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Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

Predictions of hydrologic system response to natural and anthropogenic forcing are highly uncertain due to the heterogeneity of the land surface and subsurface. Landscape heterogeneity results in spatiotemporal variability of hydrological states and fluxes, scale-dependent flow and transport properties, and incomplete process understanding. Recent community activities, such as Prediction in Ungauged Basins of International Association of Hydrological Sciences, have recognized the impasse current catchment hydrology is facing and have called for a focused research agenda toward new hydrological theory at the watershed scale. This new hydrological theory should recognize the dominant control of landscape heterogeneity on hydrological processes, should explore novel ways to account for its effect at the watershed scale, and should build on an interdisciplinary understanding of how feedback mechanisms between hydrology, biogeochemistry, pedology, geomorphology, and ecology affect catchment evolution and functioning.


1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

Predictions of hydrologic system response to natural and anthropogenic forcing are highly uncertain due to the heterogeneity of the land surface and subsurface. Landscape heterogeneity results in spatiotemporal variability of hydrological states and fluxes, scale-dependent flow and transport properties, and incomplete process understanding. The current paradigm of hydrological theory is that small-scale (local) process understanding (e.g. Richards’ equation to describe flow in porous media) is sufficient to quantify large-scale processes (e.g. baseflow recession at the outlet of a catchment), as long as some ‘equivalent’ or ‘effective’ values of the scale-dependent (flow and transport) parameters can be identified. However, many observations are invalidating this approach. Hillslope experiments and numerical simulations have revealed that subsurface flow is mainly governed by matrix flow when water storage is below some threshold value, but quickly switches to macropore flow when that threshold is crossed (e.g. Blöschl and Zehe 2005; Nieber et al. 2006; Tani 1997; Tromp-van Meerveld and McDonnell 2006). The hillslope thus acts as a non-linear filter that enhances water availability during low-flow conditions, but increases water release during high flow. At the catchment scale, several studies have shown that travel time distributions of non-reactive solutes have ‘heavy’ tails, suggesting that their transport through the subsurface is governed by large conductivity contrasts related to soil matrix porosity, macropores, fracture networks, and other large-scale heterogeneities (Kirchner et al. 2000).

Advances toward a unifying theory of watershed hydrology are hampered by our inability to quantify the heterogeneity of surface and subsurface landscape properties and how they control the hydrologic response (McDonnell et al. 2007). Figure 1 provides an example of how local subsurface heterogeneity (in this case, bedrock topography variability and associated soil depth) give rise to emergent hillslope flow properties, such as saturation connectivity, threshold-like subsurface stormflow, and corresponding flashy outflow behavior at small catchment scales (C. Harman, personal communications).

image

Figure 1. Local subsurface heterogeneity, such as bedrock microtopography and associated soil depth, gives rise to emergent properties in saturated connectivity, threshold-like subsurface stormflow and flashy hydrograph response at small catchment scales (C. Harman, personal communications).

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Aquifer heterogeneity and its effect on flow and transport has been studied in great detail. Analyses of well observations during steady and unsteady pumping tests have revealed the scale-dependent nature of flow and transport variables (such as hydraulic conductivity and dispersivity) in aquifers exhibiting non-stationary stochastic properties (Endres et al. 2007). Our ability to deploy similar techniques to study the effect of landscape heterogeneity on catchment-scale hydrologic responses is, however, limited due to technological constraints on imposing hydrological drivers (e.g. precipitation) and observing state variables (e.g. subsurface flow). Developing new hydrological theory that can explain observed phenomena, such as threshold-like flow response at hillslope scales or fractal filtering of inert tracers in precipitation and streamflow, requires thus a more holistic approach (Zehe et al. 2007). It is the objective of this article to review recent progress in understanding and modeling of catchment hydrological processes, and to present short descriptions of innovative ideas in the hydrologic literature that attempt to formulate a unifying theory at the catchment scale.

