Satellites provide important information on many meteorological and oceanographic variables. State-space models are commonly used to analyse such data sets with measurement errors. In this work, we propose to extend the usual linear and Gaussian state-space to analyse time series with irregular time sampling, such as the one obtained when keeping all the satellite observations available at some specific location. We discuss the parameter estimation using a method of moment and the method of maximum likelihood. Simulation results indicate that the method of moment leads to a computationally efficient and numerically robust estimation procedure suitable for initializing the Expectation-Maximisation algorithm, which is combined with a standard numerical optimization procedure to maximize the likelihood function. The model is validated on sea surface temperature (SST) data from a particular satellite. The results indicate that the proposed methodology can be used to reconstruct realistic SST time series at a specific location and also give useful information on the quality of satellite measurement and the dynamics of the SST.