HYPER SPECTRAL IMAGING F. SIGERNES

Transcription

1 HYPER SPECRAL IMAGING F. SIGERNES AGF-33: Remoe Sensing and Specroscop lecure noes

2 CONENS BASIC SPECROSCOPY. Ligh as waves. Inerference.3 Diffracion.4 he GRISM SPECRAL DESIGNS. he Eber-Fasie specromeers. he Cern-urner configuraion.3 he FICS concave graing specrograph.4 he GRISM specrograph.5 he Auroral Specrophoomeer number 3 SP3 3 SYSEM OPICS 3. Specromeer opical diagram 3. Fron opics 3.3 f/value of a specromeer 3.4 Magnificaion of he slis 3.5 Bandpass and resoluion 4 HROUGHPU AND EENDUE 4. Definiions 4. Eendue 4.3 Flu 4.4 hroughpu 4.5 Sra ligh 5 IMAGING SPECROSCOPY 5. Shor bacground 5. A low cos hperspecral imager 5.3 Hperspecral image formaion and spaial resoluion 5.4. Eample applicaions 5.5. Wha s ne? 6 CALIBRAION 6. Specral narrow field of view calibraion 6. Bandpass filer insrumen calibraion 6.3 Color camera calibraion AIUDE CORRECIONS 7. Inroducion Leas suares adjusmen 7.3 Kalman filering Airborne aiude esimaion 59 ii

3 8 FUURE 7.5 Aiude applicaions 63 References Appendi A: Appendi B: Appendi C: Basic euaions for an Eber-Fasie specromeer Basic calculaions for a nm GRISM specrograph Spaial resoluion calculaions wih differen fron opics iii

4 . BASIC SPECROSCOPY In order o eplain how a specromeer wors, i is necessar o refresh some basic conceps of phsics lie inerference and diffracion., he e opical elemen of a specromeer is he diffracion graing. In his chaper we will derive he graing euaion, resolving power, angular dispersion, and graing efficienc.. Ligh as waves Firs, le us define ha ligh is in paces called phoons wih wave lie properies. Fig.. shows a sinusoidal wave ha moves wih veloci v along he -ais. he ampliude of he wave can hen be defined as simpl E, E0 sin ± ω,. where E 0 is he maimum ampliude of he wave. he wave repeas iself periodicall in ime b λ/ v, where λ is he wavelengh and v is eual o he speed of ligh, c. π/λ is called he propagaion number or wave number, and angular freuenc is ωπ/λ. A more convenien wa of represening a wave is o use comple mahemaics. A wave can be described as he real par of a comple number see Fig.. i. E0cosφ isinφ. Furhermore, Euler s formula saes ha φ e i cosφ isinφ,.3 where is φ ± ω. he real par of is hen Re E E0 cosφ..4 his is a ver useful form when dealing wih waves due o rules of comple mahemaics. In 3D we ma epress he wave as iφ E r, E0e.5 where r is he posiion and he phase is defined as φ r ω ξ..6 ξ is he iniial phase of he wave. Figure.. Sinusoid wave. Figure.. Comple number.

5 . Inerference.. wo wave sources Le use consider he simple case when jus wo waves S and S ac ogeher. According o euaion.5 and.6, he waves ma be epressed as iφ E E0e iφ E E0e,.7 where he phases are φ r ω ξ. φ r ω ξ.8 Figure.3. wo waves. A poin P he ne resul mus be he sum of he vecors E E E. he inensi is hen * iφ iφ iφ iφ I E E E e E e E e E e E E E E E 0 E i φ φ cos φ φ i φ φ E0 E0 E0 E0 cosσ he phase difference σ r r ξ ξ is a sum of wo erms. he firs erm is a resul of he pah difference beween he wo waves and he second due o he iniial phase difference. Consrucive and desrucive inerference occurs when cos σ euals or -, respecivel. E 0 0 E 0 e 0 0 e.9 nπ σ n π Consrucive inerference Desrucive inerference.0 In euaion.0, n is a posiive or negaive ineger. Noe ha in he above we assumed ha he waves were linearl polaried plane waves... Several wave sources Figure.4. N number waves. Ne we consider N number of waves ha are emied b S N coheren and monochromaic sources. Each source is separaed b a disance of a. i r ω E E e E E 3 E E 0 e i r ω 0 e i r3 ω 0. i rn ω EN E0e Coheren means ha he phase difference beween he waves are consan. Wave numbers and ampliudes are he same for all waves considered.

6 he phase difference beween he waves is due o he pah difference π σ a sin β.. λ In Fig..4 we assume ha he disance o poin P is much larger han a. I is common o use a mehod called he roaing vecor sum mehod o obain he oal vecor E a poin P. In Fig..5 O is he cenre of circle wih a radius of ρ. I is circular since σ and E 0 E 0 are he same for each source. E is hen found as E CP QP.3 ρ sin Nσ. From he riangle QCR we ge E 0 ρ sin σ..4 Figure.5. Roaing vecor sum. Eliminaing ρ from euaions.3 and.4 b division we finall obain sin Nσ E E 0.5 sin σ he inensi is proporional o E sin Nσ sin Nπa sin β / λ I I 0 I0 sin sin sin /..6 σ πa β λ Fig..6 shows a plo of euaion.6 as a funcion of wavelengh λ and angle β. he funcions have a maima eual o N I 0 when σ nπ. Or, in oher words n λ asin β..7 Zero inensi occurs when / N δ n π, where n is an ineger ecep n [ 0, N,N,... ]. here are N- minima beween maima. As a conseuence, σ π / N corresponds o he phase change from pea down o ero inensi, or he half widhs of he inerference peas. Figure.6. Inerference as a funcion of angle β and wavelengh λ from N0 sources. a nm. I Noe ha as he wavelengh increases, he profiles shifs furher awa from ero order, n 0. In addiion, he separaion beween wavelenghs increases wih increased order number, n. As N increases, he widh of he peas decreases. A disance of a nm corresponds o a graing wih 600 lines or grooves per mm. 3

7 .3 Diffracion Inerference is he resul of individual sources ineracing wih each oher. Diffracion, on he oher hand, is noiceable when a wave is disored b an obsacle ha has dimensions comparable o he wavelengh of he wave. On eample is shown in Fig..7 where ocean waves hi he enrance wall o a por or harbour. Diffracion ma be seen as he inerference of a finie wave iself. Figure.7. Eample of diffracion: Ocean waves hiing a por enrance wall..3. he single sli A single sli ma be used o diffrac ligh in he same manner as described above. he widh of he sli should be in he order of 00µm or less o see an effecs in he visible range of he specrum nm. Hugens principle sae ha ever poin of a wave fron can be hough of as he source of secondar waveles. See red dos in Fig..8. hese new waves are defined as he diffraced waves, each wih ampliude de 0. he phase difference σ beween CC and AA is π π sin β σ CD..8 λ λ he same mehod we used in secion 3. is shown in Fig..9. he lengh of E becomes E QP ρ sin α..9 When β 0 hen πb sin β E 0 ρα ρ,.0 λ since all de 0 vecors are parallel for observaions perpendicular o he sli. Figure.8. he single sli. Figure.9. Roaing vecor sum. From euaion.9 and.0 we ge sin πbsin β / λ E E0,. πbsin β / λ and he inensi is proporional o he suare of he ampliude sin πbsin β / λ sinu I I0 0 sin / I,. πb β λ u where u π bsin β / λ. 4

8 Fig..0 shows he diffracion paern as a funcion of wavelengh λ and angle β. Zero order, n0, has he larges inensi. he disance beween maima for a specific wavelengh increases as λ increases. hus, red wavelenghs are more diffraced han blue. Noe ha he maimum inensi is eual o I 0 since sinu / u for u 0. Zero inensiies occur when u nπ, or b sin β nλ..3 Figure.0. Diffracion as a funcion of angle β and wavelengh λ. b nm. I he diffracion graing We now have he ools o undersand he diffracion graing. A diffracion graing consiss of N single slis aligned as shown in Fig... he spacing beween he slis is a, and he widh of each sli is b. Each sli will ac a source wih an inensi given b euaion.. hese N emiers will produce inerference according o euaion.6. he ne inensi is hen simpl inerference caused b N slis, modulaed b he diffracion paern of one single sli sin Nπasin β / λ sin πbsin β / λ I I0 sin sin / sin /..4 πa β λ πb β λ Fig.. plos euaion.4 as a funcion of wavelengh λ and angle β. Noe ha for each wavelengh we obain maima a differen angles ecep for ero order. his is he same for all wavelenghs. Or in oher words, we obain specra on each side of ero order. Also noe ha he locaion of he inensi maima for each order increases for increasing wavelengh. his effec is he opposie of wha happens wih a prism, where here are no specral orders and blue ligh is more refraced han red. Figure.. he diffracion graing. Figure.. he diffracion graing paern as a funcion of angle β and wavelengh λ. b nm and a3b. I and N..3.3 Reflecive graings In he previous secion we looed a graings ha ransmi ligh. he plane reflecive graing 3 ma be hough of as a polished surface ha has parallel grooves which have been made b a sharp diamond. he graing consiss of narrow parallel mirrors, where each mirror acs a source of inerference in he same manner as described above. See panel of Fig..3. 5

9 he phase difference beween each source due o pah difference is seched in panel of Fig..3 π σ BC AD λ..5 π asinα asin β λ Maimum inerference occurs a σ πn, or n λ asinα sin β..6 Euaion.6 is nown as he graing euaion, where n is he specral order, α he inciden angle, and β he diffraced angle of he graing. If α β, hen he condiion is called he Lirow configuraion. Noe ha so far mos of he inensi is diffraced ino he ero order. If he reflecive surfaces are iled a an angle ω b o he graing normal, we have a blaed graing. ω b is nown as he blae angle. Fig..4 shows how a blaed graing ma loo lie. Noe ha he phase difference beween he ras S and S, as a resul of he grooves, is he same as in euaion.5. his means ha he inerference paern is no shifed in angle, and ha he graing euaion sill holds. Figure.3. Panel : Ruled reflecive graing. Panel : Ras wih phase difference red solid lines. Figure.4. Blaed graing. Noe ha β<0 due o sign convenion. his is no he case for ras S 0 and S. he blae angle ω b will inroduce a shif of he diffracion paern awa from ero order. he phase difference is now wrien as π π π σ BC AD d sinε d sinτ acosωb[ sin α ωb sin ωb β ]..7 λ λ λ he diffracion paern of he blaed graing can now be found b he same mehod as described above bu using updaed he phase differences for inerference and diffracion, euaions.5 and.7, respecivel. he resul is ploed in Fig..5. o As seen in Fig..5, a small blae angle ω b 0 will shif he diffracion paern and he maimum inensi from ero o firs order ver effecivel. Zero order is almos no visible. Noe from Fig..4 ha he graing becomes mos efficien whenα ωb ωb β, since hen each surface will ac as a small mirror, reflecing he ras wih eual angles from he groove normal. Figure.5. Blaed diffracion paern. 6

10 An epression for he blae wavelengh λb can now be derived from he graing euaion.6 nλ asinα sin β nλ asin α β cos α β α ω ω β α β ω b b b α β ω β α ω a λb sinωb cos α ωb n o A 600 grooves / mm graing wih blae angle ωb α gives a blae wavelengh λ b nm for specral order n. b b Overlapping specral orders One imporan behavior of a graing is ha i produces muliple wavelenghs a fied diffracion angles as a funcion of specral order. he overlapping wavelenghs are a muliple of he facor /n from he graing euaion.6. Fig. 6 shows he seup. A beam of whie ligh his he graing a an angle α o he graing normal. hree diffraced ras are shown as a funcion of wavelengh and diffracion angle β. Air a ground level absorbs ligh below ~90 nm. Wavelenghs below he blue line in Fig..6 are herefore no deeced. Figure.6. Specral overlapping orders of a reflecive graing G. is graing normal and α inciden angle. he hree diffraced ras have angles β, β and β 3. If ou, for eample, need o mae a firs order measuremen in he visible specral range 400 o 600 nm, hen ou will need o bloc ou second order overlapping wavelenghs from 00 o 300 nm. his is in he ulraviole par UV of he elecromagneic specrum. A cu off glass filer will be needed in order o bloc ou his range, if our deecor is sensiive o UV..3.5 Resolving power he resolving power R is a heoreical concep defined as λ R..9 λ I is a measure of a graing s abili o separae adjacen specral lines. wo specral lines are considered resolved if he disance beween hem is such ha he maimum of one falls on he firs minimum of he oher. his is nown as he Raleigh crierion. he pea inensi of a graing occurs when he phase difference isσ πn. he closes minimum occurs when σ π / N. Differeniaing euaions. and.7 gives π dσ acos β dβ.30 λ 7

