compressional wave and an upgoing and a downgoing shear wave in each sohd layer and 

 downgoing compressional and shear waves in the bottom half-space. We can arbitrarily 

 set the coefficient Aj^+| = 1, as shown, so that there are 4n + 3 unknown coefficients 

 (An, Bq, Aj, B|,C], ..., Cj^+i), where n is the number of layers. There are also 4n + 3 

 interface conditions. Three conditions at the first interface (continuity of vertical com- 

 ponents of stress and strain and zero horizontal stress) and four conditions at all other 

 interfaces (continuity of vertical and horizontal stress and strain). Since the interface con- 

 ditions can be written as a set of linear homogeneous algebraic equations, the solution can 

 be done using standard matrix inversion algorithms. This is not a practical method of 

 solution when n is large because it is necessary to invert a matrix of (4n + 3) elements and 

 because of loss of accuracy problems. The number of terms in the problem can be kept 

 under control by using transfer matrices that move the stress and strain at one interface of 

 a layer to the other interface. This method was developed by Thomson (1950). To solve 

 the problem of sound transmission through plates, Bucker (Bucker et al, 1965) extended 

 the method to include wave attenuation for the problem of bottom reflection. A serious 

 drawback of the transfer matrix method is that it also suffers from loss of accuracy. 



Fortunately, the accuracy problems can be solved using methods developed by the 

 geophysicists for earthquake problems (Thrower, 1965;Dunkin, 1965;Watson, 1970; 

 Schwab, 1970). For a layered structure of the same form that we have for the bottom re- 

 flection problem, there are natural vibrations at frequencies corresponding to zeroes of a 

 determinant, |A|^|, called the Rayleigh determinant. The geophysicists have developed 

 very fast and accurate methods for calculating |Aj^|. We show that the reflection coeffi- 

 cient can be written as R = (pj r^Q |Aj^i -pQ |A5l)/(p| r^Q |Aj^| + pq [A^D, where |A§| is 

 the same as |Aj^| except for row 1. Thus, the sophisticated methods of the geophysicists 

 can be used to solve our problem. We do have to generalize the equations to account for 

 attenuation which is neglected at earthquake frequencies. 



CALCULATION OF R FOR MANY SOLID LAYERS USING 

 KNOPOFF'S METHOD 



The standard methods of solution are not usable for the solid multilayer model 

 because of accuracy, computer storage and computer run time problems. Fast and accu- 

 rate methods developed in earth wave problems can be modified for calculation of R, in 

 particular, the fast algorithm of Schwab (1970), which is based on Knopoffs formulation 

 (Knopoff, 1964). The method has been adapted to our solid multilayer model by Bucker 

 (Bucker and Morris, 1 975). The notation used here is that of Haskell (1953). 



The upgoing and downgoing compressional waves in a liquid layer or liquid half- 

 space can be represented by the potential function 



0,-, = — [i Aj^ cos Pj^ + Bj^ sin Pj^] exp [i(a;t - kx)] , 

 where 



Pn "^ '^ ^an ^n • 



Also, choose the potential function representing the upgoing and downgoing shear waves 

 in a solid layer or solid half-space to be 



53 



