A calibration method for fully polarimetric microwave radiometers

588 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 3, MARCH 2003 A Calib a ion Me hod o Fully Pola ime ic Mic owa e Radiome e s Janne Lah inen, S uden Membe , IEEE, A. J. Gasiewski, Fellow, IEEE, Ma ian Klein, and Ignasi S. Co bella, Membe , IEEE Abs ac —A echnique o absolu e end- o-end calib a ion o a ully pola ime ic mic owa e adiome e is p esen ed. The echnique is based on he ipola ime ic calib a ion echnique o Gasiewski and Kunkee, bu is ex ended o p o ide a means o calib a ing all ou S okes pa ame e s. The ex ension is acili a ed using a biaxial phase- e a ding mic owa e pla e o p o ide a p ecisely known ou h S okes signal om he Gasiewski–Kunkee (GK) linea ly pola ized s anda d. The ela ions needed o de- e mine he S okes ec o p oduced by he augmen ed s anda d a e p esen ed, and he e ec s o nonideali ies in he a ious componen s a e discussed. The applica ion o he ex ended s anda d o de e mining he comple e se o adiome e cons an s ( he calib a ion ma ix elemen s) o he Na ional Oceanic and A mosphe ic Adminis a ion Pola ime ic Scanning Radiome e in a labo a o y en i onmen is illus a ed. A calib a ion ma ix in e sion echnique and e o analysis a e desc ibed, as well. The unce ain ies associa ed wi h p ac ical implemen a ion o he ully pola ime ic s anda d o spacebo ne wind ec o measu emen s a e discussed ela i e o e o h esholds an icipa ed o wind ec o e ie al om he U.S. Na ional Pola -O bi ing En i on- men al Sa elli e Sys em. Index Te ms—Calib a ion, dielec ic de ices, e o analysis, mi- c owa e adiome y, pola ime y, emo e sensing, wind. I. INTRODUCTION DURING he pas decade, he e has been an inc easing in e es in passi e pola ime ic mic owa e emo e sensing o ai bo ne and spacebo ne ea h applica ions, in pa icula o ma i ime wind ec o measu emen . Se e al s udies ha e elucida ed his capabili y, beginning wi h ai bo ne expe imen s e ealing ocean su ace emission aniso opies by E kin e al. [1], ollowed la e by co obo a ing measu emen s om I iso Manusc ip ecei ed Augus 26, 2001; e ised Oc obe 17, 2002. This wo k was suppo ed in pa by he Na ional Technology Agency o Finland (Tekes) unde Con ac 40206/98, he G adua e School in Elec onics, Telecommunica- ion and Au oma ion (GETA), he Founda ion o Technology, he Vilho, Y jö and Kalle Väisälä Founda ion, he U.S. Na ional Pola O bi ing En i onmen al Sa elli e Sys em In eg a ed P og am O ice, and by he Na ional Ae onau ics and Space Adminis a ion unde G an NAGW-4191. J. Lah inen was wi h he Labo a o y o Space Technology, Helsinki Uni e - si y o Technology, 02015 HUT, Finland. He is now wi h he Eu opean Space Agency, ESTEC, TOS-ETP, 2200 AG Noo dwijk ZH, The Ne he lands (e-mail: [email p o ec ed]). A. J. Gasiewski is wi h he Na ional Oceanic and A mosphe ic Adminis- a ion, En i onmen al Technology Labo a o y, Boulde , CO 80305-3328 USA (e-mail: al.gasie[email p o ec ed]). M. Klein is wi h he Uni e si y o Colo ado, Coope a i e Ins i u e o Resea ch in En i onmen al Science (CIRES), Boulde , CO 80305-3328 USA (e-mail: [email p o ec ed]). I. Co bella is wi h he Depa men o Signal Theo y and Communica ions, Uni e si a Poli ècnica de Ca alunya, E.T.S.E. Telecommunica ió, 08071 Ba celona, Spain (e-mail: [email p o ec ed]). Digi al Objec Iden i ie 10.1109/TGRS.2003.810203 e al. [2] and Yueh e al. [3]. The in e sion o pola ime ic ocean mic owa e emission aniso opies o one- and wo-dimensional ocean wind ec o imaging was i s demons a ed by Piepmeie and Gasiewski [4] ia using ipola ime ic measu emen s a 10.7 and 37 GHz. F om hese s udies, i has become clea ha new and use ul in o ma ion can be ob ained on ocean su ace aniso opies using measu emen s o he hi d and ou h S okes pa ame e s wi hin mic owa e window channels. Thi d and ou h S okes pa ame e measu emen s a e also po en ially aluable o e ical sounding o mesosphe ic he mal s uc u e [5], an applica ion ha is an icipa ed o p o ide aluable clima ic in o ma ion, and o in e e ence de ec ion in passi e mic owa e adiome y. The comple e second-o de spec al cha ac e iza ion o elec omagne ic wa es equi es a o al o ou pa ame e s a any gi en equency: wo pa ame e s o ep esen he ms powe wi hin