Linear Form of the Radiative Transfer Equation Revisited
Bormin Huang
Cooperative Institute for Meteorological Satellite Studies,
Space Science and Engineering Center
University of Wisconsin–Madison
The 15 th International TOVS Study Conference (ITSC-15)
Maratea, Italy, 4-10 October 2006
Outline
• Smith’s linear form (Applied Optics, 1991) vs. its dual form.
• The linear form with exact analytical Jacobians vs. the tangent-linear and
adjoint models.
• The linear form with exact analytical Jacobians vs. the linear form with
inexact analytical Jacobians.
• Why use the forward model with exact analytical Jacobians for physical
retrieval and data assimilation?
• Summary
• Future work
Bill Smith’s Achievement in Physical Retrieval
•
Bill has been pioneering the physical retrieval of temperature and absorbing
constituent profiles from the radiance spectra since the late ’60s.
•
His most recent linear form of the RTE was published in the landmark paper
in Applied Optics (1991).
•
This monochromatically approximate linear form and its variant have been
still used in the physical retrieval [e.g. Ma et al., 2000, Li et al., 2001].
•
His work inspired Huang et al. (Applied Optics, 2002) to successfully derive
the linear form with exact analytical Jacobians for the widely-used McMillinFleming-Eyre-Woolf type of forward models (e.g. RTTOV, RTIASI).
•
In this talk I will prove that there exists the dual representation of Smith’s
linear form (1991), and show some mathematically interesting outcomes
derived from this dual form.
Smith’s Linear Form (1991) vs. its Dual Form
obs
Rv
R
0
v
= Bv ( ps ) τ v ( ps ) −
= B ( ps )τ ( ps ) −
0
v
0
v
∫
∫
PS
0
ps
0
Bv ( p ) dτ v ( p )
B v0 ( p ) d τ v0 ( p )
δ Rv ≡ Rvobs − Rv0
δ Bv ( p ) ≡ Bv ( p ) − Bv0 ( p )
δ τ v ( p ) ≡ τ v ( p ) − τ v0 ( p )
Smith’s Linear Form
Its Dual Form
δ Rv = Bv ( ps )δτv ( ps ) + δ Bv ( ps )τv0 ( ps )
δ Rv = δ Bv ( ps )τv ( ps ) + Bv0 ( ps )δτv ( ps )
− ∫ Bv ( p) d [δτv ( p)] − ∫ δ Bv ( p)dτ ( p)
ps
ps
0
0
0
v
− ∫ δ Bv ( p)dτv ( p) − ∫ Bv0 ( p)d [δτv ( p)]
ps
ps
0
0
δ Rv = Bv ( ps ) δ τ v ( ps ) + δ Bv ( ps ) τ v0 ( ps )
δ Rv = δ Bv ( ps )τv ( ps ) + Bv0 ( ps )δτv ( ps )
− ∫ Bv ( p) d [δ τ v ( p)] − ∫ δ Bv ( p) dτ v0 ( p)
ps
ps
0
0
δ Bv ( p) =
