Radiometric sensitivity computation in aperture synthesis interferometric radiometry

680 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 (a) (b) Fig. 3. (a) Obse a ion o " 00 change o ice slab on du aluminum mold a 13.5 GHz. Poin s (1–3) co espond o he ice ime exis ence in one day; (4) wo days; (5) 32 days; (9–10) 47 days; (11) 66 days; and (12) 85 days. (b) The measu emen s analogous o (a) a 37.5 GHz. he o he hand, quick eezing may p oduce a speci ic dis ibu ion o sal concen a ion in ice olume. Howe e , he main esul o he measu emen s, as i ollows om Figs. 2 and 3, is he hys e esis o " 00 in he case o slow- cycle empe a u e change o many days. Ou o he expe imen s wi h di e en empe a u e condi ions and wi h long- ime measu emen s also indica ed he e ec o ambigui y o elec omagne ic losses. The discussion on hys e esis o he eal pa o ice dielec ic pe mi i i y a 1 kHz was published ea lie in [7]. I was obse ed o he ice ha o med he 2% NaCl solu ion. This expe imen las ed o abou an hou . Howe e , his obse a ion was in e p e ed as he in luence o he hea o phase ansi ion. In ou expe imen s, he analogous explana ion is no ue because he e was a small amoun o sal and a long pe iod o keeping ice a a low-ambien empe a u e. We conclude ha he hys e esis e ec in ou expe imen s may be connec ed wi h he exis ence o supe cooling sal mic oscopic inclusions a empe a u es lowe han eu ec ic poin , whe e hey a e in a liquid s a e. The possible supe cooling empe a u e is de e mined by he shape and he size o liquid inclusions. I seems ha hei me amo phism de e mine he olume-liquid concen a ion and, consequen ly, he imagina y pa o dielec ic pe mi i i y. The de e mina ion o he exac o igin o elec omagne ic-loss ambigui y equi es a mo e de ailed in es iga ion o eshwa e ice s uc u e and physical and chemical peculia i ies o inclusions. V. CONCLUSIONS The e a e signi ican ime changes o he imagina y pa o dielec- ic pe mi i i y e en o cons an - alue sal -impu i y concen a ion in eshwa e ice. Di e en alues o elec omagne ic loss we e expe imen ally obse ed a he iden ical empe a u e. Thus, o he de e mina ion o ice dielec ic loss, we mus ake in o accoun he ice- ime exis ence a e wa e is ozen, he ice- empe a u e his o y, and he inclusions cha ac e is ics. Dis ega ding hese condi ions in p e ious pape s led no so much o measu emen e o s, bu o he desc ip ion o ice wi h di e en s uc u es. The e o e, he c yosphe e emo e sensing equi es aking in o accoun he ime changeabili y o ice elec omagne ic p ope ies. REFERENCES [1] A. S og yn, “A s udy o he mic owa e b igh ness empe a u e o snow om he poin o s ong luc ua ion heo y,” IEEE T ans. Geosci. Remo e Sensing, ol. GE-24, pp. 220–231, Ma . 1986. [2] S. C. Wa en, “Op ical cons an s o ice om he ul a iole o he mic owa e,” Appl. Op ., ol. 23, no. 8, pp. 1206–1225, 1984. [3] C. Ma zle and U. Wegmulle , “Dielec ic p ope ies o eshwa e ice a mic owa e equencies,” J. Phys. D, Appl. Phys., ol. 20, pp. 1623–1630, 1987; E a a in J. Phys. D, Appl. Phys., ol. 21, p. 1660, 1988. [4] L. Le i and L. Luba , “On he elec ic p ope ies o ice doped wi h NH 4 F,” Phys. Kondens. Ma e ie, ol. 7, pp. 368–371, 1968. [5] I. G. Young and R. E. Salomon, “Dielec ic beha io o ice wi h HCl impu i y,” J. Chem. Phys., ol. 48, no. 4, pp. 1635–1644, 1968. [6] G. S. Bo donski and S. D. K ylo , “Radio b igh ness a ia ion o eshwa e ice co e in win e pe iod,” Iz es ya Academii