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Mathematical Models |
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| Introduction |
The least squares adjustment of a surveying network or
traverse requires that mathematical models be used to
define the relationships of the measurements made in
that network to the coordinates of the stations and
other optional "auxiliary" parameters.
For example, a simple 2D plane mathematical model that relates a distance
measurement to the coordinates of the two points between which that
distance is measured could be defined as follows (where x and y are plane
coordinates, and s is a measured distance):
[(x2 - x1)2 + (y2 - y1)2]1/2 - s1,2 = 0
Basically then, the mathematical models define how the measurements can be
calculated from the coordinates (and other optional auxiliary parameters).
We can then use these models to calculate the coordinates from the
measurements in a network adjustment. The above simple 2D model for
distances is not used by GeoLab because it is only an approximate model.
As the distance between points becomes larger, this model becomes very
inaccurate.
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| Mathematical
Models and Matrix Solutions |
Please note that this article describes GeoLab's mathematical models that
relate the measurements made in a survey to the coordinate and auxiliary
parameters we wish to solve for in a network adjustment. A related
consideration in the design of a professional-level least squares
adjustment software package such as GeoLab is the method used to solve the
resulting system of matrix equations required by the adjustment
calculations. A great deal of research has been carried out over the last
50 or so years in this branch of mathematics, and GeoLab has been designed
to especially take advantage of the results of that research as applied to
correctly solving surveying and geodetic network adjustments (I have
performed research myself in this very interesting subject). Several
methods of solving matrix equations, such as the Cholesky method, QR
factorization, Householder transformations, etc, have been developed, and
each is appropriate for various types of problems. Based on all the
research done, GeoLab uses the efficient and stable "Block Cholesky"
method to solve network matrix equations. This method has been shown to be
very stable and accurate, and it takes full advantage of the positive
definite normal equations resulting from a surveying or geodetic network
or traverse. To further optimize the matrix solution, GeoLab also uses a
proven parameter reordering method (the reverse Cuthill-McKee method). If
you wish to discuss this subject in more detail, please
contact me.
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All mathematical models used by GeoLab are geometrically rigorous (exact,
with no approximations). The extent of your network can be global and
GeoLab's models are still exact.
This article summarizes the mathematical models used in GeoLab, and
explains why they are geometrically exact and preferable in the context of
simpler models used by other less accurate software packages.
Historically, mathematical models used for network adjustments
were formulated either on a plane or on the surface of a reference
ellipsoid. To account for the fact that our measurements are made on,
or near, Earth's surface, these older mathematical models required
that we "reduce" our measurements to the surface of either a reference
ellipsoid or a map projection plane.
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As we can visualize in the simple diagram at the right, measurements
between two points 1 and 2 on the Earth's surface will not be the
same as the corresponding reduced measurements on either the map
projection plane or the surface of the reference ellipsoid.
Approximate formulas were developed to make these reductions so that
"simpler" 2D mathematical models could then be used for network
computations. |
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Examples of reductions that were required
in the older models are such things as the Earth curvature correction,
the skew normal correction, and the Laplace correction for azimuths.
In a 3D model these corrections are not required because the model
itself contains all of this information.
So, the classical method of adjusting horizontal geodetic networks
involved the reduction of measured distances, directions, angles, and
azimuths to an adopted reference ellipsoid or map projection plane.
This was done so that network computations could be done on a simpler
2D surface.
However, with the advent of 3D measurements such as GPS, along with
the development of 3D mathematical models, the apparent advantages of
the older 2D mathematical models disappeared. Not only are the 3D
models more accurate, they are simpler both conceptually and
computationally. In the case where we have a mixture of GPS and
conventional (angle, distance, azimuth, and leveled elevation
difference) measurements, any (non-GPS) station with no measurement
information for the height of that station, the height coordinate can
be held fixed in the adjustment. With the 3D mathematical models, we
can just as easily calculate the coordinates of the station markers
themselves and not the corresponding coordinates of the projection of
that point on a reference ellipsoid or map projection plane.
As our measurements become more and more accurate, and as the value of
surveyed property increases, the apparent advantages of a seemingly
simpler 2D mathematical model do not make sense any longer. The 3D
models do not use any approximations, and they are simpler. There is
therefore no need to compromise accuracy for simplicity as was done in
the past.
In the remainder of this article, we will give a brief summary of the
3D mathematical models used in GeoLab. For all the details of these
models and their development, please see the paper "Mathematical
Models for Use in the Readjustment of the North American Geodetic
Networks" by Robin R. Steeves, Technical Report Number 1, Geodetic
Survey of Canada. You can get a copy of this report by contacting the
Geodetic Survey of Canada.
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| The 3D Coordinate Systems |
Before discussing the 3D mathematical models used by GeoLab, it will
be helpful to review some of the 3D coordinate system fundamentals.
