Stochastic models in agricultural watersheds (original) (raw)
Estimating Long-Term Drainage at a Regional Scale Using a Deterministic Model
Soil Science Society of America Journal, 1997
This study discusses the use of a deterministic model, at a regional scale, when the soils have small heterogeneity. Long-term soil drainage was simulated on 3500 ha in Beauce (France), taking into account variation in soil thickness, agricultural practices, and climate characteristics for a 32-yr period. The model was based on a numerical solution of Richards' equation, using a finite element method. Hydraulic properties were determined at four sites that are 4 km apart. The results show a small variability of hydraulic parameters, with coefficient of variation (CV) ranges from 1 to 16% for the a parameter, the empirical constant n, the saturated water content O s , and the residual water content O,, and from 60 to 75% for the saturated hydraulic conductivity K,. Hence, the spatial and temporal variability of drainage are mainly related to soil thickness, agricultural practices, and climate characteristics. Assigning small spatial variability to hydraulic properties, we analyzed the role of soil thickness and initial water content (IWC), induced by agricultural practices and climate characteristics, on drainage prediction. Five IWC were defined: 100, 90,80,70, and 55% of field capacity (FC). Four soil thicknesses were studied: 30, 60,100, and 150 cm. The IWC appeared as a significant factor in drainage prediction for 60% of the 32 yr, regardless of soil thickness. Soil thickness became the most significant factor (20% of the 32 yr) for low IWC. The model was spatially extrapolated by combining the calculated drainage (which varied with soil thickness and IWC) and the spatial distribution of crops and soil thickness classes. Simulated drainage was closely related (r = 0.90) to groundwater recharge during the 32 yr, indicating that the model could be applied to other areas with appropriate parameters.
A stochastic approach for the description of the water balance dynamics in a river basin
Hydrology and Earth System Sciences Discussions, 2008
The present paper introduces an analytical approach for the description of the soil water balance dynamics over a schematic river basin. The model is based on a stochastic differential equation where the rainfall forcing is interpreted as an additive noise in the soil water balance. This equation can be solved assuming known the spatial distribution of the soil moisture over the basin transforming the twodimensional problem in space in a one dimensional one. This assumption is particularly true in the case of humid and semihumid environments, where spatial redistribution becomes dominant producing a well defined soil moisture pattern. The model allowed to derive the probability density function of the saturated portion of a basin and of its relative saturation. This theory is based on the assumption that the soil water storage capacity varies across the basin following a parabolic distribution and the basin has homogeneous soil texture and vegetation cover. The methodology outlined the role played by the soil water storage capacity distribution of the basin on soil water balance. In particular, the resulting probability density functions of the relative basin saturation were found to be strongly controlled by the maximum water storage capacity of the basin, while the probability density functions of the relative saturated portion of the basin are strongly influenced by the spatial heterogeneity of the soil water storage capacity. Moreover, the saturated areas reach their maximum variability when the mean rainfall rate is almost equal to the soil water loss coefficient given by the sum of the maximum rate of evapotranspiration and leakage loss in the soil water balance. The model was tested using the results of a continuous numerical simulation performed with a semi-distributed Correspondence to: S. Manfreda (manfreda@unibas.it) model in order to validate the proposed theoretical distributions.
Soil Science Society of America Journal, 1988
Solute leaching at the field scale can show considerable variability because of variations in the hydraulic properties of the soil. A simple mechanistic-stochastic model for predicting solute transport on this scale during intermittent flood irrigation is described and a method of calibrating the model independent of any leaching measurements is presented. The model assumes that the pore water velocity varies horizontally across the field but is uniform in the vertical direction. Explicit expressions for chemical diffusion and hydrodynamic dispersion are not included in the model. Instead, horizontal variations in the velocity serve to disperse the solute when measured on a field wide basis. The model requires knowledge of the probability density function for the velocity and estimates of the function parameters. These parameters were estimated from variations in infiltration rates measured within 63 ring infiltrometers and from water retention data. Model simulations of Br leaching agreed well with most measured data for both position of the tracer peak and the degree of solute spreading, indicating that at least under flood irrigation, variations in infiltration as controlled by soil hydraulic properties explain most of the observed variations in Br' leaching over a field. U.S. Water Conservation Lab., 4331 E. Broadway, Phoenix, AZ 85040. Third author is now at Dep. Geoscience, New Mexico Inst. of Mining and Technology, Socorro, NM 87801. Contribution of the USDA ARS, U.S. Water Conservation Lab., Phoenix, AZ 85040.
