Slowness estimation from noisy time delays observed on non-planar arrays

To estimate the slowness vector of infrasound waves propagating across an infrasound array, it is often considered that the sensors are located in a perfectly horizontal plane (2-D approximation). However, the arrays are not planar and differences between sensor altitudes cannot be neglected without introducing biases when estimating the parameters of interest (backazimuth and incidence). A closed-formula of this error depending on the geometry of the array and the wave incidence is presented. Since the unbiased 3-D estimation of the slowness parameters results in a significant alteration of the variance, a metric based on the mean square error is proposed. We found it useful to compare the 3-D to the 2-D processed results. We show the 3-D estimator does not always give the best estimates. We propose also a formulation of the boundary that allow us to choose between these two estimators depending on the situation (arrays, incidence and signal-to-noise ratios). Finally, we compare these two approaches (2-D/3-D) to all International Monitoring System arrays with synthetic data, and perform a comparative estimation of backazimuth for real data.