Ene gy-Based Recons uc ion o In e nal
S uc u al Con ibu ions (ECM)
Abs ac
This wo k de elops a gene alized o mula ion o he Ene gy Con i-
bu ion Me hod (ECM), a amewo k ha econs uc s in e nal s uc-
u al ene gy e ec s o an unknown sys em using only ex e nally mea-
su able ene gy di e ences. Ins ead o assuming speci ic mic oscopic
mechanisms (such as chemical bonding), ECM de ines a uni e sal in-
e nal con ibu ion Ee ep esen ing he sys em’s in insic s uc u al
esponse o injec ed ene gy. By obse ing changes ac oss expe imen al
condi ions, ECM es ima es bo h Ee and he emaining non-in insic
componen s. The app oach is ully mac oscopic, model-independen ,
and applicable o gases, liquids, solids, plasmas, composi es, o any
medium whe e in e nal in e ac ions lea e a epea able ene gy oo -
p in . A ull case classi ica ion (0–4) desc ibes iden i iabili y limi s
unde di e en in o ma ion le els, and nume ical examples illus a e
each egime.
1 In oduc ion
In many physical sys ems, in e nal p ocesses a e no di ec ly obse able.
Expe imen al access is o en limi ed o bounda y-le el quan i ies such as
an injec ed ene gy Ein and a measu ed ou going ene gy Eou . All mic o-
1
scopic e en s—con e sion, s o age, dissipa ion, s uc u al econ igu a ion—
a e comp essed in o he esidual
E es =Ein −Eou .(1)
T adi ional analyses a emp o decompose his esidual in o speci ic com-
ponen s (e.g. bonding, ic ion, coupling). Howe e , such a decomposi ion
usually equi es s ong mic oscopic assump ions ha a e no jus i ied o
many sys ems (especially gases, plasmas, o complex he e ogeneous media).
ECM adop s a di e en iewpoin . Ra he han asking which mic oscopic
mechanism c ea ed he esidual, i de ines an in insic s uc u al con ibu ion
ha cap u es how he sys em, as a whole, consis en ly modi ies he ene gy
ha passes h ough i . This in insic pa is deno ed by Ee and is sepa a ed
om ex insic, se up-dependen e ec s Eex . The cen al ques ions a e:
1. How can Ee be es ima ed om mac oscopic measu emen s alone?
2. Unde wha condi ions is such an es ima e unique, app oxima e, o
only pa ially iden i iable?
ECM as a black-box s uc u al me hod
We ea he sys em as a black box: ene gy is injec ed, in e ac s wi h in e nal
s uc u e, and eme ges.
The ole o ECM is o in e om epea ed measu emen s which pa o
E es is in insic o he sys em and which pa is due o ex e nal o inciden al
e ec s.
2
2 Basic ECM Fo mula ion
We decompose he esidual as
E es =Ee +Eex ,(2)
whe e:
•Ee :in insic s uc u al con ibu ion (sys em-speci ic),
•Eex :ex insic con ibu ion, including:
–equipmen loss Eloss,
–c oss-e ec s Ec oss,
–condi ion-dependen de ia ions Econd.
Thus
Eex =Eloss +Ec oss +Econd.(3)
Concep ually, Ee is he pa o he esidual ha :
1. is epea able ac oss iden ical expe imen al uns,
2. emains s able unde mode a e scaling o Ein,
3. o igina es om he sys em’s in e nal s uc u e, no om ex e nal de-
ices.
3 Ex ac ion P inciples
Suppose we pe o m Nexpe imen s indexed by k, possibly wi h a ying
condi ions (in ensi y, equency, geome y). Fo each:
E(k)
es =Ee +E(k)
ex .(4)
3
3.1 A e aging and baseline es ima ion
I E(k)
ex luc ua es a ound a mean alue Ea g
ex , hen:
b
Ee =1
N
N
X
k=1
E(k)
es −Ea g
ex .(5)
When Ea g
ex is calib a ed o negligible, he sample mean o E(k)
es becomes a
di ec es ima o o Ee .
3.2 In ensi y sweep
Va ying he inpu ene gy Ein allows us o sepa a e con ibu ions ha scale
wi h in ensi y om hose ha do no . I we deno e in ensi y by Iand model
Eex (I) = aI +bI2+c, (6)
hen eg ession on (I, E es) pai s can es ima e (a, b, c) and isola e Ee as he
emaining cons an componen .
3.3 Pa e n ex ac ion ia SVD
Fo mo e complex da a (mul iple condi ions, equencies, geome ies), we can
o m a ma ix Rwhose en ies a e esiduals:
Rkℓ =E(k,ℓ)
es ,(7)
wi h kindexing mac oscopic condi ions and ℓindexing con ol pa ame e s.
Applying singula alue decomposi ion (SVD):
R=UΣV⊤,(8)
4
e eals how many dominan in e nal modes a e equi ed o ep esen he da a
and how much o he s uc u e can be asc ibed o a low-dimensional in insic
esponse.
