A derivation is given for the factorization of the generally anisotropic,
elastic wave equation into two operators which are 1st-order wrt
to a preferred direction of propagation.
One factor gives the "one-way" equation for forward-going body
waves. A 3x3 matrix notation is used which
emphasizes the role of the Christoffel equation (or what might be called its
"preferred-coordinate projection") familiar from ray theory.
To recognize this it helps to obtain first an *exact* factorization of the
wave equation for homogeneous anisotropic media. When inhomogeneities
exist it is necessary to use a Fourier representation for
differential operators
wrt the transverse coordinates, but still the Christoffel equation
is the key. The so-called "square root operator" required in the
factorization becomes more correctly the "root of a matrix operator
quadratic equation" intimately connected to the algebraic
Christoffel equation.

The factorization does not assume narrow-angle propagation or involve an
*a priori* reference phase. It includes the P and two S waves and
the forward coupling
between them. It remains valid at slowness-surface singularities
like conical points (a.k.a acoustic axes -- a most
stringent test) and for wavefronts which fold.
In the frequency-domain this wide-angle one-way equation involves
Fourier synthesis wrt transverse position/slowness and its time-domain
analogue involves 2D slant stacking (Radon transformation),
multiplication by a 3x3 "propagator" matrix and an inverse 2D slant stack.

The one-way equation can be solved "analytically" by using "Path Integrals". These in turn may be reduced by stationary-phase arguments to standard ray theory, Maslov and Kirchhoff representations. In this sense, the "gap" between ray theory and the numerical solution of the full wave equation is bridged or filled out, at least for forward propagation.

Alternatively, the one-way wave equation can be solved by numerical "forward-stepping" algorithms and here again a number of choices are available. Narrow-angle Taylor or rational approximations to the matrix propagator in the Fourier/slant-stack represention lead to the anisotropic analogues of the classic 15° and 45° partial differential equations. The integral wide-angle form is also reminiscent of the "split-step" and "elastic phase screen" approaches of Tappert, Wu and others, suggesting other implementations.

Body waves, anisotropy, one-way wave equations, ray theory, finite-difference methods.

Thomson, C.J.:
The "gap" between seismic ray theory and "full" wavefield extrapolation.
*Geophys. J. int., 137 (1999), 364-380.
*

In: Seismic Waves in Complex 3-D Structures, Report 7, pp. 319-347, Dep. Geophys., Charles Univ., Prague, 1998.

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