Localization, navigation and tracking form a special application domain of Bayesian filtering, where the position and velocity of a mobile (and possibly additional hyper-parameters) should be estimated based on (i) a prior model for the possible displacement of the mobile, (ii) noisy measurements provided by a sensor, and (iii) a georeferenced information source (digital map, reference data base, etc.), providing for each spatial position an estimate of the quantity measured by the sensor. For

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... xample in terrain-aided navigation (TAN) a radio-altimeter combined with an inertial navigation system (INS) provides an estimation of the terrain height below the platform, which can be correlated with the terrain height at each horizontal position, as read on a digital map. In wireless communications, the signal power received by the mobile from an access point (WiFi) or from a base station (GSM, UMTS) and measured by the mobile itself, can be correlated with another estimation of the signal power received at each spatial position, as read on a digital attenuation map or from a reference data base. Values read on a digital map are usually subject to errors which are in general spatially correlated and modeled as Gaussian random fields, with a known correlation function. This results in a temporal correlation of measurement noises, which should be accounted for in evaluating the likelihood function, an essential step in the derivation of the equation for the Bayesian filter. The method described below shows how to perform this evaluation in an optimal way, using classical properties of Gaussian random vectors, and how to implement numerically the resulting Bayesian filter in terms of a particle filter.