Now modify the basis vectors so that the interface conditions can be within in the following 

 matrix form 



1 







-1 







+ 1 







(71-1) 







71 















-71 







(71-1) 





^0''aO 











Al 





-Pq/PI 



X 



^1 '"al 

 *^1 ^|31 



= 





 



- 



Dl 











_A2 









Now solve for Bq using Cramer's Rule 

 



-1 



+ 1 



-pq/pi r 



I 



I 



Bnr, 



O'aO 



+ 1 



The elements inside the dashed areas designated A|^ are the elements of the Rayleigh deter- 

 minant. Fast and accurate methods are finding lAj^i have been developed (as mentioned 

 before) because the zeroes of |Aj^| determine the phase velocity of earthquake waves. 

 Finally, we can write 









-1 







1 1 









-71 







(7i-l)| 



^oraO 









J 



' 



(Pq/Pi) 



71-1 







71 



° i 









-71 







(r,-l)J 



In the above |Aj^| is the Rayleigh determinant and |Ag| is the same except for the first 

 row. The fast methods developed for calculation of |Aj^| can be used to evaluate |Ag|. It 

 follows then that the plane wave reflection coefficient can be written as 



R ^ (Pi ^ao I^rI -PO l^sl)/(Pl '"aO 1%I + ^0 l^s') • 



55 



