Uncertainty-scaled detection floors versus length gates for road descents from noisy elevation profiles

Let a road segment be sampled at horizontal positions $s_1 < s_2 < \cdots < s_n$ with elevation observations $z_i = h(s_i) + \varepsilon_i$, where $h$ is the true road surface profile and $\varepsilon_i$ are independent, zero-mean errors with known standard deviation $\sigma_z$, the vertical accuracy of the elevation model. Let $\Delta s$ denote horizontal distance and define the estimated grade over an interval $[a,b]$ of the segment by $\hat{g} = (z(b) - z(a))/\Delta s$.

A detector declares a descent (a rideable downhill run) on an interval when the estimated grade clears a rideability threshold $g_0 > 0$. Two families of detection floors are compared. An uncertainty-scaled floor declares a descent when $\hat{g} \ge g_0 + k \cdot \mathrm{se}(\hat{g})$, where $k > 0$ is a fixed multiplier and $\mathrm{se}(\hat{g})$ is the standard error of the estimated grade under the error model. A length gate instead requires, in addition to $\hat{g} \ge g_0$, that the run length satisfy $\Delta s \ge L_{\min}$ for a fixed minimum $L_{\min} > 0$.

Conjecture: for every length gate $L_{\min}$, there exists a multiplier $k$ such that the uncertainty-scaled floor attains a false-discovery rate and recall at least as good on road networks whose elevation comes from a raster DEM with known vertical accuracy, so the length gate contributes nothing that an uncertainty-scaled floor cannot provide.