2 Seismic Wave Propagation and Earth models

2. Seismic Wave Propagation and Earth models
Fig. 2.26 Ray through a multi-layered model with constant velocity within the layers but
increasing velocity with depth of the layers. The ray angle i increases accordingly with depth.
This can be generalized by modeling a velocity gradient with depth as a stack of many thin
layers with constant velocity. Rays and travel times for this case are sketched in Fig. 2.27.
The plot of arrival times t versus distance x is generally called the travel-time curve. The
tangent dti/dxi on the travel-time curve at any distance xi corresponds to the inverse of the
horizontal wave propagation velocity 1/vappi and thus to the ray parameter pi of that ray which
comes back to the surface at xi. Because of sin i = sin 90° = 1 at the turning point of the ray,
we can determine the velocity vtp at the turning point of the ray either from the gradient of the
travel-time curve at xi via pi = dti/dxi = 1/vtp or by knowing the sub-surface velocity voi at
station xi and measuring the incidence angle ioi at that station (vtp = voi /sin ioi).
Fig. 2.27 Raypaths (middle) and travel-time curve (right) for a model with velocity v
gradually increasing with depth z (left). The incidence angle i increases continuously until it
reaches 90°at the turning point tp, then the rays turn up again to reach the surface at xi. On the
travel-time curve each point comes from a different ray with a different slowness and ray
parameter p. The gradient of the tangent on the travel time curve at xi is the ray parameter pi =
dti/dxi. In the considered case of modest velocity increase with depth the distance x increases
with decreasing p. The related travel-time curve is termed prograde.
2.5.3.2 Effect of a sharp velocity increase
Next we consider the effect of a sharp velocity increase, which may be an increase in gradient
(second-order discontinuity) or an instantaneous velocity jump (first-order discontinuity). Fig.
28