A physical basis of GOES SST retrieval Prabhat Koner Andy Harris Jon 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 -> (K T S e -1 K+S a -1 ) -1 K T S e -1 ΔY  ML -> (K T S e -1 K) -1 K T S e -1 ΔY  RTLS -> (K T K+λR)K T ΔY (Regularized Total Least square) 3.9 & 10.7 μm -> 2 channel 3.9, 10.7 & 13.3 μm -> 3 channel Two variables of SST & TCWV

Nomenclature X -> state space parameters Y -> Measurements ΔY -> Residual (Measurements – forward model output) ΔX -> Update of state space variables δ Y -> Measurement noise δ X -> Retrieval error S a -> a priori/ background covariance S e -> 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=C 0 +∑C i T i ; 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 observation A priori First we will see Pros-con of OEM

Global SST distributions match quite well, but… …large differences between 1 st 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 =(K T S e -1 K+S a -1 ) -1 K T S e -1 ΔY Information comes from measurement and a priori covariance. COV= (√S a -1 ) -1 ΔY S a = By accident the choice S a, COV retrieval is just adding residual of 3.9 μm channel with FG. Covariance has no physical meaning Add measurement increase noise into retrievals *TCWV 2 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 δ x <= κ (K) δy K=randn(2); (κ(K) = 118) x=randn(2,1); [ ] y=K*x For ii=1:100 error(:,ii)=0.01*randn(2,1); x rtv (:,ii)=K -1 (y+error(:,ii)); End Apart from model physics and measurement errors, the error due to κ(K) plays a role. (K T S e -1 K+S a -1 ) -1 K T S e -1 ΔY 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 ECMWF It is not an instrument calibration problem or bias Fast Forward Model Error Need improvement of fast forward model.

Simulated Retrieval δ x<= κ (K) δy

Quality Flagging Algorithm K=U∑V T -> Singular vector decomposition V T ∑ 1 -1 U ΔY -> Principle Component solution V T ∑ n -1 U 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. (Y δ -KX) T S e -1 (Y δ -KX) + (X-X a ) T S a -1 (X-X a ) does not apply if Y ≠ KX

Few Definitions Model: a simplified description of how the ‘real world’ process behaves Forward 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 output Inverse 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” “Absorption coefficient”

Forward Model Spectral intensity observed at L “Absorption coefficient” “number density” “Cross section” “Volume Mixing Ratio” “Line shape”“Line strength” GENSPEC: line shape: Voigt: line Strength: HITRAN 2004