principles and applications of microearthquake networks

% 4. Seismic Ray Tracing for Minimum Time Path
4.4.4. One-Dimensional Multilayer Velocity Models
The most commonly used velocity model in microearthquake studies
consists of a sequence of horizontal layers of constant velocity, each layer
having a higher velocity than the layer above it. For the source and the
station at the same elevation, the ray paths and travel times are well
known (e.g., Officer, 1958, pp. 239-242). For an earthquake source at
depth, the specific formulas involved have been given by Eaton (1969, pp.
2638). We now derive an equivalent set of formulas by a systematic
approach using results from Section 4.4.1.
Let us first consider the simple case of a single layer of thickness h, and
velocity v1 over a half-space of velocity vz (Fig. 20). For a surface source
at point A, we consider two possible ray paths to a surface station at point
B. The travel time of the direct wave, Td, along p atha is
(4.88) Td = A/u1
where A is the epicentral distance between the source and the station, i.e.,
A = AT. The travel time of the refracted wave, T,, along pathACDB is
(4.89) T, = (AT/vl) + (m/u,) + (m/v,)
Using Snell's law, we have
(4.90) sin 8/v, = sin 90°/u2 = sin 4/v1
so that 8 = qb (Fig. 20), and AT = m. Consequently, Eq. (4.89) can be
written in terms of 8, hl, and A as (see Officer, 1958, p. 240),
(4.91)
A 2h,
T, = - + - cos 8
v2 1' 1
Using Eqs. (4.88) and (4.91), we can plot travel time versus epicentral
A
I
A(source)
-
B(station)
h, Velocity=V,
-
r
A' C D B'
Fig. 20.
half-space.
Velocity = V2
Diagram of direct and refracted paths in a velocity model of one layer over a