On Bandlimited Spatiotemporal Field Sampling with Location and Time Unaware Mobile Sensors

Sampling of a spatiotemporal field for environmental sensing is of interest. Traditionally, a few fixed stations or sampling locations aid in the reconstruction of the spatial field. Recently, there has been an interest in mobile sensing and location-unaware sensing of spatial fields. In this work, the general class of fields governed by a constant coefficient linear partial differential equations is considered. This class models many physical fields including temperature, pollution, and diffusion fields. The analysis is presented in one dimension for a first exposition. It is assumed that a mobile sensing unit is available to sample the spatiotemporal field of interest at unknown locations and unknown times -- both generated by independent and unknown renewal processes. Based on measurement-noise affected samples, a spatial field reconstruction algorithm is proposed. It is shown that if the initial condition on the field is bandlimited, then the mean squared error between the original and the estimated field decreases as $O(1/n)$, where $n$ is the average sampling density of the mobile sensor. The field reconstruction algorithm is universal, that is, it does not require the knowledge of unknown sampling locations' or unknown time instants' statistical distributions.