The bat was flying the other way!
After spending some more time looking at the bat data I noticed that the delay measurements are not constant across a single call. The first part of the chirp looks good and there’s a pretty constant delay measurement as the volume increases, but in the tail, the delays become noisier. No doubt this is partly a signal-to noise issue but I think it might also be an instrumental effect, possibly some sort of self-interference. Here’s what I mean:
Spectrogram showing the delay measured between the microphone pair for a single bat call. The smaller panels show amplitude and delay as a function of frequency and time for the two cross-sections indicated by the red lines.
The pale green region on the left-hand side of the spectrogram is the delay measured during the rising phase of the call. The upper panels of the figure show the amplitude and delay against time at the frequency indicated by the horizontal red line. The delay looks nice and constant until the sound level peaks, then it begins to get noisy. In the frequency domain, the lower frequencies tend to behave better for longer.
Is this a systematic effect or is it something to do with the bat? Well, here’s a similar plot for an artificial signal, generated by an ultrasonic range-finder:
A delay spectrogram for an artificial source. The delay cross-section plots show additional lines representing delays with +/- a cycle of phase.
The behaviour is pretty similar, so I’m thinking that the effect is instrumental and SNR-related rather than anything to do with the bat.
Anyway, this made me realise that I could get better direction estimates if I just restricted the analysis to data on the rising amplitude phase of the call. When I did this, I noticed two things. Firstly, the quality of the delay measurements improved a lot and secondly, I realised that my previous analysis was completely wrong! It’s pretty clear that the bat was flying in the opposite direction. My initial model, which I needed to use to correct for phase-turn ambiguities and was based on a single frequency channel only, was so far off that I’d forced a solution in the wrong direction. So here’s how it looks now:
Spectrogram of the bat observation with a revised rising-phase analysis technique. Cross-sections in time and frequency for SNR (amplitude), Direction vs time indicated by vectors and in the panel below amplitude. Phase vs frequency is also shown on the right and again shows how the lower frequency end is better behaved.
This can also be modelled if I assume a bat moving in a straight line at constant speed:
Modelled bat trajectory. This is not a unique solution. A constant height has also been assumed in this case.
The above solution is by no means unique as it’s quite poorly constrained with only two mics. For example, an alternative solution could have the bat moving faster but at a higher altitude. Nevertheless, the fitted parameters look plausible.
Adding two more microphones will allow a true direction to be measured and also help greatly in resolving the phase turn ambiguities. So maybe it’s time to start thinking about more hardware! :-)