Interferometric imaging by deconvolution: theory and numerical examples Ivan Vasconcelos and Roel Snieder † † Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA Summary Seismic interferometry is a field of growing interest in exploration seismology. In this paper we pro- vide the theoretical basis for performing interfer- ometry by deconvolution. We argue that for gen- eral models, deconvolution interferometry gives only the causal scattering response between any two re- ceivers, as opposed to cross-correlation which gives both causal and acausal scattering responses. De- convolution interferometry also gives rise to a spuri- ous event not present in cross-correlation. Through a simple model, we gain physical insight about the meaning of each term in deconvolution interferome- try. We also show deconvolution interferometry can also be accomplished after summation over sources. We demonstrate the feasibility of deconvolution in- terferometry with numerical examples on with im- pulsive sources, and show that the deconvolution in- terferometry artifacts are not mapped onto the image space. Finally, we show that deconvolution interfer- ometry can successfully image drill-bit source data without independent estimates of the source function with quality comparable to impulsive source data. Introduction Seismic interferometry consists in extracting the Green’s function between two receivers by correlat- ing the wavefields recorded at these two receivers, excited by incoherent sources. Different proofs on how Green’s functions emerge from correlations are available in the literature: by representation theo- rems (Weaver and Lobkis, 2004), by time-reversal (Bakulin and Calvert, 2004), by stationary-phase analysis (Snieder et al., 2005) are just some exam- ples. In drill-bit seismic imaging, 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 SWD im- age could be uninterpretable. Deconvolution inter- ferometry can potentially not only be a viable option to multidimensional interferometric imaging, but it may also prove to be a key technology in some appli- cations, such as drill-bit imaging. Deconvolution versus cross-correlation Let the frequency-domain wavefield u(r A , s,ω) recorded at r A be the superposition of the impul- sive direct and scattered wavefields u 0 (r A , s,ω) and u S (r A , s,ω) convolved with a source function S s (ω) associated with an excitation at s. It is important to note here that the wavefields u(r A , s,ω) may be elastic and attenuative, and u S (r A , s,ω) may contain higher-order scattering and inhomogeneous waves. Also, S s (ω) can be arbitrarily complex, and may vary as a function of s. Given that the deconvolution of a wavefield recorded at r A by another recorded at r B is represented by D AB , then to perform interferome- try by deconvolution it is necessary to integrate D AB over some closed surface Σ containing the sources: Z Σ D AB (ω) ds = Z Σ u(rA, s,ω)u * (rB, s,ω) |u(r B , s,ω)| 2 ds , (1) where * denotes complex-conjugation. It follows form right-hand side of equation [1] that the source func- tion SS (ω) gets canceled in the integrand. As in the cross-correlation approach (Snieder et al., 2006), the numerator in equation [1] is not zero-phase and ex- pands to four wavefield cross-terms from recordings at rA and rB. The denominator is strictly zero-phase, but it is oscillatory and contains cross-terms between the scattered and direct wavefields at rB. To identify the leading order terms in equation [ 1] we assume |u 0 | 2 >> |u S | 2 to drop terms which are quadratic on scattered wavefields. The denominator can be fac- tored in |u 0 (r B , s,ω)| 2 , then we expand it first-order Taylor series. When multiplying the numerator by the expansion of the denominator we again ignore terms quadratic in scattering, which leads us to Z Σ D AB (ω) ds = Z Σ u 0 (r A , s,ω)u * 0 (r B , s,ω) |u0(rB, s,ω)| 2 ds | {z } D1 + Z Σ uS (rA, s,ω)u * 0 (rB, s,ω) |u 0 (r B , s,ω)| 2 ds | {z } D2 - Z Σ u S (r B , s,ω)u 0 (r A , s,ω)u * 0 (r B , s,ω) u0(rB, s,ω) ds | {z } D3 , (2) which shows the three leading-order terms in per- forming deconvolution interferometry before stack- ing over sources. Note that if u 0 (r B , s,ω) is com- prised of a single impulsive arrival, the oscillatory character of integrands in D1 and D2 (equation [2]) is due to the behavior of the numerators alone. This turns our attention back to interferometry by cross- correlations. Assuming a source function S(ω) that is slowly-varying with s, then Z Σ CAB(ω) ds = Z Σ u(rA, s,ω)u * (rB, s,ω) ds ≈ |S(ω)| 2 (C1+ C2+ C3+ C4) , (3) where C1 through C4 are the same cross-terms as for the numerator in the integrand of equation [3], respectively: direct-direct † , scattered-direct † , direct- scattered † and scattered-scattered † . In correlation interferometry, the term C2 gives rise to a causal re- sponse corresponding to uS (rA, rB,ω) (Snieder et al., † - this is used to denote time-reversal. The terms associated with the deconvolution/correlation filters are time-reversed.