Computational Timereversal Imaging A J Devaney Department of

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: Tech. Sat 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 1

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 Intervening medium Without time-reversal compensation February 23, 2000 A. J. Devaney--BU presentation Final illumination Intervening medium With time-reversal compensation 2

Computational 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 narrow band wave fields February 23, 2000 A. J. Devaney--BU presentation 3

Array Imaging 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 4

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 5

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. Single element Illumination February 23, 2000 Single element Measurement Arbitrary Illumination A. J. Devaney--BU presentation Applied array excitation vector e Array output = K e 6

Mathematics of Time-reversal Arbitrary Illumination Applied array excitation vector e Array output = K e Multi-static response matrix = K Array excitation vector = e Array output vector = v v=Ke 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 7

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

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

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

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

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

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

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

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

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 16

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

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

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

Down Going Green Function z=z 0 February 23, 2000 A. J. Devaney--BU presentation 20

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 21

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

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

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

Time-reversal Schemes Two different types of time-reversal experiments 1. Start iteration from surface array 2. Start iteration from borehole array. Multi-static data matrix no longer square Two possible image formation schemes 1. Image eigenvectors of Tt Image eigenvectors of Tr 2. February 23, 2000 A. J. Devaney--BU presentation 25

Singular Value Decomposition Surface to 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 26

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

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

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 29