Download Inverse problem - Montclair State University

Seismic imaging using an
inverse scattering
algorithm
Montclair State University
Chapter of SIAM
Bogdan G. Nita
Dept. of Mathematical Sciences
Montclair State University
March 24, 2010
Contents
• Describe the diversity of physical sciences
applications for inverse problems
• Describe the inverse scattering approach to
imaging and inversion of seismic data
• Describe an imaging algorithm and recent
results
Acknowledgements
• This work is in collaboration with Ashley
Ciesla and Gina-Louise Tansey
Direct (forward) and Inverse
Problems
• Direct problem: given the information about
a medium, describe the propagation of a
wave (acoustic, elastic, EM etc), path of an
object etc. in that medium (find the answer
given some hypothesis)
• Inverse problem: given measurements of
amplitude (e.g. velocity etc) and phase
(arrival time) for the wave (or the object)
determine the properties of the medium
(given the answer, determine the hypothesis
of the problem)
Examples of FP - Pre-calculus
• Drop a stone into a well. Given the
depth of the well, how long it will take
the stone to hit the water?
Examples of IP - Pre-calculus
• Drop a stone into a well, and measure
the time when you hear the splash.
How deep is the well?
Inverse Problems
"Can you hear the shape of a drum?"
Marc Kac, 1966
Examples of IP solvers
• Our brain solves inverse problems all
the time: which direction to go, where
are surrounding objects located (useful
in designing robots)
• Blind people use signals and noise to
guide themselves
• Whales, bats, dolphins use sounds for
guidance
Inverse problems in life sciences
• Medical imaging – magnetic resonance
imaging (MRI), x-rays imaging, computer
tomography (CAT scan), ultrasound.
• Ground penetrating radar (GPR):
engineering, archeology, mines detection.
• Underwater sonar (acoustics), submarine
sonar
• Military radar scattering
• Deep earth seismology, seismic exploration
Medical Imaging - MRI
• Magnetic Resonance Imaging (MRI) uses a property
of hydrogen atoms to visualize soft tissues in the
body. The nucleus of hydrogen spins like a wobbling
spinning top. In a strong magnetic field, the
'wobbles' line up. If a brief radio signal is sent
through the body, the atoms get knocked out of
alignment. As the atoms flip back, they emit radio
waves which are detected and analyzed by
computer. Different signal strengths represent
different tissues, depending on how much hydrogen
is in them as water or fats. The signals are combined
to form a 'slice' image through the body, and many
slices may be combined to give a 3D view.
Medical Imaging - MRI
Axial head
MRI images
The global response to
holding one's breath for
15 seconds. The entire
gray matter volume is
activated by the breathholding task.
Medical Imaging – X-rays
imaging
• High energy electromagnetic radiation (Xrays) passes through the human body
and is recorded on photographic film
placed behind the patient. The image that
appears is due to the different absorption
levels between soft tissue and bones.
Downsides of this procedure include poor
resolution of the soft tissue and possible
risks of radiation contamination.
Medical Imaging – X-rays
imaging
Medical Imaging – CAT scan
• CAT scans take the idea of conventional
X-ray imaging to a new level. Instead of
finding the outline of bones and organs,
a CAT scan machine forms a full threedimensional computer model of a
patient's insides. Doctors can even
examine the body one narrow slice at a
time to pinpoint specific areas.
Medical Imaging – CAT scan
Medical Imaging - Ultrasound
•
•
•
•
•
•
•
Ultrasound or ultrasonography is a medical imaging technique that
uses high frequency sound waves and their echoes. The technique is
similar to the echolocation used by bats, whales and dolphins, as well
as SONAR used by submarines. In ultrasound, the following events
happen:
The ultrasound machine transmits high-frequency (1 to 5 megahertz)
sound pulses into your body using a probe.
The sound waves travel into your body and hit a boundary between
tissues (e.g. between fluid and soft tissue, soft tissue and bone).
Some of the sound waves get reflected back to the probe, while some
travel on further until they reach another boundary and get reflected.
The reflected waves are picked up by the probe and relayed to the
machine.
The machine calculates the distance from the probe to the tissue or
organ (boundaries) using the speed of sound in tissue (5,005 ft/s
or1,540 m/s) and the time of the each echo's return (usually on the
order of millionths of a second).
The machine displays the distances and intensities of the echoes on
the screen, forming a two dimensional image.
Medical Imaging - Ultrasound
Ground penetrating Radar- GPR
•
Ground penetrating radar (GPR, sometimes called
ground probing radar, georadar, subsurface radar or
earth sounding radar) is a noninvasive electromagnetic
geophysical technique for subsurface exploration,
characterization and monitoring. It is widely used in
locating lost utilities, environmental site characterization
and monitoring, agriculture, archaeological and forensic
investigation, unexploded ordnance and land mine
detection, groundwater, pavement and infrastructure
characterization, mining, ice sounding, permafrost, void,
cave and tunnel detection, sinkholes, subsidence, karst,
and others). It may be deployed from the surface by
hand or vehicle, in boreholes, between boreholes, from
aircraft and from satellites. It has the highest resolution
of any geophysical method for imaging the subsurface,
with centimeter scale resolution sometimes possible.
GPR – engineering and construction
• Pipes and crack detection using GPR
GPR - archeology
Conducting a Ground Penetrating Radar (GPR) survey in
area of a suspected slave cemetery
GPR – mines detection
GPR – other structures
Investigation of the dynamics of the dune field in far southern Utah
Underwater sonar - acoustics
• Use high frequency sound waves to
locate objects in the water
Sonar – locating wrecks
Soviet submarine S7 on 40-45 m
depth off the Swedish east coast.
Hertha, sunk off the Swedish
coast in 1922, on 65 m depth.
Sonar – fishing
Ice fishing
Loch Ness Monster art installation in
Death Valley National Park, CA, USA.
Sonar – submarine
• To locate a target, a submarine uses active and passive
SONAR (sound navigation and ranging). Active sonar
emits pulses of sound waves that travel through the
water, reflect off the target and return to the ship. By
knowing the speed of sound in water and the time for the
sound wave to travel to the target and back, the
computers can quickly calculate distance between the
submarine and the target. Whales, dolphins and bats
use the same technique for locating prey (echolocation).
Passive sonar involves listening to sounds generated
by the target. Sonar systems can also be used to realign
inertial navigation systems by identifying known ocean
floor features .
Sonar – submarine
Sonar station onboard the USS La Jolla
nuclear-powered attack submarine
Digital Art of a Submarine Using
Sonar For Location
Military radar scattering
Long range radar antenna
• RADAR is a system used to detect, range (determine the distance
of), and map objects such as aircraft, ships, and rain, that was first
suggested as a "ship finder" by Dr. Allen B. DuMont in 1932. Coined
in 1941 as an acronym for Radio Detection and Ranging, it has
since entered the English language as a standard word, losing the
capitalization in the process.
Military radar scattering
Radar Image of a Fighter aircraft
The B-2 Spirit bomber uses Stealth
technology to avoid radar detection
Deep earth seismology
• Science which studies data collected
from earthquakes to determine the
source of the earthquake (location),
and structures which the waves have
interacted with before being recorded
(inner core, mantle etc)
Deep earth seismology
Raypaths for p and s waves in a typical earthquake
Deep earth seismology
Simulated earthquake and global wavefield propagation throughout Earth.
Seismic exploration
• Earth’s shallow subsurface investigation for
finding natural resources (hydrocarbon, natural
gas, coal etc)
Marine experiments: air guns
Seismic exploration
• Acoustic wave propagating: complex waves arrivals
even for simple geometries
Typical seismic data
Components of the data
•
•
•
•
Direct arrival
Free surface multiples
Internal multiples
Primary reflections
Data after FS multiples removal
Typical seismic data
Forward and Inverse
Scattering Algorithms
What is scattering theory?
• Scattering theory is a form of perturbation theory
LG  
L0G0  
L0  L  V
G  G0  G0VG
Lippman-Schwinger Eq.
• L-S equation relates differences in media to
differences in wavefield
Scattering Theory (cont’d.)
G  G0  G0VG0  G0VG0VG0  
(1)
Inverse Series, V as power series in data
V  V1  V2  V3  
(2)
Substitute (2) into (1) and evaluate on the
measurement surface, m
(G  G0 ) m  G0V1G0
G0V2G0  G0V1G0V1G0
G0V3G0  G0V1G0V1G0V1G0  G0V1G0V2G0  G0V2G0V1G0

