Mathematics of Kirchhoff Inversion
One-day Short Course
presented by
Norman Bleistein
University Emeritus Professor
Department of Geophysics
Center for Wave Phenomena
Colorado School of Mines
norm@dix.mines.edu
cwp.mines.edu/~norrm
Presented
Sunday, September 11, 2005
International Meeting
Sociedada Brasileira de Geofísica
Salvador, Brazil
Preface
In 2001, Mathematics of Multidimensional Seismic Imaging, Migration and Inversion
(MMSIMI), by Jack K. Cohen, John W. Stockwell and me, was published by Springer-Verlag.
Since that time, I have continued doing research in this general area, largely as a consequence of
a consulting relationship with Veritas, DGC.
Not surprisingly, my viewpoint about the underlying theory has evolved since the publication of
that book. In particular, I prefer starting the inversion derivation from an asymptotic form of the
forward model of the reflected wave as a volume integral with a Reflectivity function “nearly”
completely identified in that representation.
Furthermore, the two-and-one-half-dimensional theory would seem to have been
overtaken by the speed and memory capacity of computers, so that I have now relegated it to a
last part of the current presentation.
Taking higher priority in my mind is an extension of the Kirchhoff theory that starts from
an integral inversion formula with the variables of integration being coordinates at the image
point. The formula starts relatively compactly stated, although implementation presents
problems. However, when the integral is recast in source/receiver coordinates, all sorts of
salutary consequences arise. Among them:
i. Division by Green’s function amplitudes in traditional Kirchhoff inversion formulas now
is replaced by multiplications by adjoint Green’s functions. This provides for more stable
numerical integration formulas.
ii. The inversion is a sum over all sources and receivers that provides an output in a suite of
common angle panels, covering both opening angles and azimuthal angles between
specular rays to sources and receivers from the image point.
iii. The sum over available source/receiver pairs—available traces—lends itself to relatively
straightforward processing of data collected from surveys using multi-streamer receiver
sets. The classical Kirchhoff inversion method does not provide a formula to process
data acquired in this manner.
The seminal ideas for this new work must be credited to Sheng Xu and Yu Zhang of
Veritas, DGC, with Samuel H. Gray (Veritas) and Guanquan Zhang providing challenges,
inspiration and astute criticism and oversight as these ideas developed. (An incomplete
bibliography is listed below.)
Preparing these slides was a useful exercise for me. First, it forced me to acknowledge
my changed priority order and point of view. Second, it forced me to attempt a uniformity of
notation that certainly has not been my hallmark in the past. Some of the figures addressing
ideas of the text have been revised and there are new figures developed along with the new ideas
mentioned above.
I just hope there is enough time in the day!
Norm Bleistein
July 28, 2005
Bleistein, N., Zhang, Y., Xu, S., Gray, S. H., Zhang, G., 2005, Migration/Inversion: Think Image
Point Coordinates, Process in Acquisition Surface Coordinates: preprint,
http://cwp.mines.edu/~norm/Papers/MIUpperSurf.pdf
Xu, S., Chauris, H, Lambaré G. and Noble, M. S., 2001, Common-angle migration: A strategy
for imaging complex media: Geophysics, 66, 6, 1877-1894.
Zhang, Y., Xu, S., Zhang, G., Bleistein, N., 2004, How to obtain true amplitude common-angle
gathers from one-way wave equation migration?: Expanded Abstracts, International Meeting
of the Society of Exploration Geophysicists , Tulsa..
i. Short Course, Brazil
Outline
1. Motivation—forward modeling and imaging
o Hagedoorn modeling
o Hagedoorn imaging
 Analytical description
o Mathematical description of a reflector
 Singular function of a surface
o Wave equation
 Time domain
 Frequency domain
 Green’s functions
o Kirchhoff modeling, single reflector, exact
o Plan for deriving asymptotic inversion
o Effect of band limiting on imaging
2. Ray theory, propagation, reflection
o “Ansatz”
 Eikonal equation (travel time)
 Transport equation (Amplitude)
 Initial data on rays
• Green’s function
• Reflection
3. Asymptotic modeling of reflected wave
o Method of stationary phase
o Kirchhoff approximations applied to exact Kirchhoff formula
o Recast surface integral formula as a volume integral
4. Deriving Kirchhoff inversion
o Formal inversion via approximate (local) Fourier transform
 Beylkin determinant
 Properties of local Fourier interpretation
 Special cases
• Common shot
• Common offset
5. Full wave form Kirchhoff inversion
o Motivation: Better Green’s function ⇒ better imaging
o Review of “true amplitude” concept
o Approximate pseudo-inverse applied to Kirchhoff exact modeling formula
 Kirchhoff approximation applied to better Green’s functions
 Modeling formula recast as a volume integral
• Interpretation as a modeling operator
o Inversion
 Adjoint of modeling operator
 Pseudo-inverse of modeling operator
 Asymptotic normalization of operator

