Standing proud: a response to ‘Soil-erosion models: where do we really stand?’ by Smith et al.

EARTH SURFACE PROCESSES AND LANDFORMSEarth Surf. Process. Landforms 35, 1349–1356 (2010)Copyright © 2010 John Wiley & Sons, Ltd.Published online 28 June 2010 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/esp.2047

Exchanges

Standing proud: a response to ‘Soil-erosion models: where do we really stand?’ by Smith et al.John Wainwright,1,2* Anthony J. Parsons,1 Eva N. Müller,2 Richard E. Brazier3 and D. Mark Powell4

1 Sheffi eld Centre for International Drylands Research, Department of Geography, University of Sheffi eld, Sheffi eld, UK2 Institut für Geoökologie, Universität Potsdam, Potsdam, Germany3 Department of Geography, University of Exeter, Exeter, UK4 Department of Geography, University of Leicester, Leicester, UK

Received 30 July 2009; Revised 15 April 2010; Accepted 29 April 2010

*Correspondence to: John Wainwright, Sheffi eld Centre for International Drylands Research, Department of Geography, University of Sheffi eld, Winter Street, Sheffi eld, S10 2TN, UK. E-mail: J.Wainwright@sheffi eld.ac.uk

Our original papers (Wainwright et al., 2008a, 2008b, 2008c) which prompted the comment by Smith et al. identifi ed a need to revisit soil-erosion modelling concepts and applications. Our argument was based on a critique that demonstrated misunderstandings and untested assumptions in the literature over the last 35 years or so. We feel that the response by Smith et al. further underlines the need to be openly critical about the models developed in the discipline (including our own) and transparent about the quality of the process representation therein. The comment of Smith et al. is based around fi ve issues, which we will address in turn, before replying to their critique of MAHLERAN and fi nally some of the more general issues that they address.

Suspended Sediment on Hillslopes

Smith et al. fail to note that we present a theoretical argument based on the work of Van Rijn (1984), as well as a selection of eight papers providing empirical evidence to support the point that transport in suspension will rarely occur on hill-slopes. Their counterexample is based on the basis of ‘sub-stantial empirical evidence’, presented as citations to two papers. The argument presented depends on empirical obser-vations of enrichment, and they state that if transport of primary particles were as aggregates, ‘less particle enrichment would occur’. This statement is problematic on at least two levels. First, as Smith et al. note, enrichment may be due to both preferential transport of fi nes and preferential deposition of coarse particles (not either/or as they state). Given that the travel-distance approach on which MAHLERAN is based provides a representation of both of these processes, this argument cannot be used to contradict the approach. Indeed, in a sub-sequent application of MAHLERAN (Turnbull, 2008; Turnbull et

al., 2010), clay and silt fractions always show enrichment ratios >1, and frequently sand-sized particles do also. Secondly, standard methods of particle-size analysis [e.g. BS1377 (BSI, 1990); ASTM D6913 – 04e2 (ASTM, 2009)] will produce results in terms of primary particles and not aggregates. There is no clear-cut evidence of the mode of transport of particles from such data. Furthermore, the evidence based on the size of eroded sediment is not as straightforward as Smith et al. suggest [see discussion in Parsons et al. (1991, pp. 143–144) and in Foster (1982, pp. 326–327)]. For example, Poesen and Savat (1980), Ellison (1944) and Parsons et al. (1992) all found sand-sized particles to be the dominant material in splash, which is pertinent because raindrop impact is the main form of detachment in interrill areas. The Hjulstrøm and Shields curves for fl ow detachment also have a minimum at about 0·1 mm. The evidence for enrichment, in cases where it does occur, would appear to be strongly in favour of differential transport distance rather than dominant mode of movement. We do agree with the comments about the lack of detailed process understandings of aggregate behaviour in erosion, a point we make on both p. 819 and p. 1124 of our papers (Wainwright et al., 2008a, 2008c).

The remainder of this section in Smith et al. is concerned with statements regarding Bennett (1974). They say that ‘this is not so much a “model” proposed by Bennett as it is a simple differential statement of sediment[-]mass continuity’. It is hard to see how a continuity equation is not a model, unless one believes that only simulation or numerical approaches are worthy of the term model. Bennett (1974) clearly intended to produce a model from his title onwards. A broader discussion of what constitutes a model can be found in Kirkby (1987) and Mulligan and Wainwright (2003). Bennett made an assumption of the relationship between water and sediment movement, but was very careful to state that the assumption was made in rela-tion to suspended sediment (Bennett, 1974, p. 486) and that

