Clear Sky Forward Model & Its Adjoint Model

MURI Review April 27, 2004

leslie moy, dave tobin, paul van delst, hal woolf

Reproduce and Upgrade existing GIFTS/IOMI Fast Model

• Coefficients promulgated 2003.

• Greatly improved the dependent set statistics (esp. water vapor).

• SVD regression and optical depth weighting incorporated.

• Written in flexible code with visualization capabilities. Under CVS control.

Accomplishments:

Write the Corresponding Tangent Linear and Adjoint Code

• Tested to machine precision accuracy.

• User friendly “wrap-around” code complete.

Fast Model Production Flowchart:

Lineshapes& Continua

Spectral lineparameters

Layering, l

Fast Model

Predictors, Qi

Reduce to sensor’sspectral resolution

Fast Model

Coefficients, ci

Compute monochromaticlayer-to-spacetransmittances

ConvolvedLayer-to-Space

Transmittances, z (l)

Fast ModelRegressions

Effective Layer

Optical Depths, keff

ProfileDatabase

Fixed GasAmounts

keff = -ln (eff ) = i=1:N ci Qi

Dependent Set Statistics: RMS(LBL-FM)

current model MURI version MURI model w/ OD weighted SVD

AIRS model c/o L. Strow, UMBC

------- GIFTS NeDT@296K------- OSS RMS upper limit*

OSS model c/o Xu Liu, AER, Inc. OPTRAN, AIRS 281 channel setc/o PVD

User Input:

Profile of temperature, ozone, water vapor

at 101 levels

Forward 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 calculatelevel to space transmittance

Trans_to_Rad.m - calculate radiance

User Output:

Radiance Spectrum

User Output:

Profile perturbationof temperature, ozone,

water vapor at 101 levels

Adjoint Model:

Layer_AD.m - layer to level sensitivitiesPredictor_AD.m -

level to predictor sensitivitiesCalc_Trans_AD.m - predictor to

transmittance sensitivitiesTrans_to_Rad_AD.m - transmittance to

radiance sensitivities

User Input:

Radiance Spectrumperturbation

Compare to observations

Use to adjustinitial profile

• Forward (FWD) model. The FWD operator maps the input state vector, X, to the model prediction, Y, e.g. for predictor #11:

211 T

WP

TTW

WT

TT

PW

W

PP

32

111111

21

Tangent-linear (TL) model. Linearisation of the forward model about Xb, the TL operator maps changes in the input state vector, X, to changes in the model prediction, Y,

Or, in matrix form:

1

1132

21

11

100

010

0

n

T

W

T

n

T

W

P

T

W

P

Simple Example: One Line Forward Model

Using the same procedure as in the Simple Model, build up the Tangent Linear Model.

Forward Model:

Layer.m - input: 101 level values of T,w,oz output: 100 layer values

Predictor.m - input: layer valuesoutput: predictor values

Calc_Trans.m - input: predictors, and coefficientsoutput: transmittances

Trans_to_Rad.m - input: transmittancesoutput: TOA radiance

TL Model:

Layer_TL.m - input: d(level value)output: d(layer value)

Predictor_TL.m - input d(layer value),output: d(predictor),

Calc_Trans_TL.m - input: d(predictor) output: d(transmittance)

Trans_to_Rad_TL.m - input: d(transmittance)output: d(radiance)

Testing subroutines:A range of Perturbations are added to a baseline input value. Plus&Minus 25% The Forward output (less the baseline value) is compared to the TL output.

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.

Input Temperature at Layer 44 were varied 25%.

TL results = blue, FWD-T0 results = red

Difference between TL and FWD

TL testing for Dry Predictor #6 vs Temp at all layers.Similar plots made for each subroutine’s variables.

Layer no.D(temp), %

D(d

ry. p

red#

6)

• 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,

0

1

2

111

*

*11

*2

1*

*11

*3

1*

n

nnn

nnn

P

WPT

W

TPT

WT

Expanding this into separate equations:

n

T

WT

n

T

W

P

T

W

P

*

*

11*

32

21

1

*

*

11*

10

01

000

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.

Output variable layer Input variable layer

AD

- T

Lt r

esid

ual

Adjoint for US Standard Profile, d(radiance)/d(ozone)

Wavenumber, cm-1

Pre

ssur

e, -

mba

r

Adjoint for US Standard Profile, d(radiance)/d(water vapor)P

ress

ure,

-m

bar

Wavenumber, cm-1

‘Reproduce and Upgrade existing GIFTS Fast Model’

• Increase the number and quality of training profiles (48); extend satellite zenith angle range; further improve the dependent and independent set statistics.

• Breakout other gases.

• Steer research direction based on user feedback.

Future Goals:

Corresponding Tangent Linear and Adjoint Code

• Improve code for speed and ease of use. Convert to a more efficient programming language.

• Investigator testing of code.

Adjoint code can be used for sensitivity analysis.d(Dry Predictor #6) / d(layer Temperature).

Input variable layerOutput variable layer

Sens

itiv

ity,

d(P

6)/d

(T)