Model-based inverse problems in underwater acoustics

(1)

Workshop on Inverse Obstacle Problems

Instituto Superior T´

ecnico

Lisboa, Portugal

4 - 6 November 2002

by

S´

ergio M.M. Jesus

SiPLAB - FCT, Universidade do Algarve

Campus de Gambelas, PT-8000-810 Faro, Portugal

(2)

Mo

del-based

in

v

erse

problems

in

underw

ater

acoustics

S.M.

Jesus

(sjesus@ualg.pt)

SiPLAB-F

CT,

Univ

ersidade

do

Algarv

e

Campus

de

Gam

b

elas,

PT-8000-810

F

aro

P

ortugal

This

w

ork

w

as

partially

supp

orted

under

con

tract

2./2.1/MAR/1698/9

5,

PRAXIS,

F

CT,

P

ortugal.

(3)

Outline

• Mo

del-Based

Pro

cessing

(MBP)

• the

problem

• the

in

version

metho

ds

• real

w

orld

examples

• conclusions

(4)

Ocean Acoustic Tomography - synoptic

Acoustic Model

environment

new

^

X

receiver

acoustic pressure

source signal

Correlator

Optimization

s(t)

source

(range,depth)

r

C(r,r )

r

g(r)

o

y=g( r ) + n

~

(5)

The

pro

blem

• Impl

ici

t

Mo

del

Co

ndit

io

ning

Da

ta:

y

(r

o

₎

=

g

(r

o

₎

+

n

Mo

del:

˜g

(r

)

ˆr

o

=

mi

n

r

F

[y

(r

o

),

˜g

(r

)]

• Mo

del

wi

th

r

a

ndo

m

co

mp

o

nen

ts

Da

ta:

y

(r

o

₎

=

g

(r

o

₎

+

n

Mo

del:

m

(r

)

=

α

˜g

(r

)

α

an

d

r

rand

om

(6)

F

orw

ard

mo

dels

∇

2

p

+

ω

2

c

2

(z

)

_p

=

0

• Ra

y

solution

τ

=

Z

Γ

ds

(z

,r

)

c(

z)

p

=

p

o

I

X

i

=1

1

4 π

s

i

exp

(j

ω

τ

i

)

(7)

F

orw

ard

mo

dels

(con

t.)

• Normal-mo

de

solution

p

=

p

o

M

X

m

=1

φ

m

(z

o

)φ

m

(z

)

√

k

m

r

exp

(ik

m

r

−

α

m

r)

d

2

φ

m

dz

2

+

ω

2

c

2

(z

)

−

k

2 m

φ

m

=

0

(8)

P

e

rt

u

r

bati

on

in

v

er

sion

e

x

am

ple

∆

τ

≈

−

Z

Γ

δ

c(

z

)

c

2 o

(z

)

ds

∆

τ

=

E

δ

c

+

n

δ

c

=

Ψ

α

=

[Ψ

t

E

t

EΨ

]

₋

1

Ψ

t

E

t

∆

τ

INTIMA

TE

sea

tr

ial

June

96 -Naza

r´e,

P

ortug

al

0 1 2 3 4 5 0 20 40 60 80 100 120 Distância horizontal (km) Profundidadde (m)

c (m/s)

1508 1510 1512 1514 1516 1518 1520 166.8 167 167.2 167.4 0 20 40 60 80 100 120 140

Julian date

Depth (m)

(9)

F

orw

ard

mo

del-based

example

Data

mo

del:

y(

t,

r

o

₎

=

x

(t,

r

o

₎

+

n

(t

)

x

(t,

r

o

)

=

g(

t,

r

o

₎

∗

s(

t)

Mean

square

error

estimator:

ˆr

o

=

min

r

E

[k

r

−

r

o

_k

2

]

The

Matc

hed-Filter

(or

correlator-receiv

er):

H

(ω

,r

o

₎

=

H

o

G

∗

(ω

,r

o

)S

∗

(ω

)

P

nn

(ω

)

e

−

j

ω

τ

(10)

F

orw

ard

mo

del-based

example

(con

t.)

