Numerical modelling of landscape evolution: geomorphological perspectiveswpg.forestry.oregonstate.edu/sites/wpg/files/seminars/... · 2015-01-09 · Numerical modelling of landscape

Numerical modelling of landscapeevolution geomorphologicalperspectives

Yvonne Martina and Michael Churchb

aDepartment of Geography University of Calgary Calgary ABT2N 1N4 CanadabDepartment of Geography and Peter Wall Institute for Advanced StudiesThe University of British Columbia Vancouver BC V6T 1Z2 Canada

Abstract A resurgence of interest in landscape evolution has occurred as computationaltechnology has made possible spatially and temporally extended numerical modelling Wereview elements of a structured approach to model development and testing It is argued thatnatural breaks in landscape process and morphology define appropriate spatial domains forthe study of landscape evolution The concept of virtual velocity is used to define appropriatetimescales for the study of landscape change Process specification in numerical modellingrequires that the detail incorporated into equations be commensurable with the particular scalebeing considered This may entail a mechanistic approach at small (spatial) scales whereas ageneralized approach to process definition may be preferred in large-scale studies Thedistinction is illustrated by parameterizations for hillslope and fluvial transport processesbased on scale considerations Issues relevant to model implementation including validationverification calibration and confirmation are discussed Finally key developments andcharacteristics associated with three approaches to the study of landscape modelling(i) conceptual (ii) quasi-mechanistic and (iii) generalized physics are reviewed

Key words geomorphology landscape evolution model implementation numerical modellingprocess specification scale

I Introduction

The ball is now in the geomorphologistsrsquo court (Anderson and Humphrey 1989 350)

This remark appeared in an early discussion of information requirements formodelling the interaction of tectonic and surface processes at large scales The

WC Arnold 2004 1011910309133304pp412ra

Author for correspondence Tel (403) 220-6197 fax (403) 282-6561 e-mail ymartinucalgaryca

Progress in Physical Geography 283 (2004) pp 317ndash339

adoption of numerical modelling for the study of combined exogene andendogene processes represented a recent development at the time Researcherswere turning to the geomorphological literature to find information about therate of surface transport processes at large spatial and temporal scales inorder to model the lithospheric response to erosion and sedimentationAnderson and Humphrey found that existing results did not offer the infor-mation necessary for successful implementation of such models Their state-ment was made at the beginning of a resurgence of interest in landscapeevolution that had occurred in recent years (review by Thomas and Summer-field 1987) In subsequent years much progress has been made in numericalmodelling of landscape evolution (reviews by Merritts and Ellis 1994 Sum-merfield 2000) although many outstanding issues still remain concerningthe representation of geomorphological processes over large spatial and tem-poral scales

It is proposed herein that numerical modelling of landscape evolution recon-ciles lsquohistoricalrsquo and lsquoprocessrsquo studies by explaining characteristic features of his-torical geomorphological landscapes in terms of physically based processes(without necessarily being able to recreate the details of specific landscapes)Numerical modelling requires a framework within which material fluxes canbe reconciled across all resolved scales The crux of modelling from a geomor-phological perspective is how to represent the flux lsquolawsrsquo governing landform-ing processes Adequate calibration and testing of transport relations used insuch models remains elusive because of difficulties in acquiring suitable fielddata over representative timescales In turn the litmus test of a model dependson finding appropriately rigorous ways to compare model results with the evi-dence of particular landscapes The resolution of both of these problems isdominated by scale Spatial and temporal scales of a study should guide thespecification of process and selection of test procedures within numericalmodels A judgement must be made regarding lsquoflawsrsquo within the model thatmust be tolerated for the sake of computational tractability while replicatingthe essential characteristics of the phenomena under consideration The dramati-cally expanded body of tools and techniques now available including muchimproved topographic mapping (DEMs) high resolution air photographs sub-stantially improved surveying instruments and sophisticated dating techniqueshave increased our ability to resolve key issues related to process specificationand model testing

Many of the existing landscape models consider the interactions of tectonic andgeomorphological processes At intermediate timescales geomorphological deve-lopment of the landscape can be studied independently of tectonic considerationsThis approach is increasingly important in studies of lsquolong-termrsquo effects of land-scape management as well as to study the implications of geomorphological pro-cesses over extended periods An examination of methodology surrounding thenumerical representation of geomorphological phenomena at large scales can pro-vide insight into appropriate specifications of processes for a particular scale ofstudy It can also reveal the weaknesses in our ability to do so The objectiveof this paper then is to examine theoretical and methodological issues relatedto the quantitative study of geomorphological processes within the context oflandscape evolution

318 Numerical modelling of landscape evolution

II Scale

1 Scale and process

Scales are a set of natural measures that are intrinsic to a system The chosen scaleguides the appropriate specification of the system and processes within it Generallyas the spatial and temporal scales of a study decrease (in the physical rather than thecartographical sense) it is possible to resolve greater detail in the processes Devel-opment and calibration of flux relations on the basis of transport rates measured inthe field or laboratory is an easier task at smaller scales because the system configur-ation remains relatively simple Results obtained in smaller-scale studies cannot bereplicated directly at large scales since details found in process equations at smallerscales are no longer resolvable

A significant dilemma then is how to define relations and calibrate them to rep-resent the operation of geomorphological processes at large scales Considerationmust be given to the way in which scale limits the information available to derivea representation of processes Equations are required that are generalized appro-priately to match the information

To define and calibrate process equations for large scales emphasis must beplaced first on identifying dominant processes at large scales (Murray 2002) thenon establishing appropriate process equations to describe them and finally onestimating their rates To calibrate relations methods must be developed that yieldestimates of long-term transport rates Advances in radiometric isotope dating(cf Cerling and Craig 1994 Ring et al 1999) and in luminescence dating (Duller2000) have occurred that increase the potential to estimate long-term process ratesThese methods are proving critical to establishing rates of process operation inlandscape evolution (eg Bierman 1994 Garver et al 1999 Clemmensen et al2001 Nichols et al 2002 amongst many papers) Other approaches that have thepotential to provide useful information include the use of remotely sensed imagerysuch as aerial photography to generate large data bases for specific types of geomor-phological phenomena such as shallow landsliding (Guzzetti et al 2002 Martinet al 2002) or large failures in bedrock (eg Hovius et al 1997 2000) and thewell-established practice of using digital elevation models to provide a quantitativeeasily manipulated record of topography In short the potential for improved under-standing of process operation at a variety of scales has widened considerably as aresult of technological advances

2 Spatial scale

If scales represent natural measures in a system then some defining criterion for anatural measure must be selected One basis for the delineation of a natural measureis to locate natural breaks often based on process domains that partition variabilityin some relevant attribute of the system Such scalings are often presented (see forexample Schumm and Lichty 1965 in geomorphology Steyn et al 1981 in climatol-ogy Delcourt et al 1983 in ecology Ehleringer and Field 1993 in plant physiology)In landscape the primary attribute of concern is morphology Morphological breaksin the landscape can be used to guide the selection of appropriate domains for thestudy of landscape change and the extent of such domains in turn dictates the

Y Martin and M Church 319

scale of study There are two approaches to identify such breaks The first is to con-sider landscape units of traditional geomorphological interest the second is directnumerical analysis of the landscape to reveal the distribution of variance in keyproperties We first explore three alternative choices for the scale of landscapeevolution studies that are well-represented in geomorphological textbooks tectonicunits drainage basins and hillslopes

The largest customary landscape scale is based on tectonically determined topo-graphy (eg Summerfield 1991 Ahnert 1998) Geology climate and developmenthistory determine the characteristics of a particular tectonic landscape (eg Brozovicet al 1997 Montgomery et al 2001) The tectonic unit represents the fundamentalunit for definition of the balance between uplift and downwearing that exists in evol-ving landscapes and the largest landscape unit for which there appears to be greatermorphological variability between units than within a unit Studies to understandthe evolution of tectonic units constitute the currently most active area of landscapemodelling (Beaumont et al 2000) Most such studies represent attempts to under-stand generic features of tectonic landscapes though they may refer to specific land-scapes for context or for qualitative comparison Increasingly however investigators(eg Koons 1989 Tucker and Slingerland 1996 Ellis et al 1999 Anderson 2002) areattempting to use numerical models to understand aspects of the history of specifictectonic landscapes

