As explained la Scarborough (1930), Article 114., the log-transfor- 

 matloa of the dependent variable (7) causes the residuals In the least- 

 squares residual equations to be of unequal weight. In the case of a 

 power law function (which Is given as an example In Scarborough (1930) 

 and will not be reproduced here) , the weighting function Is the squared 

 dependent variable (7^). Appendix A. 2 presents an ASCII FORTRAN 77 pro- 

 gram for doing such a weighted regression anal7sls in log-log space. 

 Because of the chl-square distribution of errors associated with the 

 amplitude spectrum, however, the residuals In this case are equall7 

 weighted and no weighting function Is required. 



In fitting the envelope estimates of discrete band passes for use 

 in "province picking," there Is no requirement for log transformation of 

 the data or weighting of residuals. This Is the case because the error 

 associated with all envelope estimates is theoretlcall7 constant. In 

 order to derive properl7 the regression coefficients a and b in linear- 

 linear space, it is necessary to use an iterative method. The following 

 development is modified from Scarborough (1930), Article 115., to appl7 

 to the power law functional form. 



We can express the regression coefficients as the sum of initial 

 estimates and differences as follows: 



a - a^j + Aa 

 b - bj, + Ab 



where a^.b^ - initial estimates 

 Aa,Ab > correction factors 



126 



