The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology

This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.     

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.     

Stochastic perturbation analysis of groundwater flow. Spatially variable soils,semi-infinite domains and large fluctuations

As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.     

Development and testing of a physically based,three-dimensional model of surface and subsurface hydrology

We present a numerical, catchment-scale model that solves flow equations of surface and subsurface flow in a three-dimensional domain. Surface flow is described by the two-dimensional parabolic approximation of the St. Venant equation, using Manning’s equation of motion; subsurface flow is described by the three-dimensional Richards’ equation for the unsaturated zone and by three-dimensional Darcy’s law for the saturated zone, using an integrated finite difference formulation. The hydrological component is a dynamic link library implemented within a comprehensive model which simulates surface energy, radiation budget, snow melt, potential evapotranspiration, plant development and plant water uptake. We tested the model by comparing distributed and integrated three-dimensional simulated and observed perched water depth (PWD), stream flow data, and soil water contents for a small catchment. Additional tests were performed for the snow melting algorithm as well as the different hydrological processes involved. The model successfully described the water balance and its components as evidenced by good agreement between measured and modelled data.     

基于尺度变换随机共振的瞬变电磁弱信号检测

晚期瞬变电磁信号是大参数、多频率、且各频率分量未知的电磁信号,特别在晚期还是强噪声背景下的微弱信号.随机共振系统可以在极限信噪比的情况下提取微弱信号,但仅适用于小参数、单一频率或频率已知的高频信号的情况.为此,本文提出一种基于尺度变换的随机共振算法用于检测晚期瞬变电磁信号,该方法充分利用了随机共振检测弱信号的优势,通过引入尺度变换,消除了随机共振系统对待测信号频率的限制,在绝热近似理论下,实现了从强噪声中提取出微弱晚期瞬变电磁目标信号.理论分析和仿真结果表明:对埋在强噪声中的未知频率的晚期瞬变电磁信号,采用连续的压缩算法,以获得一个适当的输入信号到随机共振体系,根据输出信号共振谐振峰的变化,经反变换运算可得待测弱晚期瞬变电磁信号的各个未知频率,各个频率的叠加即可还原晚期瞬变电磁信号.与传统方法相比,本方法数据采集量和采集时间都减小到原来的几十分之一,能在极限信噪比(信噪比-50 dB)下,提取出目标信号,为获得深层目标信号提供了可能,从而提高了瞬变电磁探测仪器的探测深度和探测精度.     

Solution of stochastic groundwater flow by infinite series,and convergence of the one-dimensional expansion

This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series solutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random field. The standard deviation of lnK is denoted . The solution method is based on a Taylor series expansion in terms of parameter , around the value =0, of the hydraulic head (H) and gradient (J). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gradient J is given by a linear first-order recursion involving a stochastic integral operator. The convergence of the -expansion solution is not guaranteed a priori. In one dimension, however, we prove convergence by solving explicitly the hierarchical sequence of equations to all orders. An infinite-order stochastic solution is obtained in the form of a -power series that converges for any finite value of . It is pointed out that other expansion methods based on K rather than lnK yield divergent series. The infinite-order solution depends on the integration method and the boundary conditions imposed on individual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningful far-field gradient conditions. The whole approach can serve as a basis for treatment of higher-dimensional problems.     

On the Panday and Huyakorn surface–subsurface hydrology test case: analysis of internal flow dynamics

A number of previous studies using models of integrated surface‐subsurface hydrology have adopted the Panday and Huyakorn (P&H) tilted V‐catchment test case (Panday S, Huyakorn PS. 2004. A fully coupled spatially distributed model for evaluating surface/subsurface flow. Advances in Water Resources 27: 361–382) to show inter‐code comparability. The P&H test case is used to evaluate models that simulate a broad range of hydrological processes, and yet only the catchment outflow hydrograph has been presented as verification of the consistency between codes. Therefore, a more comprehensive evaluation of the surface‐subsurface hydrology of the P&H case is needed. This study explores the internal catchment functioning of the P&H case, using the popular catchment simulator MODHMS. The processes leading to streamflow generation in the model are illustrated, including separation of overland flow (OLF) and groundwater discharge to the stream. The results identify non‐physical flow processes due to the problem set‐up, and modifications to the P&H case are suggested that include changes to stream roughness and incision of the stream channel to overcome these shortcomings. A modified P&H case produced more plausible transfers between OLF and the stream, and an increased groundwater discharge to the stream (6·5% of streamflow in the modified case compared to 0·5% in the original case). Despite changes to internal flow processes, near‐identical outflow hydrographs were obtained, showing the importance of considering and comparing internal flow processes when using surface‐subsurface hydrology test cases to evaluate integrated hydrological simulators. Copyright © 2011 John Wiley & Sons, Ltd.     

