SEISMIC METHODS 727 



Application of Least Squares to AT. — After the delay of the reflected 

 wave in the low-velocity layer and the elevation corrections have been 

 applied to each recording trace of the seismogram, the next procedure is 

 to establish the value of the step-out time AT to be used in the computation 

 of the dip of the reflecting interface. 



All necessary calculations could be made by the use of the step-out 

 time of the two outside traces alone, or the data from any other pair of 

 seismometers could be separately computed. When this is done, it is gener- 

 ally found that the results using dififerent pairs recording the same reflection 

 differ somewhat because of experimental errors of the individual traces. 



It is desirable to obtain the best possible final result by averaging sep- 

 arate pairs of the step-out time measurements. It is also recognized that 

 the percentage of probable error of the more distant traces is less than the 

 corresponding errors of those traces which are nearer to the center of a 

 split spread. Thus, it is obvious that more weight should be given to the 

 outside pairs of traces, other conditions being equal, in obtaining the 

 averaged value for AT. 



The theory of least squares dictates that the weight to be given to any 

 one measurement, when averaging it with other similar measurements, is 

 inversely proportional to the probable error of the measurement. 



Since the desired result is the ratio of the step-out AT to the spread S, 

 if each trace has the same probability of error as any other trace, then the 

 probable error in any one measurement of AT/5" will be inversely propor- 

 tional to S, and the weight of the measurement. A, will be proportional to 

 the square of S. Applying this to an actual case of 12-trace record in which 

 the instrument interval is uniform over the entire length of the spread, let 

 ti be the outside step-out time, (2 the step-out time of the next inside pair, 

 etc. Let the distance 6^ be expressed in terms of trace intervals. The averag- 

 ing can be conveniently done in tabular form, as in Table 22. 



TABLE 22 



The weighted average of t/S is the total of the fourth column divided 

 by the total weights in the third column. The weighted average step-out 

 time for the total spread is obtained by multiplying this figure by 11, the 

 spread length in units of trace intervals. 



Average t = 0.307/286 • 11 = 0.0118 sec. 



