The statistical modelling of daily rainfall depth is commonly carried out independently from the rainfall occurrence process (dry/wet alternation). In this study, we investigate the effects of accounting for the dependence between rainfall depth (h) and wet spell duration (ws) using rain gauge data collected on 34 sites across Europe between 1971 and 2016. For each site, a set of conditioned haws sub-datasets was generated, and the two-parameter Polylog distribution was fitted on these sub-datasets. The results of the fittings highlight a good capability of the Polylog distribution to capture the statistical behavior of haws, as well as the presence of statistically significant differences in the various haws fittings obtained in some of the sites. This dependence has been quantified as "Weak" in about 13 sites (38%), and "Strong" in 9 sites (26%). The haws dependence was directly modelled for each site with a logarithmic function, showing how this approach is able to well reproduce the arithmetic and geometric means of the sub-samples (standard estimate errors of 0.62 and 0.42 mm, respectively). An attempt at explaining the spatial variability of the dependence through a penalized regression highlighted some key factors, such as longitude and distance from the shoreline; however, the selected features only partially explained the variability and the connection to climate factors. Overall, this research introduces a simple procedure to directly account for the haws dependence, which can provide a more robust modelling of the rainfall processes, as well as an improved understanding of the rainfall generating processes.

The modelling of the occurrence of rainfall dry and wet spells (ds and ws, respectively) can be jointly conveyed using the inter-arrival times (it). While the modelling of it has the advantage of requiring a single fitting for the description of all rainfall time characteristics (including wet and dry chains, an extension of the concept of spells), the assumption on the independence and identical distribution of the renewal times it implicitly imposes a memoryless property on the derived ws, which may not be true in some cases. In this study, two different methods for the modelling of rainfall time characteristics at station scale have been applied: i) a direct method (DM) that fits the discrete Lerch distribution to it records, and then derives ws and ds (as well as the corresponding chains) from the it distribution; and ii) an indirect method (IM) that fits the Lerch distribution to the ws and ds records separately, relaxing the assumptions of the renewal process. The results of this application over six stations in Europe, characterized by a wide range of rainfall regimes, highlight how the geometric distribution does not always reasonably reproduce the ws frequencies, even when it are modelled by the Lerch distribution well. Improved performances are obtained with the IM, thanks to the relaxation of the assumption on the independence and identical distribution of the renewal times. A further improvement on the fittings is obtained when the datasets are separated into two periods, suggesting that the inferences may benefit for accounting for the local seasonality.

<p>Analysis of daily rainfall data, and subsequent modelling of some derived variables concerning rainfall, is fundamental in different areas such as agricultural, ecological, and engineering disciplines. A way of studying the alternance of consecutive rainy days (wet spells) and no-rainy days (dry spells) is through the interarrival time (<em>IT</em>), which is the time elapsed between two consecutives rainy days. If we suppose that <em>IT</em> observations are independent and identically distributed (i.i.d.), <em>ITs</em> are usually modelled through a renewal processes. The simplest renewal process is the Bernoulli process with <em>ITs</em> geometrically distributed. The need to suppose a non-constant probability of rain brings to more sophisticated models. Previous works [Agnese et al. (2014), Baiamonte et al. (2019)] have successfully proposed the three-parameter family of the <em>Hurwitz-Lerch-Zeta</em> distribution (HLZD), which represents a forward step with respect to other commonly used <em>IT</em> distributions. In [Agnese et al. (2022)], a second successfully reached goal was to show that the HLZD is also suitable to model the rainfall depth, <em>h</em>. In literature, rainfall depths are more frequently treated as continuous, despite sometimes these models fail to account for the time discreteness of the sampled process. Indeed, daily rainfall depth measurements are usually carried out by automatic-counting how many times a small bucket corresponding to 0.2 mm is filled. Due to the abundance of ties in the data, the variable depth <em>h</em> is well suited to be considered discrete. We present results involving data never considered in literature and consisting of measures sampled along 60-70 years at 7 different stations. These stations represent different climates from the rainfall characteristics point of view and let us to infer about the great handiness of the HLZD within rainfall modelling. Current research is addressed to modelling further rainfall variables related to <em>IT</em> and <em>h</em>, such as wet and dry spells, and the cumulative rainfall depth in a wet spell. Furthermore, given the remarkable performance of the HLZD family of distributions in the univariate modelling, we aim at modelling the dependence structure between <em>IT</em> and <em>h</em>, exploiting possibly new methodological advances in the subject, such as discrete copulas.</p> <p> </p> <p><strong>References</strong></p> <ul> <li>Agnese, G. Baiamonte, E. Di Nardo, S. Ferraris, and T. Martini (2022). Modelling the frequency of interarrival times and rainfall depths with the Poisson Hurwitz-Lerch zeta distribution. <strong>Fractal and Fractional</strong>, 6(9).</li> <li>Agnese, G. Baiamonte, and C. Cammalleri (2014). Modelling the occurrence of rainy days<br />under a typical Mediterranean climate. <strong>Advances in Water Resources</strong>, 64:62–76.</li> <li>G. Baiamonte, L. Mercalli, D. C. Berro, C. Agnese, and S. Ferraris (2019). Modelling the frequency distribution of inter-arrival times from daily precipitation time-series in north-west Italy. <strong>Hydrology Research</strong>, 50(1):339–357.</li> </ul>

