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During the recent years there has been an increasing interest in multivariate frequency analysis of hydrological variables, e.g. those describing extreme events like rainfall, floods or droughts. The multivariate analysis provides a better understanding of the phenomena under investigation and an additional insight about the interrelationships between the different variables (e.g. peak, volume and duration of the flood), exploiting the complete structure of the problem and making a full use of the available data. However, while the developments on multivariate analysis of hydrological data has produced a large body of literature, a clear assessment of the use of these methods in the design and risk assessment of hydraulic structures is still a matter of debate. In the present work we illustrate a general, structure-based framework for the design and/or risk assessment of hydraulic structures in a bivariate environment; we also compare it to recently proposed methods which are based on the assumption of hydrological design events (as is customary in the univariate context). For illustration purposes, both the structure-based and the design event-based approaches are applied to the design of an idealized structure, thus exploring the differences among the methods as function of the parameters involved. Our work highlights that the return period of structure failure in a multivariate environment strictly depends on the particular structure under design, and in most cases the design of an hydraulic structure cannot be based on a single, hydrological multivariate design event. This acts as a warning for practitioners against the use of design methods based on single hydrological events, as usually done in the context of univariate hydrology, thus neglecting the interplay between the structure and the hydrological loads acting on it.

"Quantification of the various components of hydrological processes in a watershed remains a challenging topic as the hydrological system is altered by internal and external drivers. Watershed models have become essential tools to understand the behavior of a catchment under dynamic processes. In this study, a physically-based watershed model called Soil Water Assessment Tool (SWAT) was used to understand the hydrologic behavior of the Upper Tiber River Basin, central Italy. The model was successfully calibrated and validated using observed weather and flow data for the period of 1963–1970 and 1971–1978 respectively. Eighteen parameters were evaluated and the model showed high relative sensitivity to groundwater flow parameters than the surface flow parameters. Analysis of annual hydrological water balance was performed for the entire upper Tiber watershed and selected sub-basins. The overall behavior of the watershed was represented by three categories of parameters governing surface flow, sub-surface flow and the whole basin response. The base flow contribution has shown that 60% of the stream flow is from shallow aquifer in the sub-basins. The model evaluation statistics that evaluate the agreement between the simulated and observed streamflow at the outlet of a watershed and other three different sub-basins has shown coefficient of determination (R2) from 0.68 to 0.81 and Nash Sutcliffe Efficiency (ENS) between 0.51 and 0.8 for the validation period. The components of the hydrologic cycle showed variation for dry and wet period within the watershed for the same parameter sets. Based on the calibrated parameters the model can be used for prediction of the impact of climate and land use changes and water resources planning and management"

"The influence of rainfall movement on the peak and shape of basin hydrograph has. been long recognized. Early literature results highlight that a storm moving in the downstream direction. produces a greater peak discharge, compared with a storm moving upstream. Recent studies pointed out. that the assumption of uniform rainfall can lead to a potential underestimation of flood risks, and that this. can be hugely enhanced by rainstorm movement, also depending on regional characteristics of storm. dynamics.. In this study we propose an analytical approach to investigate the influence of the pattern of moving. rainstorms on the hydrological response of a river basin. We analyze the response of the basin to an. excess rain field (rain contributing to direct runoff), which is assumed to be variable in space and time;. catchment response to an instantaneous input of uniform rainfall is assumed to be fully characterized by. the time-invariant distribution of the travel time of water particles within the basin.. Our results show how, in the framework assumed herein, peak flows magnification is a consequence of a. rainstorm moving over the basin in synchrony with surface flows. This exact superposition, which is called. "resonance" condition, depends on the relative size of the rainstorm with respect to basin size. In particular,. for storm sizes much smaller than the dimension of the catchment, the resonance condition occurs for. infinite combinations of the direction and speed of the moving rainstorm. For illustration purposes, we. present and discuss a case study based on the "open-book" idealized catchment."

