The quality analysis of well-logging inversion results has always been an important part of formation evaluation. The precise calculation of hydrocarbon reserves requires the most accurate possible estimation of porosity, water saturation, and shale and rock-matrix volumes. The local inversion method conventionally used to predict the above model parameters depth by depth represents a marginally overdetermined inverse problem, which is rather sensitive to the uncertainty of observed data and limited in estimation accuracy. To reduce the harmful effect of data noise on the estimated model, we have suggested the interval inversion method, in which an increase of the overdetermination ratio allows a more accurate solution of the well-logging inverse problem. The interval inversion method inverts the data set of a longer depth interval to predict the vertical distributions of petrophysical parameters in a joint inversion procedure. In formulating the forward problem, we have extended the validity of probe response functions to a greater depth interval assuming the petrophysical parameters are depth dependent, and then we expanded the model parameters into a series using the Legendre polynomials as basis functions for modeling inhomogeneous formations. We solved the inverse problem for a much smaller number of expansion coefficients than data to derive the petrophysical parameters in a stable overdetermined inversion procedure. The added advantage of the interval inversion method is that the layer thicknesses and suitably chosen zone parameters can be estimated automatically by the inversion procedure to refine the results of inverse and forward modeling. We have defined depth-dependent model covariance and correlation matrices to compare the quality of the local and interval inversion results. A detailed study using well logs measured from a Hungarian gas-bearing unconsolidated formation revealed that the greatly overdetermined interval inversion procedure can be effectively used in reducing the estimation errors in shaly sand formations, which may refine significantly the results of reserve calculation.

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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A new inversion based Fourier transformation technique named as Legendre-Polynomials Least-Squares Fourier Transformation (L-LSQ-FT) and Legendre-Polynomials Iteratively Reweighted Least-Squares Fourier Transformation (L-IRLS-FT) are presented. The introduced L-LSQ-FT algorithm establishes an overdetermined inverse problem from the Fourier transform. The spectrum was approximated by a series expansion limited to a finite number of terms, and the solution of inverse problem, which gives the values of series expansion coefficients, was obtained by the LSQ method. Practically, results from the least square method are responsive to data outliers, thus scattered large errors and the estimated model values may be far from reality. A definitely better option is attained by introducing Steiner’s Most Frequent Value method. By combining the IRLS algorithm with Cauchy-Steiner weights, the Fourier transformation process was robustified to give the L-IRLS-FT method. In both cases Legendre polynomials were applied as basis functions. Thus the approximation of the continuous Fourier spectra is given by a finite series of Legendre polynomials and their coefficients. The series expansion coefficients were obtained as a solution to an overdetermined non-linear inverse problem. The traditional DFT and the L-IRLS-FT were tested numerically using synthetic datasets as well as field magnetic data. The resulting images clearly show the reduced sensitivity of the newly developed L-IRLS-FT methods to outliers and scattered noise compared to the traditional DFT. Conclusively, the newly developed L-IRLS-FT can be considered to be a better alternative to the traditional DFT.