Migration Deconvolution

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Overview of Seismic Imaging

• Jerry Schuster
• Geology Geophysics Dept.
• Univ. of Utah

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Outline

• Forward Acoustic Problem

• Inverse Acoustic Problem

• Seismic Experiment

• Seismic Resolution

4
ZO Seismic Section
Depth
Time
5

• Monterey Optical Image

0 km 0.1 km
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Acoustic Forward Problem
Depth
7
Goal Compute ZO Seismic Section
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Acoustic Forward Problem
Depth
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Normal Moveout Correction
Depth
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Stack
Depth
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Stacked Seismic Section
Depth
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Outline

• Forward Acoustic Problem

• Inverse Acoustic Problem

• Seismic Experiment

• Seismic Resolution

13
What is the Problem?
Events Can Originate Updip
V/2
DVT/2
Depth
Time
14
What is another Problem?
Events Originate Pt. Diffractors
Depth
Time
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0 km 30 m
0 m 30 m
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ZO Data
0 km 3 km
0 km 7 km
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Seismic Inverse Problem
Given d Lo
Find o(x,y,z)
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Review

• Inverse Acoustic Problem

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Outline

• Forward Acoustic Problem

• Inverse Acoustic Problem

• Seismic Experiment

• Seismic Resolution

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.1 1 10 100
1000
l
(km)
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Seismic Data Acquisition Parameters
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Judge Seismic Section
0 m 10 m
0 m
100 m
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N-S Vertical Section Tomograms
1.5 km/s 0 km/s
0 m 3 m
0 m 3m
1.5 km/s 0 km/s
0 m 3 m
1.5 km/s 0 km/s
0 m
12 m
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.1 1 10 100
1000
l
(km)
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Marine Experiment
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.1 1 10 100
1000
l
(km)
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1Hz Global Tomography
12000
0
(km)
6000
0
Mantle
(km/s)
13.72
10.29
6000
Core
6.858
3.429
0
12000
(km)
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Alaska
UU Strong Motion Network
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Utah Stations
0
Distance (km)
0 s 1.6 s
150
0 50 km
0 km
200 km
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Alaska Earthquake
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0 km
700 km
.1 1 10 100
1000
l
(km)
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Summary
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NutsBolts of Migration

• Jerry Schuster
• Geology Geophysics Dept.
• Univ. of Utah

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Outline

• Born Modeling

• Prestack Migration

• Least Squares Migration

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Born Modeling
x
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Born Modeling
d(xx)
o(x)
x
x
x
Scatterer
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Time-Domain Diffraction Stack Modeling
?
?
?
?
-
-
i
i
xx

e
xx
e
d(x)
o(x)
A(x,x)
A(x,x)


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Born Modeling
..
d(x,t)
d( t-? - ? )

o(x)
xx
xx
A(x,x)
A(x,x)
?/c
T
(x ,0)
(x ,0)
s
g
(x,z)
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Born Modeling
(x ,0)
(x ,0)
s
g
(x,z)
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Born Modeling
..
d(x,t)
d( t-? - ? )

o(x)
xx
xx
A(x,x)
A(x,x)
T
(x ,0)
(x ,0)
s
g
(x,z)
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Outline

• Born Modeling

• Prestack Migration

• Resolution

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Fresnel Zone
z
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ZO Migration Smear Reflections along Fat Circles
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ZO Migration Smear Reflections along Fat Circles
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ZO Migration Smear Reflections along Circles
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ZO Migration Resolution Intersection of Fresnel
Zones
Vertical Res. Near-Offset Traces
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ZO Migration Resolution Intersection of Fresnel
Zones
Horiz. Res. Far-Offset Traces
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Why is Pt. Scatterer Response of Migration a
Blurred Version of Point?
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0 km 3 km
0 km 7 km
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Conclusions

• LSM

• Far-Offset Traces Better Hor. Res.

• Near-Offset Traces Better Vert. Res.