11 and a cos β dβ ndλ..3 herefore, π π dσ ndλ..3 λ N Rearranging euaion.3 gives us λ R n N..33 λ For eample, a resolving power of R600 grooves/mmmm 700, will resul in λ0.07nm a λ500nm for n..3.6 Angular dispersion d β he angular dispersion is defined as. Differeniaing he graing euaion.6 we dλ simpl obain dβ n..34 dλ acos β I is a measure on how ras spread ou in a diffraced angle per uni wavelengh, and i plas a e role in he calculaion of he insrumenal bandpass. Noe ha he angular dispersion increases wih order..3.7 Graing efficienc n he heoreical efficienc E λ of a graing ma now be found as he inegral facor beween he inensi a specral order n and he oal diffracion paern. Or since I I λ β hen E n λ β n β λ β n β π / λ π / I β dβ I β dβ Figure.7. heoreical efficienc for a graing wih 600 grooves / mm and a blae angle ω b α 9.63 o. he blae wavelengh λ b nm for n. Fig..7 shows a pical efficienc curve E for a 600 grooves / mm graing, blaed a 9.63 o. For λ each wavelengh λ and specral order n we obain β n from he graing euaion. β is he angle of pea inerference down o firs minimum from euaions.30 and.3 λ β..35 Nacos β n he graing efficienc also depends on which coaing maerial is used. For eample, gold reflecs ligh 0-5% beer han aluminum in he near-infra red region of he specrum. 8

12 Figure.8. he GRISM configuraion wih graing sign convenion and coordinaes..4 he GRSIM A prism saced in series wih a ransmission graing forms a diffracive elemen nown as he GRISM or he Carpeners prism. Fig..8 shows a wedge prism and a ransmission graing separaed a disance D in he direcion air cap. he ra from A o B is parallel wih he - ais. I does no refrac since i his he prism head on. A poin B i ra will refrac according o Snell s law n sin ω sinα..36 p he prism inde of refracion n p is given b Cauch s formula as B n p A,.37 λ where A and B are consans depending on prism maerial. he disance from poin B o C is now calculaed as Lcos α ω,.38 Lsin α ω where L Dcosω / cosα. A poin C he refraced angle α becomes he inciden angle o he graing. he graing euaion ma now be modified b euaion.36 n λ asinα sin β a n sinω sin β..39 p he posiion of diffraced ras of lengh S from poin C become S cos ω β..40 S sin ω β Euaions form he basics for ra racing as funcion of wavelengh. Fig..9 and able. show he resul of a compuer generaed ra racing for he firs specral order in he visible par of he specrum nm. he prism has a diameer of 50 mm and a wedge angle ω 30 o. he maerial is Barium Crown BaK4. he graing has 600 lines / mm. 9

13 Noe ha he ellow - green wavelengh close o ~500 nm passes sraigh hrough he GRISM, parallel o he -ais. his on-ais effec of he GRISM is favorable due o he simplici o sac addiional opics on boh he fron and bac side of i. Image uali will be preserved due o negligible off-ais effecs. he oal spread in he diffraced angles of he specrum is also less han using a graing alone. he laer is due o he fac ha a prism disperses blue ligh more han red, whereas he graing diffracs red ligh more han blue. he ne effec is a compac specrum cenered a he sraigh hrough wavelengh λ s of he GRISM, or when α β ω. Figure.9. Compuer generaed ra racing for GRISM. P is wedged prism wih ω 30 o. G is 600 lines/mm graing. Firs order specral range: nm. Wavelengh λ [nm] Refracive inde n g BaK4 Inciden angle α [deg.] Diffraced angle β [deg.] GRISM angle ωβ [deg.] able. GRISM daa able for Fig. 9. Wedge angle ω 30 o. n g is refracive inde of BaK4. Graing has 600 lines/mm wih inciden angle α and diffracive angle β. he GRISM angle ωβ is relaive o he - ais. he modified graing euaion.39 becomes n λs a np sinω..4 Furhermore, b differeniaing euaion.39 we obain dλ dnp B dλ n asinω acos β asinω acos β.4 3 dβ dβ λ dβ Rearranging gives he angular dispersion of a GRISM 3 d β n ab sinω / λ..43 dλ a cos β 3 he erm ab sinω / λ > 0, which implies ha he GRISM has increased angular dispersion compared o using a graing alone. 0

14 . SPECRAL DESIGNS. he Eber-Fasie specromeers In 889 Herman Eber 5 described a specromeer using a spherical mirror and a plane reflecing graing. Also nown as a Plane Graing Ssem PGS. His sech is reproduced in Fig.. and shows he insrumen as a phoographic specrograph. Figure.. he Eber-Fasie specromeer: curved enrance sli, concave mirror, 3 plane reflecing graing, 4 curved ei sli, 5 collecor lens, 6 phoon couning deecor, and 7 order soring cu-off filer. Figure.. Reproducion of Eber s sech of his specromeer. l and l are ligh baffles, P phoographic plae, G graing, S enrance sli plane, K side plaes, and H spherical concave mirror. he ligh from he enrance sli S is firs refleced b he righ half par of he concave mirror H. he graing G is hen illuminaed b a parallel beam of ligh, since he enrance sli is locaed in he focus plane of he mirror. Secondl, he diffraced ras from he graing are focused b he lef half par of he mirror on o he phoographic plae P. he graing ma also be roaed o sweep an angular inerval. he phoographic plae is hen replaced b an ei sli and a phoon couning deecor. As illusraed in Fig.., when he graing urns, he image of he enrance sli is observed a he ei sli in differen wavelenghs. his opical ssem is nown as he Eber monochromaor. In 95 William G. Fasie improved he performance of he insrumen, using curved slis insead of sraigh slis o reduce aberraions from he mirror. 6, 7 Problems of asigmaism became less, and he resoluion of he insrumen was improved. he insrumen is herefore named he Eber-Fasie specromeer. Noe ha he design onl uses wo acive opical elemens, which maes i ideal for low ligh applicaions such as airglow and fain auroras. Fig..3 shows a m focal lengh auroral Eber-Fasie. I is based on he original consrucion b Fasie a John Hopins

15 Figure.3 he m Green Eber Fasie specromeer. Pelier cooler for phoomuliplier ube, Pulse Amplifier and Discriminaor PAD wih High Volage HV conroller, 3 sli widh adjuser wheel, 4 order soring filer in fron of enrance sli, 5 graing moor ssem, 6 ligh pipe, 7 power suppl o cooler, 8 power o PAD, and 9 DC suppl o he HV conroller. Universi, Marland in he beginning of he 70 s. In 978, i was ransferred b Gulamabas Sivjee and Charles Serling Deehr o he Auroral Saion in Advendalen, Svalbard, Norwa, from he Geophsical Insiue GI, Universi of Alasa, USA. wo more followed in 980. he insrumens are named m Green, ½m Blac and m Silver Bulle according o focal lengh and chassis colour. Furhermore, Ove Harang s ½m Whie was moved in 004 from he Sibon Observaor in Norwa up o Svalbard. In 007, he were all insalled a he new Kjell Henrisen Observaor KHO on Svalbard. Daa from hese insrumens are widel published and recognied. Even hough here are onl wo acive opical componens used in he Eber-Fasie specromeer, he image uali is limied due o off-ais aberraions such as coma, asigmaism, curved focal field defocusing and spherical aberraions. he effec of coma is seen as an asmmeric recorded specral line profile. he base is enlarged on one of he sides of he specral line as a funcion of wavelengh. Defocusing causes he ei sli image o be blurred. Boh will degrade he bandpass, he spaial resoluion and he signal o noise raio. he are due o off-ais geomer of he PGS ssem. Spherical aberraion of he mirror will also blur he ei sli image due he effec ha edge versus cener ras fail o focus o he same poin. I can onl be correced b he use of aspheric opics. Even worse is asigmaism ha produces wo foci, he angenial and he sagial foci, when using a spherical mirror off-ais. he ne effec is a sli heigh magnificaion a he ei sli. A poin a he enrance sli ends o be imaged as a line perpendicular o he dispersion plane. he effec is minimied b aspheric opics and curved slis. he basic euaions for an Eber-Fasie specromeer are derived in Appendi A.

16 . he Cern-urner configuraion In 930 M. Cern and A. F. urner 8 described a specromeer using wo concave mirrors insead of onl one as in he Eber-Fasie design. Fig..4 shows he opical laou. I is clearl eual o he Eber- Fasie configuraion, bu i is more fleible. he mirrors ma now have differen focal lenghs, sie and posiion compared o he more rigid design of he Eber- Fasie. he wo fla mirrors labelled and 6 in Fig..4 are jus used o mae he design more compac. he are no used in he original laou 8. Figure.4. he Cern-urner configuraion: enrance sli, fla mirror, 3 concave mirror collecor, 4 reflecive graing, 5 focusing mirror, 6 fla mirror, and 7 ei sli. Off-ais aberraions sill affec he image uali, bu he use of wo smaller mirrors insead of one large is cos efficien. he price of high uali aspheric mirrors wih minimum aberraions ends increase dramaicall wih sie or aperure. As a conseuence, he Cern-urner configuraion is one of he mos used designs in specroscop..3 he FICS concave graing specrograph Bac in 883 Henr Augusus Rowland invened he concave graing 3 a he John Hopins Universi. I boh collimaes and focuses he inciden ligh beam. No era mirrors or lenses are reuired. oda, holographic ion-beam eching echniues produce blaed concave graings wih high groove densi and ecellen image uali. Fig..5 shows a concave graing specrograph made b Oriel Insrumens. he design is compac wih no moving pars, and i has a high aperure f/value. he specral range is nm. hese ualiies mae i an ideal insrumen for laboraor wor, including lamp cerificaion and absolue calibraion of low ligh sources. Figure.5. ORIEL Fied Imaging Compac Specrograph FICS : concave holographic graing, fla mirror, 3 deecor CCD, 4 fiber bundle, and 5 enrance sli. 3

17 .4 he GRISM specrograph Figure.6. he GRISM specrograph: L fron lens, S enrance sli, L collecor lens, G graing, P prism, L3 camera lens and CCD imaging deecor. One popular design is he use of a GRISM as dispersive elemen. As menioned in secion in secion.4, his design maes i possible o obain a sraigh hrough cener wavelengh parallel he opical ais of he ssem. Fig..6 shows a pical configuraion of a GRISM specrograph. he opical elemens ma be saced head on on-ais. he fron lens L focuses ligh ono he enrance sli S. Lens L collimaes he GRISM and lens L3 focuses he diffraced ligh ono he image deecor CCD. he on-ais design reduces geomerical aberraions such as asigmaism and coma. he image uali is high compared o off-ais ssems. he graing ma even be replicaed or cemened sraigh ono he prism. No air cap is necessar as seched in Fig..8. A second prism ma also be used o opimie he opical performance and proec he graing. A prism-graing-prism PGP elemen is shown in Fig..7. An era prism maes i possible o fine une he sraigh hrough wavelengh of he ssem 9. he compan Specim Ld. from Oulu in Finland has since he 90 s produced a large number of PGP elemen based specrographs for boh indusr and science. he onl disadvanage of he PGP ssems is he relaive high number of opical elemens needed compared o an off-ais mirror ssem such as Eber-Fasie design. Figure.7. PGPelemen. Prism, Graing and 3 Prism. High aperure and minimum ransmission losses hrough a minimum of opical elemens are essenial for low ligh applicaions. 4

18 .5 he Auroral Specrophoomeer number 3 SP3 Figure.8. he SP3 combined specrograph and Lirow specromeer 0. he las design presened in his chaper is he auroral specrophoomeer number 3 SP3. he hisor of SP3 ells a sor of auroral specroscop ha sared wih Lars Vegard s specrographs and he discover of he proon aurora bac in 939. Vegard was former suden of Krisian Bireland and he founder of he Auroral Observaor in romsø. Laer on he same insiuion, Anders Omhol and Leif Harang reuesed higher specral resoluion and sensiivi han Vegard s specrographs, which used large uar prisms insead of graings as he e dispersive elemens. he resul was SP3 consruced in 960 b Will Soffregen a he Ionospheric Observaorium in Uppsala, Sweden. I was used eensivel b his suden Kjell Henrisen o sud aurora, airglow and sar specra. In 977 i measured for he firs ime Helium emissions in he polar clef from N-Ålesund, Svalbard. I has even been used o obain specra of Barium released b roces. Fig..8 shows he principal ra diagram and recording euipmen of SP3 as originall designed b Soffregen 0. he design has wo modes. In he firs mode, he collimaor lens illuminaes ligh from he enrance sli a an inciden angle eual o he diffraced angle b he reflecive graing. his is as menioned in secion.3.3 nown as he Lirow condiion. he collimaor also focuses he ligh ono he ei sli. In fron of he ei sli is a fla mirror ha can be slighl bended o accoun for asigmaism. 5

19 As he graing sweeps in angular inervals, raw daa from he cooled phoon couning deecor is ploed on he pen recorder. he resul is saced specra wih ime. In he second mode, he insrumen is used as a specrograph wih a phoographic camera ha loos down on he graing. he insrumen was found in lae spring of 993 soced awa in he garage of he Auroral Observaor in romsø. I was hen compleel covered wih old scienific papers of he aurora, which mos probabl saved i from rus and degradaion. he insrumen was resored and he opical componens realigned b he auhor under he supervision of his advisor Kjell Henrisen. oda, i is par of Svalbard Museum ehibiion of he aurora. Figure.9. he SP3 a UNIS 006. Even hough he SP3 is an old design from he 60 s, i is sill aracive for auroral sudies wih is dual modes, high specral resoluion and aperure. New conrol elecronics and deecors have been developed since i was iniiall pu o use. As wih ha he Eber- Fasie specromeers a KHO, he phoon couning and compuer elecronics have been coninuousl upgraded since he lae 80 s. Or in oher words, he opic designs sa more or less he same, while he field of opoelecronics has grown eponeniall over he las decades. 6

