# Estimation principle and forward model

Extracted from Chimot, J., Global mapping of atmospheric composition from space – Retrieving aerosol height and tropospheric NO2 from OMI, PhD book, Delft University of Technology (TU Delft), The Royal Netherlands Meteorological Institute (KNMI), July 2018.

Since an atmospheric satellite measurement is a light spectrum containing the information on our atmospheric composition, it is necessary to convert it into a geophysical parameter: e.g. estimate the abundance of pollutant gases in the troposphere. Such a process is commonly named retrieval of an atmospheric state x (i.e. state vector) from a measurement y.

The atmospheric retrieval requires a forward model F to describe the dependency between x and y:

y = F(x) + ε + ∆F(x),

∆F(x) is the forward model error, and ε is the measurement error (noise). In practice, the retrieval does not represent the true state vector x but rather a summary x ′ of our best understanding of the physics that explains most of the measurement. x ′ may include more than one geophysical parameter under consideration (e.g. simultaneous retrieval of NO2 and aerosols). Furthermore, to close the budget of the radiance spectrum y, other parameters must be considered in the forward model: pressure, temperature, surface properties but also the instrument characteristics such as the geometry angles. They form then the set (or best of estimate) of forward model pa- rameters b. It is important to notice that b is not fitted. x in the equation above becomes then x = (x′, b), such that:

y = F(x′, b) + ∆F(b) + ε,

The forward model error may combine different sources: the model inaccuracy in itself ∆F, and/or the gap between the assumed b′ and the true atmospheric state parameters b that are not fitted but transported through the forward model dF/db (b −b′). y becomes then (Eskes and Boersma, 2003):

y = F(x′, b′) + ε + ∆F + dF/db (b − b′) + ε.