Stochastic Hydrology Research Articles

Quantile-Based Approach for Improving the Identification of Preferential Groundwater Networks

Identifying preferential paths for groundwater flow is one of the basics for understanding aquifer systems. Shallow free-surface aquifers often have flow directions (locally) similar to those of their surface counterparts, especially if surface and groundwater bodies are directly connected. This work proposes a novel and simple framework to improve the identification of Preferential Groundwater Networks in free-surface aquifers. This is possible by proposing a quantile mapping procedure borrowed from stochastic hydrology, usually employed to adjust rainfall simulations (for example, achieved via climate models) upon available gauge-based data. This well-known procedure is applied to redistribute simulations of the aquifer bottom elevation for a real case study in Lombardy, Northern Italy. The result is a spatial redistribution of the elevation quantiles that leads to aquifer bottom surfaces carved with Preferential Groundwater Networks that are spatially consistent with the surface river network. This way, groundwater flow directions are redistributed to mimic their surface counterparts, but aquifer bottom elevations and slopes are far gentler as they were previously simulated from borehole data information. Furthermore, the errors in the spatial reframing of borehole data and the discrepancy of variogram structures before and after the redistribution procedure are not dramatically dissimilar.

Scaled Brownian motion with random anomalous diffusion exponent

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, characterized by anomalous diffusion through the diffusion exponent. It is a Gaussian, self-similar process with independent increments and has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process called scaled Brownian motion with random exponent (SBMRE), which retains the key features of SBM at the level of individual trajectories, but with anomalous diffusion exponents that vary randomly across trajectories.We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf) and the qth absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions for the anomalous diffusion exponent, namely the mixture of two point distributions and beta distribution, and explore the asymptotics of the corresponding characteristics. Theoretical results for SBMRE are validated through numerical simulations and compared with the analogous characteristics of SBM.

Quantifying and Classifying Streamflow Ensembles Using a Broad Range of Metrics for an Evidence‐Based Analysis: Colorado River Case Study

AbstractStochastic hydrology produces ensembles of time series that represent plausible future streamflow to simulate and test the operation of water resource systems. A premise of stochastic hydrology is that ensembles should be statistically representative of what may occur in the future. In the past, the application of this premise has involved producing ensembles that are statistically equivalent to the observed or historical streamflow sequence. This requires a number of metrics or statistics that can be used to test statistical similarity. However, with climate change, the past may no longer be representative of the future. Ensembles to test future systems operations should recognize non‐stationarity and include time series representing expected changes. This poses challenges for their testing and validation. In this paper, we suggest an evidence‐based analysis in which streamflow ensembles, whether statistically similar to and representative of the past or a changing future, should be characterized and assessed using an extensive set of statistical metrics. We have assembled a broad set of metrics and applied them to annual streamflow in the Colorado River at Lees Ferry to illustrate the approach. We have also developed a tree‐based classification approach to categorize both ensembles and metrics. This approach provides a way to visualize and interpret differences between streamflow ensembles. The metrics presented, along with the classification, provide an analytical framework for characterizing and assessing the suitability of future streamflow ensembles, recognizing the presence of non‐stationarity. This contributes to better planning in large river basins, such as the Colorado, facing water supply shortages.

Characterizing the multisectoral impacts of future global hydrologic variability

There is significant uncertainty in how global water supply will evolve in the future, due to uncertain climate, socioeconomic, and land use change drivers and variability of hydrologic processes. It is critical to characterize the potential impacts of uncertainty in future water supply given its importance for food and energy production. In this work, we introduce a framework that integrates stochastic hydrology and human-environmental systems to characterize uncertainty in future water supply and its multisector impacts. We develop a global stochastic watershed model and demonstrate that this model can generate a large ensemble of realizations of basin-scale runoff with global coverage that preserves the mean, variance, and spatial correlation of a historical benchmark. We couple this model with a well-known human-environmental systems model to explore the impacts of runoff variability on the water and agricultural sectors across spatial scales. We find that the impacts of future hydrologic variability vary across sectors and regions. Impacts are felt most strongly in the water and agricultural sectors for basins that are expected to have unsustainable water use in the future, such as the Indus River basin. For this basin, we find that the variability in future irrigation water withdrawals and irrigated cropland increase over time due to uncertainty in renewable water supply. We also use the Indus basin to show how our stochastic ensemble can be leveraged to explore the global multisector consequences of local extreme runoff conditions. This work introduces a novel technique to explore the propagation of future hydrologic variability across human and natural systems and spatial scales.

