HISTORY OF THE METHODS/FLOW CHART
ABOUT THE AUTHOR/CV
1996, 1999, 2006
Applications to Seismology
The key idea is to use Fast Marching Methods to compute the first arrivals in seismology. Because the technique so efficiently solves the Eikonal equation, it gives a very fast way of computing these answers. The ability to compute the first arrivals is an important part of what is known as three-dimensional prestack migration, which is the process by which one builds a model for the Earth's interior.
Three-dimensional (3D) prestack migration of surface seismic data is a tool for imaging the earth's subsurface when complex geological structures and velocity fields are present. The most commonly used imaging techniques applied to 3-D prestack surveys are methods based on the Kirchhoff integral, because of its flexibility in imaging irregularly sampled data and its relative computational efficiency. In order to perform this Kirchhoff migration, one approximately solves the wave equation with a boundary integral method. The reflectivity at every point of the earth's interior is computed by summing the recorded data on multidimensional surfaces; the shapes of the summation surfaces and the summation weights are computed from the Green's functions of the single scattering wave-propagation experiment
In 3-D seismic surveying, seismic waves are generated by surface sources (shots), and the reflected waves are recorded at surface receivers (geophones). The Green's function describes the energy of the wavefield back-scattered from the reflector point at all possible source and receiver combinations.
For 3-D prestack Kirchhoff depth migration, the Green's functions are represented by five-dimensional (5D) tables; these tables are functions of the source/receiver surface locations (x,y) and of the reflector position (x,y,z) in the earth's interior.
The key element of 3-D prestack Kirchhoff depth migration is the calculation of traveltime tables used to parameterize the asymptotic Green's functions. This is where the Fast Marching Method comes in; it allows one to compute these travel time tables *extremely* fast.
Computing first arrivals can be done efficiently and accurately using Fast Marching Methods. However, there are cases in which later arrivals contain important information. A good example of the important of these later arrivals is in geophysical imaging, in which one tries to predict what lies beneath the earth's surface by sending waves into the ground and recording their reflection. In this case, the first returning wave might not contain all the information, and later arrivals might contain more energy which can be used to more accurately predict what lies beneath.
equation (though with an extra dimension). We can do this by solving using a variant of fast marching methods and ordered upwind methods. Start a surface at the boundary (in three dimensional phase space): for every point on this boundary, we know the escape position and angle, since it is the same as the starting point (it's already on the exit!). Then, we can systematically march inwards, reaching back to the known escape values, and eventually cover the entire phase space cube: this is what is shown below.
Below is one result from this technique: Waves propagate from the top point through a region that contains a slowness disk in the center: you can easily see the waves propagate around the slow part, and double back on themselves, creating multiple arrivals.
New Book and Resource on Level Set and Fast Marching Methods