− ∫ δ Bv ( p)dτv ( p) − ∫ Bv0 ( p)d [δτv ( p)]
d lnτ v0i ( p)
i =1
dUi0 ( p)
∫
dτ v0 ( p ) = τ v0 ( p ) ∑ d ln τ v0i ( p )
d lnτ ( p)
dT ( p)
dT ( p) = d lnτ ( p) 0 ,
dU ( p)
dUi ( p)
0
vi
− ∑ ∫ β ( p)τ ( p)δ T ( p)d lnτ ( p)
N
ps
+∑∫
i=1
0
0
v
0
vi
dT ( p)
β ( p)τ ( p)δUi ( p) 0 d lnτ v0i ( p)
dUi ( p)
0
v
0
q i ( p ′) d p ′
τ v ( ps ) ≈ τ v0 ( ps )
0
v
dUi ( p)
d lnτ v0i ( p)
dUi0 ( p)
=
dUi ( p)
0
d
ln
τ
( p)
v
0
i
dUi ( p)
δ Rv ≈ βv0 ( ps )τ v0 ( ps ) δ Ts
δ Rv ≈ βv0 ( ps )τ v0 ( ps )δ Ts
0
p
i=1
0
vi
0
i
i=1
0
N
N
0
v
0
∂ Bv (T ( p))
δT ( p) ≡ βv0 ( p) δT ( p)
∂ T ( p)
1
U i ( p) ≡
g
ps
ps
0
δ τ v ( p) ≈ τ v0 ( p) ∑δ Ui ( p)
N
ps
N
− ∑ ∫ βv0 ( p)τ v0 ( p) δ T ( p)
ps
i=1
0
N
ps
+∑∫
i=1
0
dUi ( p)
0
ln
τ
d
vi ( p)
o
dUi ( p)
d T 0 ( p)
0
β ( p)τ ( p)
δ
τ
d
(
)
ln
( p)
U
p
v
i
0
i
dUi ( p)
0
v
0
v
δ Rv ≈ βv0 ( ps )τ v0 ( ps )δTs
N
δ Rv ≈ βv0 ( ps )τv0 ( ps )δ Ts
N
− ∑ ∫ β ( p)τ ( p) δT ( p) d lnτ ( p)
ps
i =1
0
N
ps
+ ∑∫
i =1
0
0
v
0
v
− ∑ ∫ βv0 ( p)τv0 ( p)δ T ( p)
0
vi
dT ( p)
β ( p)τ ( p)δUi ( p) 0 d lnτ v0i ( p)
dUi ( p)
0
v
ps
i =1
0
N
ps
+ ∑∫
0
v
i =1
The effective temperature profile
of the i th absorbing gas:
0
dUi ( p)
0
d
τ
ln
vi ( p)
o
dUi ( p)
dT 0 ( p)
β ( p)τ ( p) 0 δ Ui ( p) d lnτv0i ( p)
dUi ( p)
0
v
0
v
The effective temperature profile
of the i th absorbing gas:
d T ( p)
δ Ti ( p ) ≡ δ T ( p ) − δ U i ( p )
d U i0 ( p )
dUi ( p)
dT 0 ( p)
δ Ti ( p) ≡ δ T ( p) 0 −δ Ui ( p) 0
dUi ( p)
dUi ( p)
Final Linear Form
N
δ Rv ≈ β ( ps )τ ( ps ) δ Ts − ∑ ∫ βv0 ( p) δTi ( p)τ v0 ( p) d lnτ v0 ( p)
0
v
0
v
i =1
ps
0
i
d T ( p)
δ Ti ( p ) ≡ δ T ( p ) − δ U i ( p )
d U i0 ( p )
dUi ( p)
dT 0 ( p)
δ Ti ( p) ≡ δ T ( p) 0 −δ Ui ( p) 0
dUi ( p)
dUi ( p)
d U i ( p)
0
0
d
U
(
p
)
⎡
⎤
i
U i ( p) = U i0 ( p) +
[T ( p) − Ti ( p)] ⎣ T ( p ) − T ( p ) ⎦ d T 0 ( p ) + U i ( p )
d T ( p)
d U i0 ( p )
0
0
⎡
= U i ( p) +
T
( p ) − Ti ( p ) ⎤⎦
0
⎣
d T ( p)
A special case:
T ( p) = T 0 ( p)
⇒
d Ui0 ( p)
U i ( p ) = U ( p) +
[T ( p) − Ti ( p)]
d T ( p)
0
i
The general case: T ( p ) ≠ T 0 ( p )
1
Ui ( p) = 0
×
T ( p) − T ( p)
∫{
p
0
⇒
}
dUi0 ( p′) 0
dT 0 ( p′)
⎡⎣T ( p′) − Ti ( p′)⎤⎦
U ( p) +
d p′
0
dT ( p′)
d p′
0
i
The retrieval quality of absorbing gas profiles depends on the quality of
temperature first guess!
