Relevance of Radiative Transfer Model in
Physical Inversion
Xu Liu and Richard Lynch
AER Inc.
Outline of the Presentation
1. Requirements on radiative transfer model for physical inversion
2. Overview of radiative transfer calculation in inhomogeneous
atmosphere
3. Calculation of Analytical Jacobians
4. Transformation of variables using EOFs before physical
inversion
5. Overview of how to model channel radiances
6. Application of OSS forward model to CrIS and NAST-I
instruments
7. Conclusions
What is an idea Fast Radiative Transfer
Model for Physical Inversion
R = F ( x) + ε
•
δR = Kδx
Accurate
– Idea if the accuracy relative to LBL is controllable
•
Physical parameterization
– Accuracy and physical parameterization is closely coupled
– Use least non-physical assumption
•
Fast
– Modern computer technology can accommodate large model
parameters
– ILS (SFR) convolution should be done during the training
•
Perform RT calculation monochromatically
– Calculates Jacobian efficiently
– Calculates downwelling radiances efficiently
– Be able to handle multiple scattering
•
Treat Planck function and surface properties properly
– Be able to model non-localized instrument line shape function
efficiently
•
Train the forward model under variety of conditions
– Be able to handle variable observation altitude for aircraft instrument
Radiative Transfer Equation for
Infrared Spectral Region
0
Rν ≅ ε ν Bν (Θ s )Ts ,ν + ∫ Bν (Θ( p ))
ps
+ (1 − ε ν )Ts ,ν
•
•
•
•
∂Tν ( p,θ u )
dp
∂p
ps
∂Tν* ( p,θ d )
∫0 Bν (Θ( p)) ∂p dp + ρν Ts ,ν Tν ( p s ,θ sun ) F0,ν cosθ sun
The first term is the surface emission
The second term is the upwelling thermal emssion
The third term is the reflected downwelling radiation
The last term is the reflected solar radiation
Defining Atmospheric Layering
• Schematic for atmospheric layer convention
T1
T0*
T1*
T2
level 0 (TOA)
layer 1
level 1
layer 2
level 2
T2*
TN-2
level N-2
*
TN-2
TN-1
TN
*
TN-1
layer N-1
level N-1
layer N
level N (SURFACE)
Recursive Radiative Transfer
Calculations
N
Rν = ∑ (Tν ,i −1 − Tν ,i ) B
+
ν ,i
i =1
+ ε νs Tν , N B
+
ν ,s
N
+ (1 − ε νs )Tν , N ∑ (Tν*,i −Tν*,i −1 ) Bν−,i
i =1
+ ρ s Tν , N Tsun ( p s ,θ sun ) F0,v cosθ sun
Tν' ,l = (1 − ε νs )Tν , N Tν*,l
If we define:
+
ν ,i
Rν = B
N
∑ (Tν
i =1
'
−Tν ,i −1 ) + ε νs Tν , N B
'
,i
+
ν ,s
+ ρ s Tν , N Tsun ( p s ,θ sun ) F0,v cosθ sun
−
ν ,i
+B
1
∑ (Tν
i=N
,i −1
− Tν ,i )
Calculation of Analytical Jacobians
τ l0 = τ fix ( pl , Θ l ) + [k H 2O ( pl , Θ l ) + k Hself2O (q H 2O , Θ l )ω H 2O ]ω H 2O + k O 3 ( pl , Θ l )ω O 3 + .......
 l 0

Tl = exp − ∑τ i secθ obs 
 i =1



Tl* = exp − ∑τ i0 secθ d 
 i =l +1

N
∂R ∂τ l0 ∂R ∂Bl
∂R
+
=
∂X l ∂τ l0 ∂X l ∂Bl ∂X l
∂Ti − Ti secθ obs
=
0
∂τ l0
i≥l
i<l
∂Ti * − Ti * secθ d
=
∂τ l0
0
i<l
i≥l
N
N

'
'
+
+
+
−
− Tl Bl + ∑ (Ti −1 − Ti ) Bi + TN ε s Bs + ∑ (Ti − Ti −1 ) Bi  secθ obs
i =l +1
i =1


l −1



*
−
+ − (1 − ε s )TN Tl −1 Bl + ∑ (Ti ' − Ti −' 1 ) Bi−  secθ d − Rsol (secθ obs + secθ sun )
i =1



∂Bl+
∂Bl− '
+
(Tl −1 − Tl ) +
(Tl − Tl '−1 )
∂X l
∂X l
∂τ l0
∂R
=−
∂X l
∂X l
Calculation of Jacobians
 ∂R 
∂Bl
∂R ∂τ
∂R ∂B
∂R
≅
= 0
+
(Tl −1 − Tl ) + 