Nauk Rossii Se ia Fiziki A mos e y i Okeana, ol. 29, no. 6, pp. 842–847, 1993. [7] V. V. Bogo odsky and G. P. Hohlo , “In luence o some sal componen s and i s composi ion on ice elec ical p ope ies,” P oc. A c . An a c . Ins ., in Russian, ol. 295, pp. 89–95, 1970. Radiome ic Sensi i i y Compu a ion in Ape u e Syn hesis In e e ome ic Radiome y Ad iano Camps, Ignasi Co bella, Ja ie Ba ´ a, and F ancesc To es Abs ac — This pape is conce ned wi h he adiome ic sensi i i y compu a ion o an ape u e syn hesis in e e ome ic adiome e de o ed o ea h obse a ion. The impac o sys em pa ame e s and he use o simul aneous edundan measu emen s a e analyzed. The In e e o- me ic Radiome e Unce ain y P inciple is p esen ed; i quan i ies he ela ionship be ween adiome ic sensi i i y and angula esolu ion. Index Te ms—In e e ome y, adiome y, emo e sensing, sensi i i y. I. INTRODUCTION An in e e ome ic adiome e measu es he co ela ion be ween he analy ic signals collec ed by di e en an ennas [ S 1 ( ) and S 2 ( ) ]. These co ela ions p o ide he samples o he so-called isibili y Manusc ip ecei ed Decembe 27, 1995; e ised July 2, 1997. This wo k was suppo ed by he Eu opean Space Agency. wi hin he amewo k o ESA MIRAS CCN 2, Wo k O de 10, ESTEC Con ac 9777/92/NL/PB ac i i ies, wi h MATRA MARCONI SPACE as main con ac o . The au ho s a e wi h he Depa men o Signal Theo y and Communica- ions, Uni e si a Poli ` ecnica de Ca alunya, 08034 Ba celona, Spain (e-mail: [email p o ec ed]). Publishe I em Iden i ie S 0196-2892(98)00737-2. 0196–2892/98$10.00 1998 IEEE IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 681 unc ion, which has dimensions o Kel in. V 1 ; 2 ( u; )= 1 2 E [ S 1 ( ) S 3 2 ( )] =  +   1 T ( ;  )~ 12 0 u +  c 1 e 0 j 2  ( u +  ) d d (1) whe e ( u; ) is he baseline and is equal o he di e ence be ween he an enna posi ions o e he XY plane no malized o he wa eleng h; T ( ;  ) K is he so-called modi ied b igh ness empe a u e [1]. T ( ;  )= T B ( ;  ) 1 0  2 0  2 F n 1 ( ;  ) F 3 n 2 ( ;  ) (2) whe e T B ( ;  ) , dimensions o Kel in, is he b igh ness empe a u e; ( ; ) a e he di ec o cosines, wi h espec he ( X; Y ) axes, equal (sin  cos ; sin  sin  ) ; F n 1 ; 2 ( ;  ) a e he no malized an enna ol age pa e n; and ~ 12 (  ) , he inge-wash unc ion, (wi hou uni s) akes in o accoun spa ial deco ela ion e ec s [2]. In he ideal case, no deco ela ion e ec s ~ 12 (  )  1 and iden ical an enna pa e ns F n 1 = F n 2 = F n , he modi ied b igh ness empe - a u e can be eco e ed by means o a disc e e Fou ie T ans o m o he isibili y samples T ( ;  )= F 0 1 [ V ( u; )] : (3) In la ge in e e ome e s, in o de o simpli y he signal dis ibu ion ne wo k, he c oss-co ela ions a e usually pe o med a baseband by means o eal co ela o s a e in-phase and quad a u e demodula ion V 1 ; 2 / E [ i 1 ( ) i 2 ( )] + jE [ q 1 ( ) i 2 ( )] : (4) II. RADIOMETRIC SENSITIVITY COMPUTATION Radiome ic sensi i i y is de ined as he minimum inpu change ha can be de ec ed a he ou pu [2]–[4]. In a in e e ome ic adiome e , i is limi ed by he disc e iza ion and he ini e co e age o he spa ial equencies plane ( u; ) and he SNR, which can be imp o ed by inc easing he in eg a ion ime and/o he p ede ec ion bandwid h [3]. The ini e ( u; ) co e age and he disc e iza ion e o s se he sa u a ion limi ha is eached o high SNR’s. A. Disc e iza ion and Fini e ( u; ) Co e age In a o al-powe o Dicke adiome e , he measu ed