Time-reversal and Interferometry with applications to forward modeling of wave propagation and a chapter on receiver functions

In exploration seismics and non-destructive evaluation, acoustic, elastic and electro-magnetic waves sensitive to inhomogeneities in the medium under investigation are used to probe its interior. Waves multiply scattered by the inhomogeneities carry significant information but, due to their non-linear relation with the inhomogeneities, are notoriously dificult to image or invert for subsurface structure. Recently, however, this paradigm may have been broken as it was shown that high-order multiply scattered acoustic waves can be time-reversed and focused onto their original source location through arbitrary, unknown, inhomogeneous media using a so-called time-reversal mirror: in a first step, the multiply scattered waves are recorded on an array of transducers partially surrounding the medium, in the second step the recorded wavefields are time-reversed and reemitted into the medium (i.e., the time-reversal mirror acts as a linear boundary condition on the medium injecting the time-reversed, multiply scattered wave- field). The multiply scattered waves retrace their paths through the medium and focus on the original source location. In another development the full waveform Green's function between two (passive) receivers has been observed to emerge from crosscorrelation of multiply scattered coda waves. This process is called interferometry. The principal aim of this thesis is to explore the relation between time-reversal and interferometry and to apply the resulting insights to forward modelling of wave propagation in the broader context of inversion. A secondary aim is to see if the seismological receiver function method can be applied to a reflection setting in ways that are both dynamically and kinematically correct. These aims are achieved through: (1) Derivation of an integral representation for the time-reversed wavefield in arbitrary points of an inhomogeneous medium [first, for the acoustic case, based on the Kirchhoff-Helmholtz integral, then for the elastic case based on the Betti-Rayleigh reciprocity theorem]. Evaluation of these integral representations for points other than the original source point will be shown to give rise to the Green's function between the two points. Physically intuitive explanations will be given as to why this is the case. (2) Application of ordinary reciprocity to the integral representation for the time-reversed wavefield to get an expression in terms of sources on the surrounding surface only. This gives rise to an efficient and flexible forward modeling algorithm. By illuminating the medium from the surrounding surface and storing full waveforms in as many points in the interior as possible, full waveform Green's functions between arbitrary points in the volume can be computed by cross correlation and summation only. (3) Derivation of an exact, interferometric von Neumann type boundary condition for arbitrary interior perturbed scattering problems. The exact bound- ary condition correctly accounts for all orders of multiple scattering, both inside the scattering perturbation(s) and between the perturbations and the background model and thus includes all so-called higher-order, long-range interactions. (4) A comprehensive study of the receiver function method in a reflection setting, both kinematically and dynamically. All presented results are verified and illustrated by numerical (finite-difference) modelling. Overall, the results in this thesis demonstrate that, while the original instabilities associated with direct inversion remain, multiply scattered waves can be used in an industrial context { both in real-life experiments and in forward modelling { in ways that are stable. The presented advances in forward modelling are argued to have a significant impact on inversion as well, albeit indirectly

Waves in space-dependent and time-dependent materials: a systematic comparison

Waves in space-dependent and time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and time-dependent materials. After a review of reflection and transmission coefficients, we discuss Green's functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of reciprocity theorems leads to the well-known source-receiver reciprocity relation for the Green's function of a space-dependent material and a new source-receiver reciprocity relation for the Green's function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green's function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green's function retrieval in a time-dependent material. After an introduction of a matrix-vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices.Comment: 41 pages, 5 figure

Finite-difference modelling of wavefield constituents

The finite-difference method is among the most popular methods for modelling seismic wave propagation. Although the method has enjoyed huge success for its ability to produce full wavefield seismograms in complex models, it has one major limitation which is of critical importance for many modelling applications; to naturally output up- and downgoing and P- and S-wave constituents of synthesized seismograms. In this paper, we show how such wavefield constituents can be isolated in finite-difference-computed synthetics in complex models with high numerical precision by means of a simple algorithm. The description focuses on up- and downgoing and P- and S-wave separation of data generated using an isotropic elastic finite-difference modelling method. However, the same principles can also be applied to acoustic, electromagnetic and other wave equation

Closed-aperture unbounded acoustics experimentation using multidimensional deconvolution

In physical acoustic laboratories, wave propagation experiments often suffer from unwanted reflections at the boundaries of the experimental setup. We propose using multidimensional deconvolution (MDD) to post-process recorded experimental data such that the scattering imprint related to the domain boundary is completely removed and only the Green's functions associated with a scattering object of interest are obtained. The application of the MDD method requires in/out wavefield separation of data recorded along a closed surface surrounding the object of interest, and we propose a decomposition method to separate such data for arbitrary curved surfaces. The MDD results consist of the Green's functions between any pair of points on the closed recording surface, fully sampling the scattered field. We apply the MDD algorithm to post-process laboratory data acquired in a two-dimensional acoustic waveguide to characterize the wavefield scattering related to a rigid steel block while removing the scattering imprint of the domain boundary. The experimental results are validated with synthetic simulations, corroborating that MDD is an effective and general method to obtain the experimentally desired Green's functions for arbitrary inhomogeneous scatterers

Identifying Multiply-Scattered Wavepaths in Strongly Scattering and DispersiveMedia

The ability to extract information from scattered waves is usually limited to singly scattered energy even if multiple scattering might occur in the medium. As a result, the information in arrival times of higher-order scattered events is underexplored. This information is extracted using fingerprinting theory. This theory has never previously been applied successfully to real measurements, particularly when the medium is dispersive. The theory is used to estimate the arrival times and scattering paths of multiply scattered waves in a thin sheet using an automated scheme in a dispersive medium by applying an additional dispersion compensation method. Estimated times and paths are compared with predictions based on a sequence of straight ray paths for each scattering event given the known scatterer locations. Additionally, numerical modelling is performed to verify the interpretations of the compensated data. Since the source also acts as a scatterer in these experiments, initially, the predictions and the numerical results did not conform to the experimental observations. By reformulating the theory and the processing scheme and adding a source scatterer in the modelling, it is shown that predictions of all observed scattering events are possible with both prediction methods, verifying that the methods are both effective and practically achievable. Applied Geophysics and Petrophysic