# Physical Deterministic (PD) Sea Surface Temperature (SST)

## Table of Contents

## 1 - Product Summary

Unlike stochastic retrievals, where the coefficients or error covariances have been derived for “average” conditions from a (usually wide) set of measurement instances, a physical deterministic (PD) retrieval scheme obtains a mathematical inversion on single pixel measurements. The main advantage of the deterministic method is that the confidence of error estimation at the individual pixel is high, and error is not treated as definite information. Some of the advantages of PDSST algorithm over prevalent methodologies are:

- Substantially reduces SST error by adapting to “local” atmospheric information through NWP (Numerical Weather Prediction) data.
- Ensures the balance between information content of SST retrieval and retrieval noise and sensitivity is close to 1 when the initial guess (IG) is “far” from truth.
- Aerosols can be explicitly included in the forward modeling calculations as well as in the retrieval vector, to compensate for the significant source of regional error.
- Straightforward option to include both multiple channels and multi-parameter to reduce uncertainty/error in SST.
- Provides meaningful pixel-level estimates of error and information content analysis.
- Automatically adapts to improvements in RT modelling, instrument calibration, improvements in cloud detection, etc., due to case-by-case dynamic estimates of noise in the “measurement minus forward model”.

Like any physical retrieval method, the quality of PDSST retrieval is susceptible to errors in the fast- radiative transfer modeling (FRTM) and NWP data (except retrieved parameters), but these errors should be continuously reduced due to improvement of both FRTM capability and NWP data quality. Moreover, using increased numbers of both measurements (channels) and retrieved parameters in simultaneous PDSST retrieval scheme is an alternative means of reducing the SST retrieval error.

## 2 - Algorithm Description

The following sections will discuss the derivation of the PD algorithm, beginning from the LS formulation.

### Least squares (LS) method

The basic form of the inverse method conventionally stated for a nonlinear problem using the residual ($ Δy_ẟ$) between observation and model is:

$$ Δy_ẟ = ΚΔx $$

where $Δy_ẟ = yΔ − f(x_{ig}$), is the model minus observation (a.k.a. residual), x is a retrieval model state vector (i.e. parameters of interest, primarily SST, water vapor, etc. – variables that can significantly affect the observed brightness temperatures), ; x_{ig} is the initial guess of the state vector, $Δx$ is the difference between the IG for the retrieval model state and the “true” state, $y_ẟ = y + Δy$ is a vector of instrument channel BTs assuming $ẟy$ is the measurement error, $f(x_{ig}$) is the calculated BTs from RT modeling using the IG of forward model state (usually atmospheric profiles of water vapor and temperature, and a surface temperature, etc.) as an input and $Κ$ is the Jacobian (partial derivatives of channel simulated BTs with respect to retrieved parameters). The deterministic form of the least- squares (LS) solution of the linearized physical model obtained from Eqn. (1) is:

$$x_{ls} = x_{ig} + (Κ^TΚ)^{-1}Κ^T Δy_ẟ$$

Additional methods were used to derive the final PDSST method. To read more about this algorithm, refer to the paper listed below.

## References

A deterministic method for profiles retrievals from hyperspectral satellite measurements. IEEE Trans. on Geo-science & Remote Sensing 2016, 54(10), in press. DOI: 10.1109/TGRS.2016.2565722.

(2016)