Relation of result to Kirchhoff inversion with ray-theoretic Green’s
functions
6. Think image point coordinates, process in source/receiver coordinates
o Motivation
o Inversion as an integral over angular variables at the image point
o Transformation to acquisition surface coordinates
 Deconvolution processing ⇒ Correlation processing
o Inversion of data acquired from multi-streamer surveys
7. Two-and-one-half-dimensional (2.5D) inversion
o Motivation (3D processing over a 2D (cylindrical) Earth model)
o 3D inversion for identical lines of data
 Stationary phase applied to out-of-plane integration in operator
 In-plane inversion
• In-plane 3D propagation
• Application to in-plane inversion
o 2.5D inversion formula
Kirchhoff (Born) Inversion
Short course
Norm Bleistein
Center for Wave Phenomena
Colorado School of Mines
cwp.mines.edu/mmsimi
2
1
More current point of
view than in MMSIMI
Migration/Inversion: Think Image Point
Coordinates, Process in Acquisition
Surface Coordinates:
with Y. Zhang, S. Xu, S. H. Gray, G.
Zhang, to appear
• cwp.mines.edu/~norm
Kirchhoff inversion is…
• Leading order
• High frequency
• Asymptotic technique
4
2
So is …
• Wave equation migration
–(including “true amplitude”)
• Kirchhoff migration
• Stolt (f/k) migration
• Phase shift migration
5
Graphic approach to inversion
…due to Hagedoorn
3
7
8
4
Record arrival times
• Source/receiver position - ξ
• Arrival time - t ?
• Arrival time - vt/2
10
5
11
12
6
13
14
7
15
16
8
17
18
9
19
Hagedoorn construction
20
10
Kirchhoff processing
21
22
11
23
D Data at time τ on trace ξ
24
12
25
D Data at time τ on trace ξ
W Spatial weighting function
“Reflectivity”
26
13
D Data at time τ on trace ξ
D Filtered observed data
• Wave shaping
27
W Spatial weighting function
• Migration (Reflector map)
• Inversion (Parameter estimation)
• Model consistent
28
14
“Reflectivity”
To be defined later
29
Course objective:
Fill in the details
30
15
Reflector image
Singular function of a
surface
Delta function of distance
normal to the surface
Singular function of a
surface:
16
Reflector image:
image of the singular function
of the reflector
Action of the singular function
Volume integral
Surface integral
17
Tools of the trade
Forward modeling
Forward modeling
The wave equation
18
Forward modeling
The wave equation
The wave equation
Frequency domain
38
19
Green’s function
39
Point source
40
20
Green’s function
representation of solution
41
Volume part
42
21
Surface part
43
Born modeling
• Volume integral
• Pseudo-sources
– Created from
medium perturbations
22
Kirchhoff modeling
• Surface integral
– Reflector
• Ray theory/ WKBJ for data
– Generalization of the
Kirchhoff approximation
Kirchhoff modeling
23
Inversion objective
• Image the reflector
• Estimate reflection coefficient
– Plane waves, planar reflectors
– Curved reflectors?
• “High frequency concept!”
• Length scales >> wave length
“High
frequency” modeling
24
“High
frequency” inversion
“High
frequency” inversion
with offset
25
“The Plan”
• Model reflected wave field
– data acquisition
• Common shot, common offset, etc
• Identify “Reflectivity”
–
51
“The Plan”
• M reflected wave field
– data acquisition
• Common shot, common offset, etc
• Identify “Reflectivity”
–
• “Solve” for Reflectivity
52
26
Why reflectors?
Not even “steps” in the
in velocity, density
High frequency data cannot
resolve steps!
54
27
High frequency data cannot
resolve steps!
55
High frequency data cannot
resolve steps!
56
