Nuno Vieira da Silva

Accurate seabed modeling using finite difference methods

Finite difference is the most widely used method for seismic wavefield modeling. However, most finite-difference implementations discretize the Earth model over a fixed grid interval. This can lead to irregular model geometries being represented by ‘staircase’ discretization, and potentially causes mispositioning of interfaces within the media. This misrepresentation is a major disadvantage to finite difference methods, especially if there exist strong and sharp contrasts in the physical properties along an interface. The discretization of undulated seabed bathymetry is a common example of such misrepresentation of the physical properties in finite-difference grids, as the seabed is often a particularly sharp interface owing to the rapid and considerable change in material properties between fluid seawater and solid rock. There are two issues typically involved with seabed modeling using finite difference methods: firstly, the travel times of reflections from the seabed are inaccurate as a consequence of its spatial mispositioning; secondly, artificial diffractions are generated by the staircase representation of dipping seabed bathymetry. In this paper, we propose a new method that provides a solution to these two issues by positioning sharp interfaces at fractional grid locations. To achieve this, the velocity model is first sampled in a model grid that allows the center of the seabed to be positioned at grid points, before being interpolated vertically onto a regular modeling grid using the windowed sinc function. This procedure allows undulated seabed bathymetry to be represented with improved accuracy during modeling. Numerical tests demonstrate that this method generates reflections with accurate travel times and effectively suppresses artificial diffractions.

Forward modelling formulas for least-squares reverse-time migration

Reverse-time migration can be formulated in a least-squares inversion framework. This is referred to as least-squares reverse-time migration, which attempts to find an optimal model of the reflectors that fits the observed data in a least-squares sense. Based on different representations of the model, different formulas of the forward modelling for least-squares reverse-time migration can be derived. In this paper, we derive two different formulas. One formula is to recover the impedance-perturbation-related images based on Born approximation. The other is to invert the reflectivity-related images based on Kirchhoff approximation. The theoretical analysis unveils there is an iω difference between the two formulas. Consequently, the seismic image using the two formulas has different shape/phase: the one based on Born approximation produces anti-symmetric images; the other based on Kirchhoff approximation gives symmetric images. Two numerical examples demonstrate the similarities and differences between the two formulas. We derive two formulas of forward modelling for least-squares reverse-time migration based on Born and Kirchhoff approximations. Analysis unveils an iω difference exists between the two formulas. Consequently, their seismic images have different shape/phase: the Born approximation produces anti-symmetric images while the Kirchhoff approximation gives symmetric images. Numerical examples demonstrate these features between the two formulas.

A domain decomposition method for 3-D Controlled Source Electromagnetics

SUMMARY Three dimensional controlled source electromagnetics (CSEM) forward modelling in the frequency domain requires the solution of a large scale, complex and linear system. Such a system when derived from finite elements and finite differences formulations is sparse and ill-conditioned. The use of direct solvers is prohibitively expensive for realistic-sized problems since several million degrees of freedom are usually involved. To avoid inherent hardware limitations iterative solvers are an option, however their potential efficiency relies on the use of efficient pre-conditioning. We present an alternative pre-conditioner for the CSEM forward modelling problem, based on the Schur complement method, and will discuss some limitations and possible solutions.