Soil Science, 2016
The formulation of a model that can reliably simulate the temporal groundwater level fluctuations of an aquifer is important for effective water resource management and for the prevention of possible desertification effects. Mires Basin at the island of Crete, Greece, is part of a major watershed with significantly reduced groundwater resources because of overexploitation during the past 30 years. In this work, the interannual variability of groundwater level is modeled with a discrete time autoregressive exogenous variable (ARX) model that is based on physical grounds (soilwater balance equation). Precipitation surplus is used as an exogenous variable in the ARX model. Three new modified versions of the original form of the ARX model are proposed and investigated: the first considers a larger time scale; the second considers a larger time delay in terms of the groundwater level input; and the third considers the groundwater level difference between the last two hydrological years, which is incorporated in the model as a third input variable. Modeling results for the time series of the spatially averaged groundwater level show very good agreement, after an initial adaptation period, with measured data. Among the three modified versions of the original ARX model considered in this work, the third model version shows significantly better agreement with measured data.
Presenting a Mathematical Model for Estimating the Deep Percolation Due to Irrigation
Infiltration is one of the most important factors of hydrology cycle. Deep percolation is the flowing of soil water by gravity below the effective depth of the root zone, that is an important factor in filling of groundwater and design of subsurface drainage. Deep percolation can be determined by taking field data to estimate soil water depletion using water balance equation. This method is very expensive and time consuming. The goal of this research was to quantify deep percolation due to irrigation with using a mathematical model. The input variables of this model are the effective parameters on deep percolation such as, bed slope, inflow rate and coefficients of soil infiltration. These variables were measured at 16 farms in Zayandehrood basin. Comparison between estimated and measured deep percolation showed that the model’s error percentage is 1.73%.
Modeling water-table fluctuations in a sloping aquifer with random hydraulic conductivity
Environmental Geology, 2001
To prevent environmental problems like water logging and increase in soil salinity which are responsible for the degradation of the top productive soils, an optimum ditch drainage design is required. For this purpose a knowledge of the spatio-temporal variation of the water table is essential. In this study the spatio-temporal variation of the water table in a sloping ditch drainage system has been modeled from a stochastic point of view, incorporating randomness in hydraulic conductivity to get the expression for the mean and the standard deviation of the water-table height. The hydraulic conductivity has been considered to be a realization of a log-normal distribution. Application of these expressions in the prediction of mean water-table variation with the associated error bounds has been demonstrated with the help of a ditch drainage problem of a sloping aquifer. The sensitivity analysis has also been carried out to see the effect of variability in the hydraulic conductivity on the water-table¯uctuations. The error bounds quanti®ed on the water-table height will thus help in the decisionmaking process for proper drainage design.
