Chap. 9] SEISMIC METHODS 569 



{d) Calculation ffom surface profiles. Since the travel-time equation 

 contains two unknowns — depth and average velocity— a minimum of two 

 equations must be set up to determine both velocity and depth by record- 

 ing times in at least two -distances. Thus (for horizontal beds) 



_ . 7 ^2 — Xi 



(9-81c) 



In practice many distances are required to give accurate values. Squares 

 of travel times are plotted against squares of the distances. For a reliable 

 determination the curve must be a straight line. The cotangent of its 

 angle a with the abscissa is the square of the average velocity. By dif- 

 ferentiating in the travel-time formula (v ^ = x + 4c? ), the square of 

 travel time with respect to the square of the distance, we have d{t )/d{x ) — 

 1/v^ or v^ = cotan a. Since f = x N + 4rfVv^) the velocity squared is 

 the ratio of distance squared divided by times squared. The ordinate at 

 zero distance is 4d^/v^. For this calculation travel times have to be cor- 

 rected for weathered layer and elevations; they should also be reduced to 

 regional datum. 



When the reflecting beds are inchned, it is necessary to shoot an average 

 velocity proj&le in two directions. In this case the curve representing 

 the squares of travel times as function of the squares of distances is no 

 longer a straight line. Its direction of curvature depends upon whether 

 the profile is shot up dip or down dip. From formulas (9-75a) and 

 (9-756) we obtain by differentiation 



(down dip) tan a^ = — -^ = -„ I 1 H sm ^ J 



(up dip) tan a„ = —-^- = - I 1 - - sin ^ I, 



(9-81d) 



which indicates that the curvature decreases with distance ; that is, the time- 

 squared distance-squared curves approach true velocities farthest out from 

 the shot point. The arithmetic mean of two tangents to the curve at 

 identical up- and down-dip distances gives the true velocity to the depth 

 under the shot point: 



tan au + tan ad _ I (g-Sle) 



5. Field practice, (a) General procedure. In a new area where neither 

 transmission characteristics of the near-surface beds nor the subsurface 

 section is known, the most suitable shot depth and arrangement of re- 

 ceivers (shot distance and receiver interval) must be determined by experi- 



