Estimating the energy source and re ectivity byseismic

Data produced by a reproducible source contains redundant information which allows seismic inversion to simultaneously determine the high-frequency uctuation in the P-wave velocity (or re ectivity) as well as the input energy source. The seismogram model is the plane-wave convolutional model derived from the constant density, variable sound velocity acoustic wave equation. The rst step is to analyze this linearized model when the background velocity is constant. Then perturbations in the seismic data stably determine corresponding perturbations in the source and re ectivity. The stability of this determination improves as the slowness aperture over which the data is de ned increases. Further, the normal operator for the convolutional seismogram model is continuous with respect to velocity. Thus the stability result for constant background velocities may be extended to more realistic background velocity models which vary slowly and smoothly with depth. The theory above is illustrated with four synthetic numerical examples derived from marine data. The examples indicate that for a wide slowness aperture, inversion is very e ective in establishing the true shape of the re ectivity and the shape and location of the compactly supported energy source. As this aperture window narrows, the corresponding inversion-estimated model still describes the data quite accurately, but the inversion is not able to recover the original two distinct parameters.