Canadian Water Resources Journal, 2016
In regions where soils are seasonally or perennially wet, subsurface drainage represents an essential water management practice. Two hydrological models with different modeling approaches as well as different dimensional and spatial scales, DRAINMOD (1D, lumped and field-scale) and CATHY (3D, spatially distributed and watershed-scale), were compared in terms of their performance to predict tile-drain flow and to simulate evapotranspiration (ET) under field conditions. Two metrics were defined to assess the capacity of the models to represent the soil water dynamics: relative errors in simulating peak flow and drainage volume. Using different hydraulic conductivity scenarios, both models provided similar results. For the total predicted/observed tile-drain flow comparison, the two models yielded very similar results. In terms of coefficient of determination (R 2) and Nash-Sutcliffe model efficiency (NSE), their performances were low in simulating tile-drain flows for dry periods (low observed tile-drain flow). During periods with higher observed tile-drain flow, the performance of both models was good (R 2 > 0.75 and NSE mostly > 0.60), but DRAINMOD produced better results than CATHY did. The two models had similar ET values (R 2 > 0.80). Regarding the impact of the hydraulic conductivity of each soil layer on subsurface drainage outflow, this study showed that the soil layer below the tile-drain system was the most influential for the two models. Dans les régions où les sols sont saisonnièrement ou perpétuellement humides, le drainage souterrain agricole représente une pratique essentielle de gestion de l'eau. Deux modèles hydrologiques avec différentes approches de modélisation ainsi que différentes échelles dimensionnelles et spatiales, DRAINMOD (1D, localisé et à l'échelle du champ) et CATHY (3D, spatialement distribué et à l'échelle de bassin versant), ont été comparés en fonction de leur performance pour prédire l'écoulement aux drains et à simuler le processus d'évapotranspiration dans des conditions de terrain. Deux mesures ont été définies pour évaluer la capacité des modèles à représenter la dynamique de l'eau dans le sol: les erreurs relatives à simuler le débit de pointe et le volume de drainage. En utilisant différents scénarios de conductivité hydraulique, les deux modèles ont fourni des résultats similaires. Vis-à-vis de la comparaison d'écoulement total drainé simulé/ observé, les deux modèles ont donné des résultats très similaires. En termes de coefficient de détermination (R 2) et le coefficient d'efficacité de Nash-Sutcliffe (NSE), pendant les périodes sèches (faible débit observé aux drains souterrains) leurs performances ont été faibles en simulant l'écoulement aux drains souterrains. Pendant les périodes d'écoulement observé aux drains plus élevé, la performance des deux modèles était bonne (R 2 > 0,75 et NSE généralement > 0,60), mais DRAINMOD produit de meilleurs résultats que CATHY. Les deux modèles ont bien simulé le processus d'évapotranspiration (R 2 > 0,80). En ce qui concerne l'impact de la conductivité hydraulique de chaque couche de sol sur l'écoulement sortant de drains souterrains, cette étude a montré que la couche de sol sous le système des drains souterrains était la plus influente pour les deux modèles. When subsurface drains are in place, the drainable water fraction of the soil profile is converted to shortterm (detention) storage over a period of few hours, days or weeks, depending on a number of variables. These include tile drain size, depth and spacing, soil type, outlet size/condition, and whether or not under continuous rainfall or snowmelt conditions. When drainable water is removed from the soil profile, infiltration can then occur. This is due to available soil pore space allowing water that would otherwise be stored in the surface depressions to infiltrate and have a direct pathway to downstream flow via the subsurface drains (BTSAC 2012).
Water Resources Research, 1983
Many researchers have expressed concerns regarding the uniqueness of parameter estimates for conceptual rainfall-runoff (R-R) models obtained through calibration. Recent studies (Sorooshian et al., this issue; Sorooshian and Gupta, this issue) have revealed that even though stochastic parameter estimation techniques can help, the problems are not all due to inefficiencies in the calibration techniques used but are caused by the manner in which the model is structurally formulated. Thus even when calibrated under ideal conditions (simulation studies), it is often impossible to obtain unique estimates for the parameters. It is possible to resolve this problem, at least in part, by appropriate reparameterizations of the pertinent model equations. In this paper the percolation equation of the soil moisture accounting model of the National Weather Service River Forecast System (SMA-NWSRFS) will be discussed. It is shown that a logical reparameterization of this equation can result in conditions that improve the chances of obtaining unique parameter estimates. It is believed that these results have implications for other conceptual R-R models in which similar approaches are used in the representation of the percolation/infiltration process.
