Adaptive filter solution for processing lidar returns: optical parameter estimation

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. Re e ences 1. R. T. H. Collis and P. B. Russell, “Lase measu emen o pa - icles and gases by elas ic backsca e ing and di e en ial ab- so p ion,” in Lase Moni o ing o he A mosphe e,E.D. Hinkley, ed. ~Sp inge -Ve lag, New Yo k, 1976!, Chap. 4, pp. 91–102. 2. R. M. Measu es, “Lase - emo e-senso equa ions,” in Lase Remo e Sensing: Fundamen als and Applica ions ~K iege , Malaba , Fla., 1992!, Chap. 7, pp. 237–280. 3. D. K. Killinge and N. Menyuk, “Lase sensing o he a mo- sphe e,” Science 235, 37–45 ~1987!. 4. A. I. Ca swell, “Lida emo e sensing o a mosphe ic ae osols,” in P opaga ion Enginee ing: Thi d in a Se ies, L. R. Bisson- ne e and W. B. Mille , eds., P oc. SPIE 1312, 206–220 ~1990!. 5. G. J. Kunz and G. de Leeuw, “In e sion o lida signals wi h he slope me hod,” Appl. Op . 32, 3249–3256 ~1993!. 6. J. D. Kle , “S able analy ical in e sion solu ion o p ocessing lida e u ns,” Appl. Op . 20, 211–220 ~1981!. 7. R. J. Ba low, S a is ics ~Wiley, New Yo k, 1989!. 8. J. J. Mo e, “The Le enbe g–Ma qua d algo i hm: imple- men a ion and heo y,” in Nume ical Analysis, Lec u e No es in Ma hema ics 630, G. A. Wa son, ed. ~Sp inge -Ve lag, New Yo k, 1977!, pp. 105–116. 9. J. D. Kle , “Lida calib a ion and ex inc ion coe icien s,” Appl. Op . 22, 514–515 ~1983!. 10. J. D. Kle , “Lida in e sion wi h a iable backsca e yex inc- ion a ios,” Appl. Op . 24, 1638–1643 ~1985!. 11. G. J. Kunz, “P obing o he a mosphe e wi h lida ,” in P oceed- ings o Remo e Sensing o he P opaga ion En i onmen ~AGARD-CP-502!,23, 1–11 ~1992!. 12. R. E. Kalman, “A new app oach o linea il e ing and p edic- ion p oblems,” J. Basic Eng. 82, 35–46 ~1960!. 13. H. W. So enson, Kalman Fil e ing Techniques. Ad ances in Con ol Sys ems. Theo y and Applica ions ~IEEE, New Yo k, 1985!, Vol. 3. 14. R. G. B own and P. Y. C. Hwang, In oduc ion o Random Sig- nals and Applied Kalman Fil e ing ~Wiley, New Yo k, 1992!. 15. B. J. Rye and R. M. Ha des y, “Nonlinea Kalman il e ing echniques o incohe en backsca e lida : e u n powe and log powe es ima ion,” Appl. Op . 28, 3908–3917 ~1989!. 16. D. G. Lainio is, P. Papapa aske a, G. Ko hapalli, and K. Pla- anio is, “Adap i e il e applica ions o LIDAR: e u n powe and log powe es ima ion,” IEEE T ans. Geosci. Remo e Sens. 34, 886–891 ~1996!. 17. R. J. McIn y e, “Mul iplica ion noise in uni o m a alanche pho odiodes,” IEEE T ans. Elec on De ices ED-13, 164–168 ~1966!. 18. P. S. Maybeck, S ochas ic Models, Es ima ion and Con ol ~Academic, New Yo k, 1977!, Vol. 1. 19. W. B. Jones, In oduc ion o Op ical Fibe Communica ion Sys- ems ~Hol , Rineha & Wins on, New Yo k, 1988!, Chap. 7, 8. 20. A. Papoulis, P obabili y, Random Va iables and S ochas ic P ocesses ~McG aw-Hill, New Yo k, 1991!. 21. D. G. Lainio is, “Pa i ioned es ima ion algo i hms. I: Non- linea es ima ion,” J. In . Sci. 7, 203–255 ~1974!. 22. D. G. Lainio is, “Pa i ioning: a uni ying amewo k o adap i e sys ems. I: Es ima ion,” P oc. IEEE 64, 1126– 1143 ~1976!. 23. H. Koshmiede , “Theo ie de Ho izon alen Sich wei e,” Bei . Phys. F eien A mos. 12, 33–53 ~1924!. 24. P. W. K use, L. D. McGlauchlin, and R. B. McQuis on, Ele- men s o In a ed Technology: Gene a ion, T ansmission and De ec ion ~Wiley, New Yo k, 1962!. 7034 APPLIED OPTICS yVol. 37, No. 30 y20 Oc obe 1998