Full Inverse Problem Research Articles

A 3D parametric inversion algorithm for triaxial induction data

We develop a parametric inversion algorithm to determine simultaneously the horizontal and vertical resistivities of both the formation and invasion zones, invasion radius, bed boundary upper location and thickness, and relative dip angle from electromagnetic triaxial induction logging data. This is a full 3D inverse scattering problem in transversally isotropic media. To acquire sufficient sensitivity to invert for all of these parameters, we collect the data using a multicomponent, multispacing induction array. For each transmitter-receiver spacing this multicomponent tool has sets of three orthogonal transmitter and receiver coils. At each logging point single-frequency data are collected at multiple spacings to obtain information at different depths of investigation. This inversion problem is solved iteratively with a constrained regularized Gauss-Newton minimization scheme. As documented in the literature, the main computational bottleneck when solving this full 3D inverse problem is the CPU time associated with constructing the Jacobian matrix. In this study, to achieve the inversion results within a reasonable computational time, we implement a dual grid approach wherein the Jacobian matrix is computed using a very coarse optimal grid. Furthermore, to regularize the inversion process we use the so-called multiplicative regularization technique. This technique automatically determines the regularization parameter. Synthetic data tests indicate the developed inversion algorithm is robust in extracting formation and invasion anisotropic resistivities, invasion radii, bed boundary locations, relative dip, and azimuth angle from multispacing, multicomponent induction logging data.

A novel approach to scatterers localization problem

A new approach to find the number and locations of electromagnetic scatterers (sources) from one view, one frequency field data is presented. The case of scatterers (sources) with dimensions comparable to the wavelength is considered. First, in order to devise an effective technique, a new relevant property of the electromagnetic field, the point source spectral content, is introduced and its relationship with both the sliding windowed Fourier transform of the field and the local bandwidth function is discussed. To enlighten its usefulness it is also shown that a footprint of the scattering system encoding information on its geometry can be easily extracted from the scattered field by exploiting this new property. On this basis, the new localization technique is introduced. In order to restrict the search region, the minimum circular envelope enclosing the scatterers is found by purposely introducing a new technique exploiting the effective bandwidth of the radiated field. Then, the number and locations of scatterers is retrieved by using the field local quantitative feature previously introduced, without the complexities of the full inverse problem as it is usually done by the traditional approaches. In this way, not only a simplified technique is obtained but also the ill-posedness of the problem and the noise effects are significantly mitigated. The effectiveness of the proposed technique and its overall performance with respect to a singular value decomposition based approach are proved by means of a numerical analysis.

Reconstruction of the three-dimensional refractive index in electromagneticscattering by using a propagation–backpropagation method

This paper is concerned with the development of an iterative algorithm forthe reconstruction of the three-dimensional complex-valued refractiveindex of an object in electromagnetic scattering. The inverse problem ismodelled by a second-order Maxwell equation for the electric field atfixed frequency with plane, time-harmonic microwave irradiation. Thecomplex-valued scattered electric field is measured at some distance from theobject for a finite number of directions of the incident waves. Each ofthese measurements is used to improve iteratively a starting guess forthe true refractive index of the object. For each iteration one forwardand one adjoint problem of a system of a second-order equation haveto be solved. The complexity of each step is O(N3log(N)).The algorithm solves the full nonlinear inverse problem. It can be consideredas a nonlinear generalization of the algebraic reconstruction techniquein x-ray tomography. The practical usefulness of the algorithm is shown in threenumerical simulations.

The effective properties of a perforated elastic plate Numerical optimization by genetic algorithm

We consider a perforated elastic plate in which identical traction-free holes form square or hexagonal periodic arrays. The optimization problem of finding the hole shape in this structure that maximizes either of its effective moduli is posed as a full scale inverse problem of elasticity. In this context, only limiting cases of a single hole and cellular solids with thin cell walls have been studied thus far. Here, we cover the gap between these cases numerically solving the problem by the genetic algorithm approach. A new time-saving scheme of the fitness evaluation provides reliable data even near the percolation limit. The presented numerical results comprehensively describe the optimal behavior of perforated plates in terms of equivalent homogeneous medium. Similar, though less sophisticated approach was used by the author Vigdergauz, (Int. J. Solids Struct. 38 (2001) 6851) to optimize an isolated inclusion in a plate.