an enna em- pe a u e is gi en by equa ions 4.55–4.60 o [4]. The e o commi ed depends on he pa icula b igh ness empe a u e dis ibu ion being obse ed and can be minimized by maximizing he an enna main- beam e iciency (MBE), which equi es he use o an ennas wi h a ape ed illumina ion ha , in u n, educe he achie able spa ial esolu ion. On he o he hand, an in e e ome ic adiome e o ms he b igh - ness empe a u e map by a disc e e-in e se Fou ie ans o m o he isibili ies measu ed by he a ay (3). I has been shown [1], [5], [6] ha , as p oposed in [7], he op imum shape o a wo-dimensional (2-D) in e e ome ic a ay is a Y . Y -a ays gene a e he la ges egula ( u; ) co e age o e an hexagonal g id o a gi en numbe o an ennas, hus maximizing he angula esolu ion o , con e sely, minimizing he ha dwa e equi emen s [1]. The impulse esponse o he in e e ome e in he di ec ion (  0 ; 0 ) can be in e p e ed as he beam syn hesized by he a ay, and i is called he equi alen a ay ac o ( AF eq ) [3] because o i s simila i ies wi h phased a ays AF eq ( ; ;  0 ; 0 )= A n W ( u n ; n )~ n u +  o 1 e + j 2  ( u (  0  )+ (  0  )) (5) whe e A is he pixel’s a ea in he ( u; ) plane; A = d 2 o T -a ays; A = p 3 d 2 = 2 o Y -a ays; A = d o one-dimensional (1-D) a ays; and d is he spacing be ween adjacen an ennas no malized o he wa eleng h o he minimum baseline. The unc ion W ( u; ) is a window used o weigh he isibili y samples. In a simila way, he MBE can be de ined as MBE = main lobe j AF eq ( ;  ) j d  4  j AF eq ( ;  ) j d  (6) whe e he AF eq is no squa ed because i e e s o b igh ness empe a u es, a powe measu emen , howe e , om (5), he AF eq may ha e nega i e lobes. The MBE can be op imized by a p ope selec ion o he window unc ion. Table I shows he sidelobe le el (SLL) and he MBE a he SLL o i e di e en windows o an Y -a ay wi h N EL =43 an ennas pe a m spaced d =0 : 89 wa eleng hs, as p oposed o MIRAS [7]. Deco ela ion e ec s ha e been neglec ed since B= 0  2%. The sa u a ion o he adiome ic sensi i i y shown in Fig. 1 is due o he disc e iza ion o MBE e o compu ed a he cen e o he ins an aneous ield o iew (FOV) [1]. The e o dec eases wi h he a ay size and he window smoo hness. B. SNR I he eal and imagina y pa s o he isibili y unc ion a e ob ained by c oss-co ela ing he in-phase and quad a u e componen s o he signals collec ed by he an ennas once digi alized [7], sligh ly di e en esul s a e ob ained om hose p esen ed in [2], [3], and [8] o he 1-D in e e ome e ESTAR, o adioas onomy. Th ee e ec s ha now ha e been aken in o accoun a e p ede ec ion il e s’ shape ( ec angula o Gaussian), single sideband (SSB) o double sideband (DSB) ecei e s wi h he same p ede ec ion bandwid h, and co ela o ’s ype. The MIRAS inge-wash unc ion was compu ed in [5] and [6], aking in o accoun he o e all equency esponse o he ecei ing chain [7]. I was ound ha he inge-wash unc ion is be e app oxima ed by a Gaussian il e (7a) han by a ec angula il e (7b) wi h he same noise bandwid h B (7c). j H ( ) j = e 0 (( 0 ) =B ) (7a) j H ( ) j =5 0 o B (7b) B 1 = + 1 01 j H ( ) j 2 d (7c) whe e 5( x )=1 o j x j 1 = 2 and 0 elsewhe e. Consequen ly, i is expec ed ha mo e accu a e esul s a e ob ained wi h he Gaussian model. Following he p ocedu e used in [3], he s anda d de ia ion o he eal and imagina y pa s o he isibili y unc ion can be compu ed, aking in o accoun ha I/Q demodula ion is pe o med p io o he co ela ion [6, App. 