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Spatio-temporal distribution of seismic wave fields and intensities: active source focusing and energy transfer in strongly scattering media
Jaimes Caballero, Manuel Alejandro
Jaimes Caballero, Manuel Alejandro
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2023
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2024-10-18
Abstract
As waves propagate from a source to a receiver, traversing a scattering medium, they interact with the inhomogeneities present in the medium. Depending on the scattering strength of the medium, or the propagation time, different waves may be recorded at the receiver location. These waves range from direct to single scattered to weakly multiple scattered to diffuse waves, and they contain information about the medium through which they propagate. These different waves have been used to locate and image seismic sources in strongly scattering media using the time-reversal method. This method consists of recording, at discrete receivers, the wave forms generated by a source, time-reversing these signals, and sending these signals back into the propagating medium (either physical or virtual) as new sources. Once the time-reversed signals are propagated into the medium, all the way back to the excitation time of the (original) source, the wave field is concentrated at the source location, provided the wave field is known on a closed surface surrounding the medium. The resolution with which one images the source is limited by the bandwidth of the wave field and the recording aperture. To deal with these resolution limitations, I develop a weighted time-reversal method, that is aimed at optimally localizing the source in space and time, within a given area. The weights compensate for the frequency content of the wave field and the acquisition geometry of the experiment, at the expense of very large wave field amplitudes outside the area on which I localize the source.
In addition to source focusing, one may want to describe the propagation of a wave field in strongly scattering media. Such description, via wave field amplitudes and phases, is not always possible because one may not know the location or characteristics of individual scatterers. If instead of wave field amplitudes and phases, one considers wave energy, it is possible to describe the wave propagation on average using the radiative transfer equations, which are based on energy conservation. These equations, which are hard to solve numerically due to coupling between angular directions as well as wave modes, describe the propagation of energy in a scattering medium as a function of space, time, direction of propagation, and wave mode (if considering elastic waves). To solve these equations, I construct a time-stepping algorithm with which I evolve the energy over time. To validate my numerical method, I compare the solutions that I obtain with known solutions, and find good agreement.
Because my algorithm accounts for the direction of propagation and wave mode, I use my algorithm to study the distribution of energy among propagation directions (valid for acoustic and elastic waves), and among wave modes (valid only for elastic waves). I find that the distribution of energy among propagation directions and wave modes, also known as the extent of equipartitioning, strongly depends on space and time. Thus, the equipartitioning of a wave field is not a global, but a local property. This local behavior has implications for Green's function reconstructions where one assumes that the noise field that one uses in the reconstruction is equipartitioned. Because equipartitioning is a local rather than a global concept, the accuracy of the Green's function retrieval may be a function of space and time, contingent upon the length of the time windows that are used in the reconstruction of the Green's function.
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