⇒

Neyman-P

earson

detector

(W

GN)

⇒

maximizes

the

Signal-to-Noise

Ratio

(SNR)

ρ(

t,

r)

=

1

2 π

Z

Ω

H

(ω

,r

)G

(ω

,r

)S

(ω

)e

j

ω

t

dω

2

Z

Ω

|H

(ω

,r

)|

₂

P

nn

(ω

)dω

or

the

SNR

estimate

ˆρ(

t,

r)

=

E







|

Z

Ω

˜G

∗

(ω

,r

)S

∗

(ω

)Y

(ω

,r

o

)e

j

ω

(

t

−

τ

)

dω

|

₂

Z

Ω

| ˜G

∗

(ω

,r

)S

∗

(ω

)|

₂

P

nn

(ω

)dω







(11)

−100 −80 −60 −40 −20 0 20 40 60 80 100 −5 0 5 10 15 20 angulo (graus) b(t)

X

source

Acoustic Model

Correlator

y

p(r,z)

b(r,z)

(r,z)

environment

Ambiguity surface

acoustic pressure

receiver

(12)

INTIFANTE’00 Experimental Site

-9˚ 30' -9˚ 30' -9˚ 15' -9˚ 15' -9˚ 00' -9˚ 00' -8˚ 45' -8˚ 45' -8˚ 30' -8˚ 30' 38˚ 00' 38˚ 00' 38˚ 15' 38˚ 15' 38˚ 30' 38˚ 30' 38˚ 45' 38˚ 45' 39˚ 00' 39˚ 00' -9˚ 30' -9˚ 30' -9˚ 15' -9˚ 15' -9˚ 00' -9˚ 00' -8˚ 45' -8˚ 45' -8˚ 30' -8˚ 30' 38˚ 00' 38˚ 00' 38˚ 15' 38˚ 15' 38˚ 30' 38˚ 30' 38˚ 45' 38˚ 45' 39˚ 00' 39˚ 00' -9˚ 30' -9˚ 30' -9˚ 15' -9˚ 15' -9˚ 00' -9˚ 00' -8˚ 45' -8˚ 45' -8˚ 30' -8˚ 30' 38˚ 00' 38˚ 00' 38˚ 15' 38˚ 15' 38˚ 30' 38˚ 30' 38˚ 45' 38˚ 45' 39˚ 00' 39˚ 00' -9˚ 30' -9˚ 30' -9˚ 15' -9˚ 15' -9˚ 00' -9˚ 00' -8˚ 45' -8˚ 45' -8˚ 30' -8˚ 30' 38˚ 00' 38˚ 00' 38˚ 15' 38˚ 15' 38˚ 30' 38˚ 30' 38˚ 45' 38˚ 45' 39˚ 00' 39˚ 00' -9˚ 30' -9˚ 30' -9˚ 15' -9˚ 15' -9˚ 00' -9˚ 00' -8˚ 45' -8˚ 45' -8˚ 30' -8˚ 30' 38˚ 00' 38˚ 00' 38˚ 15' 38˚ 15' 38˚ 30' 38˚ 30' 38˚ 45' 38˚ 45' 39˚ 00' 39˚ 00' INTIFANTE’99 Area

LISBON

Sesimbra

Setubal

Espichel Cape

(13)

Ev

en

ts

2 a

nd

5 :

bat

h

ym

etr

y

and

runs

(14)

Em

pir

ical

Ort

hogo

nal

F

uncti

ons

c

EOF

=

ˆc

+

N

X

n

=1

α

n

u

n

ˆN

=

mi

n

N

{

P

N

n

=1

λ

2 n

P

M

m

=1

λ

2 m

>

0 .8

}

1506

1510

1515

0

20

40

60

80

100

119 Soundspeed (m/s)

Depth (m)

−0.2

−0.1

0

0.1

0

20

40

60

80

100

119 Soundspeed (m/s)

Depth (m)

1st EOF

2nd EOF

(15)

P

h

ysical

mo

del

-NW/

N

E

tr

a

c

k

0

60

119 Depth (m)

Range (km)

Source (63 m)

Subbottom

0

2

5.7 1506

1518

m/s

α

=0.8 dB/

λ

ρ

=1.9 g/cm

3 1800 m/s

VA

Sediment

2.0 m

1650 m/s

α

=0.8 dB/

λ

ρ

=1.9 g/cm

3

(16)

V

er

ti

ca

l

Line

A

r

ra

y

(17)

Radio

Buo

y:

depl

o

yme

n

t

and

se

tup

(18)

Sound

so

urce

and

e

mit

te

d

si

gnal

s

(19)