20 3. SYSEM OPICS In his chaper we will esablish a general view on how o describe a specromeer ssem. Parameers lie he f/value, magnificaion and specral bandpass will be eplained. 3. Specromeer opical diagram An opical diagram is a sandard wa o race ras hrough a specromeer in an unrolled linear fashion. I visualies a cener cross secion of he insrumen perpendicular o he slis, parallel o he ais of refracion. Fig. 3. shows he laou of an opical diagram. Figure 3.. Reproducion in color of opical diagram of a specromeer. 4 AS is aperure sop, L fron lens, L collimaor lens or mirror, G graing, L 3 focusing lens or mirror, S area of ligh source, S area of source image, S area of enrance sli, S 3 area of he diffraced sli image ei sli area, p objec disance, image disance of L, f focal lengh of L, f 3 focal lengh of L 3, and Ω half angles. he -ais is parallel o he ais of diffracion, perpendicular o he slis. Ligh from he source area S illuminaes he fron lens L wih half angle Ω and objec disance p. S is he imaged area of he source S ha is focused ono he enrance sli plane b lens L. Ω is he half angle and is he image disance of L. Ne, L collimaes he graing G wih ligh ha passes hrough he enrance sli area S wih half angle Ω. L 3 focuses he diffraced ligh from he graing ono he ei sli plane wih half angle Ω 3. S 3 is hen he area of diffraced enrance sli image. f and f 3 are defined as he enrance arm lengh of L and ei arm lengh of L 3, respecivel. In pracice, hese arm lenghs corresponds o he focal lenghs of L and L 3. Noe ha he opical elemens L, L and L 3 can be eiher lenses or mirrors as long as he are able o focus or collimae he beam. he graing ma also be eiher ransmiing or reflecing. he main poins are ha he enrance sli mus be locaed in he focus planes of L and L, and ha he ras form a parallel beam ha illuminaes he graing G. L 3 will hen focus each parallel diffraced beam as a funcion of wavelengh ono he ei sli plane. he opical diagram is in oher words, a general wa o describe ra racing hrough an specromeer. I is a useful ool o use o calculae insrumenal performance and evaluae differen specral designs. 7

21 3. Fron opics he fron opics is in he opical diagram of Fig. 3., idenified as L. he purpose is o image ligh from an objec ono he enrance sli plane. L is no resriced o be onl a single lens elemen. An objecives such elescopes or even microscopes can be used as fron opics. Neverheless, he hin lens formula holds for mos cases:. 3. f p f is he focal lengh, p he objec disance and he image disance of L. Fig. 3. shows he 3 principal ras hrough a hin lens. Magnificaion of he lens is defines as M. 3. p he speed of he lens is uanified b he raio beween focal lengh and he effecive aperure, defined as he f/value or f/# f f /#. 3.3 D Liewise, numerical aperure of L is defined as NA µ sin Ω 3.4 Noe ha µ for air. Figure 3.. Single lens diagram of L wih 3 principal ras. p is objec disance, f focal lengh and is image disance. Figure 3.3. Single lens collimaed. D is diameer of lens, f focal lengh and Ω half angle. From Fig. 3.3 he half angle of L for objecs a infini is D Ω arcan. 3.5 f Appling euaion 3.4 ields D D NA µ sin Ω µ sin arcan µ f, 3.6 f which gives µ f /# 3.7 NA Euaion 3.7 shows us ha f/# and NA are boh a measure of ligh gahering power, and he are inversel proporional, i.e. a high NA resuls in a low f/#. 3.3 f/value of a specromeer he f/value of a specromeer depends on wheher ou observe he graing from he ei sli or he enrance sli. I also depends on he projeced widhs of he graing as seen from he enrance and ei sli. Namel, W Wg cosα, 3.8 W3 Wg cos β where W g is he widh of he graing. See illusraion in Fig Figure 3.4. Illuminaed graing. 8

22 Le us define D and D 3 as he euivalen diameers as seen from he enrance and ei sli, respecivel. If he heigh of he graing is H g, hen he following euaions should hold assuming ha he projeced areas can be described as circular diss D WgH g cosα π WH g WgH g cosα D π. 3.9 D3 WgH g cos β π W3H g WgH g cos β D3 π he f/values, as seen from enrance and ei, should hen be f f f /# D Wg H g cosα π. 3.0 f3 f3 f /# 3 D3 WgH g cos β π From euaion 3.0 i is clear he f/value in and ou of a specromeer is no necessaril eual since i depends α and β, which implies ha i depends on wavelengh. 3.4 Magnificaion of he slis he graing acs as a mirror along he enrance sli of heigh h. Sli heigh magnificaion is hen simpl he resul of a lens wih objec disance f and image disance f 3. See Fig I hen follows ha h h f 3 h h f f f Sli widh magnificaion is on he oher hand cosα f 3 w w. 3. cos β f In he above euaions he area of he enrance sli is magnified or magnified ei sli area is S w h full in Chaper 4. Figure 3.5. Lens magnificaion of he enrance sli heigh h of a specromeer wih collimaor lens L and deecor lens L 3. S w h. he resuling de- 3. Euaion 3. will be derived in 3.5 Bandpass and resoluion Bandpass is he measure of an insrumen abili o separae adjacen specral lines. I is defined as he recorded Full Widh a Half Maimum FWHM of a monochromaic specral line. Fig. 3.6 shows he response of an insrumen o a discree monochromaic line. he half widh of he line profile a inensi B defines our bandpass. If F is he recorded specrum, B he source specrum and P he insrumenal line profile, hen F is given as he convoluion beween B and P F B P

23 Figure 3.6. Effec of specral resoluion of monochromaic line a wavelengh λ 0 and inensi B. 4 he insrumenal line profile is defined as P λ P λ P λ Pm λ, 3.4 where each P i λ i [ m] ma be relaed o widh of he slis, naural line widh, resoluion, alignmen, diffracion effecs, aberraions or uali opics ec. he naural line widh is negleced since i is usuall no resolved. We assume no degradaion due o aberraions and diffracion effecs. he widh of he insrumenal profile is hen deermined b eiher he image of he enrance sli or he ei sli, whichever is greaer. In oher words, we assume P λ P λ, and ha he insrumen is perfecl aligned. Figure 3.7. he insrumenal profile due o he slis a convoluion of he enrance slis image wih he ei sli. 4 he shape of he P is conrolled b he convoluion of wo recangular funcions wih widhs eual o he widh of he enrance sli image, d λ, and he widh of he ei sli d λ. he wo sli funcions are recangular as illusraed in Fig.3.7. If he slis are perfecl mached hen he line profile will be riangular and he FWHM euals half of he widh a he base of he pea. he specral bandpass BP of he insrumen is hen calculaed as he linear dispersion muliplied b he ei sli widh dλ BP FWHM w. 3.5 d Appling euaion 3., he inverse of euaion.34 and d f 3dβ, hen we finall obain he bandpass as 0

24 dλ w a cos β a cos β cosα f 3 BP w w d f n f n f f β cos β 3.6 a cosα BP w n f Noe ha bandpass depends mainl on L and he enrance sli widh. Bandpass varies as cos α, while dispersion varies as cos β.

25 4. HROUGHPU AND EENDUE he e uesion in his chaper is o find a wa o uanif how much ligh is acuall passing hrough a specromeer? o answer his uesion, we need o define eendue and hroughpu of a specromeer. 4. Definiions he number of phoons emied from a source S per uni ime ino a solid angle Q is nown as he flu Φ. Inensi I is defined as he flu per uni solid angle. Radiance B is he inensi hrough a uni surface area, or phoons flu per uni area and solid angle. According o he above, flu is given in unis of # phoons [ Φ ], 4. s and inensi I has unis # phoons [ I ]. 4. sr s Radiance B is hen in unis of # phoons [ B]. 4.3 cm sr s Fig.4. shows how he solid angle is defined. Figure 4.. Solid angle. 4. Eendue he field of view or he accepance cone where phoons are allowed o ravel ino will define how much ligh ha is deeced b an insrumen. A useful uani called he geomeric een eendue characeries he abili of an opical ssem o accep ligh. I is a funcion of he area S of an emiing source and he solid angle Q is ligh propagaes ino or ou of. I is defined as G ds dq. 4.4 In our case we ma wrie as G S Q, where A π p sin Ω Q π sin Ω. 4.5 p p he eendue is hen simpl G π S sin Ω. 4.6 In erms of numerical aperure NA µ sin Ω Figure 4.. Fron lens eendue. NA G π S. 4.7 µ Euaion 4.7 is useful when woring wih fibers. For eample, a fused silica fiber wih a diameer of 00µ m will have an eendue of mm sr NA G π S π π 4.8 µ

26 Figure 4.3. Opical diagram of a specromeer. Eendue ma be viewed as he maimum geomeric beam sie an insrumen can accep. I is a consan, and should be so hroughou he insrumen. Oherwise, we will lose ligh due o geomerical blocing ec. An insrumen is said o be opimall consruced if he eendue is consan hrough each of he componens used. Re-calling he opical diagram from Chaper 3, a specromeer will be eendue opimied when G π S sin Ω π S sin Ω π S sin Ω π S3 sin Ω he opical diagram is reproduced in Fig Noe ha he fron lens L produces an image in he enrance sli plane. Onl ligh from he image cross secion ha is defined b he widh and heigh of he enrance sli will propagae ino he insrumen. I is imporan no o overfill he field of view of L. he e eendue euaion becomes π S sin Ω π S3 sin Ω3, 4.0 and i is defined as where he inpu eendue is eual o he oupu eendue of a specromeer. Or in oher words, he eendue is eual seen from boh slis. From he enrance sli he eendue is epressed as GA cosα G w h f, 4. where G A cosα is he illuminaed area of he graing as seen from he enrance sli. Correspondingl, a he ei sli GA cos β G w h f 3 Since G G3, i follows ha G cosα GA cos β A w h w h f f Using h / h f / f hen 3 cosα f 3 w w. 4.4 cos β f his shows us ha he eendue depends on he graing euaion, sie of slis, and he opics used. hus, all pars of he specromeer mus be considered and complemen each oher. 3

27 4.3 Flu Since eendue defines he maimum geomeric cone and beam an insrumen can accep, he phoon flu hrough his cone mus be a funcion of he radiance and he eendue Φ B G. 4.5 We see ha # phoons # cm sr cm s sr phoons s [ B] [ G] [ Φ]. 4.6 Euaion 4.5 can now be used o evaluae phoon flues in and ou of a specromeer ssem. 4.4 hroughpu hroughpu is he usable flu a he ei sli, available o he deecor. If we define B o be he oal radiance of he source enering he insrumen a he enrance sli, hen he oal flu in is given as GA cosα Φ B G B w h f. 4.7 A he ei sli he flu will be n GA cos β Φ B E n G3 B E w h f λ λ λ λ λ λ λ, n where E λ is he efficienc of he graing a specral order n a wavelengh λ calculaed in Chaper. Bλ is he specral he radiance. he facor λ represens geomeric losses aberraions and ransmission facors of he opical componens. λ for a perfec insrumen 4.5 Sra ligh 4.5. Random sra ligh he radiance of random sra ligh Bd due o scaer from differen componens such as mirrors, screws ec is direcl proporional o he flu densi Φ / GA or B hw cosα B d C, 4.9 f where C epresses he uali of he opics as funcion of random scaer. he random scaer flu a he ei is hen CB0 hwcosα GA cos β h w Φ d Bd G f f 3 he raio of deeced o random flu gives us an idea of he opical signal o noise raio S/N 4

28 n Φ λ Bλ Eλ f λ S / N. 4. Φ d B0 C h w cosα he main facors ha conrol he opical signal o noise raio in euaion 4. is he n uali of he opics as defined hrough he parameers E λ, C and λ. Holographic graings have a low C. his, one should mae sure o use a graing wih maimum efficienc in he wavelengh region of ineres, and use aberraion correced opics, if i is available Direcional sra ligh here are mainl 3 pes of direcional sra ligh sources:. Incorrec illuminaion of he specromeer due o overfilled opics ec.. Re-enr specra of unwaned orders / specra ha are focused ono he ei plane. 3. Graing ghoss and sra ligh due periodic machine ruling errors ec. Soluions:. Use field lenses and aperure sops o illuminae correcl.. Use mass and side baffles on he graing. il he deecor. 3. Bu a new graing. As an ion-eched blaed holographic graing would be he recommended choice. 5

