Remote Sensing. Remote sensing is a quasi-linear estimation problem. Equation of radiative transfer: ) T B e τ T(z) (z)e τ. τ(z)

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1 Remote Sensng Remote sensng s a quas-lnear estmaton problem Equaton of radatve transfer: T B ( K) T B o T(),α() τ() B o () ) T B e τ T() ()e τ o T ( K = + α τ () = τ o = τ() α() d d nepers m - temperature profle ec22.5- D

2 Radaton from Sky-Illumnated Reflectve Surfaces T S T B 2 T G ec reflectvty R = - ε emssvty (specular surface) T ( K = B ) τ o + Re T() α( )e + T() α ()e α() d τ () 4 α() d T() α ()e d for τ >> 4 RT e s sky temperature 2 τ o 3 τ +ε T e o + d G o ground temperature d terms 4 D2

3 Temperature Weghtng Functon W(,f,T) Terms:, 2, 3 4 For τ >> : o ( ) T B (f) T o + T()W,f,T() d Alternatvely, ( ) ( ) T (f) T + T() T () W ',f,t () d Incremental weghtng functon: B o o o ( ()) W',f,T o T = B T() o T () ec Note: we have ~lnear relaton: T B (f ) T() (not Fourer) D3

4 Atmospherc Temperature T() Retrevals from Space α(ω) (sngle P) α o ω pressure P f area # molecules m 2 ω o Therefore α f(p) for P-broadenng trace consttuents o τ() T T() α ()e d for τ >> B o alttudes where ntrnsc & doppler broadenng domnate W(,f) ω ω o pressure domnates ω ω o ± ω o ± 2 ec α() D4

5 Atmospherc Temperature T() Retrevals from Space W(,f) ω ω o ω o ± alttudes where ntrnsc & doppler broadenng domnate pressure domnates ω ω o ± 2 τ() T B T() d()e d for τ α() o >> α(ω) ω ncreasng P τ scale heght ω o ω f W(f,) ec D5

6 Atmospherc Temperature T() Retrevals from Below ω o ω o ± ω o α() d W(f,) α() W( f, ) =α ()e ~decayng exponentals, rate s fastest for ω Temperature profle retrevals n sem-transparent solds or lquds where (/α) >> λ: transparency, If α() constant: frequency dependent o ec W(f,) D6

7 Atmospherc Composton Profles α () α ρ () d B = ρ ρ B + o ρ o () [ ] T () T( )e d T () ( ) W (, f ) d W(,f) f vewed from space ρ Because α() and W(,f) are strong functons of the unknown ρ(), ths retreval problem s qute non-lnear and can be sngular (e.g. f T() = constant). In ths case, good statstcs can be helpful. Incremental weghtng functons defned relatve to a nearly normal ρ() can help lneare the problem. ec D7

8 Optmum near Estmates (near Regresson) Parameter vector estmate ( ) [ ] pˆ = Dd d =,d,..., d N ec determnaton matrx t ( ˆ ) ( ˆ ) Choose D to mnme E p p p p Derve D : D j { t ( ˆ ) ( ˆ ) data vector E p p p p = = ( t t t d D )( ) j = E p Dd p = E 2d j D jd 2d p D j [ ] = [ ] Therefore D E dd E p d j j DE dd = E pd ; C D = E dp = C d t t t t t d TH row of D F

9 Optmum near Estmates (near Regresson) [ ] = [ ] Therefore D E dd E p d j j DE dd = E pd ; C D = E dp = C d t t t t t d The lnear regresson soluton s p ˆ = Dd where D = C d t t E dp ec F2

10 D s east-square-error Optmum f: ) Jontly Gaussan process (physcs + nstrument): () pr r = e 2 Λ ( 2 π ) N2 ( )( t t ) Λ = E r m r m r m r m 2 ( ) Λ ( ) m = E r, where r = r,r 2,...,r N 2) Problem s lnear: data d = Mr + n + d o parameter vector nose (JGRVZM) ec22.5- F3

11 Examples of near Regresson ˆp : p ˆ = [ D D 2] d D slope = D 3 ˆp D regresson plane slope = D 2 D slope = D 2 d d d 2 note change n slope Equvalently: ( E [] ) ( ) ˆp = p + D d d 2 ec22.5- F4

12 Regresson Informaton ) The nstrument alone (va weghtng functons) 2) Uncovered nformaton (to whch the nstrument s blnd, but whch s correlated wth propertes the nstrument does see; D retreves ths too.) 3) Hdden nformaton (nvsble n nstrument and uncorrelated wth vsble nformaton); t s lost. ec G

