János Kalmár

DTM-based surface and volume approximation: geophysical applications

Based on earlier investigations, two new methods for local surface interpolation developed by us were applied to derive regular grids, that is Digital Terrain Models (DTM) from the digitized contour lines of two surfaces. These surfaces represent two main structural boundaries of the Earths crust, the pre-Tertiary basement and the Mohorovicic discontinuity in the Pannonian Basin, Hungary. The grids were used to model the volume of sediments and mantle material above the basement and below the Moho respectively in order to build up an initial version of a 3-D model of the crust in Hungary for regional gravity field modeling. The grid representation provides an evident and elementary way for modeling a body by right rectangular prisms of Δx × Δy × z, where Δx and Δy are distances between grid knots in the X and Y directions respectively, and z represents a grid value at a grid knot. However, because of the nature of forward gravity modeling, the elementary prisms might be drawn together in a simple volume density distribution (e.g. homogeneous layer) and in such a way that the computation time can be reduced significantly. Therefore, a recursive algorithm for volume approximation by right rectangular prisms of a body determined by its DTM also was developed and applied in the determination of the 3-D crustal model in the Pannonian Basin.

Investigation of sediment compaction in the Pannonian basin using 3D gravity modelling

Abstract The great number of geophysical and geological data available in Hungary has made possible the direct modelling of the density distribution of the upper crust. Without considering lateral inhomogeneities, a simple 1D theoretical model was introduced for the relationship between Bouguer gravity anomalies and basement depths to derive a general formula describing the vertical change of density in the sediments. The validity of the correlation model was investigated by direct 3D gravity modelling. The 1D model proved a satisfactory approximation, so it was applied to the observed gravity field over a selected area of the Pannonian basin, and an extensive analysis of the sediment compaction was carried out based on a high-resolution digital depth model (DDM) and volume model of the sediments covering the Pannonian basin with an average thickness of 2 km. The derived density-depth function of the selected area differs from the generally accepted vertical change of density estimated by geological-geophysical exploration, although the effect of the sediment compaction is clearly visible. The obtained density contrast function resulted in apparent density values for the sediments higher than given by the reference model used in 3D gravity field model computations. This systematic deviation may be a consequence of the crustal thinning which is supposed to be a consequence of basin evolution. A crust-mantle model of the investigated area extending below the Moho discontinuity was used to support this assumption.

A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation

The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa–Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.

Generalization techniques to reduce the number of volume elements for terrain effect calculations in fully analytical gravitational modelling

Beyond rectangular prism polyhedron, as a discrete volume element, can also be used to model the density distribution inside 3D geological structures. The calculation of the closed formulae given for the gravitational potential and its higher-order derivatives, however, needs twice more runtime than that of the rectangular prism computations. Although the more detailed the better principle is generally accepted it is basically true only for errorless data. As soon as errors are present any forward gravitational calculation from the model is only a possible realization of the true force field on the significance level determined by the errors. So if one really considers the reliability of input data used in the calculations then sometimes the “less” can be equivalent to the “more” in statistical sense. As a consequence the processing time of the related complex formulae can be significantly reduced by the optimization of the number of volume elements based on the accuracy estimates of the input data. New algorithms are proposed to minimize the number of model elements defined both in local and in global coordinate systems. Common gravity field modelling programs generate optimized models for every computation points (dynamic approach), whereas the static approach provides only one optimized model for all. Based on the static approach two different algorithms were developed. The grid-based algorithm starts with the maximum resolution polyhedral model defined by 3–3 points of each grid cell and generates a new polyhedral surface defined by points selected from the grid. The other algorithm is more general; it works also for irregularly distributed data (scattered points) connected by triangulation. Beyond the description of the optimization schemes some applications of these algorithms in regional and local gravity field modelling are presented too. The efficiency of the static approaches may provide even more than 90% reduction in computation time in favourable situation without the loss of reliability of the calculated gravity field parameters.

Comparison of Various Gridding Methods

A comparison of various gridding procedures is carried out using a model gravity field. This model is produced by a causative body of 64 rectangular prisms, for which exact field values can be calculated everywhere. The following methods were used in the comparisons: kriging, linear interpolation along the shortest chord, least squares approximation with normal vector, surface fitting under tension, Franke’s local thin plate spline, least squares collocation, collocation with empirical covariance function and interpolation of scattered data by Franke’s weights. Two different input data sets, consisting of 800 and 480 randomly distributed points were used in the tests. For each method, computed values on a 4 km grid (676 points) were generated for comparisons. The results of the investigations show that simple numbers such as presented in the tables are not sufficient for critical comparison of the various gridding algorithms but contour maps of output are needed.