1. Introduction to Seismology
Geology 5640/6640
Introduction to Seismology
3 Apr 2015
Last time: Reflection Seismic
Equations for a reflection two-way travel-time from a
layer-over-halfspace impedance contrast can be written:
• For deeper layers, we can approximate the two-way
travel-time as:
where: ;
• From these we get velocity & thickness from a layer with
bracketing reflections via Dix’ Equations:
Read for Mon 6 Apr: S&W (§3.4–3.5)
© A.R. Lowry 2015

2. Reflections Cont’d: Dipping Layer and Tau-P
A reflection from a dipping layer will have two-way travel-
time:
so updip/downdip limbs are symmetric about a line in x2-t2
• Using ray theory we can show that travel-time relates to
slowness as:
• By writing the multi-layer reflection equations as a dot-product
of a slowness vector with a path vector, we get the
tau-P relation:
where
t(x) = px + (p)

3. But p is a constant so
We can define a variable
so that t(x) = px + (p) .
This is the tau-P relation used in the intercept-slowness
method, and one will note that it looks like the equation for a
line (except that  is changing with p)… So this is a
description of a travel-time in terms of its slope and intercept
at any given distance x!

4. Let’s convert a layer-over-halfspace reflection to τ(p) form:
Squaring this gives:
This is the equation of an ellipse with axes along the  and p
coordinate axes:
The ellipse intersects the -axis at
and the p-axis at:

5. We can look at the relationship with x(p) by recognizing:
This means that at p = 1/v0 (where τ = 0),
This makes sense if we consider that the asymptote to
a reflection hyperbola (i.e. the slope it approaches as
x  ∞) is x = 1/v0.

6. Now consider the head wave:
Our travel-time equation is:
and the intercept for the wave arrival is
Thus,
the ray parameter is
and plugging p into the equation for (p),
which corresponds to the travel-time at the critical distance xc!

7. So the
travel-times
in tau-P
look like
ellipses for
reflections;
points for
refractions
(& direct
& surface
waves!)

8. What if there are multiple layers?
For multiple layers, the reflections are all ellipses, and head
waves occur at the intersections of ellipses.

9. Multi-Channel Geometry: Much of the success of reflection
imaging techniques stems from
experiment design: Reflections
have subtle expression in data
from a single source, so signal-
to-noise ratios in the data are
increased by stacking.
Data are collected with many,
many sources and receivers,
and then data can be categorized
by gathers for a common shot,
common receiver, common
midpoint, common offset…

11. Common Offset Gathers:
Common offset gathers are particularly handy for small-scale
projects in which the objective is to find the top of a near-
surface layer (especially if it has variable depth) by using the
expectation that travel-time follows: Vw V1 V2

12. Perhaps most important for seismic reflection imaging however
are Common Midpoint (CMP) Gathers, because these
can be used to suppress energy arriving in head waves, direct
waves, surface waves, and multiple reflections (e.g. below)
to emphasize desired primary reflections.

13. The change in reflection travel-time with offset is given by
the Normal MoveOut (NMO) equation:
For a reflection from a given layer, NMO correction stacks
CMP gathers using different assumed velocities and looks for
the stacking velocity that maximizes the stacked energy