The following coordinate systems, all described below, are:
- The Conventional Terrestrial System;
- The Local Astronomic System;
- The Local Geodetic System; and
- The Ellipsoidal System.
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| The geocentric Conventional Terrestrial
(CT) System is a 3D Cartesian (X, Y, Z) system with its origin at the
geocenter (see the diagram at the right). The Z axis of the CT system
is defined by an average position of the Earth's rotation axis. The CT
X-Z plane is parallel to the Mean Greenwich Zero Meridian, and the
system is right handed.
The geocentric ellipsoidal system is also depicted in the diagram at
the right, and consists of an ellipsoid of revolution whose geometric
center is at the geocenter and whose minor semi-axis is coincident
with the CT Z-axis. The size of this reference ellipsoid is given by
its major semi-axis (equatorial radius) a = 6378137.000 m and its
minor semi-axis b = 6356752.314 m. |
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The mathematical relationship of coordinates in these two systems is
as follows:
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X
= (n
+ h) cos
f cos
l |
(1) |
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Y
= (n
+ h) cos
f sin
l |
(2) |
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Z
= (n
(1 - e2) + h)
sin f |
(3) |
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where n
= a / (1 - e2 sin2f)1/2 |
(4) |
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Note
that we are using the symbol f
for ellipsoidal latitude, l
for ellipsoidal longitude, and h for ellipsoidal
height. The
other two coordinate systems we need are the local
astronomic (LA) system and the local geodetic (LG)
system. Both of these 3D systems have their origin
at the station marker, where we make our
measurements. |
We will name the
three axes of the LA system u, v, w (as shown in the diagram at
the right), and those of the LG system x, y, z.
The LA system w-axis is coincident with the local gravity vector
at the point P, and is positive upwards. The LA u-axis points to
astronomic north, and the LA v-axis points to astronomic east to
make a left-handed 3D Cartesian system.
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The LG system is very similar to the LA system. The LG z-axis is
coincident with the ellipsoidal normal at the station marker,
and is positive upwards. The LG x-axis points to geodetic north,
and the LG y-axis points to geodetic east.
The relationship of coordinate differences in the LA system to
those in the CT system is as follows (F is astronomic latitude;
L is astronomic longitude):
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| uij =
-DXij (sinFi
cosLi) -
DYij (sinFi
sinLi) +
DZij cosFi
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(5)
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| vij =
-DXij sinLi
+ DYij cosLi |
(6)
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| wij =
DXij (cosFi
cosLi) +
DYij (cosFi
sinLi) +
DZij sinFi |
(7)
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where DXij
= Xj - Xi, DYij
= Yj - Yi, and DZij
= Zj - Zi. The
relationship of coordinate differences in the LG system to
those in the CT system is as follows:
| xij =
-DXij (sinfi
cosli) -
DYij (sinfi
sinli) +
DZij cosfi
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(8)
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| yij =
-DXij sinli
+ DYij cosli
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(9)
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zij =
DXij (cosfi
cosli) +
DYij (cosfi
sinli) +
DZij sinfi
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(10)
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As you can see, all of the above equations
are exact. Therefore we have the basis, in the above
equations, to relate a measured quantity (e.g. a distance, an
angle, a GPS vector, ISS coordinate differences, leveled
height differences, etc.) to either the LG or LA coordinate
differences between the stations involved in the measurement.
We also have the exact mathematical relationship between
either the LA or LG systems and the CT system. The above
equations form the basis from which all mathematical models
used in GeoLab are derived.
As an
example of how the mathematical models are derived, let's
consider a spatial distance measurement. The relationship
between a spatial distance Sij from point i to
point j, to the CT coordinates of the two stations is as
follows:
| {(Xj - Xi)2
+ (Yj - Yi)2 + (Zj
- Zi)2}1/2 - Sij
= 0 |
(11) |
and it's linearized version (required for
the adjustment) is given by (rS is the residual,
and mS is the misclosure):
| rS =
DXij(dXj
- dXi)/Sij+
DYij(dYj
- dYi)/Sij
+ DZij(dZj
- dZi)/Sij
+ mS |
where Sij = (DXij2
+ DYij2 +
DZij2)1/2,
and the d values are adjustment
corrections for the indicated coordinates.
Because we have the relationship between CT
coordinate differences and those in either the LA or LG
systems, we can now easily transform this observation equation
to the LG or LA systems. All the information needed to do this
has been given in this article. For the detailed development,
please have a look at the report referenced above.
As can be seen, we do not need to make reductions to our
measurements with approximate formulae at all. If you
compare the observation equations in this exact method with
the older 2D observation equations, you'll see that the 3D
models are as simple as the older, less accurate models,
especially when you consider all the approximate and involved
reduction formulae we used to use.
If
you have comments and/or suggestions regarding this article,
please
contact me.
Dr. Robin R.
Steeves
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