1 / SPHERICAL PATH AND REFERENCE ARC
For aircraft \(i\), let \(x_i(t)\in S_R^2\) be its recorded position at dimensionless normalized time \(0\leq t\leq1\), where \(S_R^2\) is the spherical Earth of radius \(R=6371.0088\) km. Its endpoint geodesic is
\[
g_i(s)=\operatorname{Exp}_{x_i(0)}\!\left(s\,\operatorname{Log}_{x_i(0)}x_i(1)\right),
\qquad 0\leq s\leq1.
\]
Here \(g_i\) is the minor great-circle arc, \(s\in[0,1]\) is dimensionless progress along it, \(\operatorname{Log}_{x_i(0)}x_i(1)\) is the tangent vector at the first position pointing toward the last, and \(\operatorname{Exp}_{x_i(0)}\) maps a scaled tangent vector back to the sphere. Thus \(g_i(s)\) is a position, while distances along or from \(g_i\) are measured in kilometres.
USE-CASE READING This constructs the direct spherical reference route against which we compare the aircraft's recorded ground track.
2 / OBSERVED DEVIATION
Let \(d_{S_R^2}(a,b)\) denote spherical distance in kilometres and let \(g_i([0,1])\) denote the set of every position on the reference arc. The cross-track displacement is
\[
\eta_i(t)=d_{S_R^2}\!\left(x_i(t),g_i([0,1])\right).
\]
Here \(\eta_i(t)\) is the shortest distance in kilometres from the recorded position to that arc. For sample times \(t_1,\ldots,t_n\), with \(n=60\), the primary histogram statistic is the root-mean-square deviation
\[
D_i=\left[\frac{1}{n}\sum_{j=1}^{n}\eta_i(t_j)^2\right]^{1/2}.
\]
Consequently, \(D_i\) is one nonnegative kilometre-valued summary for aircraft \(i\).
USE-CASE READING This reduces each 60-point flight track to its typical sideways separation from the direct reference route.
3 / LOGNORMAL HYPOTHESIS
Let \(d_0=1\) km be a fixed reference distance, making \(D_i/d_0\) dimensionless. The simplest positive, right-skewed population model assumes that many multiplicative route effects combine in logarithmic space, so
\[
\log(D_i/d_0)\sim\mathcal N(\mu,\sigma^2),
\]
\[
f_D(d)=\frac{1}{d\sigma\sqrt{2\pi}}
\exp\!\left[-\frac{(\log(d/d_0)-\mu)^2}{2\sigma^2}\right],
\qquad d>0.
\]
Here \(\mathcal N(\mu,\sigma^2)\) is a normal law with log-mean \(\mu\) and log-variance \(\sigma^2\); \(f_D(d)\) is the probability density of RMS deviation evaluated at distance \(d>0\) km, and \(\log\) denotes the natural logarithm. Below, \(\log D\) and \(\log A\) abbreviate \(\log(D/d_0)\) and \(\log(A/d_0)\).
USE-CASE READING This is the initial claim that flight-to-flight RMS deviations may follow one familiar positive, right-skewed distribution.
4 / RANDOM-SCALE BRIDGE
Separate the magnitude of a route's departure from its standardized shape:
\[
\eta_i(t)=A_i Z_i(t),\quad
D_i=A_iQ_i,\quad
Q_i=\left[\int_0^1 Z_i(t)^2\,dt\right]^{1/2}.
\]
The dimensionless bridge \(Z_i(t)\), pinned by \(Z_i(0)=Z_i(1)=0\), represents standardized route shape; \(A_i>0\) is a latent route-specific scale in kilometres; and the dimensionless functional \(Q_i\) is the continuous-time RMS magnitude of that shape. If \(\log A_i\) is Gaussian and variation in \(Q_i\) is comparatively modest, then \(\log D_i=\log A_i+\log Q_i\) is approximately Gaussian.
USE-CASE READING This asks whether large and small deviations differ mainly by overall scale while sharing broadly comparable route shapes.
5 / BROWNIAN-BRIDGE BASELINE
A tractable baseline for the standardized route shape is a standard Brownian bridge with Karhunen-Loeve expansion
\[
Z(t)=\sum_{k=1}^{\infty}\frac{\sqrt{2}\,\xi_k}{\pi k}
\sin(\pi kt),
\qquad \xi_k\overset{\mathrm{iid}}{\sim}\mathcal N(0,1).