wo o hogonal modes (e.g., e ical and ho izon al linea , as in he modi ied S okes pa ame e basis [6, p. 125]) and wo addi ional pa ame e s o ep esen he complex cohe ence be ween hese wo modes. To exp ess he pola ime ic cha ac e is ics o hese s ochas ic ans e se wa e p ocesses, we can use he ull (modi ied) S okes ec o unde he Rayleigh–Jeans app oxima ion (1) whe e is he b igh ness empe a u e o componen , he wa eleng h, Bol zmann’s cons an , impedance o he medium, and he elec ic ield o pola iza ion . The subsc ip is equal o , , 3, and 4 when e e ing, espec- i ely, o he i s , second, hi d, and ou h S okes pa ame e s, and i equals 45, 45, , and o 45 linea , 45 linea , le -handed ci cula ly, and igh -handed ci cula ly pola ized b igh ness empe a u es, espec i ely. The wo equi alen de ini ions o he hi d and ou h S okes pa ame e in (1) ha e s imula ed he de elopmen o wo undamen ally dis inc pola ime ic adiome e a chi ec u es, i.e., ha o a co ela ing pola ime e [4] and an adding pola ime e [3]. Co ela ion o wo o hogonal wa e ampli udes can u he mo e be pe o med using ei he analog [7] o digi al de ec ion ha dwa e [8]. The adding pola ime e (e.g., see [9] and [10]) equi es he inco- he en de ec ion o a leas ou o hogonal mode combina ions along wi h pos de ec ion di e encing. 0196-2892/03$17.00 © 2003 IEEE LAHTINEN e al.: CALIBRATION METHOD FOR FULLY POLARIMETRIC MICROWAVE RADIOMETERS 589 Despi e ex ensi e wo k in passi e pola ime ic applica ions, ela i ely li le has been published on he calib a ion o po- la ime ic adiome e s, wi h he publica ion by Gasiewski and Kunkee [11] (he ea e e e ed o as he GK echnique) being he seminal wo k in his a ea. In he GK s udy a p ac ical means was p oposed o accu a ely calib a ing a ipola ime ic (i.e., i s h ee S okes pa ame e s) adiome e om i s an enna h ough i s analog- o-digi al con e e s using ela i ely simple ha dwa e. Howe e , he s udy did no add ess he calib a ion o he ou h S okes pa ame e . Acco dingly, we ha e ex ended he GK echnique o end- o-end calib a ion o a ully pola i- me ic adiome e using a simila simple passi e s anda d. The s anda d is based on ha desc ibed in [11], i.e., being composed o wo blackbodies o di e en bu p ecisely known emission empe a u es along wi h a pola iza ion-spli ing wi e-g id. In o de o gene a e a p ecisely known se o alues, we inco po a e u he a mic owa e phase e a da ion pla e. We discuss he ein he gene al equi emen s o ully pola ime ic calib a ion using his sys em, along wi h an e o analysis, and demons a e a ully pola ime ic calib a ion s anda d o labo a o y usage. The easibili y o he calib a ion me hod and cons ain s on such a s anda d sui able o wind ec o pola ime y a e also discussed. II. THEORETICAL BACKGROUND A. Gene al Requi emen s o Fully Pola ime ic Calib a ion A well-designed single-pola iza ion adiome e is highly linea in i s esponse o an enna empe a u e, hus wa an ing a wo-blackbody echnique o calib a ion (e.g., see [12]). In he wo-blackbody echnique one needs o iden i y only wo unknown sys em pa ame e s ( he gain and o se ) using wo dis inc bu p ecisely known an enna empe a u es. The adiome e ’s esponse o he hi d and ou h S okes pa ame e s as well as c oss-pola iza ion leakage is gene ally neglec ed in single and dual pola iza ion sys ems, and jus i iably so, p o- ided ha he blackbody s anda ds a e hemsel es unpola ized. Bo h analog and digi al adiome e s [13] can be calib a ed in his manne . A ullypola ime ic adiome e ,incon as ,willgene allyex- hibi some sensi i i y in each channel o all ou S okes pa am- e e s, and hus equi es mo e han wo dis inc inpu s imuli o comple e calib a ion. Based on he o mula ion o a ipola i- me ic adiome e [11] he comple e ou pu esponse o a ully pola ime ic adiome e can be w i en as (2) whe e is ideo ou pu esponse ec o ; and consis o a- diome e gain and o se pa ame e s; and is he ins umen noise e e ed o he ideo ou pu s. The o -diagonal elemen s o ep esen in e channel c oss alk, which can be he esul o one o mo e ha dwa e limi a ions, including a) limi ed po- la iza ion isola ion in he an enna, b) c oss- alk in he ideo o mic owa e ci cui y, c) unbalance o c oss- alkin he co ela o , dependingon heco ela o ypeandcon igu a ion,andd)phase