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  • Author or Editor: P. Vass x
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Treating the Fourier transform as an over-determined inverse problem is a new conception for determining the frequency spectrum of a signal. The concept enables us to implement several algorithms depending on the applied inversion tool. One of these algorithms is the Hermit polynomial based Least Squares Fourier Transform (H-LSQ-FT). The H-LSQ-FT is suitable for reducing the influence of random noise. The aim of the investigation presented in the paper was to study the noise reduction capability of the H-LSQ-FT in some circumstances. Four wavelet-like signals with different properties were selected for testing the method. Examinations were completed on noiseless and noisy signals. The H-LSQ-FT provided the best noise reduction for the noisy signal having low peak frequency and wide band width. Finally, the results obtained by the H-LSQ-FT were compared to those of other traditional methods. It is showed that the H-LSQ-FT yields better noise filtering than these methods do.

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In the paper a 2D joint inversion method is presented, which is applicable for the simultaneous determination of layer thickness variation and petrophysical parameters by processing well-logging data acquired in several boreholes along the profile. The so-called interval inversion method is tested on noisy synthetic data sets generated on hydrocarbon-bearing reservoir models. Numerical experiments are performed to study the convergence and stability of the inversion procedure. Data and model misfit, function distance related to layer thickness fitting are measured as well as estimation errors and correlation coefficients are computed to check the accuracy and reliability of inversion results. It is shown that the actual inversion procedure is stable and highly accurate, which arises from the great over-determination feature of the inverse problem. Even a case study is attached to the paper in which interval inversion procedure is applied for processing of multi-borehole logging data acquired in Hungarian hydrocarbon exploratory wells in order to determine petrophysical parameters and lateral changes of layer thicknesses.

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This paper presents a new algorithm for the inversion-based 1D Fourier transformation. The continuous Fourier spectra are assumed as a series expansion with the scaled Hermite functions as square-integrable set of basis functions. The expansion coefficients are determined by solving an over-determined inverse problem. In order to define a quick and easy-to-use formula in calculating the Jacobi matrix of the problem a special feature of the Hermite functions are used. It is well-known, that the basic Hermite functions are eigenfunctions of the Fourier transformation. This feature is generalized by extending its validity for the scaled Hermite functions. Using the eigenvalues, given by this generalization, a very simple formula can be derived for the Jacobi matrix of the problem resulting in a quick and more accurate inversion-based Fourier transform algorithm. The new procedure is numerically tested by using synthetic data.

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