The OEM (Optimal Estimation Method) is generally used for retrieval of species with deep absorption lines, such as O₃. The advantage of the OEM is that all information can be extracted from the measurements, and that a complete description of the error propagation is given. The disadvantages are that OEM tends to be computationally expensive due to online radiative transfer, and that the use of a priori data makes the products more difficult to use.

In the OEM the final retrieval result is based on the measurements as well as on a priori data. The a priori data are needed as regularisation because the inversion is ill-posed, meaning that the unknowns exceed the independent information in the measurements. For the OEM, three elements are needed:

• a forward model that describes the satellite measurements for a given atmospheric state
• the a priori profiles and its error-covariance matrices
• an inversion scheme.

Forward Model

The forward model consists of a Radiative Transfer Model (RTM) and an instrument model. The radiative transfer model computes the radiances at TOA given a certain atmospheric state given by the state vector. This state vector contains the parameters that will be retrieved by the OEM, for example the ozone profile, cloud parameters and surface albedo. The RTM computes not only the radiance but also the derivative of the radiances to the state vector elements. The instrument model simulates the Level-1B as measured by the instrument. The slit function is an important part of the instrument model, as it is used to degrade the high-resolution radiances to the spectral resolution of the instrument. The output of the forward model is a simulated measured reflectance that can be compared with the measured reflectance.

A Priori Information

In the OEM, a priori information is used to regulate the ill-posed retrieval problem, making use of existing knowledge of the atmosphere and making the solution a physically meaningful result. For the OEM, an a priori state vector and co-variance matrix is needed. The covariance matrix describes the estimated uncertainty of the a priori state vector elements, as well as the correlation between the state vector elements.

Inversion Method

The inversion method computes the retrieval results using the forward model and the a priori error covariance matrix. In OEM the cost function or ?² to be minimized is given by:

?² = [y – F(x)]^T S_y^-1[y-F(x)] + (x-x_a)^TS_a^-1(x-x_a)

• y is the vector of measured reflectance containing values for the different wavelengths
• F(x) is the vector of calculated reflectances also called the forward model;
• x is the state vector containing the parameters that are to be retrieved (among them the O₃ profile)
• S_y is the error covariance matrix of the measurement
• x_a is the a priori state vector
• S_a the error covariance matrix of the a priori state vector.

Solving the equation results in the a posteriori state vector (containing the retrieved O₃ profile), its associated covariance matrix from which diagnostics like the degrees of freedom per signal (DFS) are derived.