Time reversal methods in unknown medium and inverse problems

0.3 MB·English

by  Kenrick Bingham

Time reversal methods in unknown medium
and inverse problems
7
0
0
Kenri
k Bingham
2
Yaroslav Kurylev
n
a
Matti Lassas
J

4 Samuli Siltanen

]
P January 4, 2007
A
.
h
t
a Abstra
t
m
[ A novel method to solve inverse problems for the wave equation

is introdu
ed. The method is a
ombination of the boundary
ontrol
1
v method and an iterative time reversal s
heme, leading to adaptive
3
imaging of
oe(cid:30)
ient fun
tions of the wave equation using fo
using
3
1 waves inunknown medium. The approa
h is
omputationally e(cid:27)e
tive
1 sin
e the iteration lets the medium do most of the pro
essing of the
0
data.
7
0 The iterative time reversal s
heme also gives an algorithm for
on-
/ stru
ting boundary
ontrols for whi
h the
orresponding (cid:28)nal values
h
t are as
lose as possible to the (cid:28)nal values of a given wave in a part of
a
m the domain, and as
lose as possible to zero elsewhere. The algorithm
does not assume that the
oe(cid:30)
ients of the wave equation are known.
:
v
i
X
Keywords: Inverse problems, wave equation,
ontrol, time reversal.
r
a AMS
lassi(cid:28)
ation: 35R30, 93B05.
1. Introdu
tion
We present a novel inversion method for the wave equation. Suppose that
we
an send waves from the boundary into an unknown body with spatially
c(x)
varying wave speed . Using a
ombination of the boundary
ontrol (BC)
method and an iterative time reversal s
heme, we show how to fo
us waves
x c(x ) c
0 0
near a point inside the medium and simultaneously re
over if is
1
c(x)
isotropi
. In the anisotropi

ase we
an re
onstru
t up to a
hange of

oordinates.
In the
lassi
al BC method material parameters are re
onstru
ted using
hyperboli
te
hniques, see [1,9,10,11,25,26,27℄. Straightforwardnumeri
al
implementation of the BC method is
omputationally demanding. Inspired
by Isaa
son's iterative measurement s
heme [22℄ for ele
tri
al impedan
e to-
mography (EIT), we
ombine the BC method with an adaptive time reversal
iteration that lets the medium do most pro
essing of the data.
Traditional time reversal methods re
ord waves, invert them in time, and
send them ba
k into the medium. If the re
orded signal originated from
point sour
es or was re(cid:29)e
ted from small s
atterers in the medium, the time
reversed waves fo
us at the sour
e or s
atterer points. This is useful for time
reversal mirrors in
ommuni
ation te
hnologies and medi
al therapies. For
early referen
es on time reversal (using ultrasound in air), see the seminal
works of Fink [19, 18, 17℄. For a mi
rolo
al dis
ussion of time reversal, see
Bardos [3, 4℄, and for another mathemati
al treatment see Klibanov [29℄.
Time reversal in known ba
kground medium with random (cid:29)u
tuations
has also been extensively studied [5, 6, 7, 8, 13, 14℄ and applied to medi
al
imaging, non-destru
tive testing and underwater a
ousti
s. These methods
are outside the s
ope of this paper.
Iterativetimereversalhasbeenused to(cid:28)ndbestmeasurementsforinverse
problems. By (cid:16)best(cid:17) we here mean (cid:16)optimal for dete
ting the presen
e of an
obje
t(cid:17). This distinguishability problem has been studied for (cid:28)xed-frequen
y
problems in EIT [22℄ and a
ousti
s
attering [34℄. The
onne
tion between
optimal measurements and iterative time-reversal experiments was pointed
out in [34, 35, 36℄. For the wave equation, the best measurement problem
has been studied in [16℄, where it has been shown that the optimal in
ident
(cid:28)eld for probing a half spa
e
an be found by an iterative pro
ess involving
time reversal mirrors.
M ⊂LetRu3s des
ribe our hybrid me∂thMod in a simcp(lxe)
ase. Take a
ompa
t set
with smooth boundary , and let be a s
alar-valued wave
M
speed in . Consider the wave equation
u −c(x)2∆u = 0 M ×R ,
tt +
in (1)
u| = 0,
t=0
u | = 0,
t t=0
−c(x)−2∂ u = f(x,t) ∂M ×R ,
n +
in
∂ uf =
n
where denotes the Eu
lidean normal derivative. We denote by
uf(x,t) f
the solution of (1)
orresponding to a given boundary sour
e term .
2
T > 0 M
Take larger than the diameter of in travel time metri
, so
M
that any point inside
an be rea
hed by waves sent from the boundary
T f
in time less than . We
onsider imposing a boundary sour
e on the set
∂M × (0,2T) uf
and measuring the boundary value of the resulting wave
0 < t < 2T
during the time . All su
h boundary measurements
onstitute the
Neumann-to-Diri
hlet map
Λ : f 7→ uf| ,
2T ∂M×(0,2T)
that models a variety of physi
al measurements [28℄. Instead of the full map
Λ f f
2T
we only need here the values for a spe
i(cid:28)

