Exploring first and second-order spatio-temporal structures of lightning strike impacts in the French Alps using subsampling

TL;DR

This study models cloud-to-ground lightning strike impacts in the French Alps as a spatio-temporal point process over $W\times T$ (2011–2021) to assess first- and higher-order structure while addressing computational challenges with subsampling. The authors develop and apply nonparametric kernel estimators for $\lambda_t$, $\lambda_s$, and $\lambda_{st}$, test first-order separability via $S_{st}, S_s, S_t$, and analyze higher-order structure using the space-time Ripley’s $K$-function $K_{\mathrm{inh},st}$ under an IRMS framework. Key findings show strong inhomogeneity in time and space, pronounced non-separability in the first order, and clear clustering in space and time evidenced by significant deviations from an inhomogeneous Poisson process. Subsampling (approximately $2.5\%$) enables efficient estimation and hypothesis testing, providing practical guidance for analyzing large-scale spatio-temporal event data. The work lays groundwork for incorporating covariates (elevation, atmospheric variables) and encourages development of refined intensity models for lightning activity in complex terrains.

Abstract

We model cloud-to-ground lightning strike impacts in the French Alps over the period 2011-2021 (approximately 1.4 million of events) using spatio-temporal point processes. We investigate first and higher-order structure for this point pattern and address the questions of homogeneity of the intensity function, first-order separability and dependence between events. The tuning of nonparametric methods and the different tests we consider in this study make the computational cost very expensive. We therefore suggest different subsampling strategies to achieve these tasks.

Figures (9)

• Figure 1: Locations of lightning strike impacts aggregated per day from June 2nd to June 7th 2018 (a quite active week) observed in the French Alps study domain, which includes a part of the Mediterrannean and Italian Alps. The red curve represents the Auer's climatological border between regions called 'North Alps' and 'South Alps'. Locations of impacts are superimposed on the altitude map for a better visualization.
• Figure 2: Nonparametric temporal intensity estimated by study area (Northern Alps, Southern Alps, All the French Alps) for the 2011-2021 period; Nonparametric temporal intensity estimated by study area (Northern French Alps, Southern French Alps, All the French Alps) by season (Winter=December to March, Spring=April to May, Summer=June to August, Autumn=September to November).
• Figure 3: Nonparametric spatial estimates of $\lambda_{s}$ aggregated per year from 2011 to 2021 and over the whole period 2011-2021 (thus a nonparametric estimate of $\lambda_s$). The red line for both figures marks the North/South division of the Alps according to auer2007histalp. For a better visualization, values of intensity smaller than 100 were discarded (less than 0.7% of values) and the values for the period 2011-2021 have been divided by 11 for a better comparison.
• Figure 4: Results for the first-order separability testing procedure obtained from two subsampled versions of $\textbf{X}_{st}^{\pi_0}$ (first and second rows). 95% global envelopes tests using the ERL procedure were used where permutations are used to generate data under the null. The considered summary statistics are $S_t(\cdot)$ (left, solid curve) and $S_s(\cdot)$ (right, latent image). Envelopes are depicted only for $S_t$ and red dots indicate times or pixels for which the estimated summary statistics is outside the envelopes. Light blue rectangles corresponds to summer seasons (from June to end of September). The adjusted global p-value for each of these envelope test is $p=0.05\%$.
• Figure 5: Combined global envelope test based on the summary functions $K_{\mathrm{inh},t}(\tau)$ and $K_{\mathrm{inh},s}(r)$. We have rescaled time events in $(0,1)$ to have an easier interpretation of the $x$-axis for $K_{\mathrm{inh},t}(\tau)$ (in particular 0.075 corresponds approximately to a one-month period). Observed curves have been truncated on the $y$-axis for a better visualization. Insets represent the non-truncated versions.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• proof