As a validation tool for atmospheric water vapor retrieval we selected the AMSU-B total water vapor product [Melsheimer and Heygster, 2008].
This method uses passive microwave sounding channels which have similar surface emissivity but different atmospheric absorption characteristics. It uses three channels around the water absorption line at 183 GHz to detect the low water vapour values typical for the atmosphere over sea ice in the Central Arctic. A special characteristic of this retrieval product is that uncertainty increases with increasing water vapour values from about 1 mm for retrieval values below 2 mm to an uncertainty around 3 mm for retrieval values around 14 mm. For scenes with atmospheric water vapor above 15 mm the method does not retrieve anything as all channels around the 183 GHz frequency become saturated.
Very important to the comparison effort with OEM is that the AMSU-B TWV retrieval was specially designed to function over sea ice which makes it ideal for testing the reliability of OEM atmospheric retrieval in the Arctic. For the comparison only those pixels were selected where both the optimal estimation retrieval and the AMSU-B product have valid values which means that only TWV values between 0 and 15 mm were included.
4 | Method
4.1 Optimal estimation method (maximum a posteriori solu-tion)
In this chapter the basic retrieval method is described starting from the theory of opti-mal estimation and concluding with the adaptations necessary for the specific inversion application that is the topic of this thesis. Testing and establishing a specific set-up of the retrieval method will be addressed in more detail in Section 5.
Following the basic radiative transfer theory described in Section 2 we can approximate the brightness temperatures measured at the top of the atmosphere by a passive microwave radiometer as a function of a number of geophysical parameters.
TA= F (p) (4.1)
where p is the state vector containing both surface parameters such as SST and atmospheric profiles of temperature, humidity and liquid water, surface sea ice cover and WSP vectors.
F is the forward operator (Section 4.2) that maps the functional relationship between the state vector parameters and the observed brightness temperatures. The measured brightness temperature contains information about all of these parameters. Retrieving relevant data about all of them is an under-constrained problem because several of the state vector parameters are continuous, such as temperature or pressure profiles, while the number of brightness temperature measurements is finite [Pedersen, 1994].
In order to derive the geophysical parameters from the measurements we need a forward model that can describe the measurements in terms of the required geophysical parameters and then invert it. The forward model might not be easily invertible or the errors in the
Chapter 4. Method
measurements together with errors in the forward model simulation can make it impossible to find a solution.
The inverse problem consists of inverting a known equation which relates thermal radia-tion to the state of the atmosphere and surface in order to express atmospheric parameters in terms of the measured radiation. This sort of problem is ill-posed because there is no mathematically unique solution for it. From this we follow with the estimation prob-lem which means finding the appropriate criteria for choosing the solution that is most consistent with the measurements [Rodgers, 1976].
The estimation method used here follows [Rodgers, 2000] and is called the maximum a posteriori. Through an iterative process the prior state vector pn is nudged to a new state pn+1 so that a cost function is minimized. This cost function balances the penalty of departing from the background state vector (Pa) with the penalty of deviating from the observed brightness temperatures. This ensures that at each iteration step the state vector is within a realistic distance of the background values while trying to match the measure-ments within a reasonable precision range. This process involves two sets of constraints.
We would expect to find reasonable values for the state vectors spaced within the natural variability constraints around the the background position. These constraints for the state vector space are represented by a covariance matrix with the individual parameter vari-ances on the diagonal and inter-parameter covarivari-ances in the off diagonal elements. This is called the a priori covariance matrix because it represents the level of knowledge about the geophysical state before the measurements are made. The background values represent long term means from climatological or other sources which together with the individual parameter variances in the background covariance matrix Saconstrain the retrieved param-eters to physically realistic values. The diagonal elements of the corresponding covariance matrix should represent the natural variability of each parameter in order to allow for a consistent retrieval, but this information is limited by the quality of the prior information we have about the climate system. In order to speed up the iterative process for finding the optimal solution, the method also uses a first guess point that serves as initialization state. The first guess state can also come from a background state but that is not the only source Rodgers [2000]. According to Pedersen [1994] the retrieval accuracy is influenced by the quality of the first guess data. A similar constraint is used for the observation space.
A covariance matrix contains the individual channel variances that constrain the forward model simulated brightness temperature to a reasonable accuracy. This level of accuracy is chosen based on the combined measurement, modelling and geophysical parameter errors that together influence how far from the observed brightness temperatures the simulated values are permitted to be. The shape, construction and importance of the covariance matrices that represent the constraints for the optimal estimation are discussed in further detail in Chapter 5.