olle
tion of sour
es given
by an iterative pro
ess.
L2(∂M ×[0,2T])
We need three spe
ial operators on the fun
tion spa
e .
First, the time reversal operator
Rf(x,t) = f(x,2T −t),
se
ond, the time (cid:28)lter
1 min(2T−t,t)
Jf(x,t) = f(x,s)ds,
2
Z0
and (cid:28)nally the restri
tion operator
Pf(x,t) = χ (x,t)u(x,t),
B
χ B ⊂ ∂M ×[0,2T]
B
where is the
hara
teristi
fun
tion of a set .
We de(cid:28)ne the pro
essed time reversal iteration
1
F := P(RΛ RJ −JΛ )f,
2T 2T
ω
a := Λ (h ), b := Λ (RJh ),
n 2T n n 2T n
(2)
α 1
h := (1− )h − (PRb −PJa )+F,
n+1 n n n
ω ω
f ∈ L2(∂M × [0,2T]) α,ω > 0
where and are parameters. Starting with
h = 0 h(α) = lim h
0 n→∞ n
the iteration (2)
onverges to a limit . In the
ase
B = ∂M ×[T −s,T]
we have
limuh(α)(x,T) = 1−χ (x) uf(x,T) L2(M),
N
α→0 in (3)
(cid:0) (cid:1)
N ⊂ M x ∈ M ∂M
where is the set of points with travel time distan
e to at
s N
most . In other words, is the maximal region where the waves sent from
s
the boundary
an propagate in time .
3
y ∈ ∂M γ
y,ν
Let and denote by the geodesi
(in travel time metri
)
y
starting ffrom in the normal(yd,irTe−
tsio)n∈a∂nMd p×aRrametrised by ar
lengtt≤h.TT−akse
+
asour
e
on
entratednear andvanishingfor .
uf(x,T) N
Then is supported in the union of the set and a neighbourhood
x = γ (s)
0 y,ν
of the point , and the limit (3) vanishes outside a neighbourhood
x x T
0 0
of . This way our iteration produ
es a wave lo
alized near at time .
Below we introdu
e an alternative
onstru
tion of fo
using waves for gen-
f
eral sour
es by applying the pro
essed time reversal iteration with two
B
di(cid:27)erent sets and using the di(cid:27)eren
e of the resulting boundary values.
We show also that the limits(3)
an be used to determine the travel time
x = γ (s) ∈ M
y,ν
distan
e between an arbitrary point and any boundary
z ∈ ∂M
point . It follows that the pro
essed time reversal iteration gives
an algorithm for determining the wave speed in the medium. Furthermore,
f uh(x,t)
given a sour
e , our iterationallows us to(cid:28)nd a wave havingat time
t = T χ (x)uf(x,T ) T
1 N 2 2
approximately the value for any without knowing
the material parameters of the medium. Analogous methods without the
χ (x)
N
multiplier have been developed before in [23, 31℄.
The paper is organised as follows. In Se
tion 2 we formulate our main
results. In Se
tion 3 we prove the
onvergen
e of the pro
essed time reversal
iteration. Se
tion 4 is devoted to the analysis of the fo
using properties of
the waves, and in Se
tion 5 we show how to re
onstru
t the metri
using the
boundary distan
e fun
tion. In the last se
tion we dis
uss our method in the

