THEORY OF SEISMIC PROSPECTING 35 
call an iso-time curve because the travel-time from S to each point 
to this curve is the fixed value, 7». If one takes several values of the 
time one gets several iso-time curves. 
Another type of curve that we study is the curve of critical points. 
The critical point So’ is the point at such a distance from S that the 
first corner appears on the time-distance graph. As the direction 
through S varies, So traces out a closed curve surrounding S which 
we call the critical point curve. 
Our treatment is limited to the case of two strata. One stratum is 
bounded by two planes the upper of which is the earth’s surface. The 
lower stratum is bounded only by the lower boundary of the upper 
stratum and extends to great depths below. 
The work that follows can be put into analytical form easily but 
the geometrical constructions bring out the descriptive features more 
clearly. Besides, the geometrical constructions all correspond directly 
to a physical picture, whereas the details of analysis do not have 
counterparts that are helpful in comprehending the situation. 
We also show some properties of the critical distance from the shot 
point where the direct wave and the refracted wave hit the detector 
at the same time, i.e., where the first corner appears in the refraction 
profile. 
ISO-TIME CURVES 
In Fig. 1 S is the shot point on the plane surface of the earth II’. 
R, and R; are critical points, jn the cross-section shown, where critica! 
refractions begin in the plane interface RR’. Let V; and V2 be the 
velocities in the two sections. Then as is well known V,/Vz is the sine 
of the angle between R,S and the normal to RR’. Evidently SRiR:z 
is the section of a cone with a vertex at S and axis SS’, where S’ is 
the image of S in RR’, and with a circular base in the refracting plane. 
The impulse travels, according to our present interpretation, from 
S down to R, and then along the interface and up from each point 
of the refracting interface to the right of R;. The refracted rays travel- 
ling up are parallel in our cross-section. These rays and their normals 
are indicated as dotted lines. The normals to the rays are the wave 
fronts of the disturbance. 
A wave front is an iso-time surface, i.e., two detectors at any 
two points on the same wave front would detect the disturbance at 
the same time. The refracted returning wave front is a part of the 
surface of a cone. The intersection of this cone surface with the plane 
279 
