Adap i e il e solu ion o p ocessing
lida e u ns: op ical pa ame e es ima ion
F ancesc Rocadenbosch, G ego i Va´zquez, and Adol o Come o´n
Join es ima ion o ex inc ion and backsca e simula ed p o iles om elas ic-backsca e lida e u n
signals is ackled by means o an ex ended Kalman il e ~EKF!. Fi s , we in oduced he issue om a
heo e ical poin o iew by using bo h an EKF o mula ion and an app op ia e a mosphe ic s ochas ic
model; second, i is es ed h ough ex ensi e simula ion and unde simpli ied condi ions; and, inally, a
i s eal applica ion is discussed. An a mosphe ic model including bo h empo al and spa ial co ela-
ion ea u es is in oduced o desc ibe app oxima e luc ua ion s a is ics in he sough -a e a mosphe ic
op ical pa ame e s and hence o include a p io i in o ma ion in he algo i hm. P o ided ha easonable
models a e gi en o he il e , in e sion e o s a e shown o depend s ongly on he a mosphe ic condi ion
~i.e., he isibili y!and he signal- o-noise a io along he explo a ion pa h in spi e o modeling e o s in
he assumed s a is ical p ope ies o he a mosphe ic op ical pa ame e s. This is o ad an age in he
pe o mance o he Kalman il e because hey a e o en he poin o mos conce n in iden i ica ion
p oblems. In ligh o he adap i e beha io o he il e and he in e sion esul s, he EKF app oach
p omises a success ul al e na i e o p esen -day nonmemo y algo i hms based on exponen ial-cu e
i ing o di e en ial equa ion o mula ions such as Kle ’s me hod. © 1998 Op ical Socie y o Ame ica
OCIS codes: 010.0010, 010.1290, 010.3640.
1. In oduc ion
The supe io quali ies o lase ada s, o lida s, wi h
ega d o collima ion, spa ial esolu ion, and in e ac-
ion capabili y wi h a mosphe ic species, when com-
pa ed wi h hose o con en ional mic owa e ada
sys ems o passi e isible ins umen a ion, hold
p omise ha lida s will be long-las ing al e na i e
obse a ion sys ems. In an a mosphe ic lida , he
emission o a sho lase pulse is ollowed by he
ecep ion o some adia ion sca e ed om a mo-
sphe ic cons i uen s such as molecules, ae osols, and
clouds. The in e ac ion o he inciden adia ion
wi h hese cons i uen s changes he in ensi y andyo
he wa eleng h, depending on he s eng h o his
op ical in e ac ion and he concen a ion o he in e -
ac ing species. Consequen ly, i is possible o e-
ie e in o ma ion abou he physical s a e o he
a mosphe e along he explo a ion beam pa h.
1–3
In pa icula , es ima ion o he a mosphe ic op ical
pa ame e s, namely, ex inc ion and backsca e ,
based on pulsed elas ic-backsca e lida s ~i.e., wi h
no wa eleng h shi in ecep ion!has been in es i-
ga ed in he li e a u e.
4–6
The single-sca e ing
ange- e u n powe o an elas ic-backsca e lida
sys em can be exp essed as
2
P~R!5A
R2b~R!exp
F
22
*
0
R
a~ !d
G
, (1)
whe e P~R!is he ange- ecei ed powe ~W!,b~R!is
he ange-dependen olume backsca e coe icien
o he a mosphe e ~m
21
s
21
!,a~R!is he ange-
dependen ex inc ion coe icien ~m
21
!,Ris he ange
~m!, and Ais he sys em cons an ~Wm
3
!.
Un il now, he in e sion o lida signals has been
ackled mainly by using classic p ocedu es such as
he slope me hod,
5
exponen ial-cu e i ing, and
Kle ’s me hod.
6
Ye , all hese me hods assume sim-
pli ying andyo co ela ion hypo heses ha limi he
scope o he in e sion esul s. These a e discussed
nex .
In he case o he slope-me hod algo i hm,
5
he
assump ion o a homogeneous a mosphe e is used o
e ie e cons an alues ~a,b! as es ima es o he
The au ho s a e wi h he An ennas, Mic owa es, Rada and
Op ics G oup, Depa men o Signal Theo y and Communica ions,
Uni e si a Poli e`cnica de Ca alunya, CySo Eulalia de Anzizu
syn., 08034 Ba celona, Spain.
Recei ed 29 Sep embe 1997; e ised manusc ip ecei ed 8
Ap il 1998.
0003-6935y98y307019-16$15.00y0
© 1998 Op ical Socie y o Ame ica
20 Oc obe 1998 yVol. 37, No. 30 yAPPLIED OPTICS 7019
sough -a e unc ions a~R!and b~R!. The key o he
algo i hm is a ange-co ec ed unc ion o he o m
S~R!5ln@R2P~R!#, (2)
which enables us o ind he ex inc ion and backsca -
e coe icien s om a linea eg ession o he o m
mina,biS~R!2@ln~Ab! 22aR#i2. (3)
In he case o exponen ial-cu e i ing, a simila
app oach is ollowed, bu now he R
2
-co ec ed unc-
ion is de ined wi hou he loga i hm as
F~R!5R2P~R!, (4)
so ha he no m minimiza ion equi alen o exp es-
sion ~3! akes he o m
mina,biF~R!2Abexp~22aR!i2. (5)
His o ically, his kind o i ing was in oduced la e
because i is nonlinea in aand mus be sol ed by
using nume ical me hods.
7,8
In a di e en ca ego y, he in e sion o he ange-
dependen unc ion a~R!is i s sol ed by Kle ’s
me hod.
6
The me hod assumes a powe -law co e-
la ion be ween he ex inc ion and backsca e a mo-
sphe ic p o iles as ollows:
b~R!5B0@a~R!#g, (6)
and i equi es a guess o he co ela ion cons an g
~0.67 ,g,1!and a calib a ion a he a end o he
in e sion ange in e al in e ms o S
m
5S~R
max
!
and a
m
5a~R
max
!.
