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 Volume 39, Issue 6, 1991
Geophysical Prospecting  Volume 39, Issue 6, 1991
Volume 39, Issue 6, 1991


AN ALTERNATIVE STRATEGY FOR NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS^{1}
Authors M. S. SAMBRIDGE, A. TARANTOLA and B. L. N. KENNETTAbstractA common example of a large‐scale non‐linear inverse problem is the inversion of seismic waveforms. Techniques used to solve this type of problem usually involve finding the minimum of some misfit function between observations and theoretical predictions. As the size of the problem increases, techniques requiring the inversion of large matrices become very cumbersome. Considerable storage and computational effort are required to perform the inversion and to avoid stability problems. Consequently methods which do not require any large‐scale matrix inversion have proved to be very popular. Currently, descent type algorithms are in widespread use. Usually at each iteration a descent direction is derived from the gradient of the misfit function and an improvement is made to an existing model based on this, and perhaps previous descent directions.
A common feature in nearly all geophysically relevant problems is the existence of separate parameter types in the inversion, i.e. unknowns of different dimension and character. However, this fundamental difference in parameter types is not reflected in the inversion algorithms used. Usually gradient methods either mix parameter types together and take little notice of the individual character or assume some knowledge of their relative importance within the inversion process.
We propose a new strategy for the non‐linear inversion of multi‐offset reflection data. The paper is entirely theoretical and its aim is to show how a technique which has been applied in reflection tomography and to the inversion of arrival times for 3D structure, may be used in the waveform case. Specifically we show how to extend the algorithm presented by Tarantola to incorporate the subspace scheme. The proposed strategy involves no large‐scale matrix inversion but pays particular attention to different parameter types in the inversion.
We use the formulae of Tarantola to state the problem as one of optimization and derive the same descent vectors. The new technique splits the descent vector so that each part depends on a different parameter type, and proceeds to minimize the misfit function within the sub‐space defined by these individual descent vectors. In this way, optimal use is made of the descent vector components, i.e. one finds the combination which produces the greatest reduction in the misfit function based on a local linearization of the problem within the subspace. This is not the case with other gradient methods. By solving a linearized problem in the chosen subspace, at each iteration one need only invert a small well‐conditioned matrix (the projection of the full Hessian on to the subspace). The method is a hybrid between gradient and matrix inversion methods. The proposed algorithm requires the same gradient vectors to be determined as in the algorithm of Tarantola, although its primary aim is to make better use of those calculations in minimizing the objective function.



A FILTER, DELAY AND SPREAD TECHNIQUE FOR 3D DMO^{1}
By R.‐G. FERBERAbstractThe calculation of dip moveout involves spreading the amplitudes of each input trace along the source‐receiver axis followed by stacking the results into a 3D zero‐offset data cube. The offset‐traveltime (x–t) domain integral implementation of the DMO operator is very efficient in terms of computation time but suffers from operator aliasing. The log‐stretch approach, using a logarithmic transformation of the time axis to force the DMO operator to be time invariant, can avoid operator aliasing by direct implementation in the frequency‐wavenumber (f–k) domain.
An alternative technique for log‐stretch DMO corrections using the anti‐aliasing filters of the f–k approach in the x‐log t domain will be presented. Conventionally, the 2D filter representing the DMO operator is designed and applied in the f–k domain. The new technique uses a 2D convolution filter acting in single input/multiple output trace mode. Each single input trace is passed through several 1D filters to create the overall DMO response of that trace.
The resulting traces can be stacked directly in the 3D data cube. The single trace filters are the result of a filter design technique reducing the 2D problem to several ID problems. These filters can be decomposed into a pure time‐delay and a low‐pass filter, representing the kinematic and dynamic behaviour of the DMO operator. The low‐pass filters avoid any incidental operator aliasing. Different types of low‐pass filters can be used to achieve different amplitude‐versus‐offset characteristics of the DMO operator.



TUNNELLING AND THE GENERALIZED RAY METHOD IN PIECEWISE HOMOGENEOUS MEDIA^{1}
More LessAbstractThe contributions of the tunnelled constituents to a seismic wave are analysed in two different configurations pertaining to homogeneous acoustic media: a thin high‐velocity layer, present in a plane‐layered configuration, and a thin layer in media separated by dipping interfaces. The generalized ray method in the far‐field is used to determine them. We expand around the relevant ray parameters in order to determine the characteristics of the tunnelling ray and find that the most important feature of this type of ray is a phase (in terms of asymptotic ray theory) which has a real and an imaginary part. Numerical results illustrate this.



