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`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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