Reciprocity and Representation Theorems for Flux- and Field-Normalised Decomposed Wave Fields

We consider wave propagation problems in which there is a preferred direction of propagation. To account for propagation in preferred directions, the wave equation is decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis. This decomposition is not unique. We discuss flux-normalised and field-normalised decomposition in a systematic way, analyse the symmetry properties of the decomposition operators, and use these symmetry properties to derive reciprocity theorems for the decomposed wave fields, for both types of normalisation. Based on the field-normalised reciprocity theorems, we derive representation theorems for decomposed wave fields. In particular, we derive double- and single-sided Kirchhoff-Helmholtz integrals for forward and backward propagation of decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for backward propagation of field-normalised decomposed wave fields find applications in reflection imaging, accounting for multiple scattering.

Elastic least-squares one-way wave-equation migration

Least-squares migration seeks a reflectivity model that fits the observed data. It is used to compensate for acquisition noise, poor sampling of sources and receivers on the surface, as well as poor illumination of the subsurface. To date, least-squares migration has been mainly restricted to the imaging of acoustic wavefields. We have developed an extension of one-way wave-equation least-squares migration for elastic wavefields in isotropic media. Least-squares migration is an iterative method that requires a forward and an adjoint operator. In elastic least-squares one-way wave-equation migration, the forward operator generates data components from multiparameter images by recursive wavefield decomposition, extrapolation, and recomposition. Conversely, the adjoint operator generates multiparameter images from data components by recursively applying the adjoint of the wavefield recomposition, extrapolation, and wavefield decomposition operators. We use an extended imaging condition and regularize the inversion by applying a smoothing filter on the depth-angle axes of each common image point gather to reduce the effect of source/receiver sampling, noise, and crosstalk artifacts. Elastic least-squares migration is able to compensate for irregular subsurface illumination in elastic imaging and provides an alternative approach to interpolation and wavefield separation of multicomponent seismic data.

Trace Operators on Wiener Amalgam Spaces

The paper deals with trace operators of Wiener amalgam spaces using frequency uniform decomposition operators and maximal inequalities, obtaining sharp results. Additionally, we provide the embedding between standard and anisotropic Wiener amalgam spaces.