28
High frequency data cannot
resolve steps!
57
High frequency data cannot
resolve steps!
58
29
High frequency data can
resolve delta functions!
59
Kirchhoff modeling
We don’t know
30
Kirchhoff modeling
We don’t see
in
Kirchhoff modeling
We need ray theory for
incident waves, reflected waves
31
Summary
• Hagedoorn construction
– Integral inversion formula
• Forward modeling as point of
departure
• Reflector imaging
–High frequency inversion
63
32
Tools of the Trade
Ray theory/ WKBJ
• “High
frequency” concept
– plane waves, planar reflectors
– length scales large compared to
a wave length
64
Homogeneous wave equation
Ansatz
65
1
Homogeneous wave equation
First term in an asymptotic series
66
Substitute ….
67
2
Eikonal equation
68
First order nonlinear
partial differential equation
69
3
Remember ….
70
(First) Transport equation
71
4
Tools of the Trade
Ray theory/ WKBJ
• “Ansatz”
• Eikonal equation (phase,
traveltime)
• Transport equation (amplitude)
72
Solution technique:
Method of characteristics
73
5
Solution technique:
• Solve for trajectories in space, rays
• Determine τ along the rays
74
75
6
76
Alternative ray parameter σ
Important in 2.5D
77
7
Ray parameter, s, arclength
78
An example:
— point source
— constant wave speed
79
8
Example: reflected wave
80
Example: reflected wave
• Surface:
81
9
Example: reflected wave
• Surface:
• Traveltime:
82
Example: reflected wave
• Surface:
• Traveltime:
• Gradient:
solve
83
10
84
Tools of the Trade
Ray theory/ WKBJ
• “Ansatz”
• Eikonal equation (phase,
traveltime)
• Transport equation (amplitude)
85
11
(First) Transport equation
86
Integrate over a “smart” volume
87
12
88
Preserved
on the ray
Preserved
on the ray
Preserved
on the ray
89
13
Preserved
on the ray
Preserved
on the ray
Preserved
on the ray
90
Preserved
on the ray
Preserved
on the ray
91
14
92
Amplitude propagation
93
15
Other parameters along the ray
94
Green’s function amplitude
C: depends on choice of α1, α2
95
16
Example: α1, α2, polar coordinates
96
Reflection
Formulation
97
17
Wave types
• Incident wave
• Reflected wave
• Transmitted wave
98
Principle:
Wave incident on the reflector
gives rise to reflected and
transmitted waves.
99
18
Total wave field
• Above reflector
• Below reflector
100
Objective
Find initial data on S for
reflected and transmitted wave
in terms of incident wave
Express τR, τT, AR, AT,
in terms of τI and AI
101
19
Interface conditions on S
Wave field and its normal
derivatve are continuous
102
103
20
104
Determines
tangential
components of
gradient
105
21
Normal components?
Eikonal equation
106
107
22
Amplitudes
108
Amplitudes
109
23
Important for modeling and
inversion!
110
What happens when J = 0?
Caustic!
111
24
What happens when J = 0?
Caustic!
112
Ray theory fails at caustic
•
Higher function required
– Airy function
• Phase shift through caustic
113
25
More caustics?
• KMAH index, κ
– Count of the number of caustics
114
Summary: ray theory
• Wave equation
• Describes solution with
amplitude and phase: Aeiωτ
• τ: Eikonal equation
• A: Transport equation
• Green’s function
• Reflection process
115
26
Asymptotic modeling of
reflected wave
• Method of stationary phase
• Kirchhoff approximations
(from ray theory) applied to
exact Kirchhoff formula
• Recast surface integral formula
as a volume integral
116
Further asymptotics
Method of stationary phase
117
1
Prototype integral
Formally:
118
Prototype integral
Practically: λ “large”
119
2
Prototype integral