"Vengono indagati gli effetti della variabilità spaziale dei campi di precipitazione netta sulla risposta idrologica, applicando un modello di trasformazione afflussi-deflussi geomorfologico basato sull’approccio per tempi di residenza. Sono analizzati, in particolare, diversi rapporti tra le scale spaziali caratteristiche del bacino e del campo di pioggia netta e diverse configurazioni di velocità di trasferimento. Obiettivo del lavoro è quello di determinare sotto quali condizioni la variabilità spaziale della pioggia appare in grado di produrre effetti rilevanti. Si suppone che la precipitazione netta sia rappresentata da un campo stazionario ed isotropo, di cui sono noti i momenti del primo e secondo ordine e la scala integrale. Gli sviluppi analitici presentati permettono la quantificazione degli effetti della variabilità del campo di pioggia netta per alcune condizioni asintotiche; per le condizioni intermedie (non asintotiche) si è fatto ricorso a simulazioni numeriche Monte Carlo."

Vengono indagati gli effetti della variabilità spaziale dei campi di precipitazione netta sulla risposta idrologica, applicando un modello di trasformazione afflussideflussi geomorfologico basato sull’approccio per tempi di residenza. Sono analizzati, in particolare, diversi rapporti tra le scale spaziali caratteristiche del bacino e del campo di pioggia netta e diverse configurazioni di velocità di trasferimento. Obiettivo del lavoro è quello di determinare sotto quali condizioni la variabilità spaziale della pioggia appare in grado di produrre effetti rilevanti. Si suppone che la precipitazione netta sia rappresentata da un campo stazionario ed isotropo, di cui sono noti i momenti del primo e secondo ordine e la scala integrale. Gli sviluppi analitici presentati permettono la quantificazione degli effetti della variabilità del campo di pioggia netta per alcune condizioni asintotiche; per le condizioni intermedie (non asintotiche) si è fatto ricorso a simulazioni numeriche Monte Carlo.

"The spatial distribution of rainfall can play a key role in determining the hydrological response of river basins. Circumstances under which this role is dampened or enhanced, while extensively discussed in previous literature works, have not yet received a clear assessment.. We propose an analytical approach to investigate the dependence of basin hydrograph variability on the spatial heterogeneity of rainfall, specifically considering the response of a catchment to an instantaneous unit pulse of spatially-variable excess rainfall. Catchment response to rainfall input is assumed to be completely described by the time-invariant distribution of travel times. The excess-rainfall field accounts for the variability of both precipitation and soil characteristics and is assumed to be represented as a stationary and isotropic process, defined by its variance and integral scale. The sensitivity of the hydrologic response to the rainfall spatial variability is evaluated describing the variance of the generated hydrograph as function of basin characteristics and the stochastic properties of excess-rainfall field. This analytical approach allow us to infer results that do not depend on a specific study case.. Results suggest that the variability of the hydrologic response due to spatial rainfall heterogeneity mainly depends on the ratio between the integral scale of the excess-rainfall field and basin drainage area. For basins which are small compared to the integral scale of excess rainfall, rainfall spatial heterogeneity has a minor relevance on the hydrologic response, i.e. basin sees the precipitation as uniformly distributed. In the opposite case, for relatively large basins the spatial variability of excess rainfall is averaged over the basin area and the hydrologic response is approximately equal to the travel time distribution; under such circumstances the system is ergodic. However, at the early and late branch of the travel time distribution, i.e. in areas close to the basin outlet and those far away from it, the coefficient of variation of the hydrographs may be still high even for large basins. This happens where the isochrones are small, leading to a poor averaging of the rainfall field (non-ergodic behavior).. These asymptotic limits have been confirmed by the performance of numerical simulations, based on a Monte Carlo approach and applied to a specific study basin. Numerical experiments use a simplified width-function geomorphological model. They helped us in understanding the behavior of the response function for intermediate conditions (medium size basins), when the square of the integral scale and the drainage are same order of magnitude.. "