29 5. IMAGING SPECROSCOPY his chaper eplains how we can image an objec as a funcion of wavelengh wih high specral and spaial resoluion based on wha we have learned in he previous chapers. As an eample, we will use a GRISM specrograph in hperspecral image mode. 5. Shor bacground he rapid developmen of imaging deecors, especiall he wo-dimensional silicon Charge Coupled Device CCD, has resuled in a new generaion of imaging specrographs. In conras o he use of a phoomuliplier ube as deecor, he imaging deecor provides informaion on he spaial disribuion of he phoon inensi along he enrance sli of he specrograph. he abili of he CCD o image he enrance sli in differen wavelenghs and inensi has been used o reconsruc he image of he arge objec wih high specral and spaial resoluion. Noe ha during he las decade mos of he effor in imaging specroscop hperspecral imaging has been in he area of air- and space born sensors. A grea deal of eperience has been, gained wih hese insrumens and he have proven o be powerful ools in remoe sensing. As a resul, he use of specral imagers for indusrial applicaions such as on-line specral 3, 4, 5, 6, 7 idenificaion, soring and inspecion of objecs has become increasingl imporan. 5. A low cos hperspecral imager A low cos porable insrumen named DRONESPEX has been designed o demonsrae he echniue of specral imaging. A picure of he assembled ssem is shown in Fig. 5.. Figure 5.. DRONESPEX. A small porable low cos hperspecral imager. is fron lens objecive, aluminum barrel, 3 ring mouns, 4 moun rod, 5 camera lens and 6 CCD deecor head. he ssems opical elemens are shown in Fig..6. he camera head 6 made b he compan Waec LCL conains a piel CCD deecor. he sie of he CCD is mm wih µm sied piels. he specral range is nm and he sensiivi is lu. he camera has boh auo and manual and gain seings. he eposure ime is 8.33 ms wih a readou ime of 3.67 ms CCIR sandard. he frames are sored on a digial video recorder. he dnamic range is 8 bis per piel. A handheld specroscope made b Paon Hawsle Elecronics Ld. is fied ino he aluminum barrel. I conains a 600 grooves/mm ransmiing holographic graing, a 30 degree Lirow dispersive prism, a f 4 mm focal lengh collimaor lens, and a fied sied sli. he lens is mm in diameer. he sli is h 4.5 mm high and w 5 µm wide. he graing is fied parallel o he prism. he prism and graing normal coincide. 6

30 A sandard C-moun f 3 5 mm focal lengh objecive 5 is used o focus he diffraced and dispersed ligh from he graing/prism ono he CCD. he lens has manual iris conrol used o deec he bacground level of he CCD. he field lens or he fron lens of he specroscope is direcl conneced o he aluminum barrel. he barrel has sandard inside C-hreads. he specroscope's enrance sli is locaed 7.5 mm ino he barrel - posiioned in he focal plane of he lens. he lens in Fig. 5. is a f mm focal lengh objecive including variable focus and manual iris conrol. All mouns and adapers are found in he Mi and Mach assemblies of he compan Edmund Scienific ES. Figure 5.. Basics of hperspecral imaging. Panel A: A 3 dimensional opical diagram illusraing he main principle. Each opical componen is labeled wih leers along he ais of he coordinae ssem. L is fron lens, S enrance sli, L collecor lens, P GRISM, L 3 camera lens and CCD imaging deecor. he opical ais is parallel wih he - ais. he sli is locaed parallel o he - ais. he deecor is leveled in he - plane. Panel B: Airborne carrier illusraing he push broom echniue. he sli is oriened normal o he fligh direcion - ais. Precise esimaion of he aiude of he plane reuires a GPS for posiion, speed and heading, acceleromeers for pich θ, roll φ and aw ψ, and gros for urn raes ω, ω, ω. 5. Main principle of hperspecral imaging Fig.5. panel A shows he opical componens of he GRISM specrograph as inroduced in he previous secion. he essenial pars are again mared as S enrance sli, L collecor lens, P diffracive elemen he graing / prism GRISM, L 3 camera lens and CCD he deecor. Firs, he main purpose of a specrograph is ha i generaes images of he enrance sli as a funcion of color. he number of colored sli images on he CCD depends on he widh of he enrance sli and he dispersive elemen s abili o spread diffrac he colors, and is direcl conneced o he specral resoluion of he insrumen. 7

31 Secondl, focusing ligh from a arge b a fron lens L ono he sli - plane, forces he insrumen o accep srucure of he arge along he sli. he resuling image specrogram recorded b he CCD is he inensi disribuion as a funcion of wavelengh color and posiion along he sli. he specrogram conains boh specral and spaial informaion along a hin rac of he arge objec. In our case he arge is snow cover land surface and ocean ice flaes. Finall, in order o obain he objec's full spaial een, i is necessar o sample he whole objec. his reuires ha he insrumen mus be moved relaive o he arge. he whole idea is o record specrograms for each rac of he objec as he image a he enrance plane is moved across he sli. Our movemen is creaed b he airplane iself. See Fig.5. panel B. We are in oher word pushing / recording specrograms as we fl over he ground arge. A mehod o esimae he aiude of he airplane will be presened in Chaper 7. Noe ha depending on he applicaion, he relaive movemen beween sensor and arge ma be obained b roaing he whole insrumen, use of scanning mirrors as fron opics, or moving he arge iself. he arge ma for eample be locaed on a conveor bel or on a sliding able. 5.3 Hperspecral image formaion and spaial resoluion As described above we need o push / record frames specrograms as we fl over he arge area in order o generae images as a funcion of wavelengh. In erms of mahemaics, he inensi a each piel of he specrogram as we move over he arge ma be defined as a dimensional arra u u [ λ, h], 5. where λ represens he wavelengh ais on he deecor, parallel o he -ais of Fig.5.. Correspondingl h is posiion parallel o he -ais a he deecor. he sample couner separaes he frames in ime. he firs column of he arge image is he inegraion of he specrogram over he desired wavelengh region of ineres from λ o λ λ J λ λ [ h, ] u [ λ, h] λ, 5. λ where λ λ -λ is he bandpass of he image. he ne column in he arge image is. Noe ha λ does no necessaril have o be eual o he specral bandpass BP of he insrumen. he relaion λ BP mus hold. Bandpass and spaial resoluion mus be considered in more deail. Recall he angular dispersion of a GRISM is given b euaion.43 3 d β n ab sinω / λ. 5.3 dλ a cos β 8

32 Since d f 3 dβ hen linear dispersion is defined as dλ dλ a cos β d f 3 dβ f 3 n ab sinω / λ Linear dispersion gives he spread in wavelengh per uni disance in along he -ais of he deecor. able 5. shows how he linear dispersion changes wih wavelengh for our insrumen. Wavelengh Refracive inde Diffraced angle Linear dispersion λ [nm] n g β [deg.] dλ/d [nm/mm] able 5.. Linear dispersion using a GRISM wih ω α 30 o, graing groove spacing a nm a 600 lines / mm, specral order n, and a deecor lens wih focal lengh of f 3 5 mm. Cauch s inde of refracion consans are A.553 and B nm for Borae flin glass. See Appendi B for more calculaions. I should be noed ha he linear dispersion is abou 6 nm larger a 400 nm if he prism is removed, using onl he graing. Correspondingl, a 800 nm he difference becomes onl 4 nm. he grism improves he linear dispersion compared o using onl a graing, especiall in he blue par of he specrum. he sraigh hrough wavelengh of he ssem is abou 480 nm where α - β. he specral bandpass BP of he insrumen is calculaed as he linear dispersion imes he ei sli widh. Using euaion 5.4 and 4.4 we obain w a cosω BP f n ab sinω / λ able 5. shows he sli widh magnificaion and he bandpass of our insrumen. Wavelengh λ [nm] Sli widh magnificaion Bandpass BP [nm] able 5.. Sli widh magnificaion and heoreical specral bandpass of a grism specrograph. he prism is of Borae flin glass wih ω α 30 o. he ransmiing graing has 600 lines / mm. he specral order is n. he lens beween he enrance sli and he GRISM has f 4 mm as focal lengh. he deecor lens has a focal lengh of f 3 5 mm. he enrance sli widh is w 0.05 mm. From he above i is safe o assume ha we can use nm as bandpass of he specral range from 400 o 700 nm. he number of specral channels available is hen 300 no

33 he maimum number of color planes or specral channels ha a hperspecral imager can produce depends on he bandpass and he specral range of he insrumen, no he number of piels along he wavelengh ais of he deecor. In addiion, he bandpass also affecs he spaial resoluion hrough he field of view along he -direcion of he fligh. he field of view of he sli as seen hrough he fron lens along he direcion of fligh, defines he spaial resoluion. I is illusraed in Fig he disance d defines he ground segmen seen b he insrumen a ime 0 w d, 5.6 where is he disance o he arge in meers, and f is he focal lengh in millimeers of he fron lens L. Noe ha d is conneced o specral bandpass BP hrough he widh of he sli w. f Figure 5.3. Field of view of sli as airplane moves wih veloci v. S is sli wih widh w. L is fron lens wih focal lengh f. is aliude above ground level. During he eposure he airplane moves a disance v v 0. he spaial resoluion hen becomes eual o he disance from poin A o B d v. 5.7 he eposure ime of he images does no include he readou ime of he deecor, τ. he disance moved during readou v τ mus be less han d. If no he case, he insrumen will miss samples of he arge area under sampling. his crierion ma hen be epressed using euaion 5.6 as w τ. 5.8 f v I is clear ha spaial resoluion depends on aliude, veloci of airplane, fron opics and sli widh. he laer also defines bandpass. hese parameers are all conneced and mus be evaluaed in order o achieve opimal performance. Normal o he fligh direcion he resoluion is calculaed simpl as h, 5.9 f N where h is as before he sli heigh in millimeers and N is he effecive number of piels along he sli image. able 5.3 gives he spaial resoluion of our insrumen as a funcion of aliude using a sensor ha pushes 5 frames per second, and an airplane carrier ha flies a 00 m / hr. 30

34 [m] d [m] [m] [m] SW [m] able 5.3. Spaial resoluion of for our GRISM specrograph described above. he fron lens has a focal lengh of f mm. he speed of he plane is 00 m/hr 55.5 m/s. he enrance sli widh is w 0.05 mm. he sensor is a ½ CCD piels a 5 frames per second. N30 and 8.33ms, τ ms. SW is he swah widh. Supplemenar, see Appendi C. he readou disance becomes v τ. 76 m, which means ha we have o fl on an aliude of a leas 850 m according o able 3. he aliude limi can be lowered b increasing he disance d in euaion 3 b selecing a shorer focal lengh on he fron lens L. he spaial resoluion is.3.45 m wih a swah widh of 38.5 m Eample applicaions A whole range of possible applicaions can be invesigaed using hperspecral imaging. Each specral image represens how effecivel a arge scene absorbs, reflecs or scaers ligh a he seleced cenre wavelengh. A se of specral images will produce a uniue specral finger prin for each objec wihin a arge scene. his pe of daa becomes ideal as inpu for soring of objecs b he use of image processing algorihms lie classificaion. 8 Daa from our airborne campaigns in he arcic cover applicaion such as mapping of. Ocean colour and algae s. Vegeaion and rocs 3. Sea ice and leads 4. Snow cover. In he following eamples, a commercial win engine aeroplane of pe Dornier-8, operaed b he compan Lufranspor AS wih base in Longearben, Svalbard 78 o N, 5 o E, has been used as carrier of he insrumens. he poining direcion of he imagers is 30 degrees o nadir side view. For each mission a sliding door has been mouned on he plane, which can be opened during fligh when approaching he arge area. Posiion and speed is obained b using he eernal GPS anenna of he plane. he aiude pich, roll and aw is esimaed b using real-ime daa from 3 ais acceleromeers and elecronic urn rae sensors. A Kalman esimaor is used o process he aiude daa. Fig. 5.4 shows a pical eample of daa from our laes airborne specral imager over Longearben 78 o N, 5 o E, Norwa. he wo Gra scaled image srips o he lef are generaed wih a cenre wavelengh of 546. nm. he bandpass is 5 nm. he aliude is 698 m. he ground speed is 84 m/hr. he spaial resoluion is close o m. Panel A shows he 546. nm image of he cleaning facili for coal owned b he local coal compan. A successful unsupervised classificaion wih 0 classes is presened in panel B. he classificaion is based on onl 30 colour planes images. 3

35 Figure 5.4. Hperspecral imaging of snow cover and sea ice over Longearben, Svalbard, Noe ha he GPS posiion of he plane is found in he upper lef corner of he high resoluion camera image in panel C. he ne eample is shown in Fig he insrumen provides images of ocean colour, which is relaed o he 3 basic componens ha effec he colour composiion of sea waer: Inorganic paricles e. g. mineral paricles from fresh waer run off sil, phoplanon wih differen pigmen groups, and dissolved organic maer ellow subsance. he lef panel shows hperspecral daa as a funcion of wavelengh. he colour and gre scaled bars indicae he visible par of he elecromagneic specrum and he inensi in absolue unis, respecivel. he aliude of observaion is 3000 m. he specral bandpass is 5 nm and he spaial resoluion is approimael 7 m. 3