9,10
Then he backwa d s able
solu ion o a~R!becomes
a~R!5exp@~S2Sm!yg#
am2112
g
*
R
Rmexp@~S2Sm!yg#d
. (7)
In spi e o he ac ha his algo i hm is signi i-
can ly supe io o he slope me hod and he
exponen ial-cu e i ing ~because he homogenei y
app oxima ion is no assumed!, he accu acy o he
in e ed p o ile a~R!is limi ed by ha o he calib a-
ion a
m
and he co ela ion cons an g. Fo his ea-
son he algo i hm e ie es a ep esen a i e o he
amily a~R,g,a
m
!linked o he calib a ion pai ~g,
a
m
!, which is hough o be close o he ue ex inc ion
p o ile a~R!. In o he wo ds, he in e sion o a
ange-dependen ex inc ion p o ile om he e u n
powe is a many- o-one in e sion p oblem ~see Ap-
pendix B in Re . 11!, which can be sol ed only by
adding app op ia e a p io i in o ma ion ~e.g., calib a-
ions along he obse a ion pa h and physical con-
s ain s!.
As we ha e seen, all hese algo i hms wo k wi h
he p esen ealiza ion o he lida e u n signal, so
ha co ela ion among pas in e ed e u ns e-
mains unexplo ed. Fo example, in an elas ic-
backsca e pulsed lida sys em, o each e u n-
powe da a s eam ecei ed a new in e sion, which is
comple ely independen om hose p e iously done,
is pe o med.
One o he hings ha dis inguishes he Kalman
il e
12–14
om nonmemo y es ima o s such as hose
discussed abo e is he con enien way in which i ac-
coun s o any p io knowledge h ough a ecu si e
p ocess. As long as di e en powe ealiza ions a e
coming in, he il e upda es i sel , weigh ed by he
imbalance be ween he a p io i es ima es o he op ical
pa ame e s ~i.e., pas in e sions!and he new ones.
Thus he new es ima ion o he op ical pa ame e s, o
he p ojec -ahead s ep ~a pos e io i es ima e!,isim-
p o ed based on a s a is ical minimum- a iance c i e-
ion.
In ecen and pionee ing wo k, Rye and Ha des y
15
and Lainio is e al.
16
ha e ound applica ions o he
Kalman il e o he es ima ion o he e u n powe
and he loga i hm o powe o incohe en backsca -
e lida wi h mul iplica i e noise.
17
He e we in o-
duce an applica ion o he il e o he solu ion o he
in e se p oblem o join es ima ion o he ex inc ion
and backsca e coe icien s om he e u n powe in
an elas ic-backsca e lida .
This pape is s uc u ed as ollows: In Sec ion 2
he p oblem is o mula ed om a heo e ical poin o
iew in e ms o a i s adap i e il e based on an
ex ended Kalman il e ~EKF!; Sec ion 3 desc ibes
he unde lying s a is ics o he a mosphe ic model
assumed in e ms o he s a e-noise co a iance ma-
ix o he EKF; Sec ion 4 discusses wo examples o
join in e sion o ex inc ion and backsca e p o iles
om elas ic-backsca e simula ed lida e u n sig-
nals; and Sec ion 5 e iews some o he esul s p e-
sen ed by ackling a i s eal applica ion o he il e
o he in e sion o powe e u ns om a biaxial
elas ic-backsca e 1-J Nd:YAG lida sys em.
2. P oblem Fo mula ion
I is desi able now o s udy he easibili y o he de -
i a ion o he ex inc ion and backsca e coe icien s
o e he en i e lida in e sion ange. I is desi ed,
hen, o sol e he unc ions a~R, !and b~R, ! ha ,
unde a minimum-mean-squa e-e o c i e ion, bes
i he obse able powe P~R, !a e e y ime . The
e m mean e e s he e o he ensemble a e age o e
ime .~Some e ision o he EKF algo i hm, along
wi h he no a ion used below, is summa ized in Ap-
pendix A.!
A. S a e Vec o
Gi en he acquisi ion sampling a e o he sys em,
s
,
and conside ing he wo-way pa h o he lida signal,
he powe ime samples P
i
co espond o a spa ial
sampling pe iod
DR5c
2 s
. (8)
Hence he spa ial sampling poin s become
Ri5Rmin 1~i21!DR,i51,...,N, (9)
7020 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998
whe e R
min
is some p ede e mined minimum ange o
he sys em ~which is due o, o example, he mini-
mum ange o ull o e lap be ween he lase and he
ield o iew o he ecei ing op ics o some p ede ined
minimum in e sion ange o in e es !.
The s a e ec o o be es ima ed, x
k
, is a decima ed
e sion o he ex inc ion and backsca e unc ions
a~R!and b~R!o e he whole lida ange. This is
done in o de o ha e mo e obse ables ~ he powe
samples om each obse a ion cell! han a iables o
es ima e ~ex inc ion and backsca e samples in he
es ima ion cells!, and, as a esul , i yields an o e -
de e mined sys em wi h enhanced obse abili y.
14,18
Fu he mo e, i can be shown
14
ha i ~1! he sys em
s a e ec o is assumed o be a andom cons an , ~2!
he measu emen sequence z
k
yields an o e de e -
mined se o linea equa ions, and ~3! he obse a ion
noise becomes negligible @i.e., he measu emen noise
co a iance ma ix o Eq. ~A4!,R
k
'0#, hen he il-
e ’s es ima e depends mo e and mo e on cu en
da a ~which a e ich in new in o ma ion!and less and
less on pas in e sions. Unde hese ci cums ances
he il e beha es like a de e minis ic leas -squa es
es ima o ~which pa allels he nonmemo y ap-
p oach!, and i s es ima e becomes
13
xˆk5Hk21zk, (10)
whe e H
k21
is he pseudoin e se ma ix o H
k
@see
ela ion ~A6!#. His o ically, his b idges he gul
wi h pas o mula ions o he p oblem in he o m o
exp ession ~5!.