1350 J. WAINWRIGHT ET AL.

Copyright © 2010 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 35, 1349–1356 (2010)

this assumption would subsequently need to be tested (ibid., passim). The equation can be made to hold whatever the mode of transport, but there are problems which arise as a conse-quence. Our point is that it has produced a suite of models which implicitly assume all motion is by transport in suspen-sion or that transport in other modes occurs at velocities that are demonstrably too fast. We believe that a failure to move beyond this point has been a major failing in soil-erosion research. The assertion of Smith et al. that ‘suspension is not inherent in the Bennett dynamic conservation of mass equa-tion’ is simply not true and directly contradicts Bennett’s own description of the equation [which ‘describe[s] the movement of suspended sediment particles in a one-dimensional, infi -nitely wide, free surface fl ow’ (Bennett, 1974, p. 486; and cited in Wainwright et al., 2008a, p. 814)]. Our equation 6 (Wainwright et al., 2008a, pp. 818–819):

∂∂

+∂∂

− + =ht

qx

ds s ε 0 [1 (6a in Wainwright et al., 2008a)]

describes sediment continuity in terms of sediment mass [where hs is the equivalent depth of sediment in transport (in metres), t is time (in seconds), qs is the unit discharge of sedi-ment (in m2 s−1), ε is the rate of erosion of the surface (in m s−1), x is distance in the direction of fl ow (in metres) and d is the rate of deposition (in m s−1)]. It is clearly not ‘fundamentally the same as that of Bennett’, which is:

∂( )

∂+

∂( )∂

− =hCt

qCx

e 0 [2 (3a in Wainwright et al., 2008a)]

which describes the process in terms of water mass and sedi-ment concentration [where h is the depth of water fl ow (in metres), q is the unit discharge of water (in m2 s−1), C is the sediment concentration in the fl ow (in m3 m−3) and e is the unit net erosion rate (in m s−1)]. Furthermore, qs in Equation 1 is defi ned as hs × vs where vs is the virtual velocity (in m s−1) of the sediment, while q in Equation 2 is h × v [v is mean velocity of water (in m s−1)]. Although these terms may in some cases be related (see later), they are in no way equivalent.

Assumptions of Steady State

Smith et al. suggest that we say that the equation of Foster and Meyer (1972) presented as equation 4 in Wainwright et al. (2008a) is used in the EUROSEM, LISEM and KINEROS2 models. As can be seen in the second paragraph of p. 817 in the latter paper, the statement we make is in relation to equa-tion 5. Equation 5 is indeed a conceptual development of equation 4, so on one level this difference could be consid-ered trivial, but it does again demonstrate the broader issues of misreading of the literature. The equation in question describes the net detachment rate (DF, i.e. detachment – depo-sition, equivalent to the ε + d term in Equation 1 and the e term in Equation 2) and is:

D k T GF C F= −( ) [3 (5 in Wainwright et al., 2008a)]

where k is a rate coeffi cient (in m−1) defi ned by putting DC = k TC, DC is the detachment capacity of the fl ow (in kg m−2 s−1), GF is the sediment load in the fl ow (in kg m−1 s−1) and TC is the transport capacity of the fl ow (in kg m−1 s−1). This equation can be compared with:

D wv T CF CC= −( )↓ (4)

where w is the width of fl ow (in metres), v↓ the settling velocity of the sediment (in m s−1) and TCC is the concentration of sedi-ment at transport capacity (in m3 m−3), which is equation 28 in the description of the EUROSEM model of Morgan et al. (1998: symbols modifi ed for consistency with other equations here). It can also be compared with:

D Y T qC wvF C= −( ) ↓ (5)

where Y is a ‘dimensionless effi ciency factor’, which is equa-tion 2–30 in Jetten (2003: symbols again modifi ed for consis-tency), the manual for the LISEM model. Comparison is also made with:

e k T C ACC= −( ) (6)

where A is cross-sectional area of fl ow (in m2), which appears as equation 5 in the erosion section of the online KINEROS2 manual (KINEROS2, 2009). Finally, the Smith et al. (1995) paper cited in this section which compares EUROSEM and KINEROS2 has as its equation 5:

e bv T CCC= −( )↓ . (7)

where b ‘accounts for cohesive soil resistance to hydraulic erosion’: we will return to its defi nition in the section on set-tling velocity later. Thus, the equivalence between Equation 3 and Equations 4–7 can be clearly seen.