imbalances in he p ede ec ed signals used o measu e o . Ino de oin e hean ennab igh ness empe a u e ec o om , he elemen s o and in (2) a e equi ed. Owing o ins u- men d i , hei de e mina ion gene ally needs o be pe o med pe iodically, wi h he pe iod de e mined by he gain and o se au oco ela ion ollo cha ac e is ics (e.g., see [14]). Du ing calib a ion, a a ie y o e e ence b igh ness ec o s a e p esen ed o he an enna, esul ing in he acquisi ion o a calib a ion da a ma ix (3) whe e is he numbe o dis inc obse a ions, o calib a ion “looks.” We ep esen he ela ionship be ween he adiome e esponse o one channel and he calib a ion da a ma ix by: (4) whe e is a uni y ec o o leng h ; he subsc ip can be ei- he , , 3, o 4. In o de o de e mine he elemen s in he gain ma ix and o se ec o he se o calib a ion looks mus ul ill wo equi emen s: a) he e e ence b igh ness ec o s mus be able o bede e mined a p io i wi h adequa e p ecision and ime- liness,andb) henumbe o linea lyindependen b igh ness ec- o s mus be g ea e han o equal o he numbe o gain/o se unknowns o each channel, i.e., mus be ull ank. Fo he ully pola ime ic case he minimum ank is i e, unless one o mo e o he unknown gain/o se s pa ame e s can be p ede e - mined and held ixed by ca e ul design and s abiliza ion. B. Passi e Pola ime ic Calib a ion Ha dwa e Using he GK pola ized s anda d [11] (he e o o e e e ed o as a “linea ly pola ized s anda d”), a maximum o h ee linea ly independen S okes ec o s along wi h an unpola ized S okes ec o can be gene a ed. The unpola ized ec o is ob ained, e.g., by emo ing he pola izing wi e g id. This se o ec o s acili a es calib a ion o he i s h ee S okes channels. In o de o calib a e he ou h S okes channel, a p ecision ci cula ly po- la ized signal can be gene a ed by inse ing a biaxial phase e- a da ion pla e be ween he linea ly pola ized s anda d and he adiome e an enna. The e a da ion pla e gene a es a p ede e - mined phase shi be ween he pe pendicula ield componen s o he ansmi ed wa es. We e e o his combina ion o a lin- ea ly pola ized s anda d and a e a da ion pla e as a “ ully po- la ime ic s anda d” (Fig. 1). 590 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 3, MARCH 2003 Fig. 1. Schema ic diag am o he NOAA/ETL ully pola ime ic calib a ion s anda d. The a p io i de e mina ion o he S okes ec o gene a ed by such a s anda d p oceeds by i s calcula ing he ipola i- me ic S okes ec o o he linea ly pola ized s anda d, hen mul iplying his ec o by a ans o ma ion ma ix desc ibing he in luence o he e a da ion pla e. The ipola ime ic S okes ec o is [11] (5) (6) whe e and a e he ho and cold blackbody b igh - ness empe a u es, espec i ely; is he physical empe a u e o he pola izing g id; and , , and a e e lec ion coe - icien , ansmission coe icien , and ohmic losses o he g id o he wa es pola ized pa allel o g id wi es, espec i ely. The analogous pa ame e s o wa es pola ized pe pendicula o he g id wi es a e , , and , espec i ely. The losses o he pola izing g id a e assumed o be included as in [15]. The g id wi e o ien a ion angle measu ed wi h espec o he an enna po- la iza ion basis is , wi h de ined by he g id wi es being aligned pa allel o he adiome e ’s e ical pola iza ion axis. We assume ha and , which is he case o g ids wi h close and uni o m wi e spacing. To simpli y ou analysis, we i s igno e he he mal emission con ibu ion om he e a da ion pla e. Using (5) and (6), he S okes ec o gene a ed upon inse ion o he e a da ion pla e is (7) (8) (9) whe e is he o a ion angle o he pla e ela i e o he a- diome e . He e, e e s o he case whe e he e a da ion pla e’s slow axis is pa allel o he adiome e an enna’s e ical pola iza ion. The nonze o elemen s o a e (10) (11) (12) (13) (14) (15) (16) whe e is he ela i e phase shi be ween he slow and as axes o he pla e; is he pla e hickness; and and a e wa e numbe s o elec ic ields pa allel and pe - pendicula o he slow axis o he e a da ion pla e, espec i ely. The losses o he e a da ion pla e in he slow and as axes a e (17) whe e is he (nonnega i e) powe a enua ion coe icien o he pla e o he elec ic ield pa allel and pe pendicula LAHTINEN e al.: CALIBRATION METHOD FOR FULLY POLARIMETRIC MICROWAVE RADIOMETERS 591 TABLE