1]. The main esul s a e lis ed below o Gaussian (8a) and o ec angula p ede ec ion il e s (8b)  2 ; i =1 2 p 2 B e ( T A + T R ) 2 1+ e 0  (21 = p 2 B ) + V 2 ; i ( u; )1+ e 0  (21 = p 2 B ) 0 V 2 i; ( u; ) 1 0 e 0  (21 = p 2 B ) (8a) 682 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 TABLE I SLL [dB] AND MBE FOR THE MIRAS SPACEBORNE-INSTRUMENT EQUIVALENT-ARRAY FACTOR (  2 mn = u 2 mn + 2 mn ,  max = p 3 N EL d ) Fig. 1. Radiome ic sensi i i y dBK (10 log 1 T ) e sus SNR [10 log ( T A = V )]. T A = 200 K, MIRAS ins umen . Radiome ic sensi i i y sa u a ion is due o he disc e iza ion and ini e ( u; ) co e age e o s.  2 ; i =1 2 B e ( T A + T R ) 2 1+3 21 B + V 2 ; i ( u; )1+3 21 B 0 V 2 i; ( u; ) 1 0 321 B (8b) whe e 3( x )=1 0 j x j o j x j 1 and 0 elsewhe e; V and V i a e he eal and imagina y pa s o he isibili y unc ion; T A is he an enna empe a u e; T R = T R 1 = T R 2 is he ecei e s’ noise empe a u e; 1 = o 0 lo is he di e ence be ween he il e ’s cen al equency o and he local oscilla o ’s equency lo; and  e is he e ec i e in eg a ion ime ha depends on co ela o ’s ype, i.e.,  e =  o an analog co ela o , and  e =  /2.46 o 1-bi 2 1-bi co ela o wi h IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 683 sampling equencies s =2 B [9]. A his poin , i is in e es ing o no e he ollowing. 1) The a iance  2 V =  2 +  2 i , compu ed wi h ec angula p ede ec ion il e s, is p 2 imes la ge han wi h Gaussian il e s because noise is mo e compac ed in equency and su e s less om deco ela ion e ec s. 2) Fo any kind o il e , an imp o emen by a ac o o wo is achie ed in SSB ecei e s ( j 1 j >B= 2 ), as compa ed o DSB ecei e s ( 1 =0 ), a he expense o highe speed co ela o s and highe powe consump ion. In p ac ice, he use o SSB demodula ion simpli ies ecei e ’s design when e adia ion om local oscilla o o he an enna mus be kep below he h eshold o he signals ecei ed in he p o ec ed band, i.e., 1.400–1.427 MHz [7]. I also educes local oscilla o AM noise and o se s can be easily emo ed by high-pass il e ing he signals p io o co ela ion. 3) The use o digi al co ela o s educes he in eg a ion ime  by a ac o ha depends on quan iza ion le els and sampling equency [9]. Since he b igh ness empe a u e map is ob ained by means o a disc e e Fou ie ans o m o he isibili y samples, he isibili y e o s a e ansla ed in o he empe a u e map ^ T ( ;  )= A m n W mn 1 [ V ( u mn ; mn )+ e V ( u mn ; mn ) + je V ( u mn ; mn )] 1 e j 2  ( u  +  ) (9) whe e ( e V , e Vi ) a e he e o s in he eal and imagina y pa s o he isibili y unc ion. P io o compu a ion o he adiome ic sensi i i y some conside a ions abou edundancy and he mi ici y mus be poin ed ou . 1) He mi ici y o he Visibili y Samples: Only hal o he base- lines mus be measu ed ( u ≥0, ≥0 and u <0, > 0). The o he hal is ob ained by conjuga ing he measu ed baselines. In doing so, he noise is He mi ian oo. V 3 ( u; )= 1 2 E [ S 1 ( ) S 3 2 ( )] 3 = 1 2 E [ S 2 ( ) S 3 1 ( )] = V ( 0 u; 0 ) : (10) 2) Redundancy and Co ela ion Be ween E o s: In [5], [6], [10] and [11], i is shown ha he c oss-co ela ion o he e o s o wo iden ical baselines 1–2 and 3–4 (excep o he an enna posi ions) wi h he same in eg a ion ime  , one o hem delayed  d , is gi en by ( ec angula p ede ec ion il e s) E [1 V 12 ( +  d )1 V 3 34 ( )] = R ^ V ^ V (  d ) 0 V 12 V 3 34 = V 13 V 3 24 B e sinc  d : (11) being R ^ V ^ V (  d ) , he c oss-co ela ion be ween he measu ed is- ibili ies V 12 and V 34 a =  d . No e ha , in an ac ual onboa d in e e ome e like MIRAS [7], all baselines a e measu ed in he same ime in e al and  d =0 . Gi en i s impo ance, we explici ly show ha he noise o a isibili y sample (8b) can be ob ained om (11).  