The model conside s NyMobse a ion cells, whe e
Mis he decima ion a io, so ha he il e es ima es
NyMex inc ion samples and NyMbacksca e sam-
ples. Then he e ec i e sampling pe iod becomes
MDR, which is M imes ha o he e u n powe .
~Fo simplici y, assume ha Nis a mul iple o M.!
Ma hema ically, his can be exp essed as
ai5a~Rmin 1~i21!MDR!,i51,...,N
M,
bi5b~Rmin 1~i21!MDR!,i51,...,N
M. (11)
F om hese wo hal es o NyMelemen s, we o m he
s a e ec o o be es ima ed:
xk;~a1a2··· aNyMb1b2··· bNyM!T, (12)
whe e he subsc ip kis a eminde o he disc e e
ime
k
.
The nons a iona i y o he dynamics o he s a e
ec o is desc ibed by he ansi ion ma ix F
k
@see
Eqs. ~A1!and ~A19!# and he s a e-noise co a iance
ma ix Q
k
@Eq. ~A2!#. The o me ep esen s how
he s a e ec o p ojec s ahead om ime
k
o ime
k11
, and he la e gi es he il e key in o ma ion
abou he unde lying s a is ics o each componen o
he s a e ec o ~ he op ical pa ame e s unde s udy!.
Fo mula ion o he sys em equa ions in e ms o he
ansi ion ma ix F
k
and he s a e-noise co a iance
ma ix Q
k
ga he s all he in o ma ion he il e knows
abou he a mosphe ic model. This is ackled in Sec-
ion 3.
B. Measu emen Equa ion
I , acco ding o Eq. ~9!, each powe sample co e-
sponds o a spa ial inc emen DR, so ha P
i
5P~R
i
!,
and a ec angle app oxima ion is used o compu e he
ansmi ance e m o Eq. ~1!, he obse able powe
samples become
P15A
R12b1exp~22a1Rmin!, (13)
·
·
·
PM5A
RM2b1exp$22a1@Rmin 1~M21!DR#%,
(14)
·
·
·
PM115A
RM112b2exp$22a1@Rmin 1~M21!DR#
22a2DR%, (15)
·
·
·
PN5A
RN2bNyMexp
H
22a1@Rmin 1~M21!DR#
22
(
i52
NyM
aiMDR
J
. (16)
The R
2
-co ec ed e sion @Eq. ~4!# o his se o N
equa ions builds he measu emen ec o z
k
:
zk;@F1~xk!F2~xk!··· FN~xk!#T. (17)
By using F~R! a he han P~R!, we educe he
dynamic ma gin o z
k
and, consequen ly, nume ical
e o s a e educed as we cycle h ough he Kalman
loop @Eqs. ~A13!–~A17!#. A he a anges, howe e ,
ampli ica ion o quan iza ion noise gene a ed du ing
he analog- o-digi al con e sion migh become signi -
ican and mus be accoun ed o in a desc ip ion o he
s a is ics o he obse a ion noise.
Equa ions ~13!–~16!de ine he o e de e mined se
o equa ions discussed in Subsec ion 2.A ha ela es
he measu emen ec o z
k
o he a p io i es ima e o
he s a e ec o , xˆ
k
2
.
F om Eqs. ~12!and ~17!, he N3~2NyM!obse a-
ion ma ix H
k
can be compu ed by spli ing i in wo
N3~NyM!subma ices o he o m H5~H
1
H
2
!,
whe e
Hij
~1!5]Fi
]aj
U
x5xˆk
2
,Hij
~2!5]Fi
]bj
U
x5xˆk
2.
(18)
20 Oc obe 1998 yVol. 37, No. 30 yAPPLIED OPTICS 7021
This yields
whe e H
1
and H
2
a e e alua ed a he a p io i es i-
ma e xˆ
k
2
.
F om he s uc u e o H
1
and H
2
, i eme ges ha he
o mula ion o he in e sion p oblem in ol es, how-
e e , a ade-o be ween la ge decima ion a ios ~M!
and model accu acy. F om he poin o iew o M,i
one compa es he EKF o mula ion wi h he classical
exponen ial-cu e- i ing coun e pa o exp ession ~5!,
he exponen ial i ing algo i hm wo ks wi h a e y
la ge alue o M, equal o he leng h o he in e sion
in e al, which is, in u n, o med by a single in e sion
cell. Hence he la ge he M, he mo e obus he
sys em o Eq. ~10!agains obse a ion noise and he
be e he eg ession esul s @ his is bes seen by he
p oduc s MDRF
i
in Eq. ~19!, which become la ge o
la ge M#. Ano he ad an age o inc easing Mis he
enhancemen o he il e ’s sensi i i y o low e u n
powe s and la ge alues o R
min
. Usually, la ge di -
e ences be ween he weigh ac o s 22@R
min
1~M2
1!DR#and 22MDRa e no desi able, since hen
he ajec o y o he il e would be domina ed by he
es ima ion o he i s cell. Ye , he mos isky
d awback o la ge Ma ises om he de e io a ion
o he il e ’s model @Eqs. ~13!–~16!#: Al hough he
equi alen sampling pe iod o he op ical pa ame-
e s in he il e ’s model is MDR, he ue a mo-
sphe ic spacing DR9is di e en ial in na u e. Fo
he ime being, we assume he simpli ica ion DR95
DRin he a mosphe ic op ical p o ile, so ha such
modeling e o s a e neglec ed. In o he wo ds, he
a mosphe e is assumed homogeneous inside any ob-
se a ion cell. Al hough much esea ch is being
done in his ield, which is a om he scope o his
s udy, a sensible solu ion migh be achie ed by com-
bining he esul s o an a ay o Mcoope a i e il-
e s o , pe haps, by using nonuni o m spacing in
he o mula ion o he p oblem, depending on he
a mosphe ic si ua ion a hand.