A low-dimensional model for concentration-discharge dynamics in groundwater stream systems
Water Resources Research, 1998
In this paper we investigate the physical basis and validity of a dynamical model for environmental tracer response for a groundwater-dominated stream reach or small catchments. The dynamical model is formed by volume averaging of the local equations of saturated flow and solute transport. The approach interprets the empirical concentration-discharge C-Q as a pair of integrated state variables from an underlying state space (or phase space) for the hillslope or catchment response. The inputs represent episodic, seasonal, and random recharge rate and concentration time series. A closed form solution is found for constant inputs. Results are compared with numerical solutions of the governing partial differential equations, and agreement is found for a full range of initial states and levels of forcing. For pulse-type or piecewise constant recharge events, the phase plane trajectories for C-Q are shown to exhibit looping behavior, without the need for an assumption of hysteresis in the model. The orientation and looping direction of these solutions are shown to be controlled by the dimensionless ratio of solute residence time to hydraulic relaxation time and the relative phase lag between recharge and recharge concentration. The closed form solution is extended to the case of piecewise constant, random, input sequences resulting in a "random map" for C-Q. An application is presented for seasonal storage and flushing of SO-4 in runoff for a small catchment in central Pennsylvania. 1.
Arabian Journal of Geosciences, 2016
A transient finite difference groundwater flow model has been calibrated for the Nasia sub-catchment of the White Volta Basin. This model has been validated through a stochastic parameter randomization process and used to evaluate the impacts of groundwater abstraction scenarios on resource sustainability in the basin. A total of 1500 equally likely model realizations of the same terrain based on 1500 equally likely combinations of the data of the key aquifer input parameters were calibrated and used for the scenario analysis. This was done to evaluate model non-uniqueness arising from uncertainties in the key aquifer parameters especially hydraulic conductivity and recharge by comparing the realizations and statistically determining the degree to which they differ from each other. Parameter standard deviations, computed from the calibrated data of the key parameters of hydraulic conductivity and recharge, were used as a yardstick for evaluating model non-uniqueness. All model realizations suggest horizontal hydraulic conductivity estimates in the range of 0.03-78.4 m/day, although over 70 % of the area has values in the range of 0.03-14 m/day. Low standard deviations of the horizontal hydraulic conductivity estimates from the 1500 solutions suggest that this range adequately reflects the properties of the material in the terrain. Lateral groundwater inflows and outflows appear to constitute significant components of the groundwater budgets in the terrain, although estimated direct vertical recharge from precipitation amounts to about 7 % of annual precipitation. High potential for groundwater development has been suggested in the simulations, corroborating earlier estimates of groundwater recharge. Simulation of groundwater abstraction scenarios suggests that the domain can sustain abstraction rates of up to 200 % of the current estimated abstraction rates of 12,960 m 3 /day under the current recharge rates. Decreasing groundwater recharge by 10 % over a 20-year period will not significantly alter the results of this abstraction scenario. However, increasing abstraction rates by 300 % over the period with decreasing recharge by 10 % will lead to drastic drawdowns in the hydraulic head over the entire terrain by up to 6 m and could cause reversals of flow in most parts of the terrain.
Continuous Time Stochastic Analysis of Groundwater Flow in Heterogeneous Aquifers
Water Resour Res, 1995
The problem of depth-averaged groundwater flow in heterogeneous aquifers is looked at from a stochastic point of view. The Galerkin finite element version of the linearized stochastic equation governing the flow is solved analytically in time using an eigenvalue-eigenvector technique. The stochastic solution, which is valid for small variance of aquifer log transmissivity, relates spatial and temporal variabilities of aquifer head to aquifer heterogeneity, stochastic recharge, and random initial head. The computational effort requires the evaluation of integrals of matrices whose elements are linear functions of the nodal mean heads. The solutions are obtained for the exact (i.e., continuous) and quasi-steady approximation of the mean head. Aquifer-head temporal covariances are evaluated using standard matrix operations once the storage matrix is inverted .and the associated generalized eigenvalue problem is solved. Implication of the level of spatial discretization on the performance of the solution is examined, and the influence of aquifer heterogeneity on spatial and temporal variances of hydraulic heads is investigated in the presence of a semipervious boundary. [1981] solved the governing perturbation equation directly for one-dimensional and steady groundwater flow in the presence of boundaries. Unlike previous contributions whose major thrust was restricted to steady state flow, Dagan [1982], using Green's functions, analyzed nonsteady flow in infinite hetero
Analytical assessment and parameter estimation of a low-dimensional groundwater model
Journal of Hydrology, 2009
SummaryA class of transient groundwater models exists between the simple 0-dimensional storage-discharge functions and numerically-solved 2- and 3-dimensional spatially-distributed models. This class contains analytical solutions to linear partial differential equations subjected to time-varying stress, such as recharge or pumping. These solutions are limited to homogeneous 1-dimensional (1-D) representations of an aquifer. Their use in watershed models is rare to date, even though they are computationally efficient and overcome some of the major weaknesses of single-valued storage-discharge functions. Because these 1-D models have analytical solutions for hydraulic head and groundwater discharge, they allow both for a simpler initial assessment of model suitability and for initial estimation of model parameters using analytical methods. We evaluate two analytical, graphical parameter estimation methods whereby: (1) the second time derivative of a state (in this case, a volume) is plotted against the first derivative of the same state, and (2) the first time derivative of one state (volume) is plotted against another state (piezometric height). In both cases the identification of a data "envelope" is used to estimate model parameters, or groupings thereof. We use both methods to assess the suitability of a 1-D model to an alluvial plain system in New Zealand and examine the limitations of two graphical envelope techniques for estimating aquifer parameters at the catchment scale.
