Dealing with Landscape Heterogeneity in Watershed Hydrology: A Review of Recent Progress toward New Hydrological Theory

© 2008 The AuthorsJournal Compilation © 2008 Blackwell Publishing Ltd

Geography Compass 3/1 (2009): 375–392, 10.1111/j.1749-8198.2008.00186.x

Dealing with Landscape Heterogeneity in Watershed Hydrology: A Review of Recent Progress toward New Hydrological Theory

Peter A. Troch*, Gustavo A. Carrillo, Ingo Heidbüchel, Seshadri Rajagopal, Matt Switanek, Till H. M. Volkmann and Mary YaegerDepartment of Hydrology and Water Resources, University of Arizona

AbstractPredictions of hydrologic system response to natural and anthropogenic forcingare highly uncertain due to the heterogeneity of the land surface and subsurface.Landscape heterogeneity results in spatiotemporal variability of hydrological statesand fluxes, scale-dependent flow and transport properties, and incomplete processunderstanding. Recent community activities, such as Prediction in UngaugedBasins of International Association of Hydrological Sciences, have recognized theimpasse current catchment hydrology is facing and have called for a focused researchagenda toward new hydrological theory at the watershed scale. This new hydrologicaltheory should recognize the dominant control of landscape heterogeneity onhydrological processes, should explore novel ways to account for its effect at thewatershed scale, and should build on an interdisciplinary understanding of howfeedback mechanisms between hydrology, biogeochemistry, pedology, geomorphology,and ecology affect catchment evolution and functioning.

1 Introduction

Predictions of hydrologic system response to natural and anthropogenicforcing are highly uncertain due to the heterogeneity of the land surfaceand subsurface. Landscape heterogeneity results in spatiotemporal variabilityof hydrological states and fluxes, scale-dependent flow and transportproperties, and incomplete process understanding. The current paradigmof hydrological theory is that small-scale (local) process understanding (e.g.Richards’ equation to describe flow in porous media) is sufficient toquantify large-scale processes (e.g. baseflow recession at the outlet of acatchment), 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 experimentsand numerical simulations have revealed that subsurface flow is mainlygoverned by matrix flow when water storage is below some threshold

376 Landscape heterogeneity and hydrological theory

© 2008 The Authors Geography Compass 3/1 (2009): 375–392, 10.1111/j.1749-8198.2008.00186.xJournal Compilation © 2008 Blackwell Publishing Ltd

value, but quickly switches to macropore flow when that threshold iscrossed (e.g. Blöschl and Zehe 2005; Nieber et al. 2006; Tani 1997;Tromp-van Meerveld and McDonnell 2006). The hillslope thus acts as anon-linear filter that enhances water availability during low-flow conditions,but increases water release during high flow. At the catchment scale, severalstudies have shown that travel time distributions of non-reactive soluteshave ‘heavy’ tails, suggesting that their transport through the subsurface isgoverned by large conductivity contrasts related to soil matrix porosity,macropores, fracture networks, and other large-scale heterogeneities (Kirchneret al. 2000).

Advances toward a unifying theory of watershed hydrology are hamperedby our inability to quantify the heterogeneity of surface and subsurfacelandscape properties and how they control the hydrologic response(McDonnell et al. 2007). Figure 1 provides an example of how localsubsurface heterogeneity (in this case, bedrock topography variability and

Fig. 1. Local subsurface heterogeneity, such as bedrock microtopography and associated soildepth, gives rise to emergent properties in saturated connectivity, threshold-like subsurfacestormflow and flashy hydrograph response at small catchment scales (C. Harman, personalcommunications).


Landscape heterogeneity and hydrological theory 377

associated soil depth) give rise to emergent hillslope flow properties,such as saturation connectivity, threshold-like subsurface stormflow, andcorresponding flashy outflow behavior at small catchment scales (C. Harman,personal communications).

Aquifer heterogeneity and its effect on flow and transport has beenstudied in great detail. Analyses of well observations during steady andunsteady pumping tests have revealed the scale-dependent nature of flowand transport variables (such as hydraulic conductivity and dispersivity) inaquifers exhibiting non-stationary stochastic properties (Endres et al. 2007).Our ability to deploy similar techniques to study the effect of landscapeheterogeneity on catchment-scale hydrologic responses is, however, limiteddue to technological constraints on imposing hydrological drivers (e.g.precipitation) and observing state variables (e.g. subsurface flow). Developingnew hydrological theory that can explain observed phenomena, such asthreshold-like flow response at hillslope scales or fractal filtering of inerttracers in precipitation and streamflow, requires thus a more holisticapproach (Zehe et al. 2007). It is the objective of this article to reviewrecent progress in understanding and modeling of catchment hydrologicalprocesses, and to present short descriptions of innovative ideas in thehydrologic literature that attempt to formulate a unifying theory at thecatchment scale.

