1
Chapter 2
Time-Distance Curves of Primary Reflections
Introduction

The objective of this chapter is to be able to develop time-distance (T-X) relations of
primary reflections in various elastic, isotropic media.

A primary reflection is a P-wave that has been reflected only once from a specific
subsurface interface.

Primary reflections are the only signals in petroleum seismic records, which we have
to preserve and enhance. Therefore, they are the most important parts of our seismic
data.

Everything else on the petroleum seismic record is noise, which we will attempt to
attenuate. We will study the various types of seismic noise in Chapter 4.

In this chapter, we will also introduce and discuss the short-offset assumption.

These T-X relations are important in inverting layer velocities and thicknesses since
these curves are usually the only outputs from surface seismic exploration surveys.

We will study the inversion procedure in more detail in Chapter 3.
2
Reflection in homogeneous media
Single horizontal layer

The case of a single horizontal layer with constant velocity is shown in the following
figure.
X
S
C
R
V
H
M

In the above figure: S: the source, R: the receiver, M: the reflection point,
V: layer velocity, H: layer thickness, X: offset (source-receiver distance), and
T: the two-way traveltime from the source (S) down to the reflection point (M) and
back up to the receiver (R)
Derivation of the T-X curve
 T = (SM + MR)/V = 2 SM/V,
(SM = MR why?)
 But: SM = (X2/4 + H2)½,
(1)
(2)
 Substituting eq. (2) in (1), squaring both sides, and simplifying gives:
 T2 = 4H2/V2 + X2/V2, which can be written as:
T 2 ( X )  T02 
X2
,
V2
where T0 = 2H/V
 T0: the zero-offset two-way traveltime (at X = 0).
(3)
(4)
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
Equation (3) is called the time-distance equation, time-offset equation, the T-X curve,
or the traveltime-distance equation.

Equation (3) represents a hyperbola on the seismic section, with apex (i.e., the
minimum two-way traveltime) at T(X) = T0 and X = 0.

Practically, we know X from the field layout, and we can read T0 and T(X) from the
seismic section. The unknowns are the layer thickness (H) and velocity (V).

To find H and V, we first use equation (3) to find V, and then substitute its value in
equation (4) to find H.

There are several methods to find V from equation (3). The simplest method is to
plot T2(X) versus X2, which will be a straight line with slope = 1/V2 and intercept =
T02.

More information on hyperbolas.
Derivation of the NMO correction
 The normal moveout (TNMO(X)) is defined as the time difference (on the
seismic section) between the two-way traveltime (T(X)) at an offset (X  0)
and the two-way traveltime (T0) at zero offset (X = 0):
 TNMO(X) = T(X) - T0.
 TNMO(X) provides another way to estimate V. However, the above form of
TNMO(X) is not practical because it requires reading (picking) of T(X) which
is slow and inaccurate.
 To derive a more practical formula for TNMO(X), we make the short-offset
assumption.
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 The short-offset assumption means that we assume that the offset used is
smaller than the layer thickness (X < H).
 We then proceed as follows:
2
2
(a) From eq. (3), T 2 ( X )  T02 (1  X2 2 ) , which yields: T ( X )  T0 1  X .
2 2
T0 V
T0 V
(b) Expanding the square root in Maclaurin’s series gives:
T  T0 (1 
X 2 12
X2
X4
X6
)

T
(
1



 ...).
0
T02V 2
2T02V 2 8T04V 4 16T06V 6
Exercise: Calculate the contribution of terms in this series when X=H.
(c) Retaining two terms only, we get: T ( X )  T0 (1 
(d) TNMO ( X )  T ( X )  T0  T0 (1 
X2
).
2T02V 2
X2
X2
)

T

T
(
1

 1) giving:
0
0
2T02V 2
2T02V 2
X2
TNMO ( X ) 
.
2T0V 2

(5)
Equation (5) is sometimes called the normal moveout equation and TNMO is the
NMO correction.

