Semi-blind nonstationary deconvolution: Joint reflectivity and Q estimation

Abstract

Source signature deconvolution and attenuation or inverse quality factor- (Q-) filtering are two challenging problems in seismic data analysis which are used for extending the temporal bandwidth of the data. The separate estimates of the wavelet and, especially, the Earth Q model are by themselves problematic and add further uncertainties to inverse problems which are clearly ill-conditioned. The two problems are formulated in the framework of polynomial extrapolation and a closed form solution is provided based on the Lagrange interpolation. Analysis of the stability issue shows that the errors in the estimated results grow exponentially with both the problem size N and the inverse of Q. In order to circumvent both the instability and uncertainty of the Q model, these problems are addressed in a unified formulation as a semi-blind nonstationary deconvolution (SeND) to decompose the observed trace into the least number of nonstationary wavelets selected from a dictionary via a basis pursuit algorithm. The dictionary is constructed from the known source wavelet with different propagation times, each attenuated with a range of possible Q values. Using the Horner's rule, an efficient algorithm is also provided for application of the dictionary and its adjoint. SeND is an extension of the conventional sparse spike deconvolution to its nonstationary form, which provides the reflectivity and Q models simultaneously without requiring a-priori Q information. Assuming that the wavelet and attenuation mechanism are both known, the numerical data SeND allows to estimate both the original reflectivity and the Q models with higher accuracy, especially with respect to conventional spectral ratio techniques. The application of the algorithm to field data finally indicates a substantial improvement in temporal resolution on a seismic record.