1. science
2. mathematics
3. statistics

Data Integration with
Adjustment Techniques
Dr.-Ing. Frank Gielsdorf
University of Technology Berlin
Department for Geodesy and Adjustment Techniques
Email: [email protected]
Web: www.survey.tu-berlin.de
Data Integration with Adjustment Techniques
Contents
2
Concepts and strategies for PAI ......................................................................................... 1
2.4
Data Integration with Adjustment Techniques........................................................... 1
2.4.1
Introduction ........................................................................................................ 1
2.4.2
Estimation of parameters (Least Squares Method) ............................................ 2
2.4.2.1
Arithmetic Mean ............................................................................................ 4
2.4.2.2
Weighted Arithmetic Mean............................................................................ 6
2.4.2.3
Adjustment with Several Unknown Parameters............................................. 7
2.4.3
The Law of Error Propagation ........................................................................... 9
2.4.3.1
Error Propagation for Linear Functions ....................................................... 10
2.4.3.2
The Importance of Covariances ................................................................... 11
2.4.3.3
Adjustment and Error Propagation............................................................... 13
2.4.4
PAI as Adjustment Problem............................................................................. 14
2.4.4.1
Improving Absolute Geometry..................................................................... 16
2.4.4.2
Improving Relative Geometry...................................................................... 19
2.4.5
Appendix: Information to Matrix Operations in MS Excel ............................. 19
Data Integration with Adjustment Techniques
1
2 Concepts and strategies for PAI
2.4 Data Integration with Adjustment Techniques
2.4.1 Introduction
The geometrical properties of objects in GIS are almost exclusively described by point
coordinates referring to a global reference frame. But it is impossible to measure these point
coordinates directly; they are the result of a calculation process. The input parameters of these
calculations are measured values. Evan GPS receivers do not measure coordinates directly but
calculate them from distance measurements to satellites.
There exist several types of measured values, for instance distances, directions, local
coordinates from maps or orthophotos etc. Mostly single measured values are grouped to sets
of local coordinates. So, the measured values of a total station – horizontal direction, vertical
direction and distance - can be seen as a set of spherical coordinates, row and column of
digitized pixels as Cartesian coordinates in a local reference frame etc. The final aim of the
most evaluation processes in surveying is the determination of point coordinates in a global
reference frame.
But measured values have two essential properties:
1. They are random variables. Because it is impossible to measure a value with arbitrary
accuracy, which leads to the fact that any measured value contains some uncertainty.
2. They are redundant. Commonly there exist more measured values then necessary to be
able to calculate unique point coordinates.
A function of random variables results again in a random variable. Because of point
coordinates are functions of measurement values they are like them random variables. The
uncertainties contained in measured values lead necessarily to uncertainties in point
coordinates.
For the unique determination of a number of coordinates the exact same number of measured
values is necessary. For example consider Figure 1.
N
d1
d2
A
B
Figure 1: Unique Arc Section
The positions of the control points A and B should be known. The distances d1 and d2 from the
control points to the new point N were measured. The position of the new point can be
calculated by intersection of arcs. The two unknown coordinate values of the new point xN
and yN can be determined unique with the two measured values d1 and d2.
But what happens if we measure a third distance d3 from a control point C to N?
Data Integration with Adjustment Techniques
2
C
d3
N?
d1
d2
A
B
Figure 2: Ambiguous Arc Section
A geometrical construction with a pair of compasses would yield a small triangle. The size of
this triangle depends of the magnitude of the uncertainties in the measured values. This means
for the calculation that its result depends on which measured values were used. A calculation
with the distances d1 and d2 yields another result then a calculation with the distances d2 and
d3 .
The example raises several questions. How can one get a unique result from redundant
measured values containing uncertainties? How can one quantify the accuracy of the
measured values on one and of the result on the other hand? The reply to these questions is
the objective of adjustment theory.
Objectives of Adjustment Theory:
1. Determination of optimal and unique output parameters (coordinates) from input
parameters (measured values) which are redundant random variables under
consideration of their accuracy.
2. Estimation of accuracy values for the output parameters.
3. Detection and localisation of blunders.
But where is the relevance of adjustment techniques for the positional accuracy improvement
in GIS? The coordinates in GIS result from the evaluation of measured values. In a first step
these measured values often were local coordinates of digitized analogue maps which were
transformed into a global reference frame. These so determined global coordinates describe
the geometry of the GIS objects unique whereby the coordinates have to be addressed as
random variables. During the process of PAI new measured values (even global coordinates
can be seen as special measured values) with higher accuracy are introduced. The new
measured values are redundant to the already existing coordinates. Therefore, the
determination of new global coordinates with improved positional accuracy is a typical
adjustment problem.
But before the application of adjustment techniques for PAI will be presented in detail it is
necessary to consider the basics of the adjustment theory.
2.4.2 Estimation of parameters (Least Squares Method)
This section describes the process of parameter estimation on the basis of the least square
method. Figure 3 shows a symbolic representation of an adjustment process.