The fi nal statement by Smith et al. in this section of their discussion is in relation to the WEPP model, which they agree is based on a steady-state assumption. They make the assertion that because WEPP has been calibrated on an extensive dataset for effective rainfall and runoff on an event basis, it follows that ‘it has never been shown on a storm[-]total basis that the use of a dynamic sediment[-]routing solution results in an overall improvement on soil[-]loss predictions for a series of storms’. We would interpret this statement as refl ecting the limitations of the ‘state-of-the-art’, based on erosion models which have the inbuilt limitations that we critique in Wainwright et al. (2008a). Certainly it implies a fatalistic appreciation of the discipline, which suggests the authors believe that those involved in trying to understand intra-event processes are wasting their time. In part, this probably refl ects a different philosophy of why we develop models for understanding, rather than necessarily for prediction. We rehearse these arguments in detail elsewhere (Wainwright et al., 2009), so will not elaborate on them further here. However, it has been demonstrated that the temporal pattern of rainfall does affect soil loss (De Lima et al., 2003; Parsons and Stone 2006; Armstrong and Quinton, 2009) as well as runoff and fl ow hydraulics in general (e.g. Wainwright and Parsons, 2002) with consequent implications for processes of sediment detachment, transport and deposition. Understanding the controls of dynamic sediment routing must therefore be a component of improving event erosion estimates without the calibration that destroys the process basis of those predictions.

Relative Velocities of Sediment and Water

Smith et al. start the discussion of this point with a question – ‘Can Sediment move Slower than Water?’ – and then state

STANDING PROUD: A RESPONSE 1351


that ‘The general rebuttal to this assertion is simple’. They proceed to argue that it is obvious that sediment typically travels slower than the surrounding water. Not only is this not a rebuttal of the assertion/question that they pose, but also it is exactly the point we make about the limitations of existing erosion models.

It is hard to see how the defi nition of an ‘effective fl ow concentration’ is any different in essence from that in general use and indeed it is diffi cult to envisage any other way in which this parameter might be measured. We would dispute that ‘sediment concentration is of interest in catchment erosion modelling’. What is of interest is the amount of sediment eroded. Concentrations require the amount of sediment to be measured, either from a total outfl ow or from a sample. It is thus clear that sediment concentration is a derived rather than a primary variable relating to a process of interest. Its use provides no benefi t, and indeed introduces problems through spurious correlation by the presence of water-discharge terms on both sides of any analyses of the relationship between fl ow and sediment. Further, in cases where sediment velocity is lower than fl ow velocity (which we argue are predominant on hillslopes), when a concentration-based measure is subse-quently used to estimate erosion fl ux or yield by multiplying back through by the water discharge, the result will be an overestimate. Given the relative velocities discussed later, this overestimation is likely to be signifi cant and will typically lead to further, spurious calibration of a model. We simply fail to understand the fi xation with sediment concentration in the erosion literature. It seems to be based on nothing other than habit and the fact that water fl ow is easy to measure.

Smith et al. demonstrate that conditions may exist when sediment moves at the same velocity as the fl ow, and that other conditions may exist when there is no sediment move-ment and thus sediment velocity is zero. This argument is insuffi cient for a general relationship. The relationship may be demonstrated further by using evidence in the literature of particles moving slower than fl ow. For example, Farenhorst and Bryan (1995) show that particles moving in laminar and transitional fl ows on a 4·0-m long × 0·3-m wide fl ume at a slope of 0·035 m m−1 moved at a maximum rate of about 5% of the fl ow velocity. In these experiments, the bed included fi xed grain roughness but no form roughness, so that sediment velocities are likely to be relatively high when compared to more realistic surfaces. Parsons et al. (1998) used a similar experimental set-up with a 4·8-m long × 0·5-m wide fl ume with a similar bed at slopes of up to 0·17 m m−1. They found that even for the smallest particles observed (of 1·78 mm intermediate axis) on the steepest slope, typical particles moved at about 1% of the velocity of the water. As we noted in Wainwright et al. (2008a), sediment might be assumed to move at the same velocity as the fl ow when the sediment is moving in suspension, but even this relationship breaks down when high sediment concentrations are present (Abrahams and Atkinson, 1993; Li and Abrahams 1997).

Our argument about hyperconcentrated and débris fl ows is suggested as being specious by Smith et al. They note that ‘it is unlikely that any of the developers of the critiqued erosion models ever thought that their models would be used for hyperconcentrated or debris fl ows’. We did recognize this point in stating that ‘the range of sediment concentrations produced in most applications of erosion models is insuffi cient to require the development of this broader approach, espe-cially as it would require further modifi cations to the fl ow model’ (Wainwright et al., 2008a, p. 817). We argue that there is a much more important point to be made here regarding the issue of whether transport capacity is a useful concept (Wainwright et al., 2008a).