Drainage basins are spatial units containing integrated areal and linear pathwaysfor sediment movement The particular characteristics of a drainage basin depend onthe magnitude of the basin and on geology and climate However all drainage basinsby definition exhibit functional similarity insofar as water and sediment are routedthrough the system to a single outlet Drainage basins have long been recognized asappropriate study units for hydrological research and analyses of sediment yield(eg Chorley 1969 Chorley et al 1984) They also form the study unit for researchthat examines the development of the fluvial system at large scales (eg Davis 1899Schumm 1977) hence the focus of studies in which drainage development is inves-tigated (eg Smith and Bretherton 1972 Dunne 1980 Willgoose et al 1991a bDietrich et al 1993) Terrestrial geomorphological processes with the exception ofglacial and aeolian processes (which have not generally been incorporated into land-scape models) occur within drainage basin boundaries The drainage basin is thenthe logical unit within which to model the subaerial geomorphological evolution oflandscape (eg Tucker and Bras 2000)

The smallest scale at which landscape evolution can be studied reasonably is thehillslope scale All hillslopes have a general morphological similarity by definitioninsofar as they are planar or quasi-planar features bounded by a slope base and acrest at the top When assembled together hillslopes constitute the drainage basinsurface This distinctive form indicates a distinctive process unit Sediment is trans-ported from the upper portions of the hillslope and is deposited along the slope baseby processes that in the long term are generally diffusive in character Hillslopeshave often been the study unit to consider topographic profile development(Penck 1953 Culling 1960 Kirkby 1971 1987b Martin 2000)

Mark and Aronson (1984) introduced a direct numerical (hence within thelimits of the source data objective) approach to define topographic scales Theycomputed the variogram of topography from 30 m digital elevation models(DEMs) of topographic quadrangles with 1-m resolution of elevation The


variograms (eg Figure 1) typically revealed three distinct fractal ranges (imply-ing distinct roughness characteristics) with scale breaks at less than 1 km andless than 10 km They declared that these breaks indicate important changes indominant geomorphological process regime hence natural scales The lowerscale break evidently marks the limit of the hillslope process regime in the studiedlandscapes whilst the upper break marks a limit between the roughness imposedby fluvially organized topography and larger scales imposed by structure Beyondthis break periodic (rather than fractal) modulation of the landscape is oftenobserved owing to the structural control of relief Reasons for the breaks havebeen discussed by other authors ndash the distinction between diffusive (topographicsmoothing) processes on hillslopes linear (topographic roughening) processes indrainage basins and sedimentation processes (topographic smoothing) at largescales being cited by Culling and Datco (1987) and Chase (1992) Fundamentallyone is observing the different effects of these disparate processes on the growth oftopographic variance with changing spatial scale

Analyses of this type have been surprisingly little pursued Most analysts whohave studied topographic variance have concentrated rather on the similarity(self-affinity) of landscapes over a range of scales (review of early work in Xu et al

Figure 1 Relief variograms for field areas within the AmericanAppalachian Mountains (a) Data for Aughwick Pennsylvaniaquadrangle arrows indicate significant breaks at the limit ofhillslope scales and at the limit of fluvial ridge and valley scales(b) Trend lines for several field areas the heavy line is the areashown in (a)Source Compiled data from Figure 1a and Figure 4 of Mark andAronson (1984)


1993) ndash an exercise designed to ignore multifractal signatures In part as well arte-facts of map digitization may confound results (Xu et al 1993) and of course mapscale itself limits what may be observed (Outcalt et al 1994) Most work has resolvedwell only the intermediate domain of erosional topography of drainage basins Thereremains much work to be done on the partition of topographic variance and thedefinition of process domains in the landscape

3 Temporal scale

The selection of spatial and temporal scales for a study cannot be made in isolationAs the spatial domain increases the detection of a lsquoresolvablersquo amount of changerequires significantly longer periods of observation Hence the increase in spatialdomain necessitates a change in temporal scale

Time and length are connected via a measure of velocity ie

time scale frac14length scale

virtual velocity

lsquoVirtual velocityrsquo refers to the apparent (or average) rate of movement of materialthrough the system including the time spent in storage Sediment particles infact remain at rest during most of their journey through the landscape only rarelyundergoing actual transport The timescale associated with the virtual velocity canbe thought of as the lsquotransitrsquo time of sediment through the system The transit timeis the characteristic (average) time that it takes sediment to move through the system(eg a hillslope or a drainage basin) This provides some benchmark for definingtimescales for a study Ranges of timescales that are appropriate for the study of geo-morphological processes at each of the spatially defined study units of landscapeevolution introduced in the preceding section are presented in Table 1

An understanding of sediment residence times is necessary to estimate virtual vel-ocities Given the extreme rapidity of many significant transport processes (eglandsliding debris flows and fluvial transport during significant flood events)when they actually occur it is time spent in storage that largely determines the

Table 1 Length and timescales for study units

Study unit Ranges of diameter forstudy unit (km)

Ranges of virtualvelocitya (km yr21)

Timescale (yr)

Tectonic Lower 101 1026ndash100 101ndash107

Upper 103 103ndash109

Drainage basin Lower 100 1026ndash100 100ndash106


Hillslope Lower 1021 1026ndash1022 101ndash105


NoteaVirtual velocities cover a broad range owing to the range of possible processes associated with this parameter There-fore the timescales show a broad range In order to ensure that significant landscape changes are observed the upperranges of timescales may be the most appropriate for landscape evolution studies


rate at which material is evacuated from the system Moreover accumulations ofstored sediment define many of the geomorphologically significant elements of thelandscape The dating of sedimentary surfaces and deposits the results of which canbe used to infer long-term storage times of sediment has been a major component ofhistorical studies in geomorphology However a reconciliation of the relative roles ofmovement and storage has not generally been forthcoming

Residence times need to be evaluated for sediment subsisting on hillslopes in val-ley flats and in the active fluvial channel zone Sediment may reside on hillslopes forlong periods before being transported to the valley flat If sediment then enters theactive channel and remains a part of the active sediment load it may be movedquickly through the fluvial system However if sediment entering the valley flatdoes not enter the active fluvial system it may enter long-term storage along the foot-slope Furthermore sediment that enters the active system but is then depositedduring an aggradation phase may also go into long-term storage in the floodplainSignificant lateral or vertical erosion by the river into the valley fill is required toentrain such material into the active channel system The distribution of sedimentsamongst long- and short-term reservoirs significantly influences the patterns of land-form evolution (Kelsey et al 1987) If long-term reservoirs sequester a significantproportion of the sediment moving through the landscape then rare high magni-tude events will have the dominant influence on landscape modification

Studies of the distribution of sediment storage times are relatively uncommonRiver floodplains have been most investigated (eg Everitt 1968 Nakamura et al1995) It is found that the age distribution of floodplain deposits is approximatelyexponential (Figure 2) and furthermore the rate of floodplain erosion declines expo-nentially with increasing age (Nakamura et al 1995 Nakamura and Kikuchi 1996)presumably because the remaining stored material becomes steadily more remotefrom the locus of frequent disturbances A consequence of exponential storagetimes is that the effects of large sedimentation events (which must be large erosionevents elsewhere in the landscape) may persist for a long time This is a manifes-tation of the so-called Hurst effect (see Kirkby 1987a) An important implication ofit is that wherever significant sediment stores occur no likely sequence of sedimenttransfer observations (generally such sequences are restricted to a few decadeslength or less) is likely to encompass the full possible variability of the processnor to have included the possibly most significant events in the long term

Sediment budget studies provide a framework for the study of sediment storage(eg Dietrich and Dunne 1978 Reid and Dunne 1996) Further research in thisdirection with a particular focus on the evaluation of long-term residence of storedmaterial (eg Macaire et al 2002) is required in order to improve understanding ofsediment routing and its associated timescales

III Numerical modelling of landscape evolution

1 The role of numerical modelling

There are two general purposes for constructing and testing a model realization ofsome process or system (1) as a test of understanding of the process as embodiedin the statements incorporated into the model and (2) to predict modelled system


behaviour Models are by their nature imperfect representations of reality ndashreduced and initially conjectural representations of what we envisage as the realsystem What then can numerical modelling contribute to the study of landscapeevolution Landscape models can be used to support or more thoroughly exploreideas and hypotheses that have been partly established in other ways (Oreskeset al 1994) For example transport relations which may have been shown to providereasonable results either in the field or experimentally in the laboratory can beexplored more fully in a model Of particular importance is the possibility to exam-ine the implications of long-continued operation of such processes

