoted by ($, h, h, ) and ($, h, h, ) Let their cross-covariance matrix be denoted by Q such that I PI I II Q= Q22 7 where Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-M
22、IL-STD-600003 MC W 9777733 0088424 2 W M I L- ST D-60 O O O 1 where 2 is the variance of $, etc ., is the covariance of $ and hi, etc. and Methods for the determination of the cross-covariance matrix Q will be considered. These methods, intended as guidelines only, are somewhat generalized in the se
23、nse that they are not presented in terms of any one product. Two methods are presented; the first based on the statistics output from triangulation; the second based on a comparison of positions sampled from the product to known or diagnostic positions. 5.6 Error OroDaaation relatina to trianaulatio
24、n. First consider the case involving trian- gulation. It is not within the scope of this standard to present an exhaustive development of triangulation mathematics. Hopefully, enough for clarity and understanding is presented. The condition equations are assumed to be of the form A(L + V) + BA = D w
25、here A and B are coefficient matrices, D is a vector of constants, 9 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-STD-600001 L is a vector of observations, V is a vector of residuals, and A is a vector of parameters usually referred to as the
26、state vector In addition, define Q, as the covariance matrix associated with the observational wector L and define W as the observational weight matrix, that is, W = Q:, A few words relative to the observations and state vector regarding their respective weights are in order. Assume that the unknown
27、 state vector, A, has an initial value that results from an observational reduction process and thus can be treated as part of the observations, L. Thus, any theoretical error propagation scheme used to estimate triangulation output accuracies depends heavily on apriori covariances associated with t
28、he observations or associated with parameters treated as observations. The Covariance matrices resulting from triangulation are considered acceptable if a reference variance computed from the residuals is believable. Define this reference variance as where R is the degrees of freedom associated with
29、 the least squares adjustment. Since the weight matrix is the inverse of the observational covariance matrix, the reference variance is in variance units and will be near unity in value. In fact is sometimes referred toas the unit variance. If the unit variance is not close to unity, it becomes diff
30、icult to give much credibility to the subsequent error propagation. Rearrange the condition equations so that the form is AV+BA = F with F=D-AL. The least squares solution is defined as that solution which minimizes the function 10 Provided by IHSNot for ResaleNo reproduction or networking permitted
31、 without license from IHS-,-,-M IL-STD-60000 1 4 = VTWV - 2KT(AV + BA - F) with respect to V and A. The vector K is the Lagrange multipliers which accomplishes this minimization. Therefore, to mimimize 0, must be satisfied. Thus, aqtav = VTW - KTA = o, and aqm = -28 = o along with the condition equa
32、tions forms the system of equations WV - ATK = O, AV+BA = F,and BTK = O which must be solved for V, K and A. It can be shown that the solution is given by V = QLLATK, K = (AQLLAT)- /F - BA), and A = BT( AQLLAT)- BI- BT( AQLLAT)- F. Let N = BT(AQLLAT)-B 11 Provided by IHSNot for ResaleNo reproduction
33、 or networking permitted without license from IHS-,-,-_I -_- - _-_ MIL-STD-b00003 MC 7777733 0088427 m MI L-STO-60000 I and T = BT(AQLLAT)-F. The normal equations can be written as NA=T so that A = NT. The covariance matrix associated with the parameter A is determined by using the covariance propag
34、ation rule Qu = JALQLLJL where J = amL. Since A = N- BT(AQLLAT)- (O - AL), it follows that JA, = N- BT(AQLLAT)- (-A) and QM = -N- BT(AQLLAT)” AQLL-N- BT(AQLLAT)- AlT which simplifies to Q, = N-. 12 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- MIL
