An investigation of the free surface effect

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 wavetypes, 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. INTRODUCTION In 3-c shot records, we can recognize different types of events: refractions, surface waves and shear waves along with P waves. These waves are recorded in their true amplitude. In general we can not see shear waves as clearly as P waves because the frequency band is smaller for S waves and they are attenuated more strongly in the near surface. Current methods of P and S wave separation are based on the work of Dankbaar (1985) who showed how to use the free surface theory to design a linear mode separation (or modal filter). Later Donati and Stewart (1994) showed the P-S separation filter in the tau-p domain worked well on synthetic data and was effective in eliminating noise like ground-roll. These methods work very well on synthetic data but are troubled by noise and interfering wavetypes on real data. In order to better understand the physical effects which determine the frequency band of S waves and to estimate the Vp/Vs ratio in the near surface, we investigate the theory of the free surface effect and compare it with Hamarbitan and Margrave 16-2 CREWES Research Report — Volume 8 (1996) measurements from real data of the emergence and displacement angles. We hope to be able to define the type of recorded wave for any event in a source record. THEORY Defining the wave type in a shot record is based on the value of the displacement angle (φ ) in relation to emergent angle(θ ) . For P waves, propagation is in the direction of the particle displacement, while for S waves the propagation is perpendicular to it. From these concepts, our measurements and analysis are based on computing the total displacement vector, the displacement angle and the emergent angle using the weathering velocity information which we obtained from the Blackfoot survey . A) Measurement Let x denote the inline direction of a 2-D, 3-C shot record and y the crossline direction. Then we define θ x,y,t as the emergent angle of the seismic wavefront recorded on the 3-C record at (x,y,t) (figure 1). We also define φ x,y,t as the angle defined by the 3-C particle displacement vector recorded in the geophone. (Both angles are defined with respect to the positive x axis.)