36 Figure 5.5. Hperspecral imaging of ocean colour, vegeaion and rocs over he bas of Collinderodden and Bliodden close o he mining own of Svea, Svalbard, Focusing on he sea mud ineracions urns ou o be ver ineresing. As we go up in wavelengh he sea urn blac and he mud sil source seems o appear. I ma be ha we can see he mud sie disribuion / morpholog. his effec is no possible o see from he images aen b he digial camera lef panel s wo op images. Righ panel of Fig. 5.5 shows he resul of wo RGB composiions and an unsupervised classificaion. he seleced colour planes he 3 images used in he composiions o represen Red, Green and Blue colours bring ou he areas dominaed b vegeaion and rocs. Furhermore, he classificaion reveals a more deailed map of 0 classes separaed in o sea waer, 3 pes of roc, 4 pes of sil and 3 pes of vegeaion. Noe ha a supervised classificaion would improve and opimie he above resuls, bu ha reuires in-siu observaions on he ground. In man cases his can prove o be difficul due o logisics and access o he arge area. Neverheless, i is clearl eviden ha i is indeed necessar o combine in-siu observaions o suppor and undersand he airborne daa beer. 33

37 Figure 5.6. Hperspecral poser from airborne campaign over N-Ålesund, Svalbard,

38 One more eample of an airborne applicaion is presened as a poser. See Fig A campaign was launched in Ma 004 o sud ocean color ouside N-Ålesund 79 o N, o E, Svalbard. Simulaneousl wih sampling of he airborne daa, an image was aen b he Aua - erra saellie. he saellie image ogeher wih a high resoluion color image Fujifilm SPro digial camera wih 8 mm objecive from he plane provided us wih a wide coverage and a high spaial resoluion of he arge area, respecivel. In addiion, aiude and posiion racs are shown o he lef in Fig he hperspecral daa are shown as a gre scaled image seuence hroughou he visible par of he elecromagneic specrum. he bandpass is 5 nm and he spaial resoluion is close o m. he composie RGB is made using he 480 nm Blue, 585 nm Green and 630 nm Red, respecivel. Noe from he difference beween he images a 585 nm and 730 nm, srucure on he boom of he ba is clearl seen. Furhermore, a simple classificaion Baes reveals a map of he brown algae populaion mosl Laminae sp.. Figure 5.7 Hperspecral daa of sliced seal ooh. Fron opics is microscope wih 6 imes magnificaion. he final eample applicaion demonsraes ha a specral imager can also be used wih a microscope as fron opics. he relaive movemen beween sample and insrumen is provided b a sliding able. he sample is moved slowl under he microscope. Fig. 5.7 shows he specral images from 390 o 690 nm of a sliced seal ooh. he bandpass of he images is 5 nm. 35

39 he slice is mm hic. he magnificaion was se o 6 imes on he microscope. he ligh source is locaed under he sample - ligh is ransmied hrough he sliced ooh. A processed image showing he difference beween he 450 and 570 nm images reveals srucure, and a composie RGB shows he colour image. Classificaion sors ou 8 differen classes. Noe ha here is a small mi or miss classificaion. his is mainl due o cracs in he ooh and ha he sample iself is glued o he sample glass seen as a ransparen area in he cenre of he images. Neverheless, he age rings of he ooh are clearl idenified wih he colour red. he seal was wo ears old Wha s ne? Opics is one of he oldes branches of phsics. Specroscop sared when Newon decided o celebrae he colors of life wih a prism bac in he lae 700. Even oda opics is one of he fases growing fields of phsics, and is eending is use o oher branches of science. he advances in elecronics and new maerials have resuled in deecors ha are able o sense ligh ouside he visible par of he specrum. he recen developmen of Acousic Opical unable Filers AOF and Liuid Crsal unable Filers LCF will revoluionie specroscop jus as he arrival of blaed graings did bac in he 50 s. UV imagers and near- o far infrared imagers will soon be available - even on he commercial mare o a reasonable price. hese specral imagers will open our ees o hings we have never seen before. 36

40 6. CALIBRAION he increasing number of low ligh level opical insrumens operaed in Svalbard Longearben, Barensburg and N-Ålesund for monioring auroras and airglow phenomena emphasies he need for esablishing accurae calibraion rouines of inernaional sandard for boh cameras and specral insrumens Specral narrow field of view calibraion his secion reviews he mahemaical frame wor and he eperimenal seup of absolue calibraion of narrow field of view specral insrumens. A presenaion of he opical laboraor a UNIS is also given. he resuls of secondar sandard lamp cerificaion and brighness conrol are demonsraed. 6.. Inroducion he hroughpu or he useable phoon flu a he ei of an opical insrumen depends firs of all on inpu flu. Secondl, i depends on he insrumen s abili o accep ligh geomerical een and he uali of he opical componens used Chaper 4. he efficienc of each componen, wheher he are lenses, mirrors, graings or filers, limis he sensiivi and he wavelengh region of he insrumen. he process of ransforming he ei flu o elecronic couns b he deecor also involves loss of phoons. I is herefore necessar o calibrae he insrumen agains a source of nown inensi in order o obain he raio of elecronic couns ou o he number of phoons inciden on he insrumen. he source of he calibraion could be a lamp or an oher objec wih nown specral characerisics. In addiion, a diffuse reflecive surface is needed o mae sure ha he insrumen's field of view is uniforml illuminaed. he following describes he heoreical concep and he eperimenal seup of sensiivi calibraion using a Lamberian surface and a sandard ungsen lamp. A mehod o ransfer lamp cerificaes and a procedure o regulae source brighness of he screen wihou change in specral shape is demonsraed. 6.. he Lamberian surface A surface ha has a perfecl diffuse / mae proper is Lamberian. he radian inensi refleced in an direcion is proporional o he cosine of he angle of he normal o he surface. his is nown as Lambers Cosine law 9. According o Fig. 6., he radian inensi is epressed as # phoons I λ φ I λ 0 cosφ, 6. s sr Å where λ subscribes he wavelengh and φ is he angle wih respec o he normal. he oal emission rae in he wavelengh inerval dλ is Figure 6.. he radian inensi of a Lamberian surface. 37

41 he radian inensi inegraed over he hemisphere Insering E. 6. ino 6. gives ππ / Φ dλ sinφ dφ dψ. 6. Φ λ λ I 0 0 λφ π / λ 0. 0 π / λ 0 dλ sin φ dφ. 0 π I dλ cosφ sinφ dφ 6.3 π I 6.4 Conseuenl, # phoons Φ λ π I λ 0 dλ. 6.5 s he oal emission rae for a Lamberian surface is independen of he angles φ and ψ. Our surface SR is made of Specralon and is produced b he compan Labsphere, Inc. he reflecance facors of he screen are nearl consan ρ λ 0.98 hroughou he visible and near infrared regions of he specrum Eperimenal calibraion seup Fig. 6. shows he seup for he calibraion. he ungsen lamp FEL is locaed a disance 6.9 m from he cenre of he screen and he angle beween he screen and lamp ais is given b α. M oλ is he nown radiance cerificae of he lamp in unis of [#phoons cm - s - Å - ], iniiall obained a a disance of o 0.5 m. he source for our calibraion is hen he screen, no he lamp iself. he lamp is reaed as a poin source, radiaing phoons isoropicall. he oal number of phoons ha pass hrough a sphere wih radius o mus be he same for a sphere wih radius 4 π o M oλ 4π M λ 6.6 We assume no absorpion of phoons in he space beween he spheres. he radiance ha his he screen is hen simpl o # phoons M λ M oλ. 6.7 cm s Å Correspondingl, he emission rae ha eners he screen in erms of he radian inensi I sλ is da cosα I sλ dλ ω I sλ dλ, 6.8 where ω is he solid angle and da is he illuminaed area of he screen. he reemied radiaion of he screen is hen da cosα Φ λ I s λ dλ ρλ π I 0 λ. λ d

42 From Euaion 6. we obain 6.0 I sλ da cosα I λ φ ρ λ cosφ. π he effecive illuminaed area of he screen as seen b he insrumen is da cos φ, where φ is he angle beween he opical ais and he screen normal. he radiance owards he insrumen hen becomes I λφ # phoons Bλ 6. da cosφ s sr Å cm I λ ρ λ s cosα. 6. π From he inverse suare law and E. 6.7 we now ha I sλ M λ M oλ o. 6.3 Noe ha he radian eiance of he screen is b definiion given as phoons M B. # λ π λ 6.4 cm s Å Finall, we obain from 6. Mo λ o Bλ ρλ cosα. 6.5 π E. 6.5 epresses he brighness of he screen as seen b he insrumen in erms of he lamp and screen cerificaes, he disance beween screen and lamp, and he angle of he screen o he lamp, α. Noe ha as long as he field of view of he insrumen is filled, neiher he angle φ, nor he disance of he insrumen o he screen maers. he changing sie of he field of view a he screen compensaes for he disance and he angle, φ. Figure 6.. Seup for absolue calibraion of narrow field of view insrumens he UNIS calibraion lab he described procedure reuires a dar room o eliminae eraneous scaered ligh from he screen. Fig. 6.3 shows he new calibraion laboraor a UNIS. he facili conains 3 rooms. he lamp room is separaed from he screen room b a baffled door. 39

43 he conrol room wih he lamp power suppl and a calibraion specrograph is locaed ne o he lamp room and he screen room. A fiber bundle acs as enrance opics o he specrograph, and i runs hrough a hole in he wall beween he conrol room and he screen room. he disance beween he lamp and he screen is conrolled b mouning he lamp on a mobile able. wo rails are used o obain smooh ravel and consan horional cenre disance parallel o he screen. he able is adjusable in heigh and a fied laser is used o align he lamp wih he verical cenre of he screen. In his configuraion, he maimum disance beween screen and lamp is 8.56 m. he screen normal is poining direcl owards he lamp wih α 0 degrees. he room lighs are adjusable VDC lamps. When hese lamps are urned off, he do no produce an low level bacground emissions as gas discharge ubes have a endenc o do. he lamp power suppl ORIEL KW is coupled o a Ligh Inensi Conroller LIC. he inensi of he lamp is moniored and he curren hrough he filamen is adjused o eep he inensi consan. he 000 W lamp runs a 8. Ampere. Figure 6.3. he calibraion laboraor a UNIS: 8 8 inch Lamberian surface, rails, 3 adjusable mobile able, 4 enrance fiber o specrograph, 5 door wih baffle, 6 room lighs, 7 ungsen lamp, 8 power cable o lamp filamen. he main idea is o conrol he brighness of he screen onl varing he disance. See E he lab specrograph he specrograph is made b ORIEL. I uses a concave holographic graing 30 grooves / mm. he nominal specral range is Å. he deecor is a 6-bi dnamic range hermoelecric cooled CCD from he compan Hamamasu model INSASPEC IV. Fig. 6.4 shows he opical diagram of he insrumen. A field of view of o maches he fused silica fibre bundle used as enrance opics. he F-number is.. he enrance sli is in he focal plane of he concave graing. he focal lengh varies from mm, depending on wavelengh. 40

44 he diffraced ligh from he graing is focused and dispersed ono he ei plane. his is illusraed in Fig. 6.4, where he blue, green and red ras represen he sar, cenre and sop wavelengh recorded b he CCD, respecivel. he bandpass is approimael 80 Å wih a 00 µm wide enrance sli. he specral resolving power of his insrumen is moderae, bu for our purpose i is enough. he specrum of a ungsen lamp is smooh and coninuous. here are no line srucures ha need o be resolved. Figure 6.4. Fied Imaging Compac Specrograph FICS SN 7743: A concave holographic graing, B fla mirror, C deecor CCD, D fiber bundle, E enrance sli. Wavelengh calibraion using specral gas ube lamps wih nown emission lines mus be carried ou before an sensiivi calibraion can ae place. he mos common wa is o idenif he piel value associaed wih he a priori nown wavelengh of he emission line. A leas hree lines mus be idenified in order o minimie wavelengh errors and chec for non lineari along he wavelengh scale. he wavelengh piel relaion is given as λ a0 a p a p, [Å] 6.6 where p is piel value. he resul of he procedure is he consans a 0, a and a. he FICS specrograph was calibraed in wavelengh using 3 differen gas discharge mercur Hg lamps see Fig Noe ha no order soring filer is used o bloc ou wavelengh regions originaing from higher specral orders of he graing. he srong Hg emission line a wavelengh 537 Å is repeaing iself 3 imes a 5074 Å, 76 Å and 048 Å. he laer corresponds o specral orders, 3, and 4, respecivel. his effec is used in he calibraion o obain emission lines in he deep red par of he specrum and up. able 6. shows he resul. Coefficiens Consans a a a able 6.. Wavelengh calibraion facors for he FICS specrograph according o E he deecor CCD has piels. he specral range is from 560 Å o 0945 Å ransfer of lamp cerificae Our 000 W ungsen lamp ORIEL SN7-75 is a raceable Naional Insiue of Sandards NIS source. his lamp is firs used o calibrae he specrograph in he wavelengh region Å. he calibraion facor as a funcion of wavelengh is hen simpl B # phoons λ ε λ, 000W u λ cm s Å sr cs 6.7 W where u 000 λ is he coun rae in unis of couns per second [cs]. 4

