On the Relation Between the Velocity Coefficient and Boundary Value for Solutions of the One-Dimensional Wave Equation
RICE UNIV HOUSTON TX DEPT OF MATHEMATICAL SCIENCES
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The one-dimensional acoustic wave equation is a simple model of wave propagation in layered media. As such, it is used in theoretical seismology to study the relation between sound velocity and the surface response or seismogram boundary value as a simple instance of the reflection seismology problem. Questions about the nature of the dependence of boundary values on the velocity coefficient in the wave equation arise naturally in this context, particularly in connection with perturbational techniques. We study the dependence on the velocity coefficient of boundary values for solutions of the one- dimensional wave equation. We present estimates for both the Lipschitz continuity and the linearization error for the map between velocity coefficient and boundary value. In particular, we show that this relation is Lipschitz continuous for velocities in H2 and differentiable for velocities in H3. We also discuss the anomalous smoothness of this map in oscillatory directions, which helps explain a key idea in reflection seismology.
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