2 Quantifying Subsurface Heterogeneity

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

Traditional geostatistical methods have also been used when the targetporous medium represents a hierarchical geologic structure (Ritzi et al.



2004). Non-stationary hierarchical porous media exhibit both systematic andrandom spatial and directional variations in hydraulic properties on manylength scales (Sposito 1998). At the catchment scale, the heterogeneoussubsurface most likely will exhibit such non-stationary hierarchical struc-ture, 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 traditionalgeostatistical analysis to infer the spatial covariance structure of hierarchical,multiscale porous media is an artifact of the finite sampling windowapplied to measure geologic and hydrologic variables. They demonstratethat with the aid of truncated power variograms (Di Federico and Neuman1997) to capture the variations in geostatistical parameters (variance andintegral scale), this artifact can be eliminated using only a few scalingparameters. Their new method is able to represent multiscale randomfields that are either Gaussian or have Levy probability distributions. Thisis an improvement of the traditional geostatistical methods, which usestationary spatial statistics to characterize non-stationary hierarchical media.The results of Neuman et al. (2008) are potentially very important tocatchment hydrologists who have to deal with ubiquitous non-stationaryhierarchical geologic settings. The challenge to apply this theory at thecatchment scale is to make use of existing hydrologic data and/or to designspecific field investigations to reliably estimate the scaling parameters thatcapture multiscale variations of hydraulic and transport properties.

Another method that can be used to study the scaling behavior of strongheterogeneous porous media is the renormalization group (RG) method(Hristopoulos 2003). RG is a theoretical framework for estimating probabilitydistributions of random fields when scale is changed. Application of RGmethods in subsurface hydrology includes the study of the implicit non-lineardependence of the subsurface flow and transport processes on the hydraulicconductivity and the accurate estimation of up-scaled conductivity andmacrodispersion coefficients in multiscale heterogeneous porous media(Lunati et al. 2002). The method of up-scaling relies on a procedure inwhich individual cells are replaced by blocks of cells, and the parametervalues of the new blocks are determined by some transformation of theoriginal parameter values in the block. Renormalization iteratively appliestwo transformations: coarse graining and rescaling. The end result of thisis the elimination of the finer fluctuations and the renormalization of theprobability density function parameters. A very useful application of RGresearch in catchment hydrology is focusing on ‘anomalous’ diffusion,1 resultingfrom power-law distributions of the subsurface velocity field (Benson et al.2000) with long-range correlations (Kirchner et al. 2000). RG theory isparticularly useful for systems that exhibit critical phenomena near phasetransitions, such as percolation (at the threshold of transition, conducting clustersof all sizes appear in the subsurface; see Section 3.1 for more details).

Yeh et al. (2008) argue that recently developed tomographic surveyingtechniques (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 passivecomputerized axial tomography scan technologies that exploit recurrentnatural stimuli (e.g. lightning, earthquakes, storm events, barometric variations,river-stage variations) as sources of excitations. Along with in situ sensornetworks, they provide long-term and spatially distributed monitoring ofsystem responses at the land surface and in the subsurface. This approachto basin-scale subsurface characterization is still in its infancy and willrequire intensive interdisciplinary collaboration (e.g. surface and subsurfacehydrology, geophysics, geology, geochemistry, information and sensor tech-nology, applied mathematics, atmospheric science), but seems an importantstep toward a better understanding of the effects of heterogeneity oncatchment responses.