∂Θ l ∂τ l ∂Θ l ∂B ∂Θ l ∂Θ l
 ∂Θ l  d
0
l
ï
dt/dQ can be
calculated from
the lookup
table easily
∂Bl
∂Bl
∂R ∂B
(Tl −1 − Tl ) +
(Tl ' − Tl '−1 )
=
∂Θ l
∂B ∂Θ l ∂Θ l
∂R
∂R
m
=
×
k
m = 1,K , M
l ,
∂ω lm ∂τ l0
 ∂R
∂R
+
= (−Σ l + Bl Tl ) secθ obs +  0
0
∂τ l
 ∂τ l
∂R
= TN Bs − Σ −N (1 − ε s )
∂ε s
∂Bs
∂R
= TN ε s
∂Θ s
∂Θ s
(
)
∂R = T F cosθ exp − τ 0 secθ
∑l l
N 0
sun
sun = R sol / ρ s
∂ρ
s

δR
 + sol
0
 d ∂τ l
Temperature and Moisture Jacobians
Temperature and Moisture Jacobians
Transformation of variables
• The layer derivative can be converted to level derivative by:
above
below
∂R ∂X lay
∂R ∂X lay
∂R
+
=
above
below
∂X lev
∂X lev ∂X lay
∂X lev ∂X lay
• The truncted EOF (U) obtained from background covariance
can be used compress ∆X and K:
∆~
x = U T ∆x
~
K i = K iU
Λ = U T S xU
~
~
~
~
∆~
xi +1 = ( K iT S y−1 K i + Λ ) −1 K iT S y−1 ( y 0 − y i + K i ∆~
xi )
• If the full correlation of noise covariance Sy can be compressed
using truncated EOF obtained from PCA of radiance spectra
– The inversion of the transformed matrix will be more efficient
Difficulties of Modeling Channel
Radiances or Transmittance
T∆ν (ν ) = ∫ Φ(ν − ν ' )T (ν )dν '
R∆ν (ν ) = ∫ Φ (ν − ν ' ) R (ν )dν '
∆ν
∆ν
• where Φ is normalized ILS (SFR)
• LBL calculation of monochromatic layer transmittances or
TOA radiances is very time consuming
• Convolving monochromatic radiances or transmittances
with ILS (SRF) is also time consuming
• The Beer’s Law is no longer valid
– It’s difficult to handle inhomogeneous path and multiple gases
z
z
∆ν
∆ν
φ (v )Tgas1Tgas2 d∆ν ' ≠
z
∆ν
φ (v )Tlayer1Tlayer 2 d∆ν ' ≠
φ (v )Tgas1d∆ν '
z
∆ν
z
∆ν
φ (v )Tlayer 1d∆ν '
φ (v )Tgas2 d∆ν '
z
∆ν
φ (v )Tlayer 2 d∆ν '
Comparison of Different Fast RT Model
Model Type
Characterisitic
Limitations
Band Model
Simple parameterization
Fast Curtis
Godson approximation can be used to handle
inhomogeneous atmos.
Limited accuracy
Not accurate to extend to multiple
gases
Neural net
Simple
Fast
Jacobian calculations?
Non-physical parameterization
Correlated k
Distributions
(CKD)
Monochromatic (g-v mapping)
Level to level k correlation is approximate
Overlapping gases treatment is approximate
Not perfect for inhomogeneous path
and overlapping gases
Exponential Sum
Fitting
Transmissions (
ESFT)
Monochromatic (select few k terms or v points)
Level to level k correlation is approximate
(methods exist to handle it)
Overlapping gases treatment is approximate
Standard method is perfect for
inhomogeneous path and overlapping
gases
Optran, RTTOV
,SARTA,Gastropd
….
Polychromatic
Smart way to treat overlapping gases
Teat inhomogeneous path
Effective layer optical depth depends
on layer above it
Optimal Spectral
Sampling (OSS)
Monochromatic
Treat inhomogeneous path and overlapping
gases well
Parameterization is physical
Very good treatment of
inhomogeneous path and overlapping
gases
Overview of Different Fast RT Models
• K distribution (KD)
z
N
T ( ∆v , ω ) = φ (v − v ')T (v , ω )dv ' ≅ ∑ ∆gi exp[ − ki ( gi )ω ]
∆v
i =1
N
R( ∆v , ω ) = ∑ ∆gi {Bi (Θ, g ) + [ R0,i − Bi (Θ, g )]exp[ − ki ( g )ω ]
i =1
N
∑ ∆g
i
=1
i =1
∆gi and ki obtained by grouping k(v)
• Correlated-K distribution (CKD)
– Correlation between the spectral shape and positions is different
layers is approximate
• KD made for a set of T,P, independently
• Methods exist to correct this approximation
• Use same g-v mapping for all layers (e.g Mlayer et al….)
• Reference layer add-subtract method (e.g. Oinas, Edward ….)
Overview of Different Fast RT Models
Correlated-K distribution (Continued)
– Treatment of overlapping gases is approximate
• Assume gases are uncorrelated:
z
T ( ∆v , ω 1 , ω 2 ) = φ (v − v ' )T (v , ω 1 )T (v , ω 2 )dv '
∆v
N
M
i =1
j =1
≅ T ( ∆v , ω 1 )T ( ∆v , ω 2 ) = ∑ ∆g1,i exp[ − k1,i ( g1,i )ω 1 ]∑ ∆g2 , j exp[ − k 2 , j ( g2 , j )ω 2 ]
• Introduce ωi (or functions of ωi, i=gas1, gas2….) as additional