ase of noisy measurements.
2. De(cid:28)nitions and main results
M ⊂ Rm m ≥ 1
Letus
onsiderthe
losure , ,ofanopensmoothset,ora(non-
(M,g) m

ompa
t or
ompa
t)
omplete Riemannian manifold of dimension
with a non-empty boundary. For simpli
ity, we assume that the boundary
∂M u
is
ompa
t. Let solve the wave equation
u (x,t)+Au(x,t) = 0 M ×R ,
tt +
in (4)
u| = 0, u | = 0,
t=0 t t=0
Bν,ηu|∂M×R+ = f.
f ∈ L2(∂M ×R ) A
+
Here, is a real valued fun
tion, is a formally self-adjoint
ellipti
partial di(cid:27)erential operator of the form (in lo
al
oordinates in the
M

ase when is a manifold)
m
∂ ∂v
Av = − µ(x)−1|g(x)|−21 µ(x)|g(x)|21gjk(x) (x) +q(x)v(x)
∂xj ∂xk (5)
j,k=1 (cid:18) (cid:19)
X
4
[gjk] c I ≤ [gjk(x)] ≤ c I
0 1
where is a smooth real positive de(cid:28)nite matrix, ,
c ,c > 0 |g| = ([g ]) [g ] [gjk]
0 1 jk jk
, det , where is the inverse matrix of , and
µ(x) ≥ c > 0 q(x)
2
and are smooth real valued fun
tions. Also,
B v = −∂ v +ηv
ν,η ν
η : ∂M → R
where is a smooth bounded fun
tion and
m
∂
∂ v = µ(x)gjk(x)ν v(x),
ν k∂xj
j,k=1
X
ν(x) = (ν ,ν ,...,ν ) ∂M
1 2 m
where m gjkνiνs t=he1interior
o-normal ve
tor (cid:28)eld of
normalised so that j,k=1 j k . A parti
ular example is the operator
P A = −c2(x)∆+q(x)
0
(6)
∂ v = c(x)−m+1∂ v ∂ v
ν n n
for whi
h , where is the Eu
lidean normal derivative
v
of .
We denote the solutions of (4) by
uf(x,t) = u(x,t).
For the initial boundary value problem (4) we de(cid:28)ne the response operator
Λ
(or non-stationary Robin-to-Diri
hlet map) by setting
Λf = uf|∂M×R+. (7)
Λ
T
We also
onsider the (cid:28)nite time response operator
orresponding to time
T > 0
,
Λ f = uf| .
T ∂M×(0,T)
(8)
Λ : L2(∂M × (0,T)) → H1/3(∂M × (0,T))
T
By [39℄, the map is bounded,
Hs(∂M × (0,T)) ∂M × (0,T)
where denotes the Sobolev spa
e on . Below
Λ L2(∂M ×(0,T))
T
we
onsider as a bounded operator that maps to itself.
g (x)
jk
We also need some other notation. The matrix , that is, the inverse
gjk(x) M
of the matrix , is a Riemannian metri
in that is
alled the travel
time metri
. The reason for this is the waves propagate with speed one with
ds2 = g (x)dxjdxk d(x,y)
respe
t to the metri
jk jk . We denote by the
distan
e fun
tion
orrespondinPg to this metri
. For the wave equation we
L2(M,dV )
µ
de(cid:28)ne the spa
e with inner produ
t
hu,vi = u(x)v(x)dV (x),
L2(M,dVµ) µ
ZM
5
dV = µ(x)|g(x)|1/2dx1dx2...dxm
µ
where .
t > 0 Γ ⊂ ∂M
For and , let
M(Γ,t) = {x ∈ M : d(x,Γ) ≤ t},
(9)
Γ t
be the domainLo2f(iBn(cid:29))u=en{
fe∈ofL2(a∂tMtim×eR .) : (f) ⊂ B} B ⊂ ∂M×R
+ +
We denote supp , ,
Γ ⊂ ∂M
identifyinfg ∈funL
t2i(oΓns×aRnd)their zero
ontinuations. When is open
+
set and , it is well known (see e.g. [21℄) that the wave
uf(t) = uf(·,t) M(Γ,t)
is supported in the domain ,
uf(t) ∈ L2(M(Γ,t)) = {v ∈ L2(M) : (v) ⊂ M(Γ,t)}.
supp
2.1. Results on
onvergen
e of pro
essed time reversal iteration.
f uf(x,T)
Our obje
tive is to (cid:28)nd a boundary sour
