Determination of parameters of self-potential anomalies using Hilbert transform method

Although the Hilbert Transform (HT) has been used in electrical engineering and signal analysis for a long time (Bracewell, 1965), its application in geophysical studies started in 1970's. The HT is a method of direct solution. The aim of using the HT in geophysical studies is to obtain more than one equation containing 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, a ± 1 error sampling interval was expected for the determinations. In order to minimize the error, the most appropriate sampling interval should be chosenUp to the present time, the HT has been used extensively only in magnetic and seismic studies. But in the aforementioned studies, it has been used mostly as a Fourier Transform (FT). Taner (1979), in his study, obtained the HT through convolution by using a normalized Hilbert time-domain operator truncated to 61 points The Hilbert Transform (HT) is a mathematical transform function which shifts the phase of a signal as much as π/2 without changing its amplitude. With this definition HT is a linear system, which transforms odd and even functions, with equal amplitude to each other in space or frequency field. Since HT is a linear set, the system should have an input signal, a transfer function and an output function. The HT can be applied to the Fourier Transform (FT) and convolution methods.. In this study, the model parameters of which were unsolved so far, self-potential (SP) methods were determined with HT using convolution and FT methods and the results were compared. In this study, Structural parameters were determined directly from the geophysical anomalies using analytical functions of the complex gradients and the Hilbert Transforms can be applied to reach the above-mentioned situation. The Hilbert Transforms, which can be carried out in two different ways using the Fourier Transform and convolution methods, were used to provide the convolution method between the complex gradients of the anomaly. Structural parameters (electric dipole moment, polarization angle and depth) were then determined from the solutions of the constructed equations. This method was used for two models, a sphere and a horizontal cylinder, with synthetic data without any random noise. The results of this study are as follow: (1) The parameters were determined exactly for the theoretical models using the HT method. Especially, the location of the structure, which had not been determined before, was obtained precisely and directly from the anomaly for the self-potential method. (2) Before the interpretation of the field data with the HT method, the anomaly should be refined from noise. If this procedure has not been carried out, pseudo roots could be formed in the complex gradients of anomaly.