Computational Time-reversal Imaging

1. Computational Time-reversal Imaging A.J. Devaney Department of Electrical and Computer Engineering Northeastern University email:devaney@ece.neu.edu Web: www.ece.neu.edu/faculty/devaney/ 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 A.J. Devaney--BU presentation

2. Intervening medium Intervening medium Intervening medium 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 Without time-reversal compensation With time-reversal compensation 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 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 A.J. Devaney--BU presentation

4. Illumination Measurement Back propagation Array Imaging 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 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. A.J. Devaney--BU presentation

6. Applied array excitation vector e = K e Arbitrary Illumination Single element Illumination Single element Measurement Array output 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. A.J. Devaney--BU presentation

7. Applied array excitation vector e = K e Arbitrary Illumination Array output Mathematics of Time-reversal 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 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 MUSIC Conventional A.J. Devaney--BU presentation

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

10. Time-reversal Matrix A.J. Devaney--BU presentation

11. Array Point Spread Function A.J. Devaney--BU presentation

12. Well-resolved Targets SVD of T A.J. Devaney--BU presentation

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

14. Time-reversal Imaging of W.R.T. 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: A.J. Devaney--BU presentation

16. 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 A.J. Devaney--BU presentation

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

18. Ground Reflector and Time-reversal Matrix A.J. Devaney--BU presentation

19. Earth Layer  1 A.J. Devaney--BU presentation

20. Down Going Green Function z=z0 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 A.J. Devaney--BU presentation

22. Off-set VSP Survey for DOE A.J. Devaney--BU presentation

23. Acoustic Source A.J. Devaney--BU presentation

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

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

27. Transmitter and Receiver Time-reversal Matrices A.J. Devaney--BU presentation

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

29. 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 A.J. Devaney--BU presentation