e su
h that the wave is
x
0
lo
alized in a neighbourhood of a single point . Note that in this paper
(u(x,T),u (x,T))
t
we do not fo
us the pair , but only the value of the wave,
u(x,T)
.0 < T < T Γ ⊂ ∂M f ∈ L2(Γ×R )
0 +
Let , be ohp(αen) ∈setLa2(n∂dM ×R ) α > 0. Our (cid:28)rst aim
+
is to
onstru
t boundary values , su
h that
limuh(α)(T) = χ uf(T)
α→0 M(∂M,T0) (10)
L2(M) χ (x) X
X
in . Here is the
hara
teristi
fun
tion of the set . Then
limuf−h(α)(T) = (1−χ )uf(T)
M(∂M,T0)
α→0
M(Γ,T) \ M(∂M,T ) T → T
0 0
is supported in (see Figure 1). When and
Γ → {z} M(Γ ,T)\
j j
M(∂M,T t)hen under suitable assumptions dis
fuss∈edLl2a(tΓer×theRse)t
0 j +
goes to a single point. Thus for , the waves
(1−χ )uf(T)
M(∂M,T0) lo
alize to a small neighbourhood of a single point. For
later use we will
onsider (10) below in a more general setting.
h(α)
Next we explain a pro
edure to (cid:28)nd boundary values . Denote
R f(x,t) = f(x,2T −t),
2T
(11)
J h(x,t) = J(s,t)h(x,s)ds
2T
Z[0,2T]
J(s,t) = 1χ (s,t)
where 2 L ,
L = {(s,t) ∈ R ×R : t+s ≤ 2T, s > t}.
+ +
(12)
6
M(Γ,T)\M(∂M,T )
0
Figure 1: The set . Our goal is to (cid:28)nd waves sent from
the boundary that fo
us into this area.
M(∂M,T )
0
M(Γ,T)
s s
Γ
R = R J = J
2T 2T
We
all the time reversal map and the time (cid:28)lter. We
P = P : L2(∂M ×[0,2T]) → L2(∂M ×[0,2T])
B
denote by the multipli
ation
operator
P f(x,t) = χ (x,t)f(x,t),
B B
B = J (Γ ×[T −T ,T]), Γ ⊂ ∂M 0 ≤ T ≤ T
where j=1 j j j are open sets and j .
Λ R , J L2(∂M ×[0,2T])
2T 2T 2T
Next, we
oSnsider , and as operators from to
itself.
α ∈ (0,1) ω > 0
Let and be a su(cid:30)
iently large
onstant. We de(cid:28)ne
a ,b ∈ L2(∂M×[0,2T]) h = h (α) ∈ L2(∂M×[0,2T])
n n n n
and by the iteration
a := Λ (h ), b := Λ (RJh ),
n 2T n n 2T n
(13)
α 1
h := (1− )h − (PRb −PJa )+F,
n+1 n n n
ω ω
a = 0 b = 0 h = 0
0 0 0
with , , , and
1
F = P(RΛ RJ −JΛ )f.
2T 2T
ω
We say that this iteration is the pro
essed time reversal iteration on time-
[0,2T] P f ∈ L2(∂M ×[0,2T])
interval with proje
tor and starting point .
a b
n n
Here
orrespondstothe(cid:16)iteratedmeasurement(cid:17), tothe(cid:16)time(cid:28)ltered
h h a
n+1 n n
and time-reversed measurement(cid:17) and to post-pro
essing of , and
b
n
using time reversal and time (cid:28)ltering.
7
T > 0 ∂M
Theorem 1 Let . Assume we are given and the response operator
Λ Γ ⊂ ∂M j = 1,...,J 0 ≤ T ≤ T
2T j j
. Let , be non-empty open sets, , and
B = J (Γ ×[T −T ,T]). f ∈ L2(∂M×R ) ω
j=1 j j Let + and let for large enough
α ∈ (0,1) h = h (α)
n n
and S fun
tions be de(cid:28)ned by the pro
essed time reversal
P f = f|
B 0 ∂M×(0,2T)
iteration (13) with proLje2
(∂toMr ×Ran)d starting point . These
+
fun
tions
onverge in ,
h(α) = lim h (α)
n
n→∞
and the limits satisfy
limuh(α)(x,T) = χ (x)uf(x,T)
N
α→0
L2(M) N = J M(Γ ,T ) ⊂ M
in , where j=1 j j .
S
This theorem is proven later in Se
tion 3. Before that we
onsider its