Forward modeling
120
Prototype integral
Migration/Inversion
121
3
Prototype integral
Migration/Inversion
122
Prototype integral
Stationary points
123
4
Contribution from stationary point
124
Contribution from stationary point
Count: pos e-values - neg e-values
125
5
How accurate?
126
One dimension: n = 1
127
6
Example: the Hankel
Function
Green’s function in 2D,
homogeneous medium
(within a constant)
128
Example: the Hankel
Function
129
7
Real part: solid - exact
130
Imaginary part
131
8
Percentage error
132
Return to modeling equation
133
9
Use ray theory to simplify
134
135
10
136
137
11
138
139
12
Generalization of
The Kirchhoff Approximation
140
141
13
Use ray theory to simplify
142
Use ray theory to simplify
143
14
Write as volume integral?
Use singular function γ(x)?
Not with normal derivatives
144
Write as volume integral?
Analyze: method of stationary phase
145
15
Stationary phase analysis
146
Stationary phase analysis
147
16
Stationary phase analysis
148
Stationary phase analysis
149
17
Stationary phase analysis
Snell’s law!
150
151
18
Stationary phase analysis
Snell’s law!
152
Stationary phase analysis
Snell’s law!
153
19
Stationary phase analysis
Snell’s law!
154
Revised reflection modeling
155
20
Revised reflection modeling
stationary phase approximation
156
Revised reflection modeling
Surface integral to volume integral
157
21
Revised reflection modeling
Reflectivity function!
158
“The Plan”
• Model reflected wave field
– data acquisition
• Common shot, common offset, etc
• Identify “Reflectivity”
–
159
22
Revised reflection modeling
Finally!!!!
160
Summary: arriving at
• Start from Kirchhoff integral
• Use ray theory to simplify
• Use stationary phase to simplify
• Transform surface integral to
volume integral
161
23
Deriving Kirchhoff inversion
162
Revised reflection modeling
163
1
Objective: invert for reflectivity
164
Sources and Receivers
Common shot
Common offset
165
2
Phase
166
Amplitude
167
3
Same modeling operator,
Different notation
Inverse?
168
Cascade modeling and
inversion operators
169
4
`must be a delta function!
170
`must be a delta function!
171
5
`must be a delta function!
172
‘must be a delta function!
173
6
What about the Jacobian?
174
Beylkin determinant
175
7
Beylkin determinant
176
Solve!
177
8
Recall
178
Kirchhoff inversion
h accomodates variable upper
surface
179
9
Equivalence
• Start from Born approximation
for forward modeling
• Same inversion formula
180
Micro-local geometry at the
image point
181
10
Micro-local:
changes with
source point
182
Micro-local:
changes with
receiver point
183
11
Micro-local:
changes with
image point
184
Micro-local geometry at the
image point
185
12
Micro-local geometry at the
image point
186
Beylkin determinant:
alternative form
187
13
Beylkin determinant:
alternative form
188
Beylkin determinant:
alternative form
189
14
Common Shot
(planar acquisiion surface)
xs = constant, xr = (ξ1, ξ2,0)
190
Common Shot
(planar acquisiion surface)
xs = constant, xr = (ξ1, ξ2,0)
191
15
Common Offset
• No simplification
• Computationally intensive
192
Common Offset
• Geometric computation
193
16
Summary
• Inversion formula for
reflectivity
–Unspecified source/receiver
specification
–Two acquisition surface
parameters
–Characterized by Beylkin
determinant
194
What can we prove?
• Apply inversion formula to
(Kirchhoff) model data
• Two integrals for modeling,
three for inversion
• Apply multi-dimensional
stationary phase (spatial
coordinates)
195
17