3 Dealing with Subsurface Heterogeneity in Catchment Modeling

3.1 FRACTIONAL ADVECTION AND DISPERSION

Burns (1996) describe results from a one-dimensional tracer test in alaboratory-scale (1 m) sandbox with very uniform sand and relativehomogeneous and isotropic properties. A number of tracer tests wereconducted to estimate the transport characteristics of the sand. These testsshowed non-Gaussian breakthrough curves with very heavy leading andtrailing tails. Such non-Gaussian transport in a homogeneous porousmedium 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–dispersionequation (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 dispersivityis allowed to change (increase) in an attempt to mimic the faster spreadingof the tracer plume. Benson et al. (2000) show that non-Fickian transportcan be successfully simulated by means of a transport equation that usesfractional order dispersion derivatives. Solutions of this fractional ADErepresent plumes that spread proportional to time1/α instead of time1/2 forthe classical ADE (α is a real number between 1 and 2 and defines thefractional dispersion derivative). The main advantage is that the dispersionparameter is not a function of time or distance, but a property that representsthe transport characteristics of the porous medium. These results demonstratethat a fractional-order equation is a powerful predictive tool of heavy-tailedtransport behavior. Such behavior, present even in the relatively homogeneousexperiment of Burns (1996), is known to govern flow and transportthrough the heterogeneous subsurface.

Kirchner et al. (2000) used time series of chloride concentrations inrainfall and streamflow from several catchments in Wales to demonstrate that



these catchments act as fractal filters. Although the chloride concentrationsin rainfall have a white noise power spectrum, the power spectra ofstreamflow chloride concentrations show 1/f fractal scaling, indicating thatthe travel time distributions in these catchments are power-law distributions.This implies that contaminant (e.g. nitrate) export out of the catchmentswould have heavy tails (the catchments have very long memory of pastinputs). If indeed the travel time distribution follows a power-law, that is,p(t)~t−α, then the convolution method to estimate output concentrations ofinert tracers (such as stable water isotopes) from known input concentrationsrepresents a fractional integral, suggesting that the fractional ADE is theappropriate prediction tool to handle effects of subsurface heterogeneityon catchment flow and transport. More research is needed to explore howthis approach could provide the correct theoretical framework to estimatecatchment water travel time distributions from inert tracer observations.

3.2 PERCOLATION THEORY

The previous discussion focused on the characterization of subsurfaceheterogeneity and its effect on large-scale flow and transport withoutspecial 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 tocharacterize spatial patterns, but are not able to discern between patternswith and without connectivity (Di Domenico et al. 2007; Western andGrayson 1998). The concept of connectivity is strongly linked to percolationphenomena and widely used in percolation theory. The phenomenon ofpercolation is defined as ‘the special property of a system which emerges atthe onset of macroscopic connectivity within it’ (Berkowitz and Balberg1993). Percolation theory provides the theoretical basis to determine theprobability that certain points within the system are connected. The prob-ability, p, that there is an open path from some fixed point to a distanceof r decreases polynomially, that is, p(r)~r−β, where β is some parameterthat does not depend on local properties of the porous medium, but onlyon its dimension. Lehmann et al. (2007) applied percolation theory tomodel the fill-and-spill mechanism observed by Tromp-van Meerveld andMcDonnell (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 criticalpoint refers to a state of subsurface connectivity (related to a thresholdrainfall amount) that triggers rapid drainage.

Di Domenico et al. (2007) used percolation theory to study the transitionbetween spatially random patterns to spatially connected patterns of soilmoisture (Western and Grayson 1998; Zehe and Sivalapan 2008), andaddressed the question whether this process behaves as a critical pointphenomenon. Critical point behavior refers to a state transition fromunorganized (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 thecritical point. In other words, similar to river networks (Rodriguez-Iturbe and Rinaldo 1997), soil moisture percolation networks should bescale-invariant. Di Domenico et al. (2007) used simulated soil moisturefields and applied the RG method to show that percolation clusters of anup-scaled modeling lattice is qualitatively the same as the percolationcluster of the original lattice.

Nieber et al. (2006) present an interesting numerical simulation studyin which they explore the effect of disconnected macropores on flowthrough a porous cylinder. The macropores are embedded in a homogeneousloamy soil matrix and have diameter size of 0.005–0.01 m. The saturatedhydraulic conductivity of the macropores is six orders of magnitude largerthan that of the soil matrix. Even though the macropores do not form aconnected network, Nieber et al. (2006) observe that the modeled flowincreases significantly (up to 40%) when the cylinder reaches saturation.At that time, all macropores are filled with water and the flow increasesthreshold-like. They speculate that a similar mechanism can take place ina 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. Thenumerical study of Nieber et al. (2006) further supports the idea thatpercolation theory could be used to describe the self-organization thatappears to occur in hillslope soils.