2 Quantifying Subsurface Heterogeneity

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

The use of geostatistics, often coupled with Monte Carlo simulations, is a common practice among subsurface hydrologists. Traditionally, stationary geostatistical inversion is used to assess spatial variability of parameters, such as permeability, transmissivity, and porosity (Neuman et al. 2004). In these methods, the subsurface (aquifer) is viewed as randomly heterogeneous but statistically homogenous. Flow properties, such as transmissivity, are treated as multivariate, statistically homogenous, isotropic random fields having a constant ensemble mean, variance, and integral (autocorrelation) scale. To estimate the parameters of such random fields, pumping tests are very useful. In quasi-steady-state analyses of well responses at constant discharge, for example, Monte Carlo simulations are combined with geostatistical inversion to yield detailed ‘tomographic’ estimates of how the parameters vary in space (Yeh et al. 2008). Drawbacks to such geostatistical methods are the need for multiple cross-hole pressure interference tests and the intensive computing power necessary to run the detailed simulations. Neuman et al. (2004) developed a type curve method to graphically solve the inversion problem. They found that simple graphical approaches could be used to estimate the geometric mean, integral scale, and variance of local aquifer properties using only quasi-steady-state head data.

Traditional geostatistical methods have also been used when the target porous medium represents a hierarchical geologic structure (Ritzi et al. 2004). Non-stationary hierarchical porous media exhibit both systematic and random spatial and directional variations in hydraulic properties on many length scales (Sposito 1998). At the catchment scale, the heterogeneous subsurface most likely will exhibit such non-stationary hierarchical structure, and thus its effect on flow and transport needs to be accounted for. Recent work by Neuman et al. (2008) shows that the ‘success’ of traditional geostatistical analysis to infer the spatial covariance structure of hierarchical, multiscale porous media is an artifact of the finite sampling window applied to measure geologic and hydrologic variables. They demonstrate that with the aid of truncated power variograms (Di Federico and Neuman 1997) to capture the variations in geostatistical parameters (variance and integral scale), this artifact can be eliminated using only a few scaling parameters. Their new method is able to represent multiscale random fields that are either Gaussian or have Levy probability distributions. This is an improvement of the traditional geostatistical methods, which use stationary spatial statistics to characterize non-stationary hierarchical media. The results of Neuman et al. (2008) are potentially very important to catchment hydrologists who have to deal with ubiquitous non-stationary hierarchical geologic settings. The challenge to apply this theory at the catchment scale is to make use of existing hydrologic data and/or to design specific field investigations to reliably estimate the scaling parameters that capture multiscale variations of hydraulic and transport properties.

Another method that can be used to study the scaling behavior of strong heterogeneous porous media is the renormalization group (RG) method (Hristopoulos 2003). RG is a theoretical framework for estimating probability distributions of random fields when scale is changed. Application of RG methods in subsurface hydrology includes the study of the implicit non-linear dependence of the subsurface flow and transport processes on the hydraulic conductivity and the accurate estimation of up-scaled conductivity and macrodispersion coefficients in multiscale heterogeneous porous media (Lunati et al. 2002). The method of up-scaling relies on a procedure in which individual cells are replaced by blocks of cells, and the parameter values of the new blocks are determined by some transformation of the original parameter values in the block. Renormalization iteratively applies two transformations: coarse graining and rescaling. The end result of this is the elimination of the finer fluctuations and the renormalization of the probability density function parameters. A very useful application of RG research in catchment hydrology is focusing on ‘anomalous’ diffusion,1 resulting from power-law distributions of the subsurface velocity field (Benson et al. 2000) with long-range correlations (Kirchner et al. 2000). RG theory is particularly useful for systems that exhibit critical phenomena near phase transitions, such as percolation (at the threshold of transition, conducting clusters of all sizes appear in the subsurface; see Section 3.1 for more details).

Yeh et al. (2008) argue that recently developed tomographic surveying techniques (e.g. Cassiani et al. 2006) combined with data fusion concepts will enhance basin-scale characterization of the heterogeneous subsurface. Basin-scale tomographic surveys can make use of different types of passive computerized axial tomography scan technologies that exploit recurrent natural stimuli (e.g. lightning, earthquakes, storm events, barometric variations, river-stage variations) as sources of excitations. Along with in situ sensor networks, they provide long-term and spatially distributed monitoring of system responses at the land surface and in the subsurface. This approach to basin-scale subsurface characterization is still in its infancy and will require intensive interdisciplinary collaboration (e.g. surface and subsurface hydrology, geophysics, geology, geochemistry, information and sensor technology, applied mathematics, atmospheric science), but seems an important step toward a better understanding of the effects of heterogeneity on catchment responses.

