graduate program in acoustics applied research laboratory 8-13 july 2007 maxent 20072 application of...

8-13 July 2007 MaxEnt 2007 2

Graduate Program in Acoustics

Applied Research Laboratory

Application of the Maximum Entropy method to sonar signal processing

R. Lee Culver, H. John Camin, Jeffrey A. Ballard, Colin W. Jemmott, and Leon H. Sibul

Applied Research Laboratory and Graduate Program in AcousticsThe Pennsylvania State University, P.O. Box 30

State College, PA 16804

27th International Workshop onBayesian Inference and Maximum Entropy Methods in

Science and EngineeringSaratoga Springs, NY, July 8-13, 2007

Work supported by Office of Naval Research

8-13 July 2007 MaxEnt 2007 3



Outline

• Sonar application description• Domain of existing solutions• A new Estimator-Correlator detector that makes

use of the Maximum Entropy principal• An example: the 1996 Strait of Gibraltar

Acoustic Monitoring Experiment (SGAME)• Planned extensions

8-13 July 2007 MaxEnt 2007 4


Applied Research LaboratoryProblem description

Horizontal line array(plan view)

Fan ofnarrow beams In

crea

sing

tim

e

frequency

time-frequency plotfor a single beam

Problem: Many signals are detected, but what and where are the sources?

successiveFFTs of

one beam

detectedlines

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Matched Field Processing

source

vertical line array(side view)

bottom

surface

acousticrays

8-13 July 2007 MaxEnt 2007 6


Applied Research LaboratoryMatched field processing

Bucker, H. P. (1976). “Use of calculated sound fields and matched-field detection to location sound sources in shallow water,” J. Acoust. Soc. Am. 59 (2), pp. 368-373.

Sensors

th

m

complex amplitude of j sensor

for frequency

j mc

FFT

.

.

.

.

.

.

12

.

.

.

J

*

m

cross product of of amplitude from

sensors j and k at frequency

jk m j m k ma c c

8-13 July 2007 MaxEnt 2007 7



0

Let , represent a set of cross-spectral elements

calculated using knowledge of the environment and

and acoustic propagation model for a source at location .

Let represent the set of

jk m

jk m

a

a

x

x

-1 0*

1 1,

cross-spectral elements

obtained from the array, where indicates time average.

Define the Detection Factor (DF) as follows:

DF , = N ,

(J = # of

J J

m jk m jk mj k j

a a x x

sensors, N = normalization factor)

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Bucker, H. P. (1976).J. Acoust. Soc. Am. 59 (2),

pp. 368-373.Range, kyd

Dep

th,

yd

DF , mx

8-13 July 2007 MaxEnt 2007 9


Applied Research LaboratoryOptimum Uncertain

Field Processor

Richardson, A. M. and L. W. Nolte (1991). “A posteriori probability source localization in an uncertain sound speed, deep ocean environment,” J. Acoust. Soc. Am. 89, pp. 2280-2284.

Richardson and Nolte applied a Bayesian method

to this problem.

The observations consist of signal and additive noise :

, ,

is the set of source location parameters.

is the paramete

r s n

r s S Θ Ψ n

S

Θ rs describing the transmitted waveform.

is the parameters describing the propagation medium.Ψ

8-13 July 2007 MaxEnt 2007 10



Field Processor

||

| , ,

The source location pdf can be written

|| . (Bayes' rule)

The conditional pdf of the observation is related to the noise pdf by:

| , , , , .

Usi

a posteriori

p pp

p

p p

r S SS r

r

r S Θ Ψ n

r S SS r

r

r S Θ Ψ r s S Θ Ψ

| | , , , |

| , |

ng

| | , , , | ,

the source location pdf can be written

| , , , | .

p p p d d

a posteriori

pp p p d d

p

r S r S Θ Ψ Θ Ψ S

Ψ Θ

SS r n Θ Ψ S

r Ψ Θ

r S r S Θ Ψ Θ Ψ S Θ Ψ

SS r r s S Θ Ψ Θ Ψ S Θ Ψ

r

, , r s S Θ Ψ n

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Field Processor

| , |

|

So

| , , , |

is the general expression for the optimum uncertain field

processor. Maximum (MAP) estimates of are the

values of that maximize | .

pp p p d d

p

a posteriori

p

SS r n Θ Ψ S

a Ψ Θ

S r

SS r r s S Θ Ψ Θ Ψ S Θ Ψ

a

S

S S r

Richardson and Nolte find MAP estimates of for Gaussian .pnS n

, , r s S Θ Ψ n

8-13 July 2007 MaxEnt 2007 12



Field Processor

• The optimum uncertain field processor cannot be applied to our problem for two reason.1. Our array is horizontal, not vertical. With no vertical aperture,

vertical structure (multipath) in the sound field is not observed.

2. The noise field is not necessarily Gaussian (See E. J. Wegman, S.C. Schwartz and J. B. Thomas eds., Topics in Non-Gaussian Signal Processing, Springer-Verlag, New York, 1988.).

