# Adjustment based on mediating observations

As a square adjustment method or in short form mediating adjustment method of the are adjustment calculation referred to in which the required quantities (unknowns) can not be measured directly but each observation is a function of these unknowns.

If the number of observations n is greater than that of the unknowns u , the problem is over-determined with the redundancy nu , and the unknown parameters are determined so that the residual sum of squares is minimal .

An observation equation is set up for each measurement , which contains the required quantities (unknowns) as parameters . In general, the functional relationship is non-linear, so that an approximate solution must be determined and the observation equations must be linearized. The unknowns are determined by inversion of a system of linear normal equations , which results from the minimum condition of the residuals and has the dimension u × u . As a result, additions to the approximate solution are obtained with which this can be improved. The steps of linearization, solving the normal equations and improving the approximate values ​​are repeated until the additions are small enough in relation to the accuracy of the parameters (iteration).

## example

The improvement equations for the measured path or arc length S and the azimuths A 1 , A 2 are to be derived between two points P 1 , P 2 of a surveying network on the earth's ellipsoid . Let the geographical latitudes / longitudes of the points be B 1 , B 2 , L 1 , L 2 , the normal curvature radii of the ellipsoid M 1 , M 2 , N 1 , N 2 . The change in the arc length  S due to differential coordinate changes dB, dL is then (according to B. Heck, Chapter 8.2)

dS = −cos A 1 M 1 dB 1 - sin A 1 N 1 cos B 1 dL 1 + cos A 2 M 2 dB 2 + sin A 2 N 2 cos B 2 dL 2 .

If one calls the north-south or east-west point shifts as new unknowns

dx i = M i dB i , dy i = N i cos B i dL i

and if one calculates the approximate arc length S ° or the azimuth A ° 1 from the approximate coordinates of P 1 , P 2 , then the equation for improving the arc length is obtained

v S = −cosA 1 dx 1 - sinA 1 dy 1 + cos A 2 dx 2 + sin A 2 dy 2 - ( S - S ° ),

where now the difference S - S ° is to be understood as an observation. Are obtained analogously to improve equation for the measured azimuth of P 1 to P 2 to

v A 1 = (sin A 1 dx 1 - cos A 1 dy 1 - sin A 2 dx 2 + cos A 2 dy 2 ) / S ° - ( A 1 - A ° 1 ).

The quadratic normal equation matrix is ​​then formed from the coefficients of the dx, dy of the improvement equations of all observations, the inversion of which finally results in the sought coordinate changes dx, dy of all points. Added to the approximation coordinates, the final coordinates of the measuring points follow.