An investigation of the free surface effect
Nasser S. Hamarbitan, Gary F. Margrave
When P and S seismic waves are incident on a solid-air interface at a non-normal angle, the displacement of each wave will be observed on both vertical and horizontal geophones at the interface. Since the incident waves will cause P and S reflections (the free surface effect), and since the geophones record the total displacement at the interface due to all waves, a complete understanding must consider not just the emergence angle but also the resulting reflections.
Simple considerations for body waves predict a particle displacement vector which points in the direction of propagation for P waves, and perpendicular to it for S waves. The effect of the free surface is to cause the displacement vector for P waves to deflect towards the vertical and towards the horizontal for S waves.
We derived a theoretical expression from the Zoeppritz equations, which relates the emergence and displacement angles as a function of Vp/Vs. Then a Matlab program was built to study the relationship between the particle displacement angle and emergence angle for selected events in a seismic shot record. Given the (t,x) description for the trajectory of an event, the program models by least squares each component of displacement and the wavefront shape as low order polynomials in offset. The polynomials of the displacement component were used to estimate the direction of the displacement vector and the wavefront shape polynomial together with a value for the weathering velocity, was used to estimate the emergence angle.
This method was applied on a shot record from the Blackfoot broad-band survey and we observed that there are many effects in real data which complicate these measurements including interfering wave types, near surface material variations, static anomalies, and random noise. Nevertheless our results are broadly consistent with free surface theory and suggest Vp/Vs ratios in the range of 5 or greater will characterize the Blackfoot weathering layer.