x
Stationary triple for each

x
196
197
18
198
199
19
200
201
20
202
θ stat = θ spec
203
21
 
x = xstat
204
Stationary value of gradient
Specular value of gradient
205
22
Second inversion operator
206
207
23
Other attributes: same trick
• Specular source, receiver
• Specular traveltimes
• Specular point
• Surface normal at image point
208
Summarize
• Two reflectivities
• Peak values
• Peak ratio
209
24
Accuracy 2D, 2.5D
• Peak values: 2%
• Peak ratio: .2%
210
Accuracy 3D
• Peak values: 4%
• Peak ratio: ?%
211
25
Kirchhoff inversion
• Start from forward model
– Volume integral, single reflector
• Locally, Fourier transform
– Formally invert
• Reasonable accuracy on tests
212
26
Kirchhoff inversion with full
wave form Green’s functions
To build a better image…
Use a better Green’s function!
(Better than ray theory.)
213
Forward model
•
Kirchhoff approximation
(Better Green’s function)
214
1
Inversion
•
Kirchhoff approximation
(Better Green’s function)
215
Reflection coefficient
Plane waves
Planar reflectors
Homogeneous media
2
Reflection coefficient,
High frequency asymptotics
Curved wave fronts
(but not too … .)
Curve reflectors
(but not too … .)
Heterogeneous media
(but not too … .)
(but not too … .)
• f Frequency in hz
• v “Average” wave speed
• L Length scale
218
3
(but not too … .)
L
Length scale:
– Radius of curvature of wave front
– Radius of curvature of reflector
–
219
Return to exact forward model
220
4
Return to exact forward model
Kirchhoff approximation
221
Kirchhoff approximation
222
5
Full wave form Kirchhoffapproximate modeling
223
..with some more asymptotics
224
6
…as a volume integral
225
Singular function of a
surface:
7
…as a volume integral
227
…as a volume integral
228
8
…as a volume integral
• Gaussian beams
• “True amplitude” one-way
• Full wave equation
229
…as a volume integral
230
9
…as a volume integral
231
…as a volume integral
232
10
Solve via pseudo-inverse
adjoint to operator
233
Operates on functions
of and produces
functions of
Operates on functions
of
and produces
functions of
234
11
How does it work?
235
Operates on functions
of
and produces
functions of
236
12
Note: no bandlimiting in
Replace
237
Recall
238
13
“True amplitue” principles
• Asymptotic concept
• Requires single arrivals
• Only need
asymptotically
• Ray theory, stationary phase
239
Asymptotic
Earlier inversion
240
14
Inversion for reflectivity
241
Compare to earlier inversion
Same “filter”
Same weight
New!
242
15
Inversion for reflectivity
243
Note!
New result becomes old result
244
16
Summary
• Start from exact Kirchhoff formula with
exact or “better quality asymptotic
Green’s functions
• Apply asymptotics to obtain volume
integral representation
• Derive a formal pseudo-inverse for
reflectivity
245
17
Think Image Point Coordinates
Process in Source/Receiver
Coordinates
Norman Bleistein
Yu Zhang
Sheng Xu
Guanquan Zhang
Samuel H. Gray
246
Why?
• Image point inversion for
simpler Beylkin determinant
• Acquisition surface for easier
trace acquisition
• More stable processing
247
1
Why image point coordinates?
248
Kirchhoff Inversion
deconvolution structure
(including image point Kirchhoff M/I)
Image Point Kirchhoff M/I,
upper surface processing
correlation structure
249
2
Kirchhoff Inversion
deconvolution structure
(including image point inversion)
Image Point Kirchhoff M/I,
upper surface processing
correlation structure
250
Kirchhoff Inversion
deconvolution structure
(including image point inversion
Image Point Kirchhoff inversion,
upper surface processing
correlation structure
251
3
⇒
Reflection data
Fourier domain information
about medium
252

k = ω grad traveltime
253
4

  
k = ω∇τ ( x, xs , xr )
254
Micro-local:
changes with
source point
255
5
Micro-local:
changes with
receiver point
256
Micro-local:
changes with
image point
257
6

2cosθ
k =ω
 ν̂
v( x)
258

2cosθ
k =ω
 ν̂
v( x)
259
7
Integrate in Fourier Variables
{
3
⎡ 2cosθ ⎤ ∂ν̂ ∂ν̂
dk1dk2 dk3 = ω ⎢  ⎥
×
dω dν1dν 2
v(
x)
∂ν
∂ν
⎣
⎦
1
2
2
260
Integrate in Fourier Variables
{
3
⎡ 2cosθ ⎤ ∂ν̂ ∂ν̂
dk1dk2 dk3 = ω 2 ⎢  ⎥
×
dω dν1dν 2
⎣ v( x) ⎦ ∂ν1 ∂ν 2
Beylkin determinant
261
8
Integrate in Surface Variables
(Midpoints, Receivers)
3
⎡ 2cosθ ⎤ ∂ν̂ ∂ν̂
dk1dk2 dk3 = ω ⎢  ⎥
×
dω dξ1dξ2
v(
x)
∂ξ
∂ξ
⎣
⎦
1
2
2
262
Integrate in Surface Variables
(Midpoints, Receivers)
3
⎡ 2cosθ ⎤ ∂ν̂ ∂ν̂
dk1dk2 dk3 = ω 2 ⎢  ⎥
×
dω dξ1dξ2
⎣ v( x) ⎦ ∂ξ1 ∂ξ2
Beylkin determinant
263
9
Regular surface patch,
irregular sphere patch
264
Regular sphere patch,
irregular surface patch
265
10
3
3
⎡ 2cosθ ⎤ ∂ν̂ ∂ν̂
⎡ 2cosθ ⎤
×
=