Data Integration with Adjustment Techniques
x1
x2
…
xu
l1
l2
observations
l
3
unknown
parameters
x
Adjustment
…
v1
v2
…
vn
ln
residual errors
v
Figure 3: Symbolic Representation of an Adjustment
Input parameters are measured values called observations. Output parameters are the so called
unknown parameters (mostly coordinate values) and the residual errors of the observations.
The number of observations is named n, those of the unknown parameters is named u. Typical
for an adjustment problem is the fact that the number of observations is greater than the
number of unknown parameters. The difference between both values is conterminous to the
number of supernumerary observations and is called the redundancy r.
r = n-u
Observations are signed with the symbol l, unknown parameters with the symbol x and
residual errors with the symbol v. To be able to formulate adjustment problems in a clear way
it is necessary to use the vector/matrix notation. The observations, unknown parameters and
residual errors can then be grouped in vectors.
⎛ v1 ⎞
⎛ x1 ⎞
⎛ l1 ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ v2 ⎟
⎜ x2 ⎟
⎜ l2 ⎟
l = ⎜ ⎟, x = ⎜ ⎟, v = ⎜ ⎟
M
M
M
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜v ⎟
⎜x ⎟
⎜l ⎟
⎝ n⎠
⎝ u⎠
⎝ n⎠
In that course we want only regard the adjustment with observation equations. Basis of that
model is the presentation of the observations as explicit functions of the unknown parameters.
The observation equations in matrix notation are
l + v = Ax
or transformed
v = Ax − l .
(1)
The matrix A in this observation equation system is called configuration matrix or design
matrix. It describes the linear dependency between observations and unknown parameters.
The observation equation system has an infinite number of solutions. To gain the optimal
solution it is necessary to formulate an additional constraint. This additional constraint is the
demand that the square sum of the residual errors should be minimal. In matrix notation it is
!
v T v = min .
Data Integration with Adjustment Techniques
4
With this constraint it is possible to formulate an extreme value problem and to derive an
optimal and consequently unique solution for the unknown parameters:
(
x = AT A
)
−1
AT l
(2)
The equation system (2) is called the normal equation system. After solving the normal
equations the residual errors can be calculated by insetting of x in the observation
equations (1).
2.4.2.1 Arithmetic Mean
The adjustment theory is very complex and it is impossible to impart its whole content during
this course. Therefore we will begin with very simple special examples and try to generalise
them step by step.
The simplest case of an adjustment is the calculation of an arithmetic mean value:
We consider two points A and B. The distance between these two points was measured 10
times. The results of the measurement are 10 observation values which we call l1…l10. In
demand is the unknown distance between A and B which we call x. Now, it would be possible
to solve the problem by formulation of the equation
x = l3 ,
all the other observations would be redundant in that case. For each observation we could
calculate a residual error vi.
v1 = x − l1
v2 = x − l 2
M
v10 = x − l10
(3)
The residual error v3 would have the value 0 whilst the other residuals would have values non
equal 0. The value of x depends in that case on which observation we used to its
determination. But this result is not optimal. The demand for an optimal result can be
formulated as follows:
Find exactly that solution of x for which the sum of the squares of the
residual errors vi becomes minimal!
The first step of calculating an optimal x is the formulation of observation equations.
v = Ax − l
In our special case the configuration matrix A has a very simple structure with just one
column filled with ones and the vector x has just one row.
⎛1⎞
⎜ ⎟
⎜1⎟
A = ⎜ ⎟ x = (x )
M
⎜ ⎟
⎜1⎟
⎝ ⎠
Data Integration with Adjustment Techniques
5
With A and l we are able to calculate x.
x = ( A T A) −1 A T l
We want to apply this formula now for our special problem of arithmetic mean. The matrix
product ATA is 10.
(1
1 1 1 1 1 1 1 1 1)
⎛1⎞
⎜ ⎟
⎜1⎟
⎜1⎟
⎜ ⎟
⎜1⎟
⎜1⎟
⎜ ⎟
⎜1⎟
⎜ ⎟
⎜1⎟
⎜1⎟
⎜ ⎟
⎜1⎟
⎜1⎟
⎝ ⎠
(10)
The product ATA is called the normal equation matrix and is signed with the symbol N. We
have the special case of an N-matrix with one row and one column what is conterminously to
a scalar. The inverse (ATA)-1 of ATA is equal to the reciprocal of the scalar 10.
( A T A) −1 =
1
10
The expression (ATA)-1 respectively N-1 is the cofactor matrix of the unknown parameters and
is signed with the symbol Qxx.
The expression ATl results in the sum of the Observations
AT l =
(1
1 L 1)
⎛ l1 ⎞
⎜ ⎟
⎜ l2 ⎟
⎜ M ⎟
= ∑ li
⎜ ⎟
⎜l ⎟
⎝ 10 ⎠
(l1 + l 2 + L + l10 )
so that x results in
x=
1
⋅ (l1 + l 2 + L + l10 ) .
