3.1. Case Study Data
[17] The plot under study (Figure 1) is within the Pontbren experimental catchment in Powys, midWales, United Kingdom (see Marshall et al. [2009] for a location plan). The plot is part of a field used for sheep grazing and is agriculturally improved, meaning it has been underdrained by a network of tile drains and at various unknown dates has been plowed, fertilized, and reseeded with more productive grasses and clover. Soils are silty clay loams of the Cegin and Sannan series, with a distinct upper layer of approximately 30 cm depth overlying a relatively impermeable lower layer [Marshall et al., 2009]. For example, median sampled saturated hydraulic conductivities for the upper and lower layers were 1.67 and 0.018 m/d, respectively [Wheater et al., 2008]. Tile drains are located at around 750 mm beneath the surface. There is some evidence of mole activity that together with other bioturbation effects and soil structural change, could potentially create a macropore network. The climate is wet, with average annual rainfall in the field equal to 1449 mm (as measured between 1 April 2007 and 31 March 2009). Within the field is an instrumented plot (Figure 1) with a quite well defined but not isolated catchment area. The surface water catchment of the plot has an area of 0.0044 km^{2}, with average surface slope of 12.5%, while the tile drain network implies an approximate subsurface catchment area of 0.0036 km^{2}.
[18] The plot was monitored between July 2005 and July 2009, although there are significant gaps in the data series [McIntyre and Marshall, 2010]. Measurements included rainfall, overland flow, subsurface drain flow, soil water pressure at a network of sites at multiple depths, and groundwater levels [Marshall et al., 2009]. The rainfall is monitored at 10 min resolution by a tipping bucket gauge with a nearby storage gauge (there was no significant difference in volumes between the two). The surface flow is intercepted by a gutter at the bottom edge of the plot, which transmits flow to a 45° V notch weir box in series with a tipping bucket to verify measurements. The flow from the network of tile drains collects in one main tile drain, the outlet of which is connected to a 45° V notch weir box, which is also in series with a tipping bucket. The surface and drain flows are measured at 1 min intervals, and the average flows over 5 min intervals are logged. Review of the time series flow data (e.g., Figure 2) illustrates that the flow from the plot is dominated by subsurface drain flow, with surface flow relatively intermittent and shortlived. For example, from November 2006 to October 2007, 94% of the recorded flow volume was drain flow. Flow peaks are also usually dominated by drain flow, although in wet conditions the peak surface flow can exceed the peak drain flow. Both pathways exhibit relatively flashy responses, with sudden bursts of rainfall in wet periods causing the drain flow to peak typically within 40 min of peak rainfall and the surface flow within 20 min. The response of the groundwater is, in general, a slow seasonal signal, without notable response to individual wet periods, which is assumed to mean that little water infiltrates though the relatively impermeable subsoil.
[19] While we consider the flow data to be a relatively accurate record of what is collected in the weir boxes (relative, that is, to flow data typically used in catchmentscale rainfallflow studies), it cannot easily be argued that the measurements accurately represent the total plot outflow. First, it may be assumed that the surface flow collection gutter did not always intercept all the surface flow because of intermittent interference from animals, and during the largest recorded flow event (17–18 January 2007) the surface flow downpipe (connecting the gutter to the weir box) was observed to be partly blocked by storm debris. Second, the bounds of this plot were defined by topography and the identifiable subsurface drainage network, and there may have been significant unmeasured outflow particularly from throughflow. And third, although the water table variability is generally small and slow [Marshall et al., 2009], there is likely to be some unmeasured groundwater recharge. A water balance calculation for a winter period, assuming no net storage (as supported by tensiometer measurements) and that evaporation losses are equal to potential evaporation, suggests that 18% of the outflow is unmeasured. In summer, in periods of drain flow less than 0.02 L/s, some diurnal variations of up to approximately 0.001 L/s can be observed (e.g., Figure 6), with peaks at around 4:00 A.M. every day. The same signal appears in the soil water pressure measurements; hence, it is presumed to be related to the diurnal cycle of transpiration.
[20] The data period used for the DBM model identification is 10 November 2006 to 25 January 2007. This period is chosen because the relevant variables have been continuously monitored during this period and there are many events of varied magnitude and duration but without snowfall occurrence. Marshall et al. [2009] noted that large cracks appeared in the plot's soil during the summer of 2006 and that this affected the flow response for several months afterward. This will be considered when interpreting results. Although this winter period is described here as “wet,” it contained relatively dry spells, including a 15 day period with almost zero rainfall, and soil water pressure measurements show that the plot was not continually saturated [Marshall et al., 2009]. A second, drier period of 9 May 2007 to 30 July 2007 is used for validation. Rainfallflow data are shown in Figure 2 for the winter calibration period (see Figure 6 in section 3.7 for the summer calibration period).
