A unique value function for an optimal control problem of irrigation water intake from a reservoir harvesting flash floods

Operation of reservoirs is a fundamental issue in water resource management. We herein investigate well-posedness of an optimal control problem for irrigation water intake from a reservoir in an irrigation scheme, the water dynamics of which is modeled with stochastic differential equations. A prototype irrigation scheme is being developed in an arid region to harvest flash floods as a source of water. The Hamilton–Jacobi–Bellman (HJB) equation governing the value function is analyzed in the framework of viscosity solutions. The uniqueness of the value function, which is a viscosity solution to the HJB equation, is demonstrated with a mathematical proof of a comparison theorem. It is also shown that there exists such a viscosity solution. Then, an approximate value function is obtained as a numerical solution to the HJB equation. The optimal control strategy derived from the approximate value function is summarized in terms of rule curves to be presented to the operator of the irrigation scheme.     

Author Index 1991

Abstract

Evapotranspiration (ET) is an important process in the hydrological cycle and needs to be accurately quantified for proper irrigation scheduling and optimal water resources systems operation. The time variant characteristics of ET necessitate the need for forecasting ET. In this paper, two techniques, namely a seasonal ARIMA model and Winter's exponential smoothing model, have been investigated for their applicability for forecasting weekly reference crop ET. A seasonal ARIMA model with one autoregressive and one moving average process and with a seasonality of 52 weeks was found to be an appropriate stochastic model. The ARIMA and Winter's models were compared with a simple ET model to assess their performance in forecasting. The forecast errors produced by these models were very small and the models would be promisingly of great use in real-time irrigation management.     

Stochastic and analytical approaches for sediment accumulation in river reservoirs

ABSTRACT

Sediment accumulation in a river reservoir is studied by stochastic time series models and analytical approach. The first-order moving average process is found the best for the suspended sediment discharge time series of the Juniata River at Newport, Pennsylvania, USA. Synthetic suspended sediment discharges are first generated with the chosen model after which analytical expressions are derived for the expected value and variance of sediment accumulation in the reservoir. The expected value and variance of the volume of sediment accumulation in the reservoir are calculated from a thousand synthetic time series each 38 years long and compared to the analytical approach. Stochastic and analytical approaches perfectly trace the observation in terms of the expected value and variability. Therefore, it is concluded that the expected value and variance of sediment accumulation in a reservoir could be estimated by analytical expressions without the cost of synthetic data generation mechanisms.     

Estimation of seasonal crop water consumption in a vineyard using Bowen ratio‐energy balance method

Information about seasonal crop water consumption is useful to develop the appropriate irrigation scheme. Measurements of energy balance components using the Bowen ratio method were made for a complete growing season at a vineyard in the arid region of northwest China. Vine in the experiment was furrow‐irrigated using a trellis system. The measured evapotranspiration was compared with estimates using the soil water balance method. It is shown that the Bowen ratio method provided accurate estimates of evapotranspiration from the vineyard and this requires that the Bowen ratio system is appropriately installed. The energy balance components showed typical diurnal pattern with peaks that occurred around the midday, except for the ground heat flux which delayed its peak by 2–3 h. The sensible heat flux was greater than the latent heat flux and followed the net radiation closely. The ratio of the latent heat flux to net radiation was low in the early growing season and increased over time. Under the limited irrigation experienced in the vineyard, the latent heat flux was controlled by available soil moisture and the total evapotranspiration in the growing season was 253 mm. The seasonal progression of the crop coefficient is similar to that reported in the literature, with the maximum occurring during the month of September. The crop coefficient can be estimated as a non‐linear function of day of year (DOY) and used to estimate evapotranspiration from vineyards in the region. Copyright © 2007 John Wiley & Sons, Ltd.     

