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Transcript of Remote Sensing - MIT OpenCourseWare ... Remote Sensing D1 Remote sensing is a quasi-linear...

  • Remote Sensing

    D1

    Remote sensing is a quasi-linear estimation problem

    Equation of radiative transfer:

    o o

    L

    B B (z)

    0

    ) T e T(z) (z)e −τ−τ° = + α∫

    temperature profile

    nepers m-1L

    z

    (z) = (z) ∆

    τ α∫

    ∆ τ τo )

    L

    0

    T(z),α(z)

    z ° B )

    oB T

    τ(z)

    z

    Lec22.5-1 3/27/01

    T ( K dz

    dz

    = (0

    T ( K

  • Radiation from Sky-Illuminated Reflective Surfaces

    ° = B )

    L

    z TB TS

    TG 0

    R = 1 - ε

    2

    2− α−τ ∫+ α∫ z

    o 0 L

    (z)

    0

    Re T(z) (

    reflectivity

    emissivity (specular surface)

    4

    − α∫

    −τ

    + α

    ≅ α τ⎡ ⎤⎣ ⎦

    L z

    L (z)

    0

    L (z)

    o 0

    ) (z)e

    T(z) (z)e dz for 1 4

    +1

    1

    − τo2 s

    sky temperature

    3

    3

    −τ+ε +o GT e

    ground temperature

    D2 Lec22.5-2 3/27/01

    T ( K

    dz z)e dz

    >>

    dz T(z dz

    RT e

    terms

    4

  • Temperature Weighting Function W(z,f,T)

    ( )≅ + ∫ L

    B o 0

    ) T T(z)W (z)o

    Terms: , ,

    D3

    ↔B ) T(z)�Note: we have ~linear relation:

    (not Fourier)

    ( ) ( )′≅ + −∫ L

    B o o o 0

    T (f ) T ) ) )

    ( ) ∂ = ∂ o

    B o

    )

    T(z) )Incremental weighting function:

    Alternatively,

    Lec22.5-3 3/27/01

    T (f z,f,T dz τ >> For 1:

    T (f

    T(z T (z W ' z,f,T (z dz

    T (z W ' z,f,T T(z

    1 2 3 4

  • Atmospheric Temperature T(z) Retrievals from Space

    D4

    α(ω) (single P)

    f

    ∆ω ∝

    ∝ 2l l

    αo

    ωo α ≠o f(P)�

    0

    altitudes where intrinsic & doppler broadening dominate

    z

    α(z)

    W(z,f) ωo ± ∆

    ω ≅ ωo pressure dominates ∆ω

    −τ≅ α τ⎡ ⎤⎣ ⎦∫ L

    (z) B o

    0

    T T(z) (z)e 1

    ωo ± 2∆ Lec22.5-4 3/27/01

    pressure P

    area # mo ecu es m

    Therefore for P-broadening trace constituents

    >> dz for

  • Atmospheric Temperature T(z) Retrievals from Space z

    0 α(z)

    W(z,f) ωo ± ∆

    ω ≅ ωo

    altitudes where intrinsic & doppler broadening dominate

    pressure dominates ∆ω

    −τ≅ ⎡ ⎤⎣ ⎦∫ L

    (z) B o

    0

    T ) )e 1

    W(f,z)

    τ ≅ 1

    z

    ωo ω f

    α(ω) increasing P∆ω

    D5

    ωo ± 2∆

    scale height

    Lec22.5-5 3/27/01

    τ >> T(z d(z dz for

  • Atmospheric Temperature T(z) Retrievals from Below L

    0 α(z)

    W(f,z)

    ωo ωo ± ∆

    ωo

    − α∫= α ω L 0 (z)

    oW( ) (z)e i l i

    (1/α) >> λ:

    0

    z

    L

    transparency, frequency dependentIf α(z) ≅

    W(f,z) D6 Lec22.5-6 3/27/01

    dz f, z ~decay ng exponentia s, rate s fastest for

    Temperature profile retrievals in semi-transparent solids or liquids where

    constant:

  • Atmospheric Composition Profiles

    D7

    [ ]− α∫ ρ α ′≅ ρ • = ρ ρ ⎡ ⎤ ⎢ ⎥⎣ ⎦∫ ∫

    L z

    o

    L L (z)

    B B o 0 0

    (z)T (z) T( T (z) ( ( ) (z)

    W(z,f) if viewed from space

    Because α(z) and W(z,f) are strong functions of the unknown ρ(z), this retrieval problem is quite non-linear and can be singular (e.g. if T(z) = constant). In this case, good statistics can be helpful. Incremental weighting functions defined relative to a nearly normal ρ(z) can help linearize the problem.