3.2. DBM Model Identification With TimeInvariant Parameters
[21] The DBM methods were implemented using the discrete time transfer function identification modules within the CAPTAIN toolbox [Pedregal et al., 2007; Taylor et al., 2007]. Continuous time transfer function modeling is an alternative approach, also available within CAPTAIN, which can provide better defined and less biased parameter estimates in cases where fairly rapidly sampled data are available. This is possible in this case, but the discrete time results yielded similar inferences and are reported here. The sampling interval used is the smallest possible with the available rain gauge data, min. The simplified refined instrumental variable [see Young, 2008b] option of the RIVBJ routine within the CAPTAIN Toolbox is used for transfer function parameter estimation.
[22] The model identification strategy begins with a relatively straightforward and conventional DBM model identification exercise and then progressively explores avenues aimed at extracting more information from the data. Hence, to begin with, the aggregated surface and subsurface flow is used as the observed flow, and a linear flowrouting model is serially connected with a conventional nonlinear flow generation model, equation (7). Subsequently, models are fitted to the data from the two individual flow pathways to look for new information. Information about nonlinearity is then explored using statedependent parameter estimation. It is assumed that any losses from the plot, including possible unmeasured components of inflow or outflow discussed previously, and any error in the estimated catchment area, are accounted for implicitly in the parameters of the model. This may affect only the volume scaling, for example, parameter c in equation (7), although if inflow or outflow volumes are related to the wetness of the plot, it is expected to also affect the nonlinearity parameter .
[23] Note that the unit of flow used in all the modeling is mm/, where s. Therefore, the aggregated flow is not a single measurable variable; rather, it corresponds to Q_{s}/A_{s} × 600 + Q_{d}/A_{d} × 600, where Q is flow in L/s, A is catchment area in m^{2}, and the subscripts s and d refer to the surface and subsurface catchments, respectively. For the first stage of analysis, the observed surface and drain flow are aggregated in this way for the purpose of fitting the model. The nonlinearity parameter is optimized by maximizing R_{T}^{2} for each considered transfer function structure. This included all combinations of models within the ranges m = 1–3 and n = 1–3, giving nine model structures in total. The performance and optimized parameters of two of the tested structures are given in Table 1 (models 1 and 2). Model 2, with two parallel stores, gives the best R_{T}^{2} value (0.92), and this model is about the same as model 1, a single store, in terms of YIC (−11.6 compared to −11.9). YIC is a logarithmic measure, so that a more negative value, coupled with a comparatively high R_{T}^{2}, indicates a better identified model: for a more detailed explanation; see Young [1990] and Appendix 3 of Young [2001]. The identification of two parallel stores as a suitable model for this plot is neither new nor surprising: the same result was found by McIntyre and Marshall [2010] in their broader analysis of the Pontbren streamflow data, and it is known a priori from the measurements that there are dual flow pathways. Nevertheless, this result provides a benchmark to assess whether or not the model complexity can be developed systematically by applying the DBM analysis to the separated surface and drain flow components of flow.
