reflection coefficients
DESCRIPTION
seismicsTRANSCRIPT
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 21 (2 )2 22xz x z x z
2 22 2 (2 )
2zz x zz
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:
\ /
2 12 1 2 12 1 2 12 1
V VI I P PRI I V VP P P P
\ \
22 11 12 1 2 12 1
VI PTI I V VP P P P
On occasions these equations will not add up to what you might expect…!
Reflection Coefficients
\ \/ \
22 1 12 1 2 1
22 1 12 1
I I IR TI I I IP P P P
I I II I
Reflection Coefficients
\ \/ \
22 1 12 1 2 1
22 1 12 1
2 12 1
I I IR TI I I IP P P P
I I II I
I II I
Reflection Coefficients
\ \/ \
22 1 12 1 2 1
22 1 12 1
2 12 11
I I IR TI I I IP P P P
I I II I
I II I
Reflection Coefficients
\ \
\ \
/ \
/ \
22 1 12 1 2 1
22 1 12 1
2 12 11
1
I I IR TI I I IP P P P
I I II I
I II I
R TP P P P
Reflection Coefficients
\ \
\ \
/ \
/ \
22 1 12 1 2 1
22 1 12 1
2 12 11
1
I I IR TI I I IP P P P
I I II I
I II I
R TP 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:
\ /1R
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:
\ /1R
P P
\ \/ \1R T
P P P P But,
What must: \ \?T
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:
\ /1R
P P
\ \/ \1R T
P P P P But,
\ \2T
P PSo,
Reflection Coefficients
Z+
Z-
X+
\ /\ \ /\S P SP P Pu
x x z x z
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+
\ \\ \P SP Pu
x 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 CoefficientsSo, 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 ux x
u uz z
zz zz
zx zx
Reflection CoefficientsWe get the general case of all the different types of reflection and transmission (refraction or not) coefficients at all angles of incidence :
sin cos \ \\
cos \ /\
sin 2 \ \ \
cos2 \ \ \
RP PPRP SPT
P P PT
P P S
Variation of Amplitude with angle (“AVA”) for the fluid-over-fluid case (NO SHEAR WAVES)
2
22 1
1
2
22 1
1
sincos 1
( )sin
cos 1
VI I
VR
VI I
V
(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
1
2
1 1
2
sin
sin2
sin
c
c
VV
VV
FLUID-FLUID case
What occurs at the critical angle?
2
22 1
1
2
22 1
1
sincos 1
( )sin
cos 1
VI I
VR
VI I
V
(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 /1.8V: 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 : \ \/ \1R T
P P P P
\ /\ /
2 2\ /\ \ 1 1
E RP PP P
VPE TP P VP P P
For the energy coefficients at normal incidence :
Energy Coefficients
We saw that for reflection coefficients : \ \/ \1R T
P P P P
For the energy coefficients at normal incidence :
\ /\ /
2 2\ /\ \ 1 1
E RP PP P
VPE TP P VP P P
The sum of the energy is expected to be conserved across the boundary
\ \ / \ / \ \ \ \E E E E EP P P P S P P P S
Amplitude versus Offset (AVO)
Zoeppritz’s equations can be simplied if we assume that the following ratios are much smaller than 1:
VPVPaverage
VSVSaverage
average
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:
2211 2sin2\ / 22
VVz SPRiVz P averageVP P average Paverage