H15
3
22RminF10 0 ··· 0
22~Rmin 1DR!F20 0 ··· 0
·
·
··
·
··
·
···· ·
·
·
22@Rmin 1~M21!DR#FM0 0 ··· 0
22@Rmin 1~M21!DR#FM1122DRFM110 ··· 0
·
·
··
·
··
·
···· ·
·
·
22@Rmin 1~M21!DR#FN22MDRFN22MDRFN··· 22MDRFN
4
N3~NyM!
, (19)
H25





F1
xNyM11
0 0 ··· 0
F2
xNyM11
0 0 ··· 0
·
·
··
·
··
·
···· ·
·
·
FM
xNyM11
0 0 ··· 0
0FM11
xNyM12
0 ··· 0
·
·
··
·
··
·
···· ·
·
·
0 0 0 ··· FN
x2NyM





N3~NyM!
, (20)
7022 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998
Lida measu emen s a e co up ed mainly by
Gaussian addi i e obse a ion noise
k
. Assuming
ha DR95DRand ha he obse a ion noise along
he in e sion ange can be app oxima ed by ange-
dependen s a iona y elec onic he mal noise ~Gauss-
ian addi i e noise wi h a iance s
2
!, he obse a ion
noise co a iance ma ix is compu ed as
Rk5E~ k kT!5
3
s 2~R1!R14··· 0
·
·
·····
·
·
0 ··· s 2~RN!RN4
4
. (21)
This assumes ha elec onic noise domina es obse -
a ion noise. Following Re s. 17 and 19, he ange-
dependen noise a iance can be w i en as
s 2~R!5a@P~R!1Pback#1b, (22)
whe e P~R!is he ange e u n powe de ined in Eq.
~1!,P
back
is he backg ound powe om any o he
in e e ing sou ce ~ o example, he Sun!, and aand
ba e cons an s ha depend only on speci ic pa am-
e e s o he ecei ing sys em. The i s e m ac-
coun s o he con ibu ions o he signal-induced
sho noise o he o al noise, and he second one
me ges in o he a iable b he con ibu ions o bo h
da k-cu en sho noise and he mal noise. @No e
ha s
2
~R!has uni s o squa e ol s o squa e wa s,
depending on whe he equi alen noise is compu ed
a he ecei e ’s ou pu o inpu .
In ins ances whe e o he sou ces o measu emen
noise a e p esen ~e.g., R
2
-ampli ied quan iza ion
noise!, one can inc ease pe inen e ms along he
main diagonal o R
k
o accommoda e such an ex a
a iance. Nons a iona y noise can be ackled by e-
compu ing R
k
a each succeeding s ep o he il e ,
and colo ed noise ~such as synch onized lash-lamp
in e e ences!can be modeled by also using elemen s
o he main diagonal o R
k
~see also Re . 14 o u -
he insigh !.
3. A mosphe ic Model o he Ex ended Kalman Fil e
In hei mos gene al o m, ex inc ion and backsca -
e op ical pa ame e s a e nonlinea ly ela ed. Un-
less mic oscale analysis is conside ed and plen y o
bounda y calib a ions om balloon-bo ne ins umen-
a ion o o he coope a i e sys ems a e gi en, he
s uggle o model physically he empo al and spa ial
e olu ion o he op ical pa ame e s leads o awkwa d
and cumbe some esul s. A mo e con enien al e -
na i e is o y o model he mac oscopic e ec s on
hem, a he han he unde lying mic ophysical pa-
ame e s. This is done by using he ime–space s o-
chas ic co ela ion model ske ched in Fig. 1. Each
op ical componen ~ex inc ion and backsca e !o an
in e sion cell is modeled as a s ochas ic p ocess ha -
ing bo h empo al and spa ial co ela ion. Each ou -
pu b anch ep esen s one componen o he s a e
ec o x
k
, and he ec o noise p ocess w
k
is o med by
spa ially co ela ed componen s a he ou pu o he
linea sys em A. The a mosphe ic model is d i en
by an a ay o whi e-noise unco ela ed p ocesses.
A. Tempo al Co ela ion
Tempo al co ela ion is pe haps he mos a ac i e
ad an age o he EKF lida in e sion app oach o e
he nonmemo y solu ions o exp essions ~3!and ~5!
and Eq. ~7!. This ad an age comes om elling he
il e ha i should imp o e i s p ojec ion s eps based
on he ac ha he a mosphe e usually has a long
co ela ion ime and ha , consequen ly, swi
changes in any op ical pa ame e a e no possible.
Tempo al co ela ion is achie ed by modeling each
componen o he s a e ec o x
k
~wi h k he disc e e
ime!as a Gauss–Ma ko p ocess. ~To simpli y he
no a ion, we de ine he Ma ko ian p ocess y
k
as he
i h componen o ec o x
k
, so ha y
k
5x
i,k
.!
The Gauss–Ma ko p ocess,
14
which is o en called
Ma ko ian noise, is ze o-mean low-pass il e ed
Gaussian noise, whose au oco ela ion unc ion is
gi en by
Ry~ ! 5sm2exp~2du u!, (23)
whe e s
m
2
is he powe o he p ocess y
k
and dis he
3-dB cu o equency o he low-pass colo ing il e
~IIR boxes in Fig. 1, whe e IIR s ands o in ini e
impulse esponse!.
The disc e e- ime equa ion o he p ocess can be
w i en in he o m o an au o eg essi e mo ing-
a e age scala p ocess
20
as
yk115exp~21yLc!yk1wk, (24)
whe e y
k
and w
k
a e he Ma ko ian and whi e se-
quences, espec i ely, and L
c
is he empo al co ela-
ion leng h, de ined as
Lc51yd, (25)
whe e L
c
has uni s o samples @ he spa ial pe iod has
al eady been de ined in Eq. ~8!#.
