Treatment of geophysical data as a non-stationary process
Treatment of geophysical data as a non-stationary process
Marcus P.C. Rocha; Lourenildo W.B. Leite
Department of Mathematics, Department of Geophysics, Graduate Course in Geophysics, Federal University of Pará, Brazil
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ABSTRACT
The Kalman-Bucy method is here analized and applied to the solution of a specific filtering problem to increase the signal message/noise ratio. The method is a time domain treatment of a geophysical process classified as stochastic non-stationary. The derivation of the estimator is based on the relationship between the Kalman-Bucy and Wiener approaches for linear systems. In the present work we emphasize the criterion used, the model with apriori information, the algorithm, and the quality as related to the results. The examples are for the ideal well-log response, and the results indicate that this method can be used on a variety of geophysical data treatments, and its study clearly offers a proper insight into modeling and processing of geophysical problems.
Mathematical subject classification: 49N45, 54H20, 74H50, 93C05.
Key words: stochastic process, Kalmann-Bucy filter, deconvolution, state space.
1 Introduction
A seismic signal represents the transient response of the Earth to excitation due to natural phenomena, such as eathquakes, or due to artificial sources as used in geophysical exploration, and it results in non-stationaty signals in noise. The aim of the seismic processing is to improve conditions for the interpretation of registered data. A detailed representation of a seismic signal requires a rather complicated formulation, and the processing uses a group of techniques based on stochastic properties of the model.
Two types of mathematical methods for data treatment can be used to represent a seismic signal: deterministic and stochastic. The deterministic method consists of using physical theories of wave propagation involving solutions of integral and differential equations satisfying contour and initial conditions. The stochastic method takes the statistical description of time series for the expressions of dynamic laws as statistical facts.
In this this work we apply Kalman-Bucy method to treat geophysical data. For this, we analyze the a priori necessary conditions to transform the governing integral equation of the first kind to linear and to non-linear ordinary differential equations using the state space technique. We reinforce the understanding of the limitations of the procedure for the separation of the signal message from noise in the time domain, and we make use of the property that geophysical data represent realizations that are strongly white non-stationary stochastic processes.
The importance of the Kalman method stems both conceptually in the formulation of the solution of a fundamental geophysical problem, and also from its versatility and applicability as an adaptative data processing. Some basic geophysical references followed in the present topic are Robinson (1999), Mendel (1990, 1983), Crump (1974), Bayless and Brigham (1970) and Van Trees (1968). Engeneering applications have available a heavy machinery on this subject for real-time applications, and to name a few we count with Kailath (1981), Chui and Chen (1987), Candy (1987) and Brown and Hwang (1996).
2 The Wiener-Kolmogorov problem
For motivation, consider the classical optimum time-invariant operator obtained by using the criterion of minimum error variance between the actual output, (t), and the desired output, x(t). The model is expressed by
where z(t) is the measurement, and v(t) is the additive noise. The filter operation is described by the convolution integral
where h(t) is the unknown time-invarint operator that is constrained to satisfy the commonly referred as the Wiener-Hopf integral equation
where fxz(t) and fzz(t) are, respectively, the admitted known theoretical stochastic crosscorrelation and autocorrelation functions. It this basic formulation, x(t) and v(t) are stationary random processes, and together with z(t), h(t), fxz(t) and fzz(t) are real function, continuous, time unbound, and convergent.
To specify the optimun (not necessary best) operator it is necessary to solve the integral equation 3. This situation becomes more difficult as the complexity of the problem increases, and when it does not satisfy strictly the characteristics of the geophysical problem that we have in hands that has for description to be non-stationary.
The present problem, with respect to the non-stationarity and to the data window, does not satisfy the principles underlined by the convolution integral. For this reason the equation is rewritten in the form of a moving average according to the commonly referred to as the Wiener-Kolmolgorov problem. The generalization corresponds to the integral of the Booton type, and it is expressed by the matrix integral equation
for
where (t) (...truncated)