3.3 HILLSLOPE SIMILARITY AND CLOSURE RELATIONSHIPS

At the hillslope scale, percolation theory can give great insight into theconnectivity of subsurface flow. Percolation theory efficiently explains thethreshold-like flow response of a heterogeneous hillslope by means of aconnected network. However, percolation models require extensive fieldmeasurements to be calibrated, and, hence, quantifying percolationthresholds and hydraulic parameters at hillslope scales is still a formidabletask. In order to quantify the flow response in ungaged hillslopes or basins,it is imperative that we more fully understand the cumulative effect ofsurface and subsurface topography and soil heterogeneity.

In order to explicitly account for subsurface heterogeneity at the hillslopescale, it is necessary to solve the mass, momentum, and energy balanceequations in heterogeneous porous media (see Section 2). Conservationof mass, momentum, and energy and the second law of thermodynamicscan be expressed at the hydrodynamic scale at which these dynamic relationsare best understood (e.g. Darcy-Buckingham equation to describe flowthrough porous media). The selection of model parameters at such scalesis 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 statecan be reached by many potential means. In hydrological modeling, twomodels are equifinal if they lead to an equally acceptable or behavioralrepresentation of the observed natural processes (Beven and Freer 2001).Reggiani et al. (1998) developed the idea of a representative elementarywatershed (REW) in an attempt to effectively incorporate, up-scale, andthe effects of small-scale spatial variability. Formulating the conservationequations 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 constitutiverelationship between hydrologic state variables (e.g. saturated storage) andfluxes (e.g. lateral subsurface flow) are generally unknown. Zehe et al. (2006)used a physically based hydrological model that has been extensively testedwith field observations from an experimental watershed in Germany (theWeiherbach catchment) to derive REW scale state variables and effectiveREW scale soil hydraulic functions to address the closure problem relatedto subsurface flow processes.

Another way to relate the subsurface flow response from one hillslopeto another is through hillslope similarity. Berne et al. (2005) have shownthat a dimensionless number, the hillslope Peclét number (Pe), is a hillslopesimilarity parameter that predicts the dimensionless characteristic responsefunction (CRF; the subsurface flow hydrograph resulting from an initialsteady-state storage volume, normalized by the total volume of waterdrained). Lyon and Troch (2007) applied the Pe number to real hillslopesin New Zealand and the United States, and obtained good agreementbetween observed and predicted moments of the dimensionless CRF.Estimating the Pe number for a given hillslope does not require explicitquantification of the hydraulic subsurface parameters like hydraulic con-ductivity and drainable porosity, but needs information about the averagehillslope storage (related to climate and inevitably hillslope transmissivity).Moreover, the Pe number was derived under assumptions of a homogeneoussubsurface.

Harman (2007) used the two-dimensional Boussinesq equation to studythe effect of subsurface heterogeneity (expressed as spatially variable saturatedhydraulic conductivity) and climate (different recharge regimes) on hillslopeflow similarity. Harman (2007) confirmed that effective hydraulic propertiesof the subsurface cannot represent any randomized heterogeneous distributionof hydraulic properties. Heterogeneous hillslopes discharge more waterinitially, but then have longer recession tails due to the storage and slowrelease of water from regions that are less hydraulically conductive. He alsoobserved, in accordance with regime theory (Robinson and Sivapalan1997), that when the duration of time of recharge events and the lengthof time between recharge events are both small in comparison to thecharacteristic timescale of a hillslope, a memory exists between one rechargeevent and another, and the peak discharge has a contribution from more



than one event. On the other hand, when both of these are large incomparison to the characteristic timescale of a hillslope, then the hillslopeexhibits a flashy response that is only governed by one recharge event.

Different climate regimes and heterogeneous hydraulic conductivity affectboth the peak discharge and the overall structure of a hillslope’s CRF. Inan attempt to more explicitly deal with heterogeneity and climate, Harman(2007) proposes the ‘flowpath-timescale’ as an efficient concept to provideclosure relationships between internal storage and flow. The idea of flowpath-timescales is to determine the amount of time that it would take for waterrecharging at a specific location to exit the hillslope. Harman (2007) useda one-dimensional Boussinesq model, with heterogeneous hydraulicconductivity, to observe the dimensionless time that it takes for water totravel, from recharge to outlet, as a function of dimensionless distanceupslope. He parameterized the simulated flowpath-timescale by means ofan incomplete Beta function. This flowpath-timescale function is able toreplicate very well the initial part of the hydrograph in addition to thelong recessions when compared to the full model. Harman’s work revealed(i) that no unique closure relationship exist – it depends on the nature ofsubsurface heterogeneity and flow regimes, and (ii) that the closure relationsreflect an interaction between the climate and the bottom boundaryconditions (how is the hillslope connected to the channel network?).More work is needed to relate the parameters of the flowpath-timescalefunction to observable hydrologic responses and landscape characteristics.In the next section, we review studies that make use of the integratedhydrologic response to obtain information about processes, states, andproperties at catchment scales.