Table 1. Optimized Performance and Parameter Values for Model Defined by Equation (7) Combined With Different Transfer Functions for the Winter PeriodModel  Modeled Variable^{a}  Structure n, m, δ  Flow Generation Parameters  Linear Routing Parameters  Nonlinear Routing Parameters  Time Lag δ (min)  Gain G  Number of Parameters N  Performance 

c  λ  T_{1} (h)  T_{2} (h)  split to T_{1}  β  α  YIC^{b}  R_{T}^{2} 


1  q = s + d  1, 1, 0  1.65  0.37  2.3       0.71  4  −11.9  0.87 
2  q = s + d  2, 2, 0  1.92  0.44  0.97  12.5  0.47     0.81  6  −11.6  0.92 
3  s  1, 1, 0  1.02  0.82  0.35       0.76  4  −12.4  0.91 
4  s  2, 2, 0  0.99  0.81  0.27  3.65  0.51     0.84  6  −8.0  0.92 
5  s  1, 1, 0^{c}  0.73  0.70     0.65  0.25   0.95  5   0.93 
6  s  1, 1, 0^{c}  0.68  0.68     0.69  0.28    4   0.93 
7  d  1, 1, 0  1.28  0.27  3.72       0.80  4  −12.2  0.88 
8  d  2, 2, 0  2.64  0.57  1.37  21.8  0.45     0.80  6  −10.4  0.91 
9  d  1, 1, 1^{c}  0.52  −0.06     0.54  1.04  10  1.07  6   0.96 
10  d  1, 1, 1^{c}  0.62      0.44  0.93  10   4   0.96 
[24] For each of the two flow components the same array of nine different transfer function structures was tested. At this stage, we maintained the nonlinear model of equation (7) for each flow component and, again, optimized for each tested linear transfer function. Selected results are included in Table 1. For the surface flow component, a single linear store with and (model 3 in Table 1) is selected as the best compromise between performance and parsimony, and for the drain flow, a parallel store model and (model 8) is preferred. Superimposing these two models gives the transfer function
where s is the measured surface flow, d is the measured drain flow, is the estimated effective rainfall going to surface flow, and is the estimated effective rainfall going to drain flow. Equation (10a) is equivalent to three parallel flow pathways [e.g., Young et al., 1997; Young, 2005, 2008a], but unlike in previous models, the split between the three pathways is not constant because of the presence of parallel nonlinear effective rainfall models (equations (10b) and (10c)). The aggregated surface and drain flow time series obtained from equation (10) is shown in Figure 2, along with aggregated observed flow, rainfall, and estimated effective rainfall. In Figure 2, the flow is log transformed to exaggerate the low flow errors. The R_{T}^{2} value for the aggregated flow is 0.93. Although this is only 0.01 more than the original two parallel store models and therefore, arguably, a small if not insignificant performance improvement, the more complex transfer function facilitates a simulation which can distinguish between surface and subsurface pathways. This distinction is potentially important, for example, if the DBM results are used to develop models for assessing impacts of soil and drainage management.
[25] A feature of the result in Figure 2 is that the two highest flow peaks (on the 13 December and the 18 January) are modeled well only because the estimated effective rainfall is significantly higher than (almost twice the volume of) the observed rainfall. This is mathematically possible because the constraint is not imposed upon the nonlinear model, equation (7). In general, the occurrence of > r_{k} may be considered physically reasonable because if rainfall measured from one or more rain gauges is treated as an estimate of catchment average rainfall, it is expected to contain significant spatial sampling errors. And, of course, the DBM model is inherently stochastic, so that the estimated uncertainty has to be taken into account in any evaluation of the model characteristics. In the case study, however, errors arising from sampling the catchment area are presumed to be small because the plot covers an area of only around 65 m × 65 m, with the rain gauge located almost centrally within it, and comparisons with storage gauge readings show no volume bias (although both may suffer from undercatch). Furthermore, closer examination of the hydrographs in Figure 2 (e.g., Figure 3) shows that the accumulated volume of flow over each of the two largest events (12–13 December and 17–18 January) is overestimated by the model despite the good fit to peaks, supporting the view that has been overestimated. Adding to concerns about the model of equation (10), Figure 2 illustrates persistent underestimation of low flows. Therefore, despite its impressive performance in terms of R_{T}^{2}, we do not accept the model of equation (10) on the grounds that the generated effective rainfall is not physically realistic and there is a low flow bias. Instead, we go on to more critically assess the nonlinear component of the model.
3.3. StateDependent Parameter Analysis for Drain Flow
[26] We proceed now to examine whether the nonlinear component of the drain flow model can be improved using statedependent parameter estimation. To facilitate clear signals of state dependence, the simplest of the applicable transfer function structures, equivalent to one single store, is preferred as a starting point [Young, 2003]. Adapting equation (2) to this form, we obtain
[27] Both the a_{1} and b_{0} parameters were simultaneously identified as statedependent parameters, using the estimation procedure described by Pedregal et al. [2007, chapter 5] and Young et al. [2001], assuming for now that . The default settings in the CAPTAIN Toolbox SDP function were used, except that the code for the number of iterations used to estimate the noise variance ratio smoothing parameter was set to −2 on the recommendation of Pedregal et al. [2007, p. 98] and the integrated random walk [see Pedregal et al. 2007, p. 19] was used to ensure a smooth estimate of the statedependent parameters. As well as using the observed drain flow (d) as the state on which the parameters are dependent, as specified in equation (11), the observed surface flow (s), the aggregated flow (s + d), and the plot average soil water tension (p) were also considered, with the aim of identifying the most representative catchment wetness index (y) for use in equation (6). Although state dependence was present in all cases (the results are shown in the auxiliary material in Figure S1), by far, the least noisy dependence was achieved using d as the state, and these results are presented in detail in Figure 4 and subsequent analysis.
[28] Figures 4a and 4b show the relationships b_{0}(d) and a_{