PEDERSEN, GORDON, AND WHITE: SURFACE DECOUPLING EFFECTS 



Equations 3 and 4 give x ^-^'^ ^ for the isospeed approximation. 



Here the sound speed is assumed to be a constant value, C . Z 



o 



represents the depth of the receiver, 6 the ray angle, and f the 



o 



acoustic frequency in Hertz. Equations 3 and 4 are old results. 

 These expressions are essentially the same as those developed by 

 C. Spofford of AESD and used by him in the original FACT model 

 (Spofford, 1974, and Baker and Spofford, 1974). 



Equations 5 through 7 describe two new approximations developed 



by D. Gordon of NUC. The CHI approximation is given by Equations 5 



and 6. Here At is evaluated by a ray theory integral which takes 



into account the structure of the sound speed profile. For this 



method the tangent of the ray angle is integrated from the surface 



down to the receiver depth Z. The ray parameter is C , which is the 



m 



phase velocity of the ray. It is the sound speed at which the ray 

 becomes horizontal. Equation 6 is the mathematical way of expressing 

 that Z is the value of depth for which x = tt/2. 



The linear approximation is given by Equations 5 and 7. Here 

 Z is determined by Equation 5 as in the CHI method. However, 

 X is determined by Equation 7, in which x varies linearly with Z. The 

 structure of the sound speed profile determines Z as in the CHI 

 method. However, once Z has been determined, the dependence of the 

 profile on Z is ignored and x is determined by a single linear rela- 

 tionship. Note that in the CHI method x depends only on the profile 

 from surface down to depth Z, whereas in the linear method x depends 

 on the profile from the surface down to the decoupling depth no matter 

 how shallow the receiver depth is. 



Figure 5 compares the decoupling loss of mode theory with those 



of the three ray approximations. The difference between the loss of 



mode theory, H , and the particular approximation is shown as a 

 s 



572 



