Ci a ion: Mendio oz, A.; Cas elo, A.;
Celo io, R.; Salaza , A. Ve ical
C acks Exci ed in Lock-in
Vib o he mog aphy Expe imen s:
Iden i ica ion o Open and
Inhomogeneous Hea Fluxes. Senso s
2022,22, 2336. h ps://doi.o g/
10.3390/s22062336
Academic Edi o : Giacomo Oli e i
Recei ed: 25 Feb ua y 2022
Accep ed: 16 Ma ch 2022
Published: 17 Ma ch 2022
Publishe ’s No e: MDPI s ays neu al
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Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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senso s
A icle
Ve ical C acks Exci ed in Lock-in Vib o he mog aphy
Expe imen s: Iden i ica ion o Open and Inhomogeneous
Hea Fluxes
A an za Mendio oz 1,*, Alazne Cas elo 1, Rica do Celo io 2and Agus ín Salaza 1
1Depa amen o de Física Aplicada, Escuela de Ingenie ía de Bilbao, Uni e sidad del País Vasco UPV/EHU,
Plaza Ingenie o To es Que edo 1, 48013 Bilbao, Spain; [email p o ec ed] (A.C.);
[email p o ec ed] (A.S.)
2
Depa amen o de Ma emá ica Aplicada, EINA/IUMA, Uni e sidad de Za agoza, Campus Río Eb o, Edi icio
To es Que edo, 50018 Za agoza, Spain; celo io@uniza .es
*Co espondence: a an za.mendio [email p o ec ed]
Abs ac :
Lock-in ib o he mog aphy has p o en o be e y use ul o cha ac e izing kissing c acks
p oducing ideal, homogeneous, and compac hea sou ces. He e, we app oach eal si ua ions by
add essing he cha ac e iza ion o non-compac (s ip-shaped) hea sou ces p oduced by open c acks
and inhomogeneous luxes. We p opose combining lock-in ib o he mog aphy da a a se e al
modula ion equencies in o de o ga he pene a ion and p ecision da a. The app oach consis s in
in e ing su ace empe a u e ampli ude and phase da a by means o a leas -squa es minimiza ion
algo i hm wi hou p e ious knowledge o he geome y o he hea sou ce, only assuming knowledge
o he e ical plane whe e i is con ined. We p opose a me hodology o sol e his ill-posed in e se
p oblem by including in he objec i e unc ion penal y e ms based on he expec ed p ope ies o he
solu ion. These e ms a e desc ibed in a comp ehensi e and in ui i e manne . In e sions o syn he ic
da a show ha he geome y o non-compac hea sou ces is iden i ied co ec ly and ha he con ou s
a e ounded due o he penaliza ion. Inhomogeneous smoo hly a ying luxes a e also quali a i ely
e ie ed, bu s eep a ia ions o he lux a e ha d o eco e . These indings a e con i med by
in e sions o expe imen al da a aken on calib a ed samples. The p oposed me hodology is capable
o iden i ying hea sou ces gene a ed in lock-in ib o he mog aphy expe imen s.
Keywo ds:
c ack cha ac e iza ion; lock-in ib o he mog aphy; ul asound-exci ed he mog aphy;
sonic-in a ed; in e se p oblems; nondes uc i e es ing
1. In oduc ion
The mog aphic non-des uc i e es ing (NDT) me hods ha e demons a ed a high
po en ial o su ace and subsu ace de ec de ec ion and cha ac e iza ion [
1
]. The mo-
g aphic echniques consis in gene a ing a he mal unbalance in he ma e ial and eco ding
he e olu ion o he su ace empe a u e dis ibu ion by means o an in a ed came a.
The he mal pe u ba ion can be ca ied ou by exci ing he ma e ial wi h ligh (op ically
exci ed in a ed he mog aphy (IRT)), ul asounds ( ib o he mog aphy, he mosonics,
sonic IR), o elec omagne ically (induc i e he mog aphy). The mos popula modali y
o in a ed he mog aphy uses ligh o hea he ma e ial su ace. The p esence o de ec s
pe u bs he subsequen hea di usion, gi ing ise o anomalies in he su ace empe a u e
dis ibu ion wi h espec o a sound ma e ial. Consequen ly, he signa u e o he de ec
needs o be iden i ied in a p e-exis en empe a u e ield caused by he exci a ion. In his
ega d, ib o he mog aphy has a ac ed a g ea deal o in e es in ecen imes due o i s
de ec -selec i e na u e. In ib o he mog aphy, he ma e ial is exci ed wi h high-ampli ude
ul asounds. In non- iscoelas ic ma e ials, he bulk dissipa ion is small and he mechanical
ene gy is con e ed in o hea a c acks, mainly due o ic ion be ween he c ack lips. This
Senso s 2022,22, 2336. h ps://doi.o g/10.3390/s22062336 h ps://www.mdpi.com/jou nal/senso s
Senso s 2022,22, 2336 2 o 20
he mal ene gy di uses in he ma e ial and e en ually eaches he sample su ace, p oduc-
ing a ho egion abo e he de ec , in a cold en i onmen . Fo a comp ehensi e desc ip ion
o ib o he mog aphy, see [
2
]. I compa ed o op ically exci ed he mog aphy, he e is a
double ad an age in gene a ing an in e nal hea sou ce a he de ec . Fi s , he esul ing
su ace empe a u e dis ibu ion is backg ound- ee and only due o he hea gene a ed a
he de ec . Second, he hea a els only one way o each he su ace, which allows sensing
deepe egions in he ma e ial. These ad an ages apply o any NDT me hod gene a ing hea
a de ec s, o ins ance, he iden i ica ion o me allic inclusions embedded in an elec ical
insula o when he pa s a e exci ed by eddy cu en s (induc i e he mog aphy).