From rainfed agriculture to stress-avoidance irrigation: I. A generalized irrigation scheme with stochastic soil moisture

With vast regions already experiencing water shortages, it is becoming imperative to manage sustainably the available water resources. As agriculture is by far the most important user of freshwater and the role of irrigation is projected to increase in face of climate change and increased food requirements, it is particularly important to develop simple, widely applicable models of irrigation water needs for short- and long-term water resource management. Such models should synthetically provide the key irrigation quantities (volumes, frequencies, etc.) for different irrigation schemes as a function of the main soil, crop, and climatic features, including rainfall unpredictability. Here we consider often-employed irrigation methods (e.g., surface and sprinkler irrigation systems, as well as modern micro-irrigation techniques) and describe them under a unified conceptual and theoretical framework, which includes rainfed agriculture and stress-avoidance irrigation as extreme cases. We obtain mostly analytical solutions for the stochastic steady state of soil moisture probability density function with random rainfall timing and amount, and compute water requirements as a function of climate, crop, and soil parameters. These results provide the necessary starting point for a full assessment of irrigation strategies, with reference to sustainability, productivity, and profitability, developed in a companion paper [Vico G, Porporato A. From rainfed agriculture to stress-avoidance irrigation: II. Sustainability, crop yield, and net profit. Adv Water Resour 2011;34(2):272-81].     

Estimation of the annual maximum distribution from samples of maxima in separate seasons

Quantile estimates of the annual maximum distribution can be obtained by fitting theoretical distributions to the maxima in separate seasons, e.g. to the monthly maxima. In this paper, asymptotic expressions for the bias and the variance of such estimates are derived for the case that the seasonal maxima follow a Gumbel distribution. Results from these expressions are presented for a situation with no seasonal variation and for maximum precipitation depths at Uccle/Ukkel (Belgium). It is shown that the bias is often negligible and that the variance reduction by using seasonal maxima instead of just the annual maxima strongly depends on the seasonal variation in the data. A comparison is made between the asymptotic standard error of quantile estimates from monthlymaxima with those from a partial duration series. Much attention is paid to the effect of model misspecification on the resulting quantile estimates of the annual maximum distribution. The use of seasonal maxima should be viewed with caution when the upper tail of this distribution is of interest.     

Grain yield reliability analysis with crop water demand uncertainty

A new method of reliability analysis for crop water production function is presented considering crop water demand uncertainty. The procedure uses an advanced first-order second moment (AFOSM) method in evaluating the crop yield failure probability. To determine the variance and the mean of actual evapotranspiration as the component of interest for AFOSM analysis, an explicit stochastic optimization model for optimal irrigation scheduling is developed based on the first and second-order moment analysis of the soil moisture state variables. As a result of the study, the violation probabilities of crop yield at different levels were computed from AFOSM method. Also using the optimization results and the double bounded density function estimation methodology, the weekly soil moisture density function is derived which can be used as a short term reliability index. The proposed approach does not involve any discretization of system variables. The results of reliability analysis and optimization model compare favorably with those obtained from simulation.     

Impact of water withdrawals from groundwater and surface water on continental water storage variations

Humans have strongly impacted the global water cycle, not only water flows but also water storage. We have performed a first global-scale analysis of the impact of water withdrawals on water storage variations, using the global water resources and use model WaterGAP. This required estimation of fractions of total water withdrawals from groundwater, considering five water use sectors. According to our assessment, the source of 35% of the water withdrawn worldwide (4300 km³/year during 1998–2002) is groundwater. Groundwater contributes 42%, 36% and 27% of water used for irrigation, households and manufacturing, respectively, while we assume that only surface water is used for livestock and for cooling of thermal power plants. Consumptive water use was 1400 km³/year during 1998–2002. It is the sum of the net abstraction of 250 km³/year of groundwater (taking into account evapotranspiration and return flows of withdrawn surface water and groundwater) and the net abstraction of 1150 km³/year of surface water. Computed net abstractions indicate, for the first time at the global scale, where and when human water withdrawals decrease or increase groundwater or surface water storage. In regions with extensive surface water irrigation, such as Southern China, net abstractions from groundwater are negative, i.e. groundwater is recharged by irrigation. The opposite is true for areas dominated by groundwater irrigation, such as in the High Plains aquifer of the central USA, where net abstraction of surface water is negative because return flow of withdrawn groundwater recharges the surface water compartments. In intensively irrigated areas, the amplitude of seasonal total water storage variations is generally increased due to human water use; however, in some areas, it is decreased. For the High Plains aquifer and the whole Mississippi basin, modeled groundwater and total water storage variations were compared with estimates of groundwater storage variations based on groundwater table observations, and with estimates of total water storage variations from the GRACE satellites mission. Due to the difficulty in estimating area-averaged seasonal groundwater storage variations from point observations of groundwater levels, it is uncertain whether WaterGAP underestimates actual variations or not. We conclude that WaterGAP possibly overestimates water withdrawals in the High Plains aquifer where impact of human water use on water storage is readily discernible based on WaterGAP calculations and groundwater observations. No final conclusion can be drawn regarding the possibility of monitoring water withdrawals in the High Plains aquifer using GRACE. For the less intensively irrigated Mississippi basin, observed and modeled seasonal groundwater storage reveals a discernible impact of water withdrawals in the basin, but this is not the case for total water storage such that water withdrawals at the scale of the whole Mississippi basin cannot be monitored by GRACE.     

