METRIC
MIL-STD-600001
26 FEBRUARY 1990
MILITARY STANDARD
MASTER
COPY
AMSC N/A AREA MCGT
DISTRIBUTION STATEMENT A. Approved for public release; distribution
is unlimited.
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, DefenseMappingAgency,ATTN: PR,8613LeeHighway,Fairfax,VA
22031-2137 by using the self-addressed Standardization Document
Improvement Proposal (DD Form 1426) appearing at the end of this
document or by letter.
PARAGRAPH PAGE
1. SCOPE 1
1.1 Scope 1
1.2 Purpose 1
1.3 Applicability 1
2. APPLICABLE DOCUMENTS 2
2.1 Government documents 2
2.1.1 Specifications, standards, and handbooks 2
2.1.2 Other Government documents, drawings,
and publications 2
2.2 Non-Government publications 2
2.3 Order of precedence 2
3. DEFINITIONS 3
3.1 Absolute horizontal accuracy 3
3.2 Absolute vertical accuracy 3
3.3 Accu racy 3
3.4 Datum (geodesy 3
3.5 Random error 3
3.6 Relative horizontal accuracy (point-to-point) 3
3.7 Relative vertical accuracy (point-to-point) 3
3.8 Systematic error 3
4. GENERAL REQUIREMENTS 4
4.1 Accuracy requirements 4
4.2 Intended use of accuracy 4
4.3 Accuracy requirement definition 4
4.4 Formulas (simplified) 6
4.4.1 Circular error 6
4.4.2 Linear error 6
4.5 Selection of normal distribution 6
4.6 Accuracy note 6
5. DETAILED REQUIREMENTS 7
5.1 General 7
5.2 Absolute accuracy 7
5.3 Relative accuracy
5.4 Point positions
5.5 Variance-covariance matrix
5.6 Error propagation relating to triangulation
5.7 Application of triangulation output
5.8 Error propagation from sample statistics
5.9 Sample statistics when the diagnostic and product errors are
independent
5.10 Sample statistics when the diagnostic and product errors
are dependent
5.11 Summary of sample statistics methodology
5.12 Absolute accuracy computations
5.13 Point-to-point relative accuracy computations
5.14 Alternate error propagation from sample statistics
5.15 Accuracy influenced by bias
6. NOTES
6.1 International standardization agreements
6.2 International Standardization Agreements (STANAGs)
6.2.1 Quadripartite Standardization Agreements (QSTAGs)
6.2.2 Air Standardization Coordinating Committee
6.2.3 Agreements (ASCC AIR STDslSTDs/ADV PUBs)
6.2.4 International MC&G agreements
6.2.5 Executive orders
6.2.6 Inter-Agency agreements
6.2.7 Other documentation
1.1 Scope. This standard defines MC&G product accuracy and
provides a common basis for the appropriate application of these
definitions.
1.2 Purpose. The standard accuracy definitions apply uniformly
to product designers, producers and users.
1 .3 Applicability. These standards apply to both internal and
contractual development efforts by the Military Departments, Office
of the Secretary of Defense, Organization of the Joint Chiefs
of Staff and the Defense Agencies of the Department of Defense
(DoD), collectively known as DoD Components; and to all levels
involved in the preparation, maintenance of MC&G products.
3.1 Absolute horizontal accuracy. The statistical evaluation of
all random and systematic errors encountered in determining the
horizontal position of a single data point with respectto aspecified
geodetic reference datum. Expressed as a circular error at 90
percent probability.
3.2 Absolute vertical accuracy. The statistical evaluation of
all random and systematic errors encountered in determining the
elevation of a single data point with respect to Mean Sea Level
(MSL). Expressed as a linear error at 90 percent probability.
3.3 Accuracy. The degree of conformity with which horizontal positions
and vertical values are represented on a map, chart, or related
product in relation to an established standard .
3.4 Datum (geodesy). A geodetic datum is uniquely defined by five
quantities. Latitude (f), longitude (l) and geoid height (N) are
defined at the datum origin. The other two quantities defining
the datum are the semimajor axis and flattening or the semimajor
axis and the semiminor axis of the reference ellipsoid.
3.5 Random error. Errors that are not classified as blunders,
systematic errors, or periodic errors. They are numerous, individually
small, and each is likely to be positive as negative.
3.6 Relative horizontal accuracy (point-to-point). The statistical
evaluation of all random errors encountered in determining the
horizontal position of one data point with respect to another.
Expressed as a circular error over a specified distance al 90
percent probability.
3.7 Relative vertical accuracy (point-to-point). The statistical
evaluation of all random errors encountered in determining the
elevation of one data point with respect to another. Expressed
as a linear error over a specified distance at 90 percent probability.
3.8 Systematic error. An error that occurs with the same sign,
and often with a similar magnitude, in a number of consecutive
or otherwise related observations.
