For this situation, given Re Cj, Im Cj, Re Cj^j, and making 7j real, the determination 

 of 7j and Im Cj+j is not simple. When y[ is eliminated from the real and imaginary parts of 

 eq (31), a quartic equation in Im Cj+j results. Rather than derive an algebraic solution to 

 this equation, it is solved by iteration under Newton's method. A good first guess at the 

 solution is Im Cj+j = Im Cj. Four iterations usually give an accurate root. The equation is 



Im Cj (Im Cj+i)4 + [im(Cj)^ + 2(Re Cj+i)^ Im Cjl (Im Cj+i)^ 

 + 2Re Cj+i Re(Cj)3 Im Cj+j 



+ Im Cj(ReCi+i)4 - (Re Ci+i)^ Im(C^^) = f(Im Cj+j). (32) 



The root is then found: 



(ImCj+i). =(ImCj+i)._^-f/f'. 



The gradient, 7, is next given by the relationship 



7j = 



[4 Re Cj+i Im q+j (Zj+j - Zj)] . (33) 



radient, 7, is next given by the relationship 



ImCj r(ReCj^.i)2-(ImCj+i)2l + 2ReCjReCj+i ImCj+j -lm(C^) 



Because the root of eq (32) may not be exact, Im 7j may not be exactly zero. This slight 

 error can be transferred to Cj+j by using the computed real 7j to recompute Cj+j. This is 

 done in the program by transferring to a portion of the program already designed to do this. 



When sound speed and gradient at the top of the layer are given, the parameters 

 required by the program are all given. The sound speed at the bottom of the layer is rou- 

 tinely computed, however, because it may be required to make the next layer continuous. 

 Equation (5) is used to determine the sound speed at depth Zj+j, which is the depth of the 

 bottom of the layer. This is straightforward, but several comphcations arise. Only the real 

 part of the gradient at the top of the layer is used as an input because situations have not 

 arisen that require that the imaginary part of the gradient be specified. Often the attenua- 

 tion is given at both top and bottom of the layer. That is, Re Cj, Im Cj and Re 7j are given, 

 plus a relationship between Re Cj+j and Im Cj+j . The imaginary part of the gradient, Im 7j, 

 must be determined as well as both real and imaginary parts of the sound speed at the layer 

 bottom. The derivation of this case is not trivial. 



One relationship between the real and imaginary parts of the sound speed is given by 

 eq (28) and (29). From these equations at Cj.^.j we derive 



A(T-i) = 2/Cj+i, (34) 



where 



T = ReCj+j/ImCj+i. 



Substituting this expression for Cj+j into eq (31) and equating real parts gives a quadratic 

 expression for T which has a usable root of 



14 



