Oct 23 The magic of Lamé

If geophysics requires mathematics for its treatment,
it is the earth that is responsible not the geophysicist.
Sir Harold Jeffreys

This quote was offered as a disclaimer on a course I took at the University of Calgary in 1988: Dr. Ed Krebes’ Geophysics 551 Seismic Techniques. This excellent course was pivotal in my enlightenment regarding Lamé’s parameters. I repeat the quote here as it disclaims my seemingly unnecessary obfuscation in the use of equations that follow.

The basic earth parameters in reflection seismology are P-wave velocity VP, and S-wave velocity VS. However, these extrinsic dynamic quantities are composed of the more intrinsic rock parameters of density and two moduli terms, lambda (λ) and mu (μ), introduced by the 18th-century French engineer, mathematician, and elastician Gabriel Lamé. Lamé also formulated the modern version of Hooke’s law relating stress to strain as shown here in its most general tensor form:

$$\sigma_{ij}=c_{ijkl}\epsilon_{kl}=(\lambda\delta_{ij}\delta_{kl}+\mu\delta_{ik}\delta_{jl}+\mu\delta_{il}\delta_{jk})\epsilon_{kl}$$

Here, σij is the i-th component of stress on the j-th face of an infinitesimally small elastic cube, cijkl is the fourth rank stiffness tensor describing the elasticity of material, εkl is the strain, and δij is the Kronecker delta. The adage ‘stress is proportional to strain’ was first stated by Hooke in a Latin anagram ceiiinosssttuv, whose solution he published in 1678 as Ut tensio, sic vis meaning ‘As the extension [strain], so the force [stress]. ’ Despite being interestingly reversed and non-physical, Hooke’s pronouncement is illustrated here with complete mathematical rigor, and this equation creates the basis for the science of materials, including rocks. Interestingly, and most notably, only Lamé’s moduli λ and μ, appear in this equation; not bulk modulus, Young’s modulus, Poisson’s ratio, VP , VS, or any other seismically derived attribute.

The methods to extract measurements of rocks and fluids from seismic amplitudes are based on the physics used to derive propagation velocity. This derivation starts with Hooke’s law and Newton’s second law of motion, and yields a set of partial differential equations that describe the progression of a seismic wave through a medium. It also forms the basis of AVO-based descriptions of the propagating medium.

The P-wave propagation of a volume dilatation term θ derived from Hooke’s law is:

$$\rho \frac{\partial^2\theta}{∂t^2} = (\lambda+2\mu)\nabla^2\theta$$

and the S-wave propagation of the shear displacement term ( x ush):

$$\rho \frac{\partial^2(\nabla\times u_\text{sh})}{\partial t^2}=\mu\nabla^2(\nabla\times u_\text{sh})$$

The vector calculus in these equations says that the particle or volume displacement for a travelling P-wave in the earth is parallel to the propagation direction (as ∇ x θ = 0), whereas the particle displacement imposed by a passing S-wave is orthogonal to the travelling wavefront. Consequently, the intuitively simple Lamé moduli of rigidity μ and incompressibility λ afford the most fundamental and orthogonal parameterization in our use of elastic seismic waves, thereby enabling the best basis from which to extract information about rocks within the earth. These Lamé moduli form the foundation for linking many fields of seismological investigation at different scales. Unfortunately the historical development of these fields has led to the use of a wide and confusing array of parameters, which are usually complicated functions of the Lamé moduli. None of these are inherent consequences of the wave equation, as the Lamé moduli are. This includes standard AVO attributes such as intercept and gradient or P-wave and S-wave reflectivity that are ambiguous and complex permutations of Lamé moduli λ and μ, or Lamé impedances λρ and μρ. Many other parameters such as Poisson’s ratio and Young’s modulus have arisen due to inappropriate attempts to merge the static un-bound domain of geomechanics to the dynamic bound medium of wave propagation in the earth. These attempts have resulted in the use of contradictory assumptions, which are completely removed when restating equations using the magic of Lamé moduli.