Scattering-based sensitivity kernels for time-lapse differential-waveform inversion

Time-lapse seismic surveys are a powerful reservoir monitoring tool. The basic approach in time-lapse surveys is to image the changes in the reservoir by subtracting separated-in-time seismic images of the reservoir. Recently FWI has been used to improve time-lapse monitoring. However, in practice, nonlinear gradient-based FWI is limited due to its notorious sensitivity to the choice of the starting model. Kernel decomposition based on scattering theory allows to break the acoustic-wavefield sensitivity kernels with respect to model parameters into background and singular parts, which should help to address model-convergence issues in FWI. Here we apply scattering theory to the time-lapse problem, considering the time-lapse change as a perturbation of the singular part of the model. We make use of the time-lapse differential-waveform inversion framework, with the linearized scattering-based decomposition of the sensitivity kernel. Our numerical examples demonstrate that the inclusion of the singular part into the model used for backpropagation helps to improve the perturbation estimates from FWI by taking advantage of the additional subsurface illumination provided by multiple-scattering phenomena. INTRODUCTION Over the past few years, time-lapse seismic surveys has become a powerful tool used to monitor the fluidflow in a producing reservoir. Generally speaking, such a survey consists in acquiring and analyzing multiple seismic data, repeated at the same site over time in order to look for differences from which one can infer the changes in the reservoir due to production. This is possible because, as fluid saturations and pressures in the reservoir change, the seismic reflection properties change accordingly. A seismic image contains information on reflections that depends on both the geological structure and its fluid contents. On a single-time images these contributions are coupled and difficult to separate. The basic approach in time-lapse surveys is to image the changes in the reservoir by subtracting subsequent time-lapse seismic images of the reservoir from one another. Assuming that geology is time-invariant during production and that repeatability in the seismic data acquisition is, in some way, assured, the image difference from a time-lapse survey would indicate the changes due only to the fluid-flow since, to first order, the geology part subtracts out since it is time invariant (Lumley, 2001). Recently, full waveform inversion (FWI) has been used as an alternative time-lapse monitoring tool (Queiser and Singh, 2013; Yang et al., 2011; Zheng et al., 2011). FWI allows the reconstruction of highresolution velocity models of the subsurface through the extraction of the full information content of seismic data (Tarantola, 1984; Virieux and Operto, 2009). Since the FWI approach delivers high resolution quantitative images of macro-scale physical parameter, it ought to be a good candidate for monitoring applications to reconstruct the parameter variation through time evolution (Asnaashari et al., 2012). Asnaashari et al. (2012) studied the robustness of three different FWI methods applied to time-lapse problems: differential method, parallel difference method and the sequential difference method. In the differential method, instead of minimizing the difference between the observed and modeled data, the Annual WIT report 2013 215 difference of the differential data between two sets of data is minimized to obtain the time-lapse change estimate (Watanabe et al., 2004). The parallel difference method considers independent inversion of the baseline and monitor data-sets, using a similar starting model (Plessix et al., 2010). The sequential difference method uses the recovered baseline model as a starting model for the monitor data inversion. While in principle capable of handling all aspects of wave propagation affecting the seismic data, including full nonlinearity, in practice nonlinear gradient-based FWI is limited to localizing the nearest local mimimum, leading to its notorious sensitivity to the choice of the starting model. This is so because, for narrow-offset acquisition of reflection data, the seismic wavefield is rather insensitive to high/intermediate wavelengths. In previous work, we sought to help addressing model convergence issues in FWI. For this purpose, we studied a wavefield decomposition based on scattering theory that allows to break the acoustic-wavefield sensitivity kernels (SKs) with respect to model parameters into background and singular parts (Macedo et al., 2011, 2012a,b, 2013). We were able to demonstrate that the forward kernel decomposition is successful in revealing subkernels that unravel different levels of non-linearity with respect to data and model. This, in turn, can be translated into different levels of interaction between non-, single-, and multiplescattered information that otherwise would be hidden in the full-wavefield sensitivity kernels. Moreover, we predicted that part of the answer to the problem of lacking low-frequency information on the model should lie in utilizing scattered wavefields, because these travel through the medium long enough to carry this information (see also Snieder et al., 2002). In this work we apply our scattering decomposition to the time-lapse problem, considering the timelapse change as a perturbation of the singular part of the model, i.e., perturbation of the scattering potential. Under the differential-method framework, we demonstrate that the scattering-based decomposition of the sensitivity kernels allows to take advantage of the illumination of the time-lapse change due to multiplescattering phenomena in order to improve the perturbation estimates from FWI. SCATTERING-BASED SENSITIVITY KERNELS The wavefield perturbation δp due to a medium change can be evaluated, under the assumptions of the Born approximation, using so-called secondary or adjoint sources (Tarantola, 1984). Based on scattering theory, Macedo et al. (2011) decompose the full-acoustic-wavefield perturbation δp into 13 components (one reference-wavefield perturbation δp0 and 12 parts of the scattered-wavefield perturbation δps, see Figure 1) that reveal different levels of interaction between single and multiple-scattered information within data (see also Macedo et al., 2012a,b, 2013). The reference-wavefield perturbation and all components of the scattered-wavefield perturbations have the same functional structure (time dependency is ommited for simplicity): δp0 or δps,αOγ(xg;xs) = ∫ V dxGγ(x ;xg) ∗ LO [pα(x;xs)] , (1) where the subscripts α and γ denote the involved wavefield parts, with pγ denoting the incoming sourceδp 8x U δKB U δKS S 4x 1x δp U δKB