• Therefore, we take a different approach to obtain a processor that can be applied to our problem.• Use a propagation model to predict signal parameter statistics• Look for statistical clues in the observation (received signal)

J. A. Ballard (2007). “The Estimated Signal Parameter Detector”, M.S Thesis (The Pennsylvania State University, State College, PA).

8-13 July 2007 MaxEnt 2007 13



pdf of signal from near-surface source (H1)

Source nearthe bottom

Source nearthe surface

Receive array pdf of signal fromnear-bottom source (H2)

The underlying assumption is that sources at different locations will generate different received statistics

The Estimated Ocean Detector

1

2

Form the binary hypothesis test for observation :

H : source is near the surface = { , }

H : source is near the bottom.

1 1 2

2

r

r s n S s s

r s n

p1(r)

p2(r)

8-13 July 2007 MaxEnt 2007 14



| 1 1 1

We require that the conditional pdfs belong to the exponential class

which has the following form:

| ,H ( )exp , 2p K g B r Ψ r Ψ Ψ r Ψ r

1

The source waveform is a continuously transmitted sinusoid with constant,

known frequency and phase, and time-varying amplitude.

Form the likelihood ratio ( describes the medium).

| Hln

| H

p

p

Ψ

rr

r

| 11

2 2 | 2

| ,H

ln ln 1| ,H

p p dp

p p p d

r Ψ Ψ

Ψ

r Ψ Ψ

Ψ

r Ψ Ψ Ψr

r r Ψ Ψ Ψ

S.C. Schwartz, “The Estimator-Correlator for Discrete-Time Problems,” IEEE Trans. On Information Theory, Vol. 23, No. 1, January 1977, pp. 93-100.


8-13 July 2007 MaxEnt 2007 15



| 1 1 1

1 1

| ,H ( )exp , 2

Here ( ), , and are arbitrary functions. We have

extended Schwartz's derivation somewhat, in that he specified that

the term to depended only on ,

p K g B

K g B

g

r Ψ r Ψ Ψ r Ψ r

Ψ r Ψ r

r

| 1 1 1

so that the conditional pdf was

| ,H ( )exp , 3

whereas we allow to depend upon both and .

This provides some added flexibility in finding an appropriate

conditional pdf.

p K g B

g

r Ψ r Ψ Ψ Ψ r r

r Ψ


8-13 July 2007 MaxEnt 2007 16


Applied Research LaboratoryThe Estimated Ocean Detector

1 2

1

1 | 1

The and terms have the same form, and we present the

derivation of . We differentiate:

| ,H 4

Using of the exponental form of the conditional pdf and Bayes' rul

p p

p

p p p d

r Ψ Ψ

Ψ

r r

r

r r Ψ Ψ Ψr r

1 1 1 | 1

1 1 | 1 1

e:

, | ,H

, | ,H 5

p g B p p d

g B p p d

r Ψ Ψ

Ψ

Ψ r

Ψ

r r Ψ r r Ψ Ψ Ψr r r

r Ψ r Ψ r r Ψr r

Bayes’ rule

8-13 July 2007 MaxEnt 2007 17



11 | 1 1

1

11 | 1

Simplifying, we obtain:

,1| ,H 6

We define the conditional moment function:

,| ,H conditional moment 7

which is an extention to Schwart's condition

gp p d B

p

gh p d

Ψ r

Ψ

Ψ r

Ψ

r Ψr Ψ r Ψ r

r r r r

r Ψr Ψ r Ψ

r

11 | 1

al mean:

| ,H (Schwartz's conditional mean)Schwartz gh p d

Ψ r

Ψ

rr Ψ Ψ r Ψ

r

8-13 July 2007 MaxEnt 2007 18



11

1

11 1

1 1 1 1

1 1 2 2

1Using ln and the conditional moment, we can write

ln 8

which can be integrated to obtain

exp 9

where . and are similarly defined,

and the lo

pp

p

Bp h r

p c G B

G h d G B

r

rr

r r r

rr

r r

r r r

r r r

1 1

1 1 2 222

g likelihood ratio becomes

. 10p cn l G B G B n cp

r

r r r r rr

8-13 July 2007 MaxEnt 2007 19



Calculatesignal

parameter pdf

Calculatesignal

parameter pdf

2B r

1B r

1h

Received Signal, r

Conditional Moment

Functions

1G r

2G r2h

1p

1

2

lnH

H

r

H1

H2

--


211

21 2G rG r B r cr n cB r

2p

8-13 July 2007 MaxEnt 2007 20



| 1 1 1

0

This is new result in that | ,H ( )exp ,

is not a stringent requirement, i.e. it is met by many pdfs.

Ballard (2007) investigates a Gaussian conditional pdf:

1| exp

p K g B

p kN

r Ψ r Ψ Ψ r Ψ r

r s

22 2

1 10 0 0

1

1 2exp exp

2

where M is the length of and is the coherent matched filter output:

.

M M

i i ii i

M

i ii

A A Tr s k r V

N N N

V

V r s

r

r r

r

J. A. Ballard (2007). “The Estimated Signal Parameter Detector”, M.S Thesis (The Pennsylvania State University, State College, PA).

