1 Forward modeling operator L d x x x mx dx ò Model Space G model data Integral Equation 2way time Forward Modeling 2way time Forward Modeling Sum of Weighted Hyperbolas ID: 830460
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Slide1
I.1 Diffraction Stack Modeling
1. Forward modeling operator L
d
(x) =
(x |x’)
m(x’) dx’
ò
Model
Space
G
model
data
Integral
Equation:
Slide22-way time
Forward Modeling
Slide32-way time
Forward Modeling: Sum of Weighted Hyperbolas
Slide4G(x|
x’
) =
x
x’
e
iw|x-
x’
|/c
Phase
|x-
x’
|
Geom.
Spread
GREEN’s FUNCTION
|x-
x’
|
Slide5G(x|
x’
) =
x
x’
e
iw|x-
x’
|/c
Phase
|x-
x’
|
Geom.
Spread
ASYMPTOTIC GREEN’s FUNCTION
A(x
,x’
)
x
x’
+ O( )
-1
x
x’
Slide6ASYMPTOTIC GREEN’s FUNCTION
G(x|
x’
) =
A(x
,x’
)
x
x’
i
e
x’
x’
R(x’)
reflectivity
Slide7Diffraction Stack Modeling = ZO Modeling
1-way time
Slide8Diffraction Stack Modeling = ZO Modeling
2-way time
Dipping Reflector
Slide9Diffraction Stack Modeling = ZO ModelingIf c for DS is ½ that for ZO Modeling
1-way time
Slide10ASYMPTOTIC GREEN’s FUNCTION
d(x) =
A(x,x’
)
x
x’
i
e
x’
R(x’)
reflectivity
Fourier Transform:
x
x’
i
e
(t- )
x
x’
F
~
(t- )
x
x’
~
d(x)
F
x’
R(x’)
A(x
,x’
)
Slide11QUICK REVIEW FOURIER TRANSFORM
x
x’
i
e
(t- )
x
x’
d
( - t)
+
Cos( 2 t )
+
Cos( 4 t )
+
Cos( 3 t )
Cos( t )
t
cancellation
cancellation
constructive
reinforcement @ t=0
(t)
Slide12Forward Modeling Operator
(t- )
x
x’
d(x,t) =
x’
R(x’)
A(x
,x’
)
Sum over reflectivity
Spray energy along
hyperbolas
time
(t-
)
x
x’
Slide13Forward Modeling Operator
(t- )
x
x’
d(x,t) =
x’
R(x’)
A(x
,x’
)
time
REINFORCE
CANCELLATION
W
Slide14SUMMARY
W (t- )
x
x’
d(x,t) =
x’
Exploding Reflector Modeling = Diffraction Stack Modeling
Single scattering approximation (i.e., Born)
A
(x
,x’
)
R(x’)
reflectivity
Geom. spreading
Source wavelet
Data
variables
Sum over
reflectors
2. High Frequency Approximation (i.e c(x) variations > 3*
)
3. Approximates Kinematics of ZO Sections, but not Dynamics
d
(x)
= (
x |x’)
m(x’)
dx’
ò
Model
Space
G
model
data
Integral
Equation:
Slide15MATLAB Exercise: Forward Modeling
W (t- )
x
x’
d(x,t|
x’,0
) =
x’
R(x’)
1. To account for the source wavelet W(t), we
convolve data with W(t)
(recall
(t-
)*W(t)= W() )
so that modeling equation becomes (neglect A)
A). Execute MATLAB program forw.m to generate
synthetic data for a point scatterer and a 30 Hz wavelet.
B). Execute MATLAB program forwl.m to generate
synthetic data for a dipping layer model
C). Execute MATLAB program forw.m to generate
synthetic data for a syncline model. Note diffractions
and multiple arrivals. Adjust for new models. Why the
second time derivative?
Slide16MATLAB Exercise: Forward Modeling
W (t- )
x
x’
d(x,t) =
x’
R(x’)
for ixtrace=1:ntrace;
for ixs=istart:iend;
for izs=1:nz;
r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2);
time = 1 + round( r/c/dt );
data(ixtrace,time) = migi(ixs,izs)/r + data(ixtrace,time);
end;
end;
data1(ixtrace,:)=conv2(data(ixtrace,:),rick);end;
* Src Wave
Traveltime
R(x’)
{
Loop over traces
Loop over x in model
Loop over z in model