GIFTS clear sky fast model, its adjoint,

& the neglected reflected term

MURI Hyperspectral WorkshopMadison WI, 2005 June 7

bob knuteson, leslie moy, dave tobin,

paul van delst, hal woolf

Outline of Talk

• Fast Model: Development & Status

• Tangent Linear, Adjoint: Development & Status

• Surface Reflected Term: Work in Progress

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

RMS Error: water lines Red=before, Blue=after

Why the improvement? Mainly from SVD regression & Optical Depth Weighting.

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.

Dependent Set Statistics: RMS(LBL-FM)

Yr 2002 model MURI version MURI model w/ new regressions

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, dry gases, 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. 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,

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

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

10 -18x

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

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

I surf reflect = Ttoa I(i,i) cos(i) sin(i) d(i) d(i) BDRF(r,r: i,i)

I atmos

I surf emiss

I surf reflect

r, r

we write the expression more explicitly below

current fast model

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.4: Calculating Downwelling Radiance from Upwelling Fast Model SRF I() (using LBLRTM) (T GIFTS layer to space convolved) B(templayer)

Approx.3: Resolution Reduction

SRF {Ttoa 2 I( ) d } {SRF Ttoa } {SRF 2 I( ) d }

Downwelling flux = 0 0 I(,) cos() sin() d d

= 2 0 I() cos() sin() dsubstituting = 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°) + 0.3 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 /2

/2

1

1

i=1

n

1

i=1

n

Expanding on Approx. 2:

Using two points

Using one point

Dif

f er e

n ce ,

W/ (

c m2 cm

-1 st

er)

Approx 2: Error in Gaussian Quadrature Approximations(difference from using 4 points)

Approx. 3: Convolution Error Product of Convolution Minus Convolution of Products

Dif

f er e

n ce ,

W/ (

c m2 cm

-1 st

er)

1 point Gauss. Qaud. Both approx.

ConvolutionError

Errors from 1 Point Gaussian Quad & Convolution Approximations

2 point Gauss. Qaud. Both approx.

ConvolutionError

Errors from 2 Point Gaussian Quad & Convolution Approximations

Approx. 4: Using Fast Model Upwelling Level-2-space Transmissivity

to calculate Downwelling Radiance

Comparison of Downwelling Radiance

from lblrtm

from Tran(lev2space)

Close up of the window region Differenceslblrtm - From Tran(lev2space)

TOA radiancerad = rad + 0.5 (ba+bb) (1-Tb/ Ta ) Ta

BOA radiancerad = rad + 0.5 (ba+bb) (1-Tb/ Ta ) (T1/ Tb)

layer trans level 2 space layer radiance emission

level 2 ground

Tb

Ta

T1

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.

Accomplishments:

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.