Finally, Eqs. ~24!and ~25!enable us o exp ess he
Fig. 1. Time–space EKF co ela ion model.
20 Oc obe 1998 yVol. 37, No. 30 yAPPLIED OPTICS 7023

s a e- ec o ansi ion ma ix associa ed wi h Eq.
~A19!as
Fk5exp~21yLc!I, (26)
whe e Iisa~2NyM!3~2NyM!iden i y ma ix and we
ha e used he simpli ica ions ha F
k
is cons an o e
ime
k
and ha L
c
is he same o all he cells along
he lida explo a ion pa h. Wi h F
k
a ma ix, w
k
also becomes a ec o , whose 2NyMcomponen s ep-
esen whi e sequences a each ime 5
k
.
In p ac ice, Ma ko ian noise is esponsible o he
ime d i o he ac ual alue o he op ical pa ame-
e s being es ima ed by he il e a each p ojec ion
s ep. Fo his eason i is use ul o de ine an in en-
si y pa ame e p ha enables us o adjus he powe
o he Ma ko ian p ocess o , equi alen ly, o link he
d i ing Gaussian noise powe o he ampli ude
change caused in an op ical pa ame e . F om Eqs.
~23!and ~25!and Re . 14, he Gaussian noise s an-
da d de ia ion s
w
and he Ma ko ian one, s
m
, can be
ela ed as
sw5sm
Î
12exp~22yLc!. (27)
A comp ehensi e collec ion o his og ams o a gi en
s
m
ha e shown ha o e 95% o he Ma ko ian am-
pli udes dis ibu e be ween 62.5s
m
. Wi h ha in
mind, he whi e-noise s eng h s
w
5s
a
i
ha is
needed o cause a p-pe -one change in he ampli ude
o a ec o componen o x
k
~le i be a
i
!o e a co e-
la ion leng h L
c
becomes
sai5p
2.5 ai
Î
12exp
S
22
Lc
D
,i51,...,N
M. (28)
B. Spa ial Co ela ion
Con a y o wha happened wi h nonmemo y algo-
i hms, whe e analy ical co ela ion ela ions we e
assumed @ o example, homogenei y o he slope and
exponen ial-cu e- i ing algo i hms o exp essions
~3!and ~5!, espec i ely, o he powe -law co ela ion
o Eq. ~6! o Kle ’s me hod#, he app oach p esen ed
he e is based on he co ela ion g aph o Fig. 2. I
pa es he way o he in oduc ion o loose s ochas ic
ela ions among he sough -a e op ical pa ame e s
ins ead o igh analy ical ones. I seems sensible o
guess ha , o example, any ex inc ion change in a
pa icula cell will, in u n, in luence a ia ions no
only in he in-cell backsca e componen bu also in
he ex inc ion and backsca e componen s o i s
neighbo ing cells. The unde lying physical phenom-
enon being he cause, he changes may well ex end
o e se e al cells. F om he co ela ion g aph o
Fig. 2, one can build he whi e-noise s a e- ec o co-
a iance ma ix as ollows:
Cw5
F
Caa Cab
Cba Cbb
G
. (29)
These block ma ices can be de eloped as
Caa 5
3
sa1
2 sa1sa2··· n21sa1san
··· sa2
2··· n22sa2san
··· ··· ··· ···
··· ··· ··· san
2
4
, (30)
Cab 5
3
9sa1sb1 9 sa1sb2··· 9 n21sa1sbn
··· 9sa2sb2··· 9 n22sa2sbn
··· ··· ··· ···
··· ··· ··· 9sansbn
4
, (31)
whe e C
ba
5C
ab
,C
bb
is he same as C
aa
bu wi h b
and ape mu ed, is he co ela ion coe icien be-
ween one cell and he nex one along he beam pa h
~which is due o he physical con inui y o he a mo-
sphe e!, 9 is he in-cell ex inc ion- o-backsca e co -
ela ion coe icien , and s
i
has been de ined abo e in
Eq. ~28!. On he condi ion ha u u,1, u 9u,1, i can
be p o ed ha he g aph o Fig. 2 does ep esen a
co a iance ma ix.
Assuming ha empo al and spa ial co ela ion
p ocesses a e independen , he Ma ko ian noise
s a e- ec o co a iance ma ix ~i.e., he sough -a e
co a iance ma ix o he a mosphe ic model gi en o
he il e , Q
k
!can be compu ed om Eqs. ~27!,~29!,
~30!, and ~31!as
Qk5Cw
12exp~22yLc!, (32)
whe e Q
k
and C
w
a e basically he same excep o a
scaling ac o , which could, in u n, be me ged in o an
equi alen in ensi y pa ame e p9.
Al hough he in e sion o wind ields is a om he
objec i e o his s udy, his independen hypo hesis
be ween empo al and spa ial co ela ion is, howe e ,
doub ul in si ua ions wi h a signi ican adial wind
componen ~i.e., he wind componen along he explo-
a ion pa h!. In hese ins ances adial wind
s ongly co ela es bo h space and ime luc ua ions
along he line o sigh . He e one migh conside only
he spa ial co ela ion o Fig. 1 ~i.e., Q
k
5C
w,k
!, bu
his ime a a ian one, since he obse a ion cells
along he pa h become p og essi ely a ec ed by di -
e en co ela ion links as ime goes on. In addi ion,
boos ing elemen s o he main diagonal would ell
he il e o a signi ican inc ease in he co ela ion
among neighbo ing cells. In any case he possibili y
o modeling nons a iona y s a is ics in Q
k
by ese -
ing i a each succeeding s ep o he il e o e s a
wide span o a ac i e possibili ies ye o be in es i-
ga ed.
Fig. 2. Spa ial co ela ion g aph o he s a e- ec o componen s.
