Focused blind deconvolution of interferometric Green’s functions

Abstract

We detail a novel multichannel blind deconvolution (BD) algorithm that extracts the cross-correlated or interferometric Green's functions from the records due to a single noisy source. In this framework, we perform a least-squares fit of the crosscorrelated records, rather than the raw records, which greatly reduces the indeterminacy inherent to traditional BD methods. To resolve the remaining degrees of freedom, we seek a first approximation where the Green's functions are "maximally white", and relax this requirement as the iterations progress. This requirement is encoded as the focusing near zero lag of the energy of the auto-correlated Green's functions, hence we call the method focused blind deconvolution (FBD). We demonstrate the benefits of FBD using synthetic seismic-while-drilling experiments to look around and ahead of a borehole. Here, the noise due to the operation of the drill bit is not directly usable for reflection imaging, but FBD can provide the processing needed to extract the noise signature without unrealistically assuming the drill noise to be uncorrelated. The interferometric Green's functions obtained from FBD can either be directly imaged or further processed to output the usual subsurface Green's functions. Note that FBD is designed for an acquisition where the noise is recorded for a longer time period than the propagation time of the seismic waves e.g., as could be done during normal drilling operations. Traditional seismic imaging may now be augmented by added information around and ahead of the drill bit, potentially allowing less frequent traditional surveys. * Surveys of BD algorithms in the signal and image processing literature are given in Kundur and Hatzinakos (1996) and Campisi and Egiazarian (2016).

Identifiability Conditions for Multi-channel Blind Deconvolution with Short Filters

“…In most applications, however, the support size is not known a priori but rather needs to be discovered. In such applications, other ways of encoding the short channels assumption should be employed (see for example [4]). What happens when the conditions of Prop.…”

“…In SWD, the objective is to perform subsurface imaging on a drilling site without disrupting drilling operations. One proposed way of achieving this is to use the vibrations generated by the drill-bit in the subsurface as the input signal s. Multiple receivers recording the seismic traces at several locations would then allow to recover the "channel impulse responses of the Earth" {w n } N −1 n=0 using multi-channel blind deconvolution [4]. Other applications of blind deconvolution include medical imaging [5], astronomical imaging [6] and computer vision [7].…”