1. Computational Time-reversal Imaging
A.J. Devaney
Department of Electrical and Computer Engineering
Northeastern University
Web:
Talk motivation: TechSat 21 and GPR imaging of buried targets
Talk Outline
Overview
Review of existing work
New simulations
Reformulation
Future work and concluding remarks
February 23, 2000
A.J. Devaney--BU presentation

2. Experimental Time-reversal
Goal is to focus maximum amount of energy on target
for purposes of target detection and location estimation
In time-reversal imaging a sequence of illuminations is used such that each
incident wave is the time-reversed replica of the previous measured return
First illumination
Intermediate illumination
Final illumination
Intervening medium
Intervening medium
Intervening medium
Time-reversal imaging is a standard method used in ultrasound medical and industrial imaging. It is customarily performed experimentally where a target is illuminated (insonified) in a first experiment with a plane wave (all transducers pulsed in phase) and then the target is re-illuminated with the transducers now pulsed using the time-reversed measured waves, etc. The net result after such a sequence of experiments is a set of waveforms, one for each transducer element, such that if these waveforms are used to excite the transducers in the array the transmitted wave will focus on the target—irrespective of the properties of the intervening medium between transducer array and target.
Although the name “time-reversal” connotes the use of broad band wave fields (short time pulses) this is not necessary and the method works equally well with narrow band wave fields. For narrow band illumination the time-reversal process is identical to phase conjugation.
Time-reversal imaging can be performed computationally without actually performing the sequence of experiments outlined above. To do this the multi-static response matrix K must be measured. This quantity is an NXN matrix, where N is the number of antenna (or transducer) elements. From this quantity the time-reversal matrix T=K*K (* standing for complex conjugate) is computed and each eigenvalue, eigenvector pair correspond to a different target. The eigenvalues give the scattering strength of the targets and the eigenvectors give the required exciting waveforms to focus on the associated target.
Without time-reversal compensation
With time-reversal compensation
February 23, 2000
A.J. Devaney--BU presentation

3. Computational Time-reversal
Time-reversal compensation can be performed without actually
performing a sequence of target illuminations
Multi-static data
Time-reversal processor
Computes measured returns that would
have been received after time-reversal compensation
Target detection
Target location estimation
Multi-static data: Measured returns at each antenna element for pulse transmission from each element. Normally obtained by sequentially pulsing each antenna and measuring returns at all other elements. In TechSat 21 it is proposed to measure the multi-static data array K=Ki,j by simultaneously pulsing all elements using orthogonal coded waveforms or similar means. It is important to note that time-reversal processing does not require broad band waveforms and works equally well with narrow band or broad band signals.
Time-reversal processor: Computes eigenvalues and eigenvectors of time reversal matrix which is equal to K*K. Eigenvalues are used for target detection and eigenvectors for target location estimation and imaging.
Doppler: Used to isolate return from moving ground or air targets.
Time reversal processing requires only the measured multi-static array K and, thus, needs no apriori knowledge of propagation medium (e.g., ionosphere), array geometry, etc.
Target detection also requires no knowledge of propagation medium or array geometry, etc. However, target location estimation and other imaging tasks do require such knowledge. In practice (e.g., in ultrasound time-reversal imaging) it is customary to use homogeneous medium estimates of the required background Green functions for these purposes.
Return signals from
targets
Time-reversal processing requires no knowledge
of sub-surface and works for sparse
three-dimensional and irregular arrays and
both broad band and narrow band wave fields
February 23, 2000
A.J. Devaney--BU presentation

4. Array Imaging In conventional scheme it is necessary to scan the
Illumination
Measurement
Back propagation
Focus-on-transmit
Focus-on-receive
High quality image
In conventional scheme it is necessary to scan the
source array through entire object space
Time-reversal imaging provides the focus-on-transmit without scanning
Also allows focusing in unknown inhomogeneous backgrounds
February 23, 2000
A.J. Devaney--BU presentation

5. Experimental Time-reversal Focusing
Single Point Target
Illumination #1
Measurement
Phase conjugation
and re-illumination
Intervening Medium
Repeat …
If more than one isolated point scatterer present procedure will converge
to strongest if scatterers well resolved.
February 23, 2000
A.J. Devaney--BU presentation

6. Multi-static Response Matrix
Scattering is a linear process:
Given impulse response can compute response to arbitrary input
Kl,j=Multi-static response matrix = impulse response of medium
output from array element l for unit amplitude
input at array element j.
= K e
Applied array
excitation vector e
Arbitrary
Illumination
Array output
Single element
Illumination
Measurement
February 23, 2000
A.J. Devaney--BU presentation

