Roland Karcol

The effect of topography in calculating the atmospheric correction in gravimetry

Although the gravitational effect of Earth’s atmosphere has relatively small values it is generally recommended to account for it in precision gravimetry. Since the effect is height-dependent, it is especially worth considering when the survey covers a broad range of gravity station heights and where the survey is performed close to a continental coast. Previously, the Earth’s topography was not considered significant when calculating the atmospheric correction for subtraction from the theoretical ellipsoidal gravity at the station. In fact the Earth’s surface is not flat over the continents and this variation in height must produce an additional influence upon the values of such a correction. We show using several examples that accounting for the Earth’s topography significantly changes the values from those calculated in the conventional way. The necessary calculations can be efficiently performed using a newly derived formula for the gravitational effect of a spherical shell with variable density.

The gravitational potential and its derivatives of a right rectangular prism with depth-dependent density following an n-th degree polynomial

The direct gravity problem and its solution belong to the basis of the gravimetry. The solutions of this problem are well known for wide class of the source bodies with the constant density contrast. The non-uniform density approximation leads to the relatively complicated mathematical formalism. The analytical solutions for this type of sources are rare and currently these bodies are very useful in the gravimetrical modeling. The solution for the vertical component of the gravitational attraction vector for the 3D right rectangular prism is known in the geophysical literature for the density variations described by the 3-rd degree polynomial. We generalized this solution for an n-th degree, not only for the vertical component, but for the horizontal components, the second-order derivatives and the potential as well. The 2D modifications of all given formulae are presented, too. The presented general solutions, which involve a hypergeometric functions, can be used as they are, or as an auxiliary tool to derive desired solution for the given degree of the density polynomial as a sum of the elementary functions. The pros-and-cons of these approaches (the complexity of the programming codes, runtimes) are discussed, too.

Normal Earth Gravity Field Versus Gravity Effect of Layered Ellipsoidal Model

We will discuss the following question: is the present day normal gravity field calculation suitable from the aspect of the applied geophysics/gravimetry? Today, the normal field or theoretical gravity is on a regular basis strictly related to a model of rotational biaxial ellipsoid with one important property, namely the constant potential on its surface. This condition, however, would require a specific density distribution within the model. Another issue is the treating the Earth atmosphere and topography. These masses are simply moved into the interior of such ellipsoid and “dissolved” there. In short, this approach does not fit the geophysical reality very well. Instead, in our contribution, we attempt to calculate the normal field as the gravity effect of a layered rotational ellipsoid, namely the gravitational effect of a suitable set of homeoid shells bounded by two similar ellipsoids having a constant ratio of axes, plus the centrifugal component. We believe that this approach will fit the structure and density distribution within the Earth much better. We analyze and present the primary differences between the two approaches, and we also discuss the relation of the normal field to the free-air correction.

Density function evaluation from borehole gravity meter data – regularized spectral domain deconvolution approach

ABSTRACT We present a new method of transforming borehole gravity meter data into vertical density logs. This new method is based on the regularized spectral domain deconvolution of density functions. It is a novel alternative to the “classical” approach, which is very sensitive to noise, especially for high‐definition surveys with relatively small sampling steps. The proposed approach responds well to vertical changes of density described by linear and polynomial functions. The model used is a vertical cylinder with large outer radius (flat circular plate) crossed by a synthetic vertical borehole profile. The task is formulated as a minimization problem, and the result is a low‐pass filter (controlled by a regularization parameter) in the spectral domain. This regularized approach is tested on synthetic datasets with noise and gives much more stable solutions than the classical approach based on the infinite Bouguer slab approximation. Next, the tests on real‐world datasets are presented. The properties and presented results make our proposed approach a viable alternative to the other processing methods of borehole gravity meter data based on horizontally layered formations.

Towards the measurement of zero vertical gradient of gravity on the Earth’s surface

It is well known that the vertical gradient of gravity measured on the Earth’s surface depends strongly on nearby topographical shapes. We simply inverted the problem and posed the question whether a zero vertical gradient can be observed using relative gravity meters and the classical tower method of measurement in appropriate terrain conditions. Extensively using the model of a vertical cone to simulate the real in-field conditions, we have found that reversed-cone-shaped topographic depressions represent the most perspective forms, which can contribute to extremely small values of the resulting vertical gradient. In one such form, namely a karstic sinkhole, we measured the value of −0.071 mGal/m (10−5 s−2). In addition, we successfully modeled this value using a detailed local digital elevation model. We thus conclude that zero vertical gradient of gravity should be observable by common means, also on the Earth’s surface, and not only underground within very dense rocks as some ores can be. Once this is verified it could represent a contribution to the theory of the Earth’s gravity field and its geophysical as well as geodetic applications.

Density function evaluation from borehole gravity meter data based on a regularized deconvolution algorithm - a synthetic model study

Summary In our study we propose a model study of a new processing method for the borehole gravity meter (BHGM) data transformation into vertical density logs. The method uses a regularized deconvolution approach in the spectral domain with the aim to estimate the density function of a horizontal circular plate, which is crossed by a synthetic vertical borehole profile. The proposed method allows to recognize linear and even higher polynomial changes of the density with the depth. When synthetic noise is added to the modeled data, the regularization approach gives much more stable results in comparison with the classical approach, resulting from the infinite Bouguer slab approximation.

Depth Estimation of Microgravity Anomalies Sources by Means of Regularized Downward Continuation and Euler Deconvolution

A powerful toll in the estimation of potential field source depths is given by the analytical downward continuation of the measured field - down to the depth of the first important shallow sources. On the other hand, analytical downward continuation is an highly instable problem and one effective way for its solution is Tikhonov regularization. Combination with the Derivative Euler Deconvolution can effectively help in the estimation of the depths to the centres of researched near-surface microgravity anomaly sources. This was presented on one selected synthetic model studies and one real data application. In some situations the estimations from Euler deconvolution are deeper, in some shallower, on the present we are not able to explain this aspect. Experiences with the regularized downward continuation show its very low dependence on grid extent and the grid cells sizes. Derivative Euler Deconvolution has showed large sensitivity to the precise evaluation of the initial vertical derivative – it has to be smoothed or damped in the case of real data interpretation (where noise and acquisition errors are present).

Computation of the atmospheric gravity correction in New Zealand

Abstract The proper treatment and the accurate modelling of atmospheric gravity correction and other environmental effects on gravity measurements are indispensable in precise gravimetric applications such as detailed gravity surveys, micro-gravimetry and other geophysical studies which require high accuracy. In this study we apply a novel approach to compute the atmospheric gravity correction. This approach is based on an analytical integration which utilises a newly derived closed expression for the gravitational effect of a truncated spherical shell having a density varying in the radial direction. The atmospheric gravity correction is evaluated as the atmospheric component of the normal gravity at the computation point from which the gravitational effect of the topography-bounded atmosphere is subtracted. The gravitational effect of the topography-bounded atmosphere is computed by volumetric integration over atmospheric masses, considering topography as the lower atmospheric bound. This new approach is applied to compute the atmospheric gravity correction in New Zealand. The numerical results reveal that the gravitational effect of the topography-bounded atmosphere varies from −0.009 mGal (offshore) up to 0.203 mGal (in the region of Mt Cook). The corresponding values of the atmospheric gravity correction vary between 0.671 and 0.884 mGal. We also demonstrate that the errors in computed values of the atmospheric gravity correction exceed ~0.1 mGal when disregarding topography as the lower atmospheric bound.