682 EXPLORATION GEOPHYSICS 



A large amount of velocity-depth data have been made available to the 

 geophysicist by cooperative well-shooting among the various geophysical 

 operators. A study of these measured velocity data reveals that it is imprac- 

 tical to attempt to predict a quantitative value for the variation of velocity 

 with depth. It is necessary, therefore, to obtain this value from measure- 

 ments made in the area or to estimate it from prior measurements made 

 under similar geological conditions in the nearest possible vicinity. 



The simplest relationship of the variation of seismic velocity and depth 

 is obviously a linear increase of velocity with increasing depth. It is indeed 

 fortunate that, in practice, it has been found that the velocity-depth data 

 in many areas closely approximate this simple linear relationship. Perhaps 

 next to this relationship in simplicity is the variation of the velocity pro- 

 portional to the travel-time along the vertical (Z) axis. This linear time 

 law also has been found to approximate a large number of sets of measured 

 data. From a mathematical point of view the linear time law has some 

 advantages over the linear depth law, but the simplicity of the form of the 

 wave paths and wave fronts of the linear depth law offsets some of these 

 advantages in practice. 



By the direct method of least squares, the linear time relationship can 

 be closely fitted to the experimentally measured data, whereas this is not 

 the case with the linear depth relationship. A practical method of fitting 

 the linear depth law to the measured data is prosecuted by using two chosen 

 points on the experimentally measured curve to calculate a second theo- 

 retical curve. The resulting curve will intersect the experimental curve at 

 the chosen points, but in general it will not coincide with it. By choosing 

 other values of the slope constant, new theoretical curves may be computed 

 which may more closely fit the observed data. By this method of successive 

 approximations, a curve may be finally computed which fits the observed 

 data quite closely. Obviously this method is an indirect one and involves 

 considerable smoothing of the observed curves. The desirability of this 

 smoothing seems to be somewhat debatable. However, in special cases, 

 it appears that the smoothing of the observed data is justified by the fact 

 that the observed data are based upon approximations which lie within the 

 limits of the smoothing effects. 



In some cases the measured increase of velocity data fits a parabolic 

 increase of velocity. There is little doubt that there are many mathematical 

 relationships which will fit the observed data for various areas, but the 

 three variations of (1) lineal increase with depth, (2) lineal increase with 

 time, and (3) parabolic increase with depth, are the most practical to use, 

 if a good approximation to the measured data can be procured. 



If a satisfactory fit to the velocity data cannot be obtained with a simple 

 analytical expression, the theoretical considerations in such cases are more 

 difficult. But such a situation is not hopeless. If it is not convenient to 

 express a velocity-depth function in an analytical form, it is not necessary 

 to do so, because the computations can be carried out graphically with a 



