$\kappa$: A Geometry-Quality Metric Complementary to GDoP for Closed-Form TDoA Multilateration

Abstract

The Geometric Dilution of Precision (GDoP) characterizes the noise sensitivity of a Time-Difference-of-Arrival (TDoA) localization system, but does not capture every way the analytical multilateration solution can become ill-conditioned. We introduce a complementary geometry-quality metric $\kappa$, the leading coefficient of the closed-form TDoA solver's quadratic, and derive its $N$-dimensional generalization through a vectorized formulation. Two closed-form algebraic identities relate $\kappa$ to the Jacobian determinant of the measurement model and to the quadratic's discriminant, establishing that the system exhibits exactly two distinct singularity loci: branch divergence and the Jacobian/branch-merge locus flagged by GDoP. A Cram\'er--Rao-bound-linked closed form for the noise sensitivity $\sigma_\kappa$ under the standard Gaussian ToA model is validated against Monte Carlo to 2% median relative error. An empirical atlas over a dimensionless geometry parameter space confirms both identities at machine precision and shows that $\kappa$-bad regions and GDoP-bad regions are non-trivially disjoint in target space, establishing the two metrics as genuinely complementary. A case study on a four-node operational array, with per-sensor time of arrival (ToA) noise estimated empirically from Automatic Dependent Surveillance Broadcast (ADS-B)-paired over-the-air captures, shows that the theory-predicted threshold and a Monte-Carlo-measured operational threshold agree on the per-subsystem ordering at the deployment noise level. Their ratio is approximately constant across the three two-dimensional subsystems, serving as a deployment-specific calibration constant between the algebraic $\kappa$-noise floor and the downstream operational threshold, analogous in spirit to the standard relation linking GDoP to the circular error probable.