10
As we can see the solution for the least squares approach is the same as for the arithmetic
mean.
Exercise 1
- Name the given array of observations l.
- Enter the array with the configuration matrix A and name it A.
Data Integration with Adjustment Techniques
6
- Calculate the normal equation matrix N = ATA with =mmult(mtrans(A);A) and name
the resulting array N.
- Calculate the cofactor matrix of parameters by inverting the normal equation matrix
Qxx = N-1 with =minv(N) and Name the resulting array Qxx.
- Calculate the matrix product ATl with =mmult(mtrans(A);l) and name the resulting
array Atl.
- Calculate the parameter vector x = QxxATl with =mmult(Qxx;Atl) and name the
resulting array (one cell) x.
- Calculate the residual vector v = Ax-l with mmult(A;x)-l.
2.4.2.2 Weighted Arithmetic Mean
In the arithmetic mean example we assumed that all observations have the same accuracy. But
in reality mostly this assumption is not applicable. In the general case, the ingoing
observations have different accuracies e.g. because of different measuring devices. For that
case it is necessary to modify the adjustment model of the preceding section. We have now to
introduce a weight for each observation. The observation weight steers the influence of each
observation to the resulting unknown parameters. The higher the weight the higher is the
influence of the corresponding observation.
The weight of an observation is a function of its standard deviation. The standard deviation
quantifies the accuracy of a value. The standard deviations of the observations are mostly
known before the adjustment. Usually a standard deviation is signed with the symbol σ. The
square of a standard deviation σ2 is called variance. The meaning of a standard deviation will
be explained in following sections. The formula for a weight pi of an observation is
σ 02
.
σ i2
pi =
In that formula means
pi
–
weight of observation i
σi2
–
variance of observation i
σ02
–
variance of unit weight
The variance of the unit weight is a freely selectable constant. Commonly one chooses the
value for σ0 in that way that the weights get an average value of 1.
If one introduces weighted observations in an adjustment the least squares constraint changes
from
!
v T v = min
to
!
v T Pv = min .
(4)
In expression (4) P is the weight matrix which is a diagonal matrix of dimension n × n with
the observation weights on the principle diagonal.
Data Integration with Adjustment Techniques
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⎛ p1
⎞
⎜
0 ⎟
p2
⎟
P=⎜
⎜
⎟
O
⎜ 0
⎟
pn ⎠
⎝
The formula for the calculation of the unknown parameters x changes to
(
x = A T PA
)
−1
A T Pl
and the formula for the cofactor matrix of the unknown parameters Qxx to
(
Q xx = A T PA
)
−1
.
Exercise 2
- Define an arbitrary value for the standard deviation of unit weight σ0.
- Calculate the weight of each observation on the basis of its standard deviation
σ 02
pi = 2 .
σi
- Fill an array of the size 10 × 10 with zeros. Enter the weights in the principal diagonal
of the array. Name the array P.
- Calculate the normal equation matrix N = ATPA.
(
- Calculate the cofactor matrix of the unknown parameters Q xx = A T PA
)
−1
.
- Calculate the vector ATPl.
- Calculate the parameter vector (one cell) x = QxxATPl.
- Calculate the residual vector v = Ax-l.
2.4.2.3 Adjustment with Several Unknown Parameters
The arithmetic mean represents a special case of adjustment with just one unknown
parameter. In the general case the number of unknown parameters is greater then one. This
case should be explained with a simple example again.
d2
d1
0
A
1
d3
2
d4
3
d5
4
d6
5
d7
6
d8
7
d9
8
d10
9
B
x
Figure 4: Points in a One Dimensional Coordinate System
Figure 4 shows points in a one dimensional coordinate system. The coordinates xA and xB of
the control points A and B are known. The distances between the points d1…d10 were
measured. In demand are the coordinates of the new points x1…x9. The standard deviations are
