Assessment of Atmospheric
Profiles Retrieved from Satellite
Theory and Case Study
Nikita Pougatchev
Gail Bingham, Stanislav Kireev, and David Tobin
ITSC-15
Acquafredda Maratea, Italy
October 3 - 10, 2006
Assessment=End-to-End Error Modeling
Atmosphere, Signal, Retrieval and Validation
True
Profile
xsat
Radiance
ysat
Parameter Error
Noise
Instrument
Smoothing
Parameter
Forward model
SDR yˆ
SDR & EDR
Assessment
True
Profile
xval
Noise
Retrieval
x̂
Assessment Model
ŷ val
xval
yval
EDR
x̂ val
Validation System
SDR
- Sensor Data Records – Radiances/Spectra
EDR – Environmental Data Records – Retrieved Profiles (in this presentation)
Linear Assessment Model
Concept
• Atmospheric, Instrument, Forward Model, and Retrieval
parameters and their errors are random variables
• Variations and errors are characterized by Covariances
• Vertical resolution is characterized by Averaging Kernels
• Variations and errors propagate Linearly through
Atmosphere – Instrument – Signal – Retrieval-Validation
EDR Assessment Model – EDRAM
• Linear mathematical error model for the Post-launch/Validation
assessment of atmospheric profile retrievals.
• Assessment ≠ Comparison/Book-Keeping
• Assessment = Scientifically accurate relation between true state of
the atmosphere and measurements
• Validated and validating data differ by:
– Time and location
– vertical resolution and grid
– absolute accuracy and noise level
x2
x1
d
x2
d
x1
x2
x1
d
•EDRAM makes the assessment accurate by allowing for the
difference.
EDRAM Concept
• The validated system performs a set of measurements on an ensemble of true
states x
x̂ = r(x) + e
• r(x) is a nominal retrieval with the absence of any errors in the measured
signal and in the forward model
• e represents retrieval errors characterized by its mean value E{e} = Δ (Bias)
and covariance Se (retrieval Noise)
• The goal of the EDRAM is to assess actual Bias and Noise of validated
system by simulating its nominal retrieval based on validating data and to
estimate the error of the assessment.
Linear Assessment Model
Atmosphere, Signal, Retrieval and Validation
Clive Rodgers
Smoothing δxˆ = (A − I)(x − x )
+ G K δb
Parameter Error - δb δyˆ = Kδb + ε Parameter
+ G ΔFM
Forward model
Noise – ε
+G ε
Noise
ˆ
y
SDR
Instrument
Retrieval EDR x̂
a
True
Profile
xsat
y
Radiance
ysat
y
y
SDR & EDR
Assessment
True
Profile
xval
δyˆ expected
Assessment Model δxˆ expected
ŷ val
xval
yval
b
Validation System
x̂ val
EDRAM – Data Flow
Statistical Characteristics
of true states
{x1 S x1 }; {x2 S x2 }; S12
Estimation of
Validated Retrieval x̂1
a
Characterization A1 ; S ε1
Validating Data x̂2
Characterization A2 ; Sε2
Bias
Δxˆ 1 ± Err
Noise
Sε1
EDRAM -Theoretical background
Retrieval Model
xˆ i = x ai + A i (x i - x ai ) + Δxˆ i + ε i
Retrieval/Measurements
Averaging
Kernel
True
state
Bias
A priori
i=1 Validated data
i=2 Validating Data
The goal of the EDRAM is to assess actual
Bias Δx
ˆ and Noise S of validated system.
1
ε1
Noise
EDRAM - Theoretical background
Relation between Atmospheric States
i=1 Validated data
i=2 Validating Data
x1 δx1 and x2 δx 2
S x1 S x2
S12 = S T21
Mean and variation about mean of true states
Auto-covariance of true states
Cross-covariance between true states
Relation between true states
Variation at
validated point
δx1 = Bδx 2 + ξ
Correlated
Un-Correlated
S x1 = BS x2 B T + Sξ
cov(x 2 ,ξ) = 0
B = S 12 S -1
x2
Theoretical background
(Continued)
Simulating
x̂1
with
x̂ 2
Analyzed Difference
xˆ 12 = A1Bxˆ 2 = A1B(I - A 2 )x a2 + A1BA 2x 2 + A1Bε 2
δxˆ ≡ xˆ 1 - xˆ 12
Validated Measurement
Simulated Validated Measurement!
δxˆ ≡ xˆ 1 - xˆ 12 = [(I - A1 )xa1 - A1B(I - A 2 )xa2 ] + A1 x1 − A1BA 2 x2 + Δxˆ 1
Mean Difference
Mean Expected Difference
e
δxˆ
Bias
Sδxˆ = (A1B(I - A 2 ))S x2 (A1B(I - A 2 ))T + A1Sξ A1T + Sε1 + (A1B)Sε2 (A1B)T
Covariance of the Analyzed Difference
Case Study
• Validation Data Set – radiosondes at ARM
Southern Great Plain (SGP) site; July –
December 2002 (416 sondes).
