FORMULA FOR WEATHERING CORRECTION 119 
The various integrals can be evaluated by making the substitution 
u=(a\/y+5b). The final result is 
t = (4V/a?);sin [(aVZ + b)/V] — sin“ (6/V) 
+ O/V)in[(V + VV? = (ava + OPEV + VV BD} 
_ (4/a2V) { [(V2/2) sin-! [(aV/Z + 6)/V | 
+ [(b — aVZ)/2]/V* — (avV/Z + 0B)? 
— [(V2/2) sin (6/V) + (6/V)V/V? = 87 ]} + D/V. 
The above formula is general and holds for depths for any high-speed 
bed if the overburden has a velocity-depth function of the form 
given. The formula for the case of reflections is given in its parametric 
form by merely substituting for V the parameter K, and writing the 
corresponding formulas for time and distance. 

AN EXAMPLE 
The following formula was found to hold for some weathering data 
in Arkansas 
X = 29.7224? + 0.039 + 0.002 second 
and for the high-speed bed 
X = 6.3 — 0.315 + 0.0009 second. 
In the above formulas X is given in units of 1,000 feet and ¢ in seconds. 
The velocity-depth function was found to be approximated by 
V = 16.917\/y — 0.0035 + 0.0018. 
The values represented in this equation are in units of 1,000 feet. It 
will be noticed that in the above the fits are exceedingly good and well 
within experimental error. The depth computed by use of the above 
formula gives Z=50 feet. Computation of the depth by rectilinear 
propagation-paths leads to a figure of Z=40 feet, though this figure 
varied depending on the manner the lines were drawn on the graph. © 
The accompanying figure shows the data and the computations. 
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