1. GIFTS clear sky fast model, its adjoint, & the neglected reflected term
MURI Hyperspectral Workshop
Madison WI, June 7
bob knuteson, leslie moy, dave tobin,
paul van delst, hal woolf

2. Outline of Talk Fast Model: Development & Status
Tangent Linear, Adjoint: Development & Status
Surface Reflected Term: Work in Progress

3. Compute monochromatic
Fast Model Production Flowchart:
Profile
Database
Fixed Gas
Amounts
Spectral line
parameters
Lineshapes
& Continua
Layering, l
Compute monochromatic
layer-to-space
transmittances
Reduce to sensor’s
spectral resolution
Effective Layer
Optical Depths, keff
Convolved
Layer-to-Space
Transmittances, tz (l)
Fast Model
Predictors, Qi
Fast Model
Regressions
Fast Model
Coefficients, ci

4. RMS Error: water lines Red=before, Blue=after
Why the improvement?
Mainly from SVD regression & Optical Depth Weighting.

5. RMS Error: water continuum Red=before, Blue=after
Why the improvement?
Mainly from regressing nadir only Optical Depths and
applying constant factors to off-nadir values.

6. MURI model w/ new regressions OPTRAN, AIRS 281 channel set
GIFTS
OSS RMS upper limit*
Dependent Set Statistics: RMS(LBL-FM)
Yr 2002 model
MURI version
MURI model w/ new regressions
AIRS model c/o L. Strow, UMBC
OSS model c/o Xu Liu, AER, Inc.
OPTRAN, AIRS 281 channel set
c/o PVD

7. Profile of temperature,
User Input:
User Output:
Profile of temperature,
dry gases, water vapor
at 101 levels
Profile perturbation
of temperature, ozone,
water vapor at 101 levels
Use to adjust
initial profile
Forward Model:
Adjoint Model:
Layer.m - convert 101 level values to
100 layer values
Predictor.m - convert layer values to
predictor values
Calc_Trans.m - using predictors and
coefficients calculate
level to space transmittance
Trans_to_Rad.m - calculate radiance
Layer_AD.m - layer to level sensitivities
Predictor_AD.m -
level to predictor sensitivities
Calc_Trans_AD.m - predictor to
transmittance sensitivities
Trans_to_Rad_AD.m - transmittance to
radiance sensitivities
Ill posed
User Output:
User Input:
Compare to
observations
Radiance Spectrum
Radiance Spectrum
perturbation

8. Simple Example: One Line Forward Model
Forward (FWD) model. The FWD operator maps the input state vector, X, to the model prediction, Y, e.g. for predictor #11:
Tangent-linear (TL) model. Linearization of the forward model about Xb, the TL operator maps changes in the input state vector, X, to changes in the model prediction, Y,
iteration
Or, in matrix form:

9. TL testing for Dry Predictor #6 (T2) vs Temp at layer 44
TL testing for Dry Predictor #6 (T2) vs Temp at layer 44. * TL results must be linear. * TL must equal (FWD-To) at dT=0.
TL results = blue, FWD-T0 results = red
Difference between TL and FWD
Input Temperature at Layer 44 were varied 25%.

10. TL testing for Dry Predictor #6 vs Temp at all layers
TL testing for Dry Predictor #6 vs Temp at all layers. Similar plots made for each subroutine’s variables.
D(dry.pred#6)
Layer no.
D(temp), %

11. Adjoint (AD) model. The AD operator maps in the reverse direction where for a given perturbation in the model prediction, Y, the change in the state vector, X, can be determined. The AD operator is the transpose of the TL operator. Using the example for predictor #11 in matrix form,
Expanding this into separate equations:

12. Adjoint code testing for Dry Predictor #6 vs Temperature layer
Adjoint code testing for Dry Predictor #6 vs Temperature layer. AD - TLt residual must be zero. Similar plots are produced for every subroutine’s variables. x
10 -18
AD - TLt residual
Output variable layer
Input variable layer

13. Clear Sky Top of Atmosphere Radiance
I TOA = I atmos I surf emiss I surf reflect
= I atmos + Ttoa B(tempsurf ) surf Ttoa Fluxsurf Reflectivity
I atmos
I surf emiss
I surf reflect
r, r
current fast model
we write the expression
more explicitly below
I surf reflect = Ttoa   I(i,i) cos(i) sin(i) d(i) d(i)  BDRF(r,r: i,i)
This Term is often ignored because Refl < 10%. IF the term is calculated accurately enough, it can be exploited to derive surf and hence Tsurf

14. Approximations made & Their Associated Errors
I surf reflect = Ttoa   I(i,i) cos(i) sin(i) d(i) d(i)  BDRF(r,r: i,i)
Approx.1: Lambertian surface (reflection is independent of incident angle):
BDRF = R(r,r) = 1- surf (r,r)
Approx.2: Low Order Gaussian Quadrature technique for calculating flux
# quadrature points needed?
which table to use? (Abramowitz and Stegun, 1972)
Approx.3: Resolution Reduction
SRF  {Ttoa  2   I( )  d }  {SRF  Ttoa }  {SRF  2   I( )  d }
Slow, smoothing varying radiance as a function of cos theta
Approx.4: Calculating Downwelling Radiance from Upwelling Fast Model
SRF  I() (using LBLRTM) 
(T GIFTS layer to space convolved) B(templayer)

15. Downwelling flux = 0 0 I(,) cos() sin() d d
Expanding on Approx. 2: 2
/2
Downwelling flux = 0 0 I(,) cos() sin() d d
= 2 0 I() cos() sin() d
substituting  = cos (),
= 2 0  I() d
Diffusivity approximation, Low Order Gaussian Quadrature technique
0  I() d =  wi I(i ) p.921, Abramozwitz & Stegun, 1972
n=1,  0.5 I(=48)
n=2,  0.2 I(1=69°) I(2 =32°)
In contrast to the 2-stream model application:
-1 I() d =  wi I(i ) p.916, Abramozwitz & Stegun, 1972
n=1,  I(=54.73)
/2 1 1 n
i=1
Azmuthially symmetric to get rid of the phi 1 n
i=1

16. Approx 2: Error in Gaussian Quadrature Approximations
(difference from using 4 points)
Using two points
Difference, W/(cm2 cm-1 ster)
Differences in window regions
Using one point

17. Approx. 3: Convolution Error
Product of Convolution Minus Convolution of Products
Difference, W/(cm2 cm-1 ster)

18. Errors from 1 Point Gaussian Quad & Convolution Approximations
1 point Gauss. Qaud.
Both approx.
Why convolution error is a positive bias?

19. Errors from 2 Point Gaussian Quad & Convolution Approximations
Both approx.
2 point Gauss. Qaud.

20. Approx. 4: Using Fast Model Upwelling Level-2-space Transmissivity to calculate Downwelling Radiance
TOA radiance
rad = rad (ba+bb) (1-Tb/ Ta ) Ta
layer trans
level 2 space
layer radiance emission Ta Tb
BOA radiance
rad = rad (ba+bb) (1-Tb/ Ta ) (T1/ Tb) T1
level 2 ground
Close up of the window region Differences
lblrtm - From Tran(lev2space)
Comparison of Downwelling Radiance
Built in directionality
from Tran(lev2space)
from lblrtm

21. Accomplishments: • Coefficients promulgated 2003.
Reproduce and Upgrade existing GIFTS/IOMI Fast Model
• Coefficients promulgated 2003.
• Greatly improved the dependent set statistics (esp. water vapor).
• Water continuum regression made at nadir and applied to all angles.
• SVD regression and optical depth weighting incorporated.
• Written in flexible code with visualization capabilities. Under CVS control.
Write the Corresponding Tangent Linear and Adjoint Code
• Tested to machine precision accuracy.
• User friendly “wrap-around” code complete.
• Transferred code to FSU.
Investigate Surface Reflected Radiance
• Great improvement with two point Gaussian Quadrature (over 1 point).
• Convolution order causes large errors – may be overcome with regression algorithm?
• Depending on the application (micro-window or on/off line) using upwelling transmissivity for
downwelling radiance may be reasonable.