The Yalin (1972) Equation

The next element of the critique of Smith et al. also relates to the issue of transport capacity. Indeed, the discussion of the Yalin (1972) equation occurs only in the same paragraph with the clearly fl agged opening statement as is noted earlier. Our discussion is only in terms of the meaning(lessness) of the transport-capacity concept, and the EUROSEM, KINEROS2 and LISEM models are not even mentioned in this paragraph. The claim by Smith et al. that a particular transport-capacity equation is not critical is an odd one, as relatively low rain-drop-detachment rates will mean that the (TC – GF) term in Equation 3 (or equivalents in Equations 4–7) is large and thus a sensitive parameter. An implication of this statement is that transport capacity is not well constrained, which we would argue is a function of the conceptual diffi culties in its defi ni-tion. However, in contradistinction to their opening statement about being unaware of an equation developed and evaluated for very shallow fl ows, the papers by Abrahams (2003), Abrahams et al. (2001) and Li and Abrahams (1999) should be noted in this regard.

The reason we focus on the Yalin equation in this paragraph is simply that it has been more rigorously defi ned compared to the other transport-capacity equations in the literature, which tend to be of a highly empirical nature. We suggest that it is illogical to claim that a statement by Ferro (1998), which is a logical consequence of the Yalin equation, cannot be used as a critique of the Yalin equation itself. It certainly does not then support a priori the use of multiple particle sizes, which we are aware are used in KINEROS2 as stated. No further information has led us to doubt the validity of our original interpretation of transport capacity as a problematic variable. Smith et al. conclude their discussion of the point by stating that ‘the Yalin equation contains a threshold for transport capacity . . . but so does the model that they [Wainwright et al.] propose’. MAHLERAN contains no such threshold or term The whole purpose of our discussion on pp. 815–817 (Wainwright et al., 2008a) is to justify why MAHLERAN does not contain such a parameter given that its use is otherwise so well entrenched in the literature.

The Use of Settling Velocity

We have no problem with models that use settling velocity to describe the process of settling, notwithstanding the issues of producing closed forms of Stokes’ equation (e.g. Ferguson and Church, 2004) or of applications of the theory to particle mixes in fl ows (e.g. Allen, 1985). Smith et al. note that in an earlier paper ‘[i]t has been shown . . . that the settling velocity is theoretically proportional to k in the authors’ [Wainwright et al., 2008a] equation 5 [reproduced as Equation 3 earlier]’. It is informative to revisit the earlier paper to see exactly how this point was demonstrated. They fi rst defi ne ‘a concentra-tion, Cmx’ which uses ‘previously developed transport[-]capac-ity relations’ (Smith et al., 1995, p. 518). Then

Cmx is assumed to represent a state of dynamic equilib-rium between rate of erosion from a loose soil surface, eq, and rate of deposition of soil particles in suspension, ed. Deposition rate for any concentration C of particles with settling velocity vs is Cvs. Thus, at equilibrium, assuming a reversible process at the soil surface, for a given particle size,

e e C vq d mx s= = .



At any concentration, eh = eq − ed, so the equation for net erosion eh as a function of concentration C is

e bv C Ch s mx= −( )

where b accounts for cohesive soil resistance to hydraulic erosion: b is one for negative eh (deposition), and is a factor less than one for positive eh. [Smith et al. (1995, pp. 518–519; the second equation has the term en in the original, but this appears to be a straightforward typo-graphical error)]

This extract contradicts the claim of the Smith et al. critique that the ‘EUROSEM . . . [and] KINEROS2 . . . models are strictly dynamic, treating unsteady fl ow, and the only assump-tion is that at any location and time both processes of entrain-ment and deposition are occurring’ because of the explicit assumption of equilibrium conditions. Moreover, we suggest that it is based on a fundamentally incorrect assumption about reversibility of erosion and deposition. We can think of no process-based reason why erosion and deposition are revers-ible; they are clearly based on different controls. Making the assumption about reversibility – whether ‘imperfect’ or not, whatever this qualifi cation may mean – might provide a means of interpreting k in Equation 3, but it is a fl awed assumption and shows a lack of understanding of process in so doing. In other words, Smith et al. assume k = vs (i.e. b = 1) for deposition, which is reasonable if, and only if, the process of deposition is simple settling from suspension in a still fl uid. We point out both in the original paper and above that this assumption often does not hold for hillslope erosion because other modes of transport and conditions may be dominant. They also assume k = b vs (with b < 1) and so appear to believe erosion is a function of settling velocity, for which there is no physical justifi cation.

The Critique of MAHLERAN by Smith et al.

Smith et al. claim that ‘much of the workings of this model [MAHLERAN] are obscured in the description’. We focus on eight specifi c points that seem to be being made.

