An algorithm computes the direction of arrival (both azimuth and elevation angles) of a lightning-induced electromagnetic signal from differences among the times of arrival of the signal at four antennas in a Y-shaped array on the ground. In the original intended application of the algorithm, the baselines of the array are about 90 m long and the array is part of a lightning- detection-and- ranging (LDAR) system. The algorithm and its underlying equations can also be used to compute directions of arrival of impulsive phenomena other than lightning on arrays of sensors other than radio antennas: for example, of an acoustic pulse arriving at an array of microphones.

The underlying equations express the differences among the times of arrival as functions of the inner products of (1) the unit vector of the direction of arrival and (2) the unit vectors along the baselines of the array. To obtain a solution for the unit vector (and thus, equivalently, the azimuth and elevation angles) of the direction of arrival, the underlying equations are combined and modified into a matrix formulation that is amenable to a least-squares solution.

The value of the inner product calculated from the measured difference between the times of arrival of the signal at the two antennas on each baseline specifies a circle in the sky upon which nominally lies the apparent point in the sky from which the signal came. Thus, from the time-of-arrival measurements on all three baselines of the Y-shaped array, it is possible to specify three circles in the sky, all of which nominally contain the apparent point in the sky from which the signal came. Nominally, all three circles would intersect at a single point in the sky corresponding to the direction of arrival. In practice, random measurement errors prevent the three circles from intersecting at a single point; instead, they intersect to define a small trianglelike patch of sky. The least-squares-error solution corresponds to a point near the triangle, such that sum of squares of distances between the solution point and each circle in the sky is the least possible value.

This algorithm has been verified on both synthetic data and measurement data recorded by a prototype short-baseline LDAR system. An analysis of errors revealed that the azimuth error depends only on elevation and that the elevation error is small except near the horizon. Further analysis showed that the addition of a vertical baseline (two additional antennas mounted on the top and bottom of a tower) would add little of value to the measurements and calculations since the LDAR source points are typically above 20° elevation.

The primary source of error in the algorithm is the simplifying assumption that the signal originates at an infinite distance. While this assumption is never strictly true, it provides an acceptable approximation as long as the distance of the signal source is much greater than the length of the baselines. The least-squares approach also reduces this error.

An additional equation that takes account of the curvature of a wavefront arriving from a source at a finite distance has been derived and found to be accurate at close range. An algorithm that would solve iteratively for azimuth, elevation, and range of the source has been proposed but not tested.

This work was done by Stan Starr of Kennedy Space Center. For further information, access the Technical Support Package (TSP) free on-line at under the Information Sciences category. KSC-12059.

NASA Tech Briefs Magazine

This article first appeared in the March, 2003 issue of NASA Tech Briefs Magazine.

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