Water Resources Research, 1998
A new two-step stochastic modeling approach based on stochastic parameter inputs to a deterministic model system is presented. Step I combines a Stratified sampling scheme with a deterministic model to establish a deterministic response surface (DRS). Step II combines a Monte Carlo sampling scheme with the DRS to establish the stochastic model response. The new two-step approach is demonstrated on a one-dimensional unsaturated water flow problem at field scale with a dynamic surface flux and two spatially variable and interdependent parameters: The Campbell [1974] soil water retention parameter (b) and the saturated hydraulic conductivity (Ks). The new two-step stochastic modeling approach provides a highly time efficient way to analyze consequences of uncertainties in stochastic parameter input at field scale. The new two-step approach is competitive in analyzing problems with time consuming deterministic model runs where the stochastic problem can be adequately described by up to two spatially variable parameters.
Comparison of univariate and transfer function models of groundwater fluctuations
Water Resources Research, 1993
Seasonal autoregressive integrated moving average (SARIMA) univariate models and single input-single output transfer function (SARIMA with externalities or SARIMAX) models of groundwater head fluctuations are developed for 21 Upper Floridan aquifer observation wells in northeast Florida. These models incorporate empirical relationships between rainfall input and head response based on historical correlations and cross correlations between these two time series. The magnitude of the forecast error terms indicates that the SARIMA and SARIMAX models explain an average of 84-87% of the variation observed in the monthly piezometric head levels for 1-month lead forecasts. Thus the models account for the dominant processes which affect temporal groundwater fluctuations. Both the SARIMA and SARIMAX models provide unbiased forecasts of piezometric head levels; however, the SARIMAX models produce more accurate forecasts (i.e., smaller forecast probability limits) than the SARIMA models, particularly as lead time increases. Modeling efforts reveal consistent model structures over the study region, with local hydrologic and geologic conditions causing site-specific variability in the time series model parameters. INTRODUCTION The management of surface water and groundwater resources requires the use of modeling techniques which recognize the variability and uncertainty of hydrologic inputs. Rainfall, streamflow, evapotranspiration, and groundwater flow are all unpredictable processes which affect the design, operation, and management of water resource systems. Time series modeling techniques have been shown to provide a systematic empirical method for simulating and forecasting the behavior of uncertain hydrologic systems and for quantifying the expected accuracy of the forecasts. Time series modeling of suspended sediment concentration in rivers has been conducted by Sharma et al. [1979], Sharma and Dickinson [1980], Fitzgerald and Karlinger [1983], Gurne!l and Fenn [1984], Caroni et al. [1984], La Barbera et al. [1985], and Lemke [1990]. Their studies have shown that time series models provide an improved methodology for predicting suspended sediment concentrations in comparison with traditional simple regression models. Lernke [1991] developed single input-single output and multiple input-single output transfer function models for predicting daily suspended sediment concentrations and found that these models provided a good representation of dynamic fluvial processes. Matalas and Wallis [1976], O'Connell [1977], Stedinger [1981], Stedinger and Vogel [1984], and $tedinger et al. [1985] used multivariate time series models to simulate synthetic streamflows that exhibit long-term persistence. Jackson et al. [1973], Law [1974], Houston [1983], and Changnon et al. [1988] used time series analysis to examine climatological and hydrogeological variables associated with groundwater fluctuations in a variety of shallow unconfined aquifer systems. More recently, Ada
Journal of Hydrology, 1999