Modified sediments and subsurface hydrology in natural and recreated salt marshes and implications for delivery of ecosystem services

Recreation or restoration of salt marsh through the deliberate removal of flood defences (managed realignment or de‐embankment) is a common practice across Europe and the USA, with potential to enhance delivery of ecosystem services. However, recent research suggests that physical, chemical and ecological processes may be impaired in recreated sites as a result of the modified morphology, sediment structure and hydrology associated with both the restoration process and historic land use. This paper compares physical sediment properties and subsurface water levels recorded in paired natural and de‐embanked (recreated) salt marshes in SE England. Using a combination of statistical and time‐series modelling, significant differences between the natural and recreated marshes are identified. Sediment properties (bulk density, moisture content and organic content) within each marsh were statistically different and imply that de‐embanked sediments are compacted, which may affect subsurface water movement. Analysis of hydrological time series reveals that the de‐embanked salt marsh is characterized by a damped response to tidal flooding with elevated and less variable water levels. This, combined with analysis of hydrographs and hysteresis patterns over individual tidal cycles, suggests that fast, horizontal near‐surface flows enhanced by the relict land surface may play a greater role in drainage of the de‐embanked salt marsh. The importance of hydrological functioning in governing many important physical and biogeochemical processes in salt marshes suggests any modifications would have significant implications for the delivery of ecosystem services. Copyright © 2014 John Wiley & Sons, Ltd.     

Mountain hydrology of Canada and China: A case study in comparative hydrology

Comparative hydrology attempts to identify regions with similar environmental attributes and then compares the hydrological activities in these regions to facilitate the transfer of knowledge and exchange of techniques. As a case study, the mountainous regions of Canada and China were compared. Vertical zonation is a characteristic feature of such regions, generating contrasts in microclimate, vegetation, soil and the resultant hydrological responses. An increase in precipitation with altitude is modified by aspect and other local factors. In temperate and colder latitudes, snow and ice are important, especially at higher elevations. Both countries have conducted many hydrological studies on snow and glaciers, but evaporation in mountainous terrain merits more attention. Runoff from mountainous catchments is often generated by a range of processes including rainfall, snowmelt and glacier melt, and runoff response to water input tends to be rapid. This study identifies some of the strengths and weaknesses of our current knowledge of mountain hydrology in both countries, and suggests that a more comprehensive treatment of the subject will improve the prediction of the hydrological behaviour of such regions.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

The interaction between hydrology and geomorphology in a landscape simulator experiment

An experimental landscape simulator has been developed which uses a rainfall simulator to create overland flow and erosion. The simulator uses rainfall sprinklers that eliminate rainsplash and an artificial soil which has little cohesion. Experimental landscapes developed in the simulator evolved according to Howard's headward growth model. Elements of Glock's model could be identified during evolution (i.e. initiation and maximum extension), but other stages of this model were not observed (i.e. extension and integration). The Horton concept of cross‐grading and micropiracy and stream piracy was not observed despite the dominance of overland flow, nor the groundwater headward growth mechanism proposed by Dunne, the latter due to experimental design, which eliminated any perched groundwater table. The experimental apparatus produced model landscapes that are scaled‐down analogues of real world processes. Copyright © 2001 John Wiley & Sons, Ltd.     

Reporting negative results to stimulate experimental hydrology: discussion of “The role of experimental work in hydrological sciences – insights from a community survey”*

ABSTRACT

Experimental work in hydrology is in decline. Based on a community survey, Blume et al. showed that the hydrological community associates experimental work with greater risks. One of the main issues with experimental work is the higher chance of negative results (defined here as when the expected or wanted result was not observed despite careful experimental design, planning and execution), resulting in a longer and more difficult publishing process. Reporting on negative results would avoid putting time and resources into repeating experiments that lead to negative results, and give experimental hydrologists the scientific recognition they deserve. With this commentary, we propose four potential solutions to encourage reporting on negative results, which might contribute to a stimulation of experimental hydrology.     