The probability density function (pdf) of solute concentration is a useful tool for modeling transport of contaminants in heterogeneous aquifers that is increasingly used in risk assessment and, more generally, as a mean to quantify uncertainty in transport modeling. In order to be effective the pdfs should be linked in a simple manner to the spatial variability model of hydraulic properties, pore-scale (local) dispersion, and a suitable parametrization of the geochemical processes. We analyze the pdf and concentration moments of two aqueous species in equilibrium with their precipitate reacting upon mixing in two-and three-dimensional geological formations. The speciation equations, resulting from application of the chromatographic theory, provide the link between concentration pdfs (and moments) of the aqueous species and that of a passive tracer. Within this framework, we investigate the role of pore-scale dispersion and macrodispersion in enhancing mixing, and thus reaction between the aqueous species, in the case of an instantaneous injection of a water with contrasting concentrations with respect to the ambient water under the constraint that in both waters the two aqueous species are in equilibrium with their precipitate. The main conclusion of our analysis is that for the pore-scale dispersion typically observed in natural formations, the local concentration pdfs of both species are far from being Gaussian, and therefore the first two concentration moments provide very limited information of the underlying transport dynamics. Instead, the pdfs provide crucial information for applications, such as the probability of exceeding a given concentration, for example, the regulatory limit, at a particular location within the domain of interest. Furthermore, by analyzing the second-order moments of the concentration, we show that mixing is strongly affected by space dimensionality and that the two-dimensional approach, often used for computational convenience, may severely underestimate reaction rates in real settings.

"Hydrological events are often characterized by the joint behavior of several correlated random variables (e.g. storm duration and intensity). As a consequence, the critical events should be defined in terms of two or more variables; that avoids the over- or underestimation of the hydrological risk, that might be produced by the univariate probability analysis.. In the multivariate analysis, the choice of an appropriate return period for structure design leads to infinite combinations of values (isoline of the return period), each of them could be used to evaluate the effects of different hydrological loads on the structure. This work presents a new methodology to identify a limited subset of design events from the set of the infinite combinations, on the basis of their probability of occurrence. This methodology uses the copula function to represent the joint probability distribution of the considered variables.. The use of copulas enables to model the correlation structure between the variables independently of their marginal distributions. It also allows to adopt marginal distributions not belonging to the same probability family or complex marginal distributions such as finite mixtures of distributions. Furthermore copulas yield analytical expressions for the isolines of the return periods.. This work takes into consideration the case of design events characterized by the joint behavior of two variables, even if the proposed approach can be easily generalized to the multivariate case.. The proposed methodology is illustrated and discussed through the case study of the Tiber River, where copulas have been used to model the joint behavior of the annual maximum peak flows and the corresponding volumes of the floods that exceeded the threshold of 1800 m3\/s at Ripetta (Rome) gauge. In this case the joint probability distribution of peak flows and volumes has been evaluated using all the information available, including: 1) the daily flow series available starting from 1782, and 2) the censored series of the peak flows of the exceptional floods that inundated the town of Rome starting from the XV century. The application of the proposed methodology allowed to select limited sets of flood events to be used in the design of flood control reservoirs on the Tiber River.. "

Many details about the flow of water in soils in a hillslope are unknowable given current technologies. One way of learning about the bulk effects of water velocity distributions on hillslopes is through the use of tracers. However, this paper will demonstrate that the interpretation of tracer information needs to become more sophisticated. The paper reviews, and complements with mathematical arguments and specific examples, theory and practice of the distribution(s) of the times water particles injected through rainfall spend traveling through a catchment up to a control section (i.e., "catchment" travel times). The relevance of the work is perceived to lie in the importance of the characterization of travel time distributions as fundamental descriptors of catchment water storage, flow pathway heterogeneity, sources of water in a catchment, and the chemistry of water flows through the control section. The paper aims to correct some common misconceptions used in analyses of travel time distributions. In particular, it stresses the conceptual and practical differences between the travel time distribution conditional on a given injection time (needed for rainfall-runoff transformations) and that conditional on a given sampling time at the outlet (as provided by isotopic dating techniques or tracer measurements), jointly with the differences of both with the residence time distributions of water particles in storage within the catchment at any time. These differences are defined precisely here, either through the results of different models or theoretically by using an extension of a classic theorem of dynamic controls. Specifically, we address different model results to highlight the features of travel times seen from different assumptions, in this case, exact solutions to a lumped model and numerical solutions of the 3-D flow and transport equations in variably saturated, physically heterogeneous catchment domains. Our results stress the individual characters of the relevant distributions and their general nonstationarity yielding their legitimate interchange only in very particular conditions rarely achieved in the field. We also briefly discuss the impact of oversimple assumptions commonly used in analyses of tracer data.