proportional to the corresponding distance:
B
σ i = 0.01 ⋅ d i
(5)
Data Integration with Adjustment Techniques
8
In a first step we have to formulate the observation equations:
d1 + v1
d 2 + v2
= + 1 ⋅ x1
= − 1 ⋅ x1
+ v3
+ v4
+ v5
+ v6
+ v7
=
=
=
=
=
d 8 + v8
d 9 + v9
d10 + v10
=
=
=
d3
d4
d5
d6
d7
− xA
+ 1 ⋅ x2
− 1 ⋅ x2
+ 1 ⋅ x3
− 1 ⋅ x3
+ 1 ⋅ x4
− 1 ⋅ x4
+ 1 ⋅ x5
− 1 ⋅ x5
+ 1 ⋅ x6
− 1 ⋅ x6
+ 1 ⋅ x7
− 1 ⋅ x7
+ 1 ⋅ x8
− 1 ⋅ x8
+ 1 ⋅ x9
− 1 ⋅ x9
+ xB
If we shift the observations to the right side of the equation system we get:
v1
v2
= + 1⋅ x1
= − 1⋅ x1
v3
v4
v5
v6
v7
=
=
=
=
=
v8
v9
v10
=
=
=
− x A − d1
− d2
+ 1⋅ x 2
− 1⋅ x 2
+ 1⋅ x 3
− 1⋅ x 3
+ 1⋅ x 4
− 1⋅ x 4
+ 1⋅ x5
− 1⋅ x5
+ 1⋅ x 6
− 1⋅ x 6
− d3
− d4
− d5
− d6
− d7
+ 1⋅ x 7
− 1⋅ x7
+ 1⋅ x8
− 1 ⋅ x8
+ 1⋅ x9
− 1⋅ x 9
− d8
− d9
+ x B − d 10
From this equation system we can directly derive the structure of our matrices:
⎞
⎛1
⎟
⎜
⎟
⎜ −1 1
⎟
⎜
1
1
−
⎟
⎜
−1 1
⎟
⎜
⎟
⎜
−1 1
⎟
⎜
A=
⎟
⎜
−1 1
⎟
⎜
−1 1
⎟
⎜
⎟
⎜
−1 1
⎟
⎜
−1 1 ⎟
⎜
⎜
− 1⎟⎠
⎝
⎛ x1 ⎞
⎜ ⎟
⎜ x2 ⎟
⎜x ⎟
⎜ 3⎟
⎜ x4 ⎟
x = ⎜⎜ x5 ⎟⎟
⎜ x6 ⎟
⎜ ⎟
⎜ x7 ⎟
⎜ x8 ⎟
⎜⎜ ⎟⎟
⎝ x9 ⎠
⎛ d1 + x A ⎞
⎟
⎜
⎜ d2 ⎟
⎟
⎜ d
3
⎟
⎜
⎜ d4 ⎟
⎟
⎜ d
5
⎟
⎜
l=
⎜ d6 ⎟
⎟
⎜
⎜ d7 ⎟
⎜ d8 ⎟
⎟
⎜
⎜ d9 ⎟
⎜d − x ⎟
B⎠
⎝ 10
For the calculation of the parameter vector x the weight matrix P is still needed. The weights
can be calculated by formula (5).
Data Integration with Adjustment Techniques
⎛ p1
⎜
⎜
⎜
⎜
⎜
⎜
P=⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
p2
p3
p4
p5
p6
p7
p8
p9
9
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
p10 ⎟⎠
Now the parameter vector and the residual vector can be calculated:
(
x = A T PA
)
−1
A T Pl
v = Ax − l
Exercise 3
- Calculate the weight of each observation on the basis of its standard deviation.
- Build the configuration matrix A.
- Build the weight matrix P.
- Calculate the parameter vector x.
- Calculate the residual vector v.
2.4.3 The Law of Error Propagation
As already mentioned above adjustment techniques have two main objectives. One task is the
estimation of optimal parameter values but the second task is the estimation of the parameter
accuracy. The Basis for that second task is the law of error propagation.
In mathematical statistics as well as in the adjustment theory the measure for the scattering of
a random value is its standard deviation. The meaning of that value should be explained with
an example:
If we measured a distance d with an observation value of 123.45m then this value has to be
seen as a random sample of an infinite population of observations. The average value of this
infinite quantity of observations is the so called true value. A standard deviation of
σd = ± 2cm means that the true but unknown value of the distance d is with a probability of
67% in the interval (123.45m-2cm)…(123.45m+2cm). The standard deviation is a measure of
the average deviation of the observed value from the theoretical true value. Usually the
standard deviation is signed with σ. The square of the standard deviation is called variance
and signed with σ2.
The standard deviation is not identical to the maximal deviation from the theoretical true
value. But there exist a rule of thumb for the ratio of both values:
maximal deviation ≈ 3 * standard deviation
The input parameters of an adjustment are observations which are in the sense of
mathematical statistics random samples of a population. The true value λi of an observation li
Data Integration with Adjustment Techniques
10
is unknown and exist only in theory but its standard deviation σi is mostly known. The output
parameters of an adjustment are the parameters xj and the residual errors vi. Parameters as
well as residual errors are functions of the observations and therefore also random variables.
The law of error propagation describes the propagation of accuracies for linear functions of
random variables. Applying this law to an adjustment it is possible to calculate the standard
deviations of the unknown parameters and those of the residual errors.
