Ray‐path Curvature in Shallow Seismic Investigations

Equations for seismic ray‐path curvature are presented for both linear and nonlinear velocity versus depth models. Techniques are then developed for correcting cross‐hole results for ray‐path curvature and for determining the model constants for seismic refraction based on data from two source‐to‐receiver spacings. The principal advantage of the techniques is that corrections are independent of the soil properties and stress conditions above and below the test elevation, not including the relatively thin zone through which the ray path passes. The linear model is of the form $V=V_o+\mathrm{mz}\text{,}$ where $V_o$ and m are constants to be determined. Since the propagation velocity function needs only to be linear in the thin zone through which the ray path passes, this model is valid in many situations. The nonlinear model, $V=a(z+z_o{)}^b\text{,}$ has three unknowns, a, $z_o\text{,}$ and b. Since two receiver locations will only provide sufficient information to solve uniquely for two unknowns, an assumption for the third constant must be made. If $z_o=0$ is assumed, the model becomes useful for refraction studies in cohesionless soil. If $b=1/4$ is assumed, cross‐hole test results in shallow, normally consolidated, uncemented sand can be corrected.