2 V =  2 V +  2 V = E [ j 1 V 12 j 2 ]= R 1 V 1 V (0) = V 11 V 3 22 B e = j V (0 ; 0) j 2 B e =( T A + T R ) 2 B e (12) F om (11), i can be seen ha , wi h ideal noise- ee ecei e s, e o s be ween simul aneous measu emen s (  d = 0) o di e en isibili y samples a e s ongly co ela ed i he spacing be ween he an enna pai s 1–2 and 3–4 is much smalle han he down all o he ampli ude o he isibili y unc ion. This si ua ion holds o scenes consis ing on poin sou ces [2], [10], and a e aging simul aneous measu emen s does no imp o e SNR signi ican ly. On he con a y, o a smoo h empe a u e dis ibu ion, as in he case o ea h obse a ion, he isibili y unc ion decays apidly, e o s a e only pa ially co ela ed, and a e aging educes noise powe . On he o he hand, i he ecei e ’s noise empe a u e is much highe han he b igh ness empe a u e o be measu ed, he a e - aging o simul aneous measu emen s imp o es he SNR, due o he educ ion o ecei e ’s noise. This is no he case wi h ea h obse a ion a low mic owa e equencies, in which ecei e ’s noise empe a u e ( T R  80 K) is usually lowe han he a e age b igh ness empe a u e ( T A  250 K). In any case, he imp o emen shown by (13), ep oduced om [3], will always be lowe han he uppe bound ound o a linea a ay, which akes in o accoun unco ela ed e o s 1 T no edundancy = T B + T R p B p N V ! 1 T edundancy unco e o s = T B + T R p B p c +ln N V (13) whe e N V s ands o he o al numbe o isibili ies and c is he Eule ’s cons an . A de ailed analysis o edundancy and i s imp o emen on adio- me ic sensi i i y equi es a speci ic a ay con igu a ion and scene unde obse a ion. Howe e , o Y -a ays, which p o ide a e y low deg ee o edundancy [5], [7], [10], [11], his imp o emen can be app oxima ely ound i we ealize ha only baselines ela ing an ennas on he same a m can be edundan . By he ze o baseline i is unde s ood ha he one co esponding o u = =0 , which in MIRAS is non edundan , since i is measu ed by a dedica ed Dicke adiome e . Recall also ha when he He mi ian p ope y is conside ed e e y ( u; ) -poin is ac ually duplica ed. Fo he Y -a ay wi h h ee a ms, each wi h N EL =43 elemen s, plus a cen al elemen , he e a e 3 N EL (3 N EL +1) / 2 + 1 = 8386 baselines [ he ex a one co esponding o V (0 ; 0) ], 3 N 2 EL +3 N EL +1= 5551 non edundan baselines o non edundan ( u; ) poin s, and 3( N EL 0 1) = 126 edundan ( u; ) poin s wi h di e en deg ees o edundancy. I means ha 8386 0 5551 = 2709 edundan complex co ela ions ( isibili ies) lead o only 126 edundan ( u; ) poin s. This leads o an imp o emen o a 1% o a 43 an ennas pe a m Y -a ay [10], [11], e en in he case in which e o s be ween hese isibili y samples a e assumed o be comple ely unco ela ed. 3) Snapsho Radiome ic Sensi i i y: As shown in he p e ious sec ion, isibili y e o s a e He mi ian and, o compu a ional pu - poses, unco ela ed om sample o sample. Wi h hese conside a ions, he snapsho adiome ic sensi i i y, ha is, he a e age e o in each b igh ness empe a u e map ob ained a e an in eg a ion ime o  seconds, is T ( ;  )= A m n W mn 1 [ e V ( u mn ; mn )+ je V ( u mn ; mn )] 1 e j 2  ( u  +  ) 1 T ( ;  )= E [ T ( ;  ) T ( ;  ) 3 ] = A 