The hea gene a ed a c acks in ib o he mog aphy is gene ally non-uni o m. Ac ually,
in su ace-b eaking open c acks, he egion whe e he c ack lips a e no in con ac does no
p oduce hea (unless an induced b ea hing mode b ings he wo su aces in o con ac [
3
])
and, close o he c ack bo de , he closu e s esses migh lock he c ack aspe i ies, hus
p e en ing hea p oduc ion [
4
]. In e media e egions whe e he lips a e in con ac and
in ela i e mo ion p oduce hea , gene ally wi h a non-uni o m dis ibu ion. Acco dingly,
he geome y o his lux dis ibu ion is he in o ma ion accessible om empe a u e da a
measu ed a he su ace, a he han he c ack geome y.
The iden i ica ion o he shape o in e nal hea sou ces om su ace empe a u e da a
is a se e ely ill-posed in e se p oblem due o he di usi e na u e o hea p opaga ion. The
s a egies o sol e his p oblem in a gene al o m can be oughly ca ego ized in o leas -
squa es minimiza ion me hods, s a is ical me hods, and he new “ i ual wa e concep ”
me hod. In leas -squa es minimiza ion, he cause o he obse ed empe a u e dis ibu ion
(he e, he inne hea sou ce dis ibu ion) is iden i ied by minimizing he squa ed L2-no m
o he di e ence be ween he da a and he p edic ion o he model ( esidual). The ill-posed
cha ac e o he in e se p oblem makes his minimiza ion uns able, and in o de o ind a
sensible solu ion, he in e sion needs o be egula ized. An e icien s a egy o s abilize
he in e sion and inco po a e in o ma ion on he cha ac e is ics o he solu ion consis s
in adding one o se e al e ms o he esidual ha p o ide s abili y o he minimiza ion.
The minimiza ion can be ca ied ou by ei he global me hods (neu al ne wo ks [
5
], gene ic
algo i hms [
6
], pa icle swa m op imiza ion [
7
]), which sea ch o he solu ion o e la ge
anges o pa ame e alues, o local me hods (Gauss- ype o conjuga e g adien [
8
]), which
modi y he s a ing pa ame e alues in a con olled way. Global me hods a e aimed a
inding he ough global minimum bu a e less p ecise in inding he op imum solu ion
and en ail a high compu a ional cos , whe eas local me hods may ind he minimum mo e
p ecisely bu isk ge ing apped a local minima.
In s a is ical me hods [
9
–
11
], he solu ion is cha ac e ized by ea u ing he highes
p obabili y om a s a is ical poin o iew. Knowledge o he s a is ical unce ain y o he
da a se is equi ed, as well as ha ing a o wa d model in o de o calcula e he p obabili y
dis ibu ion o ind he solu ion. Las ly, he ecen ly de eloped i ual wa e concep [
12
] is
con igu ed as a wo-s ep p oblem. The i s p oblem consis s in calcula ing he so-called
i ual wa e, which can be unde s ood as he wa e equa ion solu ion coun e pa o he
ue hea di usion p oblem. Once ound, in he second s ep, back-p ojec ion echniques
allow inding he hea sou ce dis ibu ion.
The main di e ence be ween leas -squa es minimiza ion and s a is ical me hods e sus
he i ual wa e concep is ha he o me need a physical model o desc ibe he di ec
o o wa d p ocess (calcula ion o he su ace empe a u e om knowledge o he hea
sou ces), whe eas he la e does no need modeliza ion o he di ec p oblem.
So a , s a is ical me hods and he i ual wa e concep ha e been applied o cha -
ac e ize olume ic hea sou ces [
12
,
13
]. Leas -squa es minimiza ion app oaches ha e
been implemen ed o cha ac e ize ideal, compac , and homogeneous e ical plana hea
sou ces om lock-in ib o he mog aphy da a [
14
,
15
]. Howe e , he hea gene a ed by eal
c acks does no ollow ideal, compac , and homogeneous dis ibu ions, unlike he sou ces
ea ed in hese p e ious wo ks [
14
,
15
]. Wi h he idea o app oaching p ac ical si ua ions,
in his wo k, we add ess he cha ac e iza ion o hea sou ces ypically gene a ed by eal
Senso s 2022,22, 2336 3 o 20
su ace-b eaking e ical c acks wi h hal -penny shape, as well as inhomogeneous hea
sou ces in ib o he mog aphy expe imen s. We con ine ou s udy o he he mal di usion
p oblem, lea ing aside he mechanisms ha gi e ise o he hea gene a ion. We ocus
on ampli ude-modula ed exci a ion and lock-in de ec ion, as his modali y is aimed a
educing he noise in he da a, which is c ucial in ill-posed in e se p oblems. In Sec ion 2,
we p esen he solu ion o he di ec p oblem o he geome ies add essed. In Sec ion 3, we
p esen a comp ehensi e o e iew o a egula ized leas -squa es minimiza ion app oach,
in o de o gi e some insigh on he meaning o egula iza ion, and we desc ibe he in-
e sion algo i hm. The po en ial and limi a ions o Lasso (L1) [
16
,
17
] and To al Va ia ion
(TV) [
18
,
19
] egula iza ions o iden i y open and inhomogeneous hea sou ces is shown
in Sec ion 4by in e ing syn he ic da a wi h added noise. In Sec ion 5, we p esen he
expe imen al se -up and in e sions o expe imen al da a, discussing he esul s. Finally, in
Sec ion 6, we summa ize and conclude.
2. Di ec P oblem
The di ec p oblem consis s in calcula ing he su ace empe a u e dis ibu ion gene -
a ed by a ce ain dis ibu ion o modula ed hea sou ces (a equency ,
ω
= 2
π
) loca ed in
plane
Π
(x= 0) pe pendicula o he sample su ace (z= 0). We conside ha he sample is
semi-in ini e in he zdi ec ion and in ini e in xand ydi ec ions, wi h he mal conduc i i y
Kand di usi i y D. The geome y is depic ed in Figu e 1a.
