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

Computational Time-reversal Imaging

A.J. DevaneyDepartment of Electrical and Computer Engineering

Northeastern University

email: devaney@ece.neu.eduWeb: 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

February 23, 2000 A.J. Devaney--BU presentation 2

Experimental Time-reversal

Intervening mediumIntervening medium

Without time-reversal compensation With time-reversal compensation

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 Final illuminationIntermediate illumination

Intervening medium

Goal is to focus maximum amount of energy on target for purposes of target detection and location estimation

February 23, 2000 A.J. Devaney--BU presentation 3

Computational Time-reversal

Time-reversal processorComputes measured returns that would

have been received after time-reversal compensation

Multi-static data

Return signals from targets

Target detection

Time-reversal compensation can be performed without actually performing a sequence of target illuminations

Target location estimation

Time-reversal processing requires no knowledgeof 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

High quality image

Illumination

Measurement

Back propagation

In conventional scheme it is necessary to scan the source array through entire object space

Time-reversal imaging provides the focus-on-transmit without scanningAlso allows focusing in unknown inhomogeneous backgrounds

Focus-on-transmit Focus-on-receive

February 23, 2000 A.J. Devaney--BU presentation 5

Experimental Time-reversal Focusing

Illumination #1

Measurement

Phase conjugation and re-illumination

If more than one isolated point scatterer present procedure will converge to strongest if scatterers well resolved.

Repeat …

Intervening Medium

Single Point Target

February 23, 2000 A.J. Devaney--BU presentation 6

Multi-static Response MatrixScattering 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 elementIllumination

Single elementMeasurement

= K e

Applied arrayexcitation vector e

Arbitrary

Illumination Array output

February 23, 2000 A.J. Devaney--BU presentation 7

Mathematics of Time-reversalMulti-static response matrix = K

Array excitation vector = eArray output vector = v

v = K e

T = time-reversal matrix = K† K = K*K

K is symmetric (from reciprocity) so that K†=K*

= K eApplied array

excitation vector e

Arbitrary

Illumination Array output

Each isolated point scatterer (target) associated with different m valueTarget strengths proportional to eigenvalueTarget 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

Time-reversal processorcomputes eigenvalues and eigenvectors

of time-reversal matrix

Multi-static data

Return signals from targets

Eigenvalues Eigenvectors

Standard detection schemeImaging

Conventional MUSIC

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Multi-static Response Matrix

Specific target

Green Function Vector

Assumes a set of point targets

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Time-reversal Matrix

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Array Point Spread Function

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Well-resolved Targets

SVD of T

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Vector Spaces for W.R.T.

Signal Subspace Noise SubspaceWell-resolved Targets

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Time-reversal Imaging of W.R.T.

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Non-well Resolved Targets

Signal SubspaceNoise Subspace

Eigenvectors are linear combinations of complex conjugate Green functions

Projector onto S: Projector onto N:

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MUSIC

Pseudo-Spectrum

Cannot image N.R.T. using conventional methodNoise eigenvectors are still orthogonal to signal space

Use parameterized model for Green function: STEERING VECTOR

February 23, 2000 A.J. Devaney--BU presentation 17

GPR Simulation

x

z

Antenna Model

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6x 10

-3 single element radiation pattern-ideal and ideal and computed with blackman harris

normalized transverse wavenumber K/k=sin

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

Earth Layer

1

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Down Going Green Function

z=z0

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Non-collocated Sensor Arrays Current Theory limited to collocated active sensor arrays

Experimental time-reversal not possible for such cases

ActiveTransmit Array

PassiveReceive Array

Reformulated computational time-reversal based on SVD is applicable

February 23, 2000 A.J. Devaney--BU presentation 22

Off-set VSP Survey for DOE

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Acoustic Source

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Formulation

We need only measure K (using surface transmitters) to deduce K+

Surface to Borehole

Borehole to Surface

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Time-reversal SchemesTwo different types of time-reversal experiments1. Start iteration from surface array2. Start iteration from borehole array.

Multi-static data matrix no longer square

Two possible image formation schemes1. Image eigenvectors of Tt

2. Image eigenvectors of Tr

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Singular Value Decomposition

Normal Equations

Time-reversal matrices

Start from surface array

Start from borehole array

Surface to Borehole

Borehole to Surface

Surface eigenvectors

Borehole eigenvectors

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Transmitter and Receiver Time-reversal Matrices

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Well-resolved Targets

Well-resolved w.r.t. receiver array

Well-resolved w.r.t. transmitter array

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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