In the hydrologic literature, to model water flow in unsaturated soils, the Richards equation is usually applied, allowing the main components of the hydrologic cycle, as rainfall partitioning into surface runoff and infiltration, to be determined. The Richards equation is highly nonlinear, making it very challenging to derive analytical solutions. Recently, for constant rainfall intensity, under the simplified hypothesis of gravity-driven infiltration, and by assuming a capacitance framework, a simplified solution of the Richards equation that considers the Brooks and Corey hydraulic conductivity function, was suggested. By maintaining the assumption that the infiltration process is dominated by gravity, the objective of this paper is to relax the capacitance sketch applied to only one reservoir, by replicating the internal water content travel times through the discretization of the soil profile into multiple tanks connected in series. First, the previous analysis is briefly summarized, which is useful for the further developments. Then, for a fixed soil pore connectivity index (c = 0.5), which could be assumed for sandy soils, the same approach is extended to multiple non-linear reservoirs to model water balance components, achieving more reliable conditions. The presented approach could be not affected by uncertainty, since it is simply hydraulic, provided the hypotheses c = 0.5 and the assumption of a gravity-driven infiltration are satisfied. The suggested solution was compared with that derived by the Richards equation (via Hydrus-1D), where the gravity-driven hypothesis is relaxed the effect of the number of reservoirs was analyzed, and applications for both constant and time-variable rainfall intensity were performed.

The Poisson-stopped sum of the Hurwitz–Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz–Lerch zeta distribution and the one inflated Hurwitz–Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz–Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates.

The discrete three-parameter Lerch distribution is used to analyse the frequency distribution of inter-arrival times derived from 26 daily precipitation time-series, collected by stations located throughout a 28,000 km(2) area in North-West Italy (altitudes ranging from 113 m to 2,170 m a.s.l.). The precipitation regime of these Alpine regions is very different (latitude 44.5 to 46.5 N) from the typical Mediterranean precipitation regime of the island of Sicily (latitude 37 to 38 N), where the Lerch distribution has already been tested and whose results are compared. In order to verify the homogeneity of the precipitation time series, the Pettitt test was preliminarily performed. In this work, a good fitting of the Lerch distribution to NW Italy is shown, thus evidencing the wide applicability of this kind of distribution, also allowing to jointly model dry spells and wet spells. The three parameters of the Lerch distribution showed rather different values than the Sicily ones, likely due to the very different precipitation regimes. Finally, a relevant spatial variability of inter-arrival times in the study area was revealed from the regional scale application of the probability distribution here described. The outcomes of this study could be of interest in different hydrologic applications.

In this work, the probability distribution of peak discharge at the hillslope bottom is determined hypothesizing a prevalent Hortonian mechanism of runoff production for a given rainfall duration. As is well known, the probability distribution of peak discharge depends on the probability of both the rainfall event as well as that of the antecedent soil moisture conditions. In particular, the probability of the rainfall event is calculated according to the familiar rainfall duration-intensity-frequency approach, whereas the ecohydrological method from the literature is used here to define the probability of the antecedent soil moisture conditions. The latter depends on a set of parameters describing the dynamic interactions between average climate, soil and vegetation. By using the Monte Carlo procedure, the peak discharge is derived for a given rainfall duration and for each antecedent moisture condition/rainfall intensity pair from a physical-based model from the literature, by coupling the analytical solution of the overland flow equations over a hillslope with an established model that accounts for the infiltration process. Thus, the probability of peak discharge is evaluated via the typical multivariate probability distribution procedure. The methodology proposed was applied to three soil classes, i. e., silty clay loam (SCL), silty clay (SC), and silty loam (SL), for three climatically diverse Sicilian localities, namely Acireale (eastern Sicily, along the Ionian coast), Enna (central mountainous region), and Trapani (westernmost coast). For each of these places, precipitation and temperature data sets are widely available. (C) 2015 American Society of Civil Engineers.

