Modelling streamflow trends for a watershed with limited data: case of the Litani basin,Lebanon

Abstract

Streamflow variability in the Upper and Lower Litani basin, Lebanon was modelled as there is a lack of long-term measured runoff data. To simulate runoff and streamflow, daily rainfall was derived using a stochastic rainfall generation model and monthly rainfall data. Two distinct synthetic rainfall models were developed based on a two-part probabilistic distribution approach. The rainfall occurrence was described by a Markov chain process, while the rainfall distribution on wet days was represented by two different distributions (i.e. gamma and mixed exponential distributions). Both distributions yielded similar results. The rainfall data were then processed using water balance and routing models to generate daily and monthly streamflow. Compared with measured data, the model results were generally reasonable (mean errors ranging from 0.1 to 0.8?m³/s at select locations). Finally, the simulated monthly streamflow data were used to investigate discharge trends in the Litani basin during the 20th century using the Mann-Kendall and Sen slope nonparametric trend detection methods. A significant drying trend of the basin was detected, reaching a streamflow reduction of 0.8 and 0.7 m³/s per decade in January for the Upper and Lower basin, respectively.

Editor D. Koutsoyiannis; Associate editor Sheng Yue

Citation Ramadan, H.H., Beighley, R.E., and Ramamurthy, A.S., 2012. Modelling streamflow trends for a watershed with limited data: case of the Litani basin, Lebanon. Hydrological Sciences Journal, 57 (8), 1516–1529.     

A daily rainfall occurrence process

A model for the periodic (annual cycle), discrete rainfall occurrence process is presented. Using this model the probabilistic properties of the process in -day intervals can be investigated. In such an interval the rainfall occurrence process is approximated by some stationary processa _t ,tIN. The processa _t ,tIN is described by the distributions of the lengths of wet and dry sequences. It is assumed that the lengths of successive wet and dry sequences are independent. For this process the distribution of the number of wet days in -day intervals is calculated. The model is fitted to 50-year rainfall data from Wroclaw, Poland. Rainfall amounts of 0.1, 1.0 and 2.0 mm are considered as thresholds defining a wet day. To estimate the distribution of the length of wet and dry sequences the family of Pascal distribution is chosen.     

A stochastic model for daily subsurface CO2 concentration and related soil respiration

Near-surface soil CO₂ gas-phase concentration (C) and concomitant incident rainfall (P_i) and through-fall (P_t) depths were collected at different locations in a temperate pine forest every 30 min during the 2005 and 2006 growing seasons (and then averaged to the daily timescale). At the daily scale, C temporal variations were well described by a sequence of monotonically decreasing functions interrupted by large positive jumps induced by rainfall events. A stochastic model was developed to link rainfall statistics responsible for these jumps to near-surface C dynamics. The model accounted for the effect of daily rainfall variability, both in terms of timing and amount of water, and permitted an analytical derivation of the C probability density function (pdf) using the parameters of the rainfall pdf. Given the observed positive correlation between daily C and soil CO₂ fluxes to the atmosphere (F_s), the effects of various rainfall regimes on the statistics of F_s can be deduced from the behavior of C under different climatic conditions. The predictions from this analytical model are consistent with flux measurements reported in manipulative experiments that varied rainfall amount and frequency.     

THE RHONE GLACIER SURVEYS

Abstract

This paper describes a stochastic rainfall model which has been developed to generate synthetic sequences of hourly rainfalls at a point. The model has been calibrated using data from Farnborough in Hampshire, England. This rainfall data series was divided into wet and dry spells; analysis of the durations of these spells suggests that they may be represented by exponential and generalized Pareto distributions respectively. The total volume of rainfall in wet spells was adequately fitted by a conditional gamma distribution. Random sampling from a beta distribution, defining the average shape of all rainfall profiles, is used in the model to obtain the rainfall profile for a given wet spell. Results obtained from the model compare favourably with observed monthly and annual rainfall totals and with annual maximum frequency distributions of 1, 2, 6, 12, 24 and 48 hours duration at Farnborough. The model has a total of 22 parameters, some of which are specific to winter or summer seasons.     