13 Nature of Instrument-Provded Informaton Assume lnear physcs: d= WT Where d T = = [ 2 N ] le [ T,T,...,T ] data vector,d,d,...,d temperature prof 2 M W = weghtng functon matrx W = th row of W Clam: N If T = a W and nose n =, = ( ) then T ˆ = Dd = T, perfect retreval f W not sngular ec G2

14 Clam: Proof for Contnuous Varables N If T = a W and nose n =, = ( ) then T ˆ = Dd = T, perfect retreval f W not sngular W(h ) = b φ (h) W 2 (h) = b 2 φ (h) + b 22 φ 2 (h) W 3 (h) = b 3 φ (n) + b 32 φ 2 (h) + b 33 φ 3 (h) φ (h) φ j ( h)dh =δ j = j j j et: Where: φ ;b are known a pror ( from phys cs). Then: d = T( h)w ( h)dh ec G3

15 Then: Proof for Contnuous Varables j j W(h ) = b φ (h) N = k j W j(h) = N j = ( ) j t = t t = t = N N N = k b m φ m (h) b jn φn (h) ( dh) = k Q j = m = n = = d = T( h)w ( h)dh If we force T(h) W (h) = b φ (h) + b φ (h) Then: d k W ( h) W ( h)dh d k W W k Q Therefore d = Qk where Q s a known square matrx So let ˆk = Q d = k where Q s non-sngular ec G4

16 Clam: Then: Proof for Contnuous Varables N If T = a W and nose n =, = ( ) then T ˆ = Dd = T, perfect retreval f W not sngular So let ˆk = Q d = k where Q s non-sngular N N T(h) ˆ = k ˆ W (h) = k W (h) = T(h) (exact) Q.E.D. = = ( ) ( ) Equvalently: d T = Wk = W Q = WQ d = Dd ec So: D = WQ = "mn mum nformaton" soluton Whch s exact f T = Wk, n = G5

17 To what s an nstrument blnd? W(h) = b φ (h) W (h) = b φ (h) + b φ (h) An nstrument s blnd to T(h) components outsde the space spanned by φ,φ 2,,φ N or, equvalently, by ts W, W 2,,W n. By defnton, the nstrument s blnd to any φ j W, for all. ec H

18 Statstcal Methods Can Reveal Hdden Components N seen by N nstrument channels In general, T(h) = k W (h) + a φ (h) = N + all hdden components Extreme case: suppose φ (h) always accompaned by N N k W = = Then our present soluton: T(h) ˆ = (h) = a φ (h) Would become: N ( ) φ + φ + a 2 N = 2 shrnks wth decorrelaton ˆT(h) = a + φ φ 2 N+ (h). ec Thus hdden components can be uncovered f correlated wth vsble ones. H2

19 General near Estmate ˆT = β WQ a j j j= N + mnmum nformaton uncovered nformaton = Dd where D = + φ Thus retreval can be drawn only from the space spanned by N φ, φ 2,..., φ N ; β, β 2,..., β N dmensonal ty s N; T ˆ = d D That s, N channels contrbute N orthogonal bass functons to the mnmum-nformaton soluton, plus N more bass functons whch are orthogonal but correlated wth the frst N. = ec H3

20 General near Estmate As N ncreases, the fracton of the hdden space whch s uncovered by statstcs s therefore lkely to ncrease, even as the hdden space shrnks. In general: a pror varance = observed + uncovered + varance lost due to nose and decorrelaton. ec Example: 8 channels of AMSU versus 4 channels of MSU AMSU and MSU are passve mcrowave spectrometers n earth orbt soundng atmospherc temperature profles from above wth ~-km wde weghtng functons peakng at alttudes from 3 to 26 km. Note the larger rato of uncovered/lost power for AMSU. H4

21 Example: 8-Channel AMSU vs 4-Channel MSU MID-ATITUDES (TOTA POWER = 222 K 2, 5 EVES) OBSERVED POWER UNCOVERED POWER OST POWER MSU - 55 INCIDENCE ANGE, AND MSU - ANGE, NADIR AMSU - NADIR AMSU - NADIR 3 km WEIGTHING FUNCTION TROPICS (TOTA POWER = 84 K 2 ) MSU NADIR, AND UNCOVERED OST (WINDOW) AMSU - NADIR ec PERCENT POWER H5