Generating probabilistic estimates of hydrological response for Irish catchments using a weather generator and probabilistic climate change scenarios

In accounting for uncertainties in future simulations of hydrological response of a catchment, two approaches have come to the fore: deterministic scenario‐based approaches and stochastic probabilistic approaches. As scenario‐based approaches result in a wide range of outcomes, the role of probabilistic‐based estimates of climate change impacts for policy formulation has been increasingly advocated by researchers and policy makers. This study evaluates the impact of climate change on seasonal river flows by propagating daily climate time series, derived from probabilistic‐based climate scenarios using a weather generator (WGEN), through a set of conceptual hydrological models. Probabilistic scenarios are generated using two different techniques. The first technique used probabilistic climate scenarios developed from statistically downscaled scenarios for Ireland, hereafter called SDprob. The second technique used output from 17 global climate models (GCMs), all of which participated in CMIP3, to generate change factors (hereafter called CF). Outputs from both the SDprob and the CF approach were then used in combination with WGEN to generate daily climate scenarios for use in the hydrological models. The range of simulated flow derived with the CF method is in general larger than those estimated with the SDprob method in winter and vice versa because of the strong seasonality in the precipitation signal for the 17 GCMs. Despite this, the simulated probability density function of seasonal mean streamflow estimated with both methods is similar. This indicates the usefulness of the SDprob or probabilistic approach derived from regional scenarios compared with the CF method that relies on sampling a diversity of response from the GCMs. Irrespective of technique used, the probability density functions of seasonal mean flow produced for four selected basins is wide indicating considerable modelling uncertainties. Such a finding has important implications for developing adaptation strategies at the catchment level in Ireland. Copyright © 2011 John Wiley & Sons, Ltd.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

Application of linear stochastic models for drought forecasting in the Büyük Menderes river basin, western Turkey

In the present study, a seasonal and non-seasonal prediction of the Standardized Precipitation Index (SPI) time series is addressed by means of linear stochastic models. The methodology presented here is to develop adequate linear stochastic models known as autoregressive integrated moving average (ARIMA) and multiplicative seasonal autoregressive integrated moving average (SARIMA) to predict drought in the Büyük Menderes river basin using SPI as drought index. Temporal characteristics of droughts based on SPI as an indicator of drought severity indicate that the basin is affected by severe and more or less prolonged periods of drought from 1975 to 2006. Therefore, drought prediction plays an important role for water resources management. ARIMA modeling approach involves the following three steps: model identification, parameter estimation, diagnostic checking. In model identification step, considering the autocorrelation function (ACF) and partial autocorrelation function (PACF) results of the SPI series, different ARIMA models are identified. The model gives the minimum Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC) is selected as the best fit model. Parameter estimation step indicates that the estimated model parameters are significantly different from zero. Diagnostic check step is applied to the residuals of the selected ARIMA models and the results indicated that the residuals are independent, normally distributed and homoscedastic. For the model validation purposes, the predicted results using the best ARIMA models are compared to the observed data. The predicted data show reasonably good agreement with the actual data. The ARIMA models developed to predict drought found to give acceptable results up to 2 months ahead. The stochastic models developed for the Büyük Menderes river basin can be employed to predict droughts up to 2 months of lead time with reasonably accuracy.     

