Demo — Information Map

Demo — Information Map#

L14 showed you a PDOP tile and a DOP map; L33 names them. This demo draws the Cramér–Rao bound of a small sensor network over a whole map, so you can see — before anyone takes a bearing — where the network fixes a target tightly, where it fixes it badly, and where it cannot fix it at all. Read it from either side: if the sites are theirs, the red is where their fix on you is poor; if they are yours, the red is where you cannot fix them.

The model#

Each bearing-only site \(k\), at range \(R_k\) from a target with cross-range unit vector \(\mathbf{n}_k\) and bearing noise \(\sigma_\theta\) (radians), contributes information across its line of sight only; a radar adds a range row along the line of sight \(\mathbf{u}_k\):

\[ \mathbf{J} = \sum_k \frac{\mathbf{n}_k\mathbf{n}_k^\top}{R_k^2\sigma_\theta^2} \;+\; \sum_{k\,\in\,\text{radars}} \frac{\mathbf{u}_k\mathbf{u}_k^\top}{\sigma_R^2}, \qquad \operatorname{Cov}(\hat{\mathbf{p}}) \succeq \mathbf{J}^{-1}. \]

The probe’s readouts come straight from \(\mathbf{J}\)’s eigenvalues: the long semi-axis \(a = 1/\sqrt{\lambda_{\min}}\), the short one \(b = 1/\sqrt{\lambda_{\max}}\), the condition number \(\kappa = \lambda_{\max}/\lambda_{\min}\), and

\[ \text{DRMS} = \sqrt{\operatorname{tr}\mathbf{J}^{-1}} = \text{PDOP}\cdot\bar R\cdot\sigma_\theta , \]

where PDOP is computed exactly as the L14 Geolocation Explorer computes it — rows \((\bar R/R_k)\,\mathbf{n}_k^\top\) with \(\bar R\) the mean range — so the identity holds to every digit. PDOP is a bearing-only geometry factor, so the demo reports it only while every site is bearing-only. Coordinates are North–East in km, bearings clockwise from North, and every number is notional.

Interactive demo#

Open in full screen

Walkthrough#

The map is 200 km East by 150 km North. Drag any site, either end of the straight leg (A and B), or the probe; click anywhere on the map to move the probe there, or click the chart under the map to put the probe on the leg at that point.

  1. Read the default. Two bearing-only sites, S1 and S2, with \(\sigma_\theta = 1^\circ\). The color field is the CRLB DRMS at every point — light is a tight fix, red a poor one — and the probe on the leg reads DRMS 3.34 km with a long, tilted ellipse. The tiles show \(\lambda_{\max}\), \(\lambda_{\min}\), \(\kappa\), both semi-axes, the S1–S2 cut, DRMS, PDOP and \(\bar R\), and the line under them checks \(\text{PDOP}\cdot\bar R\cdot\sigma_\theta\) against DRMS.

  2. Find the seam. A dark ridge runs through S1 and S2 and on past both — and between them. The chart under the map flies the leg from A to B and spikes to the top where the leg crosses that line, 90 km along: the leg note reports DRMS \(\infty\) there. Click the spike. The probe jumps onto the seam, \(\lambda_{\min}\) reads 0, \(a\) and \(\kappa\) read \(\infty\), and the ellipse collapses into a line along the baseline — the direction two bearings cannot measure.

  3. Step off the seam. Drag the probe along the leg away from the ridge and watch \(\kappa\) fall — tens of thousands a kilometer off it, a few hundred at ten kilometers, under twenty back at the default probe — as the ellipse fattens. Near the ends of the map every cut closes, so DRMS climbs even away from the seam: distance is its own kind of blindness.

  4. Check L14’s formula. Hit Symmetric pair. The probe now sits at equal range from both sites with a 53° cut; PDOP reads 1.77, which is \(\sqrt{2}/\sin 53.13^\circ\), and \(\kappa = 4.00\), which is \(\cot^2(26.6^\circ)\).

  5. Turn on PDOP view. The color field becomes L14’s DOP map, with the same color stops, and it no longer changes with the \(\sigma_\theta\) slider: PDOP is pure geometry. Turn it back off and drag \(\sigma_\theta\) from 0.2° to 3°: every DRMS scales with it, but the seam stays exactly where it was.

  6. Add a third site. Click 3. S3 looks from the east and supplies the missing direction: the ridge fades, the chart’s spike is gone, and the worst point on the leg is now about 4 km, at its western end. Collinear three puts S3 on the S1–S2 line instead — three sites, and the seam survives.

  7. Give one site a radar. Back on Seam pair, choose S1 radar. S1 now measures range as well, adding information along its line of sight — exactly the direction the seam was missing — and the ridge vanishes. Slide \(\sigma_R\) from 0.1 km to 5 km to see how good the range has to be to matter. Then return to Bearing-only: that is what a noise jammer does to a radar, and the seam comes back.

Key observations#

  • A bearing measures across, never along. Every ellipse’s long axis points in the direction the lines of sight agree on.

  • The small eigenvalue makes the long axis. \(a = 1/\sqrt{\lambda_{\min}}\), and \(\kappa\) tells you how close the geometry is to failing.

  • PDOP is the CRLB with the range scaled out. \(\text{DRMS} = \text{PDOP}\cdot\bar R\cdot\sigma_\theta\) holds exactly, and \(\sqrt{2}/\sin\beta\) for a symmetric pair.

  • A seam is geometric. Better bearings shrink every ellipse but cannot close a seam; a sensor looking from another direction, or a range measurement, can — and jamming the range reopens it.

  • Both directions. For your receivers, fly where the field is light; against their passive network, a route through their red is a route they cannot fix. Final Project task T4 asks for exactly this chart along your route.

Source#

MATLAB bundle · L33_InformationMap.m↓

The companion script builds the Fisher information of the demo’s default two-site network at the probe point and reads the ellipse, DRMS and PDOP from it; maps DRMS over the demo’s 2 km grid and marks the seam; and flies the demo’s straight leg, once with the two sites and once with the third site added, printing where their fix on the leg is best and where it fails.

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