SEISMIC METHODS 



685 



-^ sinh -y- whose center is 



at the depth — I cosh ^ — 11. 



As a result of carrying out the integration of Equations 37 and 39 by us- 

 ing the type of velocity-depth function, V = Vi + aZ, and rearranging the 

 result, it is found that a reflected wave, recorded at the shot-point at time T 



comes from a bed which is tangent to a sphere of radius — smh — , 



whose center is at a depth | —[ cosh -^ — 1 



e( 



aT 



, below the datum surface.f 



This condition is illustrated by Fig- 

 ure 422, giving the position of the re- 

 flecting interface by the coordinates H 

 and Z, in the plane of the incident ray. 



From this result it is obvious that 

 a computation method for reflection 

 data may be carried out by the use of 

 the so-called wave front chart, which 

 depicts ray paths and wave fronts radi- 

 ating from a shot-point. Figure 423 

 illustrates one of these wave front 

 charts, 



A wave front chart may be visual- 

 ized as representing a vertical section 

 through the plane of incidence with 

 the origin at the shot-point. These two 

 sets of curves then represent the ray 

 paths and the successive positions of 

 the wave front surface as it moves for- 

 ward through the earth. Since the time 

 of reflection is twice the travel-time to 



the reflecting bed, and the chart gives the time of reflection, the apparent 

 velocity observed on the chart is one-half of the actual wave velocity. The 

 vertical depth and the horizontal distance are represented on the chart by 

 cartesian coordinates. For each observed pair of values of T and sin ai 

 obtained from reflection seismograms, one pair of values oi H and Z, which 

 give the position of the reflection, can be read from the chart. The chart 

 can be regarded as the superposition of two orthogonal sets of curves, one 

 the cartesian system of constant H and Z, the other the plotted system of 

 constant T and sin ai. Coincident points in the two systems give solutions 

 of Equations 37 and 39 and hence the desired positions of the reflecting 

 interfaces. 



Fig. 422. 



Position of reflecting interface in 

 the plane of the angle of incidence. 



t A derivation of this result is found in L. L. Nettleton's "Geophysical Prospecting for Oil," 

 p. 355, McGraw-Hill, New York, 1940. 