2 m n W 2 mn (  2 mn +  2 imn ) + u > 0 ;  0 u  0 ; > 0 W 2 mn (  2 mn +  2 imn ) 1 cos[4  ( u mn  + mn  )] (14) 684 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 TABLE II BASIC PARAMETERS OF MIRAS INSTRUMENT which can be app oxima ed by 1 T ( ;  ) ' AT A + T R B e  w  ol  p N V (15) whe e A = p 3 d 2 = 2 , he local oscilla o ac o is gi en by  ol = p 2=1 : 41 o DSB ecei e s and  ol =1 o SSB ecei e s; he il e ac o is gi en  = 4 p 2=1 : 19 o Gaussian il e s and  =1 o ec angula il e s; and he windowing ac o  w is de ined as  w = m n W 2 mn =N V (16) whe e he numbe o isibili y samples, including he He mi ian ones, is N V =6 N 2 EL +6 N EL +1 o Y -a ays. In he MIRAS case, N EL =43 and he windowing ac o  w = 1 ; 0 : 5212 ; 0 : 5717 ; 0 : 5446 ; and 0 : 4517 o he ec angula , iangula , Hamming, Hanning, and Blackmann windows, espec i ely (Table I). No e ha he weighing unc ion a enua es isibili y samples be ween dis an an ennas, whe e SNR is wo se; hus, he adiome ic sensi i i y is imp o ed a he expense o a loss in he angula esolu ion. MIRAS spacebo ne snapsho adiome ic sensi i i y can be compu ed om (15) and (16) wi h he pa ame e s lis ed in Table II [7]. Fig. 1 shows he snapsho adiome ic sensibili y in decibels 10 log ( 1 T ) [dBK] g e sus he SNR. Fo an SNR in he MIRAS ange 31.6–33.2 dB, he adiome ic sensi i i y is bounded by 7.1–15.0 K and 3.2–6.8 K o he ec angula and Blackmann windows, espec i ely. 4) Radiome ic Sensi i i y Imp o emen by Pixel A e aging: Radiome ic sensi i i y can be imp o ed in a 2-D in e e ome ic adiome e by means o “pixel a e aging.” Tha is, since a pixel emains in he FOV o a long ime, he eco e ed alues can be a e aged a e p ope co ec ion o he dependence wi h he angle o incidence. In he MIRAS case, a pixel emains in he FOV o abou 22 s (  FOV = FOV wid h/pla o m eloci y = 165 Km/7 Km/s =22 s), om which 11 s co espond o each pola iza ion. The imp o emen on he adiome ic sensi i i y in each pola iza ion is hen 1 T pixel a g. = 1 T snap-sho = (11 s= 0 : 3 s )=6 o 6 p 2 in a single pola iza ion ins umen . This imp o emen is achie ed because unsimul aneous measu emen s a e independen and he e o is educed by he squa e oo o he numbe o measu emen s, o equi alen ly, he in eg a ion ime is inc eased o he o al ime he pixel emains in he FOV  FOV. A e pixel a e aging, o he MIRAS ins umen (dual pola iza ion ins umen ), he expec ed adiome ic sensi i i y alues a e hen 1 T MIRAS  2 : 5 and 1 : 1 K o he ec angula and Blackmann windows, espec i ely, and T A  200 K. III. RADIOMETRIC SENSITIVITY IN INTERFEROMETRIC RADIOMETERS AND TOTAL POWER RADIOMETERS: THE INTERFEROMETRIC RADIOMETER UNCERTAINTY PRINCIPLE In o de o compa e in a homogeneous way he adiome ic sensi i i ies o in e e ome ic adiome e s and ideal o al-powe adiome e s, we mus ake in o accoun all he a ailable in eg a ion ime. No e ha a 2-D in e e ome ic adiome e images all he space simul aneously, while a o al-powe adiome e images only he pixel poin ed by he an enna beam. Tha is, he MIRAS spacebo ne ins umen will image ( 3 N EL +1) 2 = 130 2 =16 : 900 pixels simul aneously [1] e e y  = 0.3 s, om which he e a e 8.689 in he alias- ee FOV. An ideal o al-powe adiome e imaging only he alias- ee FOV pixels wi h he same angula esolu ion would ha e a maximum in eg a ion ime o  pixel = = 8 : 689 = 0 : 3 s = 8 : 689 = 34 : 5  s, leading o a wo s -case adiome ic sensi i i y o 1 T TPRad pixel = T sys = ( B pixel )  14 : 5 K, which is e y close o he snapsho adiome ic sensi i i y o he in e e ome e adiome e when he ec angula window is used (Sec ion II-B3). The adiome ic sensi i i y imp o emen achie ed by windowing can be now unde s ood as he spa ial a e aging o he pixel’s alue wi h i s neighbo s. In ac , he sensi i i y imp o emen by windowing is app oxima ely ela ed o he hal -powe syn hesized beamwid hs gi en in [6] and [12] by 1 T In Rad ec angula 1 T In Rad W =1  W ' 1  0 3dB W 1  0 3dB ec angula : (17) IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 685 In [6] and [12], he angula esolu ion o Y -a ays is analyzed, and o he ec angula window 1  0 3dB ec .  = (4 p 3 N EL d ) . Fo an a bi a y window W , he p oduc 1 T W 1  0 3dB ;W is ound o be 1 T W 1  2 0 3dB ;W ' p 3 2 d 2 T A + T R B e  w  ol  F p N V   w 4 p 3 N EL d ' p 3  24 T A + T R B e  ol  F ; ( N EL > 1) (18) which can be app oxima ed by 1 T 1  2 0 3dB ;W ' T A + T R B e  ol  F d: (19) Equa ion (19) is he new In e e ome ic Radiome e Unce ain y P inciple. I s a es ha he p oduc o he adiome ic sensi i i y 1 T by he 2-D angula esolu ion 1  2 0 3dB is a cons an ha depends only on ecei e s and co ela o s pa ame e s, and i is independen on he window used o p ocess he isibili y samples. I can be iewed as he in e e ome ic adiome e e sion o he o al-powe adiome e unce ain y equa ion gi en in (6.149) o [4]. IV. CONCLUSIONS The adiome ic sensi i i y o a gene al 2-D in e e ome ic a- diome e has been compu ed in his pape . The impac o he il e s’ shape has been analyzed and quan i ied as well as he ype o demod- ula ion (SSB o DSB) and he kind o co ela o . The imp o emen achie ed by means o pixel a e aging has been discussed and esul s ha e been pa icula ized o he MIRAS ins umen , a Y -shaped in e e ome ic adiome e wi h 43 an ennas pe a m, cu en ly unde s udy a he Eu opean Space Agency. I has been shown ha a e pixel a e aging, adiome ic sensi i i ies a e expec ed o be abou 2.5 o 1.1 K, depending on he weighing unc ion used o ape he isibili y samples. Finally, The new in e e ome ic adiome e unce ain y p inciple has been s a ed: i es ablishes ha he p oduc o he adiome ic sensi i i y by he angula esolu ion is a cons an ha depends only on he kind o ecei e s, co ela o s, and minimum baselines. REFERENCES [1] A. Camps, J. Ba ´a, I. Co bella, and F. To es, “The p ocessing o hexago- nally sampled signals wi h s anda d ec angula echniques: Applica ion o ape u e syn hesis in e e ome e adiome e s,” IEEE T ans. Geosci. Remo e Sensing, ol. 35, pp. 183–190, Jan. 1997. [2] R. Thompson, J. Mo an, and G. Swenson, In e e ome y and Syn hesis in Radio As onomy. New Yo k: Wiley, 1986. [3] C. S. Ru , C. T. Swi , A. B. Tanne , and D. M. LeVine, “In e e ome ic syn he ic ape u e adiome y o he emo e sensing o he Ea h,” IEEE T ans. Geosci. Remo e Sensing, ol. 26, pp. 597–611, Sep . 1988. [4] F. T. Ulaby, R. K. Moo e, and A. K. Fung, Mic owa e Remo e Sensing, ol. I. No wood, MA: A ech House, 1981. [5] J. Ba ´a, I. Co bella, F. To es, and A. Camps, “Two-dimensional disc e e o mula ion o ape u e syn hesis adiome e s,” ESA-ESTEC, Final Rep., CNN 2 o Wo k O de 10 o ESTEC Con ac 9777/92/NL/PB, Jan. 1996. [6] A. Camps, “Applica ion o in e e ome ic adiome y o Ea h ob- se a ion,” Ph.D. disse a ion, Uni e si a Poli ` ecnica de Ca alunya, Ba celona, Spain, No . 