Figu e 1 illus a es a gene ic singula alue spec um.
i
σiσ1
σ2
noise le el
Figu e 1: Schema ic singula alue spec um. Dominan modes lie abo e he
noise loo .
4 Case Classi ica ion wi h Nume ical Exam-
ples
We classi y all possible in o ma ion egimes in o i e cases (0–4) and illus a e
each wi h a conc e e nume ical example.
Case 0: Single in e nal mode, ully calib a ed losses
This is he mos a o able egime: a single in insic mode and pe ec ly
calib a ed ex e nal loss. We ha e
E es =Ee +Eloss.(9)
Nume ical example. Assume
Ein = 50 J, Eou = 33 J,
5
so
E es = 50 −33 = 17 J.
I he equipmen loss is known o be
Eloss = 2 J,
hen
Ee =E es −Eloss = 17 −2 = 15 J.
He e Ee is de e mined exac ly.
Case 1: Known in insic s uc u e, pa ially known ex-
e nal a ia ion
We assume ha he in insic con ibu ion Ee is ixed, bu ex e nal losses
a y sligh ly ac oss condi ions. Then
E(k)
es =Ee +E(k)
loss,(10)
wi h E(k)
loss pa ially known.
Nume ical example. Conside wo condi ions A and B:
Condi ion Ein Eou Eloss
A 40 25 1.0
B 40 23.6 1.4
Residuals:
EA
es = 40 −25 = 15, EB
es = 40 −23.6 = 16.4.
Thus
EA
e = 15 −1.0 = 14, EB
e = 16.4−1.4 = 15.
6
A simple a e age yields
b
Ee = 14.5 J.
This case illus a es pa ial iden i iabili y wi h small unce ain y.
Case 2: Unknown s uc u e alues, known in e nal mode
coun
Suppose we know ha he e a e nin insic modes bu no hei s eng hs.
We w i e
E(k)
e =
n
X
i=1
αiu(k)
i,(11)
whe e u(k)
idesc ibes how mode i esponds unde condi ion k.
Nume ical example. Assume n= 2 in insic modes and h ee condi ions
A, B, C. Measu ed esiduals and losses:
Cond. E es Eloss E es −Eloss
A 20 2 18
B 26 3 23
C 32 4 28
Model:
18 = α1+α2,
23 = 2α1+α2,
28 = α1+ 2α2.
Sol ing: om he i s wo equa ions,
5=α1⇒α1= 5,
hen
18 = 5 + α2⇒α2= 13.
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ECM he e inds unique mode s eng hs (α1, α2) by using mul iple condi ions.
Case 3: Unknown mode coun , s uc u e es ima ed by
SVD
In his egime we do no e en know how many in insic modes exis . We con-
s uc a esidual ma ix and apply SVD o es ima e he numbe o e ec i e
modes.
Nume ical example. Take a 3 ×3 esidual ma ix
R=
15 16 17
22 24 26
29 32 35
.
A nume ical SVD yields singula alues
σ1≈48.3, σ2≈2.1, σ3≈0.4.
The la ge gap be ween σ2and σ3sugges s ha wo in insic modes domina e,
while he hi d is nea noise le el. Thus, ECM in e s an in e nal dimension-
ali y o app oxima ely 2, wi hou di ec mic oscopic access.
Case 4: Fully unknown sys em (closes consis en es i-
ma e)
He e we only ha e esidual da a and no eliable knowledge o losses, modes,
o ex e nal pa e ns. ECM hen seeks he closes consis en in insic alue
ha minimizes esidual a ia ion:
b
Ee = a g min
EX
kE(k)
es −E.(12)
8
The minimize is he median o {E(k)
es }.
Nume ical example. Gi en ou measu emen s:
E(k)
es ∈ {15,16,17,19},
he median is
b
Ee = 16.5 J.
No exac econs uc ion is possible, bu ECM s ill p o ides he mos s able
in insic es ima e consis en wi h he da a.
5 Summa y o Nume ical Beha io
Table 1 summa izes he nume ical examples o Cases 0–4.
Case In o le el Example ou come Iden i iabili y
0 Single mode, calib a ed loss Ee = 15 J exac
1 Known s uc u e, a iable loss Ee ≈14.5 J high
2 Known mode coun , unknown alues (α1, α2) = (5,13) exac (w/ da a)
3 Unknown mode coun modes ≈2 s uc u al only
4 Fully unknown b
Ee = 16.5 J closes es ima e
Table 1: Summa y o ECM beha io in Cases 0–4.
6 Discussion: Ad an ages and Limi s
ECM does no a emp o econs uc e e y mic oscopic con ibu ion sepa-
a ely. Ins ead, i ocuses on wha can be obus ly eco e ed om bounda y
measu emen s:
•an in insic, s uc u ally epea able componen Ee ,
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