Passive source localization from spatially correlated angle-of-arrival data

Bias and variance equations are presented for two-dimensional location estimators of a nonmoving point source of radiation in an isotropic, stationary random medium. The estimators are calculated from spatially correlated angle-of-arrival data which are collected simultaneously at two sensor positions, and assumed to consist of true (unbiased) source angles plus zero-mean angular noise with equal variances at both sensors and negligible higher moments. Under these assumptions the square of the estimator bias is, in general, a quadratic function and the estimator variance a linear function of the spatial data correlation coefficient. However, for source ranges much larger than sensor separation, both the bias and the variance tend to increase linearly with decreasing correlation coefficient, whereas they tend to decrease with increasing sensor separation. The combined effect for a distant source in a stationary random medium, when evaluated for typical spatial wavefront autocorrelation functions, is a significant reduction in the estimator bias and variance dependence on sensor separation, as compared to the uncorrelated case. With minor modifications, the same results apply to the equivalent problem of using time-of-arrival data from three colinear sensor positions.