Figu e 1.
(
a
) Geome y o he p oblem, wi h hea sou ces in ed; (
b
) de ail o he geome y o he hea
sou ce, ep esen ing an open hal -penny c ack; (c) geome y o a ec angula hea sou ce.
Neglec ing hea losses by con ec ion and adia ion, he complex empe a u e a he
su ace due o he he mal wa es launched a equency om
Ω
can be calcula ed by
in eg a ing he con ibu ion o poin -like modula ed hea sou ces in plane
Π
(con ined in
a ea Ω) [20]:
T (x,y, 0) = x
Π
Q(→
0)
4πK
e−q |→
−→
0|
→
−→
0
dS0=x
Ω
Q(→
0)
4πK
e−q |→
−→
0|
→
−→
0
dS0(1)
whe e
Q(→
0)
is he posi ion-dependen lux ampli ude (null ou side
Ω
) and
q =p2πi /D
is he he mal wa e ec o . In o de o desc ibe he hea p oduced by hal -penny su ace-
b eaking c acks, we ocus on hea sou ces ea u ing he shape o semi-ci cula bands o adii
1
and
2
(
2
>
1
). Fo he sake o gene ali y, we allow he hea sou ce o be sligh ly bu ied,
wi h he uppe side loca ed a a dep h dwi h espec o he sample su ace (Figu e 1b). The
complex su ace empe a u e dis ibu ion o his case is w i en as ollows:
T (x,y, 0)=
2
Z
1
π
Z
0
Q( 0,ϕ0)
4πK
e−q qx2+(y− 0cos ϕ0)2+(d+ 0sin ϕ0)2
qx2+(y− 0cos ϕ0)2+(d+ 0sin ϕ0)2 0d 0dϕ0(2)
This exp ession also includes he case o kissing hal -penny c acks, by making 1= 0.
Senso s 2022,22, 2336 4 o 20
Fo he sake o compa ison, we also p esen in e sions co esponding o o he ge-
ome ies. Jus o gi e an example, we deal wi h ec angula hea sou ces o wid h wand
heigh hbu ied a a dep h dbelow he su ace (Figu e 1c). In his case, he exp ession o he
su ace empe a u e dis ibu ion is w i en as ollows:
T (x,y, 0)=
w/2
Z
−w/2
−d
Z
−(d+h)
Q(x0,y0)
4πK
e−q qx2+(y−y0)2+z02
qx2+(y−y0)2+z02
dy0dz0(3)
In Sec ion 4, we p esen in e sions o syn he ic su ace empe a u e da a (ampli ude
and phase) calcula ed using Equa ions (2) and (3). Fo he in e sion, we combine da a
ob ained a modula ion equencies
k
= 0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, and 12.8 Hz,
co esponding o he mal di usion leng hs
µ =pD/π
anging om 0.3 o 5 mm:
high equencies p o ide sha p de ails, whe eas low equencies pene a e deepe in
he ma e ial.
3. In e se P oblem
The gene al solu ion o he in e se p oblem consis s in inding he hea lux dis ibu-
ion
Q(→
0)
in plane
Π
, esponsible o he obse ed (noisy) su ace empe a u e da a
Tδ
k
,
(k= 1, . . . , kmax), δbeing he noise le el in he da a (L2-no m o he noise).
This app oach en ails ha :
1.
The hea sou ces a e known o be con ined in a plane pe pendicula o he su ace
(p io knowledge).
2. No speci ic geome y o he hea sou ce is supposed.
3. The he mal p ope ies o he ma e ial a e known.
4.
The shape o he spa ial dis ibu ion o hea sou ces is una ec ed by he modula ion
equency.
Acco dingly, e en i he hea sou ces a e known o be uni o m wi hin egion
Ω
, he
in e sion is no a me e pa ame e es ima ion p oblem (Q,
1
,
2
, and din Equa ion (1); Q,w,
h, and din Equa ion (2)) bu en ails meshing plane
Π
and de e mining he alue o Qa
each mesh node. This gi es gene ali y o he solu ion and is o p ac ical in e es , because
he shape o he hea sou ce is no known be o ehand, bu inc eases he di icul y o sol ing
he p oblem.
In his con ex , he o mula ion o he in e se p oblem in a leas -squa es sense consis s
in inding he Qdis ibu ion in plane
Π
ha minimizes he L2-no m o he di e ence
be ween he da a and he calcula ed empe a u es a each equency, summed o all he
modula ion equencies k,k= 1, . . . , kmax:
R2=
kmax
∑
k=1
T k(Q)−Tδ
k
2=
kmax
∑
k=1
A kQ k−Tδ
k
2
2=
kmax
∑
k=1
I kA kQ−Tδ
k
2
2(4)
He e,
A k
is he in eg al ope a o in Equa ion (1), and we ha e in oduced a equency-
dependen hea sou ce dis ibu ion
Q k( 0)=I kQ( 0)
exp essing
Q k( 0)
as he p oduc o
wo ac o s: a no malized hea sou ce dis ibu ion,
Q( 0)
, which, acco ding o assump ion 4,
is common o all modula ion equencies, and a se o in ensi ies,
I k
, ha only depend on
he modula ion equency. This allows using di e en ul asound ampli udes depending
on he modula ion equency ( ypically, highe ampli ude a high equency, o which he
signal is weake ).