Inversion results for Event 2

289.580 289.6 289.62 289.64 289.66 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Julian Time Bartlett Power 289.58 289.6 289.62 289.64 289.66 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Julian Time Range (km) 289.58 289.6 289.62 289.64 289.66 10 20 30 40 50 60 70 80 90 100 Julian Time Depth (m) 289.5885 289.6 289.62 289.64 289.66 86 87 88 89 90 91 92 93 94 95 Julian Time Sensordepth (m) 289.58 289.6 289.62 289.64 289.66 1520 1540 1560 1580 1600 1620 1640 1660 1680 1700 Julian Time Speed in sed. (m/s) 289.58 289.6 289.62 289.64 289.66 2 4 6 8 10 12 14 Julian Time Sediment thickness (m) 289.58 289.6 289.62 289.64 289.66 1550 1600 1650 1700 1750 1800 1850 1900 Julian Time Speed in sub−bottom (m/s) 289.58 289.6 289.62 289.64 289.66 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 Julian Time

Array tilt (rad)

289.58 289.6 289.62 289.64 289.66 −20 −15 −10 −5 0 5 10 15 20 Julian Time α1 289.58 289.6 289.62 289.64 289.66 −20 −15 −10 −5 0 5 10 15 20 Julian Time α2

(20)

Inversion results for Event 5

290.94 290.96 290.98 291 291.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Julian Time Bartlett Power 290.94 290.96 290.98 291 291.02 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Julian Time Range (km) 290.94 290.96 290.98 291 291.02 10 20 30 40 50 60 70 80 90 100 Julian Time Depth (m) 290.94 290.96 290.98 291 291.02 85 86 87 88 89 90 91 92 93 94 95 Julian Time Sensordepth (m) 290.94 290.96 290.98 291 291.02 1520 1540 1560 1580 1600 1620 1640 1660 1680 1700 Julian Time Speed in sed. (m/s) 290.94 290.96 290.98 291 291.02 2 4 6 8 10 12 14 Julian Time Sediment thickness (m) 290.94 290.96 290.98 291 291.02 1550 1600 1650 1700 1750 1800 1850 1900 Julian Time Speed in sub−bottom (m/s) 290.94 290.96 290.98 291 291.02 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 Julian Time

Array tilt (rad)

290.94 290.96 290.98 291 291.02 −20 −15 −10 −5 0 5 10 15 20 Julian Time α1 290.94 290.96 290.98 291 291.02 −20 −15 −10 −5 0 5 10 15 20 Julian Time α2

(21)

Ti

de

ev

ol

uti

on

and

V

A

m

o

ving

(22)

In

v

erse

mo

delling

example

T

raining

The

pair

{

r

j

,x

(r

j

)}

allo

ws

for

calculating

co

efficien

ts

c

ij

of

the

neural

net

(RBF)

net

w

ork

parameterized

by

functions

φ

Data

In

v

ersion

Giv

en

y(

r

o

₎

calculate

ˆr

o

=

N

X

j

=1

c

ij

φ

(k

y(

r

o

)

−

x

(r

j

)k

)

(23)

In

v

er

se

mo

del

li

ng

e

x

am

ple

(con

t.)

AD

V

E

N

T

U

RE

BA

NK

se

a

tr

ial

19

94 -S

icil

y

12 05 30" 12 08 00" 12 10 30" 12 13 00" 12 15 30" 37 00 00" 37 00 30" 37 01 00" 37 01 30" 37 02 00" 37 02 30" 37 03 00" 37 03 30" 37 04 00" Longitude Latitude o Start o Stop + 9:15 _{+ 9:30} + 9:45 + 10:00 + 10:15 + 10:30 + 10:45 + 11:00 + 11:15 + 11:30

*

core 281

core 280 geophone array

p-v

e

lo

cit

y

1400 1600 1800 2000 2200 09:43 09:48 09:54 09:59 10:02 120 130 140 150

Local time during run (hh:mm)

Depth (m) (a) 0 50 100 150 200 09:43 09:48 09:54 09:59 10:02 120 130 140 150

Depth (m) (b)

s-v

e

lo

ci

ty

0 200 400 600 800 09:43 09:48 09:54 09:59 10:02 120 130 140 150

Depth (m) (a) 0 50 100 150 200 09:43 09:48 09:54 09:59 10:02 120 130 140 150

Depth (m)