onsequen
es.
γ
x,ξ
2.2. Results on fo
using of the waves Let us
onsider a geodesi
(M,g) γ (0) = x γ˙ (0) = ξ
x,ξ x,ξ
in parametrised along ar
length where , with
kξk = 1 ν = ν(z) z ∈ ∂M ∂M
g
. Let , be the unit interior normal ve
tor of
(M,g) τ(z) ∈ (0,∞] t < τ(z)
in . There is a
riti
al value , su
h that for the
γ ([0,t]) t > τ(z)
z,ν
geodesi
is the unique shortest geodesi
, and for it is no
γ (t) ∂M
z,ν
longer a shortestΓge→ode{szi}
froΓm its⊂enΓdpoint ∞ Γ t=o {z}.
We say that j if j+1 j and j=1 j .
Theorem1yieldsthefollowingresultstellTingthatwe
anprodu
efo
using
b b
waves, that is, wave that fo
us to a single point.
z ∈ ∂M 0 < T0 < T < T x = γzb,ν(T)
CΓo⊂rol∂laMryj2∈LeZt and z ∈ ∂.MLet Γ → a{nz}d
j + j
, be open neighbourhoods of su
h that
j → ∞ b b
b b
when f ∈ C.∞(∂M ×R ) h (α;T ,j)
0 + n 0 b b
Let . Let be the fun
tions obtained from
P B = (Γ ×[T −
B j
the pro
essed time reversal iteration (13) with proje
tor ,
T,T]) ∪ (∂M × [T − T ,T]) f = f|
0 0 ∂M×(0,2T)
and starting point . Similarly,
h′ (α;T ,j)
n 0
let be the fun
tions obtained from the pro
essed time reversal
b PB′ B′ = ∂M ×[T −T0,T]
iteration (13) with proje
tor , and starting point
f
0
. Let
h (α;T ,j) = f −h (α;T ,j)+h′ (α;T ,j).
n 0 0 n 0 n 0
T < τ(z)
If thene
b b (14)
1 e
lim lim lim lim uhn(α;T0,j)(T) = C0(x)uf(x,T)δxb(x)
T0→Tbj→∞α→0n→∞ (T −T0)(m+1)/2
b b
b
8
M
PSfrag repla
emenFtsigure 2: Subsets of used in the proof of Corollary 2.
x
T
b
T
b 0
∂M z Γj
b
D′(M) C (x) > 0 f T > τ(z)
0
in where does not depend on . If the limit (14)
is zero.
b
b b
M(Γ ,T) \ M(∂M,T ) xˆ
j 0
In Figure 2, the set tends to the point when
j → ∞ T → Tˆ
0
and . This turns out to be essential in the proof of Corollary
b
2.
δxb (M,g)
Above, is Dira
's delta distribution on su
h that
δxb(x)φ(x)dVµ = φ(x), φ ∈ C0∞(M).
ZM
b f,h ∈ L2(∂M ×
2.3. Inner produ
ts. Later, we show that that using
[0,2T]) Λ
2T
and the boundary measurements we may
ompute via an expli
it
integral the inner produ
t
uf(x,T)uh(x,T)dV (x) = (Kf)(x,t)h(x,t)dS (x)dt,
µ g
(15)
ZM Z∂M×[0,2T]
dS ∂M
g
where is the Riemannian surfa
e volume of and
K = K := R Λ R J −J Λ
2T 2T 2T 2T 2T 2T 2T
(16)
Λ
2T
so that terms (15)
an be found using measurement operator and simple
R J
2T 2T
basi
operations like time reversal and the time (cid:28)lter operator .
∂M
In the above formula the Riemannian surfa
e volume of
an be
om-
∂M Λ
2T
puted using the intrinsi
metri
of , whi
h is determined by the map .
Indeed, its follows from Tataru's unique
ontinuation prin
iple (see [38, 40℄,
Λ
2T
see also Theorem 4 below) that the S
hwartz kernel of is supported in
