730 EXPLORATION GEOPHYSICS 



because success of the method demands that the velocity discontinuities, 

 if present, be horizontal beneath the observations. The ideal situation would 

 be encountered when working over areas where the strata are parallel to 

 the surface. Reasons for this will become obvious from a study of refrac- 

 tion shooting methods, which are covered in a subsequent section. 



The velocity depth function is calculated by making a Herglotz-Bate- 

 man-Weichert integration of the measured time-distance function. 



The slope (— — ) of the time-distance curve is the reciprocal of the 



\dx) 



apparent velocity l/V. Under the condition of horizontal stratification of 

 the velocity discontinuities, the apparent velocity Vp is equal to the velocity 

 of the seismic wave corresponding to the depth of deepest penetration (Zp) 

 of the ray recorded at a distance {X^). If {Vp) is the velocity at distance 

 (Xp), the depth of deepest penetration, (Zp), is given by the integral. 



X 



p 



Zp = — ( cosh-i -^dx (85) 







This integration can be carried out with the use of Simpson's rule or 

 other methods of numerical integration. In the general case the mathe- 

 matical difficulties are somewhat formidable in obtaining an expression for 

 the velocity distribution. However, in the special case where velocity 

 distribution is of the type V = Vi + aZ, it is relatively simple to fit the 

 time-distance curve. In this case of a linear velocity increase with depth, 

 the travel time-distance function has the form : 



r = 2/a sinh-i^^^ (86) 



If the observed refraction travel-time distance curve can be fitted to 

 this function, then the values of the constants a and V i may be directly 

 determined. 



In ca-ses where the refraction travel-time distance curve cannot be 

 fitted to this expression 86 for T , the integration 85 must first be carried 

 out, then a velocity distribution function found to fit the calculated values 

 of Z and the measured values of V . 



In California, where there are considerable thicknesses of rather similar 

 sediments, good fits of the travel-time curve to Equation 86 have been 

 obtained. Subsequent to this early refraction work, well velocity surveys 

 have indicated that the values of a and V t established by refraction work 

 were essentially correct within the first order of magnitude. 



Experience in shooting long refraction profiles in California indicates 

 that this method is rather impractical for the investigation of depths below 

 4,000 feet, because of the long distances necessary, between shot-point 

 and receiving stations, to obtain the required penetration. A distance of 



