An impulse in the data

The Huygens diffraction takes an isolated pulse function (delta function) in (x , z)-space and makes it into a hyperbola in (x , t)-space at $z \,=\, 0$.The converse is to start from a delta function in (x , t)-space at $z \,=\, 0$.This converse refers to a seismic survey in which no echoes are recorded except at one particular location, and at that location only one echo is recorded. What earth model is consistent with such observations? As shown in Figure 6 this earth must contain a spherical mirror whose center is at the anomalous recording position.

 

semicirc Figure 6 When the seismic source S is at the exact center of a semicircular mirror, then, and only then, will an echo return to the geophone at the source. This semicircular reflector is the logical consequence of a dataset where one echo is found at only one place on the earth.

It is unlikely that the processes of nature have created many spherical mirrors inside the earth. But when we look at migrated geophysical data, we often see spherical mirrors. Obviously, such input data contains impulses that are not consistent with the wave-propagation theory being explained here. This illustrates why petroleum prospectors study reflection seismic data processing, even though they personally plan to write no processing programs. The raw data is too complex to comprehend. The processed data gives an earth model, but its reliability is difficult to know. You may never plan to build an automobile, but when you drive alone far out into the desert, you should know as much as you can about automobiles.