1996. [7] Ma a Ma coni Space, “MIRAS: Mic owa e imaging adiome e wi h ape u e syn hesis. Mic owa e adiome y c i ical echnical de elop- men ,” ESA-ESTEC, Final Rep., ESTEC Con ac 9777/92/NL/PB, Jan. 1995. [8] M. E. Tiu i, “Radio as onomy ecei e s,” IEEE T ans. An ennas P op- aga ., ol. AP-11, pp. 930–938, Dec. 1964. [9] J. B. Hagen and D. T. Fa ley, “Digi al co ela ion echniques in adio science,” Radio Sci. ol. 8, pp. 775–784, Aug./Sep . 1973. [10] J. Ba ´a, A. Camps, F. To es, and I. Co bella, “Baseline edundancy and adiome ic sensi i i y: A c i ical e iew,” in Soil Mois u e and Ocean Salini y Measu emen s and Radiome e Techniques Consul a i e Mee - ing. Noo dwijk, The Ne he lands: ESA-ESTEC, Ap . 20–22, 1995. [11] , “The co ela ion o isibili y e o s and i s impac on he adiome ic esolu ion o an ape u e syn hesis adiome e ,” submi ed o publica ion. [12] , “Angula esolu ion o wo-dimensional hexagonally sampled in e e ome ic adiome e s,” Radio Sci., o be published. Recip oci y o he Bidi ec ional Re lec ance Dis ibu ion Func ion (BRDF) in Measu emen s and Models o S uc u ed Su aces William C. Snyde Abs ac —The bidi ec ional e lec ance dis ibu ion unc ion (BRDF) is one o he mos impo an su ace p ope ies o e es ial emo e sensing, bu i s de ini ion o s uc u ed su aces is no ully unde s ood. The BRDF o la su aces has a s aigh o wa d de ini ion and is usually conside ed o be ecip ocal, which means he alue is he same when he sou ce and de ec o angles a e swi ched. S uc u ed su aces, such as o es canopies and g asslands, equi e an ex ension o he de ini ion o BRDF and some addi ional measu emen condi ions. In his pape , a de ini ion o he BRDF o s uc u ed su aces is p oposed, and i is shown ha wi h his de ini ion, he BRDF is ecip ocal. In addi ion, some o he ela ed geome ical measu emen equi emen s a e discussed. I is concluded ha ecip oci y should apply o bo h measu emen s and models o s uc u ed su aces and ha ield measu emen s iola e ecip oci y no because he BRDF i sel is non ecip ocal, bu because o unco ec ed geome ic and adiome ic ac o s. Index Te ms—Elec omagne ic sca e ing by ough su aces, adia i e ans e , adiome y, emo e sensing. I. INTRODUCTION Land-co e bidi ec ional e lec ance is o p ime impo ance in e - es ial emo e sensing. In he sola - e lec i e egion o he spec um, he bidi ec ional p ope ies a e applied o no malize he e ec s o di e en sun-senso geome ies o p o ide consis en su ace ea u es o classi ica ion and change de ec ion [1]. In he he mal in a ed egion, he bidi ec ional cha ac e is ics a e applied o accoun o he e lec ed downwelling i adiance and o compu e he angula emis- si i y [2]. The bidi ec ional e lec ance dis ibu ion unc ion (BRDF) cha ac e izes su ace bidi ec ional e lec ance o all combina ions o inciden and e lec ed zeni h and azimu h angles. BRDF is an op ical p ope y o a ma e ial ha does no depend on ex e nal ac o s, such as illumina ion o a mosphe ic ansmission. In p ac ice, BRDF can be modeled, bu canno be measu ed o applied di ec ly Manusc ip ecei ed Feb ua y 19, 1997; e ised June 4, 1997. This wo k was pe o med a he Ins i u e o Compu a ional Ea h Sys em Science, Uni e si y o Cali o nia, San a Ba ba a, and suppo ed by Ea h Obse ing Sys em P og am Con ac NAS5-31370 o he Na ional Ae onau ics and Space Adminis a ion. The au ho is wi h GDE Sys ems Inc., San Diego, CA 92150-9008 USA (e-mail: [email p o ec ed]). Publishe I em Iden i ie S 0196-2892(98)00547-6. 0196–2892/98$10.00 1998 IEEE