We answer the question: what happens when a seismic wave propagates from one region of constant velocity to another region of different velocity, through a smooth transition zone - and can we detect this with seismic reflection data?
This work presents an exact analytic solution for the case of a linear ramp velocity in the transition zone, and demonstrates that for long wavelengths, the ramp looks essentially like a jump discontinuity in the medium, with the corresponding reflection and transmission coefficients. For short wavelengths, the ramp provides essentially 100% transmission and no reflection.
We describe the mathematical details of this frequency-dependent reflection and trans-mission, and derive an estimate of the feature size of the transition zone that should be detectable. If we can detect zero crossing in the middle of the frequency range of our seismic measurements, for instance at 50Hz, we expect to be able to detect transition zones with length on the order of L = c/100s-1, where c is the velocity of propagation.
A careful consideration is given to the two cases of varying the velocity parameter, one via variations in the density of the propagation medium, the other in varying the modulus of elasticity. The results are different, in particular there is a sign difference in the reflection coefficient, and a large amplitude difference in the transmission coefficient.
We also present the numerical result for the transmission and reflection of a delta spike through the velocity ramp, and observe the reflection is a broadened "boxcar" response, whose width is directly related to the width of the transition zone. The transmission of the spike, however, also results in a spike.
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