Experiments with synthetic data have indicated that generalized linear inversion may be used to estimate compressional velocities as a function of depth with high resolution directly from band-limited, unstacked data. The ocean surface was not included in these experiments. In the presence of strong surface multiples, inversion is expected to take longer and be less accurate, because events from multiple surface reflections overlie primary events and normally have differing moveout. Existing velocity-analysis techniques rely on the ability of an observer to make the difficult distinction between multiples and primaries.
Equations are provided for adding the surface to the inversion procedure. This involves adding the surface effects to the Jacobian matrix as well as to the forward modeling procedure. To speed computation, the addition of the surface effects to the Jacobian matrix is delayed until after the matrix has been multiplied by a vector in the linear-equation solution. Absorption is added to the inversion to represent the real world more closely and to improve computation speed by reducing sampling requirements.
Realistic synthetic band-limited data with high surface reverberation content were generated from a well-log velocity profile. The inversion recovered the velocity profile to within 3 percent when a velocity increasing linearly with depth was used as a starting profile. The error in the model-generated seismogram converges from 100 percent to within 2 percent of the reference data. The positions of interfaces are located more accurately at greater depths than at shallower depths because more sensors are observing deep strata than shallow strata. Convergence is to within 0.1 percent of that of the reference data at the maximum depth. The computation required 25 iterations and a total time of 66 hours on a DEC VAX 11/780. Reducing this time should be possible. In a preliminary study of the effects of noise, additive Gaussian noise was seen to limit the accuracy of the velocity estimate monotonically as the variance of the added noise was increased.