Hugo Hidalgo

Application of constructive learning algorithms to the inverse problem

A constructive learning algorithm is used to generate networks that learn to approximate the functional of the magnetotelluric inverse problem. Based on synthetic data, several experiments are performed in order to generate and test the neural networks. Rather than producing, at the present time, a practical algorithm using this approach, the object of the paper is to explore the possibilities offered by the new tools. The generated networks can be used as an internal module in a more general inversion program, or their predicted models can be used by themselves or simply as inputs to an optimization program.

Piecewise smooth models for electromagnetic inverse problems

This paper presents a new method for constructing one-dimensional (1D) electrical conductivity models of the Earth from surface electromagnetic measurements. The construction of these models is a nonlinear inverse problem that can be approached by linearization techniques combined with iterative methods and Tikhonovs regularization. The standard application of these techniques usually leads to smooth models that represent a continuous variation of conductivity with depth. In this work, the authors describe how these methods can be modified to incorporate what is known in computer vision as the line process (LP) decoupling technique, which has the ability to include discontinuities in the models. This results in piecewise smooth models that are often more adequate for representing stratified media. They have implemented a relaxation technique to construct these types of models and present numerical experiments as well as an application to field data. These examples illustrate the performance of the combined LP and Tikhonovs regularization method.

Contextual Entropy and Text Categorization

In this paper we describe a new approach to text categorization, our focus is in the amount of information (the entropy) in the text. The entropy is computed with the empirical distribution of words in the text. We provide the system with a manually segmented collection of documents in different categories. For each category a separate empirical distribution of words is computed, we use this empirical distribution for categorization purposes. If we compute the entropy of the test document for each empirical distribution the correct category shows as a maximum. For example, if we compute the entropy of a sports document using the politics or the sports empirical word distributions then the computed entropy is higher in sports than in politics. Our text categorization approach is simple, easy to code and needs no training time (aside from histogram computations). The classification time is linear on the size of the document and the number of document categories. We support our claims with extensive experimentation

Application of the kernel method to the inverse geosounding problem

Determining the layered structure of the earth demands the solution of a variety of inverse problems; in the case of electromagnetic soundings at low induction numbers, the problem is linear, for the measurements may be represented as a linear functional of the electrical conductivity distribution. In this paper, an application of the support vector (SV) regression technique to the inversion of electromagnetic data is presented. We take advantage of the regularizing properties of the SV learning algorithm and use it as a modeling technique with synthetic and field data. The SV method presents better recovery of synthetic models than Tikhonovs regularization. As the SV formulation is solved in the space of the data, which has a small dimension in this application, a smaller problem than that considered with Tikhonovs regularization is produced. For field data, the SV formulation develops models similar to those obtained via linear programming techniques, but with the added characteristic of robustness.

Neural network applied in the geophysical inversion problem

A dynamic neural network is used to obtain the resistivity information of geologic structures. Based on synthetic data several simulations are made to train and test the designed neural network. Error figures are reported to evaluate the performance of the network.

Document analysis and visualization with zero-inflated poisson

Data visualization is aimed at obtaining a graphic representation of high dimensional information. A data projection over a lower dimensional space is pursued, looking for some structure on the projections. Among the several data projection based methods available, the Generative Topographic Mapping (GTM) has become an important probabilistic framework to model data. The application to document data requires a change in the original (Gaussian) model in order to consider binary or multinomial variables. There have been several modifications on GTM to consider this kind of data, but the resulting latent projections are all scattered on the visualization plane. A document visualization method is proposed in this paper, based on a generative probabilistic model consisting of a mixture of Zero-inflated Poisson distributions. The performance of the method is evaluated in terms of cluster forming for the latent projections with an index based on Fisher’s classifier, and the topology preservation capability is measured with the Sammon’s stress error. A comparison with the GTM implementation with Gaussian, multinomial and Poisson distributions and with a Latent Dirichlet model is presented, observing a greater performance for the proposed method. A graphic presentation of the projections is also provided, showing the advantage of the developed method in terms of visualization and class separation. A detailed analysis of some documents projected on the latent representation showed that most of the documents appearing away from the corresponding cluster could be identified as outliers.

Piecewise continuous models for resistivity soundings

A robust method is presented for constructing layered Earth models from surface resistivity data. The algorithm automatically accommodates any number of discontinuities without the need to specify a priori its number or location in the vertical profile. It further includes automatic correction factors for the common segmentation of Schlumberger soundings due to static shift effects.

Image restoration with local adaptive methods

Local adaptive processing in sliding transform domains for image restoration and noise removal with preservation of edges and detail boundaries represents a substantial advance in the development of signal and image processing techniques, thanks to its robustness to signal imperfections and local adaptivity (context sensitivity). Local filters in the domain of orthogonal transforms at each position of a moving window modify the orthogonal transform coefficients of a signal to obtain only an estimate of the central pixel of the window. A minimum mean-square error estimator in the domain of sliding discrete cosine and sine transforms for noise removal and restoration is derived. This estimator is based on fast inverse sliding transforms. To provide image processing at a high rate, fast recursive algorithm for computing the sliding sinusoidal transforms are utilized. The algorithms are based on a recursive relationship between three subsequent local spectra. Computer simulation results using synthetic and real images are provided and discussed.

Geoelectric modeling with Kernel methods

Electrical resistivity variations measured along vertical boreholes are usually very rough, with amplitudes of the shortest wavelengths being about equal to those of the longest ones. On the other hand, surface resistivity measurements are usually interpreted in terms of models that contain only long wavelengths. The usual approach consists of applying Tikhonovs regularization, incorporating model roughness penalizers for regularization. The roughness measures ordinarily considered are the first and second order derivatives of the model. Short wavelengths are avoided because they can not be recovered uniquely from the data. However, they do in fact influence the overall scale of the resulting model, so they must be taken into account somehow. In this paper we present an attempt to deal with this problem by way of constructing resistivity models that are at the same time smooth and rough. Our work is based on the kernel method which we have adapted for nonlinear inversion. We keep the original connection between regularization operators and support vector kernels, so the algorithm still possesses the regularization properties of kernel methods. By incorporating the kernel method as penalizer we are able to generate a variety of resistivity variations and include some desired properties for the resulting models.

Linear and quadratic programs for the conductivity reconstruction of subsurface

An approach for constructing vertical sections of the electrical conductivity of the Earth on the basis of surface measurements is presented. Electromagnetic measurements are effected every few meters across strike using pairs of horizontal and vertical coils, with the separation between source and receiver varying in order to sample different depths. The data provide information about the vertical and horizontal distribution of electrical conductivity of a section of the Earth. The recovering of the true conductivity distribution from the measurements is a nonlinear inverse problem that allows some useful linear approximations, which we exploit here using linear programming techniques. A regularization approach is considered using epsiv-insensitive functions for both the fitness to the data and the penalizing function. This representation enforces a model structure of blocks immersed in a homogeneous basement, an useful model in many practical instances