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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - MATRIX (MAchine for Time Reversal and Immersive wave eXperimentation)

Teaser

Summary

Seismic waves are still a main source of information when it comes to understanding both the composition of, and the dynamic processes ongoing in, the Earthâ€™s interior, including earthquakes. For example, so-called seismic â€œcodaâ€ waves, because of their large number of scattering interactions with the medium, can be indicative of slight changes in stress fields before catastrophic fracturing that might provide pre-cursory signs of earthquakes. To study such phenomena in the laboratory, current approaches employ experimentation at frequencies (and scales) that are at least 4 orders of magnitude higher (and smaller) than the frequencies at which the phenomena occur in the real-world. This is so, because reflections from the boundaries of the experimental domain would otherwise disturb and invalidate the experiments by virtue of masking the signal of interests. Because of the long wavelengths involved, it is also impractical to build the vast experimental facilities that would be required to otherwise avoid such boundary reflections. However, it is often not known if the physics governing the phenomena stays the same over those 4 orders of magnitude or more. Thus, there is a need for a radically new laboratory experimental approach for studying the interaction of seismic waves with the real complex media of the Earthâ€™s subsurface.

The MATRIX (Machine for Time-Revesal and Immersive eXperimentation) project, aims to establish a fundamentally new approach to seismic wave experimentation that involves fully immersing a physical seismic experiment within a virtual numerical environment. This enormously challenging endeavour, which is relevant to many outstanding issues in seismology, has not been previously attempted. By continuously varying the output of numerous transponders closely spaced around the physical domain using a control algorithm that takes advantage of measurements made by a scanning Laser-Doppler Vibrometer and a novel theory of immersive boundary conditions, waves travelling between the physical and numerical domains will seamlessly propagate back and forth between the two domains without being affected by reflections at the boundaries between the two domains. This will allow us to investigate diverse types of Earth materials using frequencies that are much closer to those of seismic waves propagating through the Earth than previously possible. The novel laboratory enables experimentation under highly controlled conditions. A broad range of longstanding problems in wave propagation and imaging that have eluded Earth scientists and physicists for decades can then be addressed. Fine scale heterogeneity, porosity and fluid saturation in real Earth media result in complex frequency-dependent amplitude and phase responses which can be characterized in the laboratory. Synthetically produced complex models can be used in wavefield-focusing experiments and to study complete elastic time-reversal. And finally, we can study those coda waves that can be indicative of slight changes in stress fields before catastrophic fracturing and that might provide pre-cursory signs of earthquakes. The laboratory is also highly relevant to applications such as non-destructive testing, medical imaging and lithotripsy.

Work performed

The work carried out during the first half of the project has both focused on qualifying key technical and methodological components of the Matrix laboratory and its elastic immersive boundary conditions (IBCs), as well as on fundamental wave physics research.

A state-of-the-art robotised 3D Scanning Laser Doppler Vibrometer system was purchased and initial experiments were carried on out on a 3D granite rock volume to qualitatively confirm the so-called â€œvector fidelityâ€ of the 3-component wavefield measured by the LDV. The wavefield on the five accessible sides of the rock was acquired for a piezo actuator (i.e., a source) on the top surface emitting a high frequency signal, with a frequency just above the audible range [see Figure 1]. As expected, several elastic wave types, including both sound-like compressional (P) waves and the shear (SV) waves propagating directly from the source through the block could be observed (panels 1 and 2). A high amplitude wave, known as the Rayleigh wave, could also be seen propagating along the surface (panels 2 and 3). Since the LDV system is able to measure the full particle motion vector, we were also able to verify the linear and elliptical polarisations of the P-SV waves and the Rayleigh waves, respectively. We were thus able to validate the use of this equipment for measuring the entire vector wavefield on the free surface which we need to compute the immersive boundary condition.

Detailed studies of various 3-axis Piezo (shear) actuators with different surface area and length were also carried out to determine if they represent suitable multi-component (vector) sources for our elastic immersive boundary condition. As the boundary condition requires a large number of vector sources covering the entire surface of the experimentation rock volume, as well as interleaved measurements of the wave-field, the ability to use multi-component piezo actuators is of great practical importance. A one-dimensional experimental setup involving a rigid aluminium beam of square profile was constructed, since longitudinal- and shear-wave propagation are uncoupled in such a beam. This allowed verifying that excitation of one of the source components does not result in inadvertent and undesired excitation of another source component, which would imply that they are not fully decoupled. This allowed us to identify a suitable 3-axis actuator. The beam setup also enabled a number of fundamental wave physics experiments, which are also briefly described later.

An important step in implementing immersive boundary conditions for an elastic medium is the separation of the wave field recorded at the surface of the volume into incident and free-surface reflected waves. This is because the elastic IBC (vector) source signals that are needed to cancel the reflections are directly proportional to the so-called outgoing traction (i.e., a force acting across a surface) which can be computed from the outgoing velocity or displacement wavefield measured using the LDV. The small surfaces and sharp corners of the experimentation rock volume represent particular challenges, that required the development of a novel wave field separation approach. By injecting an elastic wave field recorded at the free surface of an object into a numerical simulation for a homogeneous elastic medium, it turns out that the incident and reflected fields super-imposed in the measurements naturally separate, propagating away from the injection surface in their respective directions. This means that the presence of sharp corners on or between the free surfaces is no longer problematic and that the incident wave field can be separated, in theory, with almost perfect precision.

Finally, recently we have performed the first elastic IBC experiments on a one-dimensional beam, successfully cancelling the boundary reflections of both longitudinal and flexural waves (when excited simultaneously) using a single 3-axis piezo actuator glued to the o

Final results

While we are still constructing the elastic immersive wave experimentation laboratory, we see no reason to change the expected impacts. In addition, we have identified further expected impacts similar to the above that we briefly highlight further down below.

Briefly, the next steps and expected outcomes are as follows:

1) Validation of the novel elastic wavefield separation approach on real LDV data for a source inside the rock volume. This will allow us to compute the incident traction (vector) wavefield, whose components constitute the values to use to drive the (vector) force sources distributed all over the free surface, both to cancel the incident wavefield and to re-introduce the wavefields scattered back from the virtual domain. This step also includes determining the 3D transfer functions between the ideal IBC point force sources and the multi-component piezo-actuators, similar to what has already been done in 1D.

2) Establishing the iterative/recursive implementation of the immersive boundary condition The wavefield separation, as well as the application of the boundary condition, have to be iterated since the LDV system can only measure the wavefield on the free surface one point at at time. This step also includes implementation of the full elastic boundary condition on at least one full side of the rock volume, so that we can determine appropriate sub-sampled configurations for implementing the boundary condition on all sides from an initially oversampled (in terms of the number of piezo actuators).

3) We have made some progress on specific applications of the immersive lab. In particular, we understand how we can apply arbitrary 1D, 2D or 3D periodic boundary conditions to the experimentation (rock) volume. This should eventually enable the experimental realization of elastic phononic media, using Born-von Karman and Bloch wave conditions. Such approaches could be particularly useful for studying phononic wave phenomena for complex unit cells.