21 Nov 2025
| 21 Nov 2025
Status: this preprint is open for discussion and under review for Nonlinear Processes in Geophysics (NPG).
A tempered fractional Hawkes framework for finite-memory drought dynamics
Abstract.
Meteorological droughts emerge from nonlinear land--atmosphere feedbacks and circulation anomalies, whose persistence and recurrence are not well represented by conventional stochastic models with exponentially decaying memory. Here we introduce a Tempered Fractional Hawkes Process (TFHP) to describe drought onsets as a self-exciting point process with algebraically decaying but ultimately finite memory. In this formulation, the conditional intensity of dry-spell initiation obeys a tempered fractional differential equation in the Caputo sense, where the kernel ϕ(t) ∝ t−αe−θt combines long-range dependence with exponential tempering that enforces dynamic stability. The model parameters have clear physical meaning: µ (baseline exogenous activity), κ (self-excitation strength), α (fractional memory order), and θ−1 (finite-memory horizon). Using 43 years of daily ERA5 precipitation over continental Chile, we estimate spatial fields of (µ, κ, α) and derive an effective memory timescale τm = 1/θeff. Results reveal geographically organised persistence regimes: subtropical arid and high-latitude subpolar regions exhibit slowly decaying memory and strong endogenous reinforcement, whereas mid-latitude zones display faster relaxation and weaker feedbacks. The TFHP thus offers a parsimonious and physically interpretable representation of finite climatic memory, bridging fractional calculus, point-process theory, and nonlinear geophysical dynamics. Beyond drought analysis, it provides a general framework to quantify persistence, clustering, and resilience in non-Markovian environmental systems.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.
For a detailed report oon the article, see the document attached.
The article introduces tempered and untempered fractional Hawks processes which are self-exiting point processes with decaying memory, and derives a mean field formulation of the intensity λ(t). Given rainfall data from Chile, a log-likihood maximization is performed to fit the parameters of an untempered fractional Hawkes process to the occurrence of drought periods. Based on these parameters, the temperedness parameter θ is derived and the results are compared.
While tempered fractional calculus is established elsewhere, the tempered factional Hawkes process (TFHP) is to the best of my knowledge new. The introduction of the TFHP contains multiple mathematical inaccuracies and inconsistencies that are listed in the document provided. In particular, the interpretation of the parameter α seems reversed, thus making the interpretation of the results problematic. Additionally, the newly introduced TFHP is effectively not used in the numerical method, which almost exclusively relies on untempered fractional Hawkes processes.
Overall, these problems result in inconsistencies in both the theory and the application part of the article. The theoretical issues are certainly fixable, even though this would require rewriting most of Section 2. The problem that the TFHP is not actually used in the numerical method is a more fundamental weakness of the article. In view of the improvable methodology, the results do not seem to provide a very clear interpretation, in particular with the inconsistencies regarding the parameter α (see comment (I) in the provided document).