3.4 DOING HYDROLOGY BACKWARDS

Hydrologists have for long used the integrated response of a catchment toatmospheric forcing to estimate flow and transport parameters, as well asinitial states and dominant processes. In this section, we review somerecent advances in using the integrated hydrological response to deriveinformation about the internal structure and functioning of catchments.

Martina and Entekhabi (2006) developed a method to extract spatialinformation about runoff producing processes from the analysis of precip-itation and discharge observations at the catchment scale. They arguedthat the nonlinear co-variations between storm precipitation and stormrunoff volumes can be used to draw conclusions about the marginaldistribution of available soil water storage across a basin. That means, byobserving how a catchment reacts to hydrologic forcing, it is possible toderive the antecedent moisture conditions – not at the exact location, butas a probability density function of local soil column saturation deficitprior 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 availablethroughout the catchment. The fraction of storm runoff depends on howmuch pore space is not filled with soil water at the time when the rainfallsets in. Taking this approach one step further, Martina and Entekhabi(2006) developed a method that can predict the area of runoff producingregions within a catchment, although no information about locationis provided.

Kirchner (2008) shows that certain catchments can be considered assimple first-order non-linear dynamical systems, and the governing equationscan be derived directly from field data (streamflow fluctuations). The basicprinciple of his approach is the assumption that a unique relationshipexists between the storage within a catchment and the outflow from thiscatchment. The way a catchment reacts to changes in storage can beexpressed as a function that relates changes in storage to the amount ofdischarge that is produced. This function describes how sensitive a catchmentis to changes in storage. This sensitivity function can be linear, exponential,or hyperbolic, depending on how the catchment behaves. The sensitivityfunction can be derived from streamflow time series alone. Once thisrelation is established, catchment discharge can be modeled as a functionof evaporation and precipitation time series alone. From the form of thesensitivity function, it is possible to derive the main storage behaviorwithin a catchment. In case the function is linear, the catchment can beconsidered to behave as a linear reservoir. In case the function isexponential, it exhibits a lower storage threshold below which no dischargeis produced. In case of a hyperbolic function, there is an upper limit tostorage where any surplus storage is directly converted to discharge, aprocess that can be related to the ‘fill-and-spill’ behavior of flashy hydrologicsystems (Tromp-van Meerveld and McDonnell 2006).

4 Toward a Unifying Theory of Watershed Hydrology

There is growing recognition among hydrologists that the way towardnew hydrological theory is to approach watershed hydrology as an inter-disciplinary earth science (Lin et al. 2006; Sivapalan 2005). Despitetremendous heterogeneity of landscape properties, there is structure andorganization across spatial and temporal scales that, when understood howand 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, soilscientists, geomorphologists, and hydrologists, but prior research effortshave largely been conducted within these disciplines. In the next sections,we review recent studies that focus on complexity and structure of thelandscape, and that formulate fundamental principles that govern hydrologicsystems response and functioning.



4.1 COMPLEXITY AND STRUCTURE OF THE CRITICAL ZONE

Of particular interest to catchment hydrology is the characterization ofheterogeneity 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 determinethe availability of life-sustaining resources’ (National Research Council2001). The critical zone includes the land surface, vegetation, and waterbodies, and extends through the pedosphere, unsaturated vadose zone, andsaturated groundwater zone. It serves as a conduit for the flow of waterand the transport of solutes, as well as an interface with surface andgroundwater for the exchange of solutes (Corwin et al. 2006).