Solute transport takes place in heterogeneous porous formations, with the log conductivity, Y = ln K, modeled as a stationary random space function of given univariate normal probability density function (pdf) with mean 〈Y〉, variance σY2, and integral scale IY. For weak heterogeneity, the above mentioned quantities completely define the first-order approximation of the longitudinal macrodispersivity αL = σY2 IY. However, in highly heterogeneous formations, nonlinear effects which depend on the multipoint joint pdf of Y, impact αL. Most of the past numerical simulations assumed a multivariate normal distribution (MVN) of Y values. The main aim of this study is to investigate the impact of deviations from the MVN structure upon αL. This is achieved by using the concept of spatial correlations of different Y classes, the latter being defined as the space domain where Y falls in the generic interval [Y,Y + DY]. The latter is characterized by a length scale γ(Y), reflecting the degree of connectivity of the domain (the concept is similar to the indicator variograms). We consider both "symmetrical" and "non-symmetrical" structures, for which γ(Y′) = γ(-Y′) (similar to the MVN), and γ(Y′) ≠ γ(-Y′), respectively, where Y′ = Y - 〈Y〉. For example, large Y zones may have high spatial correlation, while low Y zones are poorly correlated, or vice versa. The impact of γ(Y) on αL is investigated by adopting a structure model which has been used in the past in order to investigate flow and transport in highly heterogeneous media. It is found that the increased correlation in the low conductive zones with respect to the high ones generally leads to a significant increase in aL, for the same global IY. The finding is explained by the solute retention occurring in low Y zones, which has a larger effect on solute spreading than hi h Y zones. Conversely, αL decreases when the high conductivity zones are more correlated than the low Y ones. Dispersivity is less affected by the shape of γ(Y) for symmetrical distributions. It is found that the range of validity of the first-order dispersivity, i.e., αL = IYσY2, narrows down for non-symmetrical structures. © 2010 by the American Geophysical Union.

The effective conductivity Kef of porous formations of spatially variable permeability K is determined for media of random two-dimensional and isotropic structures. The medium is modeled as an ensemble of multiphase circular inclusions of different Y = ln K, characterized by a pdf f(Y), and of different radii R (polydisperse medium), of pdf f(R vertical bar Y), which are implanted in a matrix of K = K(0). A large number of inclusions are embedded in a large circle, to allow for exchange of space and ensemble averaging. For symmetrical pdf f(Y) = f(-Y) and symmetrical f(R vertical bar Y), the Matheron exact relationship K(ef) = KG (the geometric mean) applies. The main aim of the article is to determine the deviation of K(ef) from K(G) for symmetrical f(Y) but nonsymmetrical f(R vertical bar Y). This is related to recent studies on the effect on Kef of connectivity of spatial domains of different K classes. The problem is solved numerically by an accurate and efficient iterative procedure and by a novel, approximate, analytical method. The two procedures are illustrated and compared for the configuration of two phases of conductivities K(1), K(2), of equal volume fractions, of different radii R(1) and R(2), respectively, within a matrix of K(0) = K(G) = (K(1) K(2))(1/2). Even for very high heterogeneity (K(1)/K(2) = 1000) it is found that the effect of variable R is relatively modest and it manifests mainly at the largest attainable volume fraction. The simple analytical approximation, valid for moderate volume fractions, is applied to investigation of K(ef) for normal f(Y), and for two values of R, for Y 0 and Y > 0, respectively. The results are of interest for similar heterogeneous media and for other physical processes governed by linear relationships between the flux and the driving potential gradient.

