Fundamentals - Seismic Amplitude - page 15

Critical angle Critical angle

1

1

0.5 0.5

Reflected S

0

0

30 60 90

30 60 0

0 90

Angle of incidence (degrees) Angle of incidence (degrees)

Critical angle

Critical angle

1

1

Transmitted P

Transmitted S

0.5 0.5

0

0

30 60

0 90 30 60 90

0

Angle of incidence (degrees)

Angle of incidence (degrees)

Figure 2.15 Partitioning of P wave energy at a shale/limestone interface as a function of incidence angle. The model uses the Zoeppritz

equations for the calculation and the various components sum to one at all angles. Note that the critical angle defines the point at which there
is a dramatic rise in reflected P wave energy and a corresponding reduction of transmitted energy to zero. Appreciable S wave energy is
generated beyond the critical angle. Not shown in the diagram is energy refracted along the boundary at the critical angle. Elastic values in the
model: shale V p ¼ 2540 m/s, V s ¼ 1150 m/s, density ¼ 2.35 g/cc; limestone V p ¼ 3750 m/s, V s ¼ 1950 m/s, density ¼ 2.4 g/cc. The model

was calculated using the Crewes Energy Ratio Explorer software available at www.Crewes.org .

V p ¼ V p2 V p1 , V s ¼ V s2 V s1 , example, the A term is commonly referred to as the

intercept or by the symbols R 0 or NI (for

Δ Δ

' normal

Δ ρ ¼ ρ 2 ρ 1 :
The first term (A) is the zero angle reflection coeffi-
cient related to the contrast of acoustic impedance,
whilst the second term (B) introduces the effect of
shear velocity at non-zero angles. A third term (C)
determines the curvature of the amplitude response
near to the critical angle ( Fig. 2.18a ). The interpreter
should be aware that there are a variety of symbols for
the first and second terms used in the literature. For

and

incidence ), whilst the B term is referred to as the

gradient (G)orslope. In the past, Shell and Hess
corporations have used the symbols L and M and R 0
and R 1 respectively to refer to intercept and gradient
( Fig. 2.18 ).
The third term can be dropped to give a two term
approximation generally accredited to Shuey ( 1985 ,
although the equation does not actually appear in the
paper):

'

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