In his amewo k, in he in e sion, he empe a u es a e no calcula ed using
Equa ions (2) o (3) (o he co esponding exp ession o a pa icula geome y) bu a e
ob ained as he supe posi ion o he poin -like con ibu ions o each mesh node in plane
Π
(Equa ion (1)). Acco dingly, he numbe o unknowns in he in e sion is signi ican ly high
(numbe o mesh nodes in plane
Π
). Gi en he ill-posed cha ac e o he in e se p oblem,
he minimiza ion o R
2
is e y uns able, and sol ing he p oblem equi es s abilizing he
Senso s 2022,22, 2336 5 o 20
in e sion. A e y popula me hod o s abilize ill-posed in e se p oblems is unca ed
singula - alue decomposi ion (SVD). We op o a di e en solu ion, which consis s in
minimizing a modi ied e sion o R2by adding s abilizing e ms o he igh hand side o
Equa ion (4), because his s a egy allows in oducing in he in e sion p io in o ma ion
abou he solu ion. In he nex sub-sec ion, ollowing [
8
], we p esen a comp ehensi e and
p og essi e in oduc ion o he penal y e ms ha we inco po a e in ou in e sion, aking
unca ed SVD as he s a ing poin : om he well-known ze o-o de Tikhono o mo e
sophis ica ed unc ionals such as Lasso (L1-no m) o To al Va ia ion (TV). We s a wi h a
qui e gene al o mula ion, and la e on, we pa icula ize o he p oblem we a e add essing.
We ha e p io i ized he smoo hness o an in ui i e desc ip ion o e igo in o malism
and no a ion.
3.1. Regula iza ion Func ionals
3.1.1. T unca ed Singula -Value Decomposi ion
We s a by w i ing he di ec p oblem in an ope a o o m:
AQ = T (5)
whe e
A
is a linea ma ix ope a o ha maps he disc e ized hea sou ce dis ibu ion
Q
in plane
Π
in o he su ace empe a u e da a
T
. The leas -squa es p oblem is w i en
as ollows:
A∗AQ = A∗T(6)
whe e A∗s ands o a complex conjuga e o A. The solu ion is:
Q = (A∗A)−1A∗T(7)
I
A
has ull column ank, (
A*A
)
−1
exis s, bu i i was ank-de icien , (
A*A
)
−1
would
no exis and
Q
could no be calcula ed using Equa ion (7). The SVD me hod allows sol ing
Equa ion (6) o ank-de icien ma ices. Jus as a eminde , in SVD, ma ix
A
(mby n) is
ac o ed in o 3 ma ices:
A = USV∗(8)
whe e
U
is an mby mma ix whose columns a e o hogonal ec o s spanning he da a
space,
V
is an nby nma ix whose columns a e o hogonal ec o s spanning he model
space, and
S
is an mby ndiagonal ma ix whose diagonal elemen s s
i
(singula alues) a e
a anged in dec easing o de . I only he i s psingula alues a e non-ze o (p<m),
S
can
w i en as
S=Sp0
0 0 (9)
and Equa ion (8) can be simpli ied o
A = UpSpVp∗
, whe e
Up
and
Vp
deno e he ma ices
whose columns a e he i s pcolumns o
U
and
V
, espec i ely. The SVD can be used o
compu e a gene alized in e se o A, he so-called Moo e–Pen ose pseudoin e se, A†,
A†=(A∗A)−1A∗= VpS−1
pU∗
p(10)
which always exis s. The pseudoin e se solu ion is hen:
Q†= A†T = VpS−1
pU∗
pT(11)
In an explici o m, he pseudo-in e se is w i en as ollows:
Q†=
p
∑
i=1
U∗
.,iT
si
V.,i(12)
whe e U.,iand V.,i ep esen each o he pcolumns o Upand Vp, espec i ely.
Senso s 2022,22, 2336 6 o 20
Equa ion (12) p esen s he solu ion as a linea combina ion o model space ec o s,
mul iplied by ac o s con aining he co esponding singula alue s
i
a he denomina o .
The summa ion may include e ms wi h e y small singula alues ha gi e ise o e y
la ge coe icien s o he co esponding high- equency model space ec o s
V.,i
, which
may e en ually domina e he solu ion, ac ing as noise ampli ie s.
A na u al way o s abilize he solu ion consis s in disca ding Equa ion (12) model
space ec o s wi h e y small associa ed singula alues. This is so-called unca ed SVD
egula iza ion. Howe e , his s abili y comes a he expense o educing he accu acy o
he solu ion. The e o e, he c i e ion o disca d model space ec o s mus be a ade-o
be ween s abili y and accu acy o he solu ion.
3.1.2. Ze o-O de Tikhono Regula iza ion
A success ul solu ion o an in e se p oblem gene ally in ol es e o mula ion as an
app oxima e well-posed p oblem. The ze o-o de Tikhono egula iza ion [
21
] modi ies
he leas -squa es equa ion by adding a smoo hing e m in o de o educe he uns able
e ec s o noise in he da a.
When he da a a e noisy, he e migh be many solu ions ha adequa ely i he da a, so
ha ||
AQ −T
||
2
is small enough. In ze o-o de Tikhono egula iza ion, he solu ions
a e sough among hose ha mee ||
AQ −T
||
2≤δ
(
δ
being a speci ic esidual mis i
alue), selec ing he one ha minimizes he L2-no m o Q:
minkQk2, subjec o kAQ −Tk2≤δ(13)
In oducing ze o-o de Tikhono egula iza ion ( o a speci ic egula iza ion pa ame e
αTK), he p oblem o mula ed in Equa ion (13) can be w i en as he minimiza ion o :
R2=kAQ −Tk2
2+αTKkQk2
2(14)
Equa ion (14) is he so-called objec i e unc ion, and he i s and second e ms on he
igh hand side a e he so-called disc epancy e m and egula iza ion e m, espec i ely. The
egula iza ion e m is he p oduc o a egula iza ion pa ame e ,
αTK
, and a egula iza ion
unc ional,
kQk2
2
in his case. The la ge he
αTK
, he mo e powe ul he egula iza ion and
he la ge he e o in he solu ion. We desc ibe ou s a egy o de e mine he op imum
alue o he egula iza ion pa ame e in Sec ion 3.2.1.
