1. Tutorial on Green’s Theorem and Forward Modeling, SRME, Interferometry, and Migration..

2. Outline
Green’s Functions
Forward Modeling

3. What is a Green’s Function?
Answer: Point Src. Response of Medium
Answer: Impulse-Point Src. Response of Medium {
(t-|r|/c) =
1 t=|r|/c
ottherwise
Listening time
Listening & source coords.
G(x,t|x ,t )= s
|r|
Geometrical Spreading r
t=1
t=2
t=3
t=4 4 T c
|r| = x + y + z 2
Propagation time =

4. What is a Green’s Function?
Answer: Point Src. Response of Medium
Answer: Impulse-Point Src. Response of Medium {
1 t-t =|r|/c
ottherwise
|r|
G(x,t|x ,0 ) = s
(t -|r|/c) =
Fourier
G(x|x ) = s ~
|r|
iwr/c e
derivative d dr
G(x|x ) = s ~
|r|
iwr/c e
iw/c
+ O(1/r ) 2
Far field approx
G(x|x ) = s
|r|
iwr/c e
iw/c
+ O(1/r ) 2 n r k

5. Wave in Heterogeneous MediumAe
i(wt - wt)
G = G
G = .. c 2
satisfies
except at origin
(2)
Geometrical speading w
kr = (kc ) r /c = wt i i r
Time taken along ray segment 1 2 3 4
Valid at high w and smooth media

6. Outline Green’s Functions Forward Modeling SRME
Acausal GF, Stationarity, Reciprocity
Interferometry
Migration

7. What is Huygen’s Principle?
Answer: Every pt. on a wavefront is a secondary src. pt.
Common tangent kinematically defines next wavefront.
P(x,T ) 5 4 T T 5

8. What is Huygen-Fresnel Principle?
Answer: Every pt. on a wavefront is a secondary src. pt.
Next wavefront is sum of weighted Green’s functions.
P(x’,T ) 5
G(x,t|x’ ,T ) s
P(x,T ) 5
G(x,t|x ,T ) s T 5

9. What is Huygen-Fresnel Principle?
Answer: Every pt. on a wavefront is a secondary src. pt.
Next wavefront is sum of weighted Green’s functions.
P(x,T ) 5
G(x,t|x ,T ) s x t T 5

10. Problem with Huygen’s Principle?
P(x,T ) 5
G(x,t|x ,T ) s x t T 5

11. Problem with Huygen’s Principle?
ZOOM
P(x,T ) 5
G(x,t|x ,T ) s x t T 5

12. Problem with Huygen’s Principle?
ZOOM + -

13. Problem with Huygen’s Principle?
P(x’,T ) 5
G(x,t|x’ ,T ) s
-G(x,t|x’ ,T ) dn
= PdG/dn
ZOOM + - + - +

14. Green’s Theorem Forward Modeling} 2 d dn
d x
G(x|x) d dn
P(x ) -
P(x )
G(x|x)
P(x ) = }

15. Outline Green’s Functions Forward Modeling SRME
Acausal GF, Stationarity, Reciprocity
Interferometry
Migration

16. Reciprocity Eqn. of Convolution Type
0. Define Problem
Given:
Find: A
G(A|B)
Free surface B
G(A|x) A B
Free surface
G(B|x) x

17. Reciprocity Eqn. of Convolution Type
1. Helmholtz Eqns: 2
+ k [ ]
G(A|x) =- (x-A); 2
+ k [ ]
P(B|x) =- (x-B) *
2. Multiply by G(A|x) and P(B|x) and subtract * * 2
+ k [ ]
G(A|x) =- (x-A)
P(B|x)
P(B|x) =- (x-B)
G(A|x)
P(B|x) =
G(A|x)
P(B|x) 2
G(A|x) [ ] *
G(A|x)
P(B|x)
- G(A|x) 2
= (B-x)G(A|x) - (A-x)P(B|x) *
G(A|x) =
P(B|x)
G(A|x) 2
{ }
P(B|x) *
G(A|x)
P(B|x)
- G(A|x)
{ }
= (B-x)G(A|x) - (A-x)P(B|x) * *

18. Reciprocity Eqn. of Convolution Type 3. Integrate over a volume
G(A|x)
P(B|x)
- G(A|x)
d x 3
= G(A|B) - P(B|A)
{ } *
4. Gauss’s Theorem
G(A|x)
P(B|x)
- G(A|x)
d x 2
= G(A|B) - P(B|A)
{ } n *
Source line
G(A|B)
Free surface x B A
Integration at infinity vanishes

19. Reciprocity Eqn. of Convolution Type 3. Integrate over a volume
G(A|x)
P(B|x)
- G(A|x)
d x 3
= G(A|B) - P(B|A)
{ }
4. Gauss’s Theorem
Hence, the two integrands above are opposite
and equal in far-field hi-freq. approximation
G(A|x)
P(B|x)
- G(A|x)
d x 2
= G(A|B) - P(B|A)
{ } n
Layer interface A B =
If boundary is far away the
major contribution is specular reflection
(B-x) n
|B-x|
(A-x) n -
|A-x| n x

20. Reciprocity Eqn. of Convolution Type
G(A|x)
P(B|x)
d x 2
= G(A|B) - P(B|A) n
1st interface
n r =1
Far field
G(A|x)
P(B|x)
d x 2
+ P(B|A) = G(A|B)
2ik
Freeze B and hide it
1st interface
G(A|x)
p(x)
d x 2
=- p(A) + g(A)
2ik
Remember, g(A) is a ½ space shot gather
with src at B and rec at A
Reflected waves {
Ghost direct-Total pressure {
Ghost direct-Total pressure {
1st interface
B A
Free surface
p(x)
G(A’|x)
(1)
Dashed yellow ray is just a primary and ghost
reflection off of sea-floor. .Higher-order multiples will
also contribute..but we assume they are weak
p(x)
G(A|x)
(0)
p(x) = p(x) + p(x) + p(x) +…
(0)
(1)
(2)
= p(x)
(0)
Primary reflection

21. { { Summary Green’s theorem, far-field, hi-freq. approx.
G(A|x)
p(x)
d x 2
= p(A)
2ik
n r
scatt.
G(A|x)
p(x)
d x 2
=- p(A) + g(A)
2ik
Reflected waves {
Scattered pressure field {
obliquity
1st interface
B A
Free surface
Dashed yellow ray is just a primary and ghost
reflection off of sea-floor. .Higher-order multiples will
also contribute..but we assume they are weak
p(x)
G(A|x)
(0)
p(x) = p(x) + p(x) + p(x) +…
(0)
(1)
(2)
= p(x)
(0)
Primary reflection
If A=B and p(x)=cnst. Then we reduce to intuitive
ZO modeling formula