CHEBYSHEV EXPANSIONS FOR THE SOLUTION OF THE FORWARD GRAVITY PROBLEM^{1}
Authors S. K. ZHAO and M. J. YEDLINAbstractChebyshev expansions are used to solve the 3D forward gravity problem. Since the matrix factorization method is used to solve the coefficient equation system and the fast Fourier transform (FFT) technique is used to compute the forward and backward Chebyshev expansions, this method is very fast. Multipole expansions are used to calculate approximate boundary conditions (BCs) for realistic problems. When the length of any source‐body dimension is less than 70% of the minimum dimension of the computational domain, the relative error caused by the approximate BCs is about 1%. A cell‐average discretization method is suggested. The accuracy obtained by the cell‐average discretization is much better than that obtained by the traditional point‐injection discretization. The Chebyshev expansion technique was applied to four density models including a complex geological structure consisting of two normally faulted layers. Models which were finely sampled had a maximum relative error of about 1%.



ASPECTS OF CHARGE ACCUMULATION IN d. c. RESISTIVITY EXPERIMENTS^{1}
Authors YAOGUO LI and DOUGLAS W. OLDENBURGAbstractWhen an electric current is introduced to the earth, it sets up a distribution of charges both on and beneath the earth's surface. These charges give rise to the anomalous potential measured in the d. c. resistivity experiment. We investigate different aspects of charge accumulation and its fundamental role in d. c. experiments. The basic equations and boundary conditions for the d. c. problem are first presented with emphasis on the terms involving accumulated charges which occur wherever there is a non‐zero component of electric field parallel to the gradient of conductivity. In the case of a polarizable medium, the polarization charges are also present due to the applied electric field, yet they do not change the final field distribution. We investigate the precise role of the permittivity of the medium. The charge buildup alters the electric fields and causes the refraction of current lines; this results in the well‐known phenomenon of current channelling. We demonstrate this by using charge density to derive the refraction formula at a boundary. An integral equation for charge density is presented for whole‐space models where the upper half‐space is treated as an in‐homogeneity with zero conductivity. The integral equation provides a tool with which the charge accumulation can be examined quantitatively and employed in the practical forward modelling. With the aid of this equation, we investigate the effect of accumulated charges on the earth's surface and show the equivalence between the half‐space and whole‐space formulations of the problem. Two analytic examples are presented to illustrate the charge accumulation and its role in the d. c. problem. We investigate the relationship between the solution for the potential via the image method and via the charge density. We show that the essence of the image method solution to the potential problem is to derive a set of fictitious sources which produce the same potential as does the true charge distribution. It follows that the image method is viable only when the conductivity structure is such that the effect of the accumulated charge can be represented by a set of point images. As numerical examples, we evaluate quantitatively the charge density on the earth's surface that arises because of topography and the charge density on a buried conductive prism. By these means, we demonstrate the use of the boundary element technique and charge density in d. c. forward modelling problems.



DAMPED LEAST‐SQUARES INVERSION OF TIME‐DOMAIN AIRBORNE EM DATA BASED ON SINGULAR VALUE DECOMPOSITION^{1}
Authors H. HUANG and G. J. PALACKYAbstractAirborne electromagnetic (AEM) methods are increasingly being used as tools of geological mapping, groundwater exploration and prospecting for coal and lignite. In such applications, quantitative interpretation is commonly based on the layered‐earth model. A new approach, a damped least‐squares inversion with singular value decomposition, is proposed for interpretation of time‐domain, towed‐bird AEM data. Studies using theoretical and field AEM data indicate that inversion techniques are dependable and provide fast converging solutions. An analysis has been made of the accuracy of model parameter determination, which depends on resistivity and thickness distribution. In the common case of conductive overburden, upper‐layer resistivity and thickness are usually well determined, although situations exist where their separation becomes difficult. In the case of a resistive layer overlying a conductive basement, the layer thickness is the best‐determined parameter. In both cases, estimates of basement resistivity are the least reliable. Field data obtained with the Chinese‐made M‐l AEM system in Dongling, Anhui Province, China, were processed using the described inversion algorithm. The survey area comprised fluvial Cenozoic clays and weathered Mesozoic sediments. Inversion of AEM data resulted in accurate depth‐to‐bedrock sections and realistic estimates of the resistivities of overburden and bedrock which agree with the results of drilling and resistivity sounding.



DETECTION OF AN AIR‐FILLED DRAINAGE GALLERY BY THE VLF RESISTIVITY METHOD^{1}
Authors R. D. OGILVY, A. CUADRA, P. D. JACKSON and J. L. MONTEAbstractVLF wave impedance measurements have been successfully used to detect air‐filled drainage galleries near the town of Alcala, Spain. The galleries are detectable by H‐polarization electric field measurements due to the electric field anomalies associated with the galleries and overlying gravel deposits. The forced deviation of the primary current flowlines around the 2D void results in a higher‐than‐normal apparent resistivity and a relative phase low above the gallery.
The findings support earlier theoretical predictions that at very low frequencies (VLFs), galvanic current effects may dominate over vortex currents in moderately conductive terrains. Theoretical modelling confirmed that for a resistive target no detectable E‐polarization response can be expected from either magnetic or electric field measurements since current line deviations and vortex effects are negligible under such circumstances.
The results demonstrate the importance of using at least two orthogonal VLF transmitters in order that anomalies arising from both galvanic and inductive effects may be identified, irrespective of orientation.

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