It is widely recognized that the Hortonian mechanism of runoff generation occurs in arid and semi-arid regions, generally characterized by high rainfall intensity on soils exhibiting low infiltrabilities. Differently, in steeply sloping forested watersheds in humid climates, by infiltrating through a highly permeable upper soil horizon, water moves beneath the soil surface determining a slow response. However, in most real cases, for example when in arid regions mountain forested areas take place, both (quick and slow) runoff generation processes coexist and together contribute to the hydrologic hillslope response. In this paper, based on analytical solutions of the hydrologic response, instantaneous response functions of both quick and slow components are defined, depending on parameters characterizing geometrical and dynamical features at the hillslope scale of immediate physical meaning. For each response component, two characteristic time-scales are defined, the mean holding time spent by a particle in its motion through the hillslope and the so-called time to equilibrium, accounting for a critical rainfall duration generating a maximum discharge at the bottom of the hillslope. Approximated instantaneous response function (IRFs) in the usual form of gamma probability density functions (pdfs) are proposed, which allowed reducing the many parameters appearing in both quick and slow response models to the scale and shape parameters, also incorporating the effect of different antecedent flow conditions. At the basin scale, the collective response of all the hillslopes is described by a representative hillslope, which combined response accounts for the relative dominance of quick and slow response.(C) 2016 American Society of Civil Engineers.

Studies have shown that the footprint of a single eddy covariance (EC) system may not yield representative measurements of the turbulent fluxes at the field scale for sparse vegetated surfaces, whereas scintillometry, due to its larger footprint, may be more suitable for this purpose. However, the latter approach strongly relies on the Monin-Obukhov similarity theory (MOST) that strictly applies in the inertial sub-layer only. This work aims at experimentally confirm the reliability of displaced-beam laser scintillometer (DBLS) measurements over an olive orchard against two EC systems during summer and autumn months of 2007 through 2009. It was found that the DBLS underestimated both the momentum and sensible heat fluxes by 15 to 20% when established retrieval procedures were applied. A new method to determine the sensible heat flux from the DBLS based on the addition of a single-height wind speed measurement was tested, yielding estimates that compare well with the EC observations, with discrepancies in sensible heat fluxes of about 30 to 40 W m(-2).

The statistical inference of the alternation of wet and dry periods in daily rainfall records can be achieved through the modelling of inter-arrival time-series, IT, defined as the succession of times elapsed from a rainy day and the one immediately preceding it. In this paper, under the hypothesis that ITs are independent and identically distributed random variables, a modelling framework based on a generalisation of the commonly adopted Bernoulli process is introduced. Within this framework, the capability of three discrete distributions, belonging to the Hurwitz-Lerch-Zeta family, to reproduce the main statistical features of IT time-series was tested. These distributions namely Lerch-series (Lerch), polylogarithmic-series (Polylog) and logarithmic-series (Log) were selected thanks to their capability to describe some peculiar properties usually observed in IT series derived from daily rainfall records: very high standard deviation and skewness, relatively high frequency associated to the unitary IT, monotonically decreasing frequencies with a slow decay. Both Polylog and Log distributions are special cases of the 3-parameter Lerch distribution with a decreasing number of free parameters (2- and 1-parameter, respectively). The analysis, performed on 55 raingauges located in Sicily (Italy) under a typical Mediterranean climate, suggests that a reliable statistical representation of IT can be attained with the 3-parameter Lerch distribution. Despite the marked seasonality of rainfall in the study area, a simple subdivision of the year into two 6-month periods, roughly corresponding to the dry "semester" (D-sem) and the wet "semester" (W-sem), allows a satisfactory reproduction of IT, as well as of wet spells (WS) and dry spells (DS), separately. It was also noticed that the 2-parameter Polylog distribution could be successfully used to reconstruct the DS frequency distribution only. This result suggests that the additional parameter of the Lerch distribution is required by the inclusion of WS into the analysis. Finally, considering that Polylog outperforms the commonly adopted Log, a noteworthy step forward in DS modelling can be achieved by using Polylog distribution rather than Log one. (C) 2013 Elsevier Ltd. All rights reserved.

The daily scale is the maximum time-scale at which both clustering and intermittency characters of rain are still evident. These features can be viewed as a natural tendency to the persistence in time of the two opposite atmospheric states, i.e. rainy and dry (not rainy) states. In this work, the probabilistic structure of inter-arrival times, T, is analysed with the aim to account for the statistics of both clustering of rainfall and persistence of dry periods in a single distribution. A discrete probability distribution, the three-parameter Lerch distribution, has been applied to some inter-arrival time series derived from daily rainfall data recorded in Sicily and in Piedmont and in Aosta Valley, representing two different climatic environments. Statistical analyses have been carried out by considering two “seasons” of six months each: a growing season (GS) from April to September and a dormant season (DS) from October to March. Parameters have been estimated by using the maximum likelihood method and the performance of model fitting was evaluated by means of a chisquare test based on Monte Carlo procedure.