A stochastic model is developed for the analysis of the traveltime distribution f(tau) in a hillslope. The latter is described as made up from a surficial soil underlain by a less permeable subsoil or bedrock. The heterogeneous hydraulic conductivity K is described as a stationary random space function, and the model is based on the Lagrangian representation of transport. A first-order approach in the log conductivity variance is adopted in order to get closed form solutions for the principal statistical moments of the traveltime. Our analysis indicates that the soil is mainly responsible for the early branch of f(tau), i.e., the rapid release of solute which preferentially moves through the upper soil. The early branch of f(tau) is a power law, with exponent variable between -1 and -0.5; the behavior is mainly determined by unsaturated transport. The subsoil response is slower than that of the soil. The subsoil is mainly responsible for the tail of f(tau), which in many cases resembles the classic linear reservoir model. The resulting shape for f(tau) is similar to the Gamma distribution. Analysis of the f(tau) moments indicates that the mean traveltime is weakly dependent on the hillslope size. The traveltime variance is ruled by the distribution of distances of the injected solute from the river; the effect is coined as source zone dispersion. The spreading due to the K heterogeneity is less important and obscured by source zone dispersion. The model is tested against the numerical simulation of Fiori and Russo (2008) with reasonably good agreement, with no fitting procedure.

Laboratory measurements of the free surface profile were carried out over a physical model of a classical channel drop inserted in a sewer, equipped with a depressed stilling basin. This devise is still in use in several combined sewer systems and may have unfavourable consequences on the water quality of the receiving bodies. The present paper is part of a study to economically modify the existing structure in order to prevent dry weather sedimentation and heavily polluted first flush flow. The experimental free surface profiles observed in different conditions were compared with numerical simulations obtained using the computational flow dynamic software FLOW 3-D with the imposed conditions at the downstream boundary. The standard k- turbulence model is used in conjunction with the volume of fluid (VOF) free surface model. The computer simulations faithfully reproduce the water level in the upstream pipe branch and in the dissipation manhole. However, some discrepancies were observed between the measured and simulated jet, and in the position of the hydraulic jump.

A stochastic model is developed for the analysis of the traveltime distribution ft in a hillslope. The latter is described as made up from a surficial soil underlain by a less permeable subsoil or bedrock. The heterogeneous hydraulic conductivity K is described as a stationary random space function, and the model is based on the Lagrangian representation of transport. A first-order approach in the log conductivity variance is adopted in order to get closed form solutions for the principal statistical moments of the traveltime. Our analysis indicates that the soil is mainly responsible for the early branch of ft, i.e., the rapid release of solute which preferentially moves through the upper soil. The early branch of ft is a power law, with exponent variable between 1 and 0.5; the behavior is mainly determined by unsaturated transport. The subsoil response is slower than that of the soil. The subsoil is mainly responsible for the tail of ft, which in many cases resembles the classic linear reservoir model. The resulting shape for ft is similar to the Gamma distribution. Analysis of the ft moments indicates that the mean traveltime is weakly dependent on the hillslope size. The traveltime variance is ruled by the distribution of distances of the injected solute from the river; the effect is coined as source zone dispersion. The spreading due to the K heterogeneity is less important and obscured by source zone dispersion. The model is tested against the numerical simulation of Fiori and Russo (2008) with reasonably good agreement, with no fitting procedure.

Laboratory measurements of the free surface profile were carried out over a physical model of a classical channel drop inserted in a sewer, equipped with a depressed stilling basin. This devise is still in use in several combined sewer systems and may have unfavourable consequences on the water quality of the receiving bodies. The present paper is part of a study to economically modify the existing structure in order to prevent dry weather sedimentation and heavily polluted first flush flow. The experimental free surface profiles observed in different conditions were compared with numerical simulations obtained using the computational flow dynamic software FLOW 3-D with the imposed conditions at the downstream boundary. The standard k- turbulence model is used in conjunction with the volume of fluid (VOF) free surface model. The computer simulations faithfully reproduce the water level in the upstream pipe branch and in the dissipation manhole. However, some discrepancies were observed between the measured and simulated jet, and in the position of the hydraulic jump.