1. There is an implication that our defi nitions of terms are unclear and/or unusual. None of the terms used is novel and in our experience they are widely used elsewhere in the literature. Our use of concentrated rather than rill fl ow seems to have been especially picked out, yet again this term is commonly used (Morgan et al., 1998; Flanagan et al., 1995; Knapen et al., 2007; and 59 references in the ISI database where ‘concentrated fl ow’ and ‘rill’ occur together in the paper abstract since 1989), and the different thresh-olds based on fl ow Reynolds number, fl ow depth and fl ow shear velocity are clearly given in fi gure 4 of Wainwright et al. (2008a) and discussed in detail in the text in pp. 819–823.

2. Smith et al. have diffi culty appreciating that MAHLERAN would have transport by ‘bed load [sic] or in some other manner than suspension’. Our extensive discussion on pp. 814–817 (Wainwright et al., 2008a) provides the case for the impor-tance of distinguishing modes of transport in fl ows. This point further emphasizes the problems that have arisen from a blind following of the idea that transport is in suspension in the literature since Bennett (1974). We know of no other area of work in sediment transport where there is such dif-fi culty in accepting (or observing) that sediment also moves as bedload, be it in the form of sliding, rolling or saltation

(see Allen, 1985), or translating such an acceptance into process representation within a model.

3. There is apparently a lack of clarity in how transport dis-tances and virtual velocities are used in the model. Smith et al. coin the term ‘ “virtual” deposition’ which is odd since nowhere do we use or imply such a term. The use of transport distance to calculate amounts of deposition of material eroded at successive points along the fl ow path is clearly described. Specifi cally:

Deposition is modelled using the transport-distance approach. Given the estimate of the amount of erosion and knowledge of the distribution function of travel dis-tances of particles under specifi c transport mechanisms and fl ow conditions, the deposition rate can be calculated directly at each point along the transport pathway. In all cases at present, an exponential distribution function is assumed, both because this function captures the princi-pal characteristic of a declining probability of movement further from the source, and because insuffi cient data exist in order to evaluate whether other, more complex, distri-bution functions might better fi t actual distributions of transport distances. The simplicity of the exponential dis-tribution is complemented by its ease of parameterization, as it has a single parameter that relates directly to the mean or median observed travel distance. Thus, the parameter can also be interpreted in a physically mean-ingful way. (Wainwright et al., 2008a, p. 820)

Deposition is thus never ‘virtual’ but directly calculated: if one knows how far a particle will travel from its source, one knows exactly where it will be deposited. The use of the exponential function to describe the distribution func-tion of travel distances and thus of deposition is illustrated directly in fi gure 3 of the paper. There is also no mystery regarding how virtual velocities are used in the model. The way they are used to calculate sediment discharge is explicitly presented in equation 7 of the paper, and thus there is a direct link to the continuity equation underlying the model (equation 6 of the paper). The ways in which virtual velocity is calculated are presented directly as equa-tions 16–18 of the paper, depending on the specifi c mode of transport as defi ned by the fl ow conditions (and illus-trated by the fl owchart in fi gure 4 of the paper).

4. Smith et al. question the inclusion of the rainfall-energy term in the calculation of virtual velocities for unconcentrated fl ows. They ignore the fact that this term is included from theoretical development with empirical support (Parsons et al., 1998) and make two illogical extrapolations. These are fi rst that if rainfall ceases, unconcentrated fl ow virtual veloc-ities become zero ‘regardless of the value of fl ow stream power’. If the fl ow stream power were suffi ciently competent to maintain transport in such conditions, then the fl ow would not be unconcentrated by defi nition, and equation 17 of Wainwright et al. (2008a) would be used to calculate virtual velocity; in this equation, virtual velocity is only a function of (excess) fl ow stream power. The second extrapo-lation is a curious one regarding buoyant particles. Were one to wish to simulate buoyant particles (for example in the case of peat erosion: Evans and Warburton, 2001), then the limitations of existing parameterizations (as we discuss extensively in a broader context in Wainwright et al., 2008b, 2008c) would be apparent, and one would need to modify not only the calculation of virtual velocities under all fl ow conditions, but also the detachment and transport-distance components of the model.



5. Smith et al. seem surprised that as fl ow hydraulics change so that sediment transport is no longer by bedload but as suspension, the sediment particles will start to travel at a signifi cantly faster rate. While it is clear that the current simplifi cation in MAHLERAN in which this change is instan-taneous, rather than occurring over a period of accelera-tion, is unrealistic for a short period while the transition occurs, it probably has a small impact relative to the total amount of sediment movement taking place. Smith et al. also make their assertion in relation to a calculation based on D50 particles, implying that the calculated rates are then ‘applied to all particle sizes, however small’. This claim is not correct: the threshold stream power is calculated with equation 13 of Wainwright et al. (2008a), using Bagnold’s approach, which is a relative function of the actual particle size compared to the D50 size. In all cases in MAHLERAN, calculations are made as a direct function of the actual particle size under consideration. We do highlight diffi cul-ties with the existing literature on travel distances from existing sources (Wainwright et al., 2008a, pp. 821, 823; Wainwright et al., 2008c, p. 1124) and suggest that this is likely to arise from a reductionist approach to the erosion and transport problem.

