Oleg Portniaguine

Focusing geophysical inversion images

A critical problem in inversion of geophysical data is developing a stable inverse problem solution that can simultaneously resolve complicated geological structures. The traditional way to obtain a stable solution is based on maximum smoothness criteria. This approach, however, provides smoothed unfocused images of real geoelectrical structures. Recently, a new approach to reconstruction of images has been developed based on a total variational stabilizing functional. However, in geophysical applications it still produces distorted images. In this paper we develop a new technique to solve this problem which we call focusing inversion images. It is based on specially selected stabilizing functionals, called minimum gradient support (MGS) functionals, which minimize the area where strong model parameter variations and discontinuity occur. We demonstrate that the MGS functional, in combination with the penalization function, helps to generate clearer and more focused images for geological structures than conventional maximum smoothness or total variation functionals. The method has been successfully tested on synthetic models and applied to real gravity data.

3‐D magnetic inversion with data compression and image focusing

We develop a method of 3‐D magnetic anomaly inversion based on traditional Tikhonov regularization theory. We use a minimum support stabilizing functional to generate a sharp, focused inverse image. An iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method. The weighting functions are selected based on sensitivity analysis. To speed up the computations and to decrease the size of memory required, we use a compression technique based on cubic interpolation.Our method is designed for inversion of total magnetic anomalies, assuming the anomalous field is caused by induced magnetization only. The method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by ExxonMobil Upstream Research Company.

Inverse Spectral Decomposition

This paper introduces a method which spectrally decomposes a seismic trace by solving an inverse problem. In our technique, the reverse wavelet transform with a library of complex wavelets serves as a forward operator. The inversion reconstructs the wavelet coefficients that represent the seismic trace and satisfy an additional constraint. The constraint is needed as the inverse problem is non-unique. We show synthetic and real examples with three different types of constraints: 1) minimum L2 norm, 2) minimum L1 norm, and 3) sparse spike, or minimum support constraint. The sparse-spike constraint has the best temporal and frequency resolution. While the inverse approach to spectral decomposition is slow compared to other techniques, it produces solutions with better time and frequency resolution than popular existing methods.

Thin-bed Reflectivity Inversion

Summary Thin-bed reflectivity inversion is a form of spectr al inversion which produces sparse reflectivity estima tes that resolve thin layers below the tuning thickness . The process differs from other inversions in that it is driven by geological rather than mathematical assumptions, and is based on aspects of the local frequency spectrum obtained using spectral decomposition of various ty pes. The resolution of thin-bed reflectivity inversion i s far superior to the input data and so makes it very sui table for characterization of thin reservoirs.

Chapter 10 3-D focusing inversion of CSAMT data

Abstract We present a method for the solution of 3-D controlled source magnetotelluric (CSAMT) inverse problems. The inverse problem is formulated as the minimization of a Tikhonov parametric functional with a focusing stabilizer. Observed CSAMT apparent resistivities are converted to log-anomalous apparent resistivities, which are linearly connected to anomalous currents via the integral equation. We apply the Born iterative method to solve this integral equation, using a focusing regularized inversion. The focusing is based on a specially selected stabilizing functional which minimizes the area where strong model parameters variations and discontinuities occur. The method is illustrated using examples of 3-D inversion of model CSAMT data, and with a real data example.

Migration and analytic continuation in geoelectric imaging

Geoelectric imaging can be accomplished by using downward continuation or migration of the observed electromagnetic (EM) field in the lower halfspace. The method assumes a known background resistivity distribution. It is based on a finite-difference analytical continuation and migration of the EM field in the frequency domain. Vertical maps of the downward extrapolated or migrated fields (amplitudes and phases) produce useful images of the geoelectrical cross-section. We demonstrate also a simple technique of transforming the migrated field into resistivity images which correspond rat her well to the actual resistivity cross-sections. The method can be applied to surface (profile) magnetotelluric or controled source EM data and to borehole EM data as well.

Chapter 2 Compression in 3-D integral equation modeling

Abstract The integral equations (IE) method is a powerful tool for forward electromagnetic (EM) modeling. However, due to a dense matrix arising from the IE formulation, practical application of the IE method is limited to modeling of relatively small bodies. The use of a compression technique can overcome this limitation. The compression transformation is formulated as a multiplication by a compression matrix. Using this matrix as a preconditioner to an integral equation, we convert the originally dense matrix of the problem to a sparse matrix, which reduces its size and speeds up computations. Thus, compression helps to overcome practical limitations imposed on the numerical size of the anomalous domain in IE modeling. With the compression, the flexibility of the IE method approaches that of finite-difference (FD) or finite-element (FE) methods, allowing modeling of large-scale conductivity variations.

Two‐dimensional regularized inversion of magnetotelluric data

The two-dimensional inverse problem in magnetotellurics has been addressed by several authors, the most well-known approaches being developed by de Groot-Hedlin and Constable (1990) and by Smith and Booker (1991) in the search for a smooth model. However, a smooth model can not describe well the real model of the earth, which contains sharp contrasts at the boundaries of structures with different conductivities. In this paper, we present an approach that deals with the model of an arbitrary structure. This approach is based on regularization theory and the quasianalytic calculation of the Frechet derivatives. For the forward solution we use a fast and efficient finite difference formulation to the solution of the Helmholtz equation based on the balance method (Berdichevsky and Zhdanov, 1984). One of the most computationally expensive tasks in the solution of an inverse problem, the calculation of the Frechet derivative matrix is obtained as a solution to simple forward and back substitution of the LU decomposed matrix of coefficients from the forward problem with a different right hand term. The method of steepest descent is utilized to minimize the parametric functional in the search for a stable solution of the two-dimensional magnetotelluric inversion problem. Preliminary results for synthetic data of a conductive body in a resistive host are presented. The method can be applied for the inversion of MT and CSAMT data.