Statistical characterization of temporal structure of storms

The authors present a statistical procedure to estimate the probability distributions of storm characteristics. The approach uses recent advances in stochastic hydrological modeling. The temporal dynamics of rainfall are modeled via a reward alternating renewal process that describes wet and dry phases of storms. In particular, the wet phase is modeled as a rectangular pulse process with dependent random duration and intensity; the global dependence structure is described using multidimensional copulas. The marginal distributions are described by Generalized Pareto laws. The authors derive both the storm volume statistics and the rainfall volume distribution within a fixed temporal window preceding a storm. Based on these results, they calculate the antecedent moisture conditions. The paper includes a thorough discussion of the validity of the assumptions and approximations introduced, and an application to actual rainfall data. The models presented here have important implications for improved design procedures of water resources and hydrologic systems.     

Models of density and mechanical Q-factor distributions according to new data on the nutations and free oscillations of the Earth: 1. Ambiguity in the inverse problem

Ambiguity in the inverse problem of retrieval of the mechanical parameters of the Earth’s shell and core from the set of data on the velocities V _p and V _S , of longitudinal and transverse seismic body waves, the frequencies f _i and quality factors Q _i , of free oscillations, and the amplitudes and phases of forced nutation is considered. The numerical experiments show that the inverse problem of simultaneous retrieval of the density profile ρ in the mantle-liquid core system and the mechanical quality factor Q _μ of the mantle (if the total mass M and the total mean moment of inertia I of the Earth, and V _p and V _S are constant at all depths) has most unstable solutions. An example of depth distributions of ρ and Q _μ which are alternative to the well-known PREM model is given. In these distributions, the values of M and I and the velocities V _p and V _S at all depths for the period of oscillations T = 1 s exactly coincide with their counterparts yielded by PREM model (T = 1 s); however, the maximum deviations of the ρ and Q _μ profiles from those in the PREM model are about 3% and 40%, respectively; the mass and the moment of inertia of the liquid core are smaller than those for the PREM model by 0.75% and 0.63%, respectively. In this model, the root mean square (rms) deviations of all the measured values of f _i and Q _i from their values predicted by theory are half to third the corresponding values in the PREM model; the values of Δ for natural frequencies of the fundamental tone and overtones of radial oscillations, the fundamental tones of torsional oscillations, and the fundamental tones of spheroidal oscillations, which are measured with the highest relative accuracy, are smaller by a factor of 30, 6.6, and 2 than those in the PREM model, respectively. Such a large ambiguity in the solution of the inverse problem indicates that the current models of the depth distribution of density have relatively low accuracy, and the models of the depth distribution of the mechanical Q in the mantle are extremely unreliable. It is shown that the ambiguity in the models of depth distribution of density considerably decreases after the new data on the amplitudes and phases of the forced nutation of the Earth are taken into account. Using the same data, one may also refine by several times the recent estimates of the creep function for the lower mantle within a wide interval of periods ranging from a second to a day.     

A stochastic model for tropical rainfall at a single location

Modelling data that correspond to rainfall accumulated over fixed periods of time presents the challenging problem of dealing with a random variable that has a point mass at zero which corresponds to dry periods that occur with positive probability. One way to overcome this difficulty is to assume that the data correspond to a normal variate w, that has been truncated and transformed. The dry periods correspond to the (unobserved) negative values and the wet periods correspond to some power of the positive ones. The serial structure that is present in rainfall can be modelled by imposing a serial structure to w. We use a dynamic linear model on w using a Fourier representation to allow for the seasonality of the data, which in the case of tropical rainfall is very marked. The model is fitted using a Markov chain Monte Carlo method that uses latent variables to handle both dry periods and missing values. We use the model to estimate and predict both the amount of rainfall and the probability of a dry period. The method is illustrated with data collected in the Venezuelan state of Guárico.     

