Network based sensor localization in multi-media application of precision agriculture Part 2: Time of arrival

We study the problem of sensor node localization for a hybrid wireless sensor network deployed in a precision agriculture farm, with nodes located both underground and above-ground. We consider time of arrival measurements from unmodulated signals transmitted between neighboring sensor nodes and between satellite nodes and sensor nodes for ranging purposes. While this paper studies the network based localization of sensors, placed in multiple media, based on the measurements of time of arrival, in a companion paper we study the same problem based on the received signal strengths measurements. The localization problem is formulated with the goal of parameter estimation of the joint distribution of the ranging measurements. We derive the probability distribution for the measured time of arrival with the parameters governed by the sensor nodes´ location coordinates based on rigorous statistical analysis. Then, we formulate a maximum likelihood optimization problem to estimate the nodes´ location coordinates using the derived statistical models. We also present a sensitivity analysis of the estimates with respect to the soil complex permittivity and magnetic permeability.