Seismic Interferometry by Deconvolution: Theory and Examples

Seismic interferometry is a field of growing interest in exploration seismology. In this paper we provide the theoretical basis for performing interferometry by deconvolution. We argue that for arbitrarily complicated models, deconvolution interferometry gives the causal scattering response between any two receivers. Interferometry by cross-correlation, on the other hand, gives both causal and acausal scattering responses. Even with a closed surface of integration, deconvolution interferometry also gives rise to spurious events not present in its cross-correlation counterpart. These events arise from an extra boundary condition which is imposed by the deconvolution method in interferometry. We demonstrate the feasibility of deconvolution interferometry with numerical examples with impulsive sources, and show that the deconvolution interferometry artifacts typically are not mapped onto the image space. One application that can potentially benefit from deconvolution interferometry is imaging from drill-bit noise recordings. We compare the results from deconvolution interferometry to cross-correlation interferometry in numerical examples for a single-layer case and a subsalt drill-bit imaging example. Introduction Conventional seismic interferometry consists in extracting the Green’s function between two receivers by correlating the wavefields excited by incoherent sources and recorded at these two receivers (e.g., Weaver and Lobkis, 2004). Our objective here is to demonstrate that by deconvolving one recorded wavefield with the other, one can also extract the response between the receivers. Interferometry by cross-correlation is also known to preserve characteristics of the input excitation (e.g., the power spectrum of the source function). Deconvolution interferometry could then be a preferred method for cases in which the recorded data was excited by a complicated and poorly-known function. One example of this is drill-bit noise processing, where cross-correlated drill-bit noise shows a strong imprint of the drill-bit source function (Poletto and Miranda 2004). If this source imprint is not accounted for, the resulting drillbit noise image could be uninterpretable. Deconvolution interferometry can not only be a viable option to interferometric imaging, but it may also prove to be a key technology in some applications, such as imaging from drill-bit noise. Deconvolution interferometry Let the frequency-domain wavefield u(rA, s, ω) recorded at rA be the superposition of an unperturbed wavefield and a wavefield perturbation, represented respectively by u0(rA, s, ω) and uS(rA, s, ω). The source function is given by Ss(ω), associated with an excitation at s. It is important to note here that the medium may be arbitrarily heterogeneous and anisotropic, and that the recorded wavefields may contain higherorder scattering and inhomogeneous waves. Also, Ss(ω) can have a complicated character, and may vary as a function of s. Given that the deconvolution of a wavefield recorded at rA by that recorded at rB is represented by DAB , then to perform interferometry by deconvolution it is necessary to integrate DAB over a closed surface Σ containing the sources: ∫ Σ DAB(ω) ds = ∫ Σ u(rA, s, ω)u∗(rB, s, ω) |u(rB, s, ω)| ds , (1) where ∗ denotes complex-conjugation. We can see from right-hand side of equation [1] that the source function Ss(ω) cancels in the integrand. As in the cross-correlation approach (Snieder et al., 2006), the numerator in equation [1] is not zero-phase whereas the denominator is strictly zero-phase, but it is oscillatory and contains cross-terms between the scattered and direct wavefields recorded at rB . To evaluate the integral in equation [1], we expand the integrand in a power series. In this expansion, we identify the most prominent terms in the expansion of equation [1] by assuming |u0| >> |uS | to drop terms which EAGE 69th Conference & Technical Exhibition, London, UK, 11 14 June 2007 are quadratic on scattered wavefield. This approximation is analogous to the Born approximation for the Lippmann-Schwinger scattering series. From the series expansion of the integrand on the right-hand side of equation [1], the terms that are of leading order in the scattered wavefield are ∫