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Forward Modeling

e
W(? )
d(x)
A(x,x)
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Finite-Difference Stencil
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Outline

• Born Modeling

• Prestack Migration

• Least Squares Migration

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Born Modeling
x
x
x
Scatterer
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115.
Diffraction Stack Migration Prestack
-
-
?
?
?
?
i
i

xx

e
xx
e
m(x)
d(x)
A(x,x)
A(x,x)
..
Narrow band case direct wave correlated with data
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116.
Prestack Migration Smears Reflections along Fat
Ellipses
?/c
T
x
g
(x,z)
Doughnut describes support of ellipse described
by with ? between T and T
?/c
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SUMMARY
117.
1. Prestack DS Migration Formula
m(x)
for just one trace! N shot gathers with N traces
per shot require N2 such sumnmations
2. High Frequency Approximation (i.e
c(x) variations gt 3? )
3. Approximates Kinematics of Data, but not
Dynamics
4. Prestack N times slower than Poststack,
no 1-D assumption, velocity sensitive, better
dynamic range. Here N shot gathers
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118.
Diffraction Stack Migration Prestack
-
-
?
?
?
?
i
i

xx

e
xx
e
d(x)
d(x)
A(x,x)
A(x,x)
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Ray vs Wave Traveltimes
( )
d
dr
t (s,g)
D
s
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Ray vs Wave Traveltimes
( )
d
dr
t (s,g)
D
s
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Migration Deconvolution

• FFT in x and y

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Outline
1. Seismic Lens
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Why does a Telescope Lens act as a Migration
Operator?
Lo
o
o
Lo
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Why Similarity?
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Seismic Section
Depth
Time
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Outline
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Seismic Section
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What is the Solution?
Smear Events along Circles.
V/2
Depth
Time
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Seismic Problem
Given d Lo
Find o(x,y,z)
Lo
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Smear Events along Circles.
V/2
Depth
Time
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Seismic Experiment
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Summary
Exploding Reflector d L o
Relocates events to correct location
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Outline
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Why Similarity?
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Why does a Telescope Lens act as a Migration
Operator?
Lo
o
o
Lo
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Outline
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Hubble Telescope Large Magellanic Cloud
o
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Broken Lens

• Sparse Coverage

• Limited Aperture

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Seismic Telescope N. Sea
o
but dL o
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Numerical Tests

• Meandering River Model

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Recording Geometry
5 X 5 Sources 21 X 21 Receivers
Wavelet frequency 50 Hz
(0, 1 km)
(0, 0)
(1 km,1 km)
(1 km, 0)
A river channel
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Depth Slices of Point Scatterers

• 0

• 0

• MD Image

• Kirch. Migration Image

• Y(km)

• Y(km)

• 1

• 1

• 0

• 0

• 1

• 1

• X(km)

• X(km)

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Meandering River Model
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Kirchhoff Migration Image
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MD Image
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Numerical Tests

• 2-D SEG/EAGE overthrust model

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Velocity Model
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Comparison of KM and MD

• 0 km

• 15 km

KM

• Poststack Migration Image

15 km
0 km
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Comparison of KM and MD
15 km
0 km
KM

• Poststack Migration Image of Half Sampled Data

• 0 km

• 15 km

0 km
MD
4 km

• Deconvolved Migration Image

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Numerical Tests

• 2-D SEG/EAGE overthrust model
• 2-D Mobil marine data from the North Sea

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Velocity Model
2500
Velocity (m/s)
1500
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Time Migration Image
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Migration Deconvolution Image
X (km)
0
25
100
Migration Deconvolution Image
X (km)
0
25
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Numerical Tests

• 2-D SEG/EAGE overthrust model
• 2-D Mobil marine data from the North Sea
• 3-D SEG/EAGE Salt Model

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Inline Velocity Model
Offset (km)
0
9.2
0
Depth (km)
3.8
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Comparison of Migration and MD Image
Y (km)
Y (km)
4
6
8
4
6
8
0
0
1
1
Depth (km)
Depth (km)
2
2
3
3
4
4
Migration Crossline Section
MD Crossline Section
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Depth (km)
KM Crossline (X,97) Section
MD Crossline (X,97) Section
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Numerical Tests