6. Smith et al. note that ‘[a]nother conceptual weakness lies with the lack of any relationship for transport capacity’. As noted earlier, they previously state ‘the Yalin equation con-tains a threshold for transport capacity . . . but so does the model that they [Wainwright et al.] propose’, and so produce an apparent contradiction. They go on to assert that in MAHLERAN ‘as soon as water carrying sediment leaves an area having a source of sediment, into a lined channel for example, simulated deposition must occur, regardless of the slope or stream power of the water in that convey-ance’. This assertion is untrue in that during the simulation of a water and sediment fl ow entering a lined channel, the probability of deposition is governed by the fl ow condi-tions therein. If the stream power is suffi ciently high, bedload or suspended load transport will continue. Furthermore, if sediment is deposited on entry to the lined channel, it could be re-entrained by raindrop or fl ow detachment, depending on the conditions outlined in fi gure 4 of Wainwright et al. (2008), and continue to be transported according to the appropriate part of the algo-rithm as presented.

7. Smith et al. make a distinction about whether the raindrop-detachment model we use (after Quansah, 1981) can explain ‘dislodgement’ (a term we do not use) selectively rather than ‘settling’ (presumably relating to the ballistic trajectory of splashed particles which is not solely a settling process). It is unclear whether their argument is whether selective detachment does occur, or just whether it should be a function of three parameters [as demonstrated statisti-cally by Quansah (1981)] or any other number of param-eters. The former argument can be directly refuted from the literature. While it is indeed diffi cult to suggest that the direct observations of splash, with a maximum rate in the fi ne sand size range (Ellison, 1944; Parsons et al., 1992) convert directly into indications of selective detachment, the results of Poesen and Savat (1981) do indeed show minimum energies required for detachment fall in the same range [although they do assume that detachment and splash are directly correlated, which is not unreasonable as noted by Van Dijk et al. (2002) as long as the splash distance is known]. The results of De Ploey and Savat (1968) using radioactive tracers are consistent with this interpretation. Wainwright (1992) also demonstrated theo-retically for larger particles on a non-cohesive surface that

detachment should be a selective process. The latter argu-ment is slightly more diffi cult in that it has not, to our knowledge, been evaluated in detail either theoretically or empirically. For the non-cohesive case, Wainwright (1992) showed detachment to be a function of raindrop mass, slope angle, angle of incidence of the raindrop, particle mass and angle of repose of the sediment particle; detach-ment from a cohesive surface would add at least one further degree of freedom to the system. Thus, even if the arguments of Torri and Poesen (1992) regarding the relative importance of microtopography in overwhelming the slope effect are accepted, it is not unreasonable to have a rela-tionship for detachment with three free parameters. We note that the three parameters used are almost certainly themselves a function of other sources of variability such as those due to soil cohesion, surface sealing, aggregate stability, organic matter content and soil compaction (Wainwright et al., 2008a, p. 819, 2008c, p. 1124). On this point, Smith et al. seem to be in agreement with us based on their statement that ‘[w]e do not possess the scientifi c understanding of how the erodibility changes during a storm for different antecedent condition, soil types, and cover’. With regard to the specifi c issue of how one might determine the values of specifi c parameters, this is a straightforward question of experimental design (as dem-onstrated by Quansah, 1981). It only becomes a problem if one believes that the parameters should be derivable by optimization or calibration, which, as we note elsewhere, is not part of our modelling approach to achieve process-based understandings.

8. Smith et al. present a spurious argument about the number of parameters in MAHLERAN. Presumably, they raise this issue on one level because we claim MAHLERAN to be a parsimoni-ous model. As we discuss in Wainwright et al. (2009), parsimony is not a reference to an absolute number of parameters, but to not introducing more parameters than necessary to explain the functional controls on a system (i.e. avoiding overdetermination). It is a simple application of Occam’s razor, as employed for example in the discus-sion of the parameterization of detachment in the preced-ing paragraph. The models they are defending could also be considered to suffer from the same problem (or worse, as for example WEPP has c. 100 parameters, many of which have been shown to be redundant in terms of sig-nifi cant infl uence on model predictions: Brazier et al., 2000), but the broader issue is the idea that modelling is about ‘fi tting’ empirical results. Modelling for prediction is certainly a valid exercise, and given the current state-of-the-art, dynamic, distributed models often do not have the best predictive capacity. But this is certainly not a reason for not using such models, and indeed in the case in which we are interested – using modelling to develop and support our understanding of the soil-erosion process system – dynamic, process-based models are vital. These points are all discussed in detail in Wainwright et al. (2009). Furthermore, the detailed model-evaluation exercise that we undertook in Wainwright et al. (2008b, 2008c) shows quite clearly how the approach can be used both to improve models and model parameterizations whilst being openly critical of the model under development.