45 Figure 6.5. Wavelengh calibraion of FICS specrograph. he specrum ploed wih he colour blac is a low pressure mercur pen lamp model MS- 46 from Acon Research Corporaion. he blue specrum is from a mercur vapour ube supplied b Edmund Opics Ld SN K he red curve represens he specrum of a baer powered fluorescen ube OSRAM F65. Each mercur emission line is mared according o wavelengh and specral order m. Ne, he primar lamp is replaced b our secondar sandard, a 00 W ungsen lamp. he lamp screen disance is 8.56 m and he Figure 6.6. Specra of Lamberian screen SR he blue curve is raw couns in unis of [CS] from he screen using he 000W ungsen lamp ORIEL SN7-75. Correspondingl, he red curve represens he 00W secondar ungsen lamp. Eposure ime is 60 msec. Screen lamp disance is 8.56 m. he 000 W lamp runs on 09.6V a 8.0A. he 00 W lamp has a filamen curren of 6.50 A a a volage of 3.7 V. he screen angle α 0 o. he blac curves are he brighness of he screen in absolue unis [#phoons / cm s Å] using he 00 W lamp. he valid wavelengh region of cerificaion is mared for specral order m. eposure ime is 60 msec for boh cases. his is imporan since he coun rae as a funcion of eposure ime is no necessaril linear. In addiion, a cuoff filer is used in fron of he fiber bundle o avoid overlapping specral orders. he opical window is made of BK-7 and blocs ou he UV par of he specrum. Fig. 6.6 shows he resuls of he cerificaion. Noe ha a cerificaion above 8000 Å reuires an order blocing filer wih cuoff wavelenghs above 4000 Å. A filer wheel will be insalled o handle his issue in he near fuure. As a emporall soluion, he doed blac curve in Fig. 6.6 represens a funcional fi o he 00 W secondar cerificae based on a mehod developed b Saunders a NIS M W λ λ ep a bλ. 6.8 he euaion has onl wo unnowns. In our case he soluion becomes: a 73.9 and b he fi is obained b using he irradiance wavelenghs beween Å Screen brighness conrol According o E. 6.5 he brighness of he screen is adjusable as a funcion disance. Anoher approach would be o var he curren hrough he lamp filamen. A disadvanage of his echniue is ha he specral shape changes wih lamp curren. I would also reuire a lamp cerificaion for each curren seing. In order o save burning ime or life ime of he 000 W ungsen lamp, he 00 W lamp is used o demonsrae he screen brighness conrol using onl disance as variable. he eposure ime of he specrograph is again ep consan a 60 msec. 4

46 Figure 6.7. Calibraed specra of Lamberian screen SR as a funcion of screen o lamp disance. he source is he 00 W ungsen lamp FRED SN0. Figure 6.8. Calibraed specra of Lamberian screen SR as a funcion of screen o lamp disance. he source is a 45 W ungsen lamp ORIEL SN he mobile able is moved owards he screen is incremenal seps of 0.5 m. As seen in Fig. 6.7 he specral shape remains he same and he inensi rises wih he inverse of he disance suared. he above procedure was repeaed using a 45 W ungsen lamp ORIEL SN Fig. 8 shows he resuls. Inensiies are now down b a facor close o 5. Noe ha he signal o noise, especiall in he blue par of he specrum, ma be improved b simpl rising he eposure ime of he specrograph. An eposure ime of 4000 insead of 60 msec a a disance of 8 m, generae raw specra ha span he whole dnamic range of he deecor 6-bi. Noe ha he lef ordinaes of Fig. 6.7 and 6.8. are given in unis of Raleigh per, Ångsrøm [R/Å]. he generalied definiion is 6 R / 4π 0 # phoons cm sec sr, 6.9 and i applies direcl o he radiance B λ. In E. 6.5, a muliplicaion facor of 4π / 0 6 is used o conver he number flu of #phoons cm - sec - sr - Å - o R/Å Eperimenal uncerain he 000 W ungsen lamp supplied b Oriel is provided b a calibraion uncerain of ±3 % in he wavelengh range Å. he life ime of he lamp cerificae is 50 hours. Afer his, he lamp should be recalibraed. Oriel repors drif raes in he range o % per 00 operaing hours over he wavelengh range 4000 o 8000 Å, respecivel. Furhermore, our secondar ransfer procedure will inroduce errors 3, 4 due o disance 0.4 %, lamp orienaion and alignmen 0.3 %, specrograph sabili 0.3 %, measuremen repeaabili 0.5 %, specrograph nonlineari 0. % and laboraor sra ligh 0. % 5. Based on he above numbers our oal calibraion ransfer uncerain is esimaed o be below ±4 %. 43

47 Figure 6.9. Resul of boh wavelengh and absolue calibraion of he DRONESPEX hperspecral imager. he wavelengh calibraion was obained b he use of a mercur vapor gas Hg lamp. he mercur emission lines a 404.7, 435.8, 546. and 577 / 579. nm are mared. uλ is he specrum of he diffuse reemiing screen illuminaed b he absolue calibraion lamp ungsen. Noe ha u is he average sli response given in raw couns [CS]. Mλ represens he absolue cerificae specrum for he 000W ungsen lamp in unis [mw m - nm - ] DRONESPEX calibraion Fig. 6.9 shows he resul of he calibraion for our hperspecral insrumen Chaper 5. I is clear from Fig. 6.9 ha he insrumen did no deec man couns in he blue range of he specrum due o he low oupu of he calibraion lamp. A beer resul would be o use he sun iself as a calibraion source. he oal radiaion on a clear da will produce a higher hroughpu in he blue par opimiing he insrumenal coun rae o be more fla across he specrum. Even an overcas da would wor if he cloud cover does no change during calibraion. he seup reuires ha we now he inensi of he screen in absolue unis. his can be obained using a second insrumen ha is alread calibraed in he laboraor b he procedure oulined above. he second insrumen does no need o have imaging or hper specral capabili a regular specrograph wih a line arra deecor sensiive in he blue will be sufficien Summar specral calibraion he calibraion laboraor a UNIS has been consruced o calibrae narrow field of view insrumens wih variable source brighness conrol. 44

48 wo ungsen lamps wih power of 00 W and 45 W are cerified in he visible par of he specrum Å wihin an uncerain of ±4 %. B varing he screen o lamp disance from 8 m down o 3.5 m, screen inensiies in he range 00 3 R/Å were deeced. he oulined calibraion procedure will be repeaed o include eended wavelengh coverage b using proper cu off filers o bloc ou higher overlapping specral orders of he FICS 7743 specrograph. In addiion, we aim o use a ungsen lamp of lower power 0 W o cover inensiies below 00 R/Å. 6. Bandpass filer insrumen calibraion his secion shows how o calibrae a bandpass filer insrumen. One eample insrumen is he phoomeer. I consiss of a phoomuliplier ube as deecor, a focusing lens and a narrow bandpass filer. he eperimenal seup for a pical bandpass filer insrumen sensiivi calibraion is seched in Fig he pe of filers used depends on cener wavelengh λ c, bandpass BP and ransmission λ. he mos radiional one is he Fabr-Pero filer design, where parallel ransparen glass plaes creae inerference due o muliple reflecions beween he plaes. Panel B of Fig. 6.0 shows a pical ransmission curve for hese filers. he raw daa couns of he deecor D is given as an inegral over wavelengh u Bλ Sd λ λ, [cs] 6.0 where S λ is defined as he specral responsivi. B λ is given b E If we assume ha B λ, lens ransmission and deecor sensiivi varies slowl in he wavelengh inerval λ, hen he specral responsivi of he insrumen ma be epressed as Figure 6.0. Panel A: Filer insrumen sensiivi calibraion seup: F bandpass filer, L focusing lens, D deecor, S Lamberian screen, FEL sandard ungsen ligh source. Panel B: Sech of a pical ransmission curve for a bandpass inerference filer. S λ ε λ. [cs R - ] 6. ε is now an insrumenal consan. I hen follows ha u Bλ Sd λ λ Bλ d B d B A c ε λ ε λ ε λ λ c λ λ, 6. c where A is he area of he filer ransmission curve. he calibraion facor is hen simpl u ε. 6.3 Bλ A c Les us assume ha λ is narrow and riangular in shape. hen A becomes 45

49 A d λ λ m BP. 6.4 m is he pea ransmission of he filer a λλc. Furhermore, we define he inensi of a discree auroral emission line a wavelengh λ c o be J J 0 δ λ λ c, where δ is he Kronecer dela and J o is he absolue inensi. he number of auroral raw daa couns is hen ua J S dλ J δ λ λc ε dλ ε J 0 δ λ λ c dλ ε J λ 0 λ λ m Finall, from Es. 6.3 and 6.4 we obain ua ua A J0 Bλ c ε m u m. [R] 6.6 ua Bλ BP c u Noe ha E. 6.6 onl applies when we assume ha he ransmission profile of he filer is narrow and riangular. his is a raher crude assumpion, especiall for medium o broad bandpass filers. 6.3 Color camera calibraion A mehod o obain he average specral piel responsivi and he uanum efficienc of Digial Single Lens Refle DSLR cameras is oulined 33, 34. As eamples, wo semiprofessional cameras, he Nion D300 and he Canon 40D, are evaluaed. Figure 6.. Eperimenal seup: fiber bundle from lamp locaed in neighboring room, moun / sand for inegraing sphere, 3 order soring filer wheel in fron of enrance sli, 4 Jobin Yvon HR30 monochromaor, 5 able, 6 ei sli plane, 7 Edmund Scienific inegraing sphere, 8 laboraor lif able, 9 fiber bundle holder, 0 camera able, fiber bundle used as inpu o specrograph, Oriel FICS 7743 specrograph, 3 opical moun rail, and 4 DSLR camera wih normal 50 mm f/.4 objecive. 46

50 6.3. Shor Bacground he main moivaion for his wor is o develop a mehod o obain he specral responsivi or he uanum efficienc of each piel in a DSLR camera. hese cameras have become valuable ools for sudies of he nigh s including phenomena such as aurora and airglow. I is herefore essenial o esablish a mehod o uanif he ligh hroughpu of he cameras. Even hough our focus is on he aurora, he resuls should be of ineres o oher fields of science such as asronom, remoe sensing and indusrial processing as well. Figure 6.. Opical diagram. [A] Leica 50W fiber illuminaor: mirror, ungsen filamen, 3 hea filer, 4 blocing wall, and 5 fiber bundle. [B] Jobin Yvon HR30 Monochromaor: 6 f-maching lens, 7 order soring filer, 8 enrance sli, 9 collimaor mirror, 0 plane reflecive graing, focusing mirror, fla surface folding mirror, and 3 ei sli. [C] Edmund Scienific General purpose 6 inch diameer inegraing sphere: 4 sphere, and 5 ransmiing diffuser eflon. [D] DSLR camera: 6 50 mm normal f/.4 objecive, and 7 CMOS / CCD deecor. [E] Oriel FICS 7743 specrograph: 8 order soring filer, 9 fiber bundle, 0 enrance sli, folding mirror, concave graing, and 3 CCD deecor Eperimenal seup Figs. 6. and 6. show he eperimenal seup and he opical diagram of he ssem, respecivel. he main componens are a fiber illuminaor Leica 50W ha is conneced o he enrance sli of a monochromaor Jobin Yvon HR30. he diffraced ligh a he ei sli of he monochromaor is hen fed ino he inegraing sphere. he oupu of he sphere is he arge for boh he DSLR cameras and he inensi calibraed FICS specrograph see secion he sphere, from Edmund Opics, is designed o inegrae radian ligh flues. he sphere is 6 inches in diameer. Is inerior walls are coaed wih Specralon ha has a diffuse reflecance facor of 0.98 hroughou he visible par of he specrum. he monochromaic ligh ha eners he inch diameer inpu por of he sphere is scaered muliple imes before i eis he.5 inch diameer oupu por. 47

51 In addiion, a ransmiing diffuser is used a he oupu por of he inegraing sphere o diffuse he ligh even more see Fig. 6.. he diffuser is 0.5 mm hic and made of eflon opal. his was found necessar due o he fac ha he firs region of illuminaion in he sphere, caused b refleced ligh direcl from he ei sli of he monochromaor, is easil seen from he oupu por. his illuminaed region is raher large compared o he diameer of he sphere. he effec was clearl seen even a relaivel small off-ais view angles o he oupu por. Finall, he ne resul is a uniform illuminaed surface ha is comparable o he Lamberian surface ha is used for calibraing narrow field of view insrumens such as our specrograph. he field of view of he specrograph is full illuminaed b an area ha is uniform in inensi, and eual o he corresponding area ha fills he field of view of each seleced piel of he cameras. As long as he field of view of he piels and he specrograph are filled neiher he loo angle nor he disance o he diffuser maers. he changing sies of he piel fields of view a he diffuser surface compensaes eacl for boh he changing disances and angles. Boh he Nion D300 and he Canon 40D cameras are operaed in manual mode wih sensiivi se a ISO 600. he Nion D300 uses a Nion 50mm f/.4 AF-D lens, while he Canon 40D uses he Canon EF 50mm f/.4 USM lens. he lenses are operaed a maimum aperure f-value se o.4. Eposure imes beween 3 and 4 seconds were chosen o avoid overeposures he specral piel responsivi he assembled wavelengh unable ssem is designed for he visible par of he specrum Å, producing monochromaic lines wih a bandpass of ~ Å. he calibraed FICS specrograph measures he inensi of he inegraing sphere oupu in unis of R/Å. he bandpass of he specrograph is λ 00 Å. As resul, he specral responsivi for each camera is calculaed using all 3 specra and images. he euaion of observaions is ˆ uˆ B S λ, [cs] 6.7 where he subscrip [ rgb,, ] labels he red, green and blue channels of he Figure 6.3. Sphere source funcions. B i λ is he se of observaions ha consiss of 3 specra from he monochromaor HR30 illuminaing he 6 inch diameer inegraing sphere from Edmund Opics. ransparen Colour Filer Mosaic filer CFM in fron of each piel of he sensor, respecivel. E. 6.7 is he same as E. 6.0 epressed in vecor form. he vecor u ˆ conains 3 numbers of averaged piel raw couns per second for each colour channel,. he mari B is given as B [ BBB 3 B3]. [R/Å] 6.8 i conains he specrum as measured b he FICS specrograph. Each vecor B i, [...3] 48