AN INVESTIGATION OF THE RELATIVE ACCURACY OF THE MOST COMMON NORMAL-MOVEOUT EXPRESSIONS IN VELOCITY ANALYSES*

Three common expressions for the normal moveout of recorded seismic events are investigated by numerical simulation procedures for accuracy in predicting the root-mean-squared (RMS) or mean, as the case may be, subsurface velocity function from seismic data. The principal investigation, for which detailed error curves are shown, was derived for a stochastic subsurface model composed of strata with thicknesses ranging up to 91.4 m (300 ft) and boundary velocity contrasts ranging up to 45.7 m/sec (150 ft/sec); there was a 95 percent chance of velocity increase with increased depth. The effects of changes in the basic statistical subsurface model are discussed. The results appear to confirm the judiciousness of the choices of to and (x/z') as plotting parameters to be used with the respective percent errors in the three expressions, where are, respectively, the zero-offset arrival time of, the offset distance of, and the mean-squared velocity encountered by a seismic ray. Out of the three normal-moveout expressions examined, the “straight-raypath” expression with the RMS velocity substituted as its velocity term proved to be the most accurate in the determination of velocities.     

Optimal intraseasonal irrigation water distribution

A dynamic programming model is presented to optimise the intraseasonal distribution of irrigation water to a single crop under the constraints of limited water available and predetermined irrigation timing. The system underlying the model is characterised by two discrete state variables: the available soil-water in the root zone and the net quantity of water to be transferred to the root zone of the crop. Transition equations from one state to another are used in response to irrigation decisions. A multiplicative yield function is employed for estimating the crop yield as it is influenced by soil moisture. A numerical example is presented to illustrate the proposed procedure.     

Direct simulation of solute recycling in irrigated areas

Solute recycling from irrigation can be described as the process that occurs when the salt load that is extracted from irrigation wells and distributed on the fields is returned to the groundwater below irrigated surfaces by deep percolation. Unless the salt load leaves the system by means of drains or surface runoff, transfer to the groundwater will take place, sooner or later. This can lead to solute accumulation and thus to groundwater degradation, particularly in areas where extraction rates exceed infiltration rates (semi-arid and arid regions). Thus, considerable errors can occur in a predictive solute mass budget if the recycling process is not accounted for in the calculation. A method is proposed which allows direct simulation of solute recycling. The transient solute response at an extraction well is shown to be a superposition of solute mass flux contributions from n recycling cycles and is described as a function of the travel time distribution between a recycling point and a well. This leads to an expression for a transient ‘recycling source’ term in the advection–dispersion equation, which generates the effect of solute recycling. At long times, the ‘recycling source’ is a function of the local capture probability of the irrigation well and the solute mass flux captured by the well from the boundaries. The predicted concentration distribution at steady state reflects the maximum spatial concentration distribution in response to solute recycling and can thus be considered as the solute recycling potential or vulnerability of the entire domain for a given hydraulic setting and exploitation scheme. Simulation of the solute recycling potential is computationally undemanding and can therefore, for instance, be used for optimisation purposes. Also, the proposed method allows transient simulation of solute recycling with any standard flow and transport code.     

Influence of time‐varying frequency content in earthquake ground motions on seismic response of linear elastic systems

Earthquake ground motion records are nonstationary in both amplitude and frequency content. However, the latter nonstationarity is typically neglected mainly for the sake of mathematical simplicity. To study the stochastic effects of the time‐varying frequency content of earthquake ground motions on the seismic response of structural systems, a pair of closely related stochastic ground motion models is adopted here. The first model (referred to as ground motion model I) corresponds to a fully nonstationary stochastic earthquake ground motion model previously developed by the authors. The second model (referred to as ground motion model II) is nonstationary in amplitude only and is derived from the first model. Ground motion models I and II have the same mean‐square function and global frequency content but different features of time variation in the frequency content, in that no time variation of the frequency content exists in ground motion model II. New explicit closed‐form solutions are derived for the response of linear elastic SDOF and MDOF systems subjected to stochastic ground motion model II. New analytical solutions for the evolutionary cross‐correlation and cross‐PSD functions between the ground motion input and the structural response are also derived for linear systems subjected to ground motion model I. Comparative analytical results are presented to quantify the effects of the time‐varying frequency content of earthquake ground motions on the structural response of linear elastic systems. It is found that the time‐varying frequency content in the seismic input can have significant effects on the stochastic properties of system response. Copyright © 2016 John Wiley & Sons, Ltd.     

