Comparação da solução completa do sistema de Lamé com espectro de sismogramas

Abstract

ABSTRACT. The questions connected with the solution of direct and inverse dynamic problems for thin layer elastic models in spectral domain are considered in this work. The solution of the corresponding direct problem is obtained using the complete Lamé system represented in two-dimensional spectral domain. Such transition is based on the Laplace transform with respect to the time variable and the Fourier-Bessel transform with respect to the spatial variables. Similar transformations are used for calculation of corresponding two-dimensional spectra by multicomponent seismograms. The possibility of compatibility of these two types of the spectra calculated theoretically and using real seismograms is studied. The matter is that the direct problem, solved in the spectral domain, defines the seismograms in infinite limits on the spatial and time variables. At the same time we have real observation on the limited aperture and in a finite time interval. Transformation of the limited discrete seismogram gives the deformed spectrum which can essentially differ from the theoretical solution. We study the influence of the Laplace parameter and smoothing filters on degree of compatibility of two specified types of spectra. There is shown the possibility to obtain comprehensible degree of compatibility of these spectra for synthetic seismograms, allowing to differentiate types of the models and to distinguish change of their parameters.Keywords: thin layer elastic models, Lamé system, direct and inverse problems, Laplace and Fourier-Bessel transforms, spectral analysis, limited aperture. RESUMO.Neste trabalho foram consideradas questões associadasà solução de problemas dinâmicos diretos e inversos para os modelos elásticos delgados no domínio espectral. A solução de problemas diretos obtém-se usando a transformação do sistema de Lamé completo no domínio espectral bidimensional utilizando a transformada de Laplace com respeitoà variável temporal e a transformada de Fourier-Bessel com respeitoàs variáveis espaciais. Tais transformadas utilizam sismogramas multicomponentes para cálculo dos espectros correspondentes. Uma questão fundamentalé a possibilidade de combinação dos espectros, calculados teoricamente e por sismogramas reais. A dificuldadeé que o problema direto, resolvido no domínio espectral, define o sismograma com limites infinitos com respeitoàs variáveis temporal e espaciais. Ao mesmo tempo as observações reais são registradas com abertura limitada e num intervalo de tempo finito. A transformação de sismogramas limitados e discretos dá um espectro deformado, que pode ser bem diferente da solução teórica. Neste trabalho estuda-se a influência do parâmetro de Laplace e dos filtros de suavização no grau de compatibilidade entre estes dois tipos de espectros. O resultado principal desta investigação foi a obtenção de um nível aceitável de compatibilidade dos espectros para sismogramas sintéticos, o que permite diferenciar vários tipos de modelos e distinguir a mudança dos parâmetros deles.Palavras-chave: modelos elást...

Structural decomposition of the wave field in the solution of inverse seismic problems

“…This is absolutely unacceptable value of the functional, because similar values are obtained by setting the wrong model of the object (Mitrofanov et al, 2009c). The fact that similar differences between theoretical spectra and calculated spectra are related to inaccurate definition of the incident pulse shape, was confirmed by the results presented in Figure 9(b).…”

“…It also shows the variation of elastic parameters V p , V s , ρ, which will be consistent with thin-layer objects (blue-color variation V p , red V s and purple ρ). Models of objects in its structure correspond to two types of thin-layer models, and were used in our previous works (Mitrofanov et al, 2009b(Mitrofanov et al, , 2009c. Also, the figure shows the values of elastic parameters for the thick layers of macro model.…”

“…Selected signals served as input data to determine the spectral characteristics. This approach fully corresponds to the dynamic inversion, widely used in practice, and can be considered in terms of the linearized solution of the inverse dynamic problem, proposed in our previous articles, see Mitrofanov et al (2009bMitrofanov et al ( , 2009c. Essential here is that in the calculation of synthetic seismograms we use a complete solution that covers all the features of the generated wave field.…”

“…In our previous articles (see Mitrofanov et al, 2009bMitrofanov et al, , 2009c we have dealt with combination of spectra, based on the theoretical solution of the direct problem, with the spectra calculated from the observed seismograms. These issues are the fundamentals for implementation of the proposed algorithms in practice.…”