pyBL: An open source Python package for stochastic high-resolution rainfall modelling based upon a Bartlett Lewis Rectangular Pulse model

<p>Stochastic rainfall modelling is an increasingly popular technique used by the water and weather risk industries. It can be used to synthesise sufficiently long rainfall time series to support hydrological applications (such as sewer system design) or weather-related risk analysis (such as excess rainfall insurance product design). The Bartlett-Lewis (BL) rectangular pulse model is a type of stochastic model that represents rainfall using a Poisson cluster point process. It is calibrated with standard statistical properties of rainfall data (e.g. mean, coefficient of variation, skewness and auto-correlation and so on), but it can well preserve extreme statistics of rainfall at multiple timescales simultaneously. In addition, it is found to be less sensitive to observational data length than the existing rainfall frequency analysis methods based upon, for example, annual maxima time series, so it provides an alternative to rainfall extremes analysis when long rainfall datasets are not available. </p><p>In this work, we would like to introduce an open source Python package for a BL model: pyBL, implemented based upon the state-of-the-art BL model developed in Onof and Wang (2020). In the pyBL package, the BL model is separated into three main modules. These are statistical properties calculation, BL model calibration and model sampling (i.e. simulation) modules. The statistical properties calculation module processes the input rainfall data and calculates its standard statistical properties at given timescales. The BL model calibration module conducts the model fitting based upon the re-derived BL equations given in Onof and Wang (2020). A numerical solver, based upon Dual Annealing optimization and Nelder-Mead local minimization techniques, is implemented to ensure the efficiency as well as to prevent from being drawn to local optima during the solving process. Finally, one can use the sampling module to generate stochastically rainfall time series at a given timescale and for any required data length, based upon a calibrated BL model.</p><p>The design of the pyBL is highly modularized, and the standard CSV data format is used for file exchange between modules. Users could easily incorporate given modules into their existing applications. In addition, a team, consisting of researchers from National Taiwan University and Imperial College London, will consistently implement the new breakthroughs in BL model to the package, so users will have access to the latest developments. The package is now undergoing the final quality check and will be available on Github (https://github.com/NTU-CompHydroMet-Lab/pyBL) in due course. </p>

Brownian Survival in a Clusterized Trapping Medium

The survival problem for a Brownian particle moving among random traps is considered in the case where the traps are correlated in a particular fashion being gathered in clusters. It is assumed that the clusters are statistically identical and independent of each other and are distributed in space according to a Poisson law. Mathematically, such a trapping medium is described via a Poisson cluster point process. We prove that the particle survival probability is increased at all times as compared to the case of noncorrelated (Poissonian) traps, which implies the slowdown of the trapping process. It is shown that this effect may be interpreted as the manifestation of the trap "attraction", thus supporting assertions on the qualitative influence of the trap "interaction" on the trapping rate claimed earlier in the physical literature. The long-time survival asymptotics (of Donsker–Varadhan type) is also derived. By way of appendix, FKG inequalities for certain functionals are proven and the limiting distribution for a Poisson cluster process, under clusters' scaling, is determined.

Markov properties of cluster processes

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

Markov properties of cluster processes

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.