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SUMMARY We present a method for imaging quasi-vertically dipping faults with surface records of reflected P waves from small earthquakes. Faults are boundaries between geological structures, such as tectonic plates, and are located in earthquake active regions such as Parkfield, California. The high degree of seismic activity enables the use of multiple seismic recordings in our fault identification algorithm. Major challenges occur because of the quasi-vertical orientation of the fault and the fact that the wave reflected by the fault and recorded by the surface receivers is not well modelled by the direct arrival of the propagating wave generated by the earthquake source. Our method uses the 2-D acoustic wave equation as the model for P-wave propagation. We assume that an approximate wave speed map on the reflection side of the fault is available and the source locations are known, for example, from traveltime tomography. We also assume that the source time function is known. The new features of our method arise because earthquake sources are located very close to the fault. This has two implications: (1) the direct arrival and the reflected wave arrive almost simultaneously, so that it is impossible to separate them on a seismogram using standard techniques, and (2) most of the reflections occur above the critical angle which introduces a distortion in the reflected wave. To overcome these difficulties we use a modelled incident wave to (1) remove the direct arrival from the data, and (2) remove the post-critical distortion from the reflected wave. We justify the distortion removal using the leading-order term of an asymptotic expansion, and an optimization procedure. To complete our algorithm we utilize some features of reverse time migration: (1) the use of full acoustic wave equation for modelling and backpropagation, and (2) zero-lag correlation of the backpropagated time reversed reflected and incident fields. We present numerical examples of fault reconstructions with synthetic data.

Transient elastography and supersonic imaging are promising new techniques for characterizing the elasticity of soft tissues. Using this method, an 'ultrafast imaging' system (up to 10 000 frames s−1) follows in real time the propagation of a low frequency shear wave. The displacement of the propagating shear wave is measured as a function of time and space. The objective of this paper is to develop and test algorithms whose ultimate product is images of the shear wave speed of tissue mimicking phantoms. The data used in the algorithms are the front of the propagating shear wave. Here, we first develop techniques to find the arrival time surface given the displacement data from a transient elastography experiment. The arrival time surface satisfies the Eikonal equation. We then propose a family of methods, called distance methods, to solve the inverse Eikonal equation: given the arrival times of a propagating wave, find the wave speed. Lastly, we explain why simple inversion schemes for the inverse Eikonal equation lead to large outliers in the wave speed and numerically demonstrate that the new scheme presented here does not have any large outliers. We exhibit two recoveries using these methods: one is with synthetic data; the other is with laboratory data obtained by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII).

We review and present new results on the transient elastography problem, where the goal is to reconstruct shear stiness properties using interior time and space dependent displacement measurements. We present unique identifiability of two parameters for this inverse problem, establish that a Lipschitz continuous arrival time satisfies the eikonal equation, and present two numerical algorithms, simulation results, and a reconstruction example using a phantom experiment accomplished by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII). One numerical algorithm uses a geometrical optics expansion and the other utilizes the arrival time surface.

The crawling wave experiment, in which two harmonic sources oscillate at different but nearby frequencies, is a development in sonoelastography that allows real-time imaging of propagating shear wave interference patterns. Previously the crawling wave speed was recovered and used as an indicator of shear stiffness; however, it is shown in this paper that the crawling wave speed image can have artifacts that do not represent a change in stiffness. In this paper, the locations and shapes of some of the artifacts are exhibited. In addition, a differential equation is established that enables imaging of the shear wave speed, which is a quantity strongly correlated with shear stiffness change. The full algorithm is as follows: (1) extract the crawling wave phase from the spectral variance data; (2) calculate the crawling wave phase wave speed; (3) solve a first-order PDE for the phase of the wave emanating from one of the sources; and (4) compute and image the shear wave speed on a grid in the image plane.

Two new experiments were created to characterize the elasticity of soft tissue using sonoelastography. In both experiments the spectral variance image displayed on a GE LOGIC 700 ultrasound machine shows a moving interference pattern that travels at a very small fraction of the shear wave speed. The goal of this paper is to devise and test algorithms to calculate the speed of the moving interference pattern using the arrival times of these same patterns. A geometric optics expansion is used to obtain Eikonal equations relating the moving interference pattern arrival times to the moving interference pattern speed and then to the shear wave speed. A cross-correlation procedure is employed to find the arrival times; and an inverse Eikonal solver called the level curve method computes the speed of the interference pattern. The algorithm is tested on data from a phantom experiment performed at the University of Rochester Center for Biomedical Ultrasound.