 ⎥ sin ν1
⎢ v( x) ⎥ ∂ν ∂ν
⎢ v( x)
⎣
⎦
⎣
⎦
1
2
266
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
267
11
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
A’s: WKBJ Green’s function
amplitudes
268
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
1
 
  
iω u( xr , xs , ω )
D( x, xr , xs ) =
∫
2π
  
• exp{−iωτ ( x, xr , xs ) + iκ sgn(ω )π / 2}dω
269
12
1
 
  
iω u( xr , xs , ω )
D( x, xr , xs ) =
∫
2π
  
• exp{−iωτ ( x, xr , xs ) + iκ sgn(ω )π / 2}dω
•
 
u( xs , xr , ω )
: Data
270
1
 
  
i
ω
u(
x
D( x, xr , xs ) =
r , xs , ω )
2π ∫
  
• exp{−iωτ ( x, xr , xs ) + iκ sgn(ω )π / 2}dω
•
 
u( xs , xr , ω )
: Data
•
  
κ ( x, xs , xr )
: KMAH index
271
13
1
 
  
iω u( xr , xs , ω )
D( x, xr , xs ) =
∫
2π
  
• exp{−iωτ ( x, xr , xs ) + iκ sgn(ω )π / 2}dω
•
 
u( xs , xr , ω )
: Data
•
  
κ ( x, xs , xr )
: KMAH index
•
  
τ ( x, xs , xr )
: Travel time
272
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
Deconvolution processing
273
14
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
Sorts in common angle gathers
274
Kirchhoff Inversion in
Image Point Coordinates
1 2 cosθ

R( x,θ , φ ) =

4π 2 v( x)
  
D( x, xr , xs )
∫ A( x, xs )A( x, xr ) sin ν1dν1dν 2
xs , xr determined by ray tracing
275
15
Change variables to
upper surface coordinates
276
ν ,ν ,two variables,
1  2
xs , xr , four variables
277
16
Trick:
θ = ∫ δ (θ '− θ )dθ '
and
φ = ∫ δ (φ '− φ )dφ '
‘ ‘
‘
278
Kirchhoff Inversion in
Image Point Coordinates
∫ R( x, θ ,φ ) =
1
4π 2
  
2 cosθ ' D( x, xr , xs )

 
 
∫ v( x)
A( x, xs )A( x, xr )
• δ (θ '− θ )δ (φ '− φ )sin ν1dν1dν 2 dθ 'dφ '
279
17
Change variables of integration
ν1 , ν 2 , θ ', φ ' ⇒ α s1 , α s 2 , α r1 , α r 2
‘ ‘
‘
280
Change variables of integration
α s1 , α s 2 ⇒ xs1 , xs 2
α r1 , α r 2 ⇒ xr1 , xr 2
‘ ‘
‘
281
18
Change variables of integration
α s1 , α s 2 ⇒ xs1 , xs 2
α r1 , α r 2 ⇒ xr1 , xr 2
Ray Jacobians!
282
Change variables of integration
α s1 , α s 2 ⇒ xs1 , xs 2
α r1 , α r 2 ⇒ xr1 , xr 2
Ray Jacobians!
Green’s function amplitudes!
283
19
Kirchhoff Inversion in
Source/Receiver Coordinates

 cos β cos β
  
R( x,θ , φ ) = 32π 2 v( x) ∫  s  r D( x, xs , xr )
v( xs ) v( xr )
 
 
• A* ( x, xs )A* ( x, xr )δ (θ '− θ )δ ( sin θ (φ '− φ ))
• dxs1dxs 2 dxr1dxr 2
284
Kirchhoff Inversion in
Source/Receiver Coordinates

 cos β cos β
  
R( x,θ , φ ) = 32π 2 v( x) ∫  s  r D( x, xs , xr )
v( xs ) v( xr )
 
 
• A* ( x, xs )A* ( x, xr )δ (θ '− θ )δ ( sin θ (φ '− φ ))
• dxs1dxs 2 dxr1dxr 2
Correlation processing!
285
20
Kirchhoff Inversion in
Source/Receiver Coordinates

 cos β cos β
  
R( x,θ , φ ) = 32π 2 v( x) ∫  s  r D( x, xs , xr )
v( xs ) v( xr )
 
