Subsurface Methods as Applied in Geophysics 1097 



methods have been described in the literature for fitting empirical data 

 to mathematical expressions of type V = Vo + kZ (linear increase of 

 velocity with depth) . Hafner ^^ has described a method in which observed 

 time versus depth curves are matched with a set of master curves, the con- 

 stants in the equation being read from the master curve that best fits the 

 observed data. Another method proposed by Legge and Rupnik ^^ is based 

 on a least-squares determination of the velocity function. It is frequently 

 observed that the rate of increase of velocity is greater at shallow depths. 

 Sparks ^° suggests that the velocity function describing a hyperbolic 

 increase of velocity with depth will best fit the observed data. He presents 

 a method for obtaining the constants for the hyperbolic relation 



(l + 26Z + a6Z2) 



A discussion of the several mathematical expressions that may best de- 

 scribe the observed velocity data in many areas is given by Mott-Smith.*^ 

 After selecting the velocity function that best fits the empirical data, 

 it becomes necessary to describe the path of the reflected ray in terms 

 of this velocity. When seismic rays pass through a medium in which the 

 velocity is variable, the phenomenon of refraction requires that they must 

 deviate from a straight line. Completely and rigorously to describe the 

 path of the ray would require a detailed knowledge of the wave velocity 

 in the medium between the reflecting beds and the surface. Obviously, 

 since we have smoothed our observed data to fit a mathematical relation- 

 ship, it becomes impossible exactly to describe the path. It should be 

 noted that curved-ray paths are described by instantaneous-velocity func- 

 tions rather than average-velocity functions. By knowledge of the ray 

 path, it becomes possible to determine the depth, dip, and horizontal 

 offset of the reflecting point. Rice has assumed iso-velocity surfaces to be 

 horizontal and an asymmetric structure to show variations of depth and 

 displacement for various computational methods.®- For a parabolic in- 

 crease of velocity with depth, the over-all results of using the curved- 

 path method and the modified straight-path method proposed by Stulken ®^ 

 are quite similar. As the computations required to obtain these data are 

 involved and tedious to perform, charts are frequently constructed from 

 which the information may be readily obtained. Simplified computational 

 techniques for the linear increase of velocity with depth {V =Vo+kZ) 



^^ Hafner, W., The Seismic Velocity Distribution in the Tertiary Basins of California: Seismological 

 Soc. America Bull., vol. 30, no. 4, pp. 309-326. Oct. 1940. 



®^ Legge, J. A., Jr., and Rupnik, J. J., Least Squares Determination of the Velocity Function 

 V=Vo + kZ for Any Set of Time-Depth Data: Geophysics, vol. 8, no. 4, pp. 356-361, Oct. 1943. 



'" Sparks, N. R., A Note on Rationalized Velocity-Depth Equation: Geophysics, vol. 7, no. 2, pp. 

 142-143, Apr. 1942. 



^^ Mott-Smith, Morton, On Seismic Paths and Velocity-Time Relations: Geophysics, vol. 4, no. 1, 

 pp. 8-23, Jan. 1939. 



°^ Rice, R. B., A Discussion of Steep Dip Computing Methods Part I: Geophysics, vol. 14, no. 2, 

 pp. 109-122, Apr. 1949, and Part II, Geophysics, vol. 15, no. 1, pp. 80-93, Jan. 1950. 



'^ Stulken, E. J., Effects of Ray Curvature Upon Seismic Interpretations : Geophysics, vol. 10, no. 4, 

 pp. 472-486, Oct. 1945. 



