Perturbation methods for two special cases of the time‐lapse seismic inverse problem

Abstract

Scattering theory, a form of perturbation theory, is a framework from within which time‐lapse seismic reflection methods can be derived and understood. It leads to expressions relating baseline and monitoring data and Earth properties, focusing on differences between these quantities as it does so. The baseline medium is, in the language of scattering theory, the reference medium and the monitoring medium is the perturbed medium. The general scattering relationship between monitoring data, baseline data, and time‐lapse Earth property changes is likely too complex to be tractable. However, there are special cases that can be analysed for physical insight. Two of these cases coincide with recognizable areas of applied reflection seismology: amplitude versus offset modelling/inversion, and imaging. The main result of this paper is a demonstration that time‐lapse difference amplitude versus offset modelling, and time‐lapse difference data imaging, emerge from a single theoretical framework. The time‐lapse amplitude versus offset case is considered first. We constrain the general time‐lapse scattering problem to correspond with a single immobile interface that separates a static overburden from a target medium whose properties undergo time‐lapse changes. The scattering solutions contain difference‐amplitude versus offset expressions that (although presently acoustic) resemble the expressions of Landro (). In addition, however, they contain non‐linear corrective terms whose importance becomes significant as the contrasts across the interface grow. The difference‐amplitude versus offset case is exemplified with two parameter acoustic (bulk modulus and density) and anacoustic (P‐wave velocity and quality factor Q) examples. The time‐lapse difference data imaging case is considered next. Instead of constraining the structure of the Earth volume as in the amplitude versus offset case, we instead make a small‐contrast assumption, namely that the time‐lapse variations are small enough that we may disregard contributions from beyond first order. An initial analysis, in which the case of a single mobile boundary is examined in 1D, justifies the use of a particular imaging algorithm applied directly to difference data shot records. This algorithm, a least‐squares, shot‐profile imaging method, is additionally capable of supporting a range of regularization techniques. Synthetic examples verify the applicability of linearized imaging methods of the difference image formation under ideal conditions.

Elastic inverse scattering for fluid variation with time-lapse seismic data

“…The baseline medium is, in the conception of elastic scattering theory, the reference medium, and the monitoring medium is corresponding to the perturbed medium. The change in the earth properties between the monitoring medium and baseline medium is the variation in the properties between the reference medium and perturbed medium (Innanen et al, 2014). Under the above hypothesis, the change in the earth properties between the monitoring medium and the baseline medium could be quantitatively associated with the variation between the baseline data and monitoring data.…”

“…To simplify the mathematical complexity, a Green's function that propagates in a homogeneous reference medium was used in this manuscript, although this is not selfconsistent (Innanen et al, 2014). Under the P-wave condition, the Green's function can be expressed as…”

“…Vavryčuk (2008) uses the first-order perturbation theory in the anisotropic viscoelastic media to derive the perturbation formula in terms of stationary slowness vector, ray and phase velocities, attenuations, and Q-factors. Innanen et al (2014) discuss two special cases of the time-lapse seismic inverse problem with the scattering theory, AVO modeling/inversion, and imaging, which gives rise to the application of the perturbation theory in time-lapse seismic data. Following Innanen et al (2014), in the time-lapse seismic inversion, we also assume the baseline medium as the reference medium and the monitoring medium as the perturbed medium in the language of scattering theory, and we assume that the velocity-change-induced time variation can be neglected.…”

“…Innanen et al (2014) discuss two special cases of the time-lapse seismic inverse problem with the scattering theory, AVO modeling/inversion, and imaging, which gives rise to the application of the perturbation theory in time-lapse seismic data. Following Innanen et al (2014), in the time-lapse seismic inversion, we also assume the baseline medium as the reference medium and the monitoring medium as the perturbed medium in the language of scattering theory, and we assume that the velocity-change-induced time variation can be neglected. Under these hypothesis, we initially derive the linearized and qualitative relation between the variation in the monitoring and baseline data and the changes of the fluid factor, shear modulus, and density of the baseline and monitoring medium with the perturbation theory.…”