2.4.3.1 Error Propagation for Linear Functions
If we have a linear function of random variables with the structure
x = f1 ⋅ l1 + f 2 ⋅ l 2 + L + f n ⋅ l n
and the standard deviations σ1…σn of the random variables l1…ln are known, then the standard
deviation of the parameter x can be calculated by the formula
σ x2 = f12 ⋅ σ 12 + f 22 ⋅ σ 22 + L + f n2 ⋅ σ n2
(6)
The simplest case of a linear function is a sum. For a sum of random variables
n
x = ∑ li = l1 + l 2 + L + l n
i =1
the formula for the calculation of its standard deviation is
n
σ x2 = ∑ σ i2 = σ 12 + σ 22 + Lσ n2 ⇒ σ x = σ 12 + σ 22 + Lσ n2
i =1
In case of a difference the application of (6) yields
x = l1 − l 2
⇒ σ x2 = σ 12 + σ 22
(7)
Consider Figure 5 to have an example.
d2
d1
0
A
1
d3
2
d4
3
d5
4
d6
5
d7
6
d8
7
d9
8
9
x
Figure 5: Addition of Distances
We see our one dimensional coordinate system of the previous section. The control point A is
fix and we want to calculate the coordinates of the new points by adding the distances.
x1 = x A + d1
x 2 = x A + d1 + d 2
L
x9 = x A + d1 + d 2 + L + d 9
We apply now the law of error propagation to these functions and get:
Data Integration with Adjustment Techniques
11
σ x21 = σ d21
σ x22 = σ d21 + σ d2 2
L
σ x29 = σ d21 + σ d2 2 + L + σ d29
Exercise 4
- Calculate the standard deviations σ x1 Kσ x 9 .
- Calculate the standard deviation for the coordinate difference x8-x7.
- Compare the standard deviation of the coordinate difference x8-x7 with that of the
distance d8.
2.4.3.2 The Importance of Covariances
In exercise 4 we calculated the standard deviation of the difference x8-x7 and compared it with
the standard deviation of the distance d8. As we could see the values are not identical. Why?
The answer is that we neglected the covariance between the random values x7 and x8. But
before we explain calculations with covariances we want to interpret the standard deviations
of the distances di and those of the coordinates xi. The standard deviations of the coordinates
σxi represent the absolute accuracy of the coordinates in relation to the reference frame. On the
other hand the standard deviations of the distances σdi represent the relative accuracy of the
coordinates related to each other.
But which meaning has the covariance? If two calculated random values are functions of
partial the same random variable arguments they are stochastically dependent. The degree of
their stochastical dependency is quantified by their covariance.
The parameters x7 and x8 are stochastically dependent because they are functions of partial the
same random variables. The distances d1…d7 are arguments of both functions; just d8 is an
argument of only one of the two functions.
But how can we regard these dependencies in the error propagation? This problem can be
solved by a generalisation of the law of error propagation. The general form of the law of
error propagation can be represented in matrix notation. If there is a system of linear
equations describing the functional dependency of parameters xj on the arguments li
x = F⋅l
(8)
and the standard deviations of the arguments li are known then the variances and covariances
of the parameters xj can be calculated by the formula
C xx = F ⋅ C ll ⋅ F T .
In this formula the functional matrix F contains the coefficients of the linear functions. The
matrix Cll is called the covariance matrix of observations and contains the variances of
observations on its principal diagonal and their covariances on its secondary diagonals. In the
most common case of stochastically independent observations Cll is a diagonal matrix. Cxx is
the covariance matrix of the unknown parameters and contains their variances and
covariances.
Data Integration with Adjustment Techniques
⎞
⎛ σ l21
⎟
⎜
2
⎟
⎜
σ l2
C ll = ⎜
⎟
O
⎟
⎜
2 ⎟
⎜
σ ln ⎠
⎝
C xx
12
⎛ σ x21
cov( x1 , x 2 ) L cov( x1 , xu ) ⎞
⎟
⎜
⎟
⎜ cov( x1 , x 2 )
σ x22
M
=⎜
⎟
M
O
M
⎟
⎜
2
⎟
⎜ cov( x , x )
L
L
σ xu
1 u
⎠
⎝
Covariance matrices are always quadratic and symmetric.
Now, we want to apply the general law of error propagation to our example. At first we have
to build the functional matrix F. Because the functions are simple sums F contains just ones
and zeros.
⎛1
⎜
⎜1
⎜1
⎜
⎜1
F = ⎜1
⎜
⎜1
⎜
⎜1
⎜1
⎜⎜
⎝1
1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
⎛ σ d21
⎞
⎜
⎟
⎜
σ d2 2
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎟ C =⎜
ll
⎜
⎟
⎜
⎟
⎜
⎟
1
⎜
⎟
⎜
⎟
1 1
⎜
⎟⎟
⎜
1 1 1⎠
⎝
σ d23
σ d2 4
σ d25
σ d26
σ d27
σ d28
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
σ d29 ⎟⎠
In the result of calculation we get the covariance matrix Cxx which is fully allocated.
Now, we can use elements of Cxx for a further calculation to get the standard deviation of the
coordinate difference x8-x7. The functional matrix F for that case is simple again:
F = (− 1 1)
The Cll matrix contains the variances of x7 and x8 as well as their covariance from the
previous calculation:
⎛ σ x27
cov( x7 , x8 ) ⎞
⎟
C ll = ⎜⎜
2
⎟
cov(
x
,
x
)
σ
7
8
x8
⎠
⎝
If we solve the matrix equation for the general law of error propagation in a symbolic way
then we get the expression
σ x28− x 7 = σ x27 + σ x28 − 2 ⋅ cov( x7 , x8 ) .
