JOURNAL OF THE BALKAN GEOPHYSICAL SOCIETY, Vol. 4, No 2, May 2001, p. 29-44, 23 figs.
Estimation of some bodies parameters from the self potential method using Hilbert transform
Dokuz Eylül Uni. Müh. Fak. Jeofizik Müh. B. Bornova, 35100 Izmir, Turkey e-mail: mustafa email@example.com (Received 4 April; accepted 3 July 2000)
Abstract: Structural parameters can be determineddirectly from the geophysical anomalies having two-dimensional distributions for the potential fields. In order to achieve this, more equations are constructed for the same parameter to be determined. The special properties of analytical functions of the complex gradients and the Hilbert Transforms can be used to reach the above-mentioned situation. Certain structural parameters of two-dimensionalbodies such as horizontal cylinder and horizontal sheets, were directly determined from the anomalies of the self-potential method. Hilbert Transforms, which can be carried out in two different ways using the Fourier Transform and convolution methods, were used to provide the conversion between the complex gradients of the anomaly. Structural parameters were then determined from the solutions ofthe constructed equations (anomaly, amplitude and phase functions). Key Words: Self Potential, Inversion, 2D Bodies, Hilbert Transformation.
INTRODUCTION Although Hilbert Transform (HT) have been used in electrical engineering and signal analysis for a long time (Bracewell, 1965), their application in geophysical studies started following 1970's. The HT is a method of direct solution. The aim ofusing the HT in geophysical studies is to obtain more than one equations that contain the same structural parameters by utilizing the complex gradients of the available data. The roots and common intersection points of the anomaly and the complex gradients of the anomaly have been used to determine the structural parameters. Therefore, an error ±1 sampling interval is expected for thedeterminations. In order to minimize the error, the most appropriate sampling interval should be chosen. The parameter solution equations are different for every structure. Therefore, the structural models belonging to the anomaly should be identified before the application of HT. The HT application can be carried out through the applications of Fourier Transform (FT) and convolution methods. However, incases where the HT is obtained through FT, the discontinuity that might occur at the terminals of the anomaly causes errors in the determination of the roots and common transaction points. In order to avoid this, this discontinuity should be eliminated before application of HT to the anomaly. The shifts occur also for the roots and common
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transaction points depending on the length of the convolution operator in the same way. Therefore, one should be careful for the selection of the convolution operator length. As known from signal analysis, the signals should be identified at definite intervals or should be zero or asymptotic to zero beyond this interval. The data, which are not asymptotic to zero or not zero beyondthe identification interval, were screened by using certain procedures (i.e. base reduction, windowing or derivation procedures) in order to obey the above mentioned conditions. Additionally, the equations were simplified for the structures having the logarithmic and arctan components, through the derivation operation with respect to x before the application of HT for the sake of proceduralsimplicity. HT, by definition, is a mathematical transform function, which shifts the phase of a signal by 90° without changing its amplitude. In other words, HT is a linear system, which transforms the odd and even functions, with the same amplitude, into each other in a space or frequency domain. HT were primarily used to investigate structural parameters in magnetic method by applying the magnetic...
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