Abstract
We show that the time-dependent wave equation in both one and two spatial dimensions possesses quantities which are globally conserved. We show how these conserved quantities can be used to determine the characteristic impedance, the rock density and the elastic constant of the rock. We also demonstrate that the conserved quantities possess the capability of determining and/or bracketing the unknown component of the direct pressure response, which is required to begin downward continuation algorithms. Further, we demonstrate that the conserved quantities are always available irrespective of the source structure in time. Numerical instability, arising if the “filtering” due to the source structure is too harsh, can then be controlled to a degree by demanding that the conserved quantities be indeed conserved.
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Lerche, I. Conserved quantities and the inverse problem of reflection seismology. PAGEOPH 132, 583–597 (1990). https://doi.org/10.1007/BF00876931
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DOI: https://doi.org/10.1007/BF00876931