• Validated parameter – Atmospheric
Temperature Vertical Profile.
• Validated System – characterized by AIRS*
averaging kernels.
Auto- and Cross-Correlation
S x 2 = E{(x 2 - x 2 )(x 2 - x 2 )T }
S12 = E{(x1 - x1 )(x 2 - x2 )T }
S12 matrix 0-3 h bin
72 ev prof
Temperature
S12 matrix 0-6 h bin
72 ev prof
Temperature
3 hours
Sx (auto-covariance) matrix
72 ev prof
Auto-Covariance
Sx2
Temperature
2
S12 matrix 0-12 h bin
72 ev prof
Temperature
6 hours
12 hours
K2
1000
1000
1000
100
1000
1000
Log P
Log P
S12 matrix 0-24 h bin
72 ev prof
Temperature
5 days
100
Log P
Log P
Log P
Log P
1000
S12 matrix 0-120 h (5 days) bin
72 ev prof
Temperature
100
1000
1000
100
100
Log P
1000
1000
Log P
4 days
100
1000
100
S12 matrix 0-96 h (4 days) bin
72 ev prof
Temperature
2 days
100
1000
Log P
S12 matrix 0-48 h bin
72 ev prof
Temperature
24 hours
1000
100
Log P
100
100
Log P
100
Log P
Log P
100
Log P
-10
-5
0
5
10
15
20
25
30
100
1000
100
Log P
1000
100
Log P
1000
Non-Coincidence Error
δx1 = Bδx 2 + ξ
Variation at
validated point
Uncorrelated/Residual error
Un-Correlated
Correlated
cov(x 2 ,ξ) = 0
T
S x1 = BS x2 B + S ξ
3 hours
TT
S ξ S= S=S
BS
B
-BSB
x2
ξ x1 x1
12 hours
6 hours
100
100
24 hours
100
100
1000
1000
100
1000
100
1000
Log P
Log P
Log P
-2
0
2
4
6
8
10
12
Log P
K2
1000
1000
100
1000
Log P
Log P
100
1000
Log P
Log P
100
100
200
200
200
200
300
300
Log P
100
Log P
100
Log P
Log P
S ξ a n d S x 2 ( d ia g o n a ls )
300
300
400
400
400
400
500
500
500
500
600
600
600
Sx2
Sξ
700
800
900
1000
0
1
2
3
4
K
5
6
700
800
900
1000
7
Sx2
Sξ
0
1
2
3
4
K
5
6
700
800
900
1000
7
Sξ
0
1
600
Sx2
2
3
4
K
5
6
7
Sx2
Sξ
700
800
900
1000
0
1
2
3
4
K
5
6
7
Non-Coincidence Error
(continued)
AIRS global estimate 0.8 K( ± 3 h o u r , ± 1 0 0 k m ) σ = (0.22τ + 0.14) K
ξ
Chahine et al., 2006
Non-Coincidence Error
400 - 800 mb
Non-Coincidence Error
4
100
Non-Coincidence Error
400 - 800 mb
3.0
2.5
400
500
RMS Error σξ (K)
Log P
3 hors
6 hors
12 hors
24 hors
2 days
5 days
300
RMS Error σξ (K)
3
200
2.0
1.5
2
1.0
1
600
0.5
700
800
900
1000
0.0
0
0
1
2
3
4
RMS Error σξ (K)
5
6
0
20
40
60
80
Time bin τ (h)
100
120
0
3
6
9
Time bin τ (h)
12
Averaging Kernels
Averaging Kernels for Temperature Profile
AIRS Spectral Channels, ILS, and SNR
Optimal Estimation (Clive Rodgers)
16
100
14
12
200
Log P
H (km)
10
8
300
400
6
500
4
600
700
800
900
1000
2
0
-0.2
0.0
0.2
0.4
0.6
Averaging Kernel
0.8
1.0
-0.2
0.0
0.2
0.4
0.6
Averaging Kernel
0.8
1.0
“Satellite Retrievals” vs. Radiosondes
RMS
Non-Coincidence 6 hours
&
Smoothing Error
Non-Coincidence 6 hours Error
δxˆ = x1 - Bx 2
δxˆ = A1x1 - A1Bx 2
100
Log P
Log P
100
Square Root
Diagonals - RMS
1000
100
1000
1000
100
Log P
1000
Log P
100
Log P
200
300
400
500
600
700
800
900
1000
0
1
2
K
3
4
5
6
Smoothing & Non-Coincidence RMS
Non-Coincidence RMS
Auto-Covariance RMS
Conclusions
•Non-Coincidence Error
analysis is applicable to
Radiances (SDR) and retrievals (EDR) assessment.
•EDRAM provides scientific basis and practical tool for
accurate comparison of atmospheric profiles of different vertical
resolution and taken at different times and locations.
•EDRAM estimates retrieval bias and noise as well as
statistical significance of the estimates based on the comparison.
•EDRAM can be used for evaluation of a satellite EDR for
Earth System and Climate studies by accurately referencing them
to other data sets with known accuracy and precision.