Log 17

Log 17

Justin Vandenbroucke

justinv@hep.stanford.edu

Stanford University

5/5/03–

This file contains log entries summarizing the results of various small subprojects of the AUTEC study. Each entry begins with a date, a title, and the names of any relevant programs (Labview .vi files or Matlab .m files – if an extension is not given, they are assumed to be .m files).

Note: Higher-resolution versions of the figures are available. If a log has URL dirname/logLL.html, Figure FF of the log should be at URL dirname/FF.jpg.

5/5/03

Direct error estimates

plotUncertainty.m

We can get a good handle on the localization precision we expect with some simple analytical considerations. Event locations are determined by minimizing a least-squares metric. We assuming 4 nearest neighbors (forming a diamond) are hit. This means in particular that the 7^th (central) phone is hit. The events are detected at times t_i, i=1,2,3,7. The four times of arrival determine three measured time-differences of arrival, T^m_i, i=1,2,3 (the m is for measured). We compare these to theoretical time-differences of arrival, T_i(r), i=1,2,3. T_i depends on the position of the source in 3D, r (as well as which SVP and hydrophone positions are being used). We sample T_i with a lattice. We then have a finite set of T_i to be compared with our single T^m_i. We find the closest T_i to T^m_i by minimizing a metric, m = sum[(T_i-T^m_i)², i=1,2,3]. With sufficient lattice spacing, the lattice T^m_i point that minimizes the metric is a good first guess for the location of the event. We can then interpolate T_i for points off the lattice, and minimize m within a neighborhood of our first guess.

If we achieve zero position uncertainty, m=0. In reality we have three sources of measurement error: SVP, timing, and hydrophone location. Uncertainty in each of these introduces uncertainty in our measured TDOA. The range of TDOA values within our uncertainty then determine a range of metrics above zero. We are not able to resolve the location to a single point at which m = 0, but to a region or regions of points for which m < m_max. m_max is determined by our various measurement uncertainties.

Now recall that a ~1500 m/s sound speed determines equivalence between 1 ms and 1.5 m. This is the basis for comparing distance and time uncertainties. 1 ms is the size of our entire captured waveform. The actual impulsive signal typically spans less than this duration, and it has some maximum/central cycle that is only ~0.1 ms wide. So we expect our timing uncertainty to be below 1 ms. 1.5 ms, on the other hand, is a small distance compared to the scale of the detector. The hydrophone location uncertainty is probably ~5 m. This uncertainty dominates the timing uncertainty. Similarly, SVP’s vary up to ~ 5 m/s from month to month. So for a sound travel time of O(1) s, we expect SVP uncertainty to contribute similarly to hydrophone location uncertainty. We expect the distance scale of these two effects, 5 m, to determine our localization precision. A deviation of 5 m (3 ms) shifts our metric by ~3*(3 ms)² = 3e-5 s². So we can only resolve the event location to the region(s) with m < ~3e-5.

We can get a feel for our practical uncertainty by looking directly at deviations in path length. According to the above considerations, we can only resolve sources that have source-receiver path lengths that differ by O(5). We can use this fact to estimate upper bounds of our vertical and horizontal uncertainty. Consider two points a horizontal distance R from a single phone. The path lengths to the phone (we assume they are not refracted – refraction does not significantly affect times of arrival, and affects TDOA’s even less) are S₁ and S₂.

First we consider vertical uncertainty. There is some maximum vertical distance we can separate the two points before their path lengths differ by more than 5 m. This distance depends on R and on the depth. This distance is our vertical resolution. Figure 13 shows this resolution vs. depth for various values of R. Dashed lines are for 10 m path-length resolution; solid are for 5 m. We see that vertical resolution is best near the sea surface and horizontally close to the hydrophone. At 1 km distance the resolution is better than 100 m for all but the bottom 100 m. At 10 km distance the resolution is better than 100 m for roughly the upper half of the sea column. Events at the edge of our 5 km – radius detection volume will have R = 3.5 to 5 km.

Figure 1

Figure 14 is the equivalent plot for horizontal uncertainty. Horizontal resolution is complementary to vertical resolution: worst near the surface and directly above the phone. However this picture is much worse than will actually occur when 4 phones are used. In that case, even if an event is directly over one phone, it will be ~1 km from others horizontally. The minimum horizontal distance from any phone is a few hundred m. So we expect horizontal resolution of at worst a few 10’s of m throughout the detector volume.