On the Determination of the Downgoing P‐waves Radiated by the Vertical Seismic Vibrator*

Abstract

It is well recognized that in order to realize the full potential of the Vibroseis technique, one needs to ensure accurate phase locking and a meaningful cross‐correlation. To achieve these two important objectives we require an accurate estimate of the compressional stress wave radiated by the vibrator into the ground. In this paper a simple method (subject of a patent application) is developed for predicting the compressional stress waves radiated by a vertical vibrator. The main feature of the proposed method is that it involves the field measurement of the acceleration of the reaction mass and the baseplate, respectively. The method is illustrated by computing the compressional stress waves generated by a typical vertical vibrator radiating into ice, chalk, sand, and mud. It is shown that for a seismic vibrator radiating into hard ground the pressure of the downgoing P‐wave is 180° out of phase with the baseplate velocity. It is also shown that when the driving force of the seismic vibrator has a flat amplitude spectrum, the amplitude spectrum of the downgoing P‐wave falls off by 6 dB/octave towards low frequencies.

How do hydraulic vibrators work? A look inside the black box

“…Groundforce is typically approximated using the mass weighted sum of the reaction mass and baseplate accelerations (Castanet and Lavergne 1965; Sallas 1984). In the absence of shear stress at the contact surface, the far‐field particle displacement downhole, due to P‐wave radiation from an acoustically small disc vertically oscillating at the surface, in the theoretical world of elastic isotropic half‐spaces, has the same spectral shape except for time delay as the groundforce (Miller and Pursey 1954; Safar 1984). For small sources, it can be shown that the distribution of the vertical stress applied over the contact area need not be uniform and that the same relationship holds between the resulting groundforce and far‐field particle motion (Baeten and Ziolkowski 1990).…”

Model parameters of the nonlinear stiffness of the vibrator-ground contact determined by inversion of vibrator accelerometer data

“…If the contact is complete ͑infi-nitely stiff͒, no deformation of the springs occurs and z 2 = z 3 . This reduces equations 1 to the linear vibroseis model ͑Lerwill, 1981; Sallas and Weber, 1982;Safar, 1984;Sallas, 1984͒. We will use two rigidity profiles. The first is the model of bimodular contact as considered by Lebedev and Beresnev ͑2004͒.…”

“…Furthermore, the contact region is typically much softer than the other parts of the interacting bodies ͑the baseplate and the consolidated material below͒; as a result, the deformations in this intermediate region are large enough to change the spectrum of radiation considerably. A model of such a contact and its effect on radiation were discussed by Lebedev and Beresnev ͑2004͒. In the equivalent scheme of the vibroseis source ͑Lerwill, 1981; Sallas and Weber, 1982;Sallas, 1984;Safar, 1984͒, this nonlinearity can be accounted for by introducing a contact spring with the rigidity K c = −dF c /dx ͑Lebedev and Beresnev, 2004͒, where x ϵ z 2 − z 3 , F c is the restoring force from the contact deformation, and z 1 , z 2 , and z 3 are the displacements of the reaction mass, baseplate, and the ground beneath the plate, respectively ͑Figure 1͒. The system of equations governing the source then becomes ͑Leb-edev and Beresnev, 2004͒…”

“…Vibroseis parameters used in the calculations. The captured parameters of the ground are as given by Safar (1984) for sandy soil, the actuator constants are as provided by Lerwill (1981). …”

Modelling and modal analysis of seismic vibrator baseplate

“…The vibrator ground force can be approximated by using a ‘weighted sum’ of accelerations of the reaction mass and the baseplate (Sallas 1984; Safar 1984; Baeten and Ziolkowski 1990). The weighted‐sum method is built on the rigid body assumption of the baseplate.…”