It pins the cross-range position — perpendicular to the line of sight — to about \(R\sigma_\theta\), and says nothing at all about the along-range position. Slide the target along the line of sight and the bearing does not change.
With \(\theta = \mathrm{atan2}(\Delta E, \Delta N)\), the row is \(\mathbf{h} = [-\Delta E,\ \Delta N]/R^2 = \mathbf{n}^\top/R\), where \(\mathbf{n}\) is the unit cross-range vector. It is the same row Project 2 called \(\mathbf{H}_k\).
\(\mathbf{J} = \mathbf{H}^\top\mathbf{R}^{-1}\mathbf{H} = \sum_k \mathbf{n}_k\mathbf{n}_k^\top/(R_k^2\sigma_\theta^2)\), with \(\sigma_\theta\) in radians. Each bearing adds a rank-one piece; information from independent measurements simply adds.
\(\operatorname{Cov}(\hat{\mathbf{p}}) \succeq \mathbf{J}^{-1}\): no unbiased estimator — however clever — can have a smaller error covariance in any direction. The geometry and the sensor noise set the floor; a good least-squares fix reaches it when the noise is small.
\(\mathbf{J}\) depends only on the sensor positions, a hypothetical target position and \(\sigma_\theta\) — not on the measured bearings. Evaluate it at every point and you have a map of how well a network can fix a target anywhere, which is exactly what L14's DOP map and the L33 demo draw.
The long semi-axis is \(a = 1/\sqrt{\lambda_{\min}}\) along the eigenvector of \(\lambda_{\min}\); the short one is \(b = 1/\sqrt{\lambda_{\max}}\) along the other. The small eigenvalue makes the long axis: little information means a large error.
The 1σ distance root-mean-square error, \(\text{DRMS} = \sqrt{a^2 + b^2} = \sqrt{\operatorname{tr}\mathbf{J}^{-1}}\) — the "Predicted RMS" tile in L14's demo. It adds the squares of the semi-axes, never the semi-axes themselves.
\(\kappa = \lambda_{\max}/\lambda_{\min} = (a/b)^2\). A value of 1 is a round error circle; a large value is a long, thin ellipse; \(\kappa \to \infty\) means one direction is unobservable — a seam.
It builds one row per receiver, scaled by the mean range: \(\tilde{\mathbf{h}}_k = (\bar R/R_k)\,\mathbf{n}_k^\top\), and takes \(\text{PDOP} = \sqrt{\operatorname{tr}((\tilde{\mathbf{H}}^\top\tilde{\mathbf{H}})^{-1})}\). The scaling strips out the overall range and leaves a pure geometry factor.
Because \(\tilde{\mathbf{H}}^\top\tilde{\mathbf{H}} = \bar R^2\sigma_\theta^2\,\mathbf{J}\): the scaled rows are the information rows times a constant. Inverting and taking the square root of the trace gives \(\text{PDOP} = \sqrt{\operatorname{tr}\mathbf{J}^{-1}}/(\bar R\sigma_\theta)\) — algebra, not an approximation.
\(\text{PDOP} = \sqrt{2}/\sin\beta\), so \(\text{DRMS} = \sqrt{2}\,R\sigma_\theta/\sin\beta\). The best is \(\sqrt{2} = 1.41\) at a 90° cut; it is infinite at 0° or 180°.
On the whole line through the two sites — beyond both ends (a 0° cut) and on the segment between them (a 180° cut). There both lines of sight are parallel or opposite, \(\mathbf{J}\) is rank one, and position along the line is unobservable.
A smaller \(\sigma_\theta\) scales every eigenvalue of \(\mathbf{J}\) up by the same factor, but zero times any factor is still zero. Only a measurement from another direction — a third site off the line, or a range measurement — adds information where the seam has none.
A term \(\mathbf{u}\mathbf{u}^\top/\sigma_R^2\) along the line of sight \(\mathbf{u}\) — exactly the direction bearings leave empty. Since \(\mathbf{u} \perp \mathbf{n}\), one range-plus-bearing radar has a full-rank \(\mathbf{J}\) on its own: no seam.
It denies range and leaves each radar a strobe — a bearing only (L17). Radars crossing strobes form a bearing-only network, with bearing-only seams along the lines joining them; jamming reopens the seams that range had filled.
Our receivers fixing their emitters (Project 2, Final Project T6): \(\lambda_{\min}\) large — open the cut, shorten the range, and pay for it in exposure. Their ES or IRST network fixing us or our jammer (Final Project T4): their \(\lambda_{\min}\) small along our route — ride their seams and keep the range long.