3 Dealing with Subsurface Heterogeneity in Catchment Modeling

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

3.1 fractional advection and dispersion

Burns (1996) describe results from a one-dimensional tracer test in a laboratory-scale (1 m) sandbox with very uniform sand and relative homogeneous and isotropic properties. A number of tracer tests were conducted to estimate the transport characteristics of the sand. These tests showed non-Gaussian breakthrough curves with very heavy leading and trailing tails. Such non-Gaussian transport in a homogeneous porous medium is likely the result of preferential flow phenomena such as fingering. Preferential flow breaks the symmetry of perfect mixing (Gaussian transport; for more details, see Blöschl and Zehe 2005). The classic advection–dispersion equation (ADE), based on Fick's dispersion law, with constant parameters (longitudinal dispersivity) is unable to predict such breakthrough curves. In order to simulate such behavior with ADE, the longitudinal dispersivity is allowed to change (increase) in an attempt to mimic the faster spreading of the tracer plume. Benson et al. (2000) show that non-Fickian transport can be successfully simulated by means of a transport equation that uses fractional order dispersion derivatives. Solutions of this fractional ADE represent plumes that spread proportional to time1/α instead of time1/2 for the classical ADE (α is a real number between 1 and 2 and defines the fractional dispersion derivative). The main advantage is that the dispersion parameter is not a function of time or distance, but a property that represents the transport characteristics of the porous medium. These results demonstrate that a fractional-order equation is a powerful predictive tool of heavy-tailed transport behavior. Such behavior, present even in the relatively homogeneous experiment of Burns (1996), is known to govern flow and transport through the heterogeneous subsurface.

Kirchner et al. (2000) used time series of chloride concentrations in rainfall and streamflow from several catchments in Wales to demonstrate that these catchments act as fractal filters. Although the chloride concentrations in rainfall have a white noise power spectrum, the power spectra of streamflow chloride concentrations show 1/f fractal scaling, indicating that the travel time distributions in these catchments are power-law distributions. This implies that contaminant (e.g. nitrate) export out of the catchments would have heavy tails (the catchments have very long memory of past inputs). If indeed the travel time distribution follows a power-law, that is, p(t)~t−α, then the convolution method to estimate output concentrations of inert tracers (such as stable water isotopes) from known input concentrations represents a fractional integral, suggesting that the fractional ADE is the appropriate prediction tool to handle effects of subsurface heterogeneity on catchment flow and transport. More research is needed to explore how this approach could provide the correct theoretical framework to estimate catchment water travel time distributions from inert tracer observations.

3.2 percolation theory

The previous discussion focused on the characterization of subsurface heterogeneity and its effect on large-scale flow and transport without special attention toward landscape connectivity. Flow and transport, however, are almost always organized in tree-like networks (McDonnell et al. 2007; Zehe et al. 2007). Geostatistical techniques provide powerful tools to characterize spatial patterns, but are not able to discern between patterns with and without connectivity (Di Domenico et al. 2007; Western and Grayson 1998). The concept of connectivity is strongly linked to percolation phenomena and widely used in percolation theory. The phenomenon of percolation is defined as ‘the special property of a system which emerges at the onset of macroscopic connectivity within it’ (Berkowitz and Balberg 1993). Percolation theory provides the theoretical basis to determine the probability that certain points within the system are connected. The probability, p, that there is an open path from some fixed point to a distance of r decreases polynomially, that is, p(r)~r−β, where β is some parameter that does not depend on local properties of the porous medium, but only on its dimension. Lehmann et al. (2007) applied percolation theory to model the fill-and-spill mechanism observed by Tromp-van Meerveld and McDonnell (2006). Percolation theory may be applied to a physical system, that is, at or close to a critical point. In Lehmann et al. (2007), the critical point refers to a state of subsurface connectivity (related to a threshold rainfall amount) that triggers rapid drainage.

Di Domenico et al. (2007) used percolation theory to study the transition between spatially random patterns to spatially connected patterns of soil moisture (Western and Grayson 1998; Zehe and Sivalapan 2008), and addressed the question whether this process behaves as a critical point phenomenon. Critical point behavior refers to a state transition from unorganized (isolated) clusters to organized (highly connected) clusters. Feder (1988) has argued that percolation clusters are statistically self-similar, and thus fractals. Self-similarity leads to invariant-scale behavior near the critical point. In other words, similar to river networks (Rodriguez-Iturbe and Rinaldo 1997), soil moisture percolation networks should be scale-invariant. Di Domenico et al. (2007) used simulated soil moisture fields and applied the RG method to show that percolation clusters of an up-scaled modeling lattice is qualitatively the same as the percolation cluster of the original lattice.

