• Geophysics and Geophysical Exploration
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• v.7 no.3
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• pp.207-212
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• 2004

This article reviews recent developments in three-dimensional (3-D) magntotelluric (MT) imaging. The inversion of MT data is fundamentally ill-posed, and therefore the resultant solution is non-unique. A regularizing scheme must be involved to reduce the non-uniqueness while retaining certain a priori information in the solution. The standard approach to nonlinear inversion in geophysis has been the Gauss-Newton method, which solves a sequence of linearized inverse problems. When running to convergence, the algorithm minimizes an objective function over the space of models and in the sense produces an optimal solution of the inverse problem. The general usefulness of iterative, linearized inversion algorithms, however is greatly limited in 3-D MT applications by the requirement of computing the Jacobian(partial derivative, sensitivity) matrix of the forward problem. The difficulty may be relaxed using conjugate gradients(CG) methods. A linear CG technique is used to solve each step of Gauss-Newton iterations incompletely, while the method of nonlinear CG is applied directly to the minimization of the objective function. These CG techniques replace computation of jacobian matrix and solution of a large linear system with computations equivalent to only three forward problems per inversion iteration. Consequently, the algorithms are efficient in computational speed and memory requirement, making 3-D inversion feasible.

• Geophysics and Geophysical Exploration
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• v.10 no.2
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• pp.147-153
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• 2007

The conjugate gradient (CG) method is one of the most efficient algorithms for solving a linear system of equations. In addition to being used as a linear equation solver, it can be applied to a least-squares problem. When the CG method is applied to large-scale three-dimensional inversion of magnetotelluric data, two approaches have been pursued; one is the linear CG inversion in which each step of the Gauss-Newton iteration is incompletely solved using a truncated CG technique, and the other is referred to as the nonlinear CG inversion in which CG is directly applied to the minimization of objective functional for a nonlinear inverse problem. In each procedure we only need to compute the effect of the sensitivity matrix or its transpose multiplying an arbitrary vector, significantly reducing the computational requirements needed to do large-scale inversion.

• Tunnel and Underground Space
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• v.19 no.5
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• pp.407-420
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• 2009

In this study geostatistical technique using indicator kriging was performed to evaluate the optimal rock mass classification by integrating the various geophysical information such as borehole data and geophysical data. To get the optimal kriging result, it is necessary to devise the suitable technique to integrate the hard (borehole) and soft (geophysical) data effectively. Also, the model parameters of the variogram must be determined as a priori procedure. Iterative non-linear inversion method was implemented to determine the model parameters of theoretical variogram. To verify the algorithm, behaviour of object function and precision of convergence were investigated, revealing that gradient of the range is extremely small. This algorithm for the field data was applied to a mountainous area planned for a large-scale tunneling construction. As for a soft data, resistivity information from AMT survey is incorporated with RMR information from borehole data, a sort of hard data. Finally, RMR profiles were constructed and attempted to be interpreted at the tunnel elevation and the upper 1D level.

• Geophysics and Geophysical Exploration
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• v.2 no.4
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• pp.167-173
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• 1999

Since the resolution of the 2-D hole-to-hole seismic traveltime tomography is affected by the limited ray transmission angle, various methods were used to improve the resolution. Linear traveltime interpolation(LTI) ray tracing method was chosen for forward-modeling method. Inversion results using the LTI method were compared with those using the other ray tracing methods. As an inversion algorithm, SIRT method was used. In the iterative non-linear inversion method, the cost of ray tracing is quite expensive. To reduce the cost, each raypath was stored and the inversion was performed from this information. Using the proposed method, fast convergence was achieved. Inversion results are likely to be affected by the initial velocity guess, especially when the ray transmission angle was limited. To provide a good initial guess for the inversion, generalized regression neural network(GRNN) method was used. When the transmitted raypath angle is not limited or the geological model is very complex, the inversion results are not affected by initial velocity model very much. Since the raypath angles, however, are limited in most geophysical tomographic problems, the enhancement of resolution in tomography can be achieved by providing a proper initial velocity model by another inversion algorithm such as GRNN.