Heat flow studies: Constraints on the distribution of uranium,thorium and potassium in the continental crust

To study the amount of heat generated by radioactive decay in the continental crust, the usual practice in the literature is to fit to the heat flow and radioactivity data a relationship of the form: Q = Q_r + D · A where Q and A are the observed heat flow and radiogenic heat production. Q_r is the “reduced” heat flow and D is a depth scale. This procedure implicitly assumes that uranium, thorium and potassium have identical distributions in the crust. We suggest that significant information may be lost as the three radioelements may in fact be affected by processes operating over different depths.Data published for four heat flow provinces throughout the world are used to estimate the distributions of uranium, thorium and potassium in the continental crust. These distributions are characterized by a depth scales defined as follows: D_i =∫0h C_i(z)C_i(0)dz where h is the thickness of the layer containing the bulk of radioactivity and C_i(z) the concentration of element i at depth z. Three depth scales are computed from a least-squares fit to the following relationship: Q = Q_r + D_U · A_U + D_T · A_T + D_K · A_T where Q is the observed heat flow and Q_r some constant (a reduced heat flow). A_i is the heat generation rate due to the radioactive decay of element i, and D_i is the corresponding depth scale.The analysis suggests that the three distributions are different and that they have the same basic features in the four provinces considered. The depth scale for potassium is large in granitic areas, that for thorium is small and that for uranium lies between the other two.We propose a simple model according to which each radioelement essentially provides a record for one process. Potassium gives a depth scale for the primary differentiation of the crust. Thorium gives the depth scale of magmatic or metamorphic fluid circulation. Finally, the uranium distribution reflects the late effects of alteration due to meteoric water. We show that the heat flow and radioactivity data are compatible with this model.Our analysis and numerical results are supported by data from deep boreholes and by geochemical evidence, such as detailed investigations of plutonic series and studies of U-Th-Pb systematics.     

Rainfall modelling using Poisson-cluster processes: a review of developments

 Over a decade ago, point rainfall models based upon Poisson cluster processes were developed by Rodriguez-Iturbe, Cox and Isham. Two types of point process models were envisaged: the Bartlett–Lewis and the Neyman–Scott rectangular pulse models. Recent developments are reviewed here, including a number of empirical studies. The parameter estimation problem is addressed for both types of Poisson-cluster based models. The multiplicity of parameters which can be obtained for a given data set using the method of moments is illustrated and two approaches to finding a best set of parameters are presented. The use of a proper fitting method will allow for the problems encountered in regionalisation to be adequately dealt with. Applications of the point process model to flood design are discussed and finally, results for a model with dependent cell depth and duration are given. Taking into account the spatial features of rainfall, three multi-site models are presented and compared. They are all governed by a master Poisson process of storm origins and have a number of cell origins associated with each storm origin. The three models differ as to the type of dependence structure between the cell characteristics at different sites. Analytical properties are presented for these models and their ability to represent the spatial structure of a set of raingauge data in the South-West of England is examined. Continuous spatial-temporal models are currently being developed and results are presented for a model in which storm centres arrive in a homogeneous Poisson process in space-time, and cells follow them in time according to a Bartlett–Lewis type cluster. Examples of simulations using this model are shown and compared with radar data from the South-West of England. The paper concludes with a summary of the main areas in which further research is required.     

Effect of the North Atlantic Oscillation on winter daily rainfall and runoff in the Abruzzo region (Central Italy)

In this paper a composite analysis was used to assess the influence of the North Atlantic Oscillation (NAO) on the winter daily rainfall and seasonal runoff at 28 stations of the Abruzzo region (Central Italy) during the period 1951–2012. Compositing was based on NAO^? and NAO⁺ phases, identified by mean winter values of the normalized NAO index (NAOI) ≤?0.75 and ≥+0.75, respectively. In accordance with previous studies, it was found that NAO^? phases determine, in general, a greater number of wet days (N _w ) and (consequently) higher seasonal rainfall amounts in comparison to NAO⁺ phases. However, the NAO influence is characterized by a certain spatial variability, that can mostly be explained by orographic effects due to the Apennine Mountains. This is particularly evident for the mean rainfall depth per event (P _e ) that, during NAO^? phases, increases for the stations to the west of the Apennines, while it decreases for most of the stations to the east. The structure of winter daily rainfall of NAO⁺ and NAO^? type, was described by a simple but effective first-order Markov process, determining the transition probabilities P01 (dry to wet) and P10 (wet to dry) and modelling the rainfall depth on wet days by a Weibull distribution. The most significant influence of NAO concerns P01 and the shape parameter of the Weibull distribution that are both higher during the NAO^? phase. This means that NAO^? phases are characterized by less persistent dry periods and less variable daily rainfall depths, in comparison to NAO⁺ phases. The effect of these differences on the winter seasonal runoff was explored by applying a Curve Number rainfall-runoff model. Significant increments of the mean seasonal runoff during NAO^? phases were observed only for few stations (mainly on the west), characterized by corresponding increments of N _w , P _tot and P _e .). NAO⁺ phases, instead, are characterized by relevant increments of the seasonal runoff variability, particularly on the eastern areas. In this context, the important regulating function of the watershed conditions was also discussed.     