Nieber et al. (2006) present an interesting numerical simulation study in which they explore the effect of disconnected macropores on flow through a porous cylinder. The macropores are embedded in a homogeneous loamy soil matrix and have diameter size of 0.005–0.01 m. The saturated hydraulic conductivity of the macropores is six orders of magnitude larger than that of the soil matrix. Even though the macropores do not form a connected network, Nieber et al. (2006) observe that the modeled flow increases significantly (up to 40%) when the cylinder reaches saturation. At that time, all macropores are filled with water and the flow increases threshold-like. They speculate that a similar mechanism can take place in a hillslope where the macropores will have a diversity of orientation, shape, size, and length. They refer to field observations of Sidle et al. (2001) where hillslope seepage flow was controlled by moisture level. The numerical study of Nieber et al. (2006) further supports the idea that percolation theory could be used to describe the self-organization that appears to occur in hillslope soils.

3.3 hillslope similarity and closure relationships

At the hillslope scale, percolation theory can give great insight into the connectivity of subsurface flow. Percolation theory efficiently explains the threshold-like flow response of a heterogeneous hillslope by means of a connected network. However, percolation models require extensive field measurements to be calibrated, and, hence, quantifying percolation thresholds and hydraulic parameters at hillslope scales is still a formidable task. In order to quantify the flow response in ungaged hillslopes or basins, it is imperative that we more fully understand the cumulative effect of surface and subsurface topography and soil heterogeneity.

In order to explicitly account for subsurface heterogeneity at the hillslope scale, it is necessary to solve the mass, momentum, and energy balance equations in heterogeneous porous media (see Section 2). Conservation of mass, momentum, and energy and the second law of thermodynamics can be expressed at the hydrodynamic scale at which these dynamic relations are best understood (e.g. Darcy-Buckingham equation to describe flow through porous media). The selection of model parameters at such scales is highly uncertain due to the heterogeneity of the surface and subsurface, leading to a phenomenon referred to as equifinality (Beven and Binley 1992). Equifinality is the principle that in open systems a given end state can be reached by many potential means. In hydrological modeling, two models are equifinal if they lead to an equally acceptable or behavioral representation of the observed natural processes (Beven and Freer 2001). Reggiani et al. (1998) developed the idea of a representative elementary watershed (REW) in an attempt to effectively incorporate, up-scale, and the effects of small-scale spatial variability. Formulating the conservation equations at the REW scale introduces, however, another problem, namely, that of closure (Beven 2006). The closure problem exists because, at the scale of an REW (a hillslope or small watershed), the constitutive relationship between hydrologic state variables (e.g. saturated storage) and fluxes (e.g. lateral subsurface flow) are generally unknown. Zehe et al. (2006) used a physically based hydrological model that has been extensively tested with field observations from an experimental watershed in Germany (the Weiherbach catchment) to derive REW scale state variables and effective REW scale soil hydraulic functions to address the closure problem related to subsurface flow processes.

Another way to relate the subsurface flow response from one hillslope to another is through hillslope similarity. Berne et al. (2005) have shown that a dimensionless number, the hillslope Peclét number (Pe), is a hillslope similarity parameter that predicts the dimensionless characteristic response function (CRF; the subsurface flow hydrograph resulting from an initial steady-state storage volume, normalized by the total volume of water drained). Lyon and Troch (2007) applied the Pe number to real hillslopes in New Zealand and the United States, and obtained good agreement between observed and predicted moments of the dimensionless CRF. Estimating the Pe number for a given hillslope does not require explicit quantification of the hydraulic subsurface parameters like hydraulic conductivity and drainable porosity, but needs information about the average hillslope storage (related to climate and inevitably hillslope transmissivity). Moreover, the Pe number was derived under assumptions of a homogeneous subsurface.

Harman (2007) used the two-dimensional Boussinesq equation to study the effect of subsurface heterogeneity (expressed as spatially variable saturated hydraulic conductivity) and climate (different recharge regimes) on hillslope flow similarity. Harman (2007) confirmed that effective hydraulic properties of the subsurface cannot represent any randomized heterogeneous distribution of hydraulic properties. Heterogeneous hillslopes discharge more water initially, but then have longer recession tails due to the storage and slow release of water from regions that are less hydraulically conductive. He also observed, in accordance with regime theory (Robinson and Sivapalan 1997), that when the duration of time of recharge events and the length of time between recharge events are both small in comparison to the characteristic timescale of a hillslope, a memory exists between one recharge event and another, and the peak discharge has a contribution from more than one event. On the other hand, when both of these are large in comparison to the characteristic timescale of a hillslope, then the hillslope exhibits a flashy response that is only governed by one recharge event.

