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The frequency-dependent eikonal equation
(4) cannot be solved separately
from the transport equation because the phase slowness is
dependent on the amplitudes.
An exact numerical solution of the combined equations is difficult to compute.
I therefore investigate
a method for computing an approximate solution.
My method extrapolates the phase-slowness function along
the frequency axis, starting from
infinite frequency where the phase slowness
function is equal to the medium slowness.
This scheme assumes that
the phase slowness is a continuous
function of frequency, and that it can be extrapolated
in frequency by solving a partial differential equation.
After the phase slowness function is computed,
the traveltimes and the amplitudes can
be computed at each frequency
by using conventional dynamic raytracing
Cervený (1987).
To extrapolate the phase slowness I derive
from equation (4)
an expression for its derivative with respect to frequency.
I start by rewriting equation (4) in terms
of the inverse of the frequency and the
square of the slowness ,

| |
(8) |

The derivative of with respect to is found
by applying the rule of implicit differentiation to the
function
| |
(9) |

Equation (9) is a non-linear partial differential
equation, that can be solved by explicitly stepping in ,starting from the initial value .For each value of , and corresponding value of ,the right hand side of the equation can be evaluated and
used for computing the value of the phase slowness for the
next . A variety of explicit integration schemes can be used
for integrating equation (9);
I used a standard second-order Runge-Kutta method.
Because equation (9) is non-linear, the analysis
of convergence and stability of its numerical solutions is
not trivial, and I have not performed it yet.
However, the numerical experiments that I have run, and that I will
describe in the next section, show that the solution
of equation (9) leads to satisfactory results.
The solution is also stable,
if the round-off errors for the high-wavenumbers components
of the Laplacian of the phase slowness are adequately constrained.

** Next:** Evaluation of derivatives of
** Up:** Biondi: Solving the frequency-dependent
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Stanford Exploration Project

11/18/1997