I ANEXAMPLE OF A FULLY POLARIMETRIC CALIBRATION SEQUENCE ALONG WITH THE CORRESPONDING APRIORI BRIGHTNESS VECTORS GENERATED FOR AN IDEAL STANDARD o he slow axis ( espec i ely). The a enua ion coe i- cien is ob ained by [16] (17) whe e is equency, and and a e he e ec i e pe meabili y and complex dielec ic cons an o he e a da ion pla e, espec i ely. The de i a ion o (10)–(16) o e ical po- la iza ion has been p esen ed ea lie in [17] o he lossless case. Fo he gene al (lossy) case, we p esen he esul s o all ou S okes pa ame e s in Appendix A, wi h he de ailed de i a ion a ailable in [18]. Va ious e a da ion pla e designs ha e been p esen ed in [19]. A p ac ical e a da ion pla e is a slab o dielec ic ma e ial wi h pa allel g oo es machined on one o bo h sides (con igu a ion “D” in [19]). Sui able ma e ials include, e.g., poly e a luo oe hylene (PTFE, also known by i s ade name as Te lon), polye hylene, o c oss-linked polys y ene (known by i s ade name as Rexoli e). The e ec i e dielec ic cons an o a g oo ed pla e is di e en along he axes pa allel (slow axis) and pe pendicula ( as axis) o he g oo es. To de e mine he dielec ic cons an s he machined g oo es o he e a da- ion pla e and idges be ween hem can be conside ed o be capaci o s illed wi h ai and dielec ic, espec i ely. Fo he elec ic ield pa allel o he g oo es he capaci o s beha e as i connec ed in pa allel; o he elec ic ield pe pendicula o he g oo es he capaci o s beha e as i connec ed in se ies. The e ec i e complex dielec ic cons an s o ields bo h pa allel and pe pendicula o he g oo es a e hus app oxima ely [17] (19) (20) whe e he subsc ip s “1” and “2” e e o he bulk dielec ic ma- e ial and he su ounding medium (e.g., ai ), espec i ely. The symbol s ands o he ill ac o o he pla e, i.e., he ela i e hickness o ma e ial be ween he g oo es. Exp essions o g oo e dep hs and ill ac o s o gi en phase shi s a e p o ided in [20] and [21]. In p ac ice, he pla e’s solid and g oo ed laye s a e op imized in hickness o minimize e- lec ions and p eclude g a ing lobes. Good es ima es o he losses and a e he p oduc s o he indi idual losses o he g oo ed and solid laye s o he slab. Howe e , his app oach does no include he seconda y e ec s o in e nal e lec ion o di ac ion (e.g., see [22]), which emain o be s udied. Upon inclusion o he b igh ness empe a u e con ibu ion o he e a da ion pla e he esul ing ully pola ime ic S okes ec o becomes (21) (22) whe e is he physical empe a u e o he e a da ion pla e. (The de i a ion o (22) is p esen ed in [18].) The ully pola ime ic s anda d, along wi h an unpola ized blackbody ( ealized, o example, by emo ing bo h he e a da ion pla e and g id) can be used o gene a e ou linea ly independen pola ized S okes ec o s along wi h an unpola ized S okes ec o . Collec i ely, his se o S okes ec o s acili a es p ecise calib a ion o all ou S okes pa ame e s p o ided ha he a ious ma e ial and componen pa ame e s o he s anda d a e adequa ely known. Changing and can p o ide an in ini e numbe o dis inc calib a ion da a ma ix ows. As a p ac ical example, onepa ic- ula ly use ul and comple e se o e e ence S okes ec o s is de- sc ibed in Table I. No e ha in o de o a oid he emo al o he e a da ion pla e du ing he calib a ion p ocess he gene a ion o mixed linea ly and ci cula ly pola ized signals is equi ed. Thus, he phase shi o he e a da ion pla e should be signi i- can ly di e en om 90 o i s mul iples. C. Calib a ion Ma ix In e sion and Unce ain ies By o a ing he linea ly pola ized s anda d and he e a da ion pla e o e a ange o angles and , espec i ely, along wi h applying unpola ized looks, a ull- ank se o S okes ec o s can be obse ed (23) whe e ep esen s he gene a ed a p io i S okes ec o se , and is a ma ix o unce ain ies in he calib a ion looks 592 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 3, MARCH 2003 caused by impe ec knowledge o he pa ame e s o a ious componen s. The elemen s o a e (24) whe e he subsc ip s ands o one o con igu a ions o he calib a ion s anda d. The s anda d de ia ions o each compo- nen pa ame e , , a e he ms pa ame e e o s, , p esumed calculable o e an ensemble o simila componen s. The a e elemen s o a Jacobian ela ing small a ia ions in hese pa ame e s