Kant, Cant or Can’t? A Discussion of ‘the Real Problems with Current Models’

Many of the issues discussed in the fi nal section of the critique by Smith et al. are ones with which we would agree. There



are indeed serious defi ciencies with the data underpinning erosion models and model parameterizations. We discuss these defi ciencies in detail ourselves in Wainwright et al. (2008c, pp. 1123–1125). However, as we also discuss there, many of those limitations have directly arisen as the result of the embedded false assumptions that have been made over the last decades in the soil-erosion literature. No data are independent of the (conceptual) models which led to their measurement. For this reason, it is absolutely fundamental to follow a conceptual model that is not demonstrably incorrect.

In their opening paragraph in this section, Smith et al. suggest that ‘the dynamic solution itself may be a source of potential model[-]prediction error associated with the intra-storm temporal variation in soil erodibility’. We agree with this statement, and note that associated errors and error propa-gation in the method are of course not limited to this source. What this statement does not imply, though, is that steady-state models of a dynamic process can ever be a better model of that dynamic process. It may be possible to calibrate such models to produce better predictions given other limitations, but that approach throws understanding to the wind. We argue strongly that a holistic approach (Wainwright et al., 2008c, pp. 1123–1126) is necessary for the improvement of soil-erosion (and other) models. Model development must go hand-in-hand with empirical understandings; one cannot suspend the former while waiting for the latter. In this respect, Smith et al. compare our papers with those of Grayson et al. (1992a, 1992b), to which Smith, Goodrich and others (Smith et al., 1994) wrote a rebuttal. These papers are landmarks in the understanding of process-based, distributed hydrological models. The underlying argument of Grayson et al. was to consider models as failing if they produced the ‘right answer’ (when compared to empirical data) for demonstrably wrong process-based reasons. Further, they noted:

the lack of full and frank discussions of a model’s capabil-ity and limitations and the reticence to publish poor results. This may ultimately diminish the opportunity to advance our understanding of natural processes because the people and institutions controlling research resources are given the impression that the answers are already known. (Grayson et al., 1992b, p. 2665)

One might add that they are given this impression by those who have their minds fi rmly wedded to existing models that they defend in the face of mounting evidence against them. Smith et al. (1994) concluded by stating that one should ‘get on with developing approaches to dealing with heterogeneity using the physical laws we know’ on the grounds that one can question whether ‘there [is] reason to expect that such equa-tions will ever exist’ (Smith et al., 1994, pp. 853–854), where ‘such equations’ relates to the alternatives to existing ‘physical laws’. Certainly had Darcy taken this viewpoint prior to 1856, St Venant prior to 1871, Richards prior to 1931, or Shields prior to 1936, this discussion would not be taking place now! Proscriptive statements about what cannot (or can) be done do not contribute to scientifi c advances.

The next two paragraphs of the Smith et al. critique are about scale dependency and implicit scale parameterizations of models. These paragraphs essentially reword our own cri-tique and highlighting of the problem (Wainwright et al., 2008c, pp. 1120–1123); Smith et al. say that we ‘clearly illus-trate’ the point. We considered it vital to highlight these issues very clearly in our paper precisely because they do tell us about limitations in understanding, and point to further work that is required to improve the situation. We believe this to be

an example of a ‘full and frank discussion’ of model limitations as suggested by Grayson et al. (1992b). Furthermore, the dis-cussion of Smith et al. suggests an inability to grasp the distinc-tion between point measurements of a continuous variable and a set of areally averaged ones. All models require this distinction to be addressed. In MAHLERAN, we attempt to do so explicitly, but as we have noted in Wainwright et al. (2008a, 2008b, 2008c, 2009) the approach is in need of further devel-opment based on more iterations between theory, experi-ments, observation and prediction. However, in developing their discussion, Smith et al. show another misreading of our paper. The incompatibility of the detachment and transport components of the model were demonstrated in relation to unconcentrated fl ows, and so cannot be related to the ability to resolve the presence or absence of rills within a cell. What is more, the incompatibility is demonstrated (Wainwright et al., 2008c, p. 1123) theoretically, in relation to homogeneous fl ows on a smooth, uniform slope. It therefore also holds independently of how point measurements are averaged across a cell.