As we can see this formula which does not neglect the covariance between dependent random
variables yields the right result.
Exercise 5
- Build the functional matrix F for the coordinates x1…x9 as functions of the distances
d1…d9.
- Build the covariance matrix Cll of the observations d1…d9.
Data Integration with Adjustment Techniques
13
- Calculate the covariance matrix Cxx of the coordinates x1…x9.
- Calculate the standard deviations σx1...σx9 of the coordinates x1…x9.
- Build the functional matrix Fd for the coordinate difference x8-x9.
- Calculate the variance and the standard deviation of the coordinate difference x8-x9.
- Compare the calculated value with σd8.
2.4.3.3 Adjustment and Error Propagation
If we consider the calculation formula for the unknown parameters in an adjustment then we
can see that the parameters are linear functions of the observations.
(
)
−1
x = A T PA A T P ⋅ l
1442443
F
(
)
−1
The expression A T PA A T P in this formula is equivalent to the functional matrix F in
formula (8). This approach allows for the application of the law of error propagation to
calculate the covariance matrix of the unknown parameters.
Basis of further calculations is the empirical standard deviation of unit weight. The formula
for its calculation is
s0 =
v T Pv
,
r
where v means the residual vector an r the redundancy. The redundancy is the number of
supernumerary observations. The value equals the difference between the number of
observations n and the number of unknown parameters u.
r = n−u
The value s0 can be interpreted as the empirical standard deviation of an observation with a
weight p = 1.
A standard deviation σ should not be mistaken with an empirical standard deviation s. A
standard deviation is a constant value and it is an input parameter of an adjustment
calculation; whereas an empirical standard deviation is a random variable which is estimated
as an output parameter of an adjustment. After an adjustment calculation the empirical
standard deviation s0 should be approximately equal to the standard deviation σ0 what means
that the quotient s0/σ0 should be in the interval 0.7…1.3. Is s0 too small that means that the a
priori supposition of observation accuracy (σi) was too pessimistic. A too large s0 often
indicates that there is a blunder among the observations.
The complete derivation of the formula for the covariance matrix of unknown parameters is
too complex to be presented in this course. Therefore, the formula is just given.
(
C xx = s02 ⋅ Q xx = s02 ⋅ AT PA
)
−1
Exercise 6
This exercise is based on the result of the adjustment of exercise 3.
- Calculate the product vTPv with =mmult(mtrans(v);mmult(P;v))
Data Integration with Adjustment Techniques
- Calculate the empirical variance of the unit weight
s 02
14
v T Pv
=
r
- Calculate the empirical standard deviation of unit weight s 0 = s 02 =
v T Pv
r
- Calculate the covariance matrix of parameters Cxx = s02Qxx
- Calculate the empirical standard deviations of the coordinates x7 and x8
- Calculate the empirical standard deviation of the coordinate difference x8-x7
2.4.4 PAI as Adjustment Problem
In this section we want to give a typical example for a PAI adjustment problem. At first we
consider the workflow from map digitalization to improved global coordinates under
adjustment aspects. Then, we simplify the example to a one dimensional problem and
reproduce the process with a concrete calculation.
Local Coordinates
xl, yl
Map Digitization
Control Points
Transformation
Global Coordinates
xg, yg
GPS Measurement
GPS Coordinates
xGPS, yGPS
PAI
Improved
Global Coordinates
xg, yg
Figure 6: PAI Workflow
Map Digitization
The result of scanning a map is a raster image with a row-column coordinate system. The
coordinates of specific points (building corners, boundary points e.t.c.) are determined in the
raster system. On the basis of the scan resolution the raster coordinates can be converted into
metrical map coordinates. If the scan resolution is measured in dots per inch (dpi) then the
converting form is
x map = x raster ⋅
0.0254 m
.
resolution[dpi]
The standard deviations respectively variances of the digitized map coordinates are dependent
on scale factor and quality of the underlying map. By a rule of thumb the standard deviation
of a map coordinate value σmap is about 0.5mm.
Data Integration with Adjustment Techniques
15
The measured coordinates are random variables and they are stochastically dependent!
Why? The point positions on a map result of real world measurements and plots. Both,
measurements and plots, have been done by the principle of adjacent points. Therefore, the
coordinate values are stochastically dependent analogue the example in exercise 5.
But in general it is impossible to reconstruct a map history in detail and therefore the
covariances between digitized coordinates are not known. The outcome of a digitalization
process is an observation vector lr with map coordinates and a covariance matrix of these
observations Cll with known principal diagonal but unknown covariances.
Transformation
Map coordinates are transformed in a higher-level global reference frame. Control points with
known coordinates in the local map coordinate system and the global reference frame are used
to determine the necessary transformation parameters. The general transformation approach is
x global = t + R ⋅ x map
with
xglobal: vector of global coordinates
t:
translation vector of transformation
R:
rotation matrix of transformation
xmap:
vector of map coordinates
As we can see the global coordinates are linear functions of map coordinates. Applying the
law of error propagation to this equation it is possible to calculate the standard deviations of
global coordinates whilst their covariances remain unknown.