• 2-D SEG/EAGE overthrust model
• 2-D Mobil marine data from the North Sea
• 3-D SEG/EAGE Salt Model
• 3-D North Sea Data

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Comparison of Prestack Migration and MD Images

• X (km)

• Prestack Kirchhoff Migration Image of
• a North Sea Data Set

• X (km)

• MD Image

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Comparison of Poststack MD Depth Slices

• Kirchhoff Image

• MD Image

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Conclusions
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Outline
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Fourier Optics
d(x,0 )
Image is Fourier Transform of Aperture Lens
m
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Image is Fourier Transform of Lens
Aperture?
1. Rapid Lens Design
Point Source
Point Spread Function
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Test 1 Zero-offset Survey
Lx 1667 m
dy80 m
Ly 1667 m
dx80 m
point scatterer z0 6667 m
33 m
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Diffraction Stack Image
5 CPU Hours
1
Migration Magnitude
0
2667
2667
Y (m)
X (m)
0
0
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Seismic Array Theorem Image
6 CPU Seconds
1
Migration Magnitude
0
2667
2667
Y (m)
X (m)
0
0
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ARCOs Land Surveys
Source Geometry
Receiver Geometry
2333
I
0
0
2333
II
III
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Survey Parameters
2333 X 2333 m Rectangular Area
2
dxs dys dxg dyg of
Traces I 13 m 80 m 80 m 13 m
27878400 II 13 m 40 m 40 m 13 m
29160000 III 7 m 40 m 40 m 7 m
29160000
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0
I
2333
2333
2333
Y (m)
0
0
II
II
2333
2333
2333
Y (m)
X (m)
0
0
III
III
Y (m)
2333
2333
2333
Y (m)
X (m)
0
X (m)
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Meandering River Channel Model
Depth 5000 m
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Migration Image
After 180 Iterations Alias Energy 0.00052
Final Survey Alias Energy 0.00051
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Conclusions
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Test 2 Source-dependent Receiver Survey
Lsx
Lgx
dys
dgy
Lgy
Lsy
dgx
dxs
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KM Image with Initial Velocity
0
18 km
0
Depth (km)
1.5
KMVA Velocity Changes in the 1st Iteration
0
50
Depth (km)
(m /s)
0
1.5
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Why does a Telescope Lens act as a Migration
Operator?
Star
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SEISMIC ARRAY THEOREM AND OPTIMAL SURVEY DESIGN
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Square Root Optics
h(x',y')
126

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Square Root Optics

h(x',y')
Image is Fourier Transform of Aperture Lens
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Seismic Array Theorem

ò
ò
b(x',y')
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3D Prestack Point Migration Scatterer
Response

i(k x' k y' k x' k y')
ò
ò
e
x
y
x
y
b(x',y',x',y')
y
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4-D Seismic Array Theorem
Source Grid
Geophone Grid


Analytical
Analytical
SINCg(X ) SINCg(Y )
SINCg(X ) SINCg(Y )
g
s
s
g
Migrated Image
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Outline

• Motivation
• Seismic Array Theorem
• Numerical Verification
• Survey Design Trial and Error
• Survey Design Optimization
• Conclusions

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D.S. Vs. SAT. Image
1
Diffraction Stack Image
Seismic Array Theorem Image
Migration Magnitude
0
X (m)
0
3333
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SAT Image Freq. 100 Hz
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Alias Energy in Migration Sections
Survey Alias Energy I 971 II
192 III 735
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Trial and Error W. Texas Survey
unit meter
dxs dys dxg dyg trace
Regular 1/4 Sampling Uniform 1/4
Sampling
440 110 73 440 1,298,517
440 110 293 440 333,564
193 193 193 193 1,309,499
277 277 277 277 317,057
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Focusing Incident Light on an Image Plane
x
137
m(r)
x
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Point Scatterer Response
Regular Survey Alias Energy 85
3333
Y (m)
0
X (m)
0
3333
Uniform Survey Alias Energy 454
3333
Y (m)
0
3333
0
X (m)
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Fourier Array Theorem

