The most common parameters measurable in seismology are VP and VS, being the propagation velocities of compressional P-waves and shear S-waves in elastic media. However, these measured quantities are composed of the more fundamental rock parameters of density and two moduli termed lambda and mu, introduced and named after the 18th century French engineer, mathematician, and elastician Gabriel Lamé. Lamé also formulated the modern version of Hooke's law relating stress to strain in its general tensor form, thereby creating the basis for the science of materials, including rocks. Interestingly and most notably, only Lamé's moduli lambda and mu appear in Hooke's law and not Young's modulus, the bulk modulus, or any other common modulus.

 

The application of seismology to measure or describe rocks and fluids is based on the physics used to derive propagation velocity that originates with the elastic wave equations. These wave equations equate Hooke's law, providing the Lamé moduli, to Newton's second law that provides density and their solutions form the basis for AVO used to describe attributes of the propagating medium. The result gives relations between propagation velocity VP and VS and the intuitively simple Lamé moduli of incompressibility, lambda, and rigidity, mu. Consequently lambda and mu afford the most fundamental and orthogonal parameterization of elastic seismic waves to extract information about rocks within the Earth.

 

The historical development of seismology at widely different scales has led to the use of a large and confusing array of parameters, which are usually complicated functions of the Lamé moduli. This includes Poisson's ratio, Young's and the bulk modulus, as well as standard AVO attributes such as intercept and gradient, that arise as a result of the media's form and its measurement environment. For example, the same rock will deform volumetrically as a function of the bulk modulus, or longitudinally as a function of Young's modulus, or as a function of the nameless P-wave propagation modulus (lambda + 2mu) in the Earth. Extracting the Lamé moduli from these mixed parameters provides insight into their physical meaning, because Lamé modulntrinsic and invariant properties of elastic media. Examples of this from the fields of earthquake seismology, AVO, and passive microseismology will be presented.