In this work we develop a simple methodology to the representation of irregularly distributed data (originally described by two dimensions) in order to map them in an unique dimension along an arc of curve (in particular as triangular functions periodically spaced). It means: it is identified, approximately, the points of a planar region to those belonging to a curve superimposed on the abovementioned region. We apply, with reasonable success, such method to the case of the two-dimensional compressional wave velocity field proceeding from the geologic model of the continental shelf break strongly marked by rugosities. Results show that not big scale variations can influence, in a sensible way, the quality of model representation.
ABSTRACT. Seismic ray tomography methods are usually associated with substantial computer processing time. The reason for this is that at each step of the iterative inversion process defined by the tomographic method the two-point ray tracing problem must be solved for each source-receiver pair. In order to resolve this, an Euclidean norm (L 2 vector norm), commonly used in error functions which are to be minimized in inversion procedures, is substituted by an L 1 integral norm, which enables the estimation of model parameters by minimizing the area between observed and calculated traveltime curves that are interpolated (or adjusted) to the data points. Relatively simple mathematical developments and numerical experiments with two-dimensional compressional seismic wave velocity field models show that L 1 integral norm saves an enormous amount of processing time with no significant loss of accuracy. Occasionally, parameters of the model can be better estimated using L 1 integral norm than the L 2 vector norm that is traditionally utilized in seismic inversion tomography.Keywords: seismic ray, tomography, polynomial parameterization, seismic velocity field, two-point ray tracing problem, L 2 vector norm, L 1 integral norm.RESUMO. O alto consumo de tempo em processamento computacionalé um problema que, geralmente, está associado aos métodos de tomografia sísmica.Isto ocorre porque, em cada passo do processo iterativo de inversão definido pelo método tomográfico, o problema da conexão de dois pontos, pela curva da trajetória do raio sísmico, deve ser resolvido para cada par fonte-receptor. A fim de reduzir a gravidade deste tipo de problema, a norma Euclideana (norma L 2 de vetor), comumente empregada nas funções de erro a ser minimizado no processo de inversão,é substituída por uma norma L 1 de função. Essa mudança permite estimar parâmetros do modelo através da minimização daárea entre curvas de tempos observados e calculados que são interpoladas (ou ajustadas) aos pontos referentes aos dados. Desenvolvimentos matemáticos e experimentos numéricos relativamente simples, com modelos bidimensionais de campos de velocidade sísmica de ondas compressionais, mostram que a norma L 1 de função permite poupar uma enorme quantidade de tempo de processamento sem uma importante perda de precisão.As vezes, os parâmetros do modelo são estimados de modo mais acurado usando-se a norma L 1 da integral em lugar da norma L 2 de vetor, tradicionalmente usada na inversão tomográfica.Palavras-chave: raio sísmico, tomografia, parametrização polinomial, campo de velocidade sísmica, problema de conexão de dois pontos por traçado de raio, norma L 2 de vetor, norma L 1 de integral.
ABSTRACT. The algorithm developed, in this work, is based on the finite difference method applied to the acoustic wave equation, assuming that the Earth has an acoustic behavior, that allow us to implement, in a numerical way, a seismic modeling employing regular grids in models representing two-dimensional geological media. Second derivatives with respect to space and time of wave equation are obtained by Taylor series expansion of fourth and second orders, respectively. The models are represented by two different kinds of parameterizations: blocks and trigonometric polynomials. Simulations of the wave propagation phenomenon are accomplished in several models represented by the two mentioned parameterizations, making possible to generate synthetic seismograms to be compared. Seismograms, obtained when the polynomial parameterization is used, show some undesired behaviors such as: production of artificial reflections, weakening of reflection events, suppression of diffractions, and a little alteration in calculated traveltimes and amplitudes. By the other hand, some advantages of the proposed polynomial parameterization are: economy of computer memory space (because a complicated velocity model can be represented by a few quantity of coefficients, it means: the model is not more stored in a file, but it is compressed, or contained, in a mathematical formula or an analytical representation); production of a smooth velocity model (useful to generate a time field to be used in seismic migration, for example, in reverse time migration); the polynomial coefficients are just the parameters of the model to be estimated by an inversion procedure (independently of the degree of geometrical complication of the model and how is varying the seismic velocity on it); realistic models are better represented by trigonometric polynomial than block parameterization; and the ambiguity commonly present in inversion techniques results, is reduced if sufficient data is available. Gibbs effects, present in polynomial representations, is avoided by the finite differences method by choosing conveniently the knots of the mesh or increasing the number of polynomial coefficients. RESUMO.O algoritmo desenvolvido neste trabalhoé baseado no método das diferenças finitas aplicadoà equação acústica da onda, assumindo que a Terra se comporta como um meio acústico, que permite implementar numericamente uma modelagem sísmica empregando malhas regulares em modelos que representam meios geológicos bidimensionais. As derivadas segundas presentes na equação da onda foram obtidas por expansões da série de Taylor de quarta ordem, para o caso espacial, e de segunda, para o temporal. Os modelos foram representados por dois tipos distintos de parametrizações: por blocos e polinomial trigonométrica. São apresentadas simulações do fenômeno de propagação de ondas em vários modelos, representados pelas duas parametrizações mencionadas, possibilitando, assim, a geração de sismogramas sintéticos para comparação. Os sismogramas obtidos, com a parametrização poli...
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