A model can of course be analysed in any situation in which the governing pro-cess equations can be integrated analytically But in landscape studies only a limitednumber of two-dimensional special cases are likely ever to be analytic Numericalmodelling permits the exploration of joint or sequential action by several processes(geomorphological andor tectonic) in a distributed field This exercise usually con-founds our analytical abilities and often our intuition Hence a modelling exerciseprovides an approach to assess the plausibility of ideas about generic or historicallandscapes Perhaps the most important role for models in this respect is as a toolfor the exploration of various lsquowhat-ifrsquo questions (Oreskes et al 1994) Various con-trolling variables can be held constant while others are allowed to vary Sensitivityanalyses can be performed by changing the nature or intensity of various processesand observing the effects on the morphological evolution of landscapes Such anexercise is often possible only within a numerical modelling framework

Finally numerical landscape models can be used to explore theoretical ideas andconceptual models about which there is much conjecture but little quantitativeresearch For example Kooi and Beaumont (1996) explored the ideas of Davis andother early conceptual modellers using their numerical model

Figure 2 Exponential decay of remaining sediment volume withage measured in several river floodplains (data compiled by JRDesloges University of Toronto) Fraser River data represent theupper Fraser River above Moose Lake Grand River data are forlocations below Caledonia Ontario


Any of these purposes implies a scale specification for landscape modellingbecause the appropriate characterization of the system is affected by scaleand because the initial data requirements are thereby set In no case can the objectivebe to replicate exactly the details of development of a particular landscape since theboundary conditions and history of contingencies that have affected the landscapecannot practically be reconstructed

2 Process specification

The geomorphological rules adopted in numerical models of landscape evolutionrest on weak foundations Many questions about both the appropriate physical rep-resentation and rates of transport processes over large scales and even smallerscales remain unanswered Moreover many geomorphological relations assumedin the literature to be true have not been subjected to rigorous evaluation Forexample a linear relation between gradient and soil creep has been posited andincorporated into landscape evolution models (eg Moretti and Turcotte 1985Koons 1989 Anderson 1994 Kooi and Beaumont 1994) but has not been demon-strated at landscape scale Indeed the scanty available evidence appears to contradictit (Kirkby 1967 Martin and Church 1997 Roering et al 1999 2001) Nor have thecircumstances in which a linear model might provide a suitable approximation formodelling purposes been considered critically This situation arises because of thedifficulty to make observations of process over extensive areas and to sustainmeasurements for an appropriate period

Investigators of geomorphological processes have either adopted a full Newtonianmechanics framework for the definition of transport equations or they have fallenback on scale correlations amongst certain system driving variates and ones describ-ing summary system responses (consider for example sediment transport ratingcurves) But as the scale of study increases it is not possible to resolve the samedegree of mechanical detail as at small scales Appropriate process specificationrequires that the level of detail incorporated into equations be adjusted for theparticular scale of a study In some cases a strictly mechanistic approach may beappropriate while in other cases generalized approaches to process definition ndasheven to the level of scale correlations ndash must be adopted

At the hillslope scale the frequency of grid points for the evaluation of elevationchanges in numerical models can be relatively dense and therefore relatively smallchanges in morphology may be resolvable The researcher may preserve a fairlydetailed representation of the mechanics of processes operating on the hillslopeand variables required for calculations The parameterization may include strictlymechanistic terms such as shear and normal stresses pore water pressures andthe like (see for example studies by Montgomery and Dietrich 1994 and Wu andSidle 1995) A mechanistic expression for the strength of surficial material on slopesis the MohrndashCoulomb equation

s frac14 c0 thorn cr thorn frac12(1m)rbgdthornm(rsat r)gd cos2 u tanf0 (1)

in which c0 is the effective material cohesion cr is pseudocohesion provided by rootstrength rb and rsat are material bulk density and density of saturated materialrespectively d is the depth of surface material above the potential failure plane u


is the hillslope angle m is fractional saturation (in effect dsatd where dsat is thedepth of the saturated zone) g is gravity and f0 is the effective angle of shear resist-ance of the material This quantity is compared with the driving force for failure theshear stress (t) on a candidate failure plane

t frac14 frac12(1m)rb thornmrsatgd sin u cos u (2)

F frac14 st indicates the likelihood for failure which becomes probable as F declinesbelow 10 Neglecting cohesion

F frac14 (1m)rb thornm(rsat r)=frac12(1m)rb thornmrsattanf0

tan u

(3)

In a landscape development model such an equation would be made operationaleither by keeping a spatially distributed water balance (entailing a hydrologicalsubmodel with considerable detail) or by using some stochastic assignment of dsat

at sites where u approaches or exceeds f0 Separate rules are required in order to dis-tribute the failed material downslope (in effect to decide the character and mobilityof the failure whether slumps debris slides or debris flows)

Minor semi-continuous processes of soil displacement (creep ravel animalactivity) might be modelled as a linear diffusive process

z

tfrac14

k2z

x2i

(4)

in which k is the diffusion coefficient (a constant)It is feasible to consider spatial variation of the phenomena described by

eqs (1)ndash(4) because of a relatively high sampling frequency (of the topographic sur-face) in a hillslope-scale study Nonetheless it may still be difficult to retain all of thetrue complexity of the problem Climate and vegetation may be treated as indepen-dent parameters that are approximately constant over relevant timescales buthydrologic and vegetation characteristics may vary in space along the hillslope pro-file (see for example Ambroise and Viville 1986) Vegetation in particular presentsdifficulties because there remains no general physical formulation of its effects evenat the scale of geotechnical modelling of hillslope condition because it is a dynamicfactor and because ndash over almost any scale of landscape interest ndash it responds tolocal climatic and hydrological variations

As the scale of study is increased to that of drainage basins local irregularities inmorphology can no longer be computationally preserved and are no longer ofparticular importance in the functioning of the system Highly detailed processequations are no longer suitable as it is impossible to achieve correspondinglydetailed knowledge of controlling variables because of difficulties associated withsparse sampling and error propagation over extended time and space scales At geo-morphologically significant timescales in sufficiently steep terrain for exampleminor soil disturbance might be ignored since the downslope displacement of surfi-cial material over significant periods of time comes to be dominated by discrete fail-ures such as debris slides (see for example Roberts and Church 1986) These eventsmodelled discretely at hillslope scale might now be simulated as a diffusive processunder the assumption that after a sufficient lapse of time they will have affected


nearly every position on a slope However a significant threshold for slope instabil-ity remains so the process cannot be considered to be linear A more suitableformulation would make k frac14 k(z zxi t) yielding the nonlinear process

z

tfrac14 =xifrac12k(z zxi t)z=xi (5)

with specific nonlinear functional dependences for k (eg Martin and Church 1997Roering et al 1999)

A similar exercise of generalization can be entertained for fluvial sedimenttransport The downslope directed force of flowing water over the surface isgiven as a local average by

t frac14 rfgd sin u (6)

in which rf is fluid density d is water depth u is channel gradient and the quantitiesare specified for a unit width of channel (The equation is simply a specification ofeq (2) for water flowing in a channel of modest gradient) It has been found thatfor the transport gb of bed material (material that may form the floor and sides ofthe channel)

gb frac14 J(t to)n (7)

in which to is a function of material properties and of channel hydraulics and n 15(commonly considerably greater in gravel-bed channels) is an exponent that varieswith the intensity of the transport process To use this formulation in a model onerequires a specification of flow depth d and channel width and to determine thethreshold condition probably a specification of the other principal hydraulicquantities in particular flow velocity One must also track material propertiesparticularly grain size This requires explicit models of both hydrology and channelhydraulics For most landscape modelling purposes these requirements are beyondthe limit of resolution

In such cases sediment transport is defined using generalized formulaeRA Bagnoldrsquos approach (Bagnold 1966 1980) is commonly adopted to estimatetotal sediment transport as

Gb frac14 J(rfgQu) (8)

in which Q is streamflow The only other physical variate required is gradient uwhich is a principal variate of any landscape model So this represents an attractivealternative provided a threshold for transport can also be specified in terms of Q Atvery large scales this approach can expediently be reduced to an empirical corre-lation

