D.1.1. Stress invariants

Invariants of stress deviation (remember J ₁ = s ₁ + s ₂ + s ₃ = 0)

D.1.2. Strain in spherical coordinates

D.1.3. Isotropic linear elastic stiffness tensor

D.1.4. Isotropic linear poro-thermo-elastic stress strain law

Solved for stresses

Solved for strains

Various forms on compact notation

D.1.5. The force balance equation

The basic force balance equation is

Assuming homogeneous, isotropic poro-thermo-elasticity, and neglecting body forces, the equation may be written in terms of the displacements as

On coordinate independent form this is

or, alternatively

D.2.1. Correction for non-laminar flow

The function F( ?) introduced in Section 5.3.1 is defined as

where

ber and bei are so-called Kelvin functions, defined by

where J ₀ is the zeroth order Bessel function of the first kind.

D.2.2. Reflection, transmission and conversion coefficients at non-normal incidence

The coefficients for reflection, transmission and conversion for an incoming P-wave are given as solutions of the equation (Aki and Richards, 1980; see also Berkhout, 1987):

where

The incoming P-wave is entering through medium 1. The amplitude of the reflected P-wave is r _pp, the amplitude of the transmitted P-wave is t _pp, the amplitude of the reflected & converted wave is r _sp, and the amplitude of the transmitted & converted wave is t