H. Szegedi

On the use of Steiner’s weights in inversion-based Fourier transformation: robustification of a previously published algorithm

In our previous paper (Dobróka et al. Acta Geod Geophys Hung 47(2):185–196, 2012) we proposed a new robust algorithm for the inversion-based Fourier transformation. It was presented that the Fourier transform and its variants responds very sensitively to any little measurement noise affected an input data set. The continuous Fourier spectra are assumed as a series expansion with the scaled Hermite functions. The expansion coefficients are determined by solving an over-determined inverse problem. Here, we use the new Steiner’s weights (previously called the weights of most frequent values or abbreviated as MFV), where the scale parameter can be determined in an internal iteration process. This method results a very efficient robust inversion method in which we calculate the Steiner weights from iteration to iteration into an IRLS procedure. The new method using the Steiner’s weights is also numerically tested by using synthetic data.

Fourier transformation as inverse problem — An improved algorithm

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.

On the Reduced Noise Sensitivity of a New Fourier Transformation Algorithm

In this study, a new inversion method is presented for performing one-dimensional Fourier transform, which shows highly robust behavior against noises. As the Fourier transformation is linear, the data noise is also transformed to the frequency domain making the operation noise sensitive especially in case of non-Gaussian noise distribution. In the field of inverse problem theory it is well known that there are numerous procedures for noise rejection, so if the Fourier transformation is formulated as an inverse problem these tools can be used to reduce the noise sensitivity. It was demonstrated in many case studies that the method of most frequent value provides useful weights to increase the noise rejection capability of geophysical inversion methods. Following the basis of the latter method the Fourier transform is formulated as an iteratively reweighted least squares problem using Steiner’s weights. Series expansion was applied to the discretization of the continuous functions of the complex spectrum. It is shown that the Jacobian matrix of the inverse problem can be calculated as the inverse Fourier transform of the basis functions used in the series expansion. To avoid the calculation of the complex integral a set of basis functions being eigenfunctions of the inverse Fourier transform is produced. This procedure leads to the modified Hermite functions and results in quick and robust inversion-based Fourier transformation method. The numerical tests of the procedure show that the noise sensitivity can be reduced around an order of magnitude compared to the traditional discrete Fourier transform.

Inversion-Based Fourier Transform as a New Tool for Noise Rejection

In this study, a new inversion method is presented for performing two-dimensional (2D) Fourier transform. The discretization of the continuous Fourier spectra is given by a series expansion with the scaled Hermite functions as square-integrable set of basis functions. The expansion coefficients are determined by solving an overdetermined inverse problem. In order to define a quick algorithm in calculating the Jacobian matrix of the problem, the special feature that the Hermite functions are eigenfunctions of the Fourier transformation is used. In the field of inverse problem theory, there are numerous procedures for noise rejection, so if the Fourier transformation is formulated as an inverse problem, these tools can be used to reduce the noise sensitivity. It was demonstrated in many case studies that the use of Cauchy-Steiner weights could increase the noise rejection capability of geophysical inversion methods. Following this idea, the two-dimensional Fourier transform is formulated as an iteratively reweighted least squares (IRLS) problem using Cauchy-Steiner weights. The new procedure is numerically tested using synthetic data.

A New Procedure of Reduction to Pole of Magnetic Data - Improved Noise Rejection Capability

In geophysical research it is an important objective to find more accurate measuring and more effective data processing methods. Measurement data always contain noise, which can mislead the interpreter, or hide useful information. The often used traditional DFT algorithm shows low noise rejection capability. On the other hand there are robust methods to solve the overdetermined inverse problem with excellent noise rejection capabilities. Therefore we suggest a new inversion based Fourier transformation method, where the continuous frequency spectrum is discretized with series expansion and the series expansion coefficients as model parameters are determined in the framework of the Iteratively Reweighted Least Squares (IRLS) algorithm using the so-called Cauchy-Steiner weights. In this paper the method was tested on noisy synthetic magnetic data generated above two magnetic bodies. The results prove the successful applicability of our inversion based S-IRLS-FT algorithm.

A New Inversion Based 2D Fourier Transformation Method and Its Use in Pole-reduction of Magnetic Data Set

Considering the Fourier Transform as an inverse problem, a new 2D Fourier transformation procedure is proposed by using the Iteratively Reweighted Least Squares method. We use the special property of the Hermite functions that they are eigen-functions of the Fourier transform. This property avoids the calculation of complex integrals and simplifies the determination of the Jacobian matrice. The new method is tested by using synthetic data set. The noise reduction capability of the new inversion-based Fourier transform method is clearly demonstrated. Its application in the pole-reduction of magnetic data shows sufficient improvement in noise rejection capability compared to the traditional algorithm using Discrete Fourier Transform.

A New Robust Inversion Method Using Cauchy-Steiner Weights – And Its Application in Data Processing

The Fourier-transform (FT) often results noisy Fourier spectrum because this operation is linear and contaminate noise in time domain dataset appear in the frequency domain also. Especially, in case of non-Gaussian nature of the noise distribution (for example outliers in the data sets) can cause appreciable distortions in the Fourier spectra. If we treat the FT as an over-determined inverse problem, the noise effect on the Fourier spectrum can be greatly reduced. The geophysical series inversion was used for the discretization. The scaled Hermite functions - note that they are eigen-functions of the FT - were chosen as basis function. During the inversion process by using the Steiner-weights we get a highly robust inversional FT method. Numerical tests show the significant noise rejection capability of the new inversion based FT algorithm.

Proving the Applicability of the Petrophysical Model Describing Acoustic Hysteresis of P and S Wave Velocities

Due to the increasing demand for hydrocarbons and the depletion of the known hydrocarbon fields there is a growing claim to predict rock physical parameters more accurate at non-conventional conditions also. It is well known that acoustic velocity in rocks strongly depends on pressure which influences the mechanical, transport and elastic properties of rocks as well as wave propagation under pressure is very nonlinear and the quasistatic elastic properties of rocks are hysteretic. Characterization of hysteretic behavior is important for mechanical understanding of reservoirs during depletion. Therefore a quantitative model - which provides the physical explanation - of the mechanism of pressure dependence is required. In this paper a petrophysical model is presented which provides the connection between the propagation velocity of acoustic waves (both P and S) and rock pressure both in case of pressurization and depressurization phases as well as explains the mechanism of acoustic hysteresis. The developed model is based on the idea that the pores in rocks close under loading and reopen during unloading. The model was applied with success to acoustic P and S wave velocity data sets.

New Petrophysical Models for Describing Pressure-dependent Acoustic Velocity and Absorption Coefficient in Porous Rocks

SUMMARY The pressure dependence of seismic velocity and absorption coefficient of acoustic waves in rock is an extensively explored rock physical problem. New petrophysical models are developed based on simple physical assumptions, through which relationships between velocity and pressure as well as absorption coefficient and pressure are set up and explained. The models are based on the idea that microcracks are opened and closed under pressure. The models were applied on acoustic velocity measurement data made on core samples and also seismic velocity and quality factor data set measured by Prasad (1997). The material parameters of the models are determined by using linearized inversion method. Laboratory measurements proved properly the accuracy of the new models.