52 he dimension of he specra is reduced o 3 b re-sampling a each cener wavelengh seing of he monochromaor. he specral responsivi Sˆ is now found b solving E. 6.7 b Singular Value Decomposiion SVD he uanum efficienc Anoher wa o characerie he specral sensiivi of a camera is o calculae he Quanum Efficienc QE. I is defined as he fracion of phoons ha will generae elecrons deecable b he phoo-reacive sensors. In our case wih monochromaic source funcions in unis of R/Å, he QE ma be epressed as 4π ui g QEi 00 6, [%] Bi λ A where Ais he piel area in unis of cm, is he eposure ime in seconds and g is he gain of he deecor defined as he conversion facor beween he number of elecrons and raw couns per piel. A simple procedure of how o measure he gain is given in Handboo of Asronomical Image Processing 35. A ISO 600 he gain is and elecrons per -bi daa coun for he 40D and he D300, respecivel Calibraion resuls Our librar of calibraed specra is shown in Fig he inensi of he oupu por of he inegraing sphere rises from ~500 R/Å a 4000 Å o a maimum of ~3300 R/Å a 5800 Å, reducing o ~500 R/Å a 7000 Å. he dips a 4600 Å and 6700 Å are due o drops in he graing efficienc of he monochromaor. Fig. 6.4 shows he Nion D300 green channel raw couns per second when he sphere is illuminaed wih ligh a 5569 Å cener wavelengh. Noe ha dar frame subracion is carried ou o reduce he bacground level. he resuling image is shaped lie a seep mounain cliff wih a sharp leveled circular plaeau. he heigh of he plaeau is used as he average piel raw coun response. he plaeau heigh is calculaed b selecing raw coun values wihin a 0000 piel suare area in he cener of he image. Using he noaion in Secion 6.3.3, he plaeau heigh is G u 964 cs/s. he corresponding sandard deviaion is σ ± 30 cs/s, which in percenage erms is close o 3% variaion across he plaeau. he σ deviaion is compaible wih Poisson saisics where noise is defined as he suare roo of he couns. Figure 6.4. Raw couns from he Nion D300. Eposure ime is 3 seconds a ISO 600 wih a 50 mm normal objecive fied a f/.4. he shaded surface represens he green channel -bis raw cs per second. he source is a monochromaic illuminaed sphere a cener wavelengh 5569 Å. he cener -ais slice of he surface is shown ogeher wih he calculaed heigh of he plaeau solid horional line and he σ sandard deviaion doed lines. 49

53 he calculaed values are visualied in Fig. 6.4 b ploing he cener -ais slice of he mounain cliff. he resul loos lie a suare pulse wih ripple noise on he op ha is wihin he calculaed deviaion. he oupu of he sphere is, in oher words, sufficienl uniform o assume ha he above 0000 piel average is he same for all piels. he ne sep is o repea he procedure o obain he average raw couns per second as a funcion of wavelengh for each color channel of he cameras. he firs hing we noiced when handling he raw daa from he Canon 40D is ha he couns are fied a 4-bis resoluion. he range is, in oher words, from 0 o 6383 cs. he Nion D300 has a -bi range cs. As a conseuence, he 40D couns are scaled down b a facor of /4 in order o be comparable o he D300. he D300 was firs operaed a 3 seconds eposures wih no overeposures occurring as we changed he wavelengh of he monochromaor. A 4 seconds he green channel became overeposed. On he oher hand, he Canon 40D did no overepose a 4 seconds eposure ime. I also urned ou ha we could no se he eposure ime of he 40D o 3 seconds. Onl.5, 3. and 4 second Figure 6.5. Camera raw daa. Solid lines are he raw couns per second from he Nion D300 camera for each color channel Red, Green and Blue. he doed lines are he corresponding daa from he Canon 40D camera. Each curve is labeled wih eposure ime seings of he cameras. Boh cameras were operaed wih idenical seings using normal objecive lenses 50mm f/.4 a ISO 600. he error bars represen he σ sandard deviaion of he coun raes. inervals were possible. he procedure was herefore repeaed o mae sure ha we obain he same cs per second for all inervals. Fig. 6.5 shows he resuling raw couns per second obained wih variable eposure inervals for boh cameras. I is hard o see an difference beween he curves as a funcion of eposure ime for each camera alone, especiall for he 40D. he onl difference appears when he D300 green channel is overeposed a 4 seconds. he effec is seen comparing he 3 and 4 second eposures. he pea of he 4 second eposure loos lie i has been cu off compared o he 3 second one in he 500 o 5500 Å wavelengh region. Also noice ha he error bars become ero in his region. his is epeced since he error bars are defined as he σ sandard deviaion of he coun raes. he D300 blue and red channel coun raes are almos idenical, wih deviaions ha are well wihin he error bars. I is herefore reasonable o assume ha he raw coun raes are consan as a funcion of eposure ime in he inerval ~ 3-4 seconds for boh cameras. he coun rae profiles of he cameras shown in Fig. 6.5 are uie similar in shape, especiall he blue and red channels. However, he 40D green channel profile is slighl more smmeric han he corresponding D300 profile. his is due o he fac ha he 40D green couns are lile bi higher han he D300 green couns in he 4800 o 500 Å wavelengh region. he same is seen in he 5400 o 5800 Å wavelengh region when comparing he red channels. 50

54 Figure 6.6. Processed camera daa. Panel A: Solid lines are he specral responsivi of he Nion D300 camera for each color channel Red, Green and Blue. he doed lines are for he Canon 40D camera. Panel B shows he corresponding calculaed uanum efficienc. Boh cameras were operaed wih idenical seings using normal objecive lenses 50mm f/.4 a ISO 600. Boh cameras have pea coun raes in he green channels. However, he D40 has a higher blue han red pea coun rae, while his is vice versa for he D300. Wha his means, in erms of difference in color balance beween he cameras, depends on he specral responsivi or he uanum efficienc. he raw coun rae profiles alone are no sufficien in order o conclude on his issue. If we define he widh of he profile for each color channel o be he wavelengh region where i has eual o, or greaer han, half of is maimum coun rae, hen i is clear ha he D40 has wider profiles han he D300 for all color channels. he D40 profile is ~50 Å wider han he D300 in he green channel. he corresponding blue and red differences in widhs are 70 and 60 Å, respecivel. he main difference beween he cameras becomes eviden in he level of he coun raes. he Nion D300 coun raes are generall higher when compared o he Canon 40D, ecep for he wavelengh regions menioned above. B inegraion we find ha he D300 s blue, green and red channels have ~6, 8 and 50 % higher coun raes compared o he D40, respecivel 5

55 he specral responsivi and he uanum efficienc ma now be solved according o euaions 6.7 and 6.9, respecivel. Based on he above findings, he 3 second eposure b he D300 and he 4 second eposure b he 40D camera are chosen o represen he raw coun raes in he calculaions. he ne resul is shown in Fig. 6.6 for boh cameras. he calculaed specral responsivi and he uanum efficienc curves are compaible in shape and ampliude. Noe ha he uanum efficienc calculaion is a more direc and robus mehod since i does no depend on he Singular Value Decomposiion. I is now clear ha boh cameras have heir pea sensiivi in he blue and minima in he red channels. he color balance beween he channels is, in oher words, he same for boh cameras. However, as epeced from he level of he coun rae profiles, he Nion D300 is he mos sensiive camera. he D300 has a pea uanum efficienc of 50% and specral responsivi of cs s - R - in he blue channel a 4600 Å. he green channel uanum efficienc peas a 48% wih cs s - R - in specral responsivi a 5300 Å, while he red channel pea is lowes a 35% and.80-3 cs s - R - a 5900 Å. Again, b inegraion we find ha he D300 s blue, green and red channels have ~7, 9 and 54% higher specral responsivi compared o he D40, respecivel. he corresponding difference in uanum efficienc is, 3 and 47%. he above is a surprising resul. We double checed our eperimenal seup o mae sure ha we acuall used he same seings on boh cameras. Also noe ha, he oal difference beween he wo ses of source funcions used o obain he specral responsivi and uanum efficienc onl varied b ± 7 R/Å. he difference is so small ha i is hard o see i if we overla he curves of he 40D source funcions in Fig One facor ha could eplain he discrepanc, especiall in he red, could be he difference in he specral ransmission of he lenses. he Nion 50mm f/.4 AF-D and he Canon EF 50mm f/.4 USM lenses are almos idenical wih he same number of opical elemens in he lens consrucion. Fig. 6.7 shows he specral ransmission profiles using he FICS specrograph as he deecor and a fla Lamberian surface as he arge reference. he surface was illuminaed b a ungsen lamp. he lenses were mouned in fron of he specrograph s enrance fiber bundle. he seup is idenical o our narrow field of view inensi calibraion procedure Figure 6.7. Normal objecive specral lens ransmissions for he Nion 50mm f/.4 AF-D red solid line and he Canon EF 50mm f/.4 USM blue solid line. he doed blac line is he difference in ransmission beween he Nion and Canon lens, respecivel. he shape of he ransmission profiles is more or less eual for boh lenses. here is a graduall increase from ~80% a 4000 Å up o a more sable region above ~90% from 4500 o 6500 Å. In he deep red par of he specrum λ > 6500 Å, he ransmission facors sar o decrease more rapidl o a level below ~50 o 60% a Å. 5

56 he Nion lens is overall more effecive han he Canon lens hroughou he visible region Å and he near infra-red Å b a facor of ~5 and 5%, respecivel. If we ae ino accoun he ransmission of he lenses, he difference beween he cameras in specral responsivi and uanum efficienc is onl changed b a few percen. he D300 s blue, green and red channels now have ~, 5 and 50% higher specral responsivi compared o he D40, respecivel. he updaed differences in uanum efficienc become ~7% for he blue, 0% for he green and 45% for he red channels. he above resul indicaes ha he main difference beween hese wo cameras is in heir deecors. Boh cameras use a CMOS Complemenar Meal Oide Semiconducor sensor and a Color Filer Mosaic CFM o separae he colors. he main difference in specral responsivi is found o be in he red channels, and could well be relaed o he ransmission of he red elemens in he CFM, he infra-red filer or in he semiconducors used. his is an ineresing opic, bu i is beond he scope of his paper Lesson learned A digial camera image could conain more informaion han jus relaive scaled inensiies or color coded piel values ha have lile phsical meaning in erms of brighness on a uaniaive scale. he recen improvemens in boh sensiivi and dnamic range enable he DSLR camera o be used as an inensi ool as well. Bu he lesson learned from his eercise is ha here seems o be a lac of a common sandard for camera sensiivi given b he manufacurers. As shown above for Nion and Canon, he ISO Inernaional Organiaion for Sandardiaion values, originall defined as he speed of phoographic film, ma no be he opimum parameer represening he sensiivi of a digial sensor. he specral responsivi or he uanum efficienc is he parameer ha should be used in he fuure o characerie he sensiivi of a camera. We hope ha he manufacurers can provide his informaion in fuure. I would increase he usage and poenial of hese fanasic devices. 53