Equation (5) can be used directly to determine the layer velocity and thickness if all
X2
X
other variables are known: V 
and H 
.
4VTNMO ( X )
2T0 TNMO ( X )

We will see in Chapter 3 how equation (5) is used more practically to find V.
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Single dipping layer (DMO)

The case of a single dipping layer with constant velocity is shown in the following
figure.
X
Earth
surface
S
R
)
V
H
C

M
H
Dipping
interface
S’

In the above figure: S: the source, R: the receiver, M: the reflection point,
V: layer velocity, X: offset (source-receiver distance),
H: perpendicular distance from S to the interface, : layer (interface) true dip,
S’: mirror image of S about the interface (i.e., SS’  interface and SC = S’C resulting
in SM = S’M why?), and T: the two-way traveltime from the source (S) to the
reflection point (M) and back to the receiver (R)
Derivation of the T-X curve
 T = (SM + MR)/V = (S’M + MR)/V = S’R/V,
(SM = S’M)
 S’R can be found by applying the law of cosines in the triangle SS’R:
(6)
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 S’R2 = X2 + (2H)2 – 2(X)(2H)Cos(,
(7)
 S’R2 = X2 + 4H2 + 4X H Sin.
(8)
 Squaring eq. (6) and substituting eq. (8) for S’R, we get:
 T2 = 4H2/V2 + X2/V2 + 4X H Sin/V2, which can be written as:
 T2 = T02 + X2/V2 + 4X H Sin/V2,
(9a)
where T0 = 2H/V is the zero-offset two-way traveltime which occurs when the
source and receiver coincide at S (X = 0).

Equation (9a) can be written in the following convenient form (show?):
 X  2 HSin  
T 2  T02 Cos 2  
 .
V


2

(9b)
Equation (9b) is an equation of a hyperbola whose apex occurs at Ta = T0 Cos and
Xa = -2H Sin (the minus sign means that it lies to the left of the source).

The dip moveout (TDMO) is defined as the difference in traveltimes for two receivers
equally distant from, but on opposite sides of, the source (X); divided by the
receiver offset (X):
TDMO 

T ( X )  T ( X )
.
X
(10)
Following an approach similar to the NMO derivation, it can be shown that:
TDMO  2Sin/V,
(11)
where the approximation becomes better at short offsets (X/H <1 ).

The reflector dip (), velocity (V), and thickness (H) can be determined from Ta, T0,
and TDMO using the following relations in sequence:
 = Cos-1(Ta/T0),
(12a)
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
V  2 Sin/ TDMO,
(12b)
H = V T0 / 2.
(12c)
Equivalently, , V, and H can be determined from Ta, T0, and Xa using the following
relations in sequence:

 = Cos-1(Ta/T0),
(13a)
H = |Xa / (-2Sin,
(13b)
V = 2H / T0.
(13c)
Here are figures that show the errors introduced in estimating the velocity if the
short-offset assumption is neglected.
Reflection in a vertically heterogeneous medium

If the medium is composed of multiple layers of different velocities, then we can
trace the path of a seismic ray by using Snell’s law at each interface (Figure).

In this case, the traveltime equation of reflections from interfaces (except the first)
will not be a hyperbola and a plot of T2 versus X2 will not be a straight line but rather
a curve.

The true traveltime equations of a medium composed of several horizontal layers are:
N
X N  2
i 1
pVi H i
1  ( pVi ) 2
N
and TN  2
i 1
Hi
Vi 1  ( pVi ) 2
, where XN is the offset of the
reflection from the Nth layer, TN is the two-way traveltime to the Nth layer recorded at
XN, and p is the ray parameter of the ray (XN,TN), which is constant along the raypath.
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
To simplify the problem, we substitute all the layers above the interface by a single
layer. This approximation is good only at small offsets, where XN/ZN < 1 (XN:
maximum offset used for Nth reflector, ZN: depth to Nth reflector).

By substituting the multiple layers with a single layer, we imply that the true T-X
curve is a hyperbola at short offsets.

The single layer, substituting multiple layers above the interface, will have an average
velocity that is called the RMS, stacking, or NMO velocity, depending on the
substitution method. We will study these velocities in Chapter 3.