7. Mathematics of Time-reversal
= K e
Applied array
excitation vector e
Arbitrary
Illumination
Array output
Multi-static response matrix = K
Array excitation vector = e
Array output vector = v
v = K e
K is symmetric (from reciprocity) so that K†=K*
T = time-reversal matrix = K† K = K*K
Each isolated point scatterer (target) associated with different m value
Target strengths proportional to eigenvalue
Target locations embedded in eigenvector
The iterative time-reversal procedure converges to the eigenvector
having the largest eigenvalue
February 23, 2000
A.J. Devaney--BU presentation

8. Processing Details Multi-static data Time-reversal processor
computes eigenvalues and eigenvectors
of time-reversal matrix
Eigenvalues
Eigenvectors
Return signals from targets
Standard detection scheme
Imaging
Conventional
MUSIC
February 23, 2000
A.J. Devaney--BU presentation

9. Multi-static Response Matrix
Assumes a set of point targets
Specific target
Green Function Vector
February 23, 2000
A.J. Devaney--BU presentation

10. A.J. Devaney--BU presentation
Time-reversal Matrix
February 23, 2000
A.J. Devaney--BU presentation

11. Array Point Spread Function
February 23, 2000
A.J. Devaney--BU presentation

12. Well-resolved Targets
SVD of T
February 23, 2000
A.J. Devaney--BU presentation

13. A.J. Devaney--BU presentation
Vector Spaces for W.R.T.
Well-resolved Targets
Signal Subspace
Noise Subspace
February 23, 2000
A.J. Devaney--BU presentation

14. Time-reversal Imaging of W.R.T.
February 23, 2000
A.J. Devaney--BU presentation

15. Non-well Resolved Targets
Signal Subspace
Noise Subspace
Eigenvectors are linear combinations of complex conjugate Green functions
Projector onto S:
Projector onto N:
February 23, 2000
A.J. Devaney--BU presentation

16. A.J. Devaney--BU presentation
MUSIC
Cannot image N.R.T. using conventional method
Noise eigenvectors are still orthogonal to signal space
Use parameterized model for Green function: STEERING VECTOR
Pseudo-Spectrum
February 23, 2000
A.J. Devaney--BU presentation

17. A.J. Devaney--BU presentation
GPR Simulation
Antenna Model x  z
Uniformly illuminated slit of width 2a with Blackman Harris Filter
February 23, 2000
A.J. Devaney--BU presentation

18. Ground Reflector and Time-reversal Matrix
February 23, 2000
A.J. Devaney--BU presentation

19. A.J. Devaney--BU presentation
Earth Layer  1
February 23, 2000
A.J. Devaney--BU presentation

20. Down Going Green Function
z=z0
February 23, 2000
A.J. Devaney--BU presentation

21. Non-collocated Sensor Arrays
Current Theory limited to collocated active sensor arrays
Active
Transmit Array
Passive
Receive Array
Experimental time-reversal not possible for such cases
Reformulated computational time-reversal based on SVD is applicable
February 23, 2000
A.J. Devaney--BU presentation

22. Off-set VSP Survey for DOE
February 23, 2000
A.J. Devaney--BU presentation

23. A.J. Devaney--BU presentation
Acoustic Source
February 23, 2000
A.J. Devaney--BU presentation

24. A.J. Devaney--BU presentation
Formulation
Surface to Borehole
Borehole to Surface
We need only measure K (using surface transmitters) to deduce K+
February 23, 2000
A.J. Devaney--BU presentation

25. Time-reversal Schemes
Two different types of time-reversal experiments
Start iteration from surface array
Start iteration from borehole array.
Multi-static data matrix no longer square
Two possible image formation schemes
Image eigenvectors of Tt
Image eigenvectors of Tr
February 23, 2000
A.J. Devaney--BU presentation

26. Singular Value Decomposition
Surface to Borehole
Borehole to Surface
Normal Equations
Surface eigenvectors
Start from surface array
Time-reversal
matrices
Start from borehole array
Borehole eigenvectors
February 23, 2000
A.J. Devaney--BU presentation

27. Transmitter and Receiver Time-reversal Matrices
February 23, 2000
A.J. Devaney--BU presentation

28. Well-resolved Targets
w.r.t. receiver array
Well-resolved
w.r.t. transmitter array
February 23, 2000
A.J. Devaney--BU presentation

29. A.J. Devaney--BU presentation
Future Work
Finish simulation program
Employ extended target
Include clutter targets
Include non-collocated arrays
Compute eigenvectors and eigenvalues for realistic parameters
Compare performance with standard ML based algorithms
Broadband implementation
Apply to experimental off-set VSP data
February 23, 2000
A.J. Devaney--BU presentation