 
• A* ( x, xs )A* ( x, xr )δ (θ '− θ )δ ( sin θ (φ '− φ ))
• dxs1dxs 2 dxr1dxr 2
Multi-streamer data acquisition
286
Full Wave Form
Kirchhoff Migration/Inversion

 cos β cos β
 
R( x,θ , φ ) = 16π v( x) ∫  s  r iω u R ( xs , xr ,ω )dω
v( xs ) v( xr )
 
 
• G * ( xs , x,ω )G * ( xr , x,ω )δ (θ '− θ )
• δ ( sin θ (φ '− φ )) dxs1dxs 2 dxr1dxr 2
287
21
Full wave form - true amplitude
• Reflection coefficient :
plane wave, WKBJ concept
288
Full wave form - true amplitude
• Reflection coefficient :
plane wave, WKBJ concept
• G* : ~ WKBJ Green’s function
289
22
Full wave form - true amplitude
• Reflection coefficient :
plane wave, WKBJ concept
• G* : ~ WKBJ Green’s function
• Full wave form Kirchhoff M/I:
true amplitude processing
290
Image Point Kirchhoff M/I
• Single-valued map to
sources/receivers
291
23
Image Point Kirchhoff M/I
• Single-valued map to
sources/receivers
• Simple transformation to surface
coordinates
292
Image Point Kirchhoff M/I
• Single-valued map to
sources/receivers
• Simple transformation to surface
coordinates
• Deconvolution processing ⇒
Correlation processing
293
24
Image Point Kirchhoff M/I
• Single-valued map to
sources/receivers
• Simple transformation to surface
coordinates
• Deconvolution processing ⇒
Correlation processing
• Relatively simple formulas for
computation
294
Image Point Kirchhoff M/I
• Output in common angle gathers
295
25
Image Point Kirchhoff M/I
• Output in common angle gathers
• Ideal for multi-streamer survey data
296
Image Point Kirchhoff M/I
• Output in common angle gathers
• Ideal for multi-streamer survey data
• Full wave form Kirchhoff inversion
297
26
Two-and-one-half
Dimensions(?!)
2.5D
A concept whose importance
has been overtaken by
computer capacity
298
Two-and-one-half
Dimensions(?!)
Single line processing
3D Green’s functions over a
“2D” earth
(Hence, 2.5D)
299
1
2.5D acquisition
300
Two-and-one-half
Dimensions(?!)
• Single line of data
• Best acquired in the dip
direction
301
2
Two-and-one-half
Dimensions(?!)
• Physics: “nearly” no variation
in out-of-(x,z)-plane direction
• Mathematics: no variation in
out-of -(x,z)-plane direction
– Velocity independent of y (or x2)
302
303
3
Two-and-one-half
Dimensions(?!)
Operator depends on
out-of- plane variable
Data does not depend on
out-of- plane variable
Precompute out-of-plane
integral asymptotically
304
Recall
Frequency domain integral
isolated
305
4
Equivalent
306
Data independent of ξ2
307
5
Data independent of ξ2
308
Velocity independent of y
(or x2)
309
6
Apply stationary phase in ξ2
Output independent of x2
310
Apply stationary phase in ξ2
Propagation is in the vertical
(x1,x3) plane containing x2
311
7
Only need in-plane values of
312
Need: in-plane 3D ray theory
313
8
Recall: ray equations with
parameter σ
314
Ray equations with
parameter σ
315
9
Ray equations with
parameter σ
316
Ray equations with
parameter σ
317
10
Ray equations with
parameter σ
318
Ray equations with
parameter σ
• Rays are in-plane
2D propagation,
3D geometrical spreading
319
11
Ray equations now 2D with
parameter σ
320
Recall: amplitude, 3D
321
12
2.5D in-plane amplitude
322
2.5D in-plane amplitude
out-of-plane spreading
in-plane spreading
323
13
2.5D in-plane amplitude
324
2.5D in-plane amplitude
325
14
Beylkin determinant
326
Beylkin determinant, 2.5D
327
15
Apply stationary phase in ξ2
328
2.5D inversion formula
329
16
The data, D2D
330
2.5D Summary
• Single line of data in dip direction
• Assume v = v(x,z)
• 3D inversion with identical lines
of data (independent of y)
• Apply stationary phase in
out-of-plane direction
• 2.5D inversion formula
331
17