factor when generating k
k(g,p,T,ω
ω)
Overview of Different Fast RT Models
•
Exponential Sum Fitting of Transmissions (ESFT)
z
N
T ( ∆v , ω ) = φ (v − v ' )T (v , ω )dv ' ≅ ∑ wi exp[ − ki (vi )ω ]
i =1
∆v
N
R( ∆v , ω ) = ∑ w{Bi (Θ, v ) + [ R0,i − Bi (Θ, v )]exp[ − ki ( g )ω ]
i =1
N
∑w
i
=1
i =1
– wi and the spectral location of ki obtained by a selection/regression process
•
Treatment of inhomogeneous atmosphere
– Use wi and νi obtained for a reference layer and scale exponential term with
appropriate function of P and T
– Include all layers in the regression and selection process (Armbruster and
fisher 1996)
TOA
L
T
N
L
i =1
l =1
( ∆v , p L , ω ) ≅ ∑ wi (vi ) exp[ − ∑ ki (vi , pl )ω ( pl )]
Overview of Different Fast RT Models
•
ESFT (continued)
– Treatment of mixing gases
• Similar to CKD (assume uncorrelated)
z
T ( ∆v , ω 1 , ω 2 ) = φ (v − v ' )[T1 + δT1 (v , ω 1 )][T2 + δT2 (v , ω 2 )]dv '
∆v
N
= T1T2 + ∑ wiδT1,i (v , ω 1 )δT2 ,i (v , ω 2 )
z
i =1
z
T1T2 = φ (v − v ' )T1 (v , ω 1 )dv ' φ (v − v ' )T2 (v , ω 2 )dv '
∆v
∆v
• Equivalent extinction (Ritter and Geleyn, Edwards…)
• Additional interpolation variable as a function of ωi (or functions of ωi,
i=gas1, gas2….)
• Frequency sampling method or radiance sampling method (Sneden et
al. 1975, Tjemkes and Schmetz, 1997)
Overview of the OSS forward model
• Optimal Spectral Sampling (OSS) approximates channel radiances (or
transmittances ) according to:
R∆ν (ν ) = ∫ Φ (ν − ν ' ) R(ν )dν ' = ∑ wi Rν i + ε
∆ν
i
• Channel radiances are a linear combination of monochromatic radiances at
pre-selected frequencies
• Spectral locations/weighting coefficients are obtained through a
selection/regression process
• The computational gain is more than 3 orders of magnitude relative to the
line-by-line calculations
• RT is done monochromatically
–
–
–
–
Monochromatic lookup table is used to calculate the transmittance for various
gases and different atmospheric layers (no approximates need to be made)
Calculates Jacobian analytically (very efficient)
Treats reflected radiance accurately
Can be easily coupled with multiple scattering codes
Application of OSS to NAST-I
Expanded View of the RMS of the
Forward Model
Number of Points Per Channel
•
Average of 2.59 monochromatic spectral calculations are needed for
each NAST-I channel
NAST-I
Spectral Band
Number of
Channels
Number of
Monochromatic
Points
Average
Points per
Channel
LWIR
2718
7464
2.75
MWIR
2946
7283
2.47
SWIR
2968
7569
2.55
Validation of OSS Forward Model
Accuracy
•
The radiance errors derived
from independent profile set
follow a Gauss distribution
Observed and Modeled NAST-I
Radiances for 7/14/01 CLAM Campaign
Expanded View of the Observed and
Calculated NAST-I Radiances
Observed vs. Calculated NAST
Radiance for Crystal-Face AIRS
Underflight
Retrieved Profiles For the CAMEXIII
Campaign Near Andros Island
CrIS
Retrieval
Algorithm
was used
Retrieved Temperature Profiles from
NAST-I Instrument
Conclusions
•
Jacobian provides sensitivity of radiance with respect to the
retrieved parameters
– Recursive RT calculation gives insights
– Most terms needed for the radiances calculation can be used for
Jacobian calculation (time saving)
•
Transformation of variable accelerate inversion process
– It also provides stability to the inversion
•
The fast radiative transfer model is best done at monochromatic
frequencies
– Physical parameterization
– Efficient in calculating Jabcobian matrix needed for inversion
– Can include multiple scattering calculations
•
•
•
It’s best to train all atmospheric layers and all major gases
simultaneous
OSS model has been developed to model NAST-I radiances and
was incorporated into NASA’s retrieval algorithm
OSS has been used to simulate the CrIS EDR retrieval performance
and the retrieval algorithm has been validated using NAST-I data