THE KIRCHHOFF IMAGING METHOD

There are two basic tasks to be done: (1) make data from a model, and (2) make models from data. The latter is often called imaging.

Imagine the earth had just one point reflector at (x₀,z₀). This reflector explodes at t=0. The data at z=0 is a function of location x and travel time t would be an impulsive signal along the hyperbolic trajectory t² = ((x-x₀)²+z₀²)/v². Imagine the earth had more point reflectors in it. The data would then be a superposition of more hyperbolic arrivals. A dipping bed could be represented as points along a line in (x,z). The data from that dipping bed must be a superposition of many hyperbolic arrivals.

Now let us take the opposite point of view: we have data and we want to compute a model. This is called ``migration''. Conceptually, the simplest approach to migration is also based on the idea of an impulse response. Suppose we recorded data that was zero everywhere except at one point (x₀,t₀). Then the earth model should be a spherical mirror centered at (x₀,z₀) because this model produces the required data, namely, no received signal except when the sender-receiver pair are in the center of the semicircle.

This observation plus the superposition principle suggests an algorithm for making earth images: For each location (x,t) on the data mesh d(x,t) add in a semicircular mirror of strength d into the model m(x,z). You need to add in a semicircle for every value of (t,x). Notice again we use the same equation t² = (x²+z²)/v². This equation is the ``conic section''. A slice at constant t is a circle in (x,z). A slice at constant z is a hyperbola in (x,t).

Examples are shown in Figure 2. Points making up a line reflector diffract to a line reflection, and how points making up a line reflection migrate to a line reflector.

 

dip
Figure 2 Left is a superposition of many hyperbolas. The top of each hyperbola lies along a straight line. That line is like a reflector, but instead of using a continuous line, it is a sequence of points. Constructive interference gives an apparent reflection off to the side. Right shows a superposition of semicircles. The bottom of each semicircle lies along a line that could be the line of an observed plane wave. Instead the plane wave is broken into point arrivals, each being interpreted as coming from a semicircular mirror. Adding the mirrors yields a more steeply dipping reflector.

Besides the semicircle superposition migration method, there is another migration method that produces a similar result conceptually, but it has more desirable results numerically. This is the ``adjoint modeling'' idea. In it, we sum the data over a hyperbolic trajectory to find a value in model space that is located at the apex of the hyperbola. This is also called the ``pull'' method of migration.