Reflection Coefficients in Wave Propagation
Reflection Coefficients
For a downward travelling P wave, for the most general case:
Where the first term on the RHS is the P-wave
displacement component and the second term is the
shear-wave displacement component
Reflection Coefficients
and where both shear stress,
and as well as normal stress is continuous across the boundary:
Reflection Coefficients
When all these conditions are met and for the
special case of normal incident conditions, we
have that Zoeppritz’s equations are:
On occasions these equations will not add up to
what you might expect…!
Reflection Coefficients
Reflection Coefficients
Reflection Coefficients
Reflection Coefficients
Reflection Coefficients
Reflection Coefficients
What happens when we have a complete reflection with a 180
degree phase shift, as we might have when a ray in water travels
upward toward a free surface and reflects completely at the
interface?
Reflection Coefficients
What happens when we have a complete reflection with a 180
degree phase shift, as we might have when a ray in water travels
upward toward a free surface and reflects completely at the
interface?
We know that in this case:
Reflection Coefficients
What happens when we have a complete reflection with a 180
degree phase shift, as we might have when a ray in water travels
upward toward a free surface and reflects completely at the
interface?
We know that in this case:
But,
What must:
Reflection Coefficients
What happens when we have a complete reflection with a 180
degree phase shift, as we might have when a ray in water travels
upward toward a free surface and reflects completely at the
interface?
We know that in this case:
But,
So,
Reflection Coefficients
Z+
Z-
X+
In layer 1, just above the boundary, at the
point of incidence:
layer 1
layer 2
Briefly, how to consider displacements at interfaces
using potentials, when mode conversion occurs:
Reflection Coefficients
Z+
Z-
X+
layer 1
layer 2
Briefly, how to consider displacements at interfaces
using potentials, when mode conversion occurs:
In layer 2, just below the boundary, at the
point of incidence:
Reflection Coefficients
So, if we consider that (1) stresses as well as (2)
displacements are the same at the point of incidence
whether we are in the top or bottom layer the
following must hold true so that (3) Snell’s Law holds
true:
Reflection Coefficients
We get the general case of all the different types of
reflection and transmission (refraction or not)
coefficients at all angles of incidence :
Variation of Amplitude with angle (“AVA”) for
the
fluid-over-fluid
case (NO
SHEAR
WAVES)
(Liner, 2004; Eq.
3.29, p.68; ~Ikelle
and Amundsen,
2005, p. 94)
Reflection Coefficients
What occurs at and beyond the critical angle?
Reflection Coefficients
FLUID-FLUID case
What occurs at the critical angle?
(Liner, 2004; Eq. 3.29,
p.68; ~Ikelle and
Amundsen, 2005, p.94)
Reflection Coefficients
Reflection
Coefficients at
all angles: pre-
and post-critical
Matlab Code
Reflection Coefficients
Case:
Rho: 2.2 /1.8
V: 1800/2500
NOTES: #1
At the critical angle, the
real portion of the RC goes
to 1. But, beyond it drops.
This does not mean that the
energy is dropping.
Remember that the RC is
complex and has two terms.
For an estimation of energy
you would need to look at
the square of the
amplitude. To calculate the
amplitude we include
both
the imaginary and real
portions of the RC.
Reflection Coefficients
NOTES: #2
For the
critical ray,
amplitude is maximum
(=1) at critical angle.
Post-critical angles
also have a maximum
amplitude because all
the energy is coming
back as a reflected
wave and no energy is
getting into the lower
layer
Reflection Coefficients
NOTES: #3
Post-critical angle
rays will experience a
phase shift, that is
the shape of the
signal will change.