\]
Here \(k=1,2,\ldots\) indexes sine-wave modes, \(\pi\) is the circle constant, and \(\xi_k\overset{\mathrm{iid}}{\sim}\mathcal N(0,1)\) means that the mode coefficients are independent standard-normal variables. Its integrated squared displacement is
\[
Q^2=\int_0^1Z(t)^2\,dt
=\sum_{k=1}^{\infty}\frac{\xi_k^2}{\pi^2k^2}.
\]
Thus \(Q^2\) is a weighted sum of independent one-degree-of-freedom chi-square variables and \(Q\) is its nonnegative square root.
USE-CASE READING This supplies a mathematically tractable null model for irregular route shapes that begin and end on the reference arc.
6 / RESULTING POPULATION LAW
For a generic route, write \(D=AQ\), and let \(f_A\) and \(f_Q\) be the probability densities of its independent positive scale \(A\) and shape factor \(Q\). Their product has density
\[
f_D(d)=\int_0^{\infty}
f_A\!\left(\frac{d}{q}\right)f_Q(q)\,\frac{dq}{q}.
\]
Here \(q>0\) is the integration variable and \(1/q\) is the product-transformation Jacobian. This integral is the geometry-driven population law for kilometre-valued \(D\); a single lognormal is recovered approximately when \(Q\) is concentrated relative to \(A\).
USE-CASE READING This combines variation in overall route scale with variation in route shape to predict the fleet-wide deviation histogram.
THEOREM
7 / LOGNORMAL SMOOTHING
Consider a sequence indexed by \(\ell\) of scale-shape populations \(D_\ell=A_\ell Q_\ell\). Let \(U_\ell=\log A_\ell\sim\mathcal N(\mu_\ell,\sigma_\ell^2)\) be independent of \(V_\ell=\log Q_\ell\), and define its mean \(m_\ell=\mathbb E[V_\ell]\), variance \(\tau_\ell^2=\operatorname{Var}(V_\ell)\), and standardized log-deviation
\[
W_\ell=\frac{\log D_\ell-(\mu_\ell+m_\ell)}{\sqrt{\sigma_\ell^2+\tau_\ell^2}}.
\]
If the bridge's share of total log-variance satisfies \(\tau_\ell^2/(\sigma_\ell^2+\tau_\ell^2)\to0\), then \(W_\ell\Rightarrow\mathcal N(0,1)\), where \(\Rightarrow\) denotes convergence in distribution. More quantitatively, if \(G_\ell\sim\mathcal N(m_\ell,\tau_\ell^2)\) is a Gaussian variable matching the first two moments of \(V_\ell\), then
\[
d_{W_2}\!\left(W_\ell,\mathcal N(0,1)\right)
\leq
\frac{d_{W_2}(V_\ell,G_\ell)}{\sqrt{\sigma_\ell^2+\tau_\ell^2}}.
\]
Here \(d_{W_2}\) is the quadratic Wasserstein distance between probability laws. Proof sketch. Replace \(V_\ell\) by \(G_\ell\); then \(U_\ell+G_\ell\) is exactly Gaussian. Convolution with the independent Gaussian \(U_\ell\) contracts Wasserstein distance, and the independent coupling bound \(d_{W_2}(V_\ell,G_\ell)\leq\sqrt{2}\,\tau_\ell\) completes the convergence argument.
USE-CASE READING Across route populations, the fleet histogram becomes lognormal when variation in overall deviation scale overwhelms the remaining variation in standardized route shape.
PROPOSITION
8 / EXACT CUMULANT ATTENUATION
For one fixed population, let \(U=\log A\), \(V=\log Q\), \(\tau_Q^2=\operatorname{Var}(V)\), and \(W=[\log D-(\mathbb E[U]+\mathbb E[V])]/\sqrt{\sigma_A^2+\tau_Q^2}\), where \(\sigma_A^2=\operatorname{Var}(U)\). If \(U\) is Gaussian and independent of \(V\), additivity of cumulants gives, for every integer \(r\geq3\),
\[
\kappa_r(W)=
\frac{\kappa_r(\log Q)}
{(\sigma_A^2+\tau_Q^2)^{r/2}}.
\]
Here \(\kappa_r(X)\) denotes the order-\(r\) cumulant of \(X\). In particular, with bridge variance share \(\rho=\tau_Q^2/(\sigma_A^2+\tau_Q^2)\),
\[
\operatorname{Skew}(W)=\operatorname{Skew}(\log Q)\rho^{3/2},
\qquad
\operatorname{ExKurt}(W)=\operatorname{ExKurt}(\log Q)\rho^2.
