# Predicting Geotechnical Parameters from Seismic Wave Velocity Using Artificial Neural Networks
## 2 Seismic Wave Velocity  
Soil allows for the propagation of different types of seismic waves. Waves that deform the material through shear are referred to as shear waves, and those that produce volumetric deformations are referred to as compressional waves. These are often referred to as S-wave and P-wave, respectively. Seismic wave velocity is related to the maximum shear modulus, bulk modulus, Young's modulus, bulk density, and Poisson's ratio of the soil [50]. The longitudinal P-wave and the transverse S-wave velocity in an infinite elastic continuum are related to the elastic properties by  
\[\mathrm{V_{p}}=\sqrt{\frac{M}{\rho}}=\sqrt{\frac{B\ +\frac{4}{3}G}{\rho}}\quad \mathrm{P-waves} \tag{1}\]  
\[\mathrm{V_{s}}=\sqrt{\frac{G}{\rho}}\quad\mathrm{S-waves} \tag{2}\]  
where M (Pa) is the constraint modulus, B (Pa) is the bulk modulus, G (Pa) is the shear modulus, and \(\rho\) (kg/m\({}^{3}\)) is the mass density of the medium. Hence, the propagation velocity increases with the material stiffness and decreases with its mass density (inertia). Velocity of S-waves is always smaller than the velocity of P-waves [50]. For fluid-filled porous media, the effective bulk modulus is provided by Gassmann [50]. \[\mathrm{B_{eff}}=\mathrm{B_{SK}}+\frac{\left(1-\frac{\mathrm{B_{SK}}}{\mathrm{B_{ f}}}\right)^{2}}{\frac{\mathrm{\Theta}}{\mathrm{B_{f}}}+\frac{1-\mathrm{\Theta}}{ \mathrm{B_{g}}}-\frac{\mathrm{B_{SK}}}{\mathrm{B_{g}}}} \tag{3}\]  
where \(\mathrm{B_{SK}}\) (Pa) is the bulk modulus of the skeleton, \(\mathrm{B_{g}}\) (Pa) is the bulk modulus of the grains, \(\mathrm{B_{f}}\) (Pa) is the bulk modulus of the fluid phase, and \(\mathrm{\Theta}\) is the porosity. In the Gassmann model, the shear modulus of the soil, \(\mathrm{G_{eff}}\), (Pa) remains unaffected by the presence of the fluid at low excitation frequencies:  
\[\mathrm{G_{eff}}=\mathrm{G_{SK}}. \tag{4}\]  
For partially saturated soils, the mass density of the mixture \(\mathrm{\rho_{mix}}\), changes due to the different densities of the saturating fluids. By ignoring granular effects, fluid substitution can be used to modify the expression for the effective bulk modulus of the soil. For soil with water saturation of \(\mathrm{S_{w}}\), the fluid bulk modulus in Equation (5) given by  
\[\frac{1}{\mathrm{B_{f}}}=\frac{\mathrm{S_{w}}}{\mathrm{B_{w}}}+\frac{1-\mathrm{ S_{w}}}{\mathrm{B_{a}}} \tag{5}\]  
where \(\mathrm{B_{w}}\) (Pa) is the bulk modulus of the liquid phase and \(\mathrm{B_{a}}\) (Pa) is the bulk modulus of the air phase. Small volumes of air produce a large decrease in the modulus of the fluid phase. However, under dynamic loading, differences in inertia, shear stiffness, and bulk compressibility can add further complexity to the analysis. Seismic wave propagation in granular soil materials is more complicated due to the complex behavior of the solid skeleton and the influence of capillary forces. The skeletons \(\mathrm{B_{SK}}\) and \(\mathrm{G_{SK}}\) depend on the \"strength\" of the grain contacts and are therefore dependent upon the applied effective stress. The concept of effective stress for soils at low saturation is still an area of active research because internal forces associated with capillary forces and electrical forces at the grain surface play an important role [50]. That is why empirical relationships are often necessary to predict seismic wave propagation in the partially saturated particulate medium. There are numerous methods for measuring seismic wave velocity in the field and the laboratory. In the laboratory, the \"time of flight\" approach is common. A seismic wave is generated using a source in contact with one end of the sample, the disturbance passes through the soil and is detected by a receiver at the opposite end of the sample. Velocity is calculated by dividing the distance (sample length) by the measured travel time. The soil samples usually consist of remolded soil or a field sample with some degree of disturbance. The frequency of the seismic wave is usually in the 10-100 kHz range.