Comparing congruency robust method and L1 norm minimization in micro geodesy networks


For calculation of the displacements of points in micro geodesy networks, it is essential to discover stable and unstable points. Without knowing stable points, calculated displacements are due to datum deficiency. In this case, calculated displacements are not valid. There are two methods to discover stable and unstable points:
a-Congruency robust method
b- L1 Norm minimization
In this study the two mentioned methods are compared and the advantages and disadvantages of both are studied. For this reason, the two methods are programmed and several networks tested by them. The results of comparing these two methods appear below:

1- The two methods similarly detect all the points moved eighteen percent. L1 norm minimization results are better than the congruency robust method by seventy four percent in detecting points moved. On the other hand, the congruency robust method detects moved points better than the other method by eight percent.
2- In the networks whose displacements of points are about a few millimeters, L1 norm minimization detects moved points much better than the other method. Some of the samples are available in the tables below. These two methods discover all points when the displacements of moved points are a few centimeters and both methods are reliable. Thus, either L1 norm minimization or congruency robust method can be used in order to detect moved points.
3- The congruency robust method is not reliable when all points or all points except one or two are moved because it cannot find all moved points in this situation. On the contrary, all points are detected by the L1 norm minimization method. Neither the norm minimization nor the congruency robust method could find moved points when we have all points moved. Generally, if we have at least two unmoved points in the network, the results are reliable. In spit of this, deformation tensors should be applied.
4- The algorithm of norm minimization is simpler and its programming is easier than that of congruency robust method.
In order to discover moved and unmoved points in the network, the study suggests that the norm minimization method should be applied. Of course it is proposed that both methods be considered and the unmoved points obtained from them considered altogether as stable points. Moved points that are erroneously detected as unmoved points are discovered by a statistical test applied after calculating the displacements of unmoved points. These points are considered as unmoved points.