directs the program to use the same imaginary part of sound speed as occurred at the bottom 

 of the previous layer. Similar flags at the bottom of a layer are discussed later. 



Absorption per Hz is given in units of decibels per km (or kiloyard). The quotient of 

 absorption over frequency is used because Hamilton (ref 8) usually considers absorption (or 

 attenuation) as proportional to frequency with a coefficient k. We use the symbol h instead. 

 That is, 



a; = hf. 



We interpret a to be in units of dB per km and f in Hz, whereas Hamilton uses dB per m and 

 kHz; but the coefficients h and k remain equal. 



The complex wave number in layer i is represented as 



kj = oj/q 



-coReCICr^-iojImCICr^. (28) 



A plane wave will be attenuated a dB per km if 

 Imkj = -a/(20 0001oge) 



= -TrAf, (29) 



where 



A - h/(20 000 TT log e). 



By equating the imaginary part of kj in eq (28) and (29), the imaginary part of Cj is found to 

 be as follows: 



ImCj=l/A-[l/A2-(Req)2]''. (30) 



If a is zero, which is the case usually used in water layers, eq (30) cannot be used; but the 

 imaginary part of C is then simply zero. These two cases are treated separately in the program. 



When sound speed is given at the top and bottom of layer i, the imaginary parts of 

 the sound speeds are determined by eq (30) and the only curve fitting task is to determine the 

 gradient yj. Solving eq (5) for 7j, 



Ti = q(cf^i-C.2)/2C?^j(Zi+i-Zi). (31) 



The gradient is a complex number since the C's here are complex. The z's are real. 



A second version of this computation arises if the gradient is required to be a real 

 number. In this case, which is used to match older versions of the program, an additional 

 parameter must be left unspecified and this parameter is Im Q+j. This is equivalent to having 

 the sound absorption at the bottom of the layer unspecified. Therefore, a negative number 

 input for this parameter is used as a flag to call for this particular fitting procedure. 



13 