ò
ò

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Square Root Optics

Image is Fourier Transform of Aperture Lens
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Migration Noise Problems

• Aliasing
• Recording Footprint

• Limited Resolution
• Amplitude Distortion

• 0 km

• 15 km

0
Footprint
Amplitude distortion
Time (s)
2
Migration noise and artifacts
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Hubble Telescope Large Magellanic Cloud
r
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Solution Deconvolve the point scatterer response
from the migrated image
Reason
Migrated Section
Data
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Migration Deconvolution

• FFT in x and y

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Seismic Section
Depth Resolution
Horiz. Resolution gt
Depth
Time
DepthvelT
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Outline

• Reflection Imaging Principles

• Case History 3D Seismic Potash

• Case History 2D Tomography

• Case History Crosswell

• Summary

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Potash Geology(Pruegger Nemeth)

• Sakatchewn Province 12 km/12 km Potash mine 1
  km depth

• Geology

Potash
Salt
Karst
Limestone
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Potash Geology

• Sakatchewn Province 12 km/12 km Potash mine 1
  km depth

• Geology

Potash
Salt
Karst
Limestone
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What is the Problem?
Events Originate Updip
V/2
Depth
Time
150
Outline

• Reflection Imaging Principles

• Case History 3D Seismic Potash

• Case History 2D Tomography

• Case History Crosswell

• Summary

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2D vs 3D
Top View
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Seismic Problem
Given d Lo
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Seismic Problem
Given d Lo
154
Seismic Problem
Given d Lo
Find o(x,y,z)
2
Soln min Lo-d
PSF decon
Lo
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Outline
1. Seismic Lens
156
Why does a Telescope Lens act as a Migration
Operator?
r
Star
157
Light Incident on an Aperture
Lo d
Integral Equation
x
x
158
Lens Diverg. Waves Conv. Waves
ò
ò
g(rr)
g(rr)
159
Lens Diverg. Waves Conv. Waves
160
Seismic Problem
Given d Lo
161
l
Wave Theories vs
/D
Ray Theory
Greens Thm
Mie Theory
D
l
162
What is the Problem?
V/2
DVT/2
Depth
Time
163
What is the Problem?
Events Can Originate Updip
V/2
DVT/2
Depth
Time
164
l
Wave Theories vs
/D
Ray Theory
Greens Thm
Mie Theory
D
l
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Seismic Problem
Given d Lo
Find o(x,y,z)
2
Soln min Lo-d
PSF decon
Lo
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2-D Seismic Survey
30 m
6 km
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3-D Seismic Survey
30 m
6 km
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.1 1 10 100
1000
l
(km)
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Recall Huygens Principle Every Pt. On Wavefront
2nd Source
ZO Seismic Section
170
Recall Greens Function Pt. Impulse response of
medium
Depth
Time
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Born Approximation

172
SUMMARY
22.
1. Exploding Reflector Modeling Diffraction
Stack Modeling
R(x)
A(x,x)
A(x,x)
Sum over reflectors
2. High Frequency Approximation (i.e c(x)
variations gt 3? )
3. Approximates Kinematics of Data, but not
Dynamics
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Problem We dont know c(x,y,z), only a smooth
estimate.
174
Depth
Time
175
Recall Huygens Principle Every Pt. On Wavefront
2nd Source
Depth
Time
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Exploding Reflector ½ Velocity
V/2
Depth
Time
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Born Forward Modeling

d(x)
g(xx)
178
Review

• Forward Acoustic Problem

179
Acoustic ZO Migration
Depth
180
Forward Modeling (d Lo)

d(x)
g(xx)
o(x)
..
181
ZO Migration Smear Reflections along Fat Circles
Where did reflections come from?
182
ZO Migration Smear Reflections along Fat Circles
183
ZO Migration Smear Reflections along Circles