• Geophysics and Geophysical Exploration
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• v.8 no.2
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• pp.163-169
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• 2005

Recently, the velocity stack domain is having an attention as a very useful domain for various processing in seismic data processing. In order to be used in many applications, the velocity stack should be obtained through an inversion method and the used inversion should have properties like the robustness to noise and the parsimony of velocity stack result. Iteratively Reweighted Least-Squares (IRLS) method is the one of the inversion methods that have such properties. This paper describes the theoretical background, implementation of the method, and examines the characteristics and limits of the IRLS method.

• Geophysics and Geophysical Exploration
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• v.8 no.2
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• pp.170-176
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• 2005

Recently the velocity stack inversion is having many attentions as an useful way to perform various seismic data processing. In order to be used in various seismic data processing, the inversion method used should have properties such as robustness to noise and parsimony of the velocity stack result. The IRLS (Iteratively Reweighted Least-Squares) method that minimizes ${L_1}-norm$ is the one used mostly. This paper introduce another method, CGG (Conjugate Guided Gradient) method, which can be used to achieve the same goal as the IRLS method does. The CGG method is a modified CG (Conjugate Gradient) method that minimizes ${L_1}-norm$. This paper explains the CGG method and compares the result of it with the one of IRSL methods. Testing on synthetic and real data demonstrates that CGG method can be used as an inversion method f3r minimizing various residual/model norms like IRLS methods.

• Geophysics and Geophysical Exploration
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• v.8 no.2
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• pp.129-136
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• 2005

The damped least-squares inversion has become a most popular method in finding the solution in geophysical problems. Generally, the least-squares inversion is to minimize the object function which consists of data misfits and model constraints. Although both the data misfit and the model constraint take an important part in the least-squares inversion, most of the studies are concentrated on what kind of model constraint is imposed and how to select an optimum regularization parameter. Despite that each datum is recommended to be weighted according to its uncertainty or error in the data acquisition, the uncertainty is usually not available. Thus, the data weighting matrix is inevitably regarded as the identity matrix in the inversion. We present a new inversion scheme, in which the data weighting matrix is automatically obtained from the analysis of the data resolution matrix and its spread function. This approach, named 'extended active constraint balancing (EACB)', assigns a great weighting on the datum having a high resolution and vice versa. We demonstrate that by applying EACB to a two-dimensional resistivity tomography problem, the EACB approach helps to enhance both the resolution and the stability of the inversion process.

• 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.

• Economic and Environmental Geology
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• v.14 no.3
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• pp.193-201
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• 1981

The problem of automatic inversion of apparent resistivity sounding curves resulting from horizontally layered earth models is solved using the least-squares technique. This method, which makes use of damped least-squares algorithm in conjunction with digital filtering technique, is found to be speedier and more accurate than the conventional curve-matching method. Four sounding curves were chosen to test the inversion scheme. The analysis of the theoretical sounding data associated with a three-layer model illustrates clear advantages over the conventional curve-matching method. The usefulness of the inversion method is also shown when applied to the actual field data. It was found that the best fit earth models coincide with the subsurface structures confirmed by drilling.

• Geophysics and Geophysical Exploration
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• v.10 no.2
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• pp.124-130
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• 2007

Least squares ($l^2$ norm) solutions of seismic inversion tend to be very sensitive to data points with large errors. The $l^1$ norm minimization gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) method gives efficient approximate solutions of these $l^1$ norm problems. I propose an efficient implementation of the IRLS method for a hybrid $l^1/l^2$ minimization problem that behaves as $l^2$ norm fit for small residual and $l^1$ norm fit for large residuals. The proposed algorithm shows more robust characteristics to the decision of the threshold value than the l1 norm IRLS inversion does with respect to the threshold value to avoid singularity.

• Economic and Environmental Geology
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• v.28 no.4
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• pp.433-438
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• 1995

The most popular minimization method is based on the least-squares criterion, which uses the $L_2$ norm to quantify the misfit between observed and synthetic data. The solution of the least-squares problem is the maximum likelihood point of a probability density containing data with Gaussian uncertainties. The distribution of errors in the geophysical data is, however, seldom Gaussian. Using the $L_2$ norm, large and sparsely distributed errors adversely affect the solution, and the estimated model parameters may even be completely unphysical. On the other hand, the least-absolute-deviation optimization, which is based on the $L_1$ norm, has much more robust statistical properties in the presence of noise. The solution of the $L_1$ problem is the maximum likelihood point of a probability density containing data with longer-tailed errors than the Gaussian distribution. Thus, the $L_1$ norm gives more reliable estimates when a small number of large errors contaminate the data. The effect of outliers is further reduced by M-fitting method with Cauchy error criterion, which can be performed by iteratively reweighted least-squares method.

• 한국지구물리탐사학회:학술대회논문집
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• 2005.05a
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• pp.227-232
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• 2005

Least squares (${\ell}^2-norm$) solutions of seismic inversion tend to be very sensitive to data points with large errors. The ${\ell}^p-norm$ minimization for $1{\le}p<2$ gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions of these ${\ell}^p-norm$ problems. I propose a simple way to implement the IRLS method for a hybrid ${\ell}^1/{\ell}^2$ minimization problem that behaves as ${\ell}^2$ fit for small residual and ${\ell}^1$ fit for large residuals. Synthetic and a field-data examples demonstrates the improvement of the hybrid method over least squares when there are outliers in the data.

• 한국지구물리탐사학회:학술대회논문집
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• 2007.06a
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• pp.118-123
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• 2007

A robust objective function in the frequency domain is applied to the acoustic full waveform inversion. The proposed objective function is defined as $l_1$-norm of residual wavefields in the frequency domain. Generally, the full waveform inversion is extremely sensitive to a number of factors such as parameterization, initial model, noise and so on. The numerical tests were performed for checking the sensitivity to attenuation and several noises. For the comparison with other objective functions, the conventional least-squares method and the logarithmic method were tested under the same condition. The synthetic data examples show that the proposed algorithm is more robust than the well-known methods.

• Journal of the Korean Data and Information Science Society
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• v.18 no.1
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• pp.51-61
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• 2007

Ridge regression (RR), principal component regression (PCR) and partial least squares regression (PLS) are among popular regression methods for collinear data. While RR adds a small quantity called ridge constant to the diagonal of X'X to stabilize the matrix inversion and regression coefficients, PCR and PLS use latent variables derived from original variables to circumvent the collinearity problem. One problem of PCR and PLS is that they are very sensitive to overfitting. A new regression method is presented by combining RR and PCR and PLS, respectively, in a unified manner. It is intended to provide better predictive ability and improved stability for regression models. A real-world data from NIR spectroscopy is used to investigate the performance of the newly developed regression method.