Unlike other natural disasters, drought events evolve slowly in time and their impacts generally span a long period of time. Such features do make possible a more effective drought mitigation of the most adverse effects, provided a timely monitoring of an incoming drought is available.Among the several proposed drought monitoring indices, the Standardized Precipitation Index (SPI) has found widespread application for describing and comparing droughts among different time periods and regions with different climatic conditions. However, limited efforts have been made to analyze the role of the SPI for drought forecasting.The aim of the paper is to provide two methodologies for the seasonal forecasting of SPI, under the hypothesis of uncorrelated and normally distributed monthly precipitation aggregated at various time scales k. In the first methodology, the auto-covariance matrix of SPI values is analytically derived, as a function of the statistics of the underlying monthly precipitation process, in order to compute the transition probabilities from a current drought condition to another in the future. The proposed analytical approach appears particularly valuable from a practical stand point in light of the difficulties of applying a frequency approach due to the limited number of transitions generally observed even on relatively long SPI records. Also, an analysis of the applicability of a Markov chain model has revealed the inadequacy of such an approach, since it leads to significant errors in the transition probability as shown in the paper. In the second methodology, SPI forecasts at a generic time horizon M are analytically determined, in terms of conditional expectation, as a function of past values of monthly precipitation. Forecasting accuracy is estimated through an expression of the Mean Square Error, which allows one to derive confidence intervals of prediction. Validation of the derived expressions is carried out by comparing theoretical forecasts and observed SPI values by means of a moving window technique. Results seem to confirm the reliability of the proposed methodologies, which therefore can find useful application within a drought monitoring system.

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The results of numerical simulations of the lattice-Boltzmann equation in three-dimensional porous geometries constructed by the random positioning of penetrable spheres of equal radii are presented. Numerical calculations of the permeability are compared with previously established rigorous variational upper bounds. The numerical calculations approach the variational bounds from below at low solid fractions and are always within one order of magnitude of the best upper bound at high solid fractions ranging up to 0.98. At solid fractions less than 0.2 the calculated permeabilities compare well with the predictions of Brinkman’s effective-medium theory, whereas at higher solid fractions a good fit is obtained with a Kozeny–Carman equation.

Abstract. Assessment of landslide-triggering rainfall thresholds is useful for early warning in prone areas.In this paper, it is shown how stochastic rainfall models and hydrological and slope stability physically based models can be advantageously combined in a Monte Carlo simulation framework to generate virtually unlimited-length synthetic rainfall and related slope stability factor of safety data, exploiting the information contained in observed rainfall records and field-measurements of soil hydraulic and geotechnical parameters. The synthetic data set, dichotomized in triggering and non-triggering rainfall events, is analyzed by receiver operating characteristics (ROC) analysis to derive stochastic-input physically based thresholds that optimize the trade-off between correct and wrong predictions. Moreover, the specific modeling framework implemented in this work, based on hourly analysis, enables one to analyze the uncertainty related to variability of rainfall intensity within events and to past rainfall (antecedent rainfall). A specific focus is dedicated to the widely used power-law rainfall intensity-duration (I -D) thresholds.Results indicate that variability of intensity during rainfall events influences significantly rainfall intensity and duration associated with landslide triggering. Remarkably, when a time-variable rainfall-rate event is considered, the simulated triggering points may be separated with a very good approximation from the non-triggering ones by a I -D powerlaw equation, while a representation of rainfall as constantintensity hyetographs globally leads to non-conservative results. This indicates that the I -D power-law equation is adequate to represent the triggering part due to transient infiltration produced by rainfall events of variable intensity and thus gives a physically based justification for this widely used threshold form, which provides results that are valid when landslide occurrence is mostly due to that part. These conditions are more likely to occur in hillslopes of low specific upslope contributing area, relatively high hydraulic conductivity and high critical wetness ratio. Otherwise, rainfall time history occurring before single rainfall events influences landslide triggering, determining whether a threshold based only on rainfall intensity and duration may be sufficient or it needs to be improved by the introduction of antecedent rainfall variables. Further analyses show that predictability of landslides decreases with soil depth, critical wetness ratio and the increase of vertical basal drainage (leakage) that occurs in the presence of a fractured bedrock.

[1] Extreme droughts may be characterized by their duration, severity (magnitude or intensity), spatial extent, and frequency or return period. Comparing the time series of water supply and water demand and analyzing droughts based on the theory of runs may determine these characteristics. This study is focused on drought analysis where the underlying water supply process is periodic stochastic, such as for monthly streamflows. The probability mass function (pmf) of drought length and associated low-order moments are derived assuming a periodic simple Markov chain. The derived pmf allows estimating the occurrence probability of droughts of a given length and its return period. The applicability of the drought formulations has been illustrated using a variety of water supply series such as monthly and weekly precipitation, monthly streamflows, the Palmer hydrologic drought index, and the standardized precipitation index. The results obtained confirm the validity of the analytical derivations for drought lengths and associated return periods. The overall conclusion of the study is that simple definitions of droughts enables one characterizing droughts using stochastic approaches and analytical derivations. They are particularly useful for drought analysis because the limited hydrologic records that are generally available do not allow observing many drought events of a particular duration and, in fact, extremely long droughts may not even be observable from the historical sample. This hinders the applicability of an inferential approach for finding the probability distributions of drought lengths and their associated return periods because it is either impractical or not feasible.

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Planning and management of water resources systems under drought conditions often require the estimation of return periods of drought events characterized by high severities. Among the several methods proposed for describing droughts, the run method is the most suitable to provide an objective identification and characterization of drought events. According to such a method, droughts are identified as consecutive intervals where the investigated hydrological variable is continuously below a fixed threshold, and may be described by means of two characteristics, namely, drought duration and drought severity. Since both characteristics are necessary to estimate water deficit risks, frequency analysis of drought events cannot be based on the same approach generally used for flood analysis, such as maximum annual series or partial duration series of a single characteristic. In particular, the evaluation of return period for drought events needs to consider both duration and severity in order to take into account the pluriannual duration of several droughts. Very often a reliable analysis of the probabilistic structure of droughts based on the observed samples, using an inferential approach, cannot be properly carried out due to the limited number of drought events which can be identified even on quite long historical series. This problem has been faced by Shiau and Shen (2001), who have determined the conditional distribution of drought severity given a drought duration on the basis of generated hydrological series. In this paper their approach is extended by deriving analytically the parameters of the probability distribution of drought severity based on the stochastic process describing the underlying hydrological variable. More specifically, a gamma distribution is adopted to model drought severity and its parameter are theoretically determined as a function of the threshold level and the coefficient of variation of annual precipitation series assumed independent and lognormal distributed. Then, the return period of drought events with severity greater than or equal to a fixed value is computed as the mean interarrival time of drought events with a certain severity or greater. Such procedure has been applied on 88 annual series of data recorded in Sicilian rainfall stations, by computing for each series the return period corresponding to