Rasmussen et al. (2005) recently presented a general theory of quantitativeenergy transfer to soil systems, embracing concepts of open-systemthermodynamics and traditional quantitative models of soil developmentto predict the critical zone evolution. Like ecological systems, soil systemsare open, non-equilibrium systems which tend to self organize to optimizethe use of energy cycling within and flowing through the system, as longas entropy is passed external to the pedon through dissipative processes(see Rasmussen et al. 2005 and references therein). According to Rasmus-sen et al. (2005), the cycling and fluxes of energy and mass is driven byinter-related environmental energy gradients that maintain the system ina non-equilibrium state. The dissipation of energy along these gradientsgoverns the structural organization within the critical zone system (e.g.soil profile development), as well as the concurrent evolution of waterflow paths. Consequently, a quantification of the energy and mass inputto a soil system should represent the soil forming environment andpotentially the developmental state of the soil system (Figure 2; Rasmussenand Tabor 2007).

A prerequisite to connect pedogenic modeling and hydrological responseand to make such efforts valuable for the advancement of understandingand predictability of hydrologic processes is to quantify the mechanismslinking soil evolution to these hydrologic processes. Numerous field studieshave qualitatively related changes in soil hydrologic properties and hydrologicprocesses due to different aspects of soil evolution and soil physicalstructure in particular, but such relationships have rarely been quantified(see Lohse and Dietrich 2005 and references therein). One example isprovided by Lohse and Dietrich (2005) who quantitatively relate thedevelopment of subsurface clay-rich horizons to a decline of subsurfacehorizon saturated hydraulic conductivity values, impeding rates of verticalwater flow but increasing the importance of shallow subsurface lateral flow.

We argue that if we can find ways to quantify the implications of soildevelopment and structure on hydrological processes, approaches to predictpedogenesis and related soil properties from more readily available informationon 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 predictionsat scales of hydrological units in distributed models. While the scale ofprediction may be relatively coarse, the properties of the predictedhomogeneous units could potentially serve as mean values providing abasis for stochastic approaches. Furthermore, heterogeneity and complexitymay change over time, with climate changes and landscape disturbancefurther impairing the rate and direction of these changes (McDonnellet al. 2007). These changes and their respective consequences could beaccounted 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 understandnature, the idea of finding the fundamental principle that explains thebehavior of a system subject to different initial and boundary conditionsover a wide range of scales. Thermodynamics has offered fundamental

Fig. 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 potas-sium (CIA–K) of the first subsurface genetic horizon. Plotted lines and equations represent thebest fit regression to the data. Data derived from pedons sampled from stable landscapepositions across four environmental gradients on basalt, andesite, and granite parent materials(from Rasmussen and Tabor 2007).



laws in the field of physics and chemistry, and by extension to most otherfields in science. Water flow in a hillslope can be studied from thethermodynamic point of view (water in motion due to potential gradients).Classical thermodynamics deals mainly with isolated closed systems, buthydrology certainly deals with open systems. Rasmussen et al. (2008) positthat hydrologic systems organize and evolve in response to open systemfluxes of energy and matter driven by gravitational, geochemical, andradiant gradients. Open system energy and mass flow facilitate internalordering and elemental cycling through development of dissipativestructures that transfer energy and matter across the system. Hence, structuralorganization should be related to these transfers through the watershed.The central importance of these fluxes is widely recognized, yet we lack theability to quantify the effective flows that are fundamental for understandingcatchment functioning and predicting rates of hydrologic processes.

Optimality principles applied to watersheds may offer a framework to studythe effects of effective mass and energy flows on catchment functioningand hydrologic response. In biology, West et al. (1997) developed a modelthat describes how essential biological materials are transported throughspace-filling fractal networks of branching tubes, by minimizing thedissipated energy and assuming that the terminal tube size is scale-invariant.Schymanski et al. (2008) used vegetation optimality principles to modelsoil moisture dynamics, root water uptake and fine root respiration in atropical savanna study site. The optimality-based model reproduced themain features of observations at the site, such as a shift of roots from theshallow soil in the wet season to the deeper soil in the dry season andsubstantial root water uptake during the dry season. Recently, Bejan andco-workers (Bejan 2000; Bejan and Lorente 2004; Lewins 2003) haveproposed a generalization of thermodynamics, known as thermodynamicsof flow systems with configuration, to understand flow structures in bothnatural and man-made systems. The fundamental underlying physicalprinciple is Bejan’s constructal theory: ‘For a finite size flow system topersist in time (to survive) it must evolve in such way that it provideseasier and easier access to the currents that flow through it’ (Bejan 2007).Constructal theory offers a principle that considers not only the final stateof equilibrium of the system but also the patterns and shapes that the flowacquires through time. Simulating flow through a heterogeneous porousmedia, Lorente and Bejan (2006) found that alternating tree architectureshave flow resistances lower than those of purely unorganized porous mediawith the same volume and internal flow volume. Inside a hillslope, twodifferent kinds of flow occurs: one through the porous media (diffusion)and one trough low resistance flow paths (preferential flow). Seepage flowand channeled flow must be balanced in a global flow architecture thatminimizes global flow resistance. Constructal theory suggests that the timespend in both phases, organized and unorganized flow, should be equal asa 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. precipitationintensity). The solution may come from an evolutionary approach, wherea model that specifically considers the (fractional) diffusion–advectionphysics runs over a random field of some particular property of the soil,according to some established probability distribution, looping throughtime, evolving (according to constructal theory) by allowing changes inflow path architecture and connectivity (percolation theory). Therefore, theapproach combines the Newtonian worldview (mass and energy balances)with the Darwinian worldview (McDonnell et al. 2007). At the end, anoptimal configuration should appear and from this dimensionless numberscan be extracted. It should be possible to derive scaling laws of flow pathconfiguration from these dimensionless numbers, opening a way to addressthe problem of hydrological prediction in ungauged basins.

