Seismic imaging using an inverse scattering

algorithmMontclair 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 - MRIMedical 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 - MRIMedical 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 breath-holding task.

Medical Imaging – X-rays Medical Imaging – X-rays imagingimaging

• High energy electromagnetic radiation (X-rays) 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 Medical Imaging – X-rays imagingimaging

Medical Imaging – CAT scanMedical 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 three-dimensional 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 scanMedical Imaging – CAT scan

Medical Imaging - UltrasoundMedical 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 - UltrasoundMedical 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

Investigation of the dynamics of the dune field in far southern Utah

GPR – other structures

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

Loch Ness Monster art installation in Death Valley National Park, CA, USA.

Ice fishing

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

• 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.

Long range radar antenna

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

• Acoustic wave propagating: complex waves arrivals even for simple geometries

Seismic exploration

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

VLL

GL

LG

0

00

VGGGG 00

Lippman-Schwinger Eq.

• L-S equation relates differences in media to

differences in wavefield

Scattering Theory (cont’d.)

Inverse Series, V as power series in data

01020020100101010030

01010020

0100 )(

GVGVGGVGVGGVGVGVGGVG

GVGVGGVG

GVGGG m

000000 VGVGGVGGGG

(2)

(1)

Substitute (2) into (1) and evaluate on the measurement surface, m

321 VVVV

(1) Remove free-surface multiples

(2) Remove internal multiples

(3) Image primaries to correct spatial location

(4) Invert for local earth properties

Inversion as a series of tasks and subseries

Goal: find an algorithm (subseries) which performs task 3 and 4 simultaneously.

1D problem

)(20 zkV

00 ck

)(1)(

2

20

zc

cz

)()()()( 321 zzzz Inverse series

0

||

210 2);|(

120

ik

ezzG

zzik

1D problem (Contd.)

Calculate:

z

dzzDz ')'(4)(1

)(')'()('

2

1)( 2

1

0

112 zdzzzzz

z zz

zz

dzdzzzzzzdzzz

dzzzdzzzzzz

''')'''()''(')'('16

1')'()('

8

1

')'()(''8

1')'()(')(

4

3)(

16

3)(

111211

2

11111313

The algorithm

)(

11 ')'()(!

2/1)(

nnznSIIn dzzz

nz

Select the following terms from the full series:

Subseries:

0

)(

11 ')'()(!

2/1)(

n

nnznSII dzzz

nz

Closed form:

0

'')''(2

1'

1 ')'()(

'

100 dkdzezez

zdzzzik

zikSII

Numerical examples

)()()( 2211 ttRttRtD

01

011 cc

ccR

12

122 cc

ccR

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

• 3 interfaces• z = 100 130 160• c= 1500 1650 1725 1800• z = 100 130 160

First model: band limited data

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 multi-dimension