1. 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:

2. Forward Modeling
2-way time

3. Forward Modeling: Sum of Weighted Hyperbolas
2-way time

4. iw|x-x’|/c G(x|x’) = e |x-x’| |x-x’| x x’ GREEN’s FUNCTION Phase Geom.
Spread x
|x-x’| x’

5.    + O( ) iw|x-x’|/c G(x|x’) = e |x-x’| x x’
ASYMPTOTIC GREEN’s FUNCTION 
xx’ 
+ O( ) -1 e
iw|x-x’|/c
Phase
G(x|x’) =
|x-x’|
Geom.
Spread
A(x,x’) x 
xx’ x’

6.  e  R(x’) G(x|x’) =  ASYMPTOTIC GREEN’s FUNCTION i xx’ x’ A(x,x’)
reflectivity  x’
G(x|x’) =
A(x,x’)

7. Diffraction Stack Modeling = ZO Modeling
1-way time

8. Diffraction Stack Modeling = ZO Modeling
2-way time
Dipping Reflector

9. Diffraction Stack Modeling = ZO Modeling
If c for DS is ½ that for ZO Modeling
1-way time

10. ~ ~  e  e    (t- ) R(x’) d(x) =  F d(x) F  
ASYMPTOTIC GREEN’s FUNCTION 
xx’  i ~ e
R(x’)
reflectivity
d(x) =  x’
A(x,x’)
Fourier Transform: 
xx’  i e
 (t- ) F ~
d(x) F  x’
R(x’)
A(x,x’)
 (t ) 
xx’

11.  ( - t) e   d  (t- ) i t + + + QUICK REVIEW FOURIER TRANSFORM
xx’  i e
 (t- )   d
( - t)  
Cos( t ) t + 
Cos( 2 t ) + 
Cos( 3 t ) + 
Cos( 4 t )
constructive
t=0
cancellation
 (t)

12.  d(x,t) =  Forward Modeling Operator  (t- )  (t-  ) R(x’) x’
time
Spray energy along
hyperbolas
A(x,x’)
Sum over reflectivity
 (t-  )
xx’

13.  d(x,t) =  Forward Modeling Operator  (t- ) R(x’) W x’ A(x,x’) time
CANCELLATION
REINFORCE

14. ò   R(x’) d(x,t) = SUMMARY W (t- ) d(x) = (x |x’) m(x’) dx’ G x’
Exploding Reflector Modeling = Diffraction Stack Modeling
Single scattering approximation (i.e., Born)  x’
W (t ) 
xx’
R(x’)
d(x,t) =
Source wavelet
Sum over
reflectors
reflectivity
Data
variables
A(x,x’)
Geom. spreading
d(x) = (x |x’) m(x’) dx’ ò
Model
Space G
model
data
Integral
Equation:
2. High Frequency Approximation (i.e c(x) variations > 3* )
3. Approximates Kinematics of ZO Sections, but not Dynamics

15.   d(x,t|x’,0) = MATLAB Exercise: Forward Modeling W (t- ) 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)
W (t ) 
xx’
d(x,t|x’,0) =  x’
R(x’)
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?

16. {   d(x,t) = MATLAB Exercise: Forward Modeling W (t- ) R(x’) x’
Loop over x in model
Loop over z in model
Loop over traces
Traveltime
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;
data1(ixtrace,:)=conv2(data(ixtrace,:),rick);
R(x’) {
* Src Wave