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5. Network adjustment with gama-local

Adjustment of local geodetic network is a classical case of adjustment of indirect observations. After estimation of approximate values of unknown parameters (coordinates of points) and linearization of functions describing relations between observations and parameters we solve linear system of equations

(1)      Ax = b + v,

where A is coefficient matrix, b is vector of absolute terms (right hand side) and v is vector of residuals. This system is (generally) overdetermined and we seek the solution x satisfying the basic criterion of Least Squares

(2)      v'Pv = min,

where P is weight matrix. This criterion unambiguously defines the shape of adjusted network.

Geodetic adjustment is traditionally computed as the solution of normal equations (2)

(a)      Nx = n,  where  N=A'PA  and  n=A'Pb.

In the case of free network, i.e. network with no fixed coordinates (or network without sufficient number of fixed coordinates), the system (1) is singular, matrix A has linearly dependent columns, and there is infinite number of solutions x.

To define a unique solution x we need to definine a set of constriant equations (inner constraints)

(b)      Cx = c

to minimize

(c)      v'Pv - 2k'(Cx - c) = min

where r is the vector of residuals, r = Ax - b, k is a vector of so called Lagrange multipliers and adjusted vector x is obtained from solution of

(d)      ( A'PA, C') (x) = (A'Pb)
         ( C,    0 ) (k) = (  c )

In gama-local a slightly different approach is used, we define a second regularization criterion as

(3)      \sum x_i^2 = min,      for all selected i

stating that at the same time with (2) we demand that the sum of squares corrections of selected parameters is minimal (corrections of unknown parameters with indexes from the set of all selected unknowns. Geometrically this criterion is equivalent to adjustment of the network according to (2) with simultaneous transformation to the selected set of fiducial points. This transformation does not change the shape of adjusted network.

Often it is advantageous to work with a homogenized system, ie. with the system of project equations in which coefficient of each row and absolute term are multiplied by square root of the weight of corresponding observation.

(4)      ~A x = ~b,

where ~A = P^1/2 A, ~b = P^1/2 A. Symbol P^1/2 denotes diagonal matrix of square roots of observation weights (or Cholesky decomposition of covariance matrix in the case of correlated observations). To criterion (2) corresponds in the case of homogenized system criterion

(5)      ~v'~v = min.

Normal equations are clearly equivalent for both systems.

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