Doubly stochastic Poisson pulse model for fine-scale rainfall

Stochastic rainfall models are widely used in hydrological studies because they provide a framework not only for deriving information about the characteristics of rainfall but also for generating precipitation inputs to simulation models whenever data are not available. A stochastic point process model based on a class of doubly stochastic Poisson processes is proposed to analyse fine-scale point rainfall time series. In this model, rain cells arrive according to a doubly stochastic Poisson process whose arrival rate is determined by a finite-state Markov chain. Each rain cell has a random lifetime. During the lifetime of each rain cell, instantaneous random depths of rainfall bursts (pulses) occur according to a Poisson process. The covariance structure of the point process of pulse occurrences is studied. Moment properties of the time series of accumulated rainfall in discrete time intervals are derived to model 5-min rainfall data, over a period of 69 years, from Germany. Second-moment as well as third-moment properties of the rainfall are considered. The results show that the proposed model is capable of reproducing rainfall properties well at various sub-hourly resolutions. Incorporation of third-order moment properties in estimation showed a clear improvement in fitting. A good fit to the extremes is found at larger resolutions, both at 12-h and 24-h levels, despite underestimation at 5-min aggregation. The proportion of dry intervals is studied by comparing the proportion of time intervals, from the observed and simulated data, with rainfall depth below small thresholds. A good agreement was found at 5-min aggregation and for larger aggregation levels a closer fit was obtained when the threshold was increased. A simulation study is presented to assess the performance of the estimation method.     

A shot noise model for the generation of soil moisture data

An attempt is made to estimate the expected contribution of rainfall to soil moisture during the irrigation season. Effective rainfall and evapotranspiration are the parameters considered in the water balance carried out in the root zone. Rainfall occurrence is simulated by a Poisson process whereas evapotranspiration is described by a simple deterministic function of potential evapotranspiration and soil moisture in the root zone. Using the theory of shot noise models a closed form solution is derived from the expected soil moisture in the root zone at the end of the time interval (0,t]. For illustration purposes the proposed model is applied to a series of data from Mikra meteorological station in Greece.List of symbols x change in water storage in the root zone during the time interval t - X infiltrated rainfall of thei th storm event - ET actual evapotranspiration during thej th day - Poisson rate - number of storm events in (0,t] - t _r duration of rainfall - t _b interarrival time - h _i rainfall depth of thei th storm event - i _m mean rainfall intensity - i(t) instantaneous rainfall intensity - x(0),x(t) available soil moisture in the root zone at time 0 andt, respectively - PET potential evapotranspiration rate - x _F available soil moisture in the root zone at field capacity - soil moisture depletion rate (=PET/x _F ) - w impulse shape of filtered Poisson processes - E[·] mean value - S _i time of thei th rainfall event - N(t) time of storm events in (0,t] - estimated standard deviation The following symbols were used in this paper     

Single and multiple dislocation models of extended earthquake sources

Summary A short review of simple theoretical models for extended earthquake sources visualized as single and multiple dislocations is presented. Formulas for the duration of the radiation t _i ^s from the starting point of the destruction up to the region forming the amplitude maxima of body waves are given. A method for determining some basic parameters of extended earthquake sources (length and depth of the dislocation, azimuth of the first dislocation and the angles between every two consecutive dislocations) from the seismogram data for t_i ^s is presented. The procedure is applied to yield some practical results.     