Estimating the major hydrological processes and their characteristic time scale is important when studying the hydrology of a catchment. However, in general, only limited data are available, namely climatic data and stream discharge. In this paper, the catchment is viewed as a system that converts the rainfall to the stream discharge through a transfer function (TF). The observed TF calculated from the rainfall and the specific stream discharge depends on the hydrological processes operating in the catchment. By comparing the observed TF to simulated TF, these processes and their time scale are identified. The simulated TF are developed from the Dupuit and linear representations of the aquifer. The identification of the TF is based on the stochastic method using a spectral representation of the rainfall and stream-discharge time series. The novelty of this work is to extend the stochastic approach to the one-order catchment hydrology and to develop a model, which takes into consideration both the aquifer discharge and the rapid flows. The method was applied to three first-order agricultural catchments. For each site, the theoretical results are in accordance with reality. These results show that the stream discharge is dominated by the aquifer flow, the fast transfer accounting for 3–8% of the total discharge depending on the catchment. The stochastic approach based on spectral analysis of the temporal variation of global observations appears useful to extract significant information about dominant processes occurring in the catchment and their characteristic time scale.
A stochastic approach to modeling the role of rainfall variability in drainage basin evolution
Water Resources Research, 2000
We develop a simple stochastic theory for erosion and sediment transport, based on the Poisson pulse rainfall model, in order to analyze how variability in rainfall and runoff influences drainage basin evolution. Two cases are considered: sediment transport by runoff in rills and channels and particle detachment from bedrock or cohesive soils. Analytical and numerical results show that under some circumstances, rainfall variability can have an impact equal to or greater than that of mean rainfall amount. The predicted sensitivity to rainfall variability is greatest when (1) thresholds for runoff generation and/or particle detachment are significant and/or (2) erosion and transport are strong nonlinear functions of discharge. In general, sediment transport capacity is predicted to increase with increasing rainfall variability. Depending on the degree of nonlinearity, particle detachment capacity may either increase or decrease with increasing rainfall variability. These findings underscore the critical importance of hydrogeomorphic thresholds and other sources of nonlinearity in process dynamics. The morphologic consequences of rainfall variability are illustrated by incorporating the pulse rainfall theory into a landscape simulation model. The simulation results imply that, all else being equal, catchments experiencing a shift toward greater climate variability will tend to have (1) higher erosion rates, (2) higher drainage density (because of increased runoff erosion efficiency), and ultimately (3) reduced relief. The stochastic approach provides a useful method for incorporating physically meaningful climate data within the current generation of landscape evolution models.
On the deterministic and stochastic use of hydrologic models
Environmental simulation models, such as precipitation-runoff watershed models, are increasingly used in a deterministic manner for environmental and water resources design, planning, and management. In operational hydrology, simulated responses are now routinely used to plan, design, and manage a very wide class of water resource systems. However, all such models are calibrated to existing data sets and retain some residual error. This residual, typically unknown in practice, is often ignored, implicitly trusting simulated responses as if they are deterministic quantities. In general, ignoring the residuals will result in simulated responses with distributional properties that do not mimic those of the observed responses. This discrepancy has major implications for the operational use of environmental simulation models as is shown here. Both a simple linear model and a distributed-parameter precipitation-runoff model are used to document the expected bias in the distributional properties of simulated responses when the residuals are ignored. The systematic reintroduction of residuals into simulated responses in a manner that produces stochastic output is shown to improve the distributional properties of the simulated responses. Every effort should be made to understand the distributional behavior of simulation residuals and to use environmental simulation models in a stochastic manner.