Fundamentals - Seismic Amplitude - page 14

the amplitude increases sharply. This is associated
with the point at which the transmitted P wave amp-
litude is reduced to zero and refractions are generated
at the boundary. The angle at which these effects
occur is called the critical angle. This may be thought
of in terms of Snell

Water -wet sands and s hales

Gas sands

7

6

s Law that defines the relationship
between incident and transmission angles ( Fig. 2.16 ).
If the upper layer has lower velocity, then the critical
angle is given by:

'

5

4

:

sin 1 V p1
V p2

3

θ c ¼ ð 2 : 14 Þ

2

Beyond the critical angle, the reflected P wave is
phase-shifted relative to the incident signal. It is pos-
sible that critical angle energy from boundaries with
high velocity contrast can be misinterpreted as a
hydrocarbon effect so it is important that the inter-
preter and processor do not include these data in
gathers for AVO analysis. A seismic gather example
of critical angle energy (at a shale/limestone interface)
is shown in Fig. 2.17 .
Unfortunately, the Zoeppritz equations are com-
plicated and do not give an intuitive feel for how
rock properties impact the change of amplitude with
angle. For this reason several authors have derived
approximations to the equations for estimating amp-
litude as a function of angle for pre-critical angles.
A popular three-term approximation was developed
by Aki and Richards ( 1980 ) . Various authors have
re-formulated the approximation depending on the
purpose, but a useful starting point for the inter-
preter is the formulation generally accredited to
Wiggins et al.( 1983 ) :

1

0

0 0.1 0.2 0.3 0.4 0.5

Poisson's Ratio

Figure 2.14 Poisson's ratio and V p /V s .

for shales and sands with different fluid fill. Sands
tend to have a lower Poisson

s ratio than shales
because quartz has a lower V p /V s ratio than most
other minerals. Rocks containing compressible fluids
(oil and, especially, gas) have lower V p and slightly
higher V s than their water-wet equivalent. This means
that hydrocarbon sands will have a lower Poisson

'

' s

ratio than water-bearing sands.

2.3.3 Offset reflectivity
The isotropic and elastic behaviour of a P wave
incident on a boundary at any angle is described
by the Zoeppritz ( 1919 ) equations (see Aki and
Richards, 1980 ) ( Fig. 2.15 ). The Zoeppritz equations
describe the partitioning of P and S wave energy
into reflected and transmitted components. The vari-
ationofthePwavereflectioncoefficientwithangle
is the key parameter for most seismic interpretation,
though S wave reflection coefficients sometimes
need to be considered, for example when interpret-
ing marine seismic acquired with cables on the sea
floor.
An example of a single boundary calculation using
the Zoeppritz equations is shown in Fig. 2.15 . The
response shown is for shale overlying dolonite, not a
typical contrast of interest in hydrocarbon explor-
ation but it illustrates an important point. The
P wave amplitude of the

A+B sin 2 +C sin 2 tan 2

R ðθÞ¼ θ θ θ , ð 2 : 15 Þ

where

V p

1
2

Δ

V p + Δ ρ

A ¼

ρ

2
Δ

2
Δ ρ
ρ

¼ Δ

V p
2V p

V s
V P

V S
V S

V S
V P

B 4 2

V p
V p

1

2 Δ

¼

and C ,

V p1 +V p2
2

V s1 +V s2
2

ρ ¼ ρ 1 +

ρ 2

where V p ¼ , V s ¼ ,

2

2 + 2

¼ ð

V s1 = V p1 Þ ð V s2 = V p2 Þ

reflection initially
decreases with increasing angle, but at a certain point

' hard '

14

2

ð V s = V p Þ ,

2