4.1 Accuracy requirements. Product accuracy requirements are directly
related to the intended use(s) of the MC&G product. All products
are generated with specific intended uses defined. The intended
uses determine both the required accuracy value and the accuracy
category (i.e., absolute horizontal, relative horizontal, absolute
vertical, or relative vertical). Products having multiple intended
uses must meet the accuracy requirement for the most stringent
intended use. However, not all maps or charts of the same type
are required to meet the same accuracy. For example, City Graphics
products that are not to be used for tactical land combat may
be produced with less accuracy than those to be used for tactical
and combat.
4.2 Intended use of accuracy. The intended uses of MC&G products
typically fall inlo the following five categories: planning, navigation,
target identification, gunfire support, and target positioning.
in most instances, these uses require a specific level of relative
accuracy in both the horizontal and the vertical planes. Absolute
accuracy is required mainly for precise navigation and target
positioning.
4.3 Accuracy requirement definition. MC&G product accuracy
requirements are defined in terms of both absolute (horizontal
and vertical) and relative (horizontal and vertical) components.
Relative horizontal accuracy is further defined as either point-to-point
to point-to-graticule. An intended use may require reporting accuracies
for any or all of these defi rlitions. Both absolute and vertical
accuracies are expressed in meters on the reference datum at ground
scale. These values are computed according to a specified probability
distribution and are reported at a specified confidence level.
4.5 Selection of normal distribution. The normal distribution
function was selected since it closely fits the actual observed
frequency distributions of many physical measurements and natural
phenomena. In addition it makes error analysis a more tractable
problem .
4.6 Accuracy Note. MC&G hard copy products shall carry in
the title block of the individual product a statement as to its
specific accuracy. Digital products shall have an accuracy statement
in the header information. If the product has varying accuracies,
an accuracy diagram shall be depicted and for digital products,
accuracy values shall be depicted and for digital products, accuracy
values shall be given in the sub-region of the header information.
Accuracy statements are not to be used on MC&G products with
a scale 1:1,000,000 or smaller.
5.1 General. This standard contains detailed requirements for
both the definitions and mathematics of absolute and point-to-point
(relative) accuracy. It is understood that these are the official
statistics for stating product accuracies and for specifying hardware/software
requirements when these specifications are stated in terms of
ground position accuracies. The emphasis of this standard is on
the development of the theory which defines accuracy. The application
of that theory to any given MC&G product is not presented.
5.2 Absolute accuracy. Absolute accuracy is defined as the statistic
which gives the uncertainty of a point with respect to the datum
required by a product specification. This definition implies that
the effects of all error sources, both random and systematic,
must be considered. Absolute accuracy is stated in terms of two
components, a horizontal component and a vertical component. The
horizontal absolute accuracy associated with a product is stated
as a circular error, CE, such that 90 percent of all positions
depicted by that product have a horizontal error with magnitude
less than CE. Likewise, the absolute vertical accuracy associated
with a product is stated as a linear error, LE, such that 90 percent
of all elevations depicted by the product have an error with magnitude
less than LE.
5.3 Relative accuracy. Relative accuracy is that statistic which
gives the uncertainty between the positions of two points after
the effects of all errors common to both points have been removed.
Relative accuracy is also called point-to-point accuracy. Relative
accuracy is seen to be independent of product datum in that it
is defined as the error in the components of the vector between
the two points; but is still stated in terms of a horizontal component
and a vertical component. As in the case with absolute accuracy,
the horizontal uncertainty is stated as a CE and the vertical
error is stated as a LE.
5.4 Point positions. Point positions derived from measurements
of photographic images are usually referenced to an earth fixed
Cartesian coordinate system. A variance-covariance matrix defining
the uncertainty of this computed position relative to this coordinate
system is determined by standard error propagation techniques
utilizing apriori estimates of errors associated with the computational
parameters. The apriori estimates of the errors associated with
these computational parameters are usually in the form of a variance-covariance
matrix and includes all of the covariances resulting from the
correlation of the parameters. The parameter variance-covariance
matrices used to assess product accuracies 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 least squares
triangulation to update exposure station positions and camera
attitudes.
5.5 Variance-covariance matrix. A primary goal of any evaluation
scheme should be the construction of the variance-covariance matrix
associated with any position depicted in the product. The generation
of such matrices will likely utilize standard error propagation
techniques and/or sample statistics resulting from the comparison
of positions extracted from the product to their known positions.
Such 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 methods 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 a covariance matrix consider two vectors, denoted by
U and V, whose components are random variables. The cross-covariance
of the two vectors is defined by
E[(U - E[U]) (V- E[V])T]
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. The covariance of U is when
U = V.
