Optimization of a local geoid using heterogeneous data

Geodesy is a physical science with a great theoretical background and of an intensive applied character. It relies exclusively upon the analysis of observed quantities. Due to its inability to determine the exact value of a quantity, the theory of Geodesy utilizes various estimation methods in order to obtain the best estimates for various gravity field parameters. This Ph.D. thesis focuses on the optimization of a local geoid, using heterogeneous data. Common techniques for geoid determination, (such as gravimetry and/or combined GPS/leveling), even though they do provide an acceptable solution, they cannot adjust errors which are referred to low and medium wavelengths of the gravity field. Thus, a combined approach of these methods is investigated in order to benefitfrom the advantages of each technique. The chapter structure is as follows: The first chapter is referred to the purposes of this Thesis and includes a brief presentation of its contents. The second chapter describes the theoretical background of the Thesis and presents all introductory terms, such as the geoid, the gravity field, the covariance functions and the latest techniques for geoid determination. Special emphasis is given on the least square collocation method, which has been used on the numerical applications of this Thesis. There is also a step-by-step description of the “remove restore” technique, which is a signal separation technique of low, medium and longwavelengths. The latest conceptions for the combination of heterogeneous data are presented and a summary report on the solutions that were applied for geoid determinations in Greece, until recently. The third chapter presents the optimization method for a local geoid applied on real data. Gravity data have been used combined with GPS/leveling measurements in two distinct areas of Greece. The first area is located in the major area of Astakos (Aitoloakarnania) and the second is a part of Central Macedonia (Northern Greece). We proceed with an extended study of the gravity data and the GPS/leveling data distribution, as well as with the local and spherical covariance function models. The gravimetric geoid heights are compared with those obtained from GPS/leveling. In the next step, these differences are minimized by a parametric transformation model. Subsequently, there is a prediction of the geoidal height signal, by entering those minimized differences in an adjustment procedure. Finally, a few comments on the results and some conclusions obtained from this application. The forth chapter presents the application of the optimization procedure in three distinct geographical areas, but with simulated data. The main reason for using a simulation procedure, is the small number of real data available in Greece and the heterogeneous distribution of these gravity data and/or GPS/leveling measurements. At first, the geoid heights were computed using the GPM98A coefficient model, to degree and order 1800x1800, considered as the “real” geoid. The “real” geoidal heights are used as a reference for computing the gravimetric and GPS/leveling geoid. In order to create the data files needed for the gravimetric and GPS/leveling geoidal heights, the systematic and random errors are calculated, and we added - back to the heights of the “real” geoid. For the first region of this study (an area within Central Greece), there are nine different case studies, differed by the error and data distribution. For the gravimetric geoid distribution heights a 5’x5’ grid has been used. The GPS/leveling geoid heights must have smooth point distribution and their density should be much smaller than that of the gravimetric data. In another case, the additional amount of information, provided by the GPS/leveling data does not contribute to the improvement of the results. In these specific applications the grid of the GPS/leveling heights has set to 10’x10’, except in one case (used to control the process) where a 5’x5’ distribution has been used. All nine case studies present a different optimization percentage-wise. Excluding the first one (which is actually a test) all the rest lead to an improvement from 10.15% to 37.35%. The surfaces grids for the second and third area (the area of Astakos and another in Central Macedonia) have a distribution of 3’x3’ for the gravimetric geoid heights and 6’x6’ for the GPS/leveling geoid heights. The application of this optimization process, leads to the gravimetric geoid improvement by 32.52% to 24.87% (in the first and second areas, respectively). The fifth chapter contains our conclusions, based on the research work herein. From the results of our numerical applications, we conclude that the improvement of an existing gravimetric geoid depends on some parameters which are connected to the local characteristics of each area and of course with the accuracy, the amount and the distribution of the available data. Therefore, if the systematic errors which affect the gravimetric geoid heights is located within an area with low frequencies of the gravity field spectrum, the first step in the optimization procedure (which is the application of the parametric transformation model), has better results compared to the cases, where the systematic errors are located on the high frequency spectrum. The reason is that the 4-parameters transformation model, used for this study, cannot describe a complex surface. Furthermore, the covariance functions are very important for the prediction of the remaining geoid height signals. The covariance model of Tscherning and Rapp is preferred in large scale applications. On the other hand, Moritz’s plane model and Markov’s 2nd order model are move suitable for applications in local scale. In all these cases, already mentioned in this study, Moritz’s plane model offers the greater improvement on the gravimetric geoid