5 Concluding Remarks

A clear paradigm shift recently took place in watershed hydrology. Thetraditional view that catchment-scale processes can be modeled andunderstood from small-scale process understanding is systematically beingreplaced by a more holistic view which explicitly accepts landscapeheterogeneity as a dominant control. Moreover, hydrologists more and moreturn to their colleagues from other earth sciences, such as geomorphology,pedology, biogeochemistry, and ecology, to better understand how hydrologicalsystems have evolved and why certain patterns and functions exist at awide range of space and time scales (Figure 3). It is clear that there is stilla 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. Animportant factor that will define the rate of success in developing a newtheory of watershed hydrology is how we will educate the next generationof hydrologists. There are no textbooks out there that adopt the new viewof catchment hydrology, and thus most teachers still follow the reductionist

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



approach when discussing landscape hydrology. A true revolution can takeplace when we abandon the traditional way of teaching hydrology, andadopt right from the start the holistic view that builds on both Newtonianand Darwinian scientific discovery.

Acknowledgements

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

Short Biographies

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

Gustavo A. Carrillo is a PhD student in the Hydrology Department atthe University of Arizona, Tucson, USA. He began his PhD studies byjoining the Surface Hydrology Group in August 2007, after being awardedas Fullbright scholarship recipient. He holds a bachelor’s degree in CivilEngineering 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 andcatchment scales, watershed classification and predictions in ungauged basins.

Ingo Heidbüchel is a PhD student at the Hydrology and WaterResources Department of the University of Arizona, Tucson, USA. Hereceived his master’s degree in Hydrology from the University of Freiburgin 2007. His research focuses on stable isotope tracers, their use in thedetermination of travel time distributions in small catchments, and thedependence of flood response on landscape characteristics. Furthermore,he is interested in the processes and quantification of aquifer rechargefrom ephemeral streams in arid regions.



Seshadri Rajagopal is a graduate student working on his PhD in theHydrology and Water Resources Department, at the University of Arizona,Tucson, USA. Seshadri holds a master’s degree in Hydrology from theUniversity of Nevada, Reno, and a bachelor’s degree in Civil Engineeringfrom the National Institute of Technology, Karnataka, India. My researchinterests include hydrologic modeling for decision support, parameterestimation in hydrology, land surface-atmosphere (boundary layer) interac-tion, and modeling.

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

Till H. M. Volkmann is a student of hydrology seeking a diplomadegree (equivalent to MSc) at the Institute of Hydrology of the Universityof Freiburg since 2004. He was a DAAD graduate student scholarshipholder at the Department of Hydrology and Water Resources at theUniversity of Arizona in 2007/2008. His current research interests areoperational flash flood forecasting, rain gauge network design, hydrologicmodelling and water flow, and solute transport processes at hillslope tocatchment scales.

Mary Yaeger is a master’s student in the Department of Hydrology andWater Resources at the University of Arizona, Tucson, USA. She receiveda Bachelor of Science degree in Ecology from Florida Atlantic University in2005. She is currently working with Dr. Jennifer Duan in the Departmentof Civil Engineering studying the effects of obstructions in open channelflow 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, AZ85704, USA. E-mail: [email protected].

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

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