# Wave velocity

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 3 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Seismic velocity is the rate at which a seismic wave travels through a medium. Different methods are available for deriving velocity from the seismic data (Robinson, 1983[1]). In addition to using laboratory measurements, we also can determine velocity from acoustic logs (Lindseth, 1979[2]), vertical seismic profiles (VSPs), or velocity analysis of seismic data. Each method can give a different value of the velocity. For example, stacking velocity, which is derived from normal-moveout (NMO) data, is different from average velocity measured from a check shot.

In addition to their velocity, waves are characterized by their particle motion, amplitude, frequency, and wavelength. Velocity depends on the elastic properties and density of the medium. Elastic properties are determined by the medium’s incompressibility (or bulk modulus) and rigidity. Compressional velocity depends on both incompressibility and rigidity. Shear velocity depends only on rigidity. For all materials, the shear velocity is always less than the compressional velocity.

What is Poisson’s ratio? Poisson’s ratio ${\displaystyle \left(\sigma \right)}$ is the ratio of the lateral contraction of an elastic body to the longitudinal extension of that elastic body when it is stretched. Poisson’s ratio is an elastic constant that is useful in relating the velocities of the two types of waves: compressional waves and shear waves.

The expression

 {\displaystyle {\begin{aligned}&V_{\rm {S}}^{\rm {2}}{\rm {=}}V_{\rm {P}}^{\rm {2}}{\frac {{\rm {l}}-{\rm {2}}\sigma }{{\rm {2}}\left({\rm {l}}-\sigma \right)}}\end{aligned}}} (1)

is a relationship between the shear velocity, ${\displaystyle V_{\rm {S}}}$, and the compressional velocity, ${\displaystyle V_{\rm {P}}}$, in terms of Poisson’s ratio. In an ideal elastic solid, Poisson’s ratio equals 0.25, and ${\displaystyle V_{\rm {S}}{\rm {=}}V_{\rm {P}}{\rm {/}}{\sqrt {\rm {3}}}}$. Steel is a good example of an ideal solid. Poisson ratios for rocks vary in the range of ~0.2 to ~0.4. The upper bound on Poisson’s ratio is 0.5. For that value, ${\displaystyle V_{\rm {S}}}$ is zero, and there are no shear waves. That is the condition that holds for fluids and gases.

Several features make rocks anisotropic. Anisotropy causes the average elastic constants to depend on direction. A common type of anisotropy is the result of fine layering. For example, the Austin Chalk has alternating hard and soft layers, each of which is only a few centimeters thick. For low-frequency seismic waves, the Austin Chalk can be described as a homogeneous but anisotropic medium. Its average properties depend on the contrast between layers. Another source of anisotropy is block fracturing. Such fracturing results from past episodes of tectonic movement. A third source of anisotropy is caused by unequal stress fields existing in the rock.

In an anisotropic medium, the velocities of compressional and shear waves depend on the direction of propagation. In an anisotropic layered medium, compressional waves generally have a velocity higher in the direction parallel to the layering than in the direction perpendicular to the layering. Two types of shear waves propagate in the direction of the layering. One type is the SV wave, which has particle motion perpendicular to the layering. The other type is the SH wave, which has particle motion parallel to the layering.