How a Surface Source Behave like a Buried Source

How a Surface Source Behaves like a Buried Source

by Darrell Connelly

We will follow the development of source models presented by Aki and Richards(1980). We begin with the the representation theorem (p 29,eq 2.41):

where is the i-th component of the at position , t is the time variable, performs the convolution of the "sourcing" functions with the solutions of the medium-boundary-value problem, i.e., the Green’s functions and derivatives thereof, and (here we use Aki and Richards convention ), are the source location, source reference time, receiver location, receiver reference time, respectively, for the Green’s function, is a "source" traction with normal to the surface, is the "source" displacement, and are the source medium’s elastic constants. This representation theorem relates the possible sources and source interpretations of displacements that one can find within and on the surface of a volume. We shall use this representation theorem to discuss the types of displacement "source" functions that can arise in practice. Any of the three terms could be used to discuss a particular source. However, we shall limit our focus on the last term, the relation between a "source" displacement and a displacement within the medium.

Therefore, we limit our representation theorem to:

We shall interpret this equation as the relationship between a displacement (caused most likely by some force, say a vibrator) on a surface and the displacement "received" within the medium. We shall bring special attention to the spatial derivative within the Green’s function later in the appendix.

Let us assume the small disk-like surface shown in Figure A1 with an upper surface, , lower surface, , normal vectors to each surface, and , and displacement vectors, and . The displacement vectors are equal in magnitude and opposite in direction, thus

, and . In our representation theorem, we utilize . From our definition which implies .

Therefore, we can represent the differences of the displacements as a single displacement. Furthermore, the displacement vectors do not need to be opposite in direction and equal in magnitude for the differences to be represented as a single displacement.

Therefore, the representation theorem becomes

where we interpret as the total of the displacement dicontinuities on the disk-like surface. Thus, our displacement "source" representation theorem has the same form regardless of whether a small two-sided, disk-like surface or a single surface exists. This is remarkable. This result implies that a surface-source displacement produces an identical displacement "source" (provided equivalent mediums and effective strengths exist) to a buried disk-like source.

October 26, 1994 - Lakewood, Colorado

References:

Aki and Richards, Quantitative Seismology: Volume 1, 1980, W.H. Freeman and Company.