More specifically we consider the following models in order of increasing complexity. 1) The time-step, double-couple source, which is known to represent a source consisting of instantaneous (tangential) slip on a small region a fault embedded in an isotropic elastic medium. We illustrate the associated far-field radiation pattern and discuss the use of the double couple as an influence (Green's) function.
2) The spring and block model. This model in its simplest form consists of a chain of blocks connected by springs. The blocks rest on a rough table and the chain of blocks is drawn along slowly by an irresistible constant-velocity drive at the leading end of the spring at the (front) end of the chain. The blocks are observed to move in a jerky fashion reminiscent of earthquakes. In fact this system emulates many features of earthquakes. If we interpret each jerk as an earthquake the number of blocks slipping in one event may represent the extent of rupture at the source. The magnitude may be represented by the drop in the potential energy of the springs (or its logarithm) and the frequency-magnitude statistics are reminiscent of the Gutenberg-Richter relation. While this model can be set up as a simple bench-top demonstration, computational models simulating it and its generalizations are richer and lend themselves to more precise analysis. For instance, by replacing the frictional resistance by a viscous resistance aftershock sequences may be generated and also analyzed statistically. The chain of blocks may be replaced by a two dimensional array of blocks, and the system may be driven in more complicated ways more like a discretization of a sliding fault driven by relative motion of the halfspaces on either side of it. Other forms may be imagined in which several faults interact.
3) A numerical simulation of an earthquake model in which the region of slip is specified and the problem is to find the amount of slip. We discuss the numerical solution of the singular integral equation and in particular the discretization of the non-integrable kernel and show illustrations of the displacements obtained both for antiplane and for plane strain.
4) An analytic solution of the two-dimensional problem of strike slip on a vertical fault driven an initial shear stress. The initial configuration is held in equilibrium by static friction until the traction at some point on the fault exceeds the static limit of friction. Thereafter a region of slip expands about the point of nucleation running initially with the shearwave speed both up and down. The downward-travelling 'crack' tip slows to a subsonic speed owing to the increasing friction with depth, and slows to a stop. The complete sliding motion also stops around this time. The full sliding displacement is calculated by solving analytically certain Abel integral equations which arise from specializing the Green function for the two-dimensional wave equation to the fault plane. The far field pulse shapes are calculated and a very strong breakout phase is identified which dominates the pulse in direction close to vertically down. The subsonic part of the downward travelling 'crack' edge is interrupted by this stopping phase and runs again briefly at the shearwave speed. This work was influenced by papers of Boris Kostrov from the 1960s.
5) A repetitive earthquake model consisting of a source of the previous type embedded in a viscoelastic halfspace driven by a slow viscous shear flow at infinity. The time scale for relaxation of the viscoelastic medium is very much greater than the time scale of the Burridge-Halliday mechanism, i.e.the time it takes for elastic waves to traverse the source region. The relative magnitudes of static friction, dynamic friction, and the stress at infinity determine the ratio of the repetition rate to the time scale of the source mechanism. Unfortunately the world is not a two-dimensional halfspace undergoing antiplane shear and fault planes are not only much more complicated but we do not know the necessary parameters such as the preexisting stress fields and strength and properties of the material at the fault to predict earthquakes in a practically useful way. However, our statistical knowledge of seismicity can aid in assessing earthquake risk.

Knowing that there are only a finitely many possible labels and by postulating a Gibbs prior on the underlying distribution of label images, we show that it is possible to recover the unknown image from only a few noisy projections.

Joint work with Gabor T. Herman.

The work is in collaboration with members of my group (M. Mahmoudi and K. Patwardhan) and also in part with Honeywell (V. Morellas).

This is joint work with David Brady, Mohan Shankar and Andrew Portnoy.