\]
Here \(\operatorname{Skew}\) is standardized third cumulant and \(\operatorname{ExKurt}\) is excess kurtosis, the standardized fourth cumulant. The product is exactly lognormal only when \(\log Q\) is Gaussian as well; otherwise these identities quantify the residual asymmetry and tail shape.
USE-CASE READING This tells us exactly how much of the flight histogram's non-lognormal shape can survive after route-scale variation is added.
BROWNIAN-BRIDGE COROLLARY
9 / AN EXPLICIT THRESHOLD
For the Brownian bridge, the spectral law of \(Q^2\) has Laplace transform, for transform parameter \(\lambda>0\),
\[
\mathbb E[e^{-\lambda Q^2}]=
\left(\frac{\sqrt{2\lambda}}{\sinh\sqrt{2\lambda}}\right)^{1/2}.
\]
Here \(\mathbb E\) denotes expectation and \(\sinh\) is the hyperbolic sine. A numerical Karhunen-Loeve evaluation of the spectral sum gives
\[
\begin{aligned}
\operatorname{SD}(\log Q)&\approx0.385,\\
\operatorname{Skew}(\log Q)&\approx0.178,\\
\operatorname{ExKurt}(\log Q)&\approx-0.348.
\end{aligned}
\]
Here \(\operatorname{SD}\) is standard deviation, and \(\sigma_A=\operatorname{SD}(\log A)\) is the log-scale spread defined above. Substitution into the exact formulas shows that \(\sigma_A\gtrsim0.50\) makes both resulting standardized cumulants smaller than approximately \(0.05\) in magnitude.
USE-CASE READING For our bridge baseline, even moderately varied route scales should make the observed fleet distribution look very nearly lognormal.
EMPIRICAL DIAGNOSTIC
10 / THE SINGLE-POPULATION MODEL FAILS
For the 1,776 aircraft with positive recorded RMS deviation \(D\) in kilometres, \(\operatorname{SD}(\log D)\approx1.465\). Under the independent Brownian decomposition, matching this log-variance would imply \(\sigma_A\approx1.413\) and hence predicted log-skewness and log-excess-kurtosis near \(0.003\) and \(-0.002\). Their sample values are instead approximately \(-2.40\) and \(8.11\).
This discrepancy rejects the single independent lognormal-scale Brownian-bridge description of the full cohort. It is concentrated in a near-geodesic boundary population: 53 aircraft have recorded RMS deviation at or below \(0.1\) km. The data therefore point toward a boundary mixture or multiple route regimes rather than one universal product law \(D=AQ\).
USE-CASE READING Real flight tracks contain a distinct nearly direct group that the otherwise strongly smoothing Brownian product model cannot explain.
SEQUENTIAL DIAGNOSTIC
11 / HOW LONG UNTIL THE BOUNDARY IS VISIBLE?
Once the boundary component is forced by the cohort, the time-series question is individual: after \(n\) one-minute samples, can we assign one aircraft to that component? Define the prefix residual energy
\[
R_n^2=\frac{1}{n}\sum_{j=1}^{n}\eta(t_j)^2.
\]
If \(C=0\) denotes the near-geodesic boundary population and \(C=1\) the ordinary bridge population, then persistent small residuals are decisive only when they are rare under \(C=1\). A small-ball bound of the form
\[
\mathbb P_1(R_n\leq\tau)\leq e^{-nI(\tau)}
\]
implies that the posterior probability \(\mathbb P(C=0\mid R_n\leq\tau)\) rises once the ordinary population has too little mass near zero. For the current hour, using \(\tau=0.1\) km and the final endpoint geodesic reference, the diagnostic crosses high posterior confidence after roughly 41 elapsed minutes.
The same stopping idea has a transport version. For \(Y_n=\log(R_n/d_0)\), let \(P_{0,n}\) and \(P_{1,n}\) be the boundary and ordinary prefix laws. The Wasserstein gap
\[
\Delta_n=W_2(P_{0,n},P_{1,n})
\]
is compared with a bootstrap uncertainty floor. Classification is most credible when individual posterior evidence and population-level Wasserstein separation are both large.
USE-CASE READING The key signal is not whether an aircraft looks straight for a few minutes; it is whether near-perfect geodesic behavior persists long enough to be unlikely under ordinary route physics.