A comparison of surface and subsurface controls on summer temperature in a headwater stream

This study compared summer stream temperature between two years in the Star Creek catchment, Alberta, a headwater basin on the eastern slopes of the Canadian Rocky Mountains. Star Creek is a subsurface water dominated stream, which represents important habitat for native salmonid species. Hydrometeorological data from May to September of 2010 and 2011 accompanied by stream energy budget calculations were used to describe the drivers of stream temperature in this small forested stream. Mean, maximum, and minimum weekly stream temperatures were lower from May to August and higher in September 2011 compared to 2010. Weekly range in stream temperature was also different between years with a higher range in 2010. Inter‐annual stream temperature variation was attributed discharge differences between years, shown to be primarily governed by catchment‐scale moisture conditions. This study demonstrates that both meteorological and hydrological processes must be considered in order to understand stream temperature response to changing environmental conditions in mountainous regions. Copyright © 2013 John Wiley & Sons, Ltd.     

The effect of tide on the hydrology and morphology of a freshwater river

How does river hydrology and morphology change due to tidal influence? We contend that this is a question of particular consequence to many earth surface disciplines, but one that has not been adequately addressed. Previous studies have relied on gradients in channel morphology and stratigraphy to infer energy regime of channels. However, in tidal rivers geomorphology influences the energy regime while the energy regime influences morphology; thus, geomorphic and stratigraphic patterns do not fully resolve the mechanisms which lead to change. We addressed this problem by comparing measurements of hydraulic energy and channel morphology along a tidal gradient to predictions of these characteristics in the absence of tides, and attributed the differences to tidal processes. Measurements of discharge, channel area, and energy dissipation (in kJ day^–1) were made over a 24·8 hour period at four sites spanning the non‐tidal to tidal freshwater Newport River, NC. We then predicted those characteristics under non‐tidal conditions using hydraulic geometry relationships and literature values from coastal plain rivers. Discharge was enhanced more than 10‐fold by tide, and this tidal effect increased from upstream to downstream along the tidal gradient. Cross‐sectional area increased three‐fold due to tide. Energy dissipation measured in the upper tidal river was four‐fold lower than predicted to occur in the absence of tide because tides decreased average velocity and discharge. Energy dissipation measured downstream was similar to that predicted to occur without tides, although there was large uncertainty in predicted values downstream. While this limited dataset does not permit us to make broad generalizations for definitive models, it does provide a proof‐of‐concept for a new approach to addressing a critical problem at the interface of fluvial and coastal morphology. Copyright © 2013 John Wiley & Sons, Ltd.     

The hydrology of glacier-bed overdeepenings: Sediment transport mechanics,drainage system morphology,and geomorphological implications

Evacuation of basal sediment by subglacial drainage is an important mediator of rates of glacial erosion and glacier flow. Glacial erosion patterns can produce closed basins (i.e., overdeepenings) in glacier beds, thereby introducing adverse bed gradients that are hypothesized to reduce drainage system efficiency and thus favour basal sediment accumulation. To establish how the presence of a terminal overdeepening might mediate seasonal drainage system evolution and glacial sediment export, we measured suspended sediment transport from Findelengletscher, Switzerland during late August and early September 2016. Analyses of these data demonstrate poor hydraulic efficiency of drainage pathways in the terminus region but high sediment availability. Specifically, the rate of increase of sediment concentration with discharge was found to be significantly lower than that anticipated if channelized flow paths were present. Sediment availability to these flow paths was also higher than would be anticipated for discrete bedrock-floored subglacial channels. Our findings indicate that subglacial drainage in the terminal region of Findelengletscher is dominated by distributed flow where entrainment capacity increases only marginally with discharge, but flow has extensive access to an abundant sediment store. This high availability maintains sediment connectivity between the glacial and proglacial realm and means daily sediment yield is unusually high relative to yields exhibited by similar Alpine glaciers. We present a conceptual model illustrating the potential influence of ice-bed morphology on subglacial drainage evolution and sediment evacuation mechanics, patterns and yields, and recommend that bed morphology should be an explicit consideration when monitoring and evaluating glaciated basin sediment export rates.     