GPS Measurement
For a number of points global coordinates with higher accuracy are determined by GPS
measurements. Each GPS point yields two redundant coordinates.
PAI
If the global coordinates of a GIS would be stochastically independent then we could just
exchange the less accurate coordinates by more accurate ones. But such a proceeding would
lead to a violation of geometrical neighbourhood relationships.
Data Integration with Adjustment Techniques
Two Data Sets Before
Integration
Neglected Correlations
16
Considered Correlations
Figure 7: PAI Neglecting and Considering Correlations
Figure 7 illustrates the necessity to consider stochastical dependencies between coordinates of
adjacent points. In the presented example coordinates with higher accuracy for the building
corners were determined but not for the adjacent tree. If the correlations remain unconsidered
and the coordinates of the building are exchanged then the tree seems to stand inside the
building after PAI. But if the stochastical dependencies are considered the tree is shifted with
the building during PAI.
2.4.4.1 Improving Absolute Geometry
But how can correlations be quantified and taken into account? One option is the introduction
of artificial covariances. These can be calculated as functions of the distances between always
two points. The smaller the distance between two points the bigger becomes their covariance.
But in practice this approach is difficult to handle. Often there are more then 100.000 points
to be processed what would lead to extremely large covariance matrices.
A more practical solution is the introduction of pseudo observations. In that approach
stochastical dependencies between GIS coordinates are modelled by coordinate differences.
Basis of the determination of these pseudo observations is a delaunay triangulation
(see Figure 8).
Figure 8: Delaunay Triangulation
Before triangulation the point positions are expressed unique by global coordinates. This
parameterization is exchanged with pseudo observations generated from coordinate
differences of points which are adjacent in the triangle network. This new parameterization is
redundant but still consistent.
Data Integration with Adjustment Techniques
17
3
x GIS
⎛ x1 ⎞
⎜ ⎟
⎜ y1 ⎟
⎜x ⎟
⎜ 2⎟
⎜ y2 ⎟
=⎜ ⎟
⎜ x3 ⎟
⎜ y3 ⎟
⎜ ⎟
⎜ x4 ⎟
⎜y ⎟
⎝ 4⎠
2
l GIS
4
1
⎛ Δx12 ⎞
⎟
⎜
⎜ Δy12 ⎟
⎜ Δx ⎟
⎜ 23 ⎟
⎜ Δy 23 ⎟
⎜ Δx ⎟
34 ⎟
=⎜
⎜ Δy34 ⎟
⎟
⎜
⎜ Δx 41 ⎟
⎜ Δy 41 ⎟
⎟
⎜
⎜ Δx 24 ⎟
⎜ Δy ⎟
⎝ 24 ⎠
Now the observation vector lGPS is extended with GPS coordinates of higher accuracy.
l GIS
⎛ Δx12 ⎞
⎟
⎜
⎜ Δy12 ⎟
⎜ Δx ⎟
⎜ 23 ⎟
⎜ Δy 23 ⎟
⎜ Δx ⎟
34 ⎟
=⎜
⎜ Δy34 ⎟
⎟
⎜
⎜ Δx 41 ⎟
⎜ Δy 41 ⎟
⎟
⎜
⎜ Δx 24 ⎟
⎜ Δy ⎟
⎝ 24 ⎠
3
GPS
2
l GIS
4
1
GPS
⎛ Δx12 ⎞
⎟
⎜
⎜ Δy12 ⎟
⎜ Δx ⎟
23
⎟
⎜
⎜ Δy23 ⎟
⎜ Δx ⎟
34 ⎟
⎜
⎜ Δy34 ⎟
⎟
⎜
Δx41 ⎟
⎜
=⎜
Δy41 ⎟
⎟
⎜
⎜ Δx24 ⎟
⎜ Δy ⎟
24
⎟
⎜
⎜ xGPS _ 1 ⎟
⎟
⎜y
⎜ GPS _ 1 ⎟
⎜ xGPS _ 3 ⎟
⎟⎟
⎜⎜
⎝ yGPS _ 3 ⎠
This leads to an adjustment problem with the improved global coordinates as unknown
parameters and the coordinate distances and GPS coordinates as observations. The
observation equations have the structure:
Δxij + v Δx
Δy ij + v Δy
M
=
=
xGPS _ i + v x
yGPS _ i + v y
M
=
=
x j − xi
y j − yi
xi
yi
M
These equations represent the functional model of the adjustment. The stochastical model is
determined by the observation weights which are on their part functions of standard
Data Integration with Adjustment Techniques
18
deviations. The standard deviations of the GPS coordinates depend on the applied
measurement procedure and on the reproduction accuracy of the measured points. A practical
value is σxy = ±2cm. The standard deviations of the coordinate differences are functions of
those of the underlying map coordinates and of the corresponding point distances.