57 7. AIUDE CORRECIONS his chaper presens he heor of Leas Suare adjusmen and Kalman filering. he heor is applied o a simple Inerial Measuremen Uni IMU mouned o a dnamic moving ssem. he aim is o compile sofware rouines ha enable us o esimae he correc aiude for our airborne specral imagers and cameras. 7. Inroducion In conras o a saic frame of reference, a bod in a dnamic or moving ssem is influenced b random acceleraions / forces from nearl all direcions. he bod s aiude obained b a saic sensor will as a conseuence produce errors. A pendulum mouned inside a car can be used as an eample o illusrae his. If he car is a res or moving wih consan speed, he pendulum s angle wih he verical represens a good measure of he car s il angle. On he oher hand, when he car is acceleraed or deacceleraed he pendulum will sar o swing as a response o he forces aced up on i. As a resul, large il errors occur. In order o obain he correc aiude i is necessar o measure boh he acceleraion and he urn rae. pical sensors are 3 ais solid-sae acceleromeers combined wih urn rae gros - ofen called Inerial Measuremens Unis IMU. hese sensors are raher nois and end o drif in ime, especiall he urn rae sensors. herefore, a filer o esimae he correc aiude is needed. here are numerous echniues ha can be used, bu Kalman filering is he onl one presened here. In order o undersand he Kalman filer, we firs inroduce he heor of Leas Suare adjusmen. he heor is based on wor done b Hofmann-Wellenhof and Lichenegger 6. he Kalman filer seuence is adoped from Brown and Hwang Leas suares adjusmen 7.. Sandard adjusmen Leas suare adjusmen is based on euaions where he observaions are epressed as a funcion of unnown parameers. In mari noaion he linear observaion model is epressed as l A, 7. where l is he vecor of observaions measuremens, A he design mari and he vecor of unnowns. In case of nonlinear funcions, a alor series epansion is usuall performed. he series is hen runcaed afer he second erm in order o obain a linear funcion. Furhermore, we inroduce he covariance mari of he measuremens 54

58 Q l E [ l l ] ρ ρ n σ σ σ σ σ n ρ σ σ ρ n σ σ n. 7. σ n Here σ i is he suared variance of he measured l i variable, and ρ ij is he correlaion facor beween l i and l j. Moreover we have ρ ij ρ ji. In our cases here is no correlaion beween he sensors, hus ρ ij 0 if i j. E[.] is he mahemaic hope or epecaion value. he inverse of he covariance mari P Q l 7.3 is called he weigh mari. If we assume n observaions and u unnown parameers, he design mari A consiss of n rows and u columns. he ssem is redundan when n > u. In mos of he cases, n u. he inconsisenc is due o observaional errors or noise. In order o solve he model and assure consisenc, he noise vecor n is added o he vecor of observaions l A n. 7.4 he soluion of 7.4 becomes uniue and consisen if we appl he leas suare principle, which sae ha he weighed sum of he suared residuals is minimied 8. In erms of our noaion he leas suare principle is given b [ n ] P n If we appl his minimum principle o he above observaional model, we obain firs he following euaion n P n l A P l A 7.6 or n P n A PA A Pl l P A l Pl. 7.7 Secondl afer derivaion [ ] see Appendi E n P n A P A A P A A Pl A P l Since P P, we ge A P A A Pl Hence A P A A P l. 7.0 he soluion is hen simpl A P A A P. 7. l 55

59 In a more general form where G A P A and g A P l. G g, 7. he above mehod gives us a pracical soluion o nois measuremens. he soluion is opimied according o he influence of noise. 7.. Seuenial adjusmen he goal of his mehod is o improve he soluion of E. 7.4 b conducing seuenial measuremens. Each measuremen leads o a new esimaion of which will be more and more accurae as we coninue. o illusrae his, le us divide he observaion model ino wo ses l A n. 7.3 l A n Focusing on he firs se, a preliminar soluion is given b he Es 7. and 7. as A P A A P l G g. 7.4 he soluion is compued o limi o influence of he noise n, l A. Ne we consider he second se of measuremens, and obain similarl A P A A P l G g. 7.5 Le us assume ha he second measuremen is a lile bi differen o he firs one l l l. 7.6 E. 7.5 becomes G A P l l. 7.7 Furhermore, using E. 7.4 G A P l. 7.8 Since l l l, G A P l l. 7.9 Finalll A, and he updaed sae euaion aes he form K l A, 7.0 where K A P A A P. he above euaion is a linear combinaion of. Conseuenl, A ma be considered as a predicion of l before an measuremens are conduced. his resul is used in Kalman filering. 56

60 7.3 Kalman filering 7.3. Basic concep he Kalman filer ma be eplained b he following analog. A man sleeping in a dar room waes up in he middle of he nigh and wans o go ou of he room. he room lighs are ou, so he has o find he ei door in oal darness. He has a priori informaion on where he door is locaed. He maes an esimaion based on his informaion and sars o wal in one direcion. Afer a shor while, he wals sraigh ono a wall. his is regarded as a measuremen. He now nows a lile bi more where he is locaed in he room. In oher words, he updaes his iniial esimae wih he measuremen. A his poin a predicion of a new direcion is possible. he process is repeaed wih measuremens and will conseuenl improve his chances of finding he door. he above is he main essence of Kalman filering. I is based on seuenial adjusmen in he saic case. All observaions up o epoch are used o obain opimal esimaions of he unnowns. Forunael, we do no need o sore an daa as we proceed o he ne esimae he ssem model In a dnamic ssem such as an airplane, he unnown parameers are he aiude roll, pich and aw, he coordinaes and he veloci. hese form he elemens of he sae vecor. he sae vecor is ime dependen, and i ma be prediced for an insan b means of ssem euaions. he prediced values are hen updaed b he use observaions, which conain informaion of he rue sae vecor. In a discree form he sae vecor is defined as [ n ]. I consiss of n unnown variables. he model for predicion is assumed o be a linear as a funcion of he previous sae [ w w n ] w. 7. w is he noise of he ssem. he noise of he sae vecor could for eample in an airplane be moor vibraions. E. 7. is ofen called he Marov process where is he ransiion mari. On he observaional side, he measuremen model is given b E. 7.4 as l A n. 7. he measured noise n is due o he sensors hemselves sensor noise he Kalman euaions he updaed esimaed sae vecor ˆ is a linear combinaion wih he a priori sae ˆ and he error of measured predicion.he relaion was esablished previousl wih E. 7.0 ˆ ˆ K l A ˆ

61 58 he main problem in E. 7.3 is o find he opimal K his is done b minimiing he race of he error covariance mari, Q wih respec o K.. his can be done since he race of Q is a sum of he mean suare errors in he esimaes of all he elemens of 0 K Q race. 7.4 B definiion he error covariance mari of he updaed sae vecor is [ ] E Q ˆ ˆ. 7.5 Correspondingl, he error covariance mari of he esimaed sae vecor is [ ] E Q ˆ ˆ. 7.6 Q can be obained using E. 7.3 A n A K ˆ ˆ ˆ 7.7 or simpl n K A K ˆ ˆ ˆ 7.8 Es. 7.7 and 7.8 are insered ino E. 7.5 and ields K R K K A Q A K Q A K Q A K K A Q Q Q. 7.9 Or simpl K R K K A I Q A K Q Q From he above epression is possible o compue E he derivaion rules are lised in Appendi E. [ ] 0 R K A Q A K A Q K Q race 7.3 he opimal K used in he E. 7.3 is hen R A Q A A Q K 7.3 his is he same gain facor ha we found in he leas suare adjusmen E Furhermore, he updaed error covariance mari is calculaed b insering E. 7.3 in 7.9 Q A K I Q 7.33 he error covariance mari has decreased. A his poin we are finall able o calculae he updaed sae vecor wih measuremens b E he prediced sae vecor is compued b he model euaion ˆ ˆ 7.34 based on he Marov process 7., ignoring he model noise. he prediced error covariance mari before he measuremen l hen becomes [ ] W Q E Q ˆ ˆ 7.35

62 7.3.4 Kalman algorihm filer seuence he filer seuence is based on 4 seps: Esimaion Measuremens Updaes and Predicion.. Prior esimaion - Iniialiaion Inpu a priori esimae and is error covariance Esablish R and W ˆ 0 Q 0. Measuremen Conduc observaions l 3. Updae a Compue Kalman gain: K A Q A R Q A b Updae esimae wih measuremen: ˆ ˆ l A ˆ K c Compue error covariance for updaed esimae: Q I K A Q. Esimaion Predicion Swap variables 4. Predicion ˆ Figure 7.. he Kalman filer floa diagram ˆ Q Q W 7.4 Airborne aiude esimaion 7.4. Ais configuraion he aim of his sud is o obain an esimae of he aiude o our airborne insrumens. he measuremens need o include boh he acceleraion and he urn rae in 3 dimensions 3D. Fig.7. shows he fied frame coordinae ssem of he aircraf. he sensors are 3 orhogonal mouned roaional rae sensors gros and 3 acceleromeers. he -ais is ou of he nose of he aircraf, he -ais poins ou he righ wing, and -ais is down hrough he cener of mass. he roll angleφ is defined posiive wih righ wing down. Figure 7.. Ais. φ is roll, θ pich and ϕ aw. ω is he angular urn rae [deg./s]. 59

63 60 he pich angleθ is defined similarl wih nose up as posiive direcion. ϕ, aw or heading, is posiive in he clocwise direcion when viewed from above. he angular bod raes are as follows. ω is he roll rae angular veloci abou he -ais wih posiive direcion as righ wing down. ω is he pich rae abou he -ais wih posiive direcion defined as nose up. Correspondingl, ω is he aw rae abou he -ais, posiive in he clocwise direcion when viewed from above Measuremens he measured aiude or he pich, roll and aw angles of he aircraf are calculaed from he acceleromeer Compass a a a a ArcSin a a Arcan l / / ϕ θ φ, 7.36 where { } i a i,, represens he 3 ais acceleromeer readings. hese angles are also nown as he Euler angles. he aiude calculaed b E assumes a nonacceleraed reference frame he sae vecor In order o obain smooh and coninuous aiude esimaes as a funcion of ime, uaernion mahemaics is used. Quaernions mae roaions in four dimensions. 3D roaions b Euler angles ma suffer from an effec called Gimbal loc. his problem occurs when he final roaional mari is creaed b mulipling he roaional marices in he, and -ais, respecivel. he order of muliplicaion becomes ver imporan. he problem is fied b he use of a uaernion as he sae vecor [ ] s A shor summar of he naure and behavior of uaernions are found in Appendi D he measuremen model he measuremens are relaed o he observaional model as follows s n A n A l ϕ θ φ he measuremen mari A is defined as s s s A ϕ θ φ ϕ θ φ ϕ θ φ ϕ θ φ 7.39

64 his mari is also called a ransiion mari since i permis us o conver from uaernion- o Euler frames of reference. he mahemaics behind he consrucion of A is non- rivial, bu doable. I is a Jacobian mari he ssem model - predicion he insananeous rae change of an airborne uaernion is given b Hughes 9. Ω, 7.40 where Ω is he uaernion omega mari for he curren roaional sensors 0 ω ω ω 0 ω ω ω Ω. 7.4 ω 0 ω ω ω ω ω 0 ω, ω, ω represen he rolling, piching and awing urn raes in he srap down inerial reference frame, as measured b he roaional rae sensors a he momen. he ransiion mari is hen given as I Ω, 7.4 where is he epoch or sampling period. he ransiion mari saisfies he relaion w, 7.43 which is he Marov process or he predicive sae of he Kalman filer. Noe ha he ransiion mari is updaed a ever epoch b he use of measuremens from he gros. he fundamenal Kalman relaions are now esablished b Es and he final filer seuence is shown in Fig he ransiion mari E. 7.4 is based on a semi-coninuous predicive approach - described in Appendi F. 6

65 . Prior esimaion - Iniialiaion Inpu a priori esimae and is error covariance Esablish R and W ˆ 0 Q 0. Measuremen Conduc observaions l 3. Updae a Compue Kalman gain: K A Q A R K l A ˆ Q A b Updae esimae wih measuremen: ˆ ˆ c Compue error covariance for updaed esimae: Q I K A Q. Esimaion Predicion Swap variables 4. Predicion ˆ ˆ Q Q W W R l K ˆ Q Q ˆ ˆ Figure 7.3. he Kalman filer floa diagram for airborne aiude conrol. Noise model covariance mari Noise measuremen covariance mari Measuremens Euler angle Kalman gain Updaed esimaion wih measuremens l Updaed error covariance mari Esimaion before measuremen l Esimaion error covariance mari before measuremen l Predicion before measuremen l Q Predicion error covariance mari before measuremen l Marov mari A Measuremen mari 6

66 7.5. Aiude applicaions 7.5. Eperimenal seup he firs es of he Kalman filer was conduced during he AIRSPEX MMVI campaign carried ou as a par of he course AGF33 - Remoe sensing and specroscop, a he Universi Cenre in Svalbard UNIS. he plaform conains an IMU sensor including urn rae sensors, acceleromeers and magneomeers acuired from he American firm Microsrain Inc. he GPS is from he compan Garmin. Fig. 7.4 shows he seup. he hperspecral imager and he aiude sensors were mouned on Dornier 8-0K airplane operaed b he commercial compan Lufranspor AS. he insrumens were looing ou of a sliding door a he side of he airplane, poining 30 degrees off nadir. he fligh was carried ou a an aliude of 500 m above sea level 3 rd of Ma 006, raversing Longearben 78 o N, 5 o E, Norwa, from souh-wes o norh-eas. Fig.7. 5 shows he resuls of image correcions wih he use of Kalman filered daa from he IMU / GPS. Figure 7.4. he GPS is lined o he lapop via a USB hub. he urn rae sensors / acceleromeers / magneomeers are conneced o he lapop wih he same hub. he sensors are powered b VDC baer. Figure 7.5. Kalman aiude correced hperspecral image over Longearben, Norwa,