Densification of geodetic control Networks under reproducing parametric and stochastic fiducial constraints

Abstract

A principal consideration in densification of geodetic systems has been the need to incorporate the stochasticity of the datum parameters in the densification process. The special question however is how to incorporate tile stochasticity of the datum parameters in tile estimation of tile new parameters while reproducing tile datum parameters together witlt their stochasticity (respectively reproducing parametric and stochastic fiducial constraints). A number of approaches addressing the above question have been proposed. These include: static-dynamic, pseudo-dynamic and sub-optimal network fusion approaches. Of these, pseudo-dynamic, staticdynamic and sub-optimal network fusion approaches reproduce the datum parameters and their stochasticity while static and dynamic approaches do not possess the reproducing quality. The aim in this study was to evaluate the practical applicability, and to establish the suitability, of two densification approaches with the reproducing quality. The static-dynamic and suboptimal fusion approaches are considered, with a view to identifying their strength and weaknesses as approaches to densification of geodetic systems. The results are compared to establish which of the approaches is best suited for recommendation to be adopted for geodetic densification and under what circumstances. For a general perspective, the non-reproducing techniques namely; static and dynamic approaches are also discussed, evaluated and compared to the two approaches. Although the pseudo-dynamic approach has the reproducing capability, it is not considered since the approach has the drawback in that, on one hand datum parameters are treated as non-stochastic entities, thus fixed, while on the other hand. they are treated as stochastic, resulting in an inconsistent estimation model. To evaluate these approaches, each of the techniques is used to adjust simulated and real test geodetic networks at two levels of densification. The simulated geodetic network consists of three first order, three second order and nine third order points while the real geodetic network points were extracted from the national geodetic- network of Kenya consisting of eleven first order, fifteen second order and ten third order points. For each approach, and at every level of densification on the two networks, the parameters, the standard errors and their corresponding error ellipses were compared against each other. The results indicate that the datum parameters in the static-dynamic and the sub-optimal fusion approaches are reproduced together with their stochasticity, that is, maintained definitive. Although the datum parameters are reproduced together with their stochasticity, the new point parameters obtained using sub-optimal fusion approach are similar to the parameters obtained using the dynamic approach. That is, it compares to adjusting the network using the dynamic approach and applying a corrective term on the datum parameters to keep them unchanged. The covariance matrices obtained through the two approaches are closer to each other as demonstrated by the confidence error ellipses. The results generally demonstrate that both the static-dynamic and sub-optimal fusion approaches give more realistic estimates of the parameters than the static and dynamic approaches.