Numerical modeling of the acoustic wave equation in a generalized coordinate system

Mar 6, 2023 | CWP Blog

Posted by Khalid Almuteri

Accurate wave equation modeling is necessary for understanding wave propagation under realistic acquisition conditions, such as a time-varying sea surface and/or moving source. Moreover, numerical wave-equation solutions form the basis of advanced seismic imaging and inversion techniques, such as reverse-time migration (RTM) and full-waveform inversion (FWI). Cartesian-based modeling methods are susceptible to numerical instabilities and modeling inaccuracies when introducing irregular or time-varying computational geometries such as time-varying sea surfaces and/or moving sources. Alternatively, using coordinate transformation, one can model the full acoustic wavefield in a time-invariant generalized coordinate system (Figure 1a) that incorporates the physical domain deformations (Figure 1b). Employing coordinate transformation to model the acoustic wavefield requires deriving the appropriate gradient and divergence operators, using the generalized definitions of those operators and the metric tensor constructed using the coordinate transformation definition.

Figure 1a: Graphical representation of the computational domain.

Figure 1b: Graphical representation of the physical domain.

A time-varying sea surface scatters the wavefield (Figure 2a) and introduces traveltime and amplitude perturbations compared to the flat sea surface response (Figure 2b). Figures 3a and 3b show the shot gathers of the pressure component recorded 20 m below the source for a time-varying and flat sea surface modeled using a stationary source, respectively. The shot gathers demonstrate the profound effects a time-varying sea surface has on seismic data. Such distortions to the seismic wavefield pose processing and imaging challenges, especially in time-lapse (4D) seismic experiments. Thus, accurately accounting for sea-surface effects in numerical wave-equation solutions is essential to developing appropriate processing and imaging solutions and understanding the impact of different sea-surface realizations on 4D seismic experiments.

Figure 2a: Wavefield snapshot in a homogeneous medium using a time-varying sea surface.

Figure 2b: Wavefield snapshot in a homogeneous medium using a flat sea surface.

Figure 3a: Shot gather of the pressure component acquired in a homogeneous medium using a time-varying sea surface.

Figure 3b: Shot gather of the pressure component acquired in a homogeneous medium using a flat sea surface.

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