Total least squares problem in seismic data deconvolution with inaccurate wavelet in the presence of noise

Seismic exploration is the science of imaging and exploring subsurface structures. Deconvolution is one of the major steps in seismic data processing, and is widely used in image processing, cosmology, medical imaging, and other applications. Recorded seismic data after some processing steps can be regarded as convolution of the Earth reflectivity series and seismic wavelet. Deconvolution of the data thus can result in the reflectivity series that consequently a high-resolution image of the underground can be constructed as the output of seismic processing. Traditional deconvolution methods based on the least squares (LS) filters, consider noise and uncertainty only on the observed data and assume the seismic wavelet to be known. In practical applications, however, these assumptions are met rarely because the seismic wavelet is usually unknown or is known only approximately. Thus, the kernel of deconvolution, which is constructed from the wavelet, is inaccurate. In this situation, the uncertainty is present in both the data and the kernel. The method of total least squares (TLS), as a modern mathematical tool, has been developed for solving such problems. In this paper, we propose a truncated TLS (T-TLS) algorithm for seismic data deconvolution. It is shown that in contrast to the conventional truncated singular-value-decomposition (T-SVD) technique, which truncates the singular values of the operator, the TTLS treats the singular values smoothly similar to the job done via the Wiener Filter. This means that the T-TLS as a truncation technique behaves as a hybrid of T-SVD and Wiener Filter. LS based deconvolution methods assume error only in data. In order to handle error in data and also in kernel, here we propose a T-TLS algorithm and compare its results with those of T-SVD and Wiener Filter solutions. The numerical example, presented in this paper, show that the performance of the T-TLS algorithm lies between these two techniques.