Inversion as a series of tasks and
subseries
(1) Remove free-surface multiples
(2) Remove internal multiples
(3) Image primaries to correct spatial location
(4) Invert for local earth properties
Goal: find an algorithm (subseries) which
performs task 3 and 4 simultaneously.
1D problem
V  k  ( z)
2
0
k0 

c0
c02
 ( z)  1  2
c ( z)
eik0 |z2  z1|
G0 ( z1 | z2 ;  ) 
2ik 0
 ( z )  1 ( z )  2 ( z )  3 ( z )  
Inverse series
1D problem (Contd.)
Calculate:
z
1 ( z )  4  D( z' )dz'

z

1
2
 2 ( z )     '1 ( z )  1 ( z ' )dz '  1 ( z ) 
2
0



3 3
3
1

 3 ( z )  1 ( z )  1 ( z ) '1 ( z )  1 ( z ' )dz '   ' '1 ( z )  1 ( z ' )dz ' 
16
4
8

 

z
z
z z
z
1
1
  '1 ( z )  12 ( z ' )dz ' 
 '1 ( z ' ) '1 ( z ' ' )1 ( z ' ' z ' z )dz ' 'dz '


8
16 

2
The algorithm
Select the following terms from the full series:






1
/
2
SII
1 ( z )  1 ( z ' )dz ' 
n ( z) 


z
n

n!
 
n (n)


 
Subseries:
z
n 





1
/
2
SII
1 ( z )  1 ( z ' )dz ' 
 ( z)  



n 0
n!

 
n (n)


 
Closed form:


 SII ( z )   eik z  1 ( z' )e
0



1 z'

ik0  z '   1 ( z '') dz '' 
2 


dz' dk0
Numerical examples
R1 
c1  c0
c1  c0
c2  c1
R2 
c2  c1
D(t )  R1 (t  t1 )  R2 (t  t2 )
First model
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: data
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: first iteration
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: sii algorithm
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: all
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: band limited data
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: first iteration
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: sii algorithm
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
First model: all
• 3 interfaces
• z = 100 130 160
• c= 1500 1650 1725 1800
• z = 100 130 160
Second model
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: data
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: first iteration
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: sii algorithm
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: all
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: band limited data
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: first iteration
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: sii algorithm
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Second model: all
• 4 interfaces
• z = 100 130 160 200
• c= 1500 1650 1725 1575 1725
• z = 100 130 160 200
Conclusions
• We found a new algorithm which performs
simultaneous imaging and inversion
• Although found as a series, the algorithm
has a closed form
• Numerical examples
• Future research generalize to multidimension