A goodness-of-fit test for the compound Poisson exponential model

On the basis of bivariate data, assumed to be observations of independent copies of a random vector (S,N), we consider testing the hypothesis that the distribution of (S,N) belongs to the parametric class of distributions that arise with the compound Poisson exponential model. Typically, this model is used in stochastic hydrology, with N as the number of raindays, and S as total rainfall amount during a certain time period, or in actuarial science, with N as the number of losses, and S as total loss expenditure during a certain time period. The compound Poisson exponential model is characterized in the way that a specific transform associated with the distribution of (S,N) satisfies a certain differential equation. Mimicking the function part of this equation by substituting the empirical counterparts of the transform we obtain an expression the weighted integral of the square of which is used as test statistic. We deal with two variants of the latter, one of which being invariant under scale transformations of the S-part by fixed positive constants. Critical values are obtained by using a parametric bootstrap procedure. The asymptotic behavior of the tests is discussed. A simulation study demonstrates the performance of the tests in the finite sample case. The procedure is applied to rainfall data and to an actuarial dataset. A multivariate extension is also discussed.

The Decision Tree Aided Neuro-Fuzzy Inference Characterization of the Stochastic Hydrology of the Tana Alluvial Aquifer

The Tana Alluvial Aquifer is the name given to the little-understood aquifer which is active in the areas bordering the River Tana Flow course as the river weaves its way through the sedimentary plains of Balambala, Garissa, Fafi and Ijara and, finally, into the Tana Delta areas, with the common denominator being the proximity to the Lower Tana catchment, especially the riparian corridor of the River itself, and beyond. The aquifer may extend to between five to fifteen kilometers away from the river channels course way, and at times, it may be felt even 20 kilometers away. The geology of the locality is heterogeneous and comprise sediments whose soil mechanics may not be easily deciphered, since some areas close to the river have very fresh water while others are saline (Bura East in Fafi Sub County easily comes to mind here). There are areas far from the river but bearing fresh water (Mulanjo comes to mind). In some areas, sites close to the river discharge low yield figures, whereas those located farther afield discharge favorably. The water quality and discharge are therefore stochastic variables, subject to chance occurrence. In view of this inconsistency, and on the account of data scarcity, the neuro-fuzzy inference algorithm was developed to map the Universe of Discourse of the Tana Alluvial Aquifer, aka the T.A.A., as it relates to the longitudes, latitudes, depths, and discharges of the aquifers in the study area. The mapping was with respect to aquifer discharge, the variable used to characterize an aquifer, in terms of Transmissivity and Hydraulic Conductivity, thereby defining aquifer recharge propensity. Membership functions were developed using the trapezoidal membership family, and fuzzy rules were appropriately evolved from the fuzzified aquifer data, before finally employing the Sugeno inference engines (in Python) to make predictions of discharge, at each of the T.A.A. aquifer subsets mapped for fresh, saline, hard and brackish water species. The accuracy in the outputs achieved in the areas mapped vindicated the power of the neuro-fuzzy inference systems, as the accuracy oscillated between 92 and 99 percent, when the discharge values predicted were compared with the actual known discharge values of the wells mapped. The water quality class characterization was then undertaken using the decision tree (DT) algorithm in python which gave rise to a 100 percent prediction accuracy. The same DT algorithm could not successfully predict the discrete values of aquifer discharge or EC values, with as much accuracy (but performed excellently with salinity class data), and that was why fuzzy logic was employed. The study vindicated the use of the DT and Fuzzy Logic Algorithms as simple, yet powerful analytical tools, in characterizing the Stochastic Hydrology of the Tana Alluvial Aquifer.