The ze o-o de Tikhono solu ion is equi alen o an o dina y leas -squa es p oblem
augmen ed acco ding o:
QαTK =a g min
Q∈Rn
A
√αTKIQ−T
0
2
2
=a g min
Q∈Rn
AaugQ−T
0
2
2
(15)
The size o
A
emains mby n, and
I
is he nby niden i y ma ix. As long as
αTK
is non-
ze o, he las n ows o ma ix
Aaug
a e linea ly independen , so Equa ion (15) ep esen s a
ull- ank leas -squa es p oblem ha can be sol ed by i s no mal equa ions:
A∗
augAaugQαTK =A∗
augT(16)
Using he SVD o
A
and ollowing he s eps indica ed in Sec ion 3.1.1, he solu ion can
be w i en as:
QαTK =
k
∑
i=1
si
si2+αTK
U∗
·,iT V·,i(17)
whe e k= min (m,n), and all non-ze o singula alues and ec o s a e included. Equa ion (17)
can be ew i en as:
QαTK =
k
∑
i=1
si2
si2+αTK
U∗
.,iT
si
V.,i=
k
∑
i=1
i
U∗
.,iT
si
V.,i(18)
Senso s 2022,22, 2336 7 o 20
whe e
i=s2
i/(s2
i+αTK)
a e he so-called il e ac o s, which con ol he con ibu ion
o he di e en e ms o he sum, in he ashion o a low-pass il e . Compa ison o
Equa ions (12) and (18) shows ha he penaliza ion o di e en model space ec o s
depends on he ela ion be ween
αTK
and hei associa ed singula alues. Acco dingly,
he deg ee o egula iza ion a ies be ween wo limi ing cases: o s
i
>>
αTK
,
i≈
1, and
he con ibu ion o he co esponding model space ec o s in Equa ion (18) emains he
same as in Equa ion (12), whe eas o s
i
<<
αTK
,
i≈
0, i.e., he associa ed model space
ec o s a e highly damped. Fo in e media e singula alues, as s
i
dec eases,
i
p oduces a
dec easing con ibu ion o he co esponding model space ec o s. The esul is a il e ing
o model space ec o s wi h small singula alues so e han applying unca ed SVD. As a
consequence, ze o-o de Tikhono egula iza ion p oduces a smoo h solu ion, since sha p,
high- equency model space ec o s a e il e ed ou .
Finally, le us men ion ha i is also possible o apply penal y e ms ha minimize
he L2-no m o he i s o second de i a i es o he solu ion, a he han he L2-no m o
solu ion i sel . These a e he so-called i s - and second-o de Tikhono unc ionals, which
a e men ioned in he nex sec ion.
3.1.3. Lasso and To al Va ia ion Regula iza ions
Focusing now on he pa icula in e se p oblem ha we a e add essing, we come
back o Equa ion (4). As men ioned a he beginning o his sec ion, ou goal is o e ie e
he e ical hea sou ce dis ibu ion Q ha minimizes a egula ized e sion o he squa ed
L2-no m in Equa ion (4). In p ac ice, his is ca ied ou by meshing plane Πwi h nnodes.
I a ze o-o de Tikhono penal y e m is applied, he egula ized e sion o Equa ion (4)
is w i en as ollows:
R2=
kmax
∑
k=1
I kA kQ−Tδ
k
2
2+αTKTK(Q)wi h TK(Q)=x
Π|Q|2dS ≈
n
∑
i=1|Qi|2∆S(19)
Ze o-o de Tikhono egula iza ion penalizes all nodes in plane
Π
equally, as i applies
he same egula iza ion pa ame e o each one, wi h no u he in o ma ion ega ding possi-
ble loca ions o he hea sou ces. Howe e , in o de o op imize he deg ee o egula iza ion,
o he non-linea egula iza ion p ocedu es based on local in o ma ion can be implemen ed,
aimed a pe o ming a posi ion-dependen penaliza ion.
Lasso (L1) [
16
,
17
] and o al a ia ion [
18
,
19
] egula iza ion me hods allow pe o ming
a posi ion-dependen penaliza ion by assigning a di e en egula iza ion pa ame e o each
node in plane
Π
, which, in u n, is made easible by implemen ing i e a i e me hods ha
make use o he hea sou ce dis ibu ion e ie ed in a p e ious i e a ion. This way, i is
possible o ha e an idea o which nodes need o be penalized mo e in a ollowing i e a ion,
in o de o o ce some o hem o emain damped and keep o he s domina ing he solu ion.
Le us conside a penal y e m based on a ze o-o de Tikhono unc ional, as he one
conside ed in Equa ion (19), bu wi h a egula iza ion pa ame e ha akes in o accoun he
solu ion in a p e ious i e a ion:
αTKi=αL1
1
QαL1
i,k−1
(20)
whe e ideno es he node in plane
Π
,kis he i e a ion, and we assume ha
QαL1
i,k−16=
0. The
explici expansion o his new disc e ized penal y e m o all nodes is w i en as ollows:
n
∑
i=1
αTKiQαL1
i,k
2∆S=n
∑
i=1
αL1QαL1
i,k
2
QαL1
i,k−1
∆S=
αL1 1
QαL1
1,k−1QαL1
1,k
2+1
QαL1
2,k−1QαL1
2,k
2+. . . +1
QαL1
n,k−1QαL1
n,k
2!∆S.
