SOLID MULTILAYER MODEL 



Sediments have rigidity and for more accurate model calculations rigidity must be 

 taken into account. An isotropic sediment layer can be described by three sediment 

 parameters: the density p and the two Lame constants, X and /i (Ewing, Jardetsky, and 

 Press, 1957). The density can be measured directly but X and n are determined by the 

 speed and attenuation of the compressional and shear waves that travel in the sediment. 

 The sediment and acoustic parameters are related (Bucker et al, 1965) as follows 



Sediment Parameters 



P , '>^ , 1^ 



Lame constants 

 Acoustic Parameters 



c„ - sound speed (compressional wave) 



c„ = sound speed (shear wave) 



ap = attenuation, dB/unit length (compressional) 



a^ = attenuation, dB/unit length (shear) 

 Constitutive Equations 



X + 2M=p(x2-y2-.2Xpyp)/(x2+y2)^ 



)/K 



where, Xp = 1/Cp , yp = ap/(8.686co) 

 Xg = 1/Cg , y5 = as/(8.686co) 



There are several approaches to the problem of modeling the sediment layers when 

 there are significant changes of the sediment properties with depth. Gupta (1966a, 1966b) 

 has developed closed solutions for the case where the compressional and shear velocity 

 varies Hnearly with depth while the density remains constant. More general variations can 

 be treated with the propagator method developed by Gilbert and Backus (1966). One 

 problem of the propagator method is loss of accuracy when sediment penetration of many 

 wavelengths occurs. In our most recent programs we have chosen to model the variable 

 sediment properties with many layers and to maintain accuracy by use of Knopoffs 

 formulation (Knopoff, 1964). 



The multilayer solid model is substantially more difficult than the multilayer Hquid 

 model for two reasons. First, there are twice as many waves (shear waves as well as com- 

 pressional waves) and twice as many interface conditions (continuity of horizontal com- 

 ponents of stress and strain, as well as continuity of vertical components of stress and 

 strain). Second, you cannot start at the bottom and work to the top. All of the layers 

 have to be considered as a group. The situation is shown in Figure 23. There are an 

 upgoing and a downgoing compressional wave in the water, an upgoing and a downgoing 



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