Error Tolerant Arrangements of sensors for sampling fields

We introduce and address the sensor arrangement problem. We ask when and where it is best to sample an unknown scalar field (say of temperatures or chemical concentrations) in order to estimate it to within a certain error tolerance. This question is necessary to decide where to place sensors in sensing phenomena which can be described as fields over large areas. We assume that the field is modeled as a linear combination of a set of basis functions and that the sensor measurements are noisy. Based on linear estimation theory, estimation error can be shown to be a function of sampling errors and of the geometric arrangement of sampling locations. We refer to the latter as the sensor arrangement. Our approach is to characterize different classes of sensor arrangements and to understand the circumstances under which the reconstructed fields for arrangements from these classes satisfy an error tolerance limit. We refer to these classes as error tolerant arrangement classes or ETAC´s. With a knowledge of the nature of ETAC´s we will have articulated constraints for placing sensors in time and space; furthermore, by identifying possible sampling locations in advance, we will also have simplified the planning of the motion of mobile sensors for that field. In this paper we discuss three types of ETAC´s for fields that are modeled as 2D trigonometric polynomials: uniform sensor arrangements, Delta-dense sensor arrangements and incrementally constructed sensor arrangements.