o elemen s o he S okes ec o (25) The subsc ip is a pa ame e index ha anges om 1 o he numbe o pa ame e s . The pa ame e e o s can be u he pa i ioned in o ei he sys ema ico andomunce ain ies.Sys ema icunce ain iesa e ime in a ian and do no change be ween calib a ions. These unce ain ies include, o example, mos o he unce ain ies o he pola izing g id and e a da ion pla e, as well as beam spillo e ande ec so adiome e passbanda e aging.Random unce ain ies include physical empe a u e e o s, g id o pla e deg ada ions, and he possible e ec s o a iable amoun s o mois u e condensa ion, backg ound b igh ness, and beam mis- alignmen (i p esen ). Since e o in he calib a ed b igh ness empe a u es due o sys ema ic unce ain ies can be compen- sa ed o a pos e io i (a leas in pa ), we conside hese wo classes o unce ain ies sepa a ely. Indeed, an imp o ed de e - mina iono calib a ion s anda dcha ac e is ics(e.g., e a da ion pla e phase shi ) and/o calib a ion using da a obse ed using o he independen s anda ds can be used o educe sys ema ic e o s. The pa ame e unce ain y ec o s o majo andom, sys em- a ic, and o al unce ain ies can hus be de ined as (26) whe e he subsc ip s and e e o andom and sys ema ic unce ain ies, espec i ely. Fo simplici y, i is assumed ha , , and in he abo e. The unce - ain ies o he linea ly pola ized s anda d a e cha ac e ized by he unce ain ies o he ho and cold (o ambien ) blackbody physical empe a u es, hei emissi i ies, he ansmissi i y, e lec i i y, ohmic losses and physical empe a u e o he pola izing g id, o a ion angle, and he phase shi be ween he e ical and ho izon al b igh ness empe a u es (desc ibed by , , , , , and , espec i ely). The unce ain ies o he e a da ion pla e a e cha ac e ized by he unce ain y o o a ion angle, phase shi , losses pa allel and pe pendicula o he pla e slow axis, and physical empe a u e (desc ibed by , , , , and , espec i ely). I is assumed ha a sepa a e unpola ized blackbody is also used, o which he unce ain y o i s physical empe a u e is . This a iable is edundan wi h o i ei he he ho o cold blackbody o he linea ly pola ized s anda d is used as an unpola ized a ge . In (26), we ha e assumed ha all blackbodies ha e iden ical emissi i ies, al hough his assump ion is no necessa y. Du ing calib a ion he esponse o a single adiome e channel is (27) Simila ly, he ully pola ime ic esponse is (28) whe e is he S okes ec o ma ix (augmen ed wi h a uni y column ec o ), and is he unknown gain-o se ma ix. The o al unce ain y consis s o he sum o bo h adiome ic in eg a ionnoise and he andome o s o he calib a ion s an- da d.Longin eg a ion imescanbeused o educe ,which alls as hein e sesqua e oo o hein eg a ion ime,bu onlyinso a as sys em d i e o s emain small. The es ima ion o he gain-o se es ima e ma ix is s aigh - o wa d in he case whe e he in e se o exis s (29) In o de o educe calib a ion unce ain ies, howe e , i is desi able o ha e an o e de e mined ma ix, i.e., o include mo e han i e independen obse a ions. In his case, es ima es o he unknown gain and o se pa ame e s can be ound by pseudoin e sion [11], [23] (30) The abo e in e se is gua an eed oexis p o ided ha a ull ank se o S okes obse a ions a e made and ha he unce ain y is small enough. Spli ing he gain-o se es ima e ma ix in o LAHTINEN e al.: CALIBRATION METHOD FOR FULLY POLARIMETRIC MICROWAVE RADIOMETERS 593 sepa a e and , he scene b igh ness empe a u es a e subse- quen ly compu ed om he adiome e esponses by (31) (32) whe e , , and ep esen he S okes ec o o he scene, unce ain ies in he measu ed b igh ness empe a u es, and adiome e esponses, espec i ely. The leng h o he uni y column ec o co esponds o he numbe o he scenes. I is assumed in he abo e app oach ha he gain-o se ma ix el- emen s a e s a is ically independen and ha no a p io i in o - ma ion is used in hei de e mina ion. I co ela ions be ween any o he elemen s o exis in be ween calib a ions (e.g., due o in e nal adiome e empe a u e d i ), hen hese co ela ions could po en ially be u ilized bene icially wi hin a s a is ical ( a he han pseudo-) in e sion. I is no ed ha he phase shi o a e a da ion pla e in- c eases app oxima ely linea ly wi h equency o e nonze o adiome ic bandwid hs. Howe e , he a p