(21)
Senso s 2022,22, 2336 8 o 20
As can be seen, despi e
αL1
being common o all e ms, each e m is a ec ed by a
di e en penaliza ion, because he
QαL1
alues a e di ided by he local alues ob ained
in he p e ious i e a ion. In his way, i
QαL1
i,k−1
is small and hus 1
/QαL1
i,k−1
is la ge, hen
QαL1
i,k
is o ced o emain small. O he wise, 1
/QαL1
i,k−1
is small and
Qδ,αL1
i,k
is ee o
inc ease o a y.
O e i e a ions, e en ually
QαL1
i,k−1≈QαL1
i,k
, and he penal y e m in Equa ion (21)
app oaches:
n
∑
i=1
αTKi,jQαL1
i,k
2∆S≈αL1QαL1
1,k+QαL1
2,k+. . . +QαL1
n,k∆S(22)
which ep esen s he L1-no m o
QαL1
mul iplied by he egula iza ion pa ame e
αL1
. Thus,
penalizing he leas -squa es minimiza ion wi h a penal y e m based on he lasso (L1)
unc ional:
L1(Q)=x
Π|Q|dS =kQk1≃lim
k→∞x
Π
|Qk|2
qε+|Qk−1|2dS (23)
can be in e p e ed as pe o ming a posi ion-dependen penaliza ion o ze o-o de Tikhono
penaliza ion. The p esence o a small cons an
ε
in he denomina o o Equa ion (23) is
aimed a a oiding compu ing e o s when
|Qk−1|≈
0. Equa ions (21) and (22) desc ibe he
lagged ix-poin i e a ions algo i hm ha can be used o app oxima e he non-quad a ic L1
penal y e m de ined in Equa ion (23).
Regula iza ion wi h a o al a ia ion penal y e m:
TV(Q)=x
Π|∇Q|dS =k∇Qk1(24)
is based on he same p inciple as L1, bu ac ing o e
|∇Q|
ins ead o
|Q|
. I can be
in e p e ed as he implemen a ion o a i s -o de Tikhono unc ional wi h a posi ion-
dependen egula iza ion pa ame e . The lasso unc ional penalizes he L1 no m o he
solu ion, and TV penalizes he L1 no m o he g adien o he solu ion. In p ac ice, he main
di e ence be ween L1 and TV o he solu ion o he in e se p oblem is ha L1 a ou s
spa se solu ions in plane
Π
(comp essi e sensing e ec ), whe eas TV a ou s solu ions
wi h a eas o null de i a i es, which yields blocky solu ions. The combina ion o bo h
is app op ia e o cha ac e ize he con ined hea sou ces ep esen ing c acks ha we a e
seeking. Simila ly o Equa ion (22), which app oxima es he L1-no m o he solu ion, since
TV is a non-quad a ic ope a o , i can be app oxima ed om i s -o de Tikhono penal y
unc ional using lagged ix-poin i e a ions:
TV(Q)≃lim
k→∞x
Π
|∇Qk|2
qε+|∇Qk−1|2dS =lim
k→∞x
Π
(∂yQk)2+ (∂zQk)2
qε+ (∂yQk−1)2+ (∂zQk−1)2dS (25)
Th oughou his sec ion, we ha e seen ha pa icula egula iza ion unc ionals p o-
duce speci ic ypes o solu ions: ze o-o de Tikhono yields smoo h solu ions, TV gene a es
blocky unc ions, and lasso p oduces a comp essi e sensing e ec . This indica es ha , in
ill-posed in e se p oblems, gi en some p io knowledge o he p ope ies o he solu ion,
he me e selec ion o he penal y unc ional is a ool o inco po a e his p io in o ma ion in
he in e sion.
Acco ding o he p e ious esul s, we s abilize ou in e sion by penalizing he mini-
miza ion wi h wo unc ionals based on TV and L1, plus an auxilia y ze o-o de Tikhono
penal y e m. The p ope ies o TV and L1 mo i a e his selec ion, as we seek con ined hea
Senso s 2022,22, 2336 9 o 20
sou ces p oduced a c acks in well-de ined a eas. The egula ized e sion o Equa ion (4)
o be minimized is w i en as ollows:
R2
α=kmax
∑
k=1
Iα
kA kQα−Tδ
k
2
2+αTKTk(Qα)+αL1L1(Qα)+αTV TV(Qα),
wi h α=(αTK,αL1,αTV)
(26)
3.2. In e sion Algo i hm
The egula iza ion pa ame e s
αTK
,
αL1
, and
αTV
in Equa ion (26) de e mine how la ge
he di e en egula iza ion e ms a e wi h espec o he disc epancy e m. The deg ee o
egula iza ion can be a ied by modi ying he alues o he egula iza ion pa ame e s: la ge
alues inc ease he s abili y o he in e sion p ocess, in he sense ha he solu ion becomes
less sensi i e o noise in he da a, bu his s abili y comes a he expense o in oducing an
e o in he solu ion.
3.2.1. Regula iza ion Pa ame e s
In o de o ind he op imum egula iza ion pa ame e s, ou choice is o s a i e a ions
wi h a he high ini ial alues,
αTK0
,
αL10
, and
αTV0
, and educe hem in each i e a ion
acco ding o di e en decay ac o s:
γTK =
0.3,
γL1=
0.75, and
γTV =
0.75, espec i ely.