The ‘real’ challenges identifi ed by Smith et al. seem to be largely related to developing empirical datasets and under-standings. Those that they identify are ones that we either address elsewhere, such as how to ‘characterize and/or parameterize the variability of the abiotic and biotic media over, and through which, the processes are occurring’ (see for example, Zhang et al., 1999; Zhang et al., 2002; Brazier et al., 2007; Müller et al., 2007, 2008; Parsons and Wainwright, 2000; Wainwright and Parsons, 2002; Okin et al., 2009; Turnbull et al., in press), or that we state specifi cally are not addressed at present in the parameterization of MAHLERAN, such as crusting (Wainwright et al., 2008c, p. 1115). In the latter case, their absence does not provide a valid criticism of the model structure or conceptual framework. A further example provided by Smith et al. – that of armouring of the surface from erosion – is actually represented in MAHLERAN, and has been demonstrated in our previous modelling of desert-pave-ment formation (Wainwright et al., 1995, 1999), as well as in other models (e.g. Sharmeen and Willgoose, 2007).

Critical evaluations of existing models have indeed been ‘patchy’ and avoided looking at internal variability, and the authors here are as guilty of this as any others. Indeed the Mati et al. (2006) paper cited by Smith et al., which is an applica-tion of EUROSEM, relies upon a standard calibration and vali-dation approach to ‘test’ the model, seemingly in ignorance of the previous work by Quinton (1994, 1997) within which he calls for erosion-model applications to be accompanied by uncertainty analysis, as ‘[n]ot doing so misleads the user into believing that the model output is more certain than is actually the case’ (Quinton, 1997, p. 115). It is interesting that the only positive experiences they cite other than our papers are in relation to the Rose model. We have addressed the evalua-tions of this model in detail in Wainwright et al. (2009). They also fail to cite the empirical, internal testing of the EUROSEM model by Parsons and Wainwright (2000). Comparison of their critique with our discussion of model evaluation (Wainwright et al., 2008c, pp. 1125–1126) also shows a sig-nifi cant difference in philosophical approach to model testing.

Smith et al. close with a number of points regarding MAHLERAN, in particular in relation to uncertainty, parameter-ization and its conceptual underpinnings. Nowhere do we state that MAHLERAN is not subject to uncertainty. We have worked in detail elsewhere on uncertainty and error propaga-tion (e.g. Zhang et al., 1999; Zhang et al., 2002; Brazier et al., 2000; Brazier et al., 2001), and we discuss uncertainty and error in MAHLERAN in Wainwright et al. (2008b, 2008c).



MAHLERAN does not have ‘numerous parameters which cannot be parameterised outside of the best controlled laboratory’. Even if some parameters do require laboratory experimenta-tion at present, this statement shows a clear misunderstanding of the rôle of laboratory experiments in scientifi c endeavour (see Hacking, 1983). The criticism also ignores the possibility that as certain parameters are considered to be important, measurement techniques are defi ned and elaborated to evalu-ate them. Advances in fi eld measurement due to technological progress have been signifi cant over the last few decades (e.g. Croft et al., 2009; Planchon et al., 2005). Smith et al. assert that MAHLERAN has ‘serious conceptual shortcomings’. It may not be perfect (as no model can be: Wainwright and Mulligan, 2003), but we do not believe that Smith et al. have demon-strated any such shortcomings in their critique. They say that we have ‘proved what was already known with prior models that they critique’. This statement not only seems to contradict their belief about hypotheses derived previously in their text from Kant, but also their critique is essentially an extended denial of this point. They close by noting that problems we identify with existing (prior) models also apply to MAHLERAN, citing an extract from the abstract of Wainwright et al. (2008c, p. 1113) in support of their case. In that article, we state that ‘[u]ntil there is a holistic understanding of these different com-ponents, the problems outlined here imply that it is not pos-sible to evaluate MAHLERAN (or indeed any other erosion model) fully’ (Wainwright et al., 2008c, p. 1126), and thus conclude exactly the same point. The point is essentially the key conclu-sion of the paper, and the reason for the rigorous testing of the model we carried out in this paper and in Wainwright et al. (2008b). However, we fi rmly believe that this understand-ing will be provided by a conceptual re-evaluation as well as improved datasets. We believe that the approach of Smith et al., which puts the focus on data collection, is bound to repeat the errors of the past in terms of affi rmation of the consequent and of the perils of induction (see Popper, 1972). We stand by our critique of existing models – including our own. It is now time to move the discussion on.

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Standing proud: a response to ‘Soil-erosion models: where do we really stand?’ by Smith et al.

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