    Lec22.5-7 3/27/01

    + ρ −dz z)e dz z) W z, f dz

  • Optimum Linear Estimates (Linear Regression)

    =ˆParameter vector estimate

    “determination matrix” data vector

    [ ]( )∆ 1 Nd =

    ( ) ( )− −⎡ ⎤⎣ ⎦ tˆ ˆi p p p p

    ( ) ( ){ ( )( )

    ∂ − − = = ∂

    ∂= − − = − ∂

    ⎡ ⎤ ⎣ ⎦

    ⎡ ⎤ ⎡ ⎤⎣ ⎦⎣ ⎦

    t

    ij

    tt t jj j i

    ij

    ˆ ˆE p p p p 0 D

    E p Dd p D

    THi

    [ ] [ ]

    =

    = =⎡ ⎤⎣ ⎦

    d

    i j i j

    t tt t t d

    = C

    ; �

    F1 Lec22.5-8 3/27/01

    p Dd 1,d ,..., d

    Choose D to min mize E

    Derive D :

    d D E 2d D d 2d p

    row of D

    ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

    Therefore D E dd E p d

    DE dd E pd C D E dp

  • Optimum Linear Estimates (Linear Regression)

    [ ] [ ]

    =

    = =⎡ ⎤⎣ ⎦

    d

    i j i j

    t tt t t d

    = C

    ; �

    F2

    1t t dˆi i l C

    − ⎡ ⎤= = ⎣ ⎦

    Lec22.5-9 3/27/01

    ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

    Therefore D E dd E p d

    DE dd E pd C D E dp

    The l near regress on so ution is p Dd where D E dp ⎡ ⎤ ⎣ ⎦

  • is Least-Square-Error Optimum if:D 1) Jointly Gaussian process (physics + instrument):

    ( ) ( )

    ( ) ( )−⎡ ⎤− ⎢ ⎥⎣ ⎦= π Λ

    11 r m r m2 r N 2

    1p r e 2

    ( )( )∆ ⎡ ⎤Λ − −⎣ ⎦ t t= E r m r m ∆

    =⎡ ⎤⎣ ⎦ 1 2 Nm =

    2) Problem is linear:

    = + + odata d n d

    parameter vector noise (JGRVZM)

    F3 Lec22.5-10 3/27/01

    − Λ −

    1 2

    E r , where r r ,r ,...,r

    Mr

  • Examples of Linear Regression

    0 d

    D11

    slope = D12

    [ ]− = ⎡ ⎤ ⎢ ⎥⎣ ⎦

    11 12ˆ D D 1

    d

    F4

    Equivalently:

    ( )

    [ ]( ) = + −12p̂ p D d d

    E

    slope = D12

    d1

    regression plane

    D11

    note change in slope

    d2

    slope = D13

    Lec22.5-11 3/27/01

    1 D : p

    • ≡ •

  • Regression Information

    G1

    but which is correlated with properties the instrument does see; i

    uncorrelated with visible information); it is lost.

    Lec22.5-12 3/27/01

    1) The instrument alone (via weighting functions)

    2) “Uncovered information (to which the instrument is blind,

    D retr eves this too.)

    3) “Hidden information (invisible in instrument and

  • Nature of Instrument-Provided Information

    Assume linear physics: =

    [ ] [ ]

    1 2 N

    1 2 M

    th i

    d T le

    W ix

    W i

    =

    =

    =

    =

    Where

    ( )

    N ii

    i 1 ˆ i l i i =

    = =

    = =

    Claim:

    G2 Lec22.5-13 3/27/01

    d WT

    data vector 1,d ,d ,...,d temperature profi T ,T ,...,T

    weighting function matr

    row of W

    If T a W and noise n 0,

    then T Dd T, perfect retr eva f W not s ngular

  • Proof for Continuous Variables

    G3

    ( )

    N ii

    i 1 ˆ i l i i =

    = =

    = =

    Claim:

    1

    2

    3

    ) = b (h)

    W (h) = b (h) b (h)

    W (h) = b (n) b (h) b (h)

    φ

    φ + φ

    φ + φ + φ

    i j ij 0

    i ij j j 0

    0 (h) (

    1

    ;b i i ( i ). d = T( (

    ∞∆

    ⎧φ = ⎨ ⎩

    φ

    Let:

    Lec22.5-14 3/27/01

    If T a W and noise n 0,

    then T Dd T, perfect retr eva f W not s ngular

    11 1

    21 1 22 2

    31 1 32 2 33 3

    W (h

    Where: h)dh

    are known a pr or from phys cs Then: h)W h)dh

    • φ = δ

  • Proof for Continuous Variables

    G4

    1

    2

    ) = b (h)

    W (h) = b (h) b (h)

    φ

    φ + φ

    j j 0

    d = T( ( ∞∆ ∫

    N j j

    i 1 = (h) ∆

    = ∑

    ( ) N tt t t

    j i i j i 0 0 N i N N

    i im m jn n i ji i 1 m 1 n 1 i 10

    d ( ( d

    k b (h) b (h) ( ) =

    =

    ∞ ∆

    = = = =

    = ⇒ = =

    ⎛ ⎞ = φ φ⎜ ⎟

    ⎝ ⎠

    ∑ ∫

    ∑ ∑ ∑ ∑∫

    1 d i ix

    k̂ Q d k i i− =

    = = Lec22.5-15 3/27/01

    11 1

    21 1 22 2

    W (hThen: h)W h)dh

    If we force T(h) k W

    Then: k W h) W h)dh k W W k Q

    dh k Q ⎞⎛ ⎟⎜ ⎠⎝

    Therefore Qk where Q s a known square matr

    So let where Q s non-s ngular

  • Proof for Continuous Variables

    G5

    1k