Ground-force- or plate-displacement-based vibrator control?

The feedback system of seismic vibrators minimizes the difference between the theoretical sweep and the seismic signal radiated into the earth. This requires the determination of the correct source signature, that is, the characteristic of the baseplate motion that is a true representation of the downgoing wave. The prevalent systems use ground-force-based vibrator control, which assumes that the amplitudes of the seismic waves are proportional to the force applied to the ground. Mathematical solution of the pressure boundary-value problem of linear elasticity is the theoretical underpinning of this method. However, the ground force has a significant disadvantage of not being a directly measurable parameter. It has to be inferred, with significant uncertainty, as a “weighted sum” of the accelerations of the baseplate and the reaction mass based on a simplified linear model of oscillating masses, springs, and dashpots. On the other hand, the baseplate displacement (acceleration) can itself serve as an alternative, model-independent source signature. To provide a theoretical justification for the displacement-based vibrator control, one has to show that the amplitudes of the outgoing waves are proportional to the plate displacement. This has to be done by solving a “mixed” boundary-value problem, in which displacements are specified under the plate and pressures everywhere else on the surface. Such a problem has historically been considered unsolvable, unlike the mathematically easier pressure-source radiation problem. However, the analysis of approximate approaches to the solution of the mixed boundary-value problem, as well as its exact solution for the static-displacement case, reveal equivalence between the source displacement and source force in being the proportionality factors controlling the amplitudes of the downgoing waves. This provides mathematical justification for the displacement being the correct source signature. The displacement-based feedback, or feedback based on a combination of the weighted-sum and displacement, may be a more accurate method.