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- Christiaan C. Stolk
- J. Comput. Physics
- 2013

Microlocal analysis of a seismic linearized inverse problem. Abstract: The seismic inverse problem is to determine the wavespeed c(x) in the interior of a medium from measurements at the boundary. In this paper we analyze the linearized inverse problem in general acoustic media. The problem is to nd a left inverse of the linearized forward map F, or,… (More)

Strong refraction of waves in the migration velocity model introduces kinematic artifacts coherent events not corresponding to actual reflectors into the image volumes produced by prestack depth migration applied to individual data bins. Because individual bins are migrated independently, the migration has no access to the bin component of slowness. This… (More)

In certain applications one is interested in the long-time behavior of systems described by a linear partial differential equation. For example, in kinetic equations one studies the decay to equilibrium of various linear and nonlinear systems. For the Kramers–Fokker–Planck equation, which will be studied here, exponential decay was shown in Talay (1999) and… (More)

A one-way wave equation is an evolution equation in one of the space directions that describes (approximately) a wave field. The exact wave field is approximated in a high frequency, microlocal sense. Here we derive the pseudodifferential one-way wave equation for an inhomogeneous acoustic medium using a known factorization argument. We give explicitly the… (More)

- Christiaan C. Stolk, Maarten V. de Hoop
- SIAM Journal of Applied Mathematics
- 2005

Seismic data are commonly modeled by a linearization around a smooth background medium in combination with a high frequency approximation. The perturbation of the medium coefficient is assumed to contain the discontinuities. This leads to two inverse problems, first the linearized inverse problem for the perturbation, and second the estimation of the… (More)

In this paper we use methods from partial differential equations, in particular microlocal analysis and theory of Fourier integral operators, to study seismic imaging in the wave-equation approach. Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth… (More)

- Kirk D. Blazek, Christiaan Stolk, William W. Symes
- 2013

This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The coefficients of these time-dependent partial differential equations respresent parametrically the spatially varying mechanical… (More)

A nonlinear singularity-preserving solution to seismic image recovery with sparseness and continuity constraints is proposed. We observe that curvelets, as a directional frame expansion, lead to sparsity of seismic images and exhibit invariance under the normal operator of the linearized imaging problem. Based on this observation we derive a method for… (More)