io i b igh ness empe a u es o he hi d and ou h S okes pa ame e s a e unc ions o and , espec i ely, and a e no linea . The esul ing nonlinea i y can lead o small e o s in de e mining he a p io i b igh ness empe a u es o e he en i e bandwid h o a adiome e unless a sui able se o calcula ed S okes ec o s is a e aged o e he adiome e band. These e o s, howe e , a e o second o de . Assuming, o example, a 400-MHz wide band cen e ed a 18.7 GHz and a calib a ion s anda d wi h 200-K ho –cold empe a u e di e ence, we calcula e ha he e o s emain below 0.01 and 0.04 K o hi d and ou h S okes pa ame e s o phase shi s up 90 and 180 , espec i ely. No e, howe e , ha he hi d and ou h S okes pa ame e s diminish o phase shi alues nea 90 and 0 , espec i ely, leading o highe calib a ion e o s nea hese ca dinal alues. Fo easons o bo h accu acy and con enience, i is hus desi able o ab ica e he e a da ion pla e o be nea in di e en ial phase delay. Among o he po en ial sou ces o e o a e he mal a ia- ions in he dimensions o he e a da ion pla e and a ge asym- me ies. Howe e , he same numbe o pola izing molecules is p esen du ing he mal expansion; hence, phase shi s along he p incipal axes emain ai ly cons an wi h empe a u e. Use o symme y in he ab ica ion o he linea ly pola ized a ge , e- a da ion pla e, and associa ed o a ion ha dwa e insu es agains pola iza ion basis skew and pola iza ion c oss alk. E o s e- sul ing om asymme y can gene ally be associa ed wi h e o s in o a ion angle and phase shi and can be analyzed as such. D. Accu acy and Sensi i i y Issues The unce ain ies in he calib a ion s anda d pa ame e s ha e a signi ican impac on he o e all absolu e accu acy o he cal- ib a ed adiome e . The impac o hese unce ain ies can be modeled as small de ia ions om he ue gain-o se ma ix, iz, (33) Simila o (30), he gain-o se unce ain y ma ix is ela ed o calib a ion noise by (34) Assuming ha he adiome e has a su icien ly long in eg a ion ime du ing calib a ion, he in eg a ion noise can be made neg- ligible compa ed o calib a ion s anda d unce ain ies, in which case (35) The co esponding unce ain y in he scene S okes ec o as a esul o he gain-o se unce ain y can be ob ained using and (31), (33), and (35) (36) whe e is he scene b igh ness ma ix acqui ed du ing op- e a ion, augmen ed wi h a uni y column ec o as ollows: (37) The elemen s o he gain-o se unce ain y ma ix exhibi in- e dependencies ha can be examined using a gain-o se e o co a iance ma ix (38) Since he in eg a ion noise and calib a ion s anda d e o s a e unco ela ed, he o al calib a ion e o co a iance ma ix is (39) Applying su icien ly long in eg a ion imes emo es he in e- g a ion noise componen , lea ing ...(40) (41) 594 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 3, MARCH 2003 The co a iance ma ix o he scene S okes ec o e o s now becomes (42) ...(43) (44) Equa ions (42)–(44) ela e he co ela ed e o co a iances as- socia ed wi h he use o he s anda d o he associa ed e o s in he scene S okes ec o s, and p o ide a means o de e mining he o e all impac on adiome ic accu acy. III. LABORATORY DEMOSTRATION OF FULLY POLARIMETRIC CALIBRATION To demons a e ully pola ime ic calib a ion an expe imen was ca ied ou in June 1999 a he acili ies o he U.S. Na ional Oceanic and A mosphe ic Adminis a ion’s (NOAA) En i onmen al Technology Labo a o y (ETL) in Boulde , CO. The ully pola ime ic 10.7-GHz ecei e o NOAA/ETL Pola ime ic Scanning Radiome e (PSR) [13] was used o his s udy. An exis ing linea ly pola ized calib a ion s anda d was upg aded in o a ully pola ime ic calib a ion s anda d by he addi ion o a phase e a da ion pla e. (A simila ully pola ime ic s anda d was de eloped also a he Helsinki Uni e si y o Technology, Labo a o y o Space Technology [21], [24].) A. Radiome ic Equipmen The NOAA PSR is an ai bo ne mul i equency pola ime ic imaging adiome e wi h o al powe ecei e s a 10.7, 18.7, 21.45, 37, and 89 GHz.1An in e nal calib a ion sys em con- sis ing o a pai o ambien and hea ed blackbody a ge s is in- eg a ed in o he PSR. Pe iodic iews o hese a ge s enable calib a ion o he PSR o hogonally pola ized channels using a con en ional wo-look wo-poin me hod. The linea ly pola - ized calib a ion s anda d used was