To ML or Not to ML? Training-Free Spatial Interpolation for Sparse Climate Monitoring

\begin{abstract} Climate understanding depends on continuous environmental fields, yet ground sensing is sparsest in the remote, high-relief terrain where conditions vary most and monitoring matters most, e.g., the Andes (55 stations over ${\sim}4.4$ million km$^2$, one per ${\sim}80{,}000$ km$^2$). Interpolation fills these gaps, but accurate interpolation has come to mean machine learning (ML), whose retraining, compute, and expertise costs are a constraint for resource-limited agencies. We show that training-free statistical interpolation, natural-neighbour (Voronoi/Sibson) tiles with a closed-form two-parameter elevation detrend, using only the 3-D coordinates $(x,y,z)$ every station already reports, works remarkably well. Across 27 leave-one-station-out benchmarks on eight sparse real mountain networks spanning three environmental variables, it is statistically indistinguishable from tuned Gaussian processes, gradient boosting, and random forests in 24 of 27 comparisons (median nominal gap $+6\%$), more robust than a neural network, and answers queries at a cost independent of network size, with no training and no per-region tuning. It even matches or beats the satellite-based ACAG CNN in PM$_{2.5}$ in three of four mountain-west regions, though ACAG is calibrated on these very monitors; the satellite leads only under wildfire smoke. The reason is physics: on sparse networks, accuracy is decided not by the model but by whether the variable obeys a vertical law. Elevation cuts RMSE by $26$--$82\%$ for temperature (the lapse rate) and $9$--$13\%$ for precipitation (orographic uplift), but not at all for air quality, and under wildfire smoke the elevation--concentration relationship even inverts ($r=+0.51$), making elevation-detrending actively harmful. Because the interpolator is essentially free to re-run, the choice is settled empirically: cross-validate once with elevation and once without. Where a network exists, accurate fields need no training pipeline and, in ordinary conditions, no satellite: only geometry. \end{abstract}