ASSESSMENT SCHEME OF WATER-RESERVOIRS IMPACT ON GROUNDWATER

Water reservoirs play a unique role in determining the mechanism of their impact on groundwater. The goal of our study was to develop a modern option of an impact mechanism and indicator scheme to assess reservoirs' impact on groundwaters. For this purpose, the methods of mathematical statistics and stochastic hydrology were used. Different types of reservoirs with different durations of exploitation were selected as study objects. As per the updated impact mechanism, the filtration regime and yield are the function of properties of water-reservoir level and the bed constituent rocks and respond to the level fluctuation with a certain delay. One of the main determinants of the delay time is the distance from the reservoir to the filtrate monitoring station. It was found that filtration decreases in proportion to the bed silting, while the time of delay increases in proportion to silting. As silting reaches its maximum, the filtration regime is determined only by the filtrate of the derivation tunnel and other structures and by the level of the reservoir. The indicator scheme selected to evaluate the reservoir impact on the groundwater is as follows: debit of streams, range of underground waters' horizon fluctuation, infrastructure affected by the landslide, and bogged plots of field. The criteria to assess the efficiency of the water reservoir are: if the filtrate is used for any purpose, its value augments the efficiency of the reservoir; if the filtrate is harmful to the infrastructure and environment, it undermines the importance of the reservoir.

Hybrid causal multivariate linear modelling (H_CMLM) method for the analysis of temporal rivers runoff

This paper describes the joint development of two different methods for temporal riveŕs runoff assessment. This is performed through a hybrid approach by means of Multivariate General Linear Models (MGLM; inspired by MLR as a statistical method), and Causal Reasoning (CR; as non-linear ones). This innovative methodological approach, named Hybrid Causal Multivariate Linear Modelling (H-CMLM), is mainly aimed to empower the analysis of temporal hydrological records behaviour. H-CMLM has been successfully applied to three different Spanish basins (Adaja, Mijares and Porma) which were chosen due to their disparate features. Results were divided in quantitative and qualitative. Numerical results show a very high level of equivalence between the average value of temporal dependence provided by MLM module and the continuous behaviour of temporal dependence computed by CR module and visualized through Dependence Mitigation Graph (DMG). This high coherent outcome from both modules makes the analysis much more robust from a stochastic hydrology point of view. Values for average temporal dependence are very useful for the optimal dimensioning of hydraulic infrastructures like reservoirs. Furthermore, given the annual scale of the analysis, water planning and management of several water uses such as domestic water supply, agriculture, industrial demands, among others, can be highly assisted by this new H_C-MLM method.

Multi-step rainfall forecasting using deep learning approach.

Rainfall prediction is immensely crucial in daily life routine as well as for water resource management, stochastic hydrology, rain run-off modeling and flood risk mitigation. Quantitative prediction of rainfall time series is extremely challenging as compared to other meteorological parameters due to its variability in local features that involves temporal and spatial scales. Consequently, this requires a highly complex system having an advance model to accurately capture the highly non linear processes occurring in the climate. The focus of this work is direct prediction of multistep forecasting, where a separate time series model for each forecasting horizon is considered and forecasts are computed using observed data samples. Forecasting in this method is performed by proposing a deep learning approach, i.e, Temporal Deep Belief Network (DBN). The best model is selected from several baseline models on the basis of performance analysis metrics. The results suggest that the temporal DBN model outperforms the conventional Convolutional Neural Network (CNN) specifically on rainfall time series forecasting. According to our experimentation, a modified DBN with hidden layes (300-200-100-10) performs best with 4.59E−05, 0.0068 and 0.94 values of MSE, RMSE and R value respectively on testing samples. However, we found that training DBN is more exhaustive and computationally intensive than other deep learning architectures. Findings of this research can be further utilized as basis for the advance forecasting of other weather parameters with same climate conditions.

