118 H. M. RUTHERFORD 
Here P(D) represents the penetration and V(D) is the velocity at the 
mid-point of the path of a ray with shot-length D. 
Now, it has been found that the time-distance data for the 
weathered zone fits a function of the form | 
X = at?+ Ol. 
An example of this is shown later. If the above function be substituted 
in Ewing’s formula for penetration, the result is given by 
P(D) = (1/4ax)[VVV2 — b? — b%ln(V/b + /V2/b? — 1)). 
The above expression is rather involved but fortunately it can, in the 
case of the weathered-zone data, be represented by an approximation, 
or 
V=aVy+b 
where V is the velocity, y is the penetration below the surface, and a 
and 6 are constants. 
The author has shown that the time-distance relationships for the 
case of a high-velocity bed whose overburden consists of material in 
which the velocity is a steadily increasing function of the depth is 
given by?® 
t= 2 | vay/[VoyVVE=V'G)] 
~ (2/V) f 7 V(y)dy//VE= VA) + D/V 
where Z is the depth of the high-speed bed below the surface, or the 
thickness of the overburden, V is the velocity of the high-speed bed, 
V(y) is the velocity-depth function, and ¢ and D are the time and 
horizontal distance on the surface for any ray which has been re- 
fracted over the high-speed bed. We now substitute for V(y) in the 
above expression its approximation, that is 
V(y) = avy +6 
and get 
t= 2 f “Vay/avy + VV = Va FO 
=o V. I) “(an/9 +: b)dy/\/ V2 aan ae 
5H. M. Rutherford, Trans. Amer. Geophys. Union, fourteenth annual meeting, 
1933, Pp. 289-303. 
778 