Non-stationary seismic response of MDOF systems by wavelet transform

A wavelet-based formulation has been presented in this paper for the stochastic analysis of a linear multi-degree-of-freedom (MDOF) classically damped system subjected to earthquake ground motion. The ground motion has been modelled as a non-stationary process (both in amplitude and frequency) using wavelets. Closed-form expressions of the moments of the instantaneous Power Spectral Density Function (PSDF) of the response have been derived and used to predict the statistics of the response peak of any desired order. For illustration of the formulation, an example torsionally coupled multistoried building has been considered along with the twenty synthetically generated time-histories corresponding to an example ground motion process. © 1997 John Wiley & Sons, Ltd.     

A theory of efficiency of storage-treatment systems

A simple expression is presented on the capability of storage-treatment systems to reduce non-point pollutant runoff load to natural waters. Their efficiency depends on the capacities of the facilities and probabilistic properties of runoff, such as interval, duration, volume, and concentration of runoff events. Assuming the compound Poisson process for runoff time series, the exact expressions of the ratio of treated load in terms of storage and treatment capacities are theoretically derived on the neighbourhoods of all boundaries of the domain on which the problem is defined. Then, an approximate expression over the whole domain is presented, of which the value and the first-order derivative coincide with those of the exact derived expressions near the boundaries. Accuracy is checked by Monte Carlo simulations.     

Seasonal evolution of meltwater generation,storage and discharge at a non‐temperate glacier in Svalbard

In glacierized catchments, meteorological inputs driving surface melting are translated into runoff outputs mediated by the glacier hydrological system: analysis of the relationship between meteorology and diurnal and seasonal patterns of runoff should reflect the functioning of that system, with the role of meltwater storage likely to be of particular importance. Daily meltwater storage is determined for a glacier at 78 °N in the Svalbard archipelago, by comparing inputs calculated from a surface energy balance model with measured outputs (proglacial discharge). Solar radiation, air temperature, wind speed and proglacial discharge are then analysed by regression and time‐series methods, in order to assess the meteorology–discharge relationship and its variation at diurnal and seasonal time‐scales. The recorded discharge time‐series can be divided into two contrasting intervals: up to early August, proglacial discharge was high and variable, mean hydrographs showed little indication of diurnal cycling, ARIMA models of discharge indicated a non‐seasonal, moving‐average generating process, and there was a net loss of meltwater from storage; from early August, proglacial discharge was low and relatively invariable, but with clearer diurnal cycles, regression models of discharge showed substantially improved correlations with air temperature and solar radiation, ARIMA models indicated a non‐seasonal, autoregressive generating process, and eventually a seasonal component, and there was a net gain in meltwater storage. The transition between the two periods is brief compared with the duration of the melt season. The runoff response to meteorology therefore lacks the strongly progressive element previously identified in mid‐latitude glacierized catchments. In particular, the glacier hydrological system only appears responsive to diurnal forcing following the depletion of the seasonal snowpack meltwater store. Copyright © 2001 John Wiley & Sons, Ltd.     

Correlation and covariance of runoff

The application of objective methods for interpolation of stochastic fields is based on the assumption of homogeneity with respect to the correlation function, i.e. only the relative distance between two points is of importance. This is not the case for runoff data which is demonstrated in this paper. Taking into consideration the structure of the river network and the related drainage basin supporting areas theoretical expressions are derived for the correlation function for flow along a river from its outlet and upstream. The results are exact for a rectangular drainage basin. For more complex basin geometry a grid approximation is suggested. The found relations are demonstrated on a real world example with a good agreement between the theoretically calculated correlation functions and empirical data.