Gb frac14 J(Q u) (9)

In some models for which the principal focus of attention is tectonic effects over longtimescales there is no explicit representation of alluvial processes and storage at allwhich translates into an assumption that material once it reaches slope base isremoved from consideration (eg Koons 1989 Anderson 1994) This strategy


amounts to an assertion that the transit time for sediment in the fluvial system issmaller than the model time step It is not clear that this actually is true even forvery large-scale models

Glacial aeolian and coastal processes have been comparatively little consideredin landscape modelling studies but it is certain that a similar scheme of appropri-ate generalization of the process representation will have to be developed in suchcases (see Ashton et al 2001 for an example that considers long-term large-scalecoastal development see MacGregor et al 2000 for an example considering longi-tudinal profile evolution of glacial valleys) Nearly all published landscape modelsadopt some variation on the scheme of equations presented above withthe addition of specific terms to cover effects such as tectonic forces and isostasyAn interesting example of such an additional essentially independent process islarge-scale rock-based failure (Hovius et al 1997 2000 Densmore et al 1998)Practically such events can be dealt with as shot noise under some suitablestochastic scheme (eg Dadson 2000)

Over long timescales it is additionally important to consider rock weathering In ageomorphological model this is the sole source of additional material that can bemobilized This topic has been little studied but a mathematical model based onfield observations was introduced by Heimsath et al (1997)

Major climatic vegetation and geological changes should also be considered atlarge scales although the large space and time steps in such models allow reco-gnition of only relatively significant changes Schumm and Lichty (1965) consideredclimate to be an independent variable at cyclic timescales (see Rinaldo et al 1995 fora model study) but research has suggested that there may be very complex feed-backs and interactions between landscape evolution and climate change (Molnarand England 1990) Furthermore variables that are considered to be independentat the hillslope scale may have to be considered dependent variables at larger scalesFor example vegetation may be an independent variable for a study bounded withinthe slope As scales increase and the possibility of spatially varying climate and long-term climate change is introduced vegetation will vary in response to the climaticforcing

3 Model implementation

A useful framework for implementing numerical models and assessing the limi-tations associated with them in the Earth sciences was presented by Oreskes et al(1994) under the headings validation verification calibration and confirmation

Validation strictly requires that a model contain no known flaws be internally con-sistent and produce results that are consistent with known prototypical instances Aflawless model would be a definitive representation of the physical world as we cur-rently understand it (but even that is not a guarantee that it faithfully represents thereal system) Models in Earth science are never definitive because of the complexityof the systems being studied Indeed the view that we are advancing here is that theart of Earth system modelling is to judge what lsquoflawsrsquo must be tolerated within themodel for the sake of computational tractability without compromising the essentialphenomena that are the object of the modelling exercise Pragmatically the necessityfor this judgement arises in any scientific exercise Whereas the object in analysis is to


come as close as possible to a definitive representation of the phenomena in thesynthesis of complex systems the constraints of scale and information compel thisjudgement to be the central step of the exercise

In practice then validation in Earth system models is about (i) maintaininginternal consistency in the model representation and (ii) ensuring that the modellaws represent some suitably lsquoaveragersquo behaviour of the truly more complex pro-cesses Practically the former amounts to ensuring that continuity is preserved influx relations which usually comes down to ensuring computational closure Thisis a purely technical matter that we will not pursue The latter remains a challengedue in large part to the paucity of field data available about a particular process overthe range of scales that would be necessary to test the requirement rigorously Inpractice lsquoall known prototypical instancesrsquo usually amount to an individualdatum or data set and data for different parts of a model are often assembledfrom different landscapes

Model verification refers to the process of determining the lsquotruthrsquo of the modelThis is a stronger condition than validation and it cannot in principle be achievedexcept in closed systems of deduction More prosaically Earth system models aregenerally considered to be lsquoopenrsquo because of incomplete knowledge of input para-meters The models are underspecified This occurs because of (i) spatial averagingof input parameters (ii) nonadditive properties of input parameters and (iii) theinferences and assumptions underlying model construction This lsquoincompletenessof informationrsquo means that model verification is not possible in open systems

An incomplete model requires calibration Model calibration involves the manipu-lation of adjustable model parameters to improve the degree of correspondencebetween simulated and reference results However arrival at consistent resultsdoes not imply that a model has been successfully validated or verified Calibrationof a model is often undertaken empirically without a theoretical basis for the adjust-ment A model that has been calibrated to a certain set of data may not performadequately for other data sets

In landscape modelling there are in general two strategies available for cali-bration The first entails initializing the model at some arbitrarily defined surfaceconfiguration and running it with set parameters to some reference state (forexample the actual contemporary configuration of the reference surface) This pro-cedure is analogous to the way in which other Earth system models (notably climatemodels) are calibrated The reference state is assumed to be an equilibrium stateof the system In most cases this would be an extremely difficult exercise becauseof the intimate interplay between immanent and contingent processes in determin-ing the state of any particular landscape To our knowledge this has rarely beenattempted An example that discusses the problem of model initialisationin some detail is given by Ferguson et al (2001) These authors simulated the rela-tively short-term development of an aggrading river reach in what amounts to alsquohillslope-scalersquo model and their principal effort was directed at model convergenceonto the current state of the prototype surface beginning from an arbitrary initialstate The drainage initiation problem is a major modelling problem in geomorpho-logy that is usually set up to follow the formal procedure discussed here but it is notrun to match a specific landscape

The second available procedure is to set process constants within the model tovalues derived from field measurements of processes Remarkably apart from the


specification of certain tectonic rates in full models published models have not beenable to constrain geomorphological process rates with any reasonable degree of cer-tainty For example many models have adopted diffusion to simulate various hill-slope processes such as landsliding or soil creep However values for thediffusion coefficients implemented in models have often been based on scarp studieswhich are probably not representative of the processes in operation in the largerregion being modelled In addition the climatic or geologic controls are likely tobe very different Some more recent studies have attempted to calibrate hillslopetransport equations adopted in landscape models For example Martin (2000)used an extensive landsliding data base (Martin et al 2002) based on aerial photo-graphic analysis and accompanying field data to calibrate transport equations in astudy of hillslope evolution for coastal British Columbia Nonetheless there remainsa lack of knowledge regarding transport rates for hillslope processes operating overmedium to long timescales

Similarly constants found in stream power relations used to simulate bedrock oralluvial processes are often not based on strong evidence Indeed in many cases thechoice of constants is not discussed and hence the equations have not necessarilybeen calibrated effectively Snyder et al (2003) attempted to overcome this short-coming by undertaking an empirical calibration of the stream power equation forbedrock incision using field data from northern California

An ostensible reason for the shortcoming of adequate calibration in landscapemodelling is the dramatic disparity between the record of almost any contemporarymeasured rates and the timescale of landscape models There can be no assurancethat contemporary processes conform with averages that may hold over landscape--forming timescales Church (1980) and Kirkby (1987a) argued that relativelyshort-term records of significant landscape forming processes are very unlikely todemonstrate the full range of variability that is ultimately experienced (see alsoKirchner et al 2001 for an empirical example) Moreover the pervasiveness ofhuman disturbance over much of the terrestrial surface today is apt to yield asuite of highly nonrepresentative values for many landforming processes Nonethe-less careful use of available data remains the best approach to model calibration thatis available

4 Testing outcomes

Whereas model verification and validation according to the strict definitions givenabove are not possible model confirmation remains an achievable objectiveConfirmation is established by examining the ability of a model to match prediction(model outcome) with observation in some formally defined manner

There has been almost no methodical study of this issue and no consensus as to whatconstitutes a sufficient test of a modelrsquos performance A number of approaches towardsmodel testing have been adopted On one end of the spectrum some models have beenused primarily as an exploratory tool to examine the behaviour of various processesand no strict attempt is made to evaluate the resemblance of model landscapes toreal landscapes For example Tucker and Bras (1998) used their model of drainagedevelopment primarily as an exploratory tool to examine the operation of several hill-slope process laws In other modelling studies a specific landscape or type of land-


scape is simulated and qualitative comparisons are made between simulated and reallandscapes In such cases qualitative consistency with the modern landscape may beachieved (eg Howard 1997) The study by Ferguson et al (2001) is a rare example ofan attempt at full quantitative comparison but it confines itself to a very local problemFinally some studies have compared statistical properties such as hypsometrybetween simulated and real landscapes (eg Anderson 1994) Willgoose and his col-leagues have made significant advances in this area of research using landscape stat-istics and experimental simulators to test model landscapes (eg Willgoose 1994Willgoose and Hancock 1998 Hancock and Willgoose 2001)