• 한국지구물리탐사학회:학술대회논문집
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• 2008.10a
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• pp.51-56
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• 2008

The waveform inversion for isotropic media has ever been studied since the 1980s, but there has been few studies for anisotropic media. We present a seismic waveform inversion algorithm for 2-D heterogeneous transversely isotropic structures. A cell-based finite difference algorithm for anisotropic media in time domain is adopted. The steepest descent during the non-linear iterative inversion approach is obtained by backpropagating residual errors using a reverse time migration technique. For scaling the gradient of a misfit function, we use the pseudo Hessian matrix which is assumed to neglect the zero-lag auto-correlation terms of impulse responses in the approximate Hessian matrix of the Gauss-Newton method. We demonstrate the use of these waveform inversion algorithm by applying them to a two layer model and the anisotropic Marmousi model data. With numerical examples, we show that it's difficult to converge to the true model when we assumed that anisotropic media are isotropic. Therefore, it is expected that our waveform inversion algorithm for anisotropic media is adequate to interpret real seismic exploration data.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.40 no.10
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• pp.150-158
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• 2003

In the beam pattern synthesis problem using line source, the relationship between source distribution function and beam pattern may be represented by Fourier transform pair. In this paper, a general method to synthesize the line source distribution for a desired lobe-like beam pattern is presented by developing the nonlinear inversion method based on an iterative sampling technique. This method can be applied to the synthesis of continuously distributed dielectric constants satisfying the desired inverse scattering coefficient patterns when illuminating by TE-polarized and TM-polarized plane waves to arbitrary dielectric material. Furthermore this method can also be applied to the synthesis of transmission line with arbitrary reflection coefficient patterns. Some bandstop spatial filter and dispersive transmission line filter are illustrated for generality.

• Geophysics and Geophysical Exploration
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• v.4 no.3
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• pp.70-79
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• 2001

The integral equation method is a powerful tool for numerical electromagnetic modeling. But the difficulty of this technique is the size of the linear equations, which demands excessive memory and calculation time to invert. This limitation of the integral equation method becomes critical in inverse problem. The conventional Born approximation, where the electric field in the anomalous body is approximated by the background field, is very rapid and easy to compute. However, the technique is inaccurate when the conductivity contrast between the body and the background medium is large. Quasi-linear, quasi-analytical and extended Born approximations are novel approaches to 3-D EM modeling based on the linearization of the integral equations for scattered EM field. These approximation methods are much less time consuming than full integral equation method and more accurate than conventional Born approximation. They we, however, still approximate methods for 3-D EM modeling. Iterative series methods such as modified Born, quasi-linear and quasi-analytical can be used to increase the accuracy of various approximation methods. Comparisons of numerical performance against a full integral equation and various approximation codes show that the iterative series methods are very accurate and almost always converge. Furthermore, they are very fast and easy to implement on a computer. In this study, extended Born series method is developed and it shows more accurate result than that of other series methods. Therefore, Iterative series methods, including extended Born series, open principally new possibilities for fast and accurate 3-D EM modeling and inversion.

• Geophysics and Geophysical Exploration
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• v.2 no.1
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• pp.17-25
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• 1999

A traveltime tomography has been carried out by transforming electromagnetic data in frequency domain to wave-like domain. The transform uniquely relates a field satisfying a diffusion equation to an integral of the corresponding wavefield. But direct transform of frequency domain magnetic fields to wave-field domain is ill-posed problem because the kernel of the integral transform is highly damped. In this study, instead of solving such an unstable problem, it is assumed that wave-fields in transformed domain can be approximated by sum of ray series. And for further simplicity, reflection and refraction energy compared to that of direct wave is weak enough to be neglected. Then first arrival can be approximated by calculating the traveltime of direct wave only. But these assumptions are valid when the conductivity contrast between background medium and the target anomalous body is low enough. So this approach can only be applied to the models with low conductivity contrast. To verify the algorithm, traveltime calculated by this approach was compared to that of direct transform method and exact traveltime, calculated analytically, for homogeneous whole space. The error in first arrival picked by this study was less than that of direct transformation method, especially when the number of frequency samples is less than 10, or when the data are noisy. Layered earth model with varying conductivity contrasts and inclined dyke model have been successfully imaged by applying nonlinear traveltime tomography in 30 iterations within three CPU minutes on a IBM Pentium Pro 200 MHz.