7024 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998
Usually, he s a e-noise co a iance ma ix o he
EKF, Q
k
, is he mos di icul inpu o assess, since i s
a mosphe ic coun e pa Q
k,a
~ he subsc ip a e e s
o a mosphe ic!is unknown. The p oblem o inding
good models o Q
k
has some imes been ackled by
using a pa i ioned app oach,
16,21,22
whe e he un-
knowns a e me ged in o a ec o o andom a iables
Q5~u
1
...u
P
!wi h known o assumed a p io i p ob-
abili y densi y unc ions ~conside , o example, Re .
16!. This app oach yields a bank o EKF’s wo king
in pa allel, each ma ched o an app op ia e alue o
u
i
, so ha he o e all ec o Qspans he space o
unknown pa ame e s cons ained by hei ela ed
possible alues. In heo y, join es ima ion o he
ex inc ion and backsca e pa ame e s Qwould i -
ually apply o all he elemen s o Q
k,a
plus, possibly,
he equi alen in ensi y pa ame e p9. In p ac ice,
his would in ol e a la ge a ay o il e s ha would
p obably exceed he amewo k o in elligen and sel -
o ganizing sys ems, and hence i would ce ainly p e-
en a s aigh o wa d o mula ion o he s udy.
Fo his eason, a his i s s age, es ima ion wi h a
single EKF is p e e ed in his expe imen al wo k,
e en hough his is done a he expense o la ge
modeling e o s and, hence, wo se pe o mance.
These model unce ain ies jus i y a o mula ion o
he a p io i e o co a iance ma ix as
P0
25mQ0,m$1. (33)
Wi h ega d o he a mosphe ic model, he simu-
la ions ha e used a se o pa ame e s Q
k,a
,L
c,a
, and
p
a
di e en om hose gi en o he EKF model, Q
k
,
L
c
, and p, o es he pe o mance o he il e unde
modeling e o s. Eigen alue decomposi ion is used
o compu e he linea co ela o ~Ain Fig. 1!and he
powe o he whi e-noise unco ela ed sequences
~n
1
...n
NyM
!, which a e he d i ing inpu s o he a -
mosphe ic simula o .
14
4. Simula ion Resul s
Th ough ex ensi e simula ion and simpli ied condi-
ions, join es ima ion o ex inc ion and backsca e
simula ed p o iles om elas ic-backsca e lida e-
u n signals ha e been in e ed by using he o mu-
la ion p esen ed abo e. Nex , wo simula ion se s
a e discussed; he i s one ~Figs. 3–7!co esponds o
a good- isibili y scene, and he second one ~Figs.
8–10!co esponds o mode a e- isibili y condi ions.
Simula ion pa ame e s a e summa ized in Table 1.
Fi s , we gi e a b ie ou line o he choice o s a is ical
pa ame e s.
The choice o
a
and 9
a
is based on c oss-examined
ime–space plo se s o syn hesized powe e u n sig-
nals wi h nonwindy ime–space eal obse a ions.
Good ag eemen be ween ypical eal da a se s and
simula ed ones has usually been achie ed o la ge
alues o 9
a
~ ypically be ween 0.8 and 0.9!and me-
dium alues o
a
~ ypically be ween 0.3 and 0.7!.
The o me esul is also in acco dance wi h Eq. ~6!,
whe e g51 is equi alen o 9
a
31. As o he
la e , i has been ound ha
a
alues close o uni y
a e no ad isable because hey yield s i spa ial p o-
iles ha a e so co ela ed ha i is di icul o ac-
commoda e e en mode a e he e ogenei ies along he
lida pa h. I has also been ound ha he in ensi y
pa ame e p
a
is he mos c i ical o all and ha i
mus be adjus ed o each pa icula scene. As a ule
o humb o low a mosphe ic ex inc ions, measu e-
men o he luc ua ions in he ange-co ec ed powe
has yielded accep able es ima ions o p.L
c
is usu-
ally de e mined om ough isual es ima ion.
The i s simula ed se is ela ed o a mean isibil-
i y o V
M
539.12 km. Such isibili y condi ions a e
ypical o s anda d clea o excep ionally clea ai .
F om Re s. 23 and 24 and unde he app oxima ion o
a homogeneous a mosphe e, he isibili y pa ame e
can oughly be linked o he a mosphe ic op ical pa-
ame e s a50.1 km
21
and b54310
23
km
1
s
21
o ,
equi alen ly, k
a
525 s and b54310
23
km
21
s
21
,
whe e k
a
is he ex inc ion- o-backsca e a io indi-
ca ed in Table 1 and he subsc ip a e e s o a mo-
sphe e. Hence one can wi hou dis inc ion alk
abou isibili y o homogeneous a mosphe ic op ical
pa ame e s ~a,b!.
To simula e an inhomogeneous p o ile app oxi-
ma ely ela ed o he isibili y V
M
, he simula o
compu es a ange-dependen hump-shaped backsca -
e p o ile wi h mean b, such as he one shown in Fig.
3~a!. Fo o he isibili y ma gins, he p o ile is
scaled acco dingly. In his way i is ensu ed ha he
syn hesized p o ile is always app oxima ely ela ed
o he a e age isibili y desi ed. We compu ed he
Table 1. Simula ion Pa ame e s
Basic pa ame e s
Op ical pa ame e s ~se 1!a50.1 km
21
,b54310
23
km
21
s
21
,V
M
'39.12 km
~se 2!a51km
21
,b53310
22
km
21
s
21
,V
M
'3.91 km
In e sion ange @Eq. ~9!# R
min
5200 m, R
max
55 km, DR5123.1 m @Eq. ~8!#
O de pa ame e s N540, M52@Eqs. ~11!#, i e a ions 5320
Sys em cons an @Eq. ~1!# A52.35 310
23
Wkm
3
Noise pa ame e s @Eq. ~22!# a51.8 310
210
W, b55310
218
W
2
,P
back
'2nW
Model pa ame e s @Eqs. ~28!–~32!#
A mosphe e k
a
5ayb,p
a
540%, L
c,a
550,
a
50.6, 9
a
50.9
EKF k50.9k
a
,p550%, L
c
5100, 50.3, 9 5 0.8
Ini ializa ion P
0
2
5Q
0
@Eq. ~33!#,xˆ
0
2
5~kb,...,kb,b,...,b!