IV Approaches to modelling landscape evolution

This section reviews key developments in modelling the geomorphological com-ponent of landscape evolution for three categories of models (1) conceptual (2)quasi-mechanistic and (3) generalized physics The list of landscape evolutionmodels (Beaumont et al 2000) is now so extensive as to preclude detailed discussionof all individual models

1 Conceptual models

Conceptual models of landscape evolution were created between the 1890s and themid-twentieth century (Davis 1899 Penck 1953 King 1962) The point was tolsquoexplainrsquo the observable end products of long-term landscape evolution andhence modelling often relied on spacendashtime substitution (Paine 1985 Thorn 1988)

WM Davis (1899) proposed that after a period of brief and episodic uplift land-scapes underwent downwearing through a series of predictable stages referred to asthe lsquocycle of erosionrsquo Numerous summaries and critiques of Davisrsquo work are avail-able in the literature (eg Chorley 1965 Flemal 1971 Higgins 1975) In contrastPenck (1953) rejected the notion of disjunct uplift and erosional events and insteadfocused on their continuous interaction He advocated the concept of slope replace-ment whereby the steep part of a slope retreats rapidly and leaves behind a lowerangle debris pile at its base

A frequent criticism of these early models is that exogene processes were treated ina superficial nonquantitative manner and that endogene processes were poorly rep-resented But these criticisms are made in light of much increased understanding oflandscape forming processes The significance of these conceptual models is thatthey provided an initial set of ideas of how landscapes might evolve at large scalesThe processes described in the models nearly always amounted to conjecture onthe part of the modellers and the observations on which the models were basedwere usually strictly morphological

2 Quasi-mechanistic models

After the mid-twentieth century an entire generation of Anglo-American geomor-phologists focused almost exclusively on investigations of erosion transport anddeposition of sediments within a mechanistic framework over relatively small scales(see Church 1996) Consequently a wealth of process knowledge became availablefor incorporation into quantitative models of geomorphological change


Accordingly the landscape models of Kirkby (1971) and Ahnert (1976) incorporatequasi-mechanistic process-equations The term quasi-mechanistic is used here to sig-nal the fact that although the equations do rely on a significant degree of empiricisman attempt is made to express process operation for individual processes in amechanistic manner often including as much detail as knowledge at the timeallowed In this kind of reductionist approach overall system operation is resolvedby summing individual behaviours of lower-level subsystems

In the models of Ahnert (1976) and Kirkby (1971 1987b) a series of geomorpholo-gical process equations including such details as slope wash and rainsplash areapplied to the landscape surface but calibration of equations and definition of keycoefficients and exponents are not discussed Ahnert developed a computer programto run his model increasing the efficiency of calculations whereas Kirkby in hisearly modelling efforts kept the mathematics tractable (and sometimes analytic)by focusing on the evolution of hillslope profiles The models were used to assesslandscape changes that would result when one process was operating or whenseveral processes were operating in combination In this manner various lsquowhat-ifrsquoquestions were addressed Both researchers continued to develop their models insubsequent years (eg Ahnert 1987 Kirkby 1992)

Despite the obvious attraction of quantitative rigor the quasi-mechanisticapproach to process constrained the scale suggesting that the models may not besuitable for long timescales Ahnert nevertheless has applied his model at a varietyof scales ranging from individual hillslopes through to entire mountain ranges(Ahnert 1987) even though it seems unreasonable to apply this one representationof the physics to such a large range of scales

3 Generalized physics models

a Overview Generalized physics models involve deliberate attempts to simplifythe representation of what are known to be a large number of individuallycomplex processes to a level of detail appropriate for extended scales of space andtime Early models of this type focused on the development of hillslope profilesand so were strictly geomorphological models For example Culling (1960 19631965) Luke (1972) and Hirano (1975) recognized the potential of diffusion-typerelations to model transport processes over large scales The analytical solution ofequations necessary at the time made them cumbersome to manipulate andrestricted model complexity

The evolution of landscape surfaces can now be examined using numerical tech-niques to solve the relevant equations very efficiently within a computer Many of therecent models are used to assess the development of landscapes strongly influencedby tectonics such as foreland basins rift escarpments and collisional or convergentmountain belts (eg Flemings and Jordan 1989 Kooi and Beaumont 1994 Willettet al 2001) But models with a particular focus on geomorphological processeshave also been developed (eg Benda and Dunne 1997 Tucker and Bras 1998)

Both hillslope and fluvial processes are incorporated in the surface processes com-ponent of most of these recent models with diffusion-type equations and streampower relations typically being used to simulate them If grid cells are of a largesize for example 1 km2 fluvial and hillslope relations are often applied across


each cell However the development of models utilizing triangulated irregular net-works (TIN) grids has allowed for greater flexibility in defining discrete hillslope andchannel cells within a model (eg Tucker et al 2001) Research has taken place inrecent years that has begun to provide the necessary underpinning for calibratinggeomorphological process equations within landscape models (eg Seidl et al1994 Heimsath et al 1997 Hovius et al 1997 2000 Martin 2000 Snyder et al2003) Defining effective ways to test models remains an outstanding issue (see Sec-tion III 4)

b Key developments The prominence of diffusion-type equations is evident inseveral of the key models that emerged in the late 1980s (eg Koons 1989Anderson and Humphrey 1989 Flemings and Jordan 1989) Anderson andHumphrey applied their model to a range of scales from small-scale hillslopedevelopment to larger-scale basin sedimentation whereas Flemings and Jordanfocused on sediment delivery to foreland basins Koons examined the evolution ofthe Southern Alps of New Zealand a collisional mountain belt Uplift wasincluded in the models of Koons and Flemings and Jordan

Several landscape evolution models (Anderson 1994 Kooi and Beaumont 1994Tucker and Slingerland 1994) were published in a special issue of the Journal ofGeophysical Research (1994) These models emphasized interactions between endo-gene and geomorphological processes and the development of particular tectonicfeatures over times scales 1 Ma was explored Kooi and Beaumont (1994) andTucker and Slingerland (1994) explored the evolution of escarpments related torifting whereas Anderson (1994) explored the evolution of the Santa Cruz Moun-tains In the model of Kooi and Beaumont the cumulative effect of hillslope pro-cesses (including slow quasi-continuous and rapid episodic mass movements) issimulated using linear diffusion Anderson (1994) and Tucker and Slingerland(1994) used linear diffusion to simulate slow mass movements with additionalalgorithms defined to model large hillslope failures Tucker and Slingerland(1994) incorporated a weathering rule in their model to monitor the supply ofsediment Variants of stream power relations are used in all of these models tosimulate fluvial transport processes for alluvial andor bedrock channels Modelsfocusing on interactions between tectonics and surface processes continue toemerge up to the present day each adopting slightly different generalizedapproaches for the geomorphological component For example Avouac andBurov (1996) emphasized the role of linear and nonlinear diffusion while Willett(1999) focused on bedrock channel incision in the evolution of intracontinentaland convergent mountain belts Other researchers continued to expand and refinethe specification of key geomorphological processes in particular tectonic settingsFor example Densmore et al (1998) included a detailed rule for deep-seatedbedrock landsliding which had been lacking in earlier models in their modelof the normal-fault-bounded mountains in the Basin and Range over timescalesup to 1 Ma

Models have also been developed and refined to examine classic long-standingconcepts in the landscape literature For example Kooi and Beaumont (1996) usedtheir landscape development model to explore the conceptual models of DavisPenck and King A significant number of studies have considered the concept of


steady state in mountain evolution (Whipple and Tucker 1999 Montgomery 2001Whipple 2001 Willett et al 2001) Willett and Brandon (2002) recognize thatwhile steady state is often invoked in combined tectonicgeomorphological modelsthe particular meaning attached to the term is not always clear They use their land-scape model to provide clarification of possible ways to invoke steady state whenconsidering the formation of mountain belts This is an interesting enquiry becauseit confronts Hackrsquos (1960) assertion about equilibrium in landscape evolution with acritical test which if confirmed could become the basis for a general strategy toobtain model confirmation (cf Section III 4)

