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Interpolation of Berezkin functions to obtain the specific points for the gravity anomalies
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, 16th International Conference on Geoinformatics - Theoretical and Applied Aspects, May 2017, Volume 2017, p.1 - 6
Abstract
Purpose. Among the methods of interpretation of gravity data for 2D bounded and unbounded bodies there are Berezkin method for finding of singular points of harmonic functions, which represent gravity field models. To calculate the singular points there are developed a number of analytical and numerical methods. In the basis of such definition there is laid an analytic continuation of given harmonic function in a semi-infinite strip. In turn, one of the methods of this continuation there is the sum of basic solutions of Laplace equation by the some additional restrictions. This method reduces the problem of analytic continuation of gravity field function in the strip to the problem for determining the coefficients of expansion of the Fourier function. The report justifies a new way of determining the mentioned Fourier coefficients.
Design / methodology / approach. In order to find a singular points by a given gravity field distribution, one have to calculate a special Berezkin function as the ratio of gradients field and to find its maximums. This is not always the true case because the numerator and denominator of the function increase within the infinity at different speeds. This is proven by the results of numerical modeling for single gravity anomalies
Findings. The proof of Berezkin method is very complicated in the general formulation (with no restrictions on the parameters of singular points). So it is numerically simulated some cases for multiple locations of singular point sources caused by horizontal material lines (i.e. endless circular cylinders). They are located at different levels from the Earth’s surface, with varying intensity of gravity fields. For these models there are calculated the maximums of Berezkin function.
Practical value / implications. On the accuracy of Berezkin function calculations there depend on the order of decreasing rate of the Fourier series coefficients. So, there is derived a new interpolating polynomial and an order of values of its coefficients is estimated. It is specified, that the coefficients an decrease with the velocity of n-2, while the coefficients bn of the series do this with the velocity of n-1, if the Berezkin function does not satisfy the additional conditions.
We found the best way to interpolate the grid function under the additional condition of vanishing Berezkin function at the ends of the calculations interval. By the results of a detailed harmonic analysis of interpolation polynomials there are obtained the expressions for analytical continuation of gravity field. Berezkin functions were received by these expressions differentiation. Herewith it is a loss of computing accuracy due to Gibbs fluctuations at the ends of the interval of differential polynomial.
In order to partially reduce the impact of these fluctuations, the procedure applied for smoothing Berezkin function using the Lanczos σ-factors. Taking into account the given procedure, it is proposed the optimized expressions for calculating of the compounds of the Berezkin function – the modulus of the gravity gradient and the norm of this gradient at a fixed level.