35、-STD-600001 MC 7777711 0088Li28 T m- MIL-STD-600001 It is often true that not all of the parameters in the state vector, A, are used for the development of a product. For example, the state vector may include both ground positions and sensor related parameters. Some products may be developed using o
36、nly the ground positions, while others may also utilize the sensor parameters. To understand this situation suppose that the state vector can be written as A = ;al and the corresponding condition equations become AV + Bi + B = F which can be written as AV+BrA =F. As before the normal equations, with
37、 . B=B BI have the form (AQLLAT)- b B To simplify the notation let We = (AQLLAT)- thus BTWeB BTWeB BTWeB BTWeB 3- BTWeF BTWeF 1 . 13 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-STD-600001 Let N=BweB , N=BW,B , - N=BWeB , f=BTWeF ,and T=BTWeF
38、, then the normal equations are Next, solve ford and and determine Qz and Qz , their respective covariance matrices. The normal equation can be written as Ni + ;i = j- (FIGURE 7) Equation (figure 7) yields = - Ni) which when substituted into equation (figure 8), yields which reduces to I.-.- .I - (F
39、IGURE 8) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-STD-b00003 MC 7779733 0088Lt30 8 MIL-STD-600001 The covariance propagation rule states that Likewise, solve for 3 using equation (figure 8), that is, - . = N-l(T - NTA) which, when substitu
40、ted into equation (figure 7), becomes - - NA + NN-l(T - NTi) = i which reduces to . - . - =(N - NN-lNT)-l(BT- NN-lBT)WeF. The covariance matrix associated with is given by thus . - - . _ Q, = (N - NN-NT)- ( BT - NN- BT)WeAQLLA - - - x (N - NN-lNT)-l (BT- NN-lBT)WJT 15 Provided by IHSNot for ResaleNo
41、 reproduction or networking permitted without license from IHS-,-,-il IL - S TD - b O O O O L MC-%7 9FE1iO G? 3 Lr E M I L- ST D-60 O O O 1 which simplifies to . - - Q; = (N-NN-NT)-. It will now be shown that these expressions for Q; and Q, correspond to the partitions of N-l. Assume that A = MT, th
42、at is - which means that -7 -1 1 O O 1 which, when expanded, gives the four equations NM + “IT= 1, ._ - NM + NM= O, -. NTM + N = O, and - - NTM + NM = I. Equation (figure 11) can be rearranged so that (FIGURE 9) (FIGURE IO) (FIGURE 11) (FIGURE 12) which, when substituted into equation (figure 9) giv
43、es NG + N(-M-INTr;/l) = 1 16 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-M IL-STD-60000 1 which, solving for M, gives Likewise, equation (figure 1 O), when rearranged, gives . - i = -N-INM which, when substituted into equation (figure 12) and sol
44、ving for M yields -. - M = (i - NTN-IN)-l. Thus it has been shown that and Q,= M. A typical method of reducing the dimension of the matrix to be inverted is to “fold“ the normal equations. This is accomplished by eliminating some of the parameters from the state vector. Assume that the normal equati
45、ons are partitioned as before, that is which when expanded gives the two equations . . - . NA+NA=T and -. NTA + NA = T. (FIGURE 13) (FIGURE 14) “Folding“ is accomplished by solving equation (figure 14) for and then substituting the resulting expression into equation (figure 13) and solving for . The
46、 resulting expression is called the folded normal equations; that is, 17 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- MIL-STD-6000NC , is given by the same expression as in the unfolded case, that is, Q, = M. Since - = M (i - NN-lT), substitution
47、 for i and 7 gives - = M (BT - NN-IBT)We(D - AL). Using the covariance propagation rule and the fact that aj M-i thus, .which reduces to as in the unfolded case. 5.7 Amlication of trianaulation outwt. For the purpose of applying this information to product evaluation, it is assumed that the vector h
48、as as its components ground location coordinates which are to be used as diagnostic control points by the production organiza- tion. The covariance matrices for and are given by Q, and Q, , respectively. It should be noted that the organization generating some specific product may not use sensor parameters, that is, the Organization will not be supplied with the vector . Those organizations that require sensor paramaters have two possible sources, either the output of triangulation in the form of aposteriori
