The Electron Density Assimilative Model (EDAM) has been developed to assimilate measurements into a background ionospheric model. This background model is provided by IRI2007 [Bilitza and Reinisch, 2008] and the majority of the input data are TEC measurements derived from International GNSS Service (IGS) stations [Beutler et al., 1999]. The assimilation is based on a weighted, damped least mean squares estimation. This is a form of minimum variance optimal estimation (also referred to as Best Linear Unbiased Estimation (BLUE)) that provides an expression for an updated estimation of the state (known as the analysis) that is dependent upon an initial estimate of the state (the background model) and the differences between the background model and the observations [Menke, 1989; Twomey, 1977]. The error covariance matrices of the background model and the observations are also used to control the relative contributions of the background and the observations to the analysis:
where xa is the analysis, xb is the background model, K is the weight matrix, y is the observation vector, B is the background error covariance matrix, and R is the error covariance matrix of the observations [Rodgers, 2000]. H is the nonlinear observation operator that relates the measurements to the state:
where ɛ is the observation error. The observation operator is nonlinear because, in EDAM, the background model is comprised of the log of the ionospheric electron density. H is the Jacobian, whose elements are given by the partial differentials of the observation operator evaluated at the background model; i.e.,
The assimilation is conducted using a tilted dipole magnetic coordinate system that remains fixed in space with respect to the sun. An assimilation time step of 15 min has been used and the electron density differences between the voxels of the analysis and the background model are propagated from one time step to the next by assuming persistence combined with an exponential decay. The time constant for this decay is set at four hours. Thus if the data feed is interrupted, the analysis will decay back to the background model.