Interferometry by deconvolution: Part 1 — Theory for acoustic waves and numerical examples

Abstract

Interferometry allows for synthesis of data recorded at any two receivers into waves that propagate between these receivers as if one of them behaves as a source. This is accomplished typically by crosscorrelations. Based on perturbation theory and representation theorems, we show that interferometry also can be done by deconvolutions for arbitrary media and multidimensional experiments. This is important for interferometry applications in which ͑1͒ excitation is a complicated source-time function and/or ͑2͒ when wavefield separation methods are used along with interferometry to retrieve specific arrivals. Unlike using crosscorrelations, this method yields only causal scattered waves that propagate between the receivers. We offer a physical interpretation of deconvolution interferometry based on scattering theory. Here we show that deconvolution interferometry in acoustic media imposes an extra boundary condition, which we refer to as the free-point or clamped-point boundary condition, depending on the measured field quantity. This boundary condition generates so-called free-point scattering interactions, which are described in detail. The extra boundary condition and its associated artifacts can be circumvented by separating the reference waves from scattered wavefields prior to interferometry. Three wavefield-separation methods that can be used in interferometry are direct-wave interferometry, dual-field interferometry, and shot-domain separation. Each has different objectives and requirements.

Near-surface weakening in Japan after the 2011 Tohoku-Oki earthquake

“…Although the widest applied algorithm in seismic interferometry is based on cross correlation [e.g., Claerbout, 1968;Wapenaar, 2004;Bakulin and Calvert, 2004;Schuster et al, 2004], we use the algorithm based on deconvolution [e.g., Trampert et al, 1993;Snieder and Şafak, 2006;Vasconcelos and Snieder, 2008]. In deconvolution interferometry, we can suppress the complicated imprint of the structure (e.g., attenuation and scattering) incurred as the waves travel from the hypocenter to the borehole seismogram [Snieder et al, 2009].…”

Directional balancing for seismic and general wavefield interferometry

“…These are referred to as T1 to T4, representing the contributions of the crosscorrelation of the direct waves with the direct waves ͑T1͒, the direct waves with the scattered waves ͑T2͒, the scattered waves with the direct waves ͑T3͒, and the scattered waves with the scattered waves ͑T4͒. Similar analyses can be found using representation theorems for perturbed media ͑Vasconcelos et al, 2009͒ and for deconvolution interferometry ͑Vasconcelos and Snieder, 2008a We first define the direct and scattered wavefields between the receiver locations x 1 and x 2 as G 0 ͑x 1 ,x 2 ͒ and G scat ͑x 1 ,x 2 ͒ and assume that we want to estimate these wavefields using interferometry. Terms T1-T4 provide the following contributions to this estimate:…”

Retrieving virtual reflection responses at drill‐bit positions using seismic interferometry with drill‐bit noise

“…Both D AB and H AB are equal to 1 when x A = x B , which results in the Dirac delta function in time at zero offset. This means that the retrieved responses by both approaches satisfy the so-called clamped boundary condition (Vasconcelos and Snieder 2008a). It can also be recognized from equation (8) that, for deconvolution SI, the denominator changes when we interchange x A and x B , whereas for crosscoherence SI, the denominator does not change.…”

“…If |S| 2 is unknown, deconvolution SI (Vasconcelos and Snieder 2008a) or cross-coherence SI (Nakata et al 2011) can be used, and they are defined in the frequency domain as…”