On a new statistical wave generator based on atmospheric circulation patterns and its applications to coastal shoreline evolution

Stochastic models have a long history in the simulation of synthetic data sequences for application to risk assessment e.g. in stochastic hydrology. Their use in coastal engineering has recently grown. In this study we illustrate some typical coastal vulnerability applications of a recently developed stochastic wave climate model that has links to synoptic scale atmospheric circulation patterns (CPs). Wave data are explicitly used to delineate the links to CPs. This modelling approach allows for simulating long continuous time series of waves that are linked to their associated CPs. The time series can be used for various types of coastal vulnerability assessments as a new management tool. The east coast of South Africa is used as a case study. The results demonstrate the robustness of the modelling technique and the potential applications in coastal vulnerability assessments. The link to CPs facilitates the application of these methods to the assessment of future climate changes on coastal vulnerability.

Stochastic hydrology: the development of main ideas in Russia

The aim of the work is to assess the current state of scientific research and the main results in the field of stochastic hydrology. Various problems of scientific and applied hydrology are solved on the basis of different types of stochastic models. On the basis of these models, methods are being developed for obtaining secure engineering runoff characteristics, or reliable hydrological forecasts of various lead times. Most of the water management, construction, environmental and other problems are solved today within the framework of probabilistic models and statistical methods. The main tasks of stochastic hydrology can be divided into several groups. Despite the discussion in hydrology on the applicability of the theory of probability and random processes to the analysis of fluctuations of hydrometeorological series, the problem of choosing a research methodology remains one of the most important, and is often discussed in connection with anomalous manifestations of hydrological phenomena (extreme floods, prolonged droughts). An important applied and theoretical problem in hydrology is the choice of a one-dimensional probability distribution. In this part, discussions are important both on the problem of the type of new distributions and on methods for estimating parameters. Ideas and hypotheses are considered that can significantly increase the reliability of statistical characteristics. Climatic changes in many cases cause disturbances in the stationarity of time series of hydrological characteristics. Progress in the development of new approaches to probabilistic modeling is associated both with the well-known problem of the sufficiency of the Markov hypothesis and with the need to introduce new, more complex models and Bayesian estimation methods. The article is devoted to the discussion of the current state of these problems in the form of a scientific review.

An ensemble genetic programming model for seasonal precipitation forecasting

Seasonal precipitation forecasting is one of the most challenging tasks in stochastic hydrology. This article proposes a new ensemble model, called EGP, to a season-ahead forecast of total seasonal precipitation. The EGP integrates evolutionary genetic programming (GP) and gene expression programming (GEP) techniques with multiple linear regression to increase forecasting accuracy of standalone GP and GEP models, while it secures their explicit structure. The EGP model was trained and validated using 88 years (1930–2017) of measured precipitation data from Muratpasa Station, Antalya, Turkey. The model performance was evaluated in terms of different statistical error measures and cross-validated with two other ensemble models as well as the state-of-the-art random forest developed in this study as the benchmark. The results showed that the proposed model can increase the forecasting accuracy of the best standalone GP and GEP models up to 30%. The EGP was also found to be superior to random forest, particularly in predicting low and high seasonal precipitation amount. This model is explicit, easy to evolve, and therefore, motivating to be used in practice.

Stochastic Hydrology: Is Scientific Understanding Growing?

Among many problems of stochastic hydrology, several major problems may be singled out. (1) The methodology problem – may fluctuation of hydro-meteorological values be considered within the framework of probabilities and random processes? Was this topic discussed after 1953? (2) One-dimensional probability distributions – is there progress? Are there new models? (3) Random Processes: Is Markovian property sufficient or more complex models with memory are needed? (4) Lack of stability resulting from climate changes: Is there progress in understanding the approaches to probabilistic forecasts?

