Determining Equivalent Dipole Number by Information ... · • 81 mm mortar • Depth = 40 cm ......

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Orlando, FloridaAugust 26, 2009

UXO/Countermine/Range Forum

L-P Songa, L R Pasionb, S D Billingsb

and D W Oldenburga

bSky Research, Inc, Ashland, Oregon

aGeophysical Inversion FacilityUniv. of British Columbia, Vancouver Canada

Determining Equivalent Dipole Number by Information Theoretic Criteria

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• Motivation and Problem Formulation • Information-Theoretical Criteria • Applications: EM63, TEMTADS• Summary

Outline

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Motivation and Problem FormulationEMI sensing:

Collect data Find target signature Identify objects?

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Find target signature Data inversion

Model parameters m:Geometrical and physical properties of a target

d

F -1 (d)

Recover parameters of a physics-based model from data

md F (m)

Motivation and Problem Formulation

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• Dipole model Polarizability tensor

y

x

z

Tx Rx

Primary field

Seco

ndar

y fie

ld

Principal polarizations:

Rod-like object:Rotational matrix


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η targets within view of M sensors

Green’s function matrix

Source positionand orientation

Source polarizations

• Observed responses d are linear combinations of the dipole polarizations qj of targets weighted by Green’s function matrix A

Scat

terin

g

Illuminating

Array of sensors

• What is the number of η ?


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Sibert EM-63

• Are these anomalies due to a single or two targets?temtads

• How to determine?


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Assume a sequence of possible models θη, η = 1, 2, …

Information theoretic criteria

• How well does a model fit the data?

Likelihood L(D| θη): probability of observing the data given a model with some free parameters

• How complex is the model?

Number of free parameters in the model θη

given spatial-temporal data: [ ])(,),( 1 NttD dd L=

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Information Theoretic Criteria

a likelihood function a penalty function

( ) ),()(IC ηθη η NPDL +=

Criterion to balance data-fitting and model complexity

( )ηηη

IC min arg ˆIC =

Determining minimum number of dipolar polarizations that represent the data sufficiently

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( ) ( ) ( )ηνθη η )log(21ˆ| NDL +=MDL

( )

MDL/BIC favors a simpler model than AIC when sample size N is larger 3.789, log(N) >2

( ) ( )ηνθη η += ˆ|DLAIC

Akaike Information Criterion:

Minimum description length/Bayesian information criterion:

The smaller the AIC/MDL, the better the model.

(Akaike, IEEE Trans. AC, 1974)

(Rissanen, Automatica, 1978; Schwartz, Ann. Stat., 1978)

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What is behind the ITC?

• Principle of Parsimony (Simplicity):A complicated model (too many parameters) is not desirable: its ‘good’ performance might be misleading.

• Data-fitting performance is overseen with model complexity.

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Information-Theoretical Criteria

Signal subspace

Data d(t) are confined to a three-dimensional subspace spanned by {a1(θ), a2(θ), a3(θ)}

Information theoretic criteria is used to measure the goodness-of-fit between the potential signal subspace and the one in the observed data.

d(t) = a1(θ)L1(t) + a2(θ)L2(t) + a3(θ)L3(t)

A single object

Data d(t) are confined to a two-dimensional subspace spanned by {a1(θ) + a2(θ), a3(θ)}

d(t) = (a1(θ) + a2(θ))L1(t) + a3(θ)L3(t)

A rod-like object

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• Data covariance matrix

EigenvectorsEigenvalues

Implementation: Detection only

Signal subspace

• Eigen-decomposition

Hypothesized model:

(Wax and Kailath, IEEE-ASSP, 33(2), 1985)

• Finding smallest number for which the signal subspace {v1, …, vη}contains or best fits the sample data.

Noise level

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Implementation: Detection only (contd.)

(Wax and Kailath, IEEE-ASSP, 33(2), 1985)

Penalty function

: sample eigenvalues

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• data covariance matrix is unknown and has to be approximated using sample covariance matrix (SCM):


Implementation: Detection only (contd.)

• Thresholding the SCM:

• How to choose α? Empirically?

• α is data-dependent, SNR and source power related.

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• The eigen-structure-based criteria can be sub-optimal or even fail due to uncertainty in SCM.

• It may be desirable to determine the number of sources and estimate their locations jointly.

• Consider that the measured data is a function of both the number of source polarizations and their locations.

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Space–time data: [ ])(,),( 1 NttD dd L=

• Assume a model number, solve for set of model parameters

Likelihood of the data at sought model

• Select one over possible models with a smaller AIC or MDL values

Simultaneous detection and Estimation:


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Application: EM-63 Synthetic Data

• Position error: σ = 1cm• IMU error: σ = 1 deg

• 81 mm mortar• Depth = 40 cm from

surface• Correlated magnetic

soil noise added• Different data densities

considered

• 60 mm mortar• Depth = 40 cm• Larger magnitude of

background magnetic soil noise

Example: Apply MDL to synthetic data

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Application: EM-63 Synthetic Data

Application of the MDL to synthetic data

TrueRecovered – 2 polRecovered – 3 pol

• Example of only 1 polarization fitting the data

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Application: Camp Sibert GPO

An empirical threshold:

• Data contain 30 intact 4.2’’ mortar and 8 partial rounds

Overestimate Better Underestimate

MDL detects correct object number in all cases and seems more robust to the value of ε.

Out of 8 partial rounds, AIC/MDL detects 5 times of three polarizations under ε =1% and ε = 2%.