Different climate regimes and heterogeneous hydraulic conductivity affect both the peak discharge and the overall structure of a hillslope's CRF. In an attempt to more explicitly deal with heterogeneity and climate, Harman (2007) proposes the ‘flowpath-timescale’ as an efficient concept to provide closure relationships between internal storage and flow. The idea of flowpath-timescales is to determine the amount of time that it would take for water recharging at a specific location to exit the hillslope. Harman (2007) used a one-dimensional Boussinesq model, with heterogeneous hydraulic conductivity, to observe the dimensionless time that it takes for water to travel, from recharge to outlet, as a function of dimensionless distance upslope. He parameterized the simulated flowpath-timescale by means of an incomplete Beta function. This flowpath-timescale function is able to replicate very well the initial part of the hydrograph in addition to the long recessions when compared to the full model. Harman's work revealed (i) that no unique closure relationship exist – it depends on the nature of subsurface heterogeneity and flow regimes, and (ii) that the closure relations reflect an interaction between the climate and the bottom boundary conditions (how is the hillslope connected to the channel network?). More work is needed to relate the parameters of the flowpath-timescale function to observable hydrologic responses and landscape characteristics. In the next section, we review studies that make use of the integrated hydrologic response to obtain information about processes, states, and properties at catchment scales.

3.4 doing hydrology backwards

Hydrologists have for long used the integrated response of a catchment to atmospheric forcing to estimate flow and transport parameters, as well as initial states and dominant processes. In this section, we review some recent advances in using the integrated hydrological response to derive information about the internal structure and functioning of catchments.

Martina and Entekhabi (2006) developed a method to extract spatial information about runoff producing processes from the analysis of precipitation and discharge observations at the catchment scale. They argued that the nonlinear co-variations between storm precipitation and storm runoff volumes can be used to draw conclusions about the marginal distribution of available soil water storage across a basin. That means, by observing how a catchment reacts to hydrologic forcing, it is possible to derive the antecedent moisture conditions – not at the exact location, but as a probability density function of local soil column saturation deficit prior to a storm event. This method is based on the analysis of the non-linear behavior that the rainfall–runoff relation exhibits. The degree of non-linearity represents a signature of the pore space that is available throughout the catchment. The fraction of storm runoff depends on how much pore space is not filled with soil water at the time when the rainfall sets in. Taking this approach one step further, Martina and Entekhabi (2006) developed a method that can predict the area of runoff producing regions within a catchment, although no information about location is provided.

Kirchner (2008) shows that certain catchments can be considered as simple first-order non-linear dynamical systems, and the governing equations can be derived directly from field data (streamflow fluctuations). The basic principle of his approach is the assumption that a unique relationship exists between the storage within a catchment and the outflow from this catchment. The way a catchment reacts to changes in storage can be expressed as a function that relates changes in storage to the amount of discharge that is produced. This function describes how sensitive a catchment is to changes in storage. This sensitivity function can be linear, exponential, or hyperbolic, depending on how the catchment behaves. The sensitivity function can be derived from streamflow time series alone. Once this relation is established, catchment discharge can be modeled as a function of evaporation and precipitation time series alone. From the form of the sensitivity function, it is possible to derive the main storage behavior within a catchment. In case the function is linear, the catchment can be considered to behave as a linear reservoir. In case the function is exponential, it exhibits a lower storage threshold below which no discharge is produced. In case of a hyperbolic function, there is an upper limit to storage where any surplus storage is directly converted to discharge, a process that can be related to the ‘fill-and-spill’ behavior of flashy hydrologic systems (Tromp-van Meerveld and McDonnell 2006).

4 Toward a Unifying Theory of Watershed Hydrology

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

There is growing recognition among hydrologists that the way toward new hydrological theory is to approach watershed hydrology as an interdisciplinary earth science (Lin et al. 2006; Sivapalan 2005). Despite tremendous heterogeneity of landscape properties, there is structure and organization across spatial and temporal scales that, when understood how and why it comes about, can help improving hydrologic predictability. The central role of energy, water, and carbon flows, driven by temperature, chemical, and gravitational gradients, and modulated by vegetation, soil, bedrock, and time, has been recognized for decades by ecologists, soil scientists, geomorphologists, and hydrologists, but prior research efforts have largely been conducted within these disciplines. In the next sections, we review recent studies that focus on complexity and structure of the landscape, and that formulate fundamental principles that govern hydrologic systems response and functioning.