20 Oc obe 1998 yVol. 37, No. 30 yAPPLIED OPTICS 7025
ange-dependen ex inc ion p o ile a e he back-
sca e by eusing he ex inc ion- o-backsca e a io
k
a
. F om he ex inc ion and backsca e ini ial p o-
iles jus compu ed, he lida ange- e u n powe o
Eq. ~1! ollows as shown in Figs. 3~b!and 3~c!.To
educe he o de o he il e , he physical p oblem
has been disc e ized by using N540 and M52,
which means N540 powe samples and NyM520
obse a ion cells. Since each cell is de ined by bo h
i s ex inc ion and backsca e pa ame e s, he e a e
40 s a e- ec o componen s, 20 o each op ical pa-
ame e .
To compu e he obse a ion noise, he simula o
uses elec ical and op ical pa ame e s om an
elas ic-backsca e lida o he Poly echnic Uni e -
si y o Ca alonia in Ba celona, Spain ~sys em speci-
ica ions a e gi en in Sec ion 5! o assess ealis ic
noise pa ame e s in Eq. ~22!. They a e ep esen a-
i e o a ypical oposphe ic lida sys em ~see Table
1!and yield he ange-dependen signal- o-noise a io
~SNR!o Fig. 3~d!.
The a mosphe ic beha io was simula ed by using
he simpli ied model desc ibed in Sec ion 3 and he
model pa ame e s o Table 1. As o he EKF, a
sligh ly misma ched model is inpu , so ha , o ex-
ample, he ex inc ion- o-backsca e a io is unde es-
ima ed by 10%, he empo al co ela ion leng h is
doubled, and he spa ial co ela ion coe icien s a e
changed as indica ed in Table 1. These modeling
e o s ansla e in o Q
k,a
ÞQ
k
, as sugges ed in Sub-
sec ion 3.B. Since we a e pa icula ly conce ned
abou he pe o mance o he il e unde di e en
isibili y condi ions and a mosphe ic modeling e -
o s, he simpli ica ion in which he e a e no mis-
ma ches in he model o R
k
, so ha bo h he
obse ables and he il e sha e he same co a iance
ma ix, has been assumed. This can be jus i ied be-
cause Q
k,a
is always he hidden pa ame e o he
a mosphe e, whe eas R
k
can ul ima ely be measu ed
om he lida sys em.
The ini ializa ion o he il e , xˆ
0
2
, may come om
any o he me hods discussed in Sec ion 1; in pa ic-
ula , Eq. ~7!would yield he bes app oxima ion.
Ye , o es he pe o mance o he il e , i has been
ini ialized in he simples possible way by using a
cons an homogeneous p o ile o he ex inc ion and
backsca e componen s o he s a e ec o , as indi-
ca ed in Table 1. Since each simula ion un akes
320 i e a ions, he il e depends mo e and mo e on
he measu emen s and less and less on he ini ial
s a e.
As ime goes on, he ac ual measu emen da a ~ob-
se ables! ecei ed o any pa icula sample un
change acco ding o he a mosphe ic s a e model
gi en by F
k
and Q
k,a
, so ha slowly a ying changes
in bo h he ex inc ion and backsca e p o iles a e
accommoda ed. The il e keeps ack o he ime-
a ying na u e o he obse ables om he beginning.
Figu es 4 and 5 illus a e he ime e olu ion o he
a mosphe ic model along wi h he EKF s a e- ec o
componen s. Recall ha componen s 1–20 ep e-
sen he ex inc ion coe icien and componen s 21–40
ep esen he backsca e coe icien along wi h he
obse a ion cells, so ha i one eads by cells, he i s
one comp ises componen s 1 and 21, he second one
Fig. 3. ~Se 1!ini ial s a e o he simula ion: ~a!syn hesized
backsca e p o ile, ~b! ange-co ec ed e u n powe , ~c! e u n
powe as ecei ed by he lida , ~d!associa ed SNR.
Fig. 4. ~Se 1! ime–space e olu ion o he ex inc ion and back-
sca e p o iles: ~a!syn hesized a mosphe ic op ical pa ame e s
~ex inc ion and backsca e !,~b!EKF in e ed op ical pa ame e s.
7026 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998
comp ises componen s 2 and 22, and so on. The
empo al e olu ion o he moun ains in Fig. 4 is
caused by he Ma ko ian noise. Spa ially, wi h 9
a
5
0.9 he in-cell ex inc ion- o-backsca e co ela ion is
so high ha he wo hal es o each plo look i ually
alike ~no e ha o illus a i e pu poses he backsca -
e hal has been escaled by k
a
!.
Con e gence o he EKF om he homogeneous
ini ial p o ile o some hing close o a eal p o ile can
easily be acked by moni o ing he ace o he e o
co a iance ma ix P
k
@see Eq. ~A18!in Appendix A
and Sec ion 5 o u he insigh #. Since P
k
in o ms
he il e abou he expec ed e o ha i is commi -
ing a each ime
k
, a good con e gence c i e ion is
whe he P
k
has eached a cons an alue. In he
plo s p esen ed, as is always he case, he shape o
he es ima ed p o iles is e ie ed as , bu hei mag-
ni udes ~especially he ex inc ion one! ake some
mo e ime o se le. In he simula ions he ace o
P
k
se les by i e a ion 150, al hough a e i e a ion 50
mos de ails om he ue a mosphe ic p o ile a
sho anges a e eco e ed qui e well.