Reflection Coefficients
Energy Coefficients
We saw that for reflection coefficients :
For the energy coefficients at
normal incidence :
Energy Coefficients
We saw that for reflection coefficients :
For the energy coefficients at
normal incidence :
The sum of the energy is expected to be conserved across the boundary
Amplitude versus Offset (AVO)
Zoeppritz’s equations can be simplified if we assume that the
following ratios are much smaller than 1:
For example, the change in velocities across a boundary is very
small when compared to the average velocities across the
boundary; in other words when velocity variations occur in small
increments across boundaries… This is the ASSUMPTION
Amplitude versus Offset (AVO)
If the changes across boundaries are relatively small, then we can make a
lot of approximations to simplify the reflection and transmission
coefficients:
( Equation, 3.242, Ikelle and Amundsen)
AVO cases for P-P
In addition, we can rearrange and further simplify this equation as
follows:
( Equation, 3.255, Ikelle and Amundsen)
AVO cases for P-P
With the much simplified calculation for reflection coefficients, we can
characterize reservoirs according to the Rpp at normal incidence (
intercept) and contrast in Poisson’s ratio across their boundaries
(~gradient or slope):
slope
intercept
CLASS 1
(Z1<Z2)
CLASS 1
(Z1<Z2)
CLASS 1
(Z1<Z2)
AVO cases for P-P-CLASS 2
Class 2 shows very little contrast between layer properties
slope
intercept
CLASS 2
Z1~Z2
AVO cases for P-P-CLASS 3
Class 3 shows the characteristic bright spot that is often interpreted to
mean shale over gas sand (Z1<Z2). The reflection coefficient becomes
more negative with offset
slope
intercept
CLASS 3
Z1<Z2
AVO cases for P-P-CLASS 4
Class 4 is like class 3 in that there is a strong negative reflection BUT the
Reflection coefficient decreases with angle (GOM)
slope
intercept
CLASS 4
Z1<Z2
CLASS 4
Z1<Z2
Reflection coefficients play a crucial role in wave propagation, particularly for P and shear waves. The equations governing reflection coefficients provide insight into wave behavior at boundaries and interfaces. By examining the conditions and special cases, we can understand how these coefficients impact wave interactions, such as complete reflections with phase shifts.
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Presentation Transcript
Reflection Coefficients For a downward travelling P wave, for the most general case: = ux x z = + uz z x Where the first term on the RHS is the P-wave displacement component and the second term is the shear-wave displacement component
Reflection Coefficients and where both shear stress, 2 2 2 1 2 = + (2 ) xz 2 2 x z x z and as well as normal stress is continuous across the boundary: 2 2 2 = + 2 (2 + ) zz 2 x z z
Reflection Coefficients When all these conditions are met and for the special case of normal incident conditions, we have that Zoeppritz s equations are: + V V + 2 2 1 1 I I I I P P 2 2 1 1 2 2 1 1 = = R V V / P P P P \ 2 V + 2 2 1 I + P 1 1 1 = = T I I V V \ P P 1 2 P P \ 2 1 On occasions these equations will not add up to what you might expect !
Reflection Coefficients + 2 2 I I I I I + 2 2 1 1 1 + = + R T I I / \ P P P P 1 \ \ + 2 1 I I I I 2 1 + 1 = I 2
Reflection Coefficients + 2 I I I I I + 2 2 1 1 1 + = + R T I I / \ P P P P 2 1 \ \ + 2 1 I I I I 2 1 + 1 = I + + 2 I I I I 2 2 1 1 =
Reflection Coefficients + 2 2 I I I I I + 2 2 1 1 1 + = + R T I I / \ P P P P 1 \ \ + 2 1 I I I I 2 1 + 1 = I + + 2 I I I I 2 2 1 1 = = 1
Reflection Coefficients + 2 I I I I I + 2 2 1 1 1 + = + R T I I / \ P P P P 2 1 \ \ + 2 I I I 2 1 + 1 = I + + I 2 1 I I I I 2 2 1 1 = = 1 + = 1 R T / \ P P P P \ \
Reflection Coefficients + 2 I I I I I + 2 2 1 1 1 + = + R T I I / \ P P P P 2 1 \ \ + 2 I I I 2 1 + 1 = I + + I 2 1 I I I I 2 2 1 1 = = 1 + = 1 R T / \ P P P P \ \
Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface?
Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: = 1 R / P P \
Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: = 1 R / P P \ + = 1 R T But, / \ P P P P \ \ = ? T What must: \ P P \
Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: = 1 R / P P \ + = 1 R T But, / \ P P P P \ \ = 2 T So, \ P P \
Reflection Coefficients Briefly, how to consider displacements at interfaces using potentials, when mode conversion occurs: Z- layer 1 X+ layer 2 Z+ In layer 1, just above the boundary, at the point of incidence: \ / \ / \ \ S P S z = + P x P P x ux z
Reflection Coefficients Briefly, how to consider displacements at interfaces using potentials, when mode conversion occurs: Z- layer 1 X+ layer 2 Z+ In layer 2, just below the boundary, at the point of incidence: \ \ \ \ = P P z ux P S x
Reflection Coefficients So, if we consider that (1) stresses as well as (2) displacements are the same at the point of incidence whether we are in the top or bottom layer the following must hold true so that (3) Snell s Law holds true: + = , u u x x + = u u z z + = = zz zz + zx zx
Reflection Coefficients We get the general case of all the different types of reflection and transmission (refraction or not) coefficients at all angles of incidence : = RP P RP S T P P T P S sin cos \ \ \ P cos \ / \ P sin2 \ \ \ P cos2 \ \ \ P
Reflection Coefficients Variation of Amplitude with angle ( AVA ) for the fluid-over-fluid case (NO SHEAR WAVES) 2 sin V V cos 1 I I 2 2 1 1 ( ) = R 2 sin V V + cos 1 I I 2 2 1 1 (Liner, 2004; Eq. 3.29, p.68; ~Ikelle and Amundsen, 2005, p. 94)
Reflection Coefficients What occurs at and beyond the critical angle? V V sin sin2 = c 1 2 V V = 1 sin 1 2 c
Reflection Coefficients FLUID-FLUID case What occurs at the critical angle? 2 sin V V cos 1 I I 2 2 1 1 ( ) = R 2 sin V V + cos 1 I I 2 2 1 1 (Liner, 2004; Eq. 3.29, p.68; ~Ikelle and Amundsen, 2005, p.94)
Reflection Coefficients Reflection Coefficients at all angles: pre- and post-critical Case: Rho: 2.2 /1.8 V: 1800/2500 Matlab Code
NOTES: #1 Reflection Coefficients At the critical angle, the real portion of the RC goes to 1. But, beyond it drops. This does not mean that the energy is dropping. Remember that the RC is complex and has two terms. For an estimation of energy you would need to look at the square of the amplitude. To calculate the amplitude we include both the imaginary and real portions of the RC.
NOTES: #2 Reflection Coefficients For the critical ray, amplitude is maximum (=1) at critical angle. Post-critical angles also have a maximum amplitude because all the energy is coming back as a reflected wave and no energy is getting into the lower layer
NOTES: #3 Reflection Coefficients Post-critical angle rays will experience a phase shift, that is the shape of the signal will change.
Energy Coefficients + = 1 R T We saw that for reflection coefficients : / \ P P P P \ \ = E RP P \ / \ / P P For the energy coefficients at normal incidence : VP V P 2 11 2 = E TP P \ / \ \ P P
Energy Coefficients + = 1 R T We saw that for reflection coefficients : / \ P P P P \ \ = E RP P \ / \ / P P For the energy coefficients at normal incidence : VP V P 2 11 2 = E TP P \ / \ \ P P The sum of the energy is expected to be conserved across the boundary = + + + E E E E E \ \ / \ / \ \ \ \ P P P P S P P P S
Amplitude versus Offset (AVO) Zoeppritz s equations can be simplified if we assume that the following ratios are much smaller than 1: VS VP average VPaverage VSaverage For example, the change in velocities across a boundary is very small when compared to the average velocities across the boundary; in other words when velocity variations occur in small increments across boundaries This is the ASSUMPTION
Amplitude versus Offset (AVO) If the changes across boundaries are relatively small, then we can make a lot of approximations to simplify the reflection and transmission coefficients: 2 S 2 V 1 2 V 1 2 z P 2 = + sin R 2 Paverage \ / i average V z P V P P average ( Equation, 3.242, Ikelle and Amundsen)
AVO cases for P-P In addition, we can rearrange and further simplify this equation as follows: ??? = ??? ?????? ????????;= 0 + 4.5 ????? ??????? ? ????? ??? 0 sin ? ??? sin(? ???) ( Equation, 3.255, Ikelle and Amundsen)
AVO cases for P-P With the much simplified calculation for reflection coefficients, we can characterize reservoirs according to the Rpp at normal incidence ( intercept) and contrast in Poisson s ratio across their boundaries (~gradient or slope): CLASS 1 (Z1<Z2) CLASS 1 (Z1<Z2) slope CLASS 1 (Z1<Z2) intercept
AVO cases for P-P-CLASS 2 Class 2 shows very little contrast between layer properties slope CLASS 2 Z1~Z2 intercept
AVO cases for P-P-CLASS 3 Class 3 shows the characteristic bright spot that is often interpreted to mean shale over gas sand (Z1<Z2). The reflection coefficient becomes more negative with offset slope CLASS 3 Z1<Z2 intercept
AVO cases for P-P-CLASS 4 Class 4 is like class 3 in that there is a strong negative reflection BUT the Reflection coefficient decreases with angle (GOM) CLASS 4 Z1<Z2 slope CLASS 4 Z1<Z2 intercept