4.1 complexity and structure of the critical zone

Of particular interest to catchment hydrology is the characterization of heterogeneity of the ‘critical zone’. The critical zone is the ‘heterogeneous, near surface environment in which complex interactions involving rock, soil, water, air, and living organisms regulate the natural habitat and determine the availability of life-sustaining resources’ (National Research Council 2001). The critical zone includes the land surface, vegetation, and water bodies, and extends through the pedosphere, unsaturated vadose zone, and saturated groundwater zone. It serves as a conduit for the flow of water and the transport of solutes, as well as an interface with surface and groundwater for the exchange of solutes (Corwin et al. 2006).

Rasmussen et al. (2005) recently presented a general theory of quantitative energy transfer to soil systems, embracing concepts of open-system thermodynamics and traditional quantitative models of soil development to predict the critical zone evolution. Like ecological systems, soil systems are open, non-equilibrium systems which tend to self organize to optimize the use of energy cycling within and flowing through the system, as long as entropy is passed external to the pedon through dissipative processes (see Rasmussen et al. 2005 and references therein). According to Rasmussen et al. (2005), the cycling and fluxes of energy and mass is driven by inter-related environmental energy gradients that maintain the system in a non-equilibrium state. The dissipation of energy along these gradients governs the structural organization within the critical zone system (e.g. soil profile development), as well as the concurrent evolution of water flow paths. Consequently, a quantification of the energy and mass input to a soil system should represent the soil forming environment and potentially the developmental state of the soil system (Figure 2; Rasmussen and Tabor 2007).

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Figure 2. Pedogenic indicators regressed against effective energy and mass transfer (EEMT): (A) pedon depth; (B) total pedon clay content; (C) free Fe oxide to total Fe oxide ratio (Fed/FeT) of the first subsurface genetic horizon; and (D) the chemical index of alteration minus potassium (CIA–K) of the first subsurface genetic horizon. Plotted lines and equations represent the best fit regression to the data. Data derived from pedons sampled from stable landscape positions across four environmental gradients on basalt, andesite, and granite parent materials (from Rasmussen and Tabor 2007).

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A prerequisite to connect pedogenic modeling and hydrological response and to make such efforts valuable for the advancement of understanding and predictability of hydrologic processes is to quantify the mechanisms linking soil evolution to these hydrologic processes. Numerous field studies have qualitatively related changes in soil hydrologic properties and hydrologic processes due to different aspects of soil evolution and soil physical structure in particular, but such relationships have rarely been quantified (see Lohse and Dietrich 2005 and references therein). One example is provided by Lohse and Dietrich (2005) who quantitatively relate the development of subsurface clay-rich horizons to a decline of subsurface horizon saturated hydraulic conductivity values, impeding rates of vertical water flow but increasing the importance of shallow subsurface lateral flow.

We argue that if we can find ways to quantify the implications of soil development and structure on hydrological processes, approaches to predict pedogenesis and related soil properties from more readily available information on climatic data and landscape position can be valuable for parameter estimation for hydrologic models, particularly in ungauged basins. Furthermore, unlike point measurements, these approaches may enable parameter predictions at scales of hydrological units in distributed models. While the scale of prediction may be relatively coarse, the properties of the predicted homogeneous units could potentially serve as mean values providing a basis for stochastic approaches. Furthermore, heterogeneity and complexity may change over time, with climate changes and landscape disturbance further impairing the rate and direction of these changes (McDonnell et al. 2007). These changes and their respective consequences could be accounted for with this type of evolutionary approaches.