In recent years several models have been published that have highlighted the geo-morphological component of landscape evolution with endogene processes ifincluded being secondary in importance (eg Howard 1997 Benda and Dunne1997 Tucker and Bras 2000) Typical timescales for application of these modelsrange from 103 to 104 years Benda and Dunne (1997) developed a model to examinethe development of the Oregon Coast Range over about an 8000-year time periodfocusing on stochastic climate-driven forcing of sediment supply to channelnetworks Howard (1997) considered the development of the Utah badlands overtimescales up to many tens of thousands of years employing diffusion andthreshold-driven equations to simulate slow mass movements and rapid failuresrespectively and stream power equations for transport in bedrock and alluvial riv-ers A stochastic rainfall component was incorporated to drive geomorphologicalprocesses in Tucker and Bras (2000)

The implementation of irregular grids in numerical models of landscape changerepresents a significant development (eg Braun and Sambridge 1997 Tuckeret al 2001) This approach allows variation of grid resolution across the landscapesurface for example a higher grid resolution is usually preferred along channelsthan on interfluves Braun and Sambridge (1997) illustrated the improved model per-formance obtained when adopting irregular spatial discretization of the grid Tuckeret al (2001) implemented triangulated irregular networks in their hydrologic geo-morphological model It appears that this will be a feasible means to overcome theproblem of different spatial resolution required to model lsquoslowrsquo or episodic processeson hillslopes in comparison with the lsquofastrsquo processes in river channels at the samescale of temporal resolution This has been one of the most significant practicalproblems to hinder landscape modelling efforts with geomorphological resolution

V Conclusions

The modelling of landscape evolution has been made quantitatively feasible by theadvent of high speed computers that permit the effects of multiple processes to beintegrated together over complex topographic surfaces and extended periods oftime The development of this capability coincided with a period of intense focusin the community of geomorphologists on process models for sediment movementA few pioneers (notably F Ahnert and M Kirkby) took up the challenge to creategeomorphological models of landscape evolution Geophysical interest in surfaceprocesses as a significant aspect of the long-term very large-scale evolution ofEarthrsquos surface has helped to renew geomorphological interest in landscape devel-


opment Today a range of diffusion-advection models exists for the study of fluviallandscapes and models for other geomorphological processes are beginning toappear

The models emphasize however the dramatic shortage of data about geomorpho-logical processes over timescales sufficient to be used for sensitive calibration orcritical testing of any model The message here is that renewed field activity willbe important in the years ahead field activity directed by the identification of criticaldata requirements for modelling This activity will require a reintegration of processstudies and historical investigations in the subject since it is clear that data sets ofsufficient length to inform even models with quite high temporal resolution arealmost completely nonexistent The ball remains in the geomorphologistsrsquo courtPlaying the ball is apt to lead to a reform of the subject that goes far beyond the adop-tion of a capability to construct numerical models of landscape evolution

Acknowledgement

The manuscript benefited from reviews at several stages Our colleague OlavSlaymaker reviewed a draft of the paper and provided sage advice on the focus ofit and also some historical details for which we are grateful

References

Ahnert F 1976 Brief description of a compre-hensive three-dimensional process-responsemodel of landform development Zeitschriftfur Geomorphologie 25 29ndash49

mdashmdash 1987 Process-response models of denudationat different spatial scales In Ahnert F editorGeomorphological models ndash theoreticaland empirical aspects Catena Supplement 1031ndash50

mdashmdash 1998 Introduction to geomorphology Chiche-ster John Wiley and Sons

Ambroise B and Viville D 1986 Spatial varia-bility of textural and hydrodynamic proper-ties in a soil unit of the Ringelbachcatchment Vosges (France) Zeitschrift furGeomorphologie Supplement 58 21ndash34

Anderson RS 1994 Evolution of the SantaCruz Mountains California through tectonicgrowth and geomorphic decay Journal of Geo-physical Research 99 20161ndash79

mdashmdash 2002 Modeling of tor-dotted crests bed-rock edges and parabolic profiles of thehigh alpine surfaces of the Wind RiverRange Wyoming Geomorphology 46 35ndash58

Anderson RS and Humphrey NF 1989Interaction of weathering and transport pro-cesses in the evolution of arid landscapes In

Cross TA editor Quantitative dynamicstratigraphy Englewood Cliffs NJ PrenticeHall 349ndash61

Ashton A Murray AB and Arnoult O 2001Formation of coastline features by large-scaleinstabilities induced by high-angle wavesNature 414 296ndash300

Avouac J and Burov E 1996 Erosion as a driv-ing mechanism of intracontinental mountaingrowth Journal of Geophysical Research 10117747ndash69

Bagnold RA 1966 An approach to the sedimenttransport problem from general physics UnitedStates Geological Survey Professional Paper422I Washington DC US Government print-ing office 37 pp

mdashmdash 1980 An empirical correlation of bedloadtransport rates in flumes and natural riversProceedings of the Royal Society of London A372 453ndash73

Beaumont C Kooi H and Willett S 2000Coupled tectonic-surface process modelswith applications to rifted margins and colli-sional orogens In Summerfield MA editorGeomorphology and global tectonics ChichesterWiley 29ndash55


Benda L and Dunne T 1997 Stochastic forcingof sediment supply to channel networks fromlandsliding and debris flow Water ResourcesResearch 33 2849ndash63

Bierman PR 1994 Using in situ producedcosmogenic isotopes to estimate rates oflandscape evolution a review from the geo-morphic perspective Journal of GeophysicalResearch 99 13885ndash96

Braun J and Sambridge M 1997 Modellinglandscape evolution on geological timescales a new method based on irregularspatial discretization Basin Research 9 27ndash52

Brozovic N Burbank DW and Meigs AJ1997 Climatic limits on landscape develop-ment in the northwestern Himalaya Science276 571ndash74

Cerling TE and Craig H 1994 Geomorphol-ogy and in-situ cosmogenic isotopes AnnualReviews of Earth and Planetary Science 22273ndash317

Chase CG 1992 Fluvial landsculpting and thefractal dimension of topography Geomorphol-ogy 5 39ndash57

Chorley RJ 1965 A re-evaluation of thegeomorphic system of WM Davis InChorley RJ and Hagget P editors Frontiersin geographical teaching London Methuen21ndash38

mdashmdash 1969 The drainage basin as the fundamen-tal geomorphic unit In Chorley RJ editorWater Earth and man London Methuen77ndash98

Chorley RJ Schumm SA and Sugden DE1984 Geomorphology London Methuen

Church M 1980 Records of recent geomorpho-logical events In Cullingford RA David-son DA and Lewin J editors Timescales ingeomorphology Chichester Wiley 13ndash29

mdashmdash 1996 Space time and the mountain ndash howdo we order what we see In Rhoads BL andThorn CE editors The scientific nature ofgeomorphology Chichester Wiley 147ndash70

Clemmensen LB Pye K Murray A and Hei-nemeier J 2001 Sedimentology stratigraphyand landscape evolution of a Holocenecoastal dune system Lodbjerg NW JutlandDenmark Sedimentology 48 3ndash27

Culling WEH 1960 Analytical theory oferosion Journal of Geology 68 336ndash44

mdashmdash 1963 Soil creep and the developmentof hillside slopes Journal of Geology 71 127ndash61

mdashmdash 1965 Theory of erosion on soil-coveredslopes Journal of Geology 73 230ndash54

Culling WEH and Datco M 1987 The fractalgeometry of the soil-covered landscape EarthSurface Processes and Landforms 12 369ndash85

Dadson S 2000 Episodic processes in fluviallandscape evolution MSc Thesis Universityof British Columbia

Davis WM 1899 The geographical cycleGeographical Journal 14 481ndash504

Delcourt HR Delcourt PA and Webb T3rd 1983 Dynamic plant ecology the spec-trum of vegetational change in space andtime Quaternary Science Reviews 1 153ndash75

Densmore AL Ellis MA and Anderson RS1998 Landsliding and the evolution of nor-mal-fault-bounded mountains Journal ofGeophysical Research 103 15203ndash19

Dietrich WE and Dunne T 1978 Sedimentbudget for a small catchment in mountainousterrain Zeitschrift fur Geomorphologie Supple-mentband 29 191ndash206