Figu e 5 is a con ou plo o Fig. 4 ep esen ing
isoex inc ion and isobacksca e cu es ~scaled by k
a
o illus a i e pu poses!along ime o bo h he a -
mosphe ic and he es ima ed s a e ec o . Bo h con-
ou s look i ually alike a e he 50 h i e a ion
excep o some sligh de e io a ion in he a - ange
ex inc ion componen s o he a mosphe e ~compo-
nen s 10–20!, whe e he EKF pe o ms mo e poo ly.
This, howe e , can easily be jus i ied by he p og es-
si e educ ion in he SNR o Fig. 3~d! o inc easing
anges.
F om he poin o iew o he ime–space co ela-
ion models in oduced in Subsec ions 3.A and 3.B,
Fig. 6 compa es he ime e olu ion o he EKF es i-
ma es in ou obse a ion cells successi ely a he
along he lida explo a ion ange ~cells 5, 10, 15, and
20 loca ed a 1307.7, 2538.5, 3769.2, and 5000 m,
espec i ely!wi h hei ue a mosphe ic coun e -
pa s. The a mosphe ic backsca e e olu ion is de-
no ed by solid cu es, and he il e ’s es ima es a e
gi en by small ci cles. The il e ollows he andom
d i o each cell ai ly well in all he cases, bu
whe eas Figs. 6~a!and 6~b!show he bes - i ed cells,
Figs. 6~c!and 6~d!show some sligh unde es ima ion
o he a mosphe ic backsca e . Ho izon al solid
lines indica e he ini ial backsca e alue in each cell
be o e he a mosphe ic simula o s a s. These al-
ues co espond o he 5 h, 10 h, 15 h, and 20 h com-
ponen s o Fig. 3~a!. As expec ed om he empo al
co ela ion model o mula ion o Subsec ion 3.A,
Ma ko ian noise ansla es in o a slow empo al d i
o he backsca e igu e. Fo example, he a mo-
sphe ic empo al co ela ion leng h ~L
c,a
550 sam-
ples!is bes seen in Figs. 6~a!and 6~c!~solid cu es!.
Thus, in Fig. 6~c!, inc easing and dec easing slopes
las o app oxima ely 50 samples on a e age, and
he same happens in Fig. 6~a!excep ha now he
plo includes some kind o ho izon al in e al. F om
he poin o iew o he spa ial co ela ion, one has o
compa e all he plo s. Since
a
50.6 and each plo
Fig. 5. ~Se 1!con ou plo s o Fig. 4 showing e y good co ela-
ion be ween he ime–space e olu ion o he a mosphe ic op ical
pa ame e s and he in e ed ones: ~a!syn hesized a mosphe ic
op ical pa ame e s, ~b!EKF in e ed op ical pa ame e s.
Fig. 6. ~Se 1! empo al e olu ion o he backsca e p o iles in
ou ep esen a i e obse a ion cells along he lida beam pa h:
~ho izon al lines!s a ing backsca e alues o he a mosphe ic
simula o , ~solid cu es!a mosphe ic backsca e e olu ion, ~ci -
cles!EKF es ima es.
20 Oc obe 1998 yVol. 37, No. 30 yAPPLIED OPTICS 7027
These exp essions b idge he gul wi h he classical
linea il e i he equi alen ma ices F
k
and H
k
a e
de ined in he ollowing way:
Fk5] k~x!
]x
U
x5xˆk
, (A7)
Hk5]hk~x!
]x
U
x5xˆk
2
. (A8)
Iden i ica ion wi h he i s -o de e ms o app oxi-
ma ions ~A5!and ~A6!yields
xk11< k~xˆk!1Fk~xk2xˆk!1wk, (A9)
zk<hk~xˆk
2!1Hk~xk2xˆk
2!1 k. (A10)
App oxima ions ~A9!and ~A10! ep esen he linea -
ized e sion o he il e and esemble hose o a linea
Kalman il e excep o he ac ha a he han
p esen ing o al quan i ies o he il e , we conside
inc emen al ones. In ela ion o app oxima ions
~A9!and ~A10!, hese a e
Dxk5xk112 k~xˆk!, (A11)
Dzk5zk2hk~xˆk
2!. (A12)
In summa y he EKF’s ecu si e equa ion se be-
comes
xˆk5xˆk
21Kk@zk2hk~xˆk
2!#, (A13)
Pk5~I2KkHk!Pk
2, (A14)
xˆk11
25 k~xˆk!, (A15)
Pk11
25FkPkFkT1Qk, (A16)
Kk5Pk
2HkT~HkPk
2HkT1Rk!21, (A17)
whe e K
k
is he Kalman gain and P
k
2
is he associa ed
e o co a iance ma ix, de ined as
Pk
25E~ek
2ek2T!5E@~xk2xˆk
2!~xk2xˆk
2!T#, (A18)
whe e e
k
2
is he a p io i es ima ion e o .
Ye , ca e ul a en ion should be d awn o he ac
ha use o he EKF is isky, as he linea iza ion
p ocess akes places abou he il e ’s es ima ed a-
jec o y o he s a e ec o a he han abou a p e-
compu ed nominal ajec o y. Tha is, he pa ial
de i a i es a e e alua ed along a ajec o y ha has
been upda ed wi h he il e ’s es ima es; hus i de-
pends on he measu emen s. As a esul , he il e is
mo e likely o di e ge.
In he EKF p oblem o mula ed in his wo k, he
sys em model is linea and he s a e-space ep esen-
a ion o he a mosphe ic s a e ec o is gi en by
xk115Fkxk1wk, (A19)
whe e F
k
is he ansi ion s a e ma ix om ime
k
o
ime
k11
. I bo h he sys em and he obse a ion
model a e linea , Eqs. ~A13!–~A17!become he same
a e we eplace F
k
by F
k
and h
k
by H
k
.
We acknowledge he sponso ship o he CICYT ~In-
e minis y Commi ee o Science and Technology!
unde g an AMB96-1144-C02-C01.
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7034 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998