Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Abstract

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional $\mathcal{I}$(l) stationary. Fermat's functional $\mathcal{I}$(l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral $\mathcal{I}$(l) = $\smallint _S^R \mathcal{L}$(x ^k,x ^k ')du, taken along curve l, where ℒ(x ^k,x ^k ') is the relevant Lagrangian, x ^k are coordinates, u is a parameter used to specify the position of points along l, and x ^k ' = dx ^k÷du. If Lagrangian ℒ(x ^k,x ^k ') is a homogeneous function of the first degree in x ^k ', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ℒ⁽¹⁾(x ^k,x ^k '). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ℒ⁽¹⁾(x ^k,x ^k ') to derive the relevant Hamiltonian ℋ⁽¹⁾(x ^k,p _k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ℒ⁽¹⁾(x ^k,x ^k '). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ℒ⁽²⁾. Parameter u along the curves is taken to correspond to travel time τ, and the second-degree Lagrangian ℒ⁽²⁾(x ^k,$\dot x$ ^k) is then introduced by the relation ℒ⁽²⁾(x ^k,$\dot x$ ^k) = [ℒ⁽¹⁾(x ^k,$\dot x$ ^k)]², with $\dot x$ ^k = dx ^k÷dτ. The second-degree Lagrangian ℒ⁽²⁾(x ^k,$\dot x$ ^k) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ℒ⁽¹⁾(x ^k,$\dot x$ ^k). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ℋ⁽²⁾(x ^k,p _k) from Lagrangian ℒ⁽²⁾(x ^k,$\dot x$ ^k), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ℒ⁽²⁾(x ^k,$\dot x$ ^k) and ℋ⁽²⁾(x ^k,p _k) can be expressed in terms of the wave propagation metric tensor g _ij(x ^k,$\dot x$ ^k), which depends not only on position x ^k, but also on the direction of vector $\dot x$ ^k. It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ℒ⁽²⁾(x ^k,$\dot x$ ^k), not to the first-degree Lagrangian ℒ⁽¹⁾(x ^k,$\dot x$ ^k). It is also shown that relations ℒ⁽²⁾(x ^k,$\dot x$ ^k) = $- \frac{1}{2}$; and ℋ⁽²⁾(x ^k,p _k) = $- \frac{1}{2}$ are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x ⁱ. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.