E = {(x,t,x′,t′) ∈ (∂M × [0,2T])2 : t − t′ ≥ d(x,x′)}
the set and that
∂E ∂E
the boundary is in the support. The set determines the distan
es
z,z′ ∈ ∂M (M,g)
of points with respe
t to the metri
of , and thus also the
(∂M,g )
∂M
distan
e with respe
t to the intrinsi
metri
of the boundary .
9
2.4. Re
onstru
tionof materialparameter fun
tions. Itiswellknown
Λ µ g
jk
that the response map
an not generally determine
oe(cid:30)
ients and
be
ause of two transformations dis
ussed below.
First, we one
an introdu
e a
oordinate transformation, that is a di(cid:27)eo-
F : M → M F|
∂M
morphism su
h that the boundary value is the identity
g = F g
∗
operator. Then the push forward metri
, that is,
m ∂xp ∂xq
g (y) = g (x), y = F(x)
jk ∂yj ∂yk pqe
p,q=1
X
e µ = µ◦F−1 q = q◦F−1 η = η
and the fun
tions , , and determine the operator
A A A
of the form (5) su
h that response operators for and are the same (for
e e e
this, see [25, 26, 27℄).
e u(x,t) →eκ(x)u(x,t)
Se
ond, one
an do the gauge transformation where
κ ∈ C∞(M)
is stri
tly positive fun
tion. In this transformation the operator
A A
κ
is transforms to the operator de(cid:28)ned be
1
A w := κA( w)
e κ e
κ
and the operator in boundarey
onditioen is transformed to Bν,ηb, with
η = η −κ−1∂ κ.
ν
Λ A Λ
κ κ
Then the response operator of
oin
ides with the response operator
A b
of .
e e κ = µ−1
By above transformations, making a gauge transformation with ,
A A
κ
we
ome to operator having the same response operator as and that
µ(x) = 1

an be represented in the form (5) with . It turns out that this is
e
the only sour
e of non-uniqueness.
µ = 1 ∂M
Corollary 3 Assume that and that we are given the boundary and
Λ
the response operator . Then using the the pro
essed time reversal iteration
(M,g)
we
an (cid:28)nd
onstru
tively the manifold upto an isometry and on it
A
the operator uMniq⊂uelRy.m m ≥ 2 A M
Moreover, if , , and is of the form (6), given the set
Λ c(x) q(x) M
and the response operator we
an determine and uniquely in .
(M,g) A
We note that the unique determinationof and iswell known, see
e.g. [11, 25, 26℄ and referen
es therein. The novelty of Corollary 3 is that in
the
onstru
tionbased on pro
essed timereversal iterationtheuse ofiterated
measurement avoid many
omputationally demanding steps used required in
[26℄. Anexampleofsu
h step isaGram-S
hmidtorthogonalisationofbasisof
L2(Γ×[t ,t ]) Γ ⊂ ∂M
1 2
spa
es , with respe
t to an inner produ
t determined
Λ
2T
by . This is a typi
al step used in traditional Boundary Control method
that is very sensitive for measurement errors.
10