Wednesday, April 30, 2014
The TPL Triangulation Works Well Even With Relatively Large Errors
I added a random error of 30% to the calculated intensity values to simulate measurement errors and increased the number of measurement per side to 101 to reduce the statistical error for the estimated position. The lines of position did not always meat at a point so for the "best estimated position" I chose the point, zb, for which the sum of its distances from the lines was a minimum. It was easier to do this with Cartesian coordinates so I converted the complex numbers that I was using with z' being the position of a minimum for the inverse intensity along a side of the triangle and e' the direction of the corresponding line of position. Pi is a projection operator that gives the distance of point zb from the line of position. The formulas for the calculating zb are rather simple.
With random errors the curves for the inverse intensity are less well defined and I used a log scale for the inverse intensity to better show the spread of the data for the smaller values.
The example shown is for an unusual event with a relatively large error for the estimated position but shows how zb fits inside a small triangle formed by the lines of position. A error of 100 meters in the estimated position would narrow down the search area considerably.
More data gives better results but if there are obstructions near the black box the data along a constant course may not be complete due to the TPL passing through shadows.