Re(cf )T = -lm(C^) - 



[Iin(C^)] +Re(ci^)B 



(35) 



where 



B = Re(C^) - 8 Re Ti(Zi+i - z^/A^ + 4 Re Cj/A^. 



(36) 



From eq (34), 



ReCj+i =2T/A(t2+1) 

 and 



ImCj+i=RCj+i/T. 



The gradient can now be evaluated by eq (31) to find its imaginary part. 



Equations (34) and (35) cannot be used if the attenuation at the bottom of the layer 

 is given as zero. Therefore an alternate form must be used. This form is much simpler than 

 the previous case, since Cj+j is real. 



^i+1 " 



Re(Cj ) / [Re q - 2 Re 7i (Zj+i - Zj)] 



V2 



Im Ti = [Im q - Im(Cp/Cr;j ] [2(Zj+i - z-^)]'^ 



Finally, if the special case, 7j real, is specified by inputting a negative value for 

 absorption, eq (31) can be used directly to give 



C?,i=cf/[q-27i(Zi+i-Zi)]. 

 To evaluate the square root, let 



(37) 

 (38) 



(39) 



C._^i = a + bi. 

 1+1 



Then 



and 



Req+i= a + (a2 + b2)''^l/2 ' 



Imq^.1 =b/2Req+i. 



(40) 



(41) 



NUMERICAL BREAKDOWN 



A situation arises frequently in which a very small depth function must be computed 

 from the difference of two large numbers. A wrong answer results if this accuracy loss 

 exceeds the word size of the computer. The best way that has been found to avoid this is to 

 check for it within the program and arbitrarily replace the wrong number. In checking for 

 this, a constant, called T-lim in the computer program, is compared to the argument of the 



15 