The Tikhono egula iza ion pa ame e
αTK0
decays much as e han
αL10
and
αTV0
, so
he e ec o Tikhono egula iza ion is basically signi ican in he i s i e a ion (i e a ion
ze o). Tikhono p o ides smoo h solu ions, which is bene icial a he beginning o he
in e sion and gua an ees ha he i s solu ion does no ge domina ed by noise, bu
sha pe solu ions a e hen sough . Mo eo e , L1 and o al a ia ion canno be implemen ed
a he beginning, because hey make use o he solu ion in a p e ious i e a ion. Theo e ical
esul s [
21
] sugges ha i is p uden o s op minimiza ion i e a ions be o e achie ing he
noise le el
δ
. Keeping his in mind, in his p oblem, we ha e ound ha s opping i e a ions
when he minimum disc epancy e m is ound deli e s good esul s. This is a heu is ic
s opping c i e ion, which p obably wo ks because we a e sol ing a highly o e de e mined
p oblem wi h qui e unco ela ed da a noise and gi es us op imum esul s o he e ie ed
no malized hea sou ce dis ibu ion. An impo an aspec ha is wo h men ioning abou
he chosen s opping c i e ion is ha he e is no o e - i ing o he da a, i.e., i ing he noise
a he han he unde lying unc ion. Rega ding he op imum alues o he decay ac o s,
he e is a lack o heo e ical esul s on his subjec . Small alues dec ease he numbe o
i e a ions needed o each he solu ion, bu educ ion ac o s below 0.5 may lead o s eps
in he disc epancy e m being oo la ge o he solu ion o be e ie ed accu a ely. The
ini ial alues o he egula iza ion pa ame e s as well as hei decay ac o s a e chosen
by pe o ming sys ema ic ba e ies o in e sions un il achie ing solu ions in a easonable
numbe o i e a ions, abou 20. Nex , we desc ibe he i e a i e p ocess implemen ed o ind
he solu ion.
3.2.2. I e a ions
Fo he in e sion p ocedu e, we use domain decomposi ion i e a ions o e ie e
he no malized hea sou ce dis ibu ion,
Qα
, and he se o in ensi ies,
Iα
k
, in successi e
i e a ions, known as non-linea Gauss–Seidel i e a ions by blocks. I is a local minimiza ion
me hod used in bi-linea p oblems such as his one.
Coming back o ou p oblem,
Tδ
k≈A khQα
ki=Iα
kA k[Qα] o k=1, . . . , kmax (27)
Senso s 2022,22, 2336 16 o 20
Figu e 9.
(
a
) Pic u e o he exci a ion sys em; (
b
) de ail o he sample closed and in con ac wi h he
sono ode o exci a ion.
Figu e 10.
(
a
) Expe imen al na u al loga i hm o ampli ude (le ) and phase ( igh ) ob ained o a
sample con aining a semi-ci cula Cu s ip o inne adius
1
= 1.2 mm and ou e adius
2
= 2 mm
bu ied a d= 0.32 mm, ob ained a 0.2 Hz; (b) i ed he mog ams.
The econs uc ion ob ained by combining da a aken in he whole equency se
(0.05 up o 12.8 Hz) is depic ed in Figu e 11, oge he wi h a econs uc ion o he same
slab bu ied a d= 0.71 mm and econs uc ions co esponding o o he open hea sou ce
geome ies.
Senso s 2022,22, 2336 17 o 20
Figu e 11.
G ey-le el ep esen a ion o he no malized hea sou ce dis ibu ion in in e sions om
expe imen al da a co esponding o (
a
) semici cula Cu bands o inne and ou e adii
1
= 1.2 mm
and
2
= 2 mm, espec i ely, bu ied a dep hs d= 0.32 and 0.71 mm; (
b
) a squa e Cu band o ou e
wid h 2.8 mm, ou e heigh 1.7 mm, and hickness 0.7 mm bu ied a a dep h d= 0.55 mm; (
c
) iangula
Cu bands o ou e wid h 3.6 mm, ou e heigh 2 mm, and hickness 0.9 mm bu ied a dep hs d= 0.36
and 0.71 mm. Real con ou s depic ed in ed, and alues o he dep h o he hea sou ces and quali y
ac o Fon op and unde o each econs uc ion, espec i ely.
The esul s con i m some o he ea u es obse ed in he in e sion o syn he ic da a.
On he one hand, ounded con ou s domina e he econs uc ions, which, as explained in
Sec ion 4.1, is due o he p esence o a TV e m in he egula iza ion penal y. Fu he mo e,
he shadowing e ec is isible, due o he s onge con ibu ion o he shallowes hea
sou ces ha domina e he econs uc ion. Ne e heless, in all geome ies, he deepe cen al
a eas co ec ly show he pa h he bands ollow, and all dep hs a e well- eco e ed. The
quali y ac o s a e in all cases abo e he cu o alue o F= 0.
Nex , we ied o ob ain expe imen al da a co esponding o inhomogeneous hea
sou ces. As men ioned abo e, ou samples a e in ended o p oduce homogeneous hea
sou ces, so we decided o ake da a combining in he same expe imen wo s ips wi h
he shape o a qua e o a ci cle o o m a semi-ci cula band wi h wo homogeneous
bu di e en hea luxes on i s wo hal es. In Figu e 12, we p esen he expe imen al
ampli ude and phase he mog ams ob ained by combining wo qua e s o ci cula s ips
made o s ainless s eel and W (bo h 25
µ
m hick) wi h inne and ou e adii
1
= 4.2 mm
and
2= 5.1 mm
, espec i ely, bu ied a a dep h o d= 0.16 mm below he su ace, a a
modula ion equency o 1.6 Hz. Un o una ely, we do no ha e an independen es ima e
o he a ios o luxes gene a ed by he wo hal es in hese combina ions.
The econs uc ion ob ained by combining ampli ude and phase da a in he whole
equency se is depic ed in Figu e 13a ( igh ), oge he wi h wo mo e econs uc ions,
om da a ob ained using o he combina ions o ma e ials: on he le , annealed and ha d
Cu oils, bo h 38
µ
m hick, and a he cen e , 25
µ
m hick ha d Cu and s ainless s eel oils.
In Figu e 13b, we display he econs uc ions ob ained o he same ma e ial combina ions
bu wi h iangula geome ies.