Dietrich W Wilson C Montgomery D andMcKean J 1993 Analysis of erosionthresholds channel networks and landscapemorphology using a digital terrain modelJournal of Geology 101 259ndash78

Duller GA 2000 Dating methods geochronol-ogy and landscape evolution Progress inPhysical Geography 24 111ndash16

Dunne 1980 Formation and controls of channelnetworks Progress in Physical Geography 4211ndash39

Ehleringer JR and Field CB editors 1993Scaling physiological processes leaf to globeSan Diego CA Academic Press 388 pp

Ellis ME Densmore AL and Anderson RS1999 Development of mountainous topogra-phy in the Basin Ranges USA Basin Research11 21ndash42

Everitt BL 1968 Use of the cottonwood in aninvestigation of the recent history of a flood-plain American Journal of Science 266 417ndash39

Ferguson RI Church M and Weatherly H2001 Fluvial aggradation in VedderRiver testing a one-dimensional sedi-mentation model Water Resources Research37 3331ndash47

Flemal RC 1971 The attack on the Davisiansystem of geomorphology a synopsis Journalof Geological Education 19 3ndash13

Flemings P and Jordan T 1989 A syntheticstratigraphic model of foreland basin devel-opment Journal of Geophysical Research 943851ndash66


Garver JI Brandon MT Roden-Tice M andKamp PJ 1999 Erosional denudation deter-mined by fission-track ages of detrital apatiteand zircon In Ring U Brandon MT Will-ett S and Lister G editors Exhumation pro-cesses normal faulting ductile flow anderosion London Geological Society ofLondon Special Publication 154 283ndash304

Guzzetti F Malamud BD Turcotte DL andReichenbach P 2002 Power-law correlationof landslid areas in central Italy Earth and Pla-netary Science Letters 6085 1ndash15

Hack JT 1960 Interpretation of erosional topo-graphy in humid temperate regions AmericanJournal of Science 258A 80ndash97

Hancock G and Willgoose G 2001 Use of alandscape simulator in the validation of theSIBERIA catchment evolution model declin-ing equilibrium landforms Water ResourcesResearch 37 1981ndash92

Heimsath AM Dietrich WE Nishiizumi Kand Finkel RC 1997 The soil productionfunction and landscape equilibrium Nature388 358ndash61

Higgins CG 1975 Theories of landscape devel-opment a perspective In Melhorn WN andFlemal RC editors Theories of landformdevelopment Boston MA Allen and Unwin1ndash28

Hirano M 1975 Simulation of developmentalprocess of interfluvial slopes with referenceto graded form Journal of Geology 83 113ndash23

Hovius N Stark CP and Allen PA 1997Sediment flux from a mountain belt derivedby landslide mapping Geology 25 231ndash34

Hovius N Stark CP Shu Hao-Tsu and LinJiun-Chuan 2000 Supply and removal ofsediment in a landslide-dominated mountainbelt Central Range Taiwan Journal of Geology108 73ndash89

Howard AD 1997 Badland morphology andevolution interpretation using a simulationmodel Earth Surface Processes and Landforms22 211ndash27

Kelsey HM Lamberson RL and MadejMA 1987 Stochastic model for the long-term transport of stored sediment in a riverchannel Water Resources Research 23 1738ndash50

King L 1962 The morphology of the Earth NewYork Hafner

Kirchner JW Finkel RC Riebe CS Gran-ger DE Clayton JL and Megahan WF2001 Episodic mountain erosion inferredfrom sediment yields over decades and mil-lennia Geology 29 591ndash94

Kirkby MJ 1967 Measurement and theory ofsoil creep Journal of Geology 75 359ndash78

mdashmdash 1971 Hillslope process-response modelsbased on the continuity equation In Bruns-den D editor Slope form and process LondonInstitute of British Geographers Special Pub-lication 3 15ndash30

mdashmdash 1987a The Hurst effect and its implicationsfor extrapolating process rates Earth SurfaceProcesses and Landforms 12 57ndash67

mdashmdash 1987b General models of long-termslope evolution through mass movementIn Anderson MG and Richards KSeditors Slope stability Chichester Wiley359ndash79

mdashmdash 1992 An erosion-limited hillslope evol-ution model Catena supplement 23 157ndash87

Kooi H and Beaumont C 1994 Escarpmentevolution on high-elevation rifted marginsinsights derived from a surface processesmodel that combines diffusion advectionand reaction Journal of Geophysical Research99 12191ndash209

mdashmdash 1996 Large-scale geomorphology classicalconcepts reconciled and integrated with con-temporary ideas via a surface processesmodel Journal of Geophysical Research 1013361ndash86

Koons PO 1989 The topographic evolution ofcollisional mountain belts a numerical lookat the Southern Alps New Zealand AmericanJournal of Science 289 1041ndash69

Luke JC 1972 Mathematical models for land-form evolution Journal of Geophysical Research77 2460ndash64

Macaire JJ Bellemlih S Di-Giovanni C DeLuca P Visset L and Bernard J 2002 Sedi-ment yield and storage variations in theNegron River catchment (south western Par-isian Basin France) during the Holoceneperiod Earth Surface Processes and Landforms27 991ndash1009

MacGregor KC Anderson RS AndersonSP and Waddington ED 2000 Numericalsimulations of glacial-valley longitudinalprofile evolution Geology 28 1031ndash34

Mark DM and Aronson PB 1984 Scale-dependent fractal dimensions of topographicsurfaces an empirical investigation withapplications in geomorphology and computermapping Mathematical Geology 16 671ndash83

Martin Y 2000 Modelling hillslope evolutionlinear and nonlinear transport relations Geo-morphology 34 1ndash21


Martin Y and Church M 1997 Diffusion inlandscape development models on thenature of basic transport relations Earth Sur-face Processes and Landforms 22 273ndash79

Martin Y Rood K Schwab J and Church M2002 Sediment transfer by shallow landslid-ing in the Queen Charlotte Islands BC Cana-dian Journal of Earth Sciences 39 189ndash205

Merritts D and Ellis M 1994 Introduction tospecial section of tectonics and topographyJournal of Geophysical Research 99 12135ndash41

Molnar P and England PC 1990 LateCenozoic uplift of mountain ranges andglobal climate change chicken or egg Nature346 29ndash34

Montgomery D 2001 Slope distributionsthreshold hillslopes and steady-statetopography American Journal of Science 301432ndash54

Montgomery DR and Dietrich WE 1994 Aphysically based model for the topographiccontrol on shallow landsliding WaterResources Research 30 1153ndash71

Montgomery DR Balco G and Willett SD2001 Climate tectonics and the mega-mor-phology of the Andes Geology 29 579ndash82

Moretti I and Turcotte DL 1985 A model forerosion sedimentation and flexure withapplications to New Caledonia Journal ofGeodynamics 3 155ndash68

Murray AB 2002 Seeking explanation affectsnumerical modeling strategies Eos (Trans-actions of the American Geophysical Union) 83418ndash19

Nakamura F and Kikuchi S 1996 Some meth-odological developments in the analysis ofsediment transport processes using age dis-tribution of floodplain deposits Geomorphol-ogy 16 139ndash45

Nakamura F Maita H and Araya T 1995Sediment routing analyses based on chrono-logical changes in hillslope and riverbed mor-phologies Earth Surface Processes andLandforms 20 333ndash46

Nichols KK Bierman PR Hooke RLeBClapp EM and Caffee M 2002 Quantifyingsediment transport on desert piedmonts using10Be and 26Al Geomorphology 45 105ndash25

Oreskes N Shrader-Frechette K and Belitz K1994 Verification validation and confir-mation of numerical models in the earthsciences Science 263 641ndash46

Outcalt SI Hinkel KM and Nelson FE1994 Fractal physiography Geomorphology11 91ndash106

Paine ADM 1985 lsquoErgodicrsquo reasoning ingeomorphology time for a review ofthe term Progress in Physical Geography 91ndash15

Penck W 1953 Morphological analysis of land-forms Translated by H Czech and KC Bos-well London Macmillan

Reid LM and Dunne T 1996 Rapid evaluationof sediment budgets Reiskirchen Catena Ver-lag

Rinaldo A Dietrich WE Rigon R VogelGK and Rodriguez-Iturbe I 1995 Geomor-phological signatures of varying climateNature 374 632ndash35