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Application: Camp Sibert GPO

Example 1 in Slide 7:

A single object

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Application: Camp Sibert EM-63 Data

Example 2 in slide 7:

MDL

Most likely two objects

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Check the inverted results:

Two-Target InversionSingle Target Inversion

Observed Data (t1)

Application: Camp Sibert EM-63 DataMDL detects the presence of two objects

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Single Object Inversion

Recovered polarizations (solid lines)

Unsuccessful



MDL detects the presence of two objects

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Recovered Polarizations

Polarizations of excavated target determined via in air TEMTADS measurements.

Polarizations of base plate

Excavated Target: Scrap

Predict a Base PlateTarget 2

Target 1



MDL detects the presence of two objectsTwo-Target Inversion

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Application: TEMTADS Data

• Tx waveform: Bipolar Pulse

• 115 Time gates: 0.042 ms - 24.35 ms

• Spatial-temporal data: 625 x 115

• 25 Txs and 25 Rxs

• Sensor footprint: 2 m x 2 m

TEMTADS array:

Source: ESTCP MM-0601 report

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Experimental geometry:(Courtesy of Daniel A. Steinhurst)

• 4.2 in mortar under array centerline

• Small shallow clutter

• Deeper bigger clutter

• Clutter positioned from 0 to 1.5 m relative to centerline

Application: TEMTADS Data

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4.2 in mortar

Half shell Nose frag

Application: TEMTADS data

Polarization spectra of items in the experiments

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Single-Target Case:

Overlapping cases:

Half shell Mortar Nose frag

Mortar + Half shell Mortar + frag

Application: TEMTADS dataSome anomaly images

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S= 0 cm S= 10 cm S= 20 cm

S= 30 cm S= 40 cm S= 50 cm

Overlapping cases: mortar + shell, mono-static 2D image

Application: TEMTADS dataSome anomaly images at various separations

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S = 60 cm S = 70 cm S = 80 cm

S = 90 cm S = 1 m S = 1.1 m

Overlapping cases: mortar + shell, mono-static 2D image

Application: TEMTADS dataSome anomaly images at various separations

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Joint detection and estimation

• LLF(D|(m1 ,m2)) < LLF(D|m1) on the same dataset, does it mean to accept a two-object model and reject a single-object model?


• LLF(D|m1) likelihood for a single-object model

• LLF(D|(m1 ,m2)) likelihood for two-object model

Mortar + HR: Sepa.

0 ~ 1.5 m Mortar + nose: Sepa. 0 ~ 1.5 m

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Joint detection and estimation (contd.)


• AIC detects 5 cases of 35 as single objects

• MDL detects 13 cases of 35as single objects

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Joint detection and estimation (contd.)


What are those 5?

What are those 13?

MDL more prefers a single-object model than AIC under small or large separation when two objects are present.

• AIC detects 5 cases of 35 as single objects.

Two are correct, other three are 4.2” mortar + nose piece at separation > 1.2 m.

• MDL detects 13 cases of 35 as single objects.

Three are correct, other three are 4.2’’ mortar + large clutter at separation > 1.5 m and separation < 10 cm.

Other seven are 4.2” mortar + small clutter at separation > 1.10 m and separation < 10 cm.

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Single-Target Inversion:


H-separation = 50 cm, 4.2 in mortar and a half shell

Multi-Target Inversion:

Check the inverted results, MDL detects two objects

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Multi-Target Inversion Recovered polarizations(Solid curves)

Single-Target Inversion Recovered polarizations(Solid curves)

(Dash curves): Polarizations from non-overlapping data


Mortar

Shell

Mortar

Shell

Application: TEMTADS dataCheck the inverted results, MDL detects two objects

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H-separation = 0 cm



Application: TEMTADS dataCheck the inverted results, MDL detects a single object


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Mortar

Shell

Mortar

Shell


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H-separation = 1.5 m, 4.2 in mortar and a half shell

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H-separation = 1.5 m, 4.2 in mortar and a half shell

Mortar

Shell

Mortar

Shell


a rod-like object

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H-separation = 10 cm


H-separation = 0 cm, 4.2 in mortar and nose frag



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Mortar

frag

Mortar

frag

Application: TEMTADS dataCheck the inverted results, MDL detect a single object

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H-separation = 50 cm, 4.2 in mortar and nose fragCheck the inverted results, MDL detects two objects

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Mortar

frag

Mortar

frag

Application: TEMTADS dataCheck the inverted results, MDL detects two objects

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H-separation = 70 cm




H-separation = 1.5 m, 4.2 in mortar and nose frag

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H-separation = 1.5 m, 4.2 in mortar and nose frag

Mortar

frag

Mortar

frag


An extra rod-like object

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Future work:

• Any possible improvement in the context of EMI sensing Using a priori available information in the criteriaThresholding sample covariance matrix

Summary

Information-theoretic criteria consider both data fitting and model complexity to estimate source number

• ITC can be implemented in detection-only mode or joint detection-estimation mode.

• Joint detection-estimation mode appears robustrelative to detection-only mode.

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Acknowledgements

• The work is supported by SERDP MM-1637 project

• We are grateful to Daniel A. Steinhurst and Herb Nelson of NRL and James Kindon and Tomas Bell of SIAC for kindly providing TEMTADSsensor information and data.

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Determining Equivalent Dipole Number by Information ... · • 81 mm mortar • Depth = 40 cm ......

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