As may be no ed, o ei he geome y, simila esul s a e ob ained ega ding he hea
lux gene a ed by each ma e ial combina ion: he annealed and ha d Cu hal es (le ) ac as
a homogeneous hea sou ce, whe eas di e ences in he e ie ed hea sou ce dis ibu ion
a e mo e signi ican o he o he wo ma e ial combina ions: Cu–s ainless s eel (cen e )
and s ainless s eel–W ( igh ). These di e ences in he e ie ed luxes a e simila o bo h
geome ies, which p o es he consis ency o he in e sions. Al hough he shadowing
e ec makes he e ie ed a eas miss he con ibu ion o he cen al deepe posi ions in he
deepes cases, he o e all geome y and he dep hs o all hea sou ces a e well- eco e ed.
These esul s p o e ha di e ences in he hea lux dis ibu ions can be quali a i ely
cha ac e ized wi h he p oposed algo i hm.
Senso s 2022,22, 2336 18 o 20
Figu e 12.
(
a
) Expe imen al na u al loga i hm o ampli ude (le ) and phase ( igh ) ob ained a
a modula ion equency o 1.6 Hz in a sample con aining wo qua e s o ci cula s ips made o
s ainless s eel and W (bo h 25
µ
m hick) wi h inne and ou e adii
1
= 4.2 mm and
2
= 5.1 mm,
espec i ely, bu ied a a dep h o d= 0.16 mm below he su ace; (b) i ed he mog ams.
Figu e 13.
G ey-le el ep esen a ion o he no malized hea sou ce dis ibu ion o in e sions om
expe imen al da a co esponding o (
a
) wo qua e s o ci cula bands o inne adius
1
= 4.2 mm and
ou e adius
2
= 5.1 mm bu ied a dep hs d= 0.2, 0.27, and 0.16 mm and (
b
) wo hal es o iangula
bands o ou e wid h 5.6 mm, ou e heigh 2.4 mm, and hickness 0.9 mm, bu ied a dep hs 0.35 and
0.36 mm. Fo bo h geome ies, he ma e ial combina ions o he le and igh hal es o he bands a e
he ollowing: 38
µ
m hick annealed Cu and ha d Cu oils (le ), 25
µ
m hick Cu and s ainless s eel
oils (cen e), and 25
µ
m hick s ainless s eel and W oils ( igh ). Real con ou s depic ed in ed and
alues o he dep h o he hea sou ces on op o each econs uc ion.
Senso s 2022,22, 2336 19 o 20
6. Summa y and Conclusions
In his wo k, we ha e demons a ed ha mul i- equency lock-in ib o he mog aphy
da a in combina ion wi h a leas -squa es minimiza ion algo i hm egula ized by TV and
lasso unc ionals allows cha ac e izing ”hollow” non-compac e ical hea sou ces ypically
gene a ed by eal open c acks in ib o he mog aphy expe imen s. We ha e ob ained
semi-analy ical exp essions o he su ace empe a u e dis ibu ion gene a ed by e ical
hea sou ces wi h he shape o semi-ci cula s ipes, ep esen ing he beha io o open
hal -penny c acks exci ed wi h ul asounds. A de ailed desc ip ion o he egula iza ion
s a egies (s a ing om unca ed SVD o Tikhono , o al a ia ion, and lasso) as well as o
he in e sion algo i hm has been p esen ed, and we ha e p oposed a c i e ion o e alua e
he quali y o he econs uc ions. The in e sions o syn he ic da a wi h added noise show
ha he algo i hm is able o iden i y “hollow” uni o m hea luxes and e eal ha when he
hea sou ce spans a la ge ange o dep hs, he econs uc ions a e a ec ed by he shadowing
e ec , which blu s he deepes pa o he hea sou ce, due o he s onge con ibu ion o
shallow loca ions. Inhomogenei ies in he hea lux a e quali a i ely iden i ied excep in he
case o adial dependence o he lux. The p edic ions o he econs uc ions wi h syn he ic
da a we e con i med by in e sions o expe imen al da a aken on calib a ed samples. The
esul s con i m ha i is possible o cha ac e ize he shape o hea sou ces gene a ed by open
c acks is lock-in ib o he mog aphy expe imen s. The lock-in p ocessing o modula ed
da a allows de ec ing signals below he NETD o he came a. The possibili y o iden i ying
he egions o he c ack ha p oduce hea and he dis ibu ion o hese hea sou ces in
lock-in ib o he mog aphy open he way o unde s anding he con igu a ion and dynamics
o c acks in his kind o expe imen .
Au ho Con ibu ions:
Concep ualiza ion, A.S. and A.M.; me hodology, R.C. and A.M.; so wa e,
R.C.; alida ion, A.C.; o mal analysis, A.M. and A.C.; in es iga ion, A.S.; esou ces, A.S.; da a cu a-
ion, A.C.; w i ing—o iginal d a p epa a ion, A.M.; w i ing— e iew and edi ing, A.S.; isualiza ion,
A.C.; supe ision, A.M.; p ojec adminis a ion, A.S.; unding acquisi ion, A.M. and A.S. All au ho s
ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This esea ch is pa o a p ojec wi h g an numbe PID2019-104347RB-I00 unded by
MCIN/AEI/10.13039/501100011033. The esea ch was also unded by Uni e sidad del País Vasco,
UPV/EHU, g an numbe GIU19/058.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The da a a e a ailable unde easonable eques o he co espond-
ing au ho .
Acknowledgmen s:
The au ho s a e hank ul o echnical and human suppo p o ided by SGIke
Compu ing Se ices (UPV/EHU/ ERDF, EU).
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1. Maldague, X.P.V. Nondes uc i e E alua ion o Ma e ials by In a ed The mog aphy; John Wiley & Sons: Hoboken, NJ, USA, 2001.
2.
Mendio oz, A.; Celo io, R.; Salaza , A. Ul asound exci ed he mog aphy: An e icien ool o he cha ac e iza ion o e ical
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