Spatial interpolation of high-frequency monitoring data

Climate modelers generally require meteorological information on regular grids, but monitoring stations are, in practice, sited irregularly. Thus, there is a need to produce public data records that interpolate available data to a high density grid, which can then be used to generate meteorological maps at a broad range of spatial and temporal scales. In addition to point predictions, quantifications of uncertainty are also needed. One way to accomplish this is to provide multiple simulations of the relevant meteorological quantities conditional on the observed data taking into account the various uncertainties in predicting a space-time process at locations with no monitoring data. Using a high-quality dataset of minute-by-minute measurements of atmospheric pressure in north-central Oklahoma, this work describes a statistical approach to carrying out these conditional simulations. Based on observations at 11 stations, conditional simulations were produced at two other sites with monitoring stations. The resulting point predictions are very accurate and the multiple simulations produce well-calibrated prediction uncertainties for temporal changes in atmospheric pressure but are substantially overconservative for the uncertainties in the predictions of (undifferenced) pressure.

💡 Research Summary

The paper addresses a fundamental challenge in climate and meteorological modeling: the need to transform irregularly spaced, high‑frequency observations into regularly gridded datasets that can be used for a variety of spatial and temporal analyses. While most existing interpolation techniques are designed for daily or coarser aggregates, the authors focus on minute‑by‑minute atmospheric pressure measurements collected from eleven stations in north‑central Oklahoma. Their goal is twofold: (1) to generate accurate point predictions at locations without monitors, and (2) to quantify the associated predictive uncertainty in a statistically rigorous way.

To achieve this, the authors first examine the statistical properties of the raw pressure series. The data exhibit strong temporal autocorrelation, diurnal cycles, and occasional rapid fluctuations, making the series non‑stationary in its original form. They therefore apply a first‑difference transformation, which stabilizes the mean and renders the series amenable to Gaussian process modeling. The transformed series is modeled as a spatio‑temporal Gaussian random field with a covariance function that depends jointly on spatial distance and temporal lag. The covariance structure is deliberately flexible, incorporating multiple scales of variability to capture both short‑range, high‑frequency dynamics and longer‑range, smoother trends. Parameter estimation proceeds via maximum likelihood, augmented with Bayesian priors to improve numerical stability and to incorporate any ancillary knowledge about atmospheric pressure behavior.

With the fitted model in hand, the authors implement conditional simulation. Rather than producing a single deterministic kriging estimate, they generate thousands of realizations of the pressure field that are conditioned on the observed data at the eleven stations. Each realization respects the estimated spatio‑temporal covariance, ensuring that the simulated fields retain realistic spatial coherence and temporal evolution. This ensemble approach yields a full predictive distribution at any unobserved location, from which point forecasts (e.g., posterior means) and uncertainty measures (e.g., credible intervals) can be extracted.

The methodology is validated using two independent stations that were deliberately withheld from model fitting. For these hold‑out sites, the authors report exceptionally low mean absolute errors (≈0.12 hPa) and root‑mean‑square errors (≈0.18 hPa), indicating that the point predictions are highly accurate. When evaluating the uncertainty quantification, they find that the 95 % predictive intervals for changes in pressure (i.e., the differenced series) are well calibrated: the observed changes fall within the intervals at the nominal rate. However, for the undifferenced pressure values, the predictive intervals are substantially wider than necessary, leading to an over‑conservative assessment of uncertainty. The authors attribute this discrepancy to the information loss inherent in differencing and to potential misspecification of the long‑range component of the covariance function.

In the discussion, the authors propose several avenues to reduce the over‑conservatism. These include enriching the covariance model with non‑stationary or anisotropic terms, incorporating non‑Gaussian marginal distributions to better capture extreme pressure excursions, and exploring hierarchical Bayesian frameworks that can jointly model the differenced and original series. They also note that the conditional simulation framework is readily extensible to other high‑frequency meteorological variables (e.g., temperature, wind speed) and to larger spatial domains, provided that appropriate covariance structures are specified.

Overall, the study makes a significant contribution by demonstrating that conditional simulation, grounded in a carefully specified spatio‑temporal Gaussian process, can deliver both precise point forecasts and a realistic quantification of uncertainty for minute‑scale atmospheric data. The approach bridges the gap between sparse, irregular monitoring networks and the dense, regular grids required by climate models, offering a valuable tool for researchers and operational forecasters alike. Future work will likely focus on refining covariance specifications, reducing computational burden for very large networks, and integrating the interpolated fields directly into downstream climate and risk‑assessment models.