Quantitative evaluation of the drought adaptation processes of crops is an important prerequisite for efficient irrigation management. Modeling the plant response under water stress conditions is crucial to identify the exact irrigation timing. Assessment of any water stress function requires the knowledge of its shape and then the estimation of critical thresholds of the soil water status, below which a strong reduction of plant transpiration occurs. In this work, the macroscopic approach is used to assess the water stress function implemented in AQUACROP for mature olive tree. In particular, after discussing about the function shape, the critical thresholds of soil water status are proposed according to experimental data. Eco-physiological measures (leaf and stem water potentials) were used as water stress indicators, whereas the relative depletion was considered as independent variable. The investigation evidenced, for the investigated crop, that a convex shape better reproduces the water stress function.

This paper compares two agro-hydrological models that are used to schedule irrigation of a typical Mediterranean crop. In particular, a comparison between the Food and Agriculture Organization (FAO) model, which uses a black box approach, and the soil-water-atmosphere-plant (SWAP) model, which is based on the numerical analysis of Richards' equation, are shown for wine grape. The comparison was carried out for the 2005 and 2006 irrigation seasons and focused on hydrological balance components and on soil water contents. Next, the ordinary scheduling parameters were identified so that the performance of the two models, which aimed to evaluate the seasonal water requirements and the irrigation times, could be assessed. In the validation phase, both of the models satisfactorily simulated the soil water content, and comparable values of cumulative evapotranspiration were obtained. With the goal of recognizing the crop water stress condition in the field, the original algorithm of the FAO model was modified. This research provided evidence of how the two agro-hydrological models, although characterized by different approaches in modeling the phenomena, showed a similar behaviour when used for scheduling irrigation under soil water deficit conditions. DOI: 10.1061/(ASCE)IR.1943-4774.0000435. (C) 2012 American Society of Civil Engineers.

Table Olive (Olea europeae L.) is an important crop for the Mediterranean countries. In the past, olive grove were mostly rain fed, due to their resilience to water scarcity; the practice of irrigation is relatively recent and it has been introduced in order to increase crop productions and to improve yield quality (Patumi et al., 2002; D’Andria et al., 2004). Several researches have been focusing on the optimization of irrigation for olive trees (Fernandez and Moreno, 1999) and it has been recognized how, maintaining olive trees under slight or moderate water stress at specific phenological stages, can contribute to optimize yield and water use efficiency (Patumi et al., 1999; Berenguer et al., 2006; Caruso et al., 2011). The impact of water stress, as well as its feasible duration and intensity, depends on crop phenological

Daily solar radiation R-s at ground level is a necessary input variable required for the evaluation of evapotranspiration and crop growth, development, and yield-simulation models. Nevertheless, it is measured in few weather stations and at many locations it is not observed; also, available R-s temporal series are generally no longer than a few years. A valid surrogate of R-s measurement is the diurnal air-temperature range (Delta T); indeed, Delta T is inversely proportional to cloudiness and therefore could be a good indicator of atmospheric transmittance. As opposed to R-s, daily maximum and minimum air temperatures are measured at many locations and their observations in developed countries began in the 19th century. For this reason, several models that permit R-s indirect evaluation from air-temperature data have been suggested in the literature. The most famous models are the simple Hargreaves-Samani (HS) formula, many times recalibrated by the authors, and the Bristow-Campbell model, which has recently been improved. In this paper, the suitability of each proposed model is tested by comparing R-s real data, recorded in 33 Sicilian agrometeorological stations in the period 2003-2008, typically representing the Mid-Mediterranean area, with R-s estimates obtained by the models from Delta T data. In addition, a regional relationship is obtained for the scale coefficient K(T) of the Hargreaves-Samani formula: the relationship found improves R-s data prediction with respect to the original HS formulation. DOI: 10.1061/(ASCE)IR.1943-4774.0000480. (C) 2012 American Society of Civil Engineers.

In the paper a comparison between radiation-aerodynamic and radiation based evapotranspiration models with an independent dataset of Alfalfa reference evapotranspiration acquired by means a scintillometer is carried out. The satisfactory performance of the selected models is comparable to that showed in earlier investigations. This study also shows that using values of model parameters locally determined, is possible to improve hourly estimation of potential evapotranspiration, as obtained with the scintillometer.