simila o ha desc ibed in [11], being comp ised o ho and cold blackbody a ge s and a pola izing wi e g id. The ho a ge was a ambien empe a u e, whe eas he cold a ge was imme sed in liquid ni ogen. The pola izing g id is a ec angula Du oid mic owa e subs a e o 0.40 mm 0.0157 hickness wi h 0.17-mm- hick (0.5 oz/ ) p in ed coppe g id lines. The line wid hs we e 0.15 mm, and he illing ac o was 0.25. The g id was bonded o a 13-mm hick s y o oam slab o mechanical s abili y. O e all g id di- mensions we e 444 mm 582 mm. The linea ly pola ized s an- da d was o a able a ound i s e ical axis o any a bi a y angle 1See h p://www.e l.noaa.go / echnology/ps . Fig. 2. NOAA ully pola ime ic calib a ion expe imen se up. (A) PSR housing, (B) PSR scanhead, (C) mic owa e e a da ion pla e, and (D) linea ly pola ized s anda d. , as eco ded using a 12-bi angula encode . Unpola ized cold and ho b igh ness empe a u es we e gene a ed by ei he e- mo ing he g id o eplacing i wi h a la aluminum e lec ing pla e, espec i ely. The ullypola ime ics anda dwasimplemen edbyinse ing a o a able e a da ion pla e o e he ape u e o he linea ly po- la ized s anda d (Fig. 2). The e a da ion pla e was ab ica ed ou o a slab o c oss-linked polys y ene (Rexoli e 1422) wi h pa allel g oo es o spacing 5.07 mm, dep h 15.12 mm, and ill ac o 0.53, and machined on bo h aces (Fig. 3). The diame e o he ape u e was 518 mm. The phase shi o he pla e was de e mined by applying he o mulas p esen ed in [20] and [21] o be 53.4 a 10.7 GHz using he dielec ic p ope ies o Rex- oli e om [25] a 9.05 GHz ( , ). The pla e’s physical empe a u e was equal o he ho abso be em- pe a u e. Using low g aph ne wo k simula ion, he powe e- lec ion o he pla e a he applied equency was es ima ed o be1.9%and1.0% o pola iza ionspa allelandpe pendicula o he slow axis, espec i ely. To minimize s ay adia ion leakage, he 140-mm-long gap be ween he an enna and he calib a ion s anda d was closed o using an aluminum oil sh oud. B. Measu emen s The expe imen consis ed o a se ies o measu emen s using calib a ion s anda d con igu a ions designed o p o ide a ull ank obse a ion ma ix. The calib a ion s anda d was obse ed a a a ie y o o a ion angles bo h wi h he e a da ion pla e (i.e., o ully pola ized obse a ions) and wi hou i (i.e., o pu ely linea ly pola ized obse a ions). Cold and ho unpola - ized obse a ions we e also made, bo h wi h and wi hou he e- LAHTINEN e al.: CALIBRATION METHOD FOR FULLY POLARIMETRIC MICROWAVE RADIOMETERS 595 Fig. 3. Re a da ion pla e o he NOAA/ETL ully pola ime ic calib a ion s anda d; he pla e is moun ed on a wooden disk. a da ion pla e. The e a da ion pla e o ien a ion angle was se o ou dis inc angles ( 45 ,0,45, and 90 ), bu o he wise emained in a ian wi h espec o he an enna du ing o a ion o he linea ly pola ized s anda d. The obse ed da a we e o - ganized in o se s ob ained du ing one PSR in e nal calib a ion cycle. Each se consis ed o se e al ull o a ions o he linea ly pola ized s anda d, wi h h ee o ou such se s collec ed a each o he ou e a da ion pla e angles. Pa ame e s o he ully pola ime ic s anda d we e de- e mined o calcula e he gene a ed a p io i S okes ec o s using (5)–(22). The emissi i y o he abso be ma e ial was conside ed o be essen ially uni y a he applied equency ange. The pola izing g id o he linea ly pola ized s anda d is e ched on a low-loss mic owa e boa d, and i was hus unclea i he heo e ical de e mina ion o ees anding wi e g id cha ac e is ics as in [26] could be applied. Acco dingly, alues o and (which included he in luence o he g id) we e di ec ly es ima ed by he o hogonal-channel da a ha we e calib a ed using PSR in e nal calib a ion a ge s. Measu emen s o he o a ion angle we e calib a ed by inding he maxima and minima in he ou pu signals o he o hogonal-channel pola iza ions. The cha ac e is ics o he pola izing g id we e s udied wi hou he e a da ion pla e by a ying . The o hog- onal-channel signals we e calib a ed using PSR in e nal calib a ion a ge s. The b igh ness empe a u es and we e de e mined by applying (5) and (6) and pseudoin- e sion o he en i e da ase . The es ima ed alue o was compa ed wi h ha ob ained using a cold unpola ized iew. No di e ence could be disce