Similar expressions are employed for the other variances. Appendix 

 A contains the transformation coefficients used to produce Figure 11, 



The successful application of the mapping equations to the other 

 shelf regions was accomplished. Tables 2 through 5 clearly indicate 

 the successive convergence of the mapped curves with respect to that 

 specified for the lower and upper east coast and the western and 

 central gulf coast, respectively. Figures 12 through 15 show the 

 fit of the trans form- generated curves and that specified after the 

 last iteration for the above regions. Appendixes B and C contain 

 the mapping coefficients used to produce Figures 12 and 13, respec- 

 tively. The bottom part of Tables 4 and 5 show the results of addi- 

 tional testing of the conformal mapping equations (9) and (10) with 

 the western and central gulf coast regions. Another, less general, 

 solution of equations (9) and (10) is possible if one minimizes the 

 least square error function defined only in terms of the Y inte- 

 grals. This solution for the transformation coefficients, hereafter 

 referred to as an alternate solution, may be obtained from equations 



s c 

 (13) through (19) with W = W = . The testing procedure was to 



continue the iterative process as outlined in the previous section 

 with the initial approximation for the coefficients being those 

 values determined from the 80th iteration. The alternate solution 

 as applied to the western and central gulf coast regions was stable 

 and, moreover, provided a better fit with N = 110 than the more 

 general one. The fit of the mapped curves with respect to that 

 specified after 1 and 40 (or, 81 and 120) iterations with the alter- 

 nate solution is shown in Figures 16 and 17, respectively, for the 

 western gulf coast and Figures 18 and 19, respectively, for the cen- 

 tral gulf coast. Appendixes D and E contain the transformation co- 

 efficients used to produce Figures 17 and 19, respectively. In test- 

 ing with the other three shelf regions, the alternate solution was 

 nonconvergent in that the successive values of the error function 

 do not decrease or approach a constant as outlined in the previous 

 section. The reason for this behavior has not been investigated. 



The total variance of x and y from a linear transform of the 

 curvilinear coordinates may be defined as: 



2 



+ a 2 (32) 



where 



and 



x y 



3 A 



c = 23X j J [x - ?] 2 d5 dn ; (33) 



-3 o 



32 