We want to explain this with an example. If map coordinates with a standard deviation of
σxy = ±1m are given then the formula for the calculation of a coordinate difference standard
deviation could be:
σ Δ = σ xy ⋅
d
d0
d 0 = 100m
with
In that case an observation weight is in inverse proportion to the square of the point distance.
The results of adjustment are improved coordinates for all points and their covariance matrix.
The point accuracy depends then on the distance to the new introduced GPS points.
Exercise 7
We consider a one dimensional coordinate system with 12 points.
0
1
2
3
4
5
6
7
8
9
10
11
12
x
The global point coordinates xGIS are known and were descended from a map digitization.
Their standard deviation is σGIS = ± 1m.
The coordinates xGPS of the points 2, 8 and 12 are new determined by GPS measurements. The
standard deviation of the GPS coordinates is σGPS = ± 0.02m.
GPS
0
1
2
GPS
3
4
5
6
7
8
GPS
9
10
11
12
x
- Calculate the coordinate differences ΔxGIS as pseudo observations.
- Calculate the standard deviations of pseudo observations by the formula
Δx
σ ΔGIS = σ xGIS ⋅ GIS with d0 = 10m.
d0
- Calculate the weights of pseudo observations and of GPS observations using a
standard deviation of unit weight of σ0 = ± 1m.
- Build the configuration matrix A for the observed coordinate differences and the
observed GPS coordinates.
- Build the observation vector l.
- Build the weight matrix P.
- Calculate the improved coordinates xPAI = (ATPA)-1ATPl.
- Calculate the residual vector v = AxPAI-l.
Data Integration with Adjustment Techniques
19
- Calculate the empirical variance and the empirical standard deviation of unit weight
v T Pv
s 02 =
, s 0 = s 02 .
r
- Calculate
the
empirical
(
C xx = s 02 ⋅ Q xx = s 02 ⋅ A T PA
)
−1
covariance
matrix
of
improved
coordinates
.
- Calculate the empirical standard deviations of improved coordinates s xi =
(C xx )ii .
- Draw a MS Excel diagram with the empirical standard deviations of coordinates sxi
over the coordinates xi.
2.4.4.2 Improving Relative Geometry
The accuracy improvement of GIS coordinates is not exclusively effected by the introduction
of precise global coordinates. There are also observations describing the relative geometry
between adjacent points. For instance is mostly known that sides of buildings are rectangular
or parcel limits are straight. Such geometrical constraints can be modelled by the introduction
of corresponding observations. Rectangularity for instance can be expressed by a scalar
product. The standard deviation of this scalar product observation can be calculated by an
error propagation of the corresponding point coordinates. Because of a mason can build a
house with an accuracy of approximately 2cm a standard deviation of coordinates in about the
same size would be feasible.
Exercise 8
We want to simulate such a case by introducing an accurate distance observation in the
adjustment model of exercise 7. Observed was the distance between the points 5 and 6 with
d56 = 10.34m. The standard deviation is σd56 = ± 0.02m.
- Extend the configuration matrix A, the observation vector l and the weight matrix P
by the distance observation.
- Repeat the adjustment calculation of exercise 7.
- Compare the results of the exercises 7 and 8.
2.4.5 Appendix: Information to Matrix Operations in MS Excel
To be able to use matrix operations in MS Excel it is necessary to activate the analyse
functions:
- Choose in the main menu Extras > Add-Ins
- Activate the check Box ‘Analyse Functions’ respectively keep it activated
- Press OK
Terminating the Entry of Matrix expressions
The entry of each matrix operation expression has to be terminated by keeping the keys
control and shift pressed and then pressing enter
Data Integration with Adjustment Techniques
20
Marking of Arrays
Option 1: Set the cell cursor in the left upper cell of the array, Keep the left mouse button
pressed and pull the cursor to right lower corner of the array.
Option 2: Enter the coordinates of the left upper and the right lower cell of the array separated
by a colon. Example: (G27:N40)
Transpose of a Matrix
Highlight the resulting array with the cursor.
Enter in command line =mtrans( <array> )
To solve the expression keep the keys control and shift pressed and press enter.
Product of Two Matrices
Highlight the resulting array with the cursor.
Enter in command line =mmult( <array1> ; <array2> )
To solve the expression keep the keys control and shift pressed and press enter.
Inverting of a Matrix
Highlight the resulting array with the cursor.
Enter in command line =minv( <array> )
To solve the expression keep the keys control and shift pressed and press enter.
Indicate an Array
There are two options to indicate an array in MS Excel:
1. By cell addresses
Enter the addresses of the upper left and the lower right cell separated by ‘:’ e.g. C7:F13
or highlight the array with the cursor
2. By naming the array
Mark the array with the cursor, click in the name field in the upper right corner of the
window and enter the name of the array e.g. A for the configuration matrix.
It is possible to nest the single commands e.g. =mmult(mtrans(A);mmult(P;A))