4.2 thermodynamics and optimality principles

The search for a unifying theory is the Holy Grail in every scientific field. This fundamental desire stands in every human endeavor to understand nature, the idea of finding the fundamental principle that explains the behavior of a system subject to different initial and boundary conditions over a wide range of scales. Thermodynamics has offered fundamental laws in the field of physics and chemistry, and by extension to most other fields in science. Water flow in a hillslope can be studied from the thermodynamic point of view (water in motion due to potential gradients). Classical thermodynamics deals mainly with isolated closed systems, but hydrology certainly deals with open systems. Rasmussen et al. (2008) posit that hydrologic systems organize and evolve in response to open system fluxes of energy and matter driven by gravitational, geochemical, and radiant gradients. Open system energy and mass flow facilitate internal ordering and elemental cycling through development of dissipative structures that transfer energy and matter across the system. Hence, structural organization should be related to these transfers through the watershed. The central importance of these fluxes is widely recognized, yet we lack the ability to quantify the effective flows that are fundamental for understanding catchment functioning and predicting rates of hydrologic processes.

Optimality principles applied to watersheds may offer a framework to study the effects of effective mass and energy flows on catchment functioning and hydrologic response. In biology, West et al. (1997) developed a model that describes how essential biological materials are transported through space-filling fractal networks of branching tubes, by minimizing the dissipated energy and assuming that the terminal tube size is scale-invariant. Schymanski et al. (2008) used vegetation optimality principles to model soil moisture dynamics, root water uptake and fine root respiration in a tropical savanna study site. The optimality-based model reproduced the main features of observations at the site, such as a shift of roots from the shallow soil in the wet season to the deeper soil in the dry season and substantial root water uptake during the dry season. Recently, Bejan and co-workers (Bejan 2000; Bejan and Lorente 2004; Lewins 2003) have proposed a generalization of thermodynamics, known as thermodynamics of flow systems with configuration, to understand flow structures in both natural and man-made systems. The fundamental underlying physical principle is Bejan's constructal theory: ‘For a finite size flow system to persist in time (to survive) it must evolve in such way that it provides easier and easier access to the currents that flow through it’ (Bejan 2007). Constructal theory offers a principle that considers not only the final state of equilibrium of the system but also the patterns and shapes that the flow acquires through time. Simulating flow through a heterogeneous porous media, Lorente and Bejan (2006) found that alternating tree architectures have flow resistances lower than those of purely unorganized porous media with the same volume and internal flow volume. Inside a hillslope, two different kinds of flow occurs: one through the porous media (diffusion) and one trough low resistance flow paths (preferential flow). Seepage flow and channeled flow must be balanced in a global flow architecture that minimizes global flow resistance. Constructal theory suggests that the time spend in both phases, organized and unorganized flow, should be equal as a natural behavior of any system (Bejan 2007).

The problem now is how to evaluate the performance of the system (i.e. global flow resistance) taking in account the heterogeneity of the media (soil) and its direct interaction with the forcing inputs (i.e. precipitation intensity). The solution may come from an evolutionary approach, where a model that specifically considers the (fractional) diffusion–advection physics runs over a random field of some particular property of the soil, according to some established probability distribution, looping through time, evolving (according to constructal theory) by allowing changes in flow path architecture and connectivity (percolation theory). Therefore, the approach combines the Newtonian worldview (mass and energy balances) with the Darwinian worldview (McDonnell et al. 2007). At the end, an optimal configuration should appear and from this dimensionless numbers can be extracted. It should be possible to derive scaling laws of flow path configuration from these dimensionless numbers, opening a way to address the problem of hydrological prediction in ungauged basins.

5 Concluding Remarks

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

A clear paradigm shift recently took place in watershed hydrology. The traditional view that catchment-scale processes can be modeled and understood from small-scale process understanding is systematically being replaced by a more holistic view which explicitly accepts landscape heterogeneity as a dominant control. Moreover, hydrologists more and more turn to their colleagues from other earth sciences, such as geomorphology, pedology, biogeochemistry, and ecology, to better understand how hydrological systems have evolved and why certain patterns and functions exist at a wide range of space and time scales (Figure 3). It is clear that there is still a long way to go, but several community-driven programs and activities, such as Prediction in Ungauged Basins and Critical Zone Observatories, are paving the way toward a unifying theory of watershed hydrology. An important factor that will define the rate of success in developing a new theory of watershed hydrology is how we will educate the next generation of hydrologists. There are no textbooks out there that adopt the new view of catchment hydrology, and thus most teachers still follow the reductionist approach when discussing landscape hydrology. A true revolution can take place when we abandon the traditional way of teaching hydrology, and adopt right from the start the holistic view that builds on both Newtonian and Darwinian scientific discovery.

image

Figure 3. Feedback mechanisms between hydrology, biology, pedology, geomorphology, ecology, and landscape heterogeneity (J. Chorover, personal communications).