ned. This indica es ha he ansmission and e lec ion cha ac e is ics o he g id we e close o ideal and, hus, e i y he easibili y o an e ched pola izing g id o linea ly pola ized calib a ion s anda ds. The e a da ion pla e losses we e examined by measu ing he unpola ized cold a ge h ough he pla e. The pla e caused less han a 2-K inc ease in b igh ness, indica ing a combined e lec- ion and abso p ion loss o less han 2%, and consis en wi h heo e ical es ima es. C. Gain-O se Es ima ion The ull gain-o se ma ix was es ima ed o each obse ed da ase using (30). In o de o ob ain a su icien numbe o lin- ea ly independen measu emen s, selec ed da a o 90 was inco po a ed wi h da a o 0 , and ice e sa. Simi- la ly, selec ed da a o 0 was inco po a ed wi h da a o 45 and 45 . The PSR ambien in e nal calib a- ion a ge was used as he unpola ized sou ce. A 1 angula esolu ion he numbe o calib a ion looks anged om 400 o 900 o each in e sion. The a p io i S okes ec o s o hese se s we e de e mined using (5)–(22). Two da ase s a e p esen ed as examples: measu emen “A” was pe o med wi h 90 , and measu emen “B” wi h 45 . Raw ol ages and calib a ed esponses o measu emen “A” a e shown in Figs. 4 and 5, e- spec i ely, and o measu emen “B” in Figs. 6 and 7, espec- i ely. Compa ing he calib a ed b igh ness empe a u es, i is seen ha he ampli ude modula ion o he o hogonally pola ized channels is much smalle o case “B” han o case “A”. This educ ion in ampli ude is a consequence o he gene a ion o quad a u e-phased e ically and ho izon ally ield componen s om he linea ly pola ized signal o he g id. Each o hese ield componen s has compa able b igh ness; he e o e, he i s wo S okes pa ame e s a e expec ed o be simila . The esidual ampli ude modula ion is a consequence o he pla e’s phase shi being 53.4 (a 90 shi would cause no a ia ion o and wi h ). Ano he clea di e ence is ha o case “B”, he hi d and ou h S okes pa ame e s exhibi maxima ha a e o se by 45 in he angle . This is a consequence o he e a da ion pla e no being pa allel wi h one o he o hogonal pola iza ions o he an enna. F om he es ima ed gain ma ix and o se ec o o case “A” V K(45) V (46) se e al obse a ions ega ding he pe o mance o he PSR 10.7-GHz adiome e (as aligned du ing his expe imen ) can be made. Fi s , he symme y o he gain elemen s , , , and indica e a signi ican 45 mixing be ween he hi d and ou h S okes channels. Al hough his le el o mixing is ela i ely la ge, i is also in e ible in so wa e (i.e., du ing calib a ion)—as e idenced by he posi i e de e minan o he subma ix consis ing o , , , and . Second, he mixing om he pola ime ic channels ( and ) in o he o hogonal channels is small, wi h b igh ness empe - a u e e o s in he o hogonal channels o o de 0.02 K o less o a ypical wind ec o signal. Thi d, a signi ican le el o o hogonal channel pola iza ion mixing ( om 15 o 17 dB) 596 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 3, MARCH 2003 Fig. 4. Pola ime ic esponse o he PSR 10.7-GHz channels as a unc ion o linea ly pola ized s anda d o a ion angle (  ) . Measu emen “A”: ' = 90 . Fig. 5. S okes pa ame e s gene a ed using he NOAA/ETL ully pola ime ic calib a ion s anda d as a unc ion o linea ly pola ized s anda d o a ion angle (  ) . Measu emen “A”: ' = 90 . The solid line ep esen s he a p io i b igh ness empe a u e, and he symbols he e ie ed b igh ness empe a u e. is appa en , bu compensa ed in so wa e by o -diagonal e ms and . D. E o Analysis To de e mine calib a ion e o s he andom and sys ema ic unce ain ies in we e de i ed om he es ima ed unce ain- ies o he calib a ion s anda d pa ame e s lis ed in Table II. The es ima ed unce ain y limi s o Rexoli e we e se conse a i ely a , . The e a da ion pla e manu ac- u ing ole ances we e es ima ed o be 25 m. The unce ain y in he e a da ion pla e phase shi was subsequen ly de e mined using s anda d p opaga ion o e o s. We no e ha an accu a e igu e o phase shi can also be ob ained by di ec measu e- men ,e.g.,as in [20].Thee ec o nonze obandwid hwascom- pu ed and de e mined o be negligible. The esul ing gain-o se unce ain y ma ix due o andom unce ain y is ob ained by (35), wi h ela i e gain and o se un- ce ain ies du ing measu emen “A” p esen ed in (47) and (48),