principles and applications of microearthquake networks

4.4. Computing Travel Time and Derivatives 95
Substituting this result into Eq. (4.82), we can solve for xc. Hence, the
center of the circular ray path, point C, is given by
(4.83)
From Eqs. (4.78)-(4.79), the travel time T from point A to point B is
given by
(4.84)
where
According to Eqs. (4.66) and (4.751, the spatial derivatives of T at the
source are
(4.86)
dT/dx]., = -sin 8,/(go + gz,)
dT/dzlA = -COS OA/(g, + ~ z A )
The take-off angle at the source with respect to the downward vertical + is
just the angle flA, i.e.,
(4.87) $ = OA
For treating the more general case in which the velocity is an arbitrary
function of depth, there have been some recent advances. An important
concept is 7(p), the intercept or delay time function introduced by Gerver
and Markushevich (1966, 1967). It is defined by 7(P) = T(p) - pX(p),
where T is the travel time, X is the epicentral distance, and p is the ray
parameter defined by Eq. (4.76). The expression for ~ (p) is analogous to
the more familiar equation Ti = T - X /u for a refracted wave along a layer
of velocity u, and TI is the intercept time. The delay time function ~ (p) has
several convenient properties. Whereas travel time T may be a multiplevalued
function of X due to triplications, the corresponding ~ (p) function
is always single valued and monotonically decreasing. Calculations based
on the ~ (p) function have the advantages that low-velocity regions can be
treated in a straightforward way, and that considerable geometric and
physical interpretation can be applied to the mathematical formalism.
This approach has been useful in calculating travel times for synthetic
seismograms (e.g., Dey-Sarkar and Chapman, 1978), and for inversion of
travel-time data for the velocity function v(z) (e.g., Garmanyet al., 1979).
For further details, readers are referred to the above-rnentioned works
and references therein.