Adjustment Computations

in geodesy, a set of mathematical operations that are performed to obtain the most probable value of the geodetic coordinates of points on the earth’s surface and to estimate the accuracy of the results of measurements. Adjustment computations are performed to eliminate discrepancies due to the presence of errors in quantities whose measured values are too large and to determine the most probable values of unknown quantities or the values closest to the most probable values. In computing adjustments, this is achieved by determining corrections to such measured quantities as angles, directions, and the lengths or overmeasures of lines. Corrections are usually determined by the method of least squares in such a way that the sum of the squares of all corrections is minimized. In this case, the calculations are said to be rigorous, and the unknowns (corrections) determined from such an adjustment computation have the most probable values.

Thus, in the simplest example of a plane triangle, the sum of the angles must be equal to 180°. Because of measurement errors, the measured angles generally do not satisfy this condition and must be adjusted by adding appropriate corrections. Of the entire infinite set of corrections that bring the sum of the measured angles to 180°, only one system of corrections has the property that the sum of the squares is the minimum; this system is considered the most probable. In this example, the sum of the squares is the minimum if the discrepancy can be divided equally among all three angles.

However, the method of least squares may be legitimately applied to the adjustment of measured values only in the case where the errors are accidental. The rigorous adjustment of geodetic networks that are especially large in size entails a number of technical and occasional difficulties. Therefore, various simplified methods of computing adjustments are often used in practice. In geodetic practice, the following two methods of compensation are widely used in both rigorous and simplified adjustment computations: the method of conditioned observations and the method of variation of coordinates. In the first method, corrections are sought directly for the measured quantities; in the second method, corrections are sought for functions—as a rule, the coordinates—of the measured quantities.

Any adjustment method consists of the following main processes: preliminary calculations, the forming of condition equations or error equations, the forming of normal equations, the solution of the normal equations, and the estimation of the precision of the measured and adjusted quantities. When a large number of normal equations is involved, the solution of the equations is the most time-consuming part of adjustment computations. Hence the equations are usually solved by computer. The equations may be solved by the method of successive elimination of unknowns (Gauss’ scheme) or by iteration (approximation). Normal equations are sometimes not formed; in this case, the unknowns are determined directly from the solution or the condition equations or from the error equations. In some cases, when data obtained by low-accuracy geodetic measurements are processed, the results are adjusted graphically.