Recovery of the elastic parameters of a layered half-space

In this article we begin The study of the problem of recovering the elastic parameters of a layered half-space from single component measurements of reflected waves, and applications to reflection seismology. We analyse the perturbational relationship between the elastic moduli (p, λ and µ and the vertical component of surface particle displacement due to a point impulsive traction. We show that this linearized problem has a unique solution and derive some stability estimates. The analysis of the linearized problem proceeds by way of the plane wave decomposition, and the uniqueness proof is constructive. To illustrate these ideas, we implement the construction numerically and recover parameter perturbations from data perturbations in a synthetic example. The solution of the linearized problem may be viewed as a step in a Newton-like scheme for solution of the full inverse problem. In considering the application of any such technique to field recorded data, however, one must confront problems such as model inadequacy, data error, source signature error, band limitation, etc. The extent to which these problems are resolvable has been explored for the simpler acoustic model using techniques compatible with those presented here. We plan to address such (practically crucial) stability considerations, and their implications for the design of useful algorithms, elsewhere. The focus of the present paper is thus the mathematical uniqueness question, which may be regarded as the most primitive stability question.

Extension of the one-dimensional scattering theory, and ambiguities

Extensions of the one-dimensional scattering theory are made in two directions. In the first, one uses the Schrodinger equation. Starting from the Fadeev class of potentials (V mod integral - infinity infinity (1+ mod x mod ) mod V(x) mod dx< infinity ) or from narrower classes, the authors introduce a special x-2 behaviour at infinity. In the second, they use the impedance equation, which reduces to the Schrodinger equation if the impedance factor alpha is positive everywhere and twice differentiable, but not if the impedance factor is only piecewise differentiable-this is the working assumption used in the present paper. Behaviour of the impedance at infinity which correspond to the x-2 behaviour of the potential are also admitted. They proceed through repeated applications of a transformation which is a limiting case of a Darboux transformation and acts on the parameters in a simple algebraic way, whereas the effect on the reflection coefficient R+(k) is an overall flip Rx to -R+. Because of these extensions of the scattering theory, ghosts, singularities, etc, may appear in the analysis. Ambiguities also appear in the inverse problem; these are related to zero-energy bound states. They are discussed in detail and the theory is shown to unify all the previously published material on potential ambiguities. The full inverse problem, however, is not treated by the authors.

Acoustic scattering by ocean irregularities: Aspects of the inverse problem

Expressions are derived for the autocorrelation and structure functions of the acoustic refractive index for ocean internal waves in the presence of fine structure. The transverse autocorrelation and structure functions required in acoustic scattering formulas are obtained from these. The spectra and correlation functions of acoustic phase, complex amplitude, and intensity are considered in the case of multiple scattering, and it is shown that the effect of fine structure on the fourth moment of the acoustic field is fundamentally different from that exerted on the second moment or the phase. The problem of deducing the correlation function of ocean irregularities from the acoustic signal is considered, and it is shown that, under certain circumstances, the full inverse problem can be solved by using the second moment of acoustic complex amplitude. Finally, the effect of tides on the phase, second moment, and the spectrum of intensity fluctuations of the acoustic signal is considered. It is shown that thi...

Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

Utilization Of The Algor Package In Solution OfThe Inverse Heat Convection Problems

This paper presents the method of application of the commercial package Algor in the inverse heat convection problems solutions. Due to the Algor functions it has to be supplemented by additional modules which allows us to solve full inverse problem. Constructed hybrid program make it possible to solve various formulations of inverse problems. Two of them are dealt with here. These are evaluating boundary heat flux from the internal temperature measurements (inverse formulation of energy equation), and estimation of inlet velocity profile from internal velocity measurements (inverse formulation of momentum equation).