8-13 July 2007 MaxEnt 2007 21



1

2

10 0

2

20 0

When the signal is a sinusoid with known frequency and phase, and

random amplitude, i.e. cos , the likelihood ratio test is:

2ln exp

2ln exp

A

A

s A t

A TAV r p A dA

N N

A TAV r p A

N N

2

1

2

ln .

H

dA

H

8-13 July 2007 MaxEnt 2007 22



2 21 1 2 2

An interesting case is when the prior distributions for amplitude A are

Gaussian with equal means but different variances, i.e.

, , ,

The log likelihood ratio is found to have the

p A N m p A N m

12

2

form

2

The detector computes the variance of the matched filter output.

H

TmV r

H

8-13 July 2007 MaxEnt 2007 23



p1(A)

p2(A)

21

2102

probability of detection; probability of false alarm;

SNR = signal-to-noise ratio; SSR = 10 log

D FAP P

performanceimproves

8-13 July 2007 MaxEnt 2007 24


Applied Research LaboratoryApplying the MaxEnt Method

• We use the MaxEnt method in two ways:– Obtain by fitting an exponential (or MaxEnt) pdf to

noise samples that do not contain signal, e.g. from another beam or at another frequency (or both).

– Use an acoustic propagation model and Monte Carlo simulation to produce samples of A under H1 and H2, and use MaxEnt to estimate the prior pdfs p1(A) and p2(A).

– In both cases, we compute sample moments from the data and apply the gradient method developed by Mohammad-Djarari (1991). (I think I need to spend some more time on this part of the approach).

|np r s

8-13 July 2007 MaxEnt 2007 25



P1

P6

P7

P4

P5

P3

P2

1996 Strait of Gibraltar Acoustic Measurement Experiment (SGAME)

Tx = projectorRx = hydrophonePn = groupings of CTD drops.

• Worcester, Send, Curnuelle and Tiemann (1997), in Shallow-Water Acoustics, Bejing, China.• Tiemann, Worcester and Cornuelle (2001), JASA 109 and 110 (2 kHz data only).

• Warm, fresh surface layer of Atlantic water moving east over salty, cool layer of Mediterranean water moving west.• Strong internal tide• East-moving tidal bores released after high tide.

8-13 July 2007 MaxEnt 2007 26



hydrophones

projector


dep

th, m

.

1000

range, km12

Acoustic propagation paths

8-13 July 2007 MaxEnt 2007 27



yearday 1996

Hei

gh

t, m

.H

eig

ht,

m.

yearday 1996


CTD (conductivity and temperature vs. depth) measurements spanned the tidal cycle.

P1, P2, etc. correspond to positions shown on Slide 3.

CTD drop times relative to tidal height

8-13 July 2007 MaxEnt 2007 28


Applied Research Laboratory1996 Strait of Gibraltar Acoustic Measurement Experiment (SGAME)

Propagation loss predicted using RAM

Collins, M. (1993). “A split-step Padá solution for the parabolic equation method,” J. Acoust. Soc. Am, 93, pp. 1736-1742.

8-13 July 2007 MaxEnt 2007 29



0 10

The entropy is

ln

The constraints are

where the are estimated from the data. We form N equations:

, ,... exp , 0,1,

Solving for the ' , we o

n n n

n

N

n N n k k nk

H p A p A dA

E A A p A dr

G A A dA n N

s

1,2 0 1 2

btain

exp ln .p A A A

8-13 July 2007 MaxEnt 2007 30



0

0

0 0

0

The equations for G are linearized around a trial value .

The resulting equations are

The system is solved for , which becomes the new , and the

iteration continues unt

T

n n n nG G G

λ λ

λ

λ λ λ λ λ

λ λ

0il becomes arbitrarily small.λ λ

Mohammad-Djafari, A. (1992). “Maximum Likelihood Estimate of the Lagrange parameters of the Maximum Entropy Distributions,” in Maximum Entropy and Bayesian Methods, Proc. 11 th International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Ed. C. R. Smith, G. J. Erickson, and P. O. Neudorfer (Kluwer Academic, Dordrecht, NL).

8-13 July 2007 MaxEnt 2007 31



Received pressure, dB re 1µPa

Rel

ativ

e oc

curr

ence

* Histogram of measured received pressure MaxEnt pdf fit to received pressure MaxEnt pdf fit to RAM predictions

8-13 July 2007 MaxEnt 2007 32



* Histogram of measured received pressure MaxEnt pdf fit to received pressure MaxEnt pdf fit to RAM predictions

8-13 July 2007 MaxEnt 2007 33



Summary

• We have used an Estimator-Correlator structure to develop a Maximum Likelihood detector that can accept any exponential class noise pdf (not just a Gaussian).

• The MaxEnt method has been used to obtain exponential class pdfs. Example shown for 1996 Strait of Gibraltar Acoustic Monitoring Experiment (SGAME).

• We are learning about the MaxEnt method.

graduate program in acoustics applied research laboratory 8-13 july 2007 maxent 20072 application of...

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