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Mathematical model of geodetic network adjustment in gama-local
is defined as a set of known real-valued differentiable functions
(11) L^* = f(X^*) |
where
L^*
is a vector of theoretical correct observations and
X^*
is a vector of correct values of parameters. For the given sample set
of observations L
and the unknown vector of residuals v
we
can express the estimate of parameters X
as a nonlinear set of
equations
(12) L + v = f(X). |
With approximate values X_0
of unknown parameters
(12) X = X_0 + x |
we can linearize the equations (12)
L + v = f(X_0) + f'(X_0)x. |
yielding the linear set of equations (1)
Unknown parameters in gama-local
mathematical model are points
coordinates and orientation angles (transforming observed directions
to bearings). The observables described by functions (12) belong into
two classes
Internally in gama-local
unknown corrections to linear
observables are computed in millimeters and corrections to angular
observables in centigrade seconds. To reflect the internal units in
used all partial derivatives of angular observables by coordinates are
scaled by factor
2000/pi.
When computing coefficients of project equations (1) we expect that approximate coordinates of points are known with sufficient accuracy needed for linearization of generally nonlinear relations between observations and unknown paramters. Most often this is true but not always and generally we have to check how close our approximation is to adjusted parameters.
Generally we check linearization in adjustment by double calculation of residuals
v^I = Ax - b, v^II = ~l(~x) - l, |
Program gama-local
similarly computes and tests differences in
values of adjusted observations once computed from residuals
and once from adjusted coordinates. For measured directions
and angles gama-local
computes in addition transverse deviation
corresponding to computed angle difference in the distance
of target point (or the farther of two targets for angle).
As a criterion of bad linearization is supposed positional
deviation greater or equal to 0.0005 millimetres.
Test of linearization error *************************** Diffs in adj. obs from residuals and from adjusted coordinates ************************************************************** i standpoint target observed r difference ================================= value = [mm|cc] = [cc] == [mm]= 2 3022184030 3022724008 dist. 28.39200 -7.070 -0.003 3 3022724002 dist. 72.30700 -18.815 -0.001 7 3000001063 dir. 286.305200 11.272 -0.002 -0.001 8 3022724008 dir. 357.800600 -23.947 0.037 0.002 |
From the practical point of view it might seem that the tolerance
0.0005 mm for detecting poor linearization is too strict. Its
exceeding in program gama-local
results in repeated adjustment
with substitute adjusted coordinates for approximate. Given tolerance
was chosen so strict to guarantee that listed output results would
never be influenced by linearization and could serve for verification
and testing of numerical solutions produced by other programs.
Implicitly coordinates of constrained points are not changed in
iterative adjustments. This feature can be changed in XML input data
by setting <parameters update-constrained-coordinates = "yes" />
(see section Network parameters).
Iterated adjustement with successive improvement of approximate coordinates converges usually even for gross errors in initial estimates of unknown coordinates. If the influence of linearization is detected after adjustment, quite often only one iteration is sufficient for recovering.
For any automatically controlled iteration we have to set up certain
stopping criterion independent on the convergence and results obtained.
Program gama-local
computes iterated adjustment three times at maximum.
If the bad linearization is detected even after three readjustments it
signals that given network configuration is somehow suspicious.
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