Nearest neighbor time series bootstrap for generating influent water quality scenarios

Understanding influent water quality variability is essential for the long-term planning of potable water systems. To quantify variability and generate realistic influent scenarios, we propose a nonparametric time series approach based on k-nearest neighbor (k-NN) bootstrap resampling. The k-NN approach resamples historical data conditioned on a “feature vector” at a given time to generate values at subsequent times. We modified this algorithm by adding random perturbations to the resampled values to generate realistic extremes unobserved in the historical record. k-NN is widely used in stochastic hydrology and hydroclimatology; however, it is adapted here for the multivariate, data-limited context of water treatment. To examine the performance of the algorithm, we applied it to an eleven-year, monthly water quality dataset of alkalinity, temperature, total organic carbon, and pH from the Cache la Poudre River in Colorado. We found that the k-NN simulations captured the relevant distributional statistics of the historical record, which suggests that the algorithm produces realistic and varied scenarios. When used in conjunction with modeling and optimization, these scenarios have the potential to improve the sustainability, resilience, and efficiency of potable water systems.

Nicholas Constantinos Matalas (1930–2019)

A modest giant in the field of stochastic hydrology.

Forecasting monthly rainfall using autoregressive integrated moving average model (ARIMA) and artificial neural network (ANN) model: A case study of Junagadh, Gujarat, India

The onset, withdrawal and quantity of rainfall greatly influence the agricultural yield, economy, water resources, power generation and ecosystem. Time series modelling has been extensively used in stochastic hydrology for predicting various hydrological processes. The principles of stochastic processes have been increasingly and successfully applied in the past three decades to model many of the hydrological processes which are stochastic in nature. Time lagged models extract maximum possible information from the available record for forecasting. Artificial neural network has been found to be effective in modelling hydrological processes which are stochastic in nature. The ARIMA model was used to simulate and forecast rainfall using its linear approach and the performance of the model was compared with ANN. The computational approach of ANN is inspired from nervous system of living beings and the neurons possess the parallel distribution processing nature. ANN has proven to be a reliable tool for modelling compared to conventional methods like ARIMA and therefore ANN has been used in this study to estimate rainfall. In this study, rainfall estimation of Junagadh has been attempted using monthly rainfall training data of 32 years (1980-2011) and testing data of 5 years (2012-2016). A number of ANN model structures were tested, and the appropriate ANN model was selected based on its performance measures like root mean square error and correlation coefficient. The correlation coefficient Seasonal ARIMA (1,0,0)(3,1,1)12 and ANN back-propagation model (5-12-1) on the testing data was found to be 0.75 and 0.79 respectively. Seasonal ARIMA (1,0,0)(3,1,1)12 and ANN back-propagation model (5-12-1) were used for forecasting rainfall of 5 years (2017-2021).

Rainfall and runoff estimation of micro watersheds of coastal Navsari

The onset, withdrawal and quantity of rainfall greatly influence the agricultural yield, economy, water resources, power generation and ecosystem. Hence, if the variations in rainfall are known well in advance, it would be possible to reduce the adverse impacts related to excess or deficient rainfall, providing us prior information about droughts and floods. The principles of stochastic processes have been increasingly and successfully applied in the past three decades to model many of the hydrological processes which are stochastic in nature. Time series modelling has been extensively used in stochastic hydrology for predicting hydrological processes like rainfall and runoff. Time lagged models extract maximum possible information from the available record for forecasting. In this study, rainfall forecast of Navsari was attempted using rainfall training data of 30 years (1983-2012) and runoff was estimated using curve number method for each micro watershed of coastal Navsari. The models used for forecasting rainfall, based on the time series data, were seasonal ARIMA model and ANN multilayer perceptron model. A seasonal ARIMA model, based on Box Jenkins method model was built by determining its appropriate parameters and the ANN model was built using Levenberg Marquardt algorithm. The seasonal ARIMA model was determined using four steps namely identification of autoregressive and moving average terms, estimating the parameters using maximum likelihood method, diagnostic checking of residuals and forecasting. A number of ANN model structures were tested and the appropriate ANN model was selected based on its performance on validation data after which the model was deployed for forecasting. It was concluded that the non linear ANN model was superior to linear ARIMA model in terms of its forecasting accuracy tested on validation data. The predicted rainfall was then utilized to estimate the runoff using Curve number method for each micro watershed of Navsari coast.

