Christiaan C. Stolk

Differential Semblance Velocity Analysis By Wave-equation Migration

Prestack wave equation migration using the double square root equation produces prestack image volumes free of artifacts, even in the presence of multipathing due to complex structure. In particular image gathers in angle or offset ray parameter are flat at correct velocity, and gathers in offset are concentrated at zero offset. Differential semblance measures the deviation from flatness or concentration, and provides a method of automatic velocity updating via optimization. The adjoint state method gives a convenient computation of the differential semblance gradient as an addendum to prestack depth extrapolation.

Kinematic artifacts in prestack depth migration

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 loss of slowness information permits events to migrate along multiple incident-reflected ray pairs, thus introducing spurious coherent events into the image volume. This pathology occurs for all common binning strategies, including common-source, common-offset, and common-scattering angle. Since the artifacts move out with bin parameter, their effect on the final stacked image is minimal, provided that the migration velocity model is kinematically correct. However, common-image gathers may exhibit energetic primary events with substantial residual moveout, even with the kinematically accurate migration velocity model.

A rapidly converging domain decomposition method for the Helmholtz equation

A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration. When combined with multifrontal methods, the solver has near-linear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains. It is to our knowledge only the second method with this property next to the moving PML sweeping method.

Modeling of seismic data in the downward continuation 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 background. The discontinuities are contained in the medium perturbation. The high-frequency part of the wavefield in the background medium is described by a geometrical optics representation. It can also be described by a one-way wave equation. Based on this we derive a downward continuation operator for seismic data. This operator solves a pseudodifferential evolution equation in depth, the so-called double-square-root equation. We consider the modeling operator based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation.

Ultra fast electromagnetic field computations for RF multi-transmit techniques in high field MRI

A new, very fast, approach for calculations of the electromagnetic excitation field for MRI is presented. The calculation domain is divided in different homogeneous regions, where for each region a general solution is obtained by a summation of suitable basis functions. A unique solution for the electromagnetic field is found by enforcing the appropriate boundary conditions between the different regions. The method combines the speed of an analytical method with the versatility of full wave simulation methods and is validated in the pelvic region against FDTD simulations at 3 and 7 T and measurements at 3 T. The high speed and accurate reproduction of measurements and FDTD calculations are believed to offer large possibilities for multi-transmit applications, where it can be used for on-line control of the global and local electric field and specific absorption rate (SAR) in the patient. As an example the method was evaluated for RF shimming with the use of 7 T simulation results, where it was demonstrated that the magnetic excitation field could be homogenized, while both the local and average SAR were reduced by 38% or more.

Smooth objective functionals for seismic velocity inversion

In seismic inverse scattering, the data are divided into subsets, each of which is used to reconstruct the medium discontinuities by linearized inversion. The reconstructions depend on an a priori unknown, smoothly varying background medium (velocity model). The semblance principle, which states that the images must agree, is the basis for the reconstruction of the background medium. Several estimators for the background medium have been proposed, based on optimization of different objective functionals. Use of local, gradient-based optimization methods requires that the functionals to be optimized are smooth. Such a smoothness requirement essentially implies that the objective functional is of differential semblance type. The proof involves a characterization of pseudodifferential operators as having L2 continuous repeated commutators with order 1 pseudodifferential operators.

A mathematical framework for inverse wave problems in heterogeneous media

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 represent parametrically the spatially varying mechanical properties of materials. Rocks, manufactured materials, and other wave propagation environments often exhibit spatial heterogeneity in mechanical properties at a wide variety of scales, and coefficient functions representing these properties must mimic this heterogeneity. We show how to choose domains (classes of nonsmooth coefficient functions) and data definitions (traces of weak solutions) so that optimization formulations of inverse wave problems satisfy some of the prerequisites for application of Newtons method and its relatives. These results follow from the properties of a class of abstract first-order evolution systems, of which various physical wave systems appear as concrete instances. Finite speed of propagation for linear waves with bounded, measurable mechanical parameter fields is one of the by-products of this theory.

A Multigrid Method for the Helmholtz Equation with Optimized Coarse Grid Corrections

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let

MICROLOCAL ANALYSIS OF THE SCATTERING ANGLE TRANSFORM

G_{\rm c}

A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory

denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement