A physical basis of GOES SST retrieval

Prabhat KonerAndy HarrisJon Mittaz

Eileen Maturi

Summary

One year match up database ( Buoy & Satellite) has been analyzed to understand the retrieval problem of satellite measurement.

Various forms of errors and ambiguities for SST retrieval will be discuused.

110,000 night only data (GOES12 & Buoy) has been considered from this database to compare results with different choices of solution processes: RGR -> Regression OEM -> (KTSe

-1K+Sa-1)-1KTSe

-1 ΔY ML -> (KTSe

-1K)-1KTSe-1ΔY

RTLS -> (KTK+λR)KT ΔY (Regularized Total Least square)3.9 & 10.7 μm -> 2 channel3.9, 10.7 & 13.3 μm -> 3 channelTwo variables of SST & TCWV

Nomenclature

X -> state space parameters Y -> Measurements ΔY -> Residual (Measurements – forward model output) ΔX -> Update of state space variables δY -> Measurement noise δX -> Retrieval error Sa -> a priori/ background covariance Se -> measurement error covariance K -> Jacobian (derivative of forward model) κ(K) -> condition number of Jacobian (highest singular value/lowest singular

value) T -> Brightness temperature in satellite output BT -> Calculated Brightness temperature SSTg-> Sea Surface Temperature First Guess SSTb -> Sea Surface Temperature from Buoy measurement SSTrgr -> Sea Surface Temperature retrieval using regression SD -> Standard Deviation RMSE -> Root Mean Square Error.

SSTrgr=C0+∑Ci Ti ; C=a+b {sec(θ)-1}+…

Coefficients are derived from in situ buoy data or L4 bulk SST.

Validate with Buoy data

Bulk SST from Satellite measurement?

Historical Regressed based SST retrieval

Alternately we can use radiative transfer physics inverse model

Statistical (OEM) Deterministic (TLS)

Data & Measurement both uncertain

It can only estimate a posteriori probability density (parameters x’: “Best Guess”) by calculating Maximum likelihood P(x|y).

Measurement only uncertain Jacobian error is condidered in

cost funtion minimization. Retrieval in pixel level

Physical Retrieval

A posteriori

observationA priori

First we will see Pros-con of OEM

Global SST distributions match quite well, but…

…large differences between 1st guess and buoy SST are real

Shortcomings of OEM

A-priori based cloud screening algorithm (CSA) in place to constrain in image data

Paradox of OEM output

OEM=(KTSe-1K+Sa

-1)-1KTSe-1ΔY

Information comes from measurement and a priori covariance.

COV= (√Sa-1)-1ΔY

Sa=

By accident the choice Sa, COV retrieval is just adding residual of 3.9 μm channel with FG.

Covariance has no physical meaning

Add measurement increase noise into retrievals

12 0

0 0.15*TCWV2

To further investigate this issue, we calculate information content

H=-0.5 ln(I-AVK)Two measurement cannot produce more than 2 pieces of information.

Big Question?

OEM may be valid for linear problem, not applicable for inherently nonlinear RTE.

Condition number of Jacobian

The condition number of jacobian of most real life problem is high. Yields

δx <= κ (K) δy

K=randn(2); (κ(K) = 118) x=randn(2,1); [-1.35 0.97]

y=K*x For ii=1:100 error(:,ii)=0.01*randn(2,1); xrtv(:,ii)=K-1(y+error(:,ii)); End

Apart from model physics and measurement errors, the error due to κ(K) plays a role.

(KTSe-1K+Sa

-1)-1KTSe-1ΔY

Remedies: Reducing the condition number of inverted matrix

Regularization, constraints, scaling, weighting etc.

We use RTLS method. Uniquely solved in simulation

study.

Difficulty solve using satellite data, drive to further investigation

Online monitor ECMWFIt is not an instrument calibration problem or bias

Fast Forward Model Error

Need improvement of fast forward model.

Simulated Retrieval

δx<= κ (K) δy

Quality Flagging Algorithm

K=U∑VT -> Singular vector decompositionVT ∑1 -1U ΔY ->Principle Component solutionVT ∑n -1U ΔY ->Lowest singular value solution

Lowest singular value solution increases error in retrieval where measurement noise is high. Difference of two solutions is able identified bad retrieval.

Comparative results

Same data and model, results are different due to choice of solution methods

Results are based on single iteration. Second iteration may further improve the results for physical retrieval.

Conclusions

Radiative transfer can be successful used in retrieval of SST from satellite data if we pursue rigorous physics (accurate RTM) and mathematics

There is an ambiguity in the cost function minimization for nonlinear problems, i.e.

(Yδ-KX)TSe-1(Yδ-KX) + (X-Xa)TSa

-1(X-Xa)

does not apply if Y ≠ KX

Few DefinitionsModel: a simplified description of how the ‘real

world’ process behavesForward Model : the set of rules (mathematical

functions) that define the behaviour of the process (e.g. a set equation)Forward model of remote sensing Well understood but mathematically complex function Analytical derivative is almost impossible. Stable: On some appropriate scale a small change in

the input produces a small change in the outputInverse Model: quantities within the model structure

that need to be quantified from observation data Inverse model of remote sensing is ill-posed Is the solution we find unique? Observational, numerical and model errors often

cause the inverse problem to be unstable: a small change in the input produces a large change in the output

Forward Model

Absorption term

Emission term

Intensity of the background source

“Transmittance”between 0 and L

Spectral intensity observed at L

“Transmittance”between l and L

“Optical depth” L

xdxxLx ')'(),(

),0(

0),(),0( )()0()(

L LxL dexTBeILI

“Absorption coefficient”

Forward Model

Spectral intensity observed at L

“Absorption coefficient”

molec

m

lmlmkll1

),(),()(

“number density”

“Cross section”

“Volume Mixing Ratio”

lines

illiml pTimATimSlmk

1, ),,,,(),,(),(

“Line shape”“Line strength”

N

l

N

lk

kkllN

l

ll eeTlBeILI1 11

1,0

GENSPEC: line shape: Voigt: line Strength: HITRAN 2004