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The purpose of the mathematical developments of the preceding five lessons was two fold. The material developed was, on the one hand, intrinsically valuable. The mathematical discussions-particularly those concerning the parametric representations of the time-distance curves for the parallel layers-were necessary and will be useful in developing our interpretation methods. We were, in fact, led anew to that much-used and oft-abused average velocity concept which plays such an important role in all depth-determination techniques of the reflection seismograph method. On the other hand and most importantly, we are now in a position to outline the mathematical developments of the so-called “curved-path” theory in its general aspects. We shall then be ready to apply the results particularly to the widely-used case in which the velocity is assumed to vary linearly with depth.

Let us assume that below the horizontal plane representing the surface of the earth, we have the velocity of seismic propagation depending only on the depth below the surface. In mathematical parlance, this is expressed by saying that this velocity, v, is a function of depth below surface, h, alone; and we write,

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