A semi-analytical model of expected areal-average infiltration under spatial heterogeneity of rainfall and soil saturated hydraulic conductivity

A semi-analytical model for the estimate of expected areal-average infiltration rate at hillslope scale is presented. It accounts for spatial heterogeneity of the saturated hydraulic conductivity, K_s, and rainfall rate, r. The K_s field is characterized by a lognormal probability density function while the rainfall rate r is represented by a uniform distribution between two extreme values. The model formulation relies upon the use of cumulative infiltration as the independent variable which is then expressed as a function of an expected time for use in practical applications. The solution is applicable for those ranges of r and K_s that allow for neglecting the infiltration of surface water running downslope into pervious soils (run-on process). The model was tested by comparisons with Monte Carlo simulations carried out for a variety of coefficients of variation of r and K_s over a clay loam soil and a sandy loam soil. The model was found to be very reliable both with coupled spatial variability of r and K_s and when only one variable is characterized by spatial heterogeneity while the other is uniform.     

Upscaling of saturated conductivity for Hortonian runoff modelling

Stochastic and deterministic upscaling techniques are developed that upscale saturated conductivity at the support of 0.04 m² to representative actual infiltration (I_b) for support units (blocks) of 10¹–10⁴ m², as a function of steady state rainfall and runon to the block, under Hortonian runoff (infiltration excess overland flow). Parameters in the upscaling techniques represent the surface runoff flow pattern and the spatial probability distribution of saturated conductivity within the 10¹–10⁴ m² block. The stochastic upscaling technique represents the spatial process of infiltration and runoff using a simple process-imitating model, estimating I_b using Monte Carlo simulation. The deterministic upscaling technique aggregates these processes by a deterministic function relating rainfall and runon to I_b. The stochastic upscaling technique is shown to be capable to upscale saturated conductivity derived from ring infiltrometers to I_b values of plots (1 m²) corresponding to measured I_b values using rainfall simulators. It is shown that both upscaling techniques can be used to estimate I_b for each time step and each block in transient rainfall–runoff models, giving better estimates of cumulative runoff from a hillslope and a small catchment than model runs that do not use upscaling techniques.     

Modélisation pluie–débit journalière par des modèles conceptuels et ȁboîte noireȁ; test d’un modèle neuroflou / Daily rainfall–runoff modelling using conceptual and black box models; testing a neuro-fuzzy model

Abstract

Abstract Four rainfall–runoff models were applied on a daily time step and tested in the Cheffia basin, situated in the northeast of Algeria. The models belong to two categories: conceptual models–the GR3j model and the CREC model with eight parameters, and ?black box? models–the ARMAX model and a neuro-fuzzy model, which combines neural structure and fuzzy logic. The models were compared over two periods, one dry and the other wet. This comparison allowed a better model for the rainfall–runoff process to be proposed, on a daily time step, by combining the conceptual approach with a neuro-fuzzy system.     

Evaluation of the Liu model for predicting rainfall interception in forests world‐wide

Simple but effective models are needed for the prediction of rainfall interception under a full range of environmental and management conditions. The Liu model was validated using data published in the literature and was compared with two leading models in the literature: the Rutter and the Gash models. The Liu model was tested against the Rutter model on a single‐storm basis with interception measurements observed from an old‐growth Douglas fir (Pseudotsuga menziesii) forest in Oregon, USA. Simulated results by the Liu model were close to the measurements and comparable to those predicted by the Rutter model. The Liu model was further tested against the Gash model on a multistorm basis. The Gash and Liu models successfully predicted long‐term interception losses from a broad range of 20 forests around the world. Results also indicated that both the Gash and the Liu models could be used to predict rainfall interception using daily rainfall data, although it was assumed in both models that there is only one storm per rain day. The sensitivity of the Liu model to stand storage capacity, canopy gap fraction and evaporation rate from wet canopy surface during rainfall was investigated. Results indicate that the Liu model has the simplest form, least data requirements and comparable accuracy for predicting rainfall interception as compared with the Rutter and the Gash models. Copyright © 2001 John Wiley & Sons, Ltd.     

