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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.

In this paper, we develop a third order accurate fast marching method for the solution of the eikonal equation in two dimensions. There have been two obstacles to extending the fast marching method to higher orders of accuracy. The first obstacle is that using one-sided difference schemes is unstable for orders of accuracy higher than two. The second obstacle is that the points in the difference stencil are not available when the gradient is closely aligned with the grid. We overcome these obstacles by using a two-dimensional (2D) finite difference approximation to improve stability, and by locally rotating the grid 45 degrees (i.e., using derivatives along the diagonals) to ensure all the points needed in the difference stencil are available. We show that in smooth regions the full difference stencil is used for a suitably small enough grid size and that the difference scheme satisfies the von Neumann stability condition for the linearized eikonal equation. Our method reverts to first order accuracy near caustics without developing oscillations by using a simple switching scheme. The efficiency and high order of the method are demonstrated on a number of 2D test problems.

In this paper, we outline two improvements to the fast sweeping method to improve the speed of the method in general and more specifically in cases where the speed is changing rapidly. The conventional wisdom is that fast sweeping works best when the speed changes slowly, and fast marching is the algorithm of choice when the speed changes rapidly. The goal here is to achieve run times for the fast sweeping method that are at least as fast, or faster, than competitive methods, e.g. fast marching, in the case where the speed is changing rapidly. The first improvement, which we call the locking method, dynamically keeps track of grid points that have either already had the solution successfully calculated at that grid point or for which the solution cannot be successfully calculated during the current iteration. These locked points can quickly be skipped over during the fast sweeping iterations, avoiding many time-consuming calculations. The second improvement, which we call the two queue method, keeps all of the unlocked points in a data structure so that the locked points no longer need to be visited at all. Unfortunately, it is not possible to insert new points into the data structure while maintaining the fast sweeping ordering without at least occasionally sorting. Instead, we segregate the grid points into those with small predicted solutions and those with large predicted solutions using two queues. We give two ways of performing this segregation. This method is a label correcting (iterative) method like the fast sweeping method, but it tends to operate near the front like the fast marching method. It is reminiscent of the threshold method for finding the shortest path on a network, [F. Glover, D. Klingman, and N. Phillips, Oper. Res., 33 (1985), pp. 65-73]. We demonstrate the numerical efficiency of the improved methods on a number of examples.

This paper considers an inverse problem for a transversely isotropic three-dimensional acoustic medium, where there is one preferred direction called the fiber direction along which the wave propagates fastest and there is no preferred wave propagation direction in the isotropic plane, which is the plane orthogonal to the fiber direction. In this medium the parameters to be recovered are (1) the wave speed for a wave propagating in the direction along the fiber; (2) the wave speed for a wave propagating in any direction which is orthogonal to the fiber direction; and (3) the unit fiber direction itself. So four scalar functions are to be recovered. The data are the positions of four distinct wave fronts as the corresponding waves propagate through the medium. The mathematical relation, which is the Eikonal equation, between the wave front locations and the four unknown functions, is nonlinear. Here it is established, perhaps surprisingly, that corresponding to the given data set, there can be up to four p...

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.

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.

In the magnetic resonance elastography experiment we consider a harmonically oscillating mechanical force applied to the boundary surface of a phantom and synchronized with the motion encoding gradient. The phantom is symmetric in the direction of the applied mechanical force and the vector component in that direction decouples from the other components and satisfies a Helmholtz equation. We present a local inversion method to determine the shear wave speed that: (1) treats the phase and amplitude of the data differently; (2) computes derivatives of the data by using statistically justified filtering; and (3) varies filters according to SNR. We test our methods on data from Mayo Clinic and recover the position and stiffness of a 3 mm diameter inclusion.