Suppose that the geographic position of two points, and their
cross-covariance matrix has been determined. Let the two positions
be denoted by (f1, l1, h1) and (f2, l2, h2 ) Let their cross-covariance
matrix be denoted by Q such that
Q11 Q12
Q =
QT12 Q22
where
s2fi sfili sfihi
Q11 = sfili s2li slihi
sfihi slihi s2hi
where
s2fi is the variance of fi, etc.,
sfili is the covariance of fi and li, etc.
and
sf1f2 sf1l2 sf1h2
Q12 = sl1f2 sl1l2 sl1h2
sh1f2 sh1l2 sh1hi2
Methods for the determination of the cross-covariance matrix Q
will be considered. These methods, intended as guidelines only,
are somewhat generalized in the sense 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 propagation relating to triangulation. First consider
the case involving triangulation. 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) + BD = D
where A and B are coefficient matrices,
D is a vector of constants,
L is a vector of observations,
V is a vector of residuals, and
D is a vector of parameters usually referred to as the state vector
In addition, define QLL as the covariance matrix associated with
the observational vector L and define W as the observational weight
matrix, that is,
W = Q-1LL
A few words relative to the observations and state vector regarding
their respective weights are in order. Assume that the unknown
state vector, D, 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 the 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
s2o = VWV
R
where R is the degrees of freedom associated with 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 s2o is sometimes referred
to as the unit variance. If the unit variance is not close to
unity, it becomes difficult to give much credibility to the subsequent
error propagation.
Rearrange the condition equations so that the form is
AV+BD = F
with
F = D-AL.
The least squares solution is defined as that solution which minimizes
the function
f = VTWV - 2KT(AV + BD - F)
with respect to V and D. The vector K is the Lagrange multipliers
which accomplishes this minimization. Therefore, to minimize f,
f/V = 0 and f/D = 0
must be satisfied. Thus,
f/V = 2VTW - 2KTA = 0,
and
f/D = -2KTB = 0
along with the condition equations forms the system of equations
WV - ATK = 0,
AV + BD = F, and
BTK = 0
which must be solved for V, K and D. It can be shown that the
solution is given by
V = QLLATK,
K = (AQLLAT)-1 (F - BD), and
D = [BT(AQLLAT)-1B]-1 BT(AQ AT)-1 F
Let
N = BT(AQLLAT)-1B
T = BT(AQLLAT)-1F.
The normal equations can be written as
ND = T
so that
D = N-1T.
The covariance matrix associated with the parameter D is determined
by using the covariance propagation rule
QDD = JDLQLLJTDL where
JDL = D/L.
D = N-1BT(AQLLAT)-1 (D -AL),
it follows that
JDL = N-1BT(AQLLAT)-1 (-A)
and
QDD = -N-1BT(AQLLAT)-1 AQLL[-N-1BT(AQLLAT)-1 A]T
which simplifies to
QDD = N-1.
Let
N = BTWeB ,
N = BTWeB ,
N = BTWeB ,
T = BTWeF , and
T = BTWeF ,
then the normal equations are
N N D T
NT N D T
Next, solve for D and D and determine QDD and QDD, their respective
covariance matrices. The normal equation can be written as
ND + ND = T (FIGURE 7)
and
NTD + ND = T (FIGURE 8)
Equation (figure 7) yields
D = N-1 (T - ND)
which when substituted into equation (figure 8), yields
NTN-1(T- ND) + ND = T
which reduces to
D = (N - NTN-1N)-1 (BT - NTN-1BT)We(D-AL)
The covariance propagation rule states that
QDD = [D/L] QLL [D/L]T
where
D/L = -(N - NTN-1N)-1 (BT- NTN-1BT)WeA
thus,
QDD = (N - NTN-1N)-1
Likewise, solve for D using equation (figure 8), that is,
D = N-1(T- NTD)
which, when substituted into equation (figure 7), becomes
ND + NN-1(T - NTD) = T
which reduces to
D = (N - NN-1NT)-1 (BT- NN-1BT)WeF .
The covariance matrix associated with D is given by
QDD = [D/L] QLL [D/L]T
where
D/L = -(N - NN-1NT)-1 (BT - NN-1BT)WeA
thus
QDD = (N - NN-1NT)-1 (BT - NN-1BT)WeAQLLA
x [(N - NN-1NT)-1 (BT - NN-1BT)We]T
which simplifies to
QDD = (N-NN-1NT)-1.
It will now be shown that these expressions for QDD and QDD correspond
to the partitions of N-1. Assume that D = MT, that is
D M M T
D M M T
or
M = N-1
which means that
N N M M 1 0
NT N MT M 0 1
which, when expanded, gives the four equations
NM + NMT= 1, (FIGURE 9)
NM + NM = 0, (FIGURE 10)
NTM + NMT = 0, and (FIGURE 11)
NTM + NM = 1. (FIGURE 12)
Equation (figure 11) can be rearranged so that
MT = N-1 NTM
which, when substituted into equation (figure 9) gives
NM + N(-N-1NTM) = 1
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 only the ground
positions, while others may also utilize the sensor parameters.
To understand this situation suppose that the state vector can
be written as
D
D =
D
and the corresponding condition equations become
AV + BD + BD = F.
which can be written as
AV + [B B] D = F.
D
As before the normal equations, with
B=[B B]
have the form
BT [~ (AQLLAT)-1 [B B] D = BT (AQLLAT)-1F.
BT D BT
To simplify the notation let
We = (AQLLAT)-1
thus
BTWeB BTWeB D BTWeF
BTWeB BTWeB D BTWeF