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4.9 Test on linearization

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.

Example

 
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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