Matlab example
The exact analytical Jacobians approach vs. the tangent-linear & adjoint
approach
1. Same numerical precision!
2. The exact analytical Jacobian
approach is several times faster !!
Example:
T
T0 A ( A+ B ) (1− Td0 )
V p (Td ) = C ( ) e
Td
Compute
δ Vp =
dV p
dTd
δ Td
Source: Paul van Delst
Why Use Exact Analytical Jacobians in Retrieval and Data Assimilation?
• Physical Retrieval (1D-Var):
C ost ( X ) = R obs − R ( X )
or
C ost ( X ) = R obs − R ( X ) + λ X − X
0
• NWP Data Assimilation (3D/4D-Var):
Cost ( X ,...) = Atmospheric Dynamic Term( X ,...) + R obs − R( X )
~ ideal choice for multispersptral sounders (e.g. HIRS, GOES, MODIS)
~ an underdetermined problem (with respect to a typical forward model)
Cost ( X ,...) = Atmospheric Dynamic Term( X ,...) + X retrieval − X
~ ideal choice for hyperspctral sounders (e.g. AIRS, IASI, GIFTS)
~an overdetermined problem (with respect to a typical forward model)
•
•
Atmospheric dynamic term is basically governed by the Navier-Stokes equation,
which has no analytical Jacobians. Thus, its tangent-linear/adjoint models are
needed for data assimilation.
The exact analytical Jacobians for the widely-used McMillin-Fleming-Eyre-Woolf
type of radiative transfer/forward models (e.g. RTTOV, RTIASI) are derivable (Huang
et al., Applied Optics, 2002).
Summary
1. The classical derivation of the linear form of the RTE by
Smith et al. (1991) is reviewed. Its dual form is derived.
2. The original linear form appears to be a special case of its
dual form when the temperature first guess happens to be
the true temperature profile.
3. Linear forms with inexact analytic Jacobians make retrieval
results unreliable!
4. The exact analytic Jacobians implementation is an efficient
alternative to the tangent-linear/adjoint models for
hyperspectral retrieval and data assimilation problems with
the widely-used McMillin-Fleming-Eyre-Woolf type of
forward models (e.g. RTTOV, RTIASI).
Future Work
The Remote Sensing GENOME (Geometrical
Exploration of Nonlinear Optimization in
Measurement Environment) Project:
Unveiling the Radiance Hyperspace
for Quantifying Geophysical Retrievals
The remote sensing GENOME project aims to solve the following
long-standing fundamental problems in passive remote sensing
Given a sensor specification, its forward model and exact Jacobians, to quantify
¾
the information content of each channel, i.e., the best expected contribution from each channel to the
retrieval of temperature and absorbing gases at each pressure level,
¾
the information content of a sensor, i.e., the best expected retrieval accuracy that sets the statistical limit for
all retrieval algorithms,
¾
the error of a fast model in terms of the degradation of the best expected retrieval accuracy, as compared to
its LBL counterpart,
¾
the impact of sensor noise level on the best expected retrieval accuracy,
¾
the “first guess tolerance” – a safety measure beyond which no retrieval algorithm can reach the best
expected retrieval, and
¾
the “retrieval efficiency” – a robustness measure for any retrieval algorithm, as compared to the best
expected retrieval.
Applications:
Lossy compression retrieval impact studies:
to conclude the retrieval degradation (due to lossy compression) by the best expected retrieval accuracy that
sets the limit for all possible retrieval algorithms.
¾
Optimal channel selection for retrieval with partial channels:
to relieve the computational burden in retrieval and data assimilation with the minimum retrieval degradation
for hyperspectral sounders (e.g. AIRS, IASI, GIFTS).
Future sensor design & trade-off study for risk reduction:
to assess the information content (the best expected retrieval accuracy) of a sensor designed with various
spectral ranges, ILS resolutions, and noise levels.
Different living species have different genomes. So do different sensors!