The vibrator-ground model and the vibroseis source wavelet

Miller and Pursey (1954) have shown that for an isotropic-homogeneous-elastic half-space, the far-field particle displacement is proportional to the surface stress if the surface stress is uniformly applied over a small disc. This relationship of force and far-field particle displacement establishes the theory of the vibroseis method. It implies that the vibrator ground force is proportional to the far field particle displacement if the ground is assumed to be an isotropic-homogeneous-elastic body and the vibrator baseplate is small enough comparing with the wavelength of interest. The true ground force is the integration of the pressures beneath the baseplate. However, this true ground force is not directly measured, and it is often estimated by a weighted sum of outputs from two accelerometers placed on the reaction mass and baseplate assemblies (Sallas and Weber, 1982; Sallas, 1984). The weighted sum estimation of the ground force, often referred as the weighted-sum ground force, has been widely used as the phase locking feedback signal on the vibrator force control since 1984. Initially, the vibroseis source wavelet embedded in correlated vibroseis data is assumed to be the autocorrelation of the pilot sweep (zero-phase with a flat spectrum).