Modelling the effects of land use and climate changes on hydrology in the Ursern Valley,Switzerland

While many studies have been conducted in mountainous catchments to examine the impact of climate change on hydrology, the interactions between climate changes and land use components have largely unknown impacts on hydrology in alpine regions. They need to be given special attention in order to devise possible strategies concerning general development in these regions. Thus, the main aim was to examine the impact of land use (i.e. bushland expansion) and climate changes (i.e. increase of temperature) on hydrology by model simulations. For this purpose, the physically based WaSiM‐ETH model was applied to the catchment of Ursern Valley in the central Alps (191 km²) over the period of 1983?2005. Modelling results showed that the reduction of the mean monthly discharge during the summer period is due primarily to the retreat of snow discharge in time and secondarily to the reduction in the glacier surface area together with its retreat in time, rather than the increase in the evapotranspiration due to the expansion of the “green alder” on the expense of grassland. The significant decrease in summer discharge during July, August and September shows a change in the regime from b‐glacio‐nival to nivo‐glacial. These changes are confirmed by the modeling results that attest to a temporal shift in snowmelt and glacier discharge towards earlier in the year: March, April and May for snowmelt and May and June for glacier discharge. It is expected that the yearly total discharge due to the land use changes will be reduced by 0.6% in the near future, whereas, it will be reduced by about 5% if climate change is also taken into account. Copyright © 2013 John Wiley & Sons, Ltd.     

“Panta Rhei—Everything Flows”: Change in hydrology and society—The IAHS Scientific Decade 2013–2022

Abstract

The new Scientific Decade 2013–2022 of IAHS, entitled “Panta Rhei—Everything Flows”, is dedicated to research activities on change in hydrology and society. The purpose of Panta Rhei is to reach an improved interpretation of the processes governing the water cycle by focusing on their changing dynamics in connection with rapidly changing human systems. The practical aim is to improve our capability to make predictions of water resources dynamics to support sustainable societal development in a changing environment. The concept implies a focus on hydrological systems as a changing interface between environment and society, whose dynamics are essential to determine water security, human safety and development, and to set priorities for environmental management. The Scientific Decade 2013–2022 will devise innovative theoretical blueprints for the representation of processes including change and will focus on advanced monitoring and data analysis techniques. Interdisciplinarity will be sought by increased efforts to connect with the socio-economic sciences and geosciences in general. This paper presents a summary of the Science Plan of Panta Rhei, its targets, research questions and expected outcomes.

Editor Z.W. Kundzewicz

Citation Montanari, A., Young, G., Savenije, H.H.G., Hughes, D., Wagener, T., Ren, L.L., Koutsoyiannis, D., Cudennec, C., Toth, E., Grimaldi, S., Blöschl, G., Sivapalan, M., Beven, K., Gupta, H., Hipsey, M., Schaefli, B., Arheimer, B., Boegh, E., Schymanski, S.J., Di Baldassarre, G., Yu, B., Hubert, P., Huang, Y., Schumann, A., Post, D., Srinivasan, V., Harman, C., Thompson, S., Rogger, M., Viglione, A., McMillan, H., Characklis, G., Pang, Z., and Belyaev, V., 2013. “Panta Rhei—Everything Flows”: Change in hydrology and society—The IAHS Scientific Decade 2013–2022. Hydrological Sciences Journal. 58 (6) 1256–1275.     

The hydrology of three high-altitude forests in Central Himalaya,India: a reconnaissance study

In this preliminary study the partitioning of rain-water into various components of the hydrological cycle in three high-altitude forests of contrasting tree physiognomies (namely, Aesculus indica, Quercus semecarpifolia and Abies pindrow) were studied in the Nandadevi Biosphere Reserve, Central Himalaya, India. The results are compared with the hydrological characteristics of low-altitude forests of this region. The study has indicated a significant role of tree physiognomy with regard to rainfall partitioning into the various components of the hydrological cycle. It is suggested that A. pindrow (an evergreen tree) should be considered superior to A. indica (a deciduous tree) with regard to soil and water conservation in this region. This work is of relevance to land management programmes pertaining to afforestation, logging and regeneration. © 1998 John Wiley & Sons, Ltd.