A general algorithm for high-resolution non-stationary deconvolution in presence of Gaussian and spike-like noises

Definition of a suitable cost function is a critical step in solving a problem based on inverse theory, and minimization of the designed cost function can also be very challenging. Non-linear cost functions are usually solved by using iterative algorithms where selection of the parameters in each iteration has a profound effect on the speed of convergence and the quality of the final solution. Therefore, a scheme for automatic determination of the parameters in a general framework can be highly effective in solving inverse problems arising in applied geophysics. Seismic nonstationary deconvolution is a highly ill-conditioned problem, which is required to be solved when improving the vertical resolution of the data is intended. The solution of this problem can be very challenging specifically when a high-resolution solution is desired and when the contaminant noise in the data is non-Gaussian or spike-like. In this paper, we consider this problem assuming that the seismic wavelet and the medium quality factor (Q) are known. Specifically, we consider the minimization of a general cost function for solution of seismic non-stationary deconvolution. The iteratively reweighted least squares (IRLS) algorithm is a common technique for solving this kind of problems in geophysics. However, automatic determination of the regularization parameter in each iteration of IRLS is not an easy task and making the algorithm inefficient. Here, we extend the recently developed method, called iterative reweighted and rrefined least squares (IRRLS) method, for treating seismic deconvolution and propose a new scheme based on the secant method for automatic update of the regularization parameter in each iteration.

According to the convolutional model of the Earth, a seismic signal can be modeled as convolution of the source generated wavelet (w) with the Earth impulse response. The Earth impulse response contains the reflectivity information of the layer boundaries and the elasticity effects of the medium such as attenuation, absorption, etc. The aim of non-stationary seismic deconvolution is to recover information about subsurface from non-stationary seismic signals by solving a multi objective non-linear optimization. Solving this optimization without any prior information about w and attenuation (Q) will be impossible. There are some methods for estimating w and Q from surface seismic data. In the case of known Q and w, the problem changes to a linear optimization with non-linear cost function. The IRLS is a popular method in geophysics for solving this kind of non-linear cost functions but it can be time consuming when there is no information about the noise bound. An alternative IRRLS has been proposed that solves non-linear cost functions iteratively. Furthermore, an automatic algorithm has been developed for updating the regularization parameter at each iteration.

We have proposed a method, based on the IRRLS and secant method of root finding, for high resolution constrained non-stationary deconvolution. Numerical examples from simulated results confirmed that the proposed method is not sensitive to the initial parameters and provide high-resolution estimates of seismic reflectivity series.