Stochastic hydrogeology's biggest hurdles analyzed and its big blind spot

Abstract. This paper considers questions related to the adoption of stochastic methods in hydrogeology. It looks at factors affecting the adoption of stochastic methods including environmental regulations, financial incentives, higher education, and the collective feedback loop involving these factors. We begin by evaluating two previous paper series appearing in the stochastic hydrogeology literature, one in 2004 and one in 2016, and identifying the current thinking on the topic, including the perceived data needs of stochastic methods, the attitude in regulations and the court system regarding stochastic methods, education of the workforce, and the availability of software tools needed for implementing stochastic methods in practice. Comparing the state of adoption in hydrogeology to petroleum reservoir engineering allowed us to identify quantitative metrics on which to base our analysis. For impediments to the adoption of stochastic hydrology, we identified external factors as well as self-inflicted wounds. What emerges is a picture much broader than current views. Financial incentives and regulations play a major role in stalling adoption. Stochastic hydrology's blind spot is in confusing between uncertainty with risk and ignoring uncertainty. We show that stochastic hydrogeology comfortably focused on risk while ignoring uncertainty, to its own detriment and to the detriment of its potential clients. The imbalance between the treatment on risk on one hand and uncertainty on the other is shown to be common to multiple disciplines in hydrology that interface with risk and uncertainty.

Simulation of Stochastic Processes Exhibiting Any‐Range Dependence and Arbitrary Marginal Distributions

AbstractHydrometeorological processes are typically characterized by temporal dependence, short‐ or long‐range (e.g., Hurst behavior), as well as by non‐Gaussian distributions (especially at fine time scales). The generation of long synthetic time series that resembles the marginal and joint properties of the observed ones is a prerequisite in many uncertainty‐related hydrological studies, since they can be used as inputs and hence allow the propagation of natural variability and uncertainty to the typically deterministic water‐system models. For this reason, it has been for years one of the main research topics in the field of stochastic hydrology. This work presents a novel model for synthetic time series generation, termed Symmetric Moving Average (neaRly) To Anything, that holds out the promise of simulating stationary univariate and multivariate processes with any‐range dependence and arbitrary marginal distributions, provided that the former is feasible and the latter have finite variance. This is accomplished by utilizing a mapping procedure in combination with the relationship that exists between the correlation coefficients of an auxiliary Gaussian process and a non‐Gaussian one, formalized through the Nataf's joint distribution model. The generality of Symmetric Moving Average (neaRly) To Anything is stressed through two hypothetical simulation studies (univariate and multivariate), characterized by different dependencies and distributions. Furthermore, we demonstrate the practical aspects of the proposed model through two real‐world cases, one that concerns the generation of annual non‐Gaussian streamflow time series at four stations and another that involves the synthesis of intermittent, non‐Gaussian, daily rainfall series at a single location.

An improved gene expression programming model for streamflow forecasting in intermittent streams

Skilful forecasting of monthly streamflow in intermittent rivers is a challenging task in stochastic hydrology. In this study, genetic algorithm (GA) was combined with gene expression programming (GEP) as a new hybrid model for month ahead streamflow forecasting in an intermittent stream. The hybrid model was named GEP-GA in which sub-expression trees of the best evolved GEP model were rescaled by appropriate weighting coefficients through the use of GA optimizer. Auto-correlation and partial auto-correlation functions of the streamflow records as well as evolutionary search of GEP were used to identify the optimum predictors (i.e., number of lags) for the model. The proposed methodology was demonstrated using monthly streamflow data from the Shavir Creek in Iran. Performance of the GEP-GA was compared to that of classic genetic programming (GP), GEP, multiple linear regression and GEP-linear regression models developed in the present study as the benchmarks. The results showed that the GEP-GA outperforms all the benchmarks and motivated to be used in practice.