Ring U Brandon MT Willett S and ListerG editors 1999 Exhumation processes normalfaulting ductile flow and erosion LondonGeological Society of America Special Publi-cation 154

Roberts RG and Church M 1986 The sedi-ment budget in severely disturbed water-sheds Queen Charlotte Ranges BritishColumbia Canadian Journal of Forest Research16 1092ndash106

Roering J Kirchner J and Dietrich W 1999Evidence for nonlinear diffusive sedimenttransport on hillslopes and implications forlandscape morphology Water ResourcesResearch 25 853ndash70

mdashmdash 2001 Experimental hillslope evolution bynonlinear creep and landsliding Geology 29143ndash46

Schumm SA 1977 The fluvial system NewYork John Wiley and Sons

Schumm SA and Lichty RL 1965 Timespace and causality in geomorphology Amer-ican Journal of Science 263 110ndash19

Seidl MA Dietrich WE and Kirchner JW1994 Longitudinal profile development intobedrock an analysis of Hawaiian channelsJournal of Geology 102 457ndash74

Smith T and Bretherton F 1972 Stability andthe conservation of mass in drainagebasin evolution Water Resources Research 81506ndash29

Snyder NP Whipple KX Tucker GE andMerritts DJ 2003 Channel response to tec-tonic forcing field analysis of stream mor-phology and hydrology in the Mendocinotriple junction region northern CaliforniaGeomorphology 53 97ndash127

Steyn DG Oke TR Hay JE and Knox JL1981 On scales in climatology and meteorol-ogy Climatological Bulletin 30 1ndash8

Summerfield MA 1991 Global geomorphologyEssex Longman Scientific and Technical


mdashmdash 2000 Geomorphology and global tectonicsintroduction In Summerfield MA editorGeomorphology and global tectonics ChichesterWiley 3ndash12

Thomas MF and Summerfield MA 1987Long-term landform development keythemes and research problems In GardinerV editor International geomorphology 1986part II Chichester Wiley 935ndash56

Thorn CE 1988 An introduction to theoreticalgeomorphology Boston MA Unwin Hyman

Tucker GE and Bras R 1998 Hillslope pro-cesses drainage density and landscape mor-phology Water Resources Research 34 2751ndash64

mdashmdash 2000 A stochastic approach to modelingthe role of rainfall variability in drainagebasin evolution Water Resources Research 361953ndash64

Tucker GE and Slingerland RL 1994 Erosionaldynamics flexural isostasy and long-lived escarpments a numerical modelingstudy Journal of Geophysical Research 9912229ndash43

mdashmdash 1996 Predicting sediment flux from foldand thrust belts Basin Research 8 329ndash49

Tucker GE Lancaster ST Gasparini NMBras RL and Rybarczyk SM 2001 Anobject-oriented framework for hydrologicand geomorphic modeling using triangulatedirregular networks Computers and Geosciences27 959ndash73

Whipple KX 2001 Fluvial landscape responsetime how plausible is steady-state denuda-tion American Journal of Science 301 313ndash25

Whipple KX and Tucker GE 1999 Dynamicsof the stream power river incision modelimplications for height limits of mountain

ranges landscape response timescales andresearch needs Journal of Geophysical Research104 17661ndash74

Willett S 1999 Orogeny and orography theeffects of erosion on the structure of mountainbelts Journal of Geophysical Research 10428957ndash81

Willett SD and Brandon MT 2002 Onsteady state in mountain belts Geology 30175ndash78

Willett SD Slingerland R and Hovius N2001 Uplift shortening and steady statetopography in active mountain belts Ameri-can Journal of Science 301 455ndash85

Willgoose G 1994 A statistic for testing theelevation characteristics of landscape simu-lation models Journal of Geophysical Research99 13987ndash96

Willgoose G and Hancock G 1998 Hypso-metric curves as an indicator of catchmentform revisited Earth Surface Processes andLandforms 23 611ndash23

Willgoose G Bras RL and Rodriguez-Iturbe I 1991a A coupled channelnetwork growth and hillslope evolutionmodel 1 Theory Water Resources Research27 1671ndash84

mdashmdash 1991b A coupled channel network growthand hillslope evolution model 2 Nondimen-sionalization and applications WaterResources Research 27 1685ndash96

Wu W and Sidle RC 1995 A distributed slopestability model for steep forested catchmentsWater Resources Research 31 2097ndash110

Xu T Moore ID and Gallant JC 1993 Frac-tals fractal dimensions and landscapes ndash areview Geomorphology 8 245ndash62


adoption of numerical modelling for the study of combined exogene andendogene processes represented a recent development at the time Researcherswere turning to the geomorphological literature to find information about therate of surface transport processes at large spatial and temporal scales inorder to model the lithospheric response to erosion and sedimentationAnderson and Humphrey found that existing results did not offer the infor-mation necessary for successful implementation of such models Their state-ment was made at the beginning of a resurgence of interest in landscapeevolution that had occurred in recent years (review by Thomas and Summer-field 1987) In subsequent years much progress has been made in numericalmodelling of landscape evolution (reviews by Merritts and Ellis 1994 Sum-merfield 2000) although many outstanding issues still remain concerningthe representation of geomorphological processes over large spatial and tem-poral scales

It is proposed herein that numerical modelling of landscape evolution recon-ciles lsquohistoricalrsquo and lsquoprocessrsquo studies by explaining characteristic features of his-torical geomorphological landscapes in terms of physically based processes(without necessarily being able to recreate the details of specific landscapes)Numerical modelling requires a framework within which material fluxes canbe reconciled across all resolved scales The crux of modelling from a geomor-phological perspective is how to represent the flux lsquolawsrsquo governing landform-ing processes Adequate calibration and testing of transport relations used insuch models remains elusive because of difficulties in acquiring suitable fielddata over representative timescales In turn the litmus test of a model dependson finding appropriately rigorous ways to compare model results with the evi-dence of particular landscapes The resolution of both of these problems isdominated by scale Spatial and temporal scales of a study should guide thespecification of process and selection of test procedures within numericalmodels A judgement must be made regarding lsquoflawsrsquo within the model thatmust be tolerated for the sake of computational tractability while replicatingthe essential characteristics of the phenomena under consideration The dramati-cally expanded body of tools and techniques now available including muchimproved topographic mapping (DEMs) high resolution air photographs sub-stantially improved surveying instruments and sophisticated dating techniqueshave increased our ability to resolve key issues related to process specificationand model testing

Many of the existing landscape models consider the interactions of tectonic andgeomorphological processes At intermediate timescales geomorphological deve-lopment of the landscape can be studied independently of tectonic considerationsThis approach is increasingly important in studies of lsquolong-termrsquo effects of land-scape management as well as to study the implications of geomorphological pro-cesses over extended periods An examination of methodology surrounding thenumerical representation of geomorphological phenomena at large scales can pro-vide insight into appropriate specifications of processes for a particular scale ofstudy It can also reveal the weaknesses in our ability to do so The objectiveof this paper then is to examine theoretical and methodological issues relatedto the quantitative study of geomorphological processes within the context oflandscape evolution


II Scale

1 Scale and process




2 Spatial scale














3 Temporal scale




virtual velocity






Timescale (yr)


































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References




























































































































II Scale

1 Scale and process




2 Spatial scale














3 Temporal scale




virtual velocity






Timescale (yr)


































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References







































































































































3 Temporal scale




virtual velocity






Timescale (yr)


































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References

































































































































3 Temporal scale




virtual velocity






Timescale (yr)


































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References





























































































































3 Temporal scale




virtual velocity






Timescale (yr)


































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References





















































































































































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References













































































































































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References








































































































































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References
































































































































tan u

(3)




z

tfrac14

k2z

x2i

(4)






z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References





























































































































z
































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References














































































































































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References






































































































































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References































































































































4 Testing outcomes







1 Conceptual models























V Conclusions





Acknowledgement


References































































































































1 Conceptual models























V Conclusions





Acknowledgement


References












































































































































V Conclusions





Acknowledgement


References




































































































































V Conclusions





Acknowledgement


References































































































































V Conclusions





Acknowledgement


References






























































































































Acknowledgement


References































































































































































































































































































































































































Documents

Numerical modelling of landscape evolution: geomorphological perspectiveswpg.forestry.oregonstate.edu/sites/wpg/files/seminars/... · 2015-01-09 · Numerical modelling of landscape