Derivation of streamflow statistics based on a filtered point process

This work presents the derivation of general streamflow cumulants from daily rainfall time series. The general streamflow cumulants can be used to compute basic streamflow statistics such as mean, variance, coefficient of skewness, and correlation coefficient. Streamflow is considered as a filtered point process where the input is a daily rainfall time series assumed to be a marked point process. The marks of the process are the daily rainfall amounts which are assumed independent and identically distributed. The number of rainfall occurrences is a counting process represented by either the binomial, the Poisson, or the negative binomial probability distribution depending on its ratio of mean to variance. The first three cumulants and the covariance function of J-day averaged streamflows are deduced based on the characteristic function of a filtered point process. These cumulants are functions of the stochastic properties of the daily rainfall process and the basin-response function representing the causal relationship between rainfall and runoff.     

A fine-scale point process model of rainfall with dependent pulse depths within cells

Abstract

A cluster point process model is considered for the analysis of fine-scale rainfall time series. The model is based on three Poisson processes. The first is a Poisson process of storm origins, where each storm has a random (exponential) lifetime. The second is a Poisson process of cell origins that occur during the storm lifetime, terminating when the storm finishes. Each cell has a random lifetime that follows an exponential distribution (or terminates when the storm terminates, whichever occurs first). During cell lifetimes, a third Poisson process of instantaneous pulses occurs. The model is essentially an extension of the well-known Bartlett-Lewis rectangular pulses model, with the rectangular profiles replaced with a Poisson process of instantaneous pulse depths to ensure more realistic rainfall profiles for fine-scale series. Model equations, derived in Cowpertwait et al. (2007 Cowpertwait, P., Isham, V. and Onof, C. 2007. Point process models of rainfall: developments for fine-scale structure. Proceedings of the Royal Society of London, Series A, 463: 2569–2587. [Crossref], [Web of Science ®] , [Google Scholar]), are used to fit different sets of properties to a 60 year record of 5-min data taken from Kelburn, New Zealand. As in the previous work, two superposed processes are used to account for two main and distinct precipitation types (convective and stratiform). By treating the within-cell pulses as dependent random variables, it is found, by simulation, that improved fits to extreme values and the proportion of dry intervals are obtained.

Citation Cowpertwait, P. S. P., Xie, G., Isham, V., Onof, C. & Walsh, D. C. I. (2011) A fine-scale point process model of rainfall with dependent pulse depths within cells. Hydrol. Sci. J. 56(7), 1110–1117.     

Development and parameterization of an infiltration model accounting for water depth and rainfall intensity

Experimental work has clearly shown that the effective hydraulic conductivity (K_e) or effective infiltration rate (f_e) on the local scale of a plot cannot be considered as constant but are dependent on water depth and rainfall intensity because non‐random microtopography‐related variations in hydraulic conductivity occur. Rainfall–runoff models generally do not account for this: models assume that excess water is uniformly spread over the soil surface and within‐plot variations are neglected. In the present study, we propose a model that is based on the concepts of microtopography‐related water depth‐dependent infiltration and partial contributing area. Expressions for the plot scale K_e and f_e were developed that depend on rainfall intensity and runon from upslope (and thus on water depth). To calibrate and validate the model, steady state infiltration experiments were conducted on maize fields on silt loam soils in Belgium, with different stages and combinations of rainfall intensity and inflow, simulating rainfall and runon. Water depth–discharge and depth–inundation relationships were established and used to estimate the effect of inundation on K_e. Although inflow‐only experiments were found to be unsuitable for calibration, the model was successfully calibrated and validated with the rainfall simulation data and combined rainfall–runon data (R²: 0.43–0.91). Calibrated and validated with steady state infiltration experiments, the model was combined with the Green–Ampt infiltration equation and can be applied within a two‐dimensional distributed rainfall–runoff model. The effect of water depth–dependency and rainfall intensity on infiltration was illustrated for a hillslope. Copyright © 2012 John Wiley & Sons, Ltd.