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Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

The first author would like to acknowledge support from the University of Illinois at Urbana-Champaign Hydrologic Synthesis Project, ‘Water Cycle Dynamics in a Changing Environment: Advancing Hydrologic Science through Synthesis’, PIs: Murugesu Sivapalan, Praveen Kumar, Don Wuebbles, and Bruce L. Rhoads, National Science Foundation Hydrological Sciences Program. Useful comments from Murugesu Sivapalan, Erwin Zehe, and an anonymous reviewer on an earlier version of this article are kindly acknowledged.

Short Biographies

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Quantifying Subsurface Heterogeneity
  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References

Peter A. Troch is a Professor of Hydrology in the Department of Hydrology and Water Resources at the University of Arizona, Tucson, USA. He holds MSc diplomas in Agricultural Engineering (1985) and Systems Control Engineering (1989) and received his PhD from the University of Ghent, Belgium, in 1993. He was assistant and associate professor in the Forest and Water Management Department at the University of Ghent from 1993 to 1999. He was professor and chair of the Hydrology and Quantitative Water Management group at Wageningen University, the Netherlands, from 1999 to 2005. His main research interests are measuring and modeling flow and transport processes at hillslope to catchment scales, and developing remote sensing and data assimilation methods to improve water resources management.

Gustavo A. Carrillo is a PhD student in the Hydrology Department at the University of Arizona, Tucson, USA. He began his PhD studies by joining the Surface Hydrology Group in August 2007, after being awarded as Fullbright scholarship recipient. He holds a bachelor's degree in Civil Engineering from the National University of Colombia (Bogota, 1996) and a Magister in Civil Engineer degree from the Los Andes University (Bogota, 1996). His main research interests are modeling flows at hillslope and catchment scales, watershed classification and predictions in ungauged basins.

Ingo Heidbüchel is a PhD student at the Hydrology and Water Resources Department of the University of Arizona, Tucson, USA. He received his master's degree in Hydrology from the University of Freiburg in 2007. His research focuses on stable isotope tracers, their use in the determination of travel time distributions in small catchments, and the dependence of flood response on landscape characteristics. Furthermore, he is interested in the processes and quantification of aquifer recharge from ephemeral streams in arid regions.

Seshadri Rajagopal is a graduate student working on his PhD in the Hydrology and Water Resources Department, at the University of Arizona, Tucson, USA. Seshadri holds a master's degree in Hydrology from the University of Nevada, Reno, and a bachelor's degree in Civil Engineering from the National Institute of Technology, Karnataka, India. My research interests include hydrologic modeling for decision support, parameter estimation in hydrology, land surface-atmosphere (boundary layer) interaction, and modeling.

Matt Switanek is a student in the Hydrology and Water Resources Department at the University of Arizona, Tucson, USA. He received a civil engineering degree from the University of Arizona in 2003. His main research interest is prediction of seasonal climate at the sub-basin scale.

Till H. M. Volkmann is a student of hydrology seeking a diploma degree (equivalent to MSc) at the Institute of Hydrology of the University of Freiburg since 2004. He was a DAAD graduate student scholarship holder at the Department of Hydrology and Water Resources at the University of Arizona in 2007/2008. His current research interests are operational flash flood forecasting, rain gauge network design, hydrologic modelling and water flow, and solute transport processes at hillslope to catchment scales.

Mary Yaeger is a master's student in the Department of Hydrology and Water Resources at the University of Arizona, Tucson, USA. She received a Bachelor of Science degree in Ecology from Florida Atlantic University in 2005. She is currently working with Dr. Jennifer Duan in the Department of Civil Engineering studying the effects of obstructions in open channel flow on the bed shear stress, turbulence intensity, and sediment transport.

Notes
  • * 

    Correspondence address: Peter A. Troch, Department of Hydrology and Water Resources, University of Arizona, 1133 E. James E. Rogers Way, Harshbarger Building, Tucson, AZ 85704, USA. E-mail: patroch@hwr.arizona.edu.

  • 1

    In ‘normal’ diffusion, the variance in the position of a particle varies linearly with time (for large times).

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  5. 3 Dealing with Subsurface Heterogeneity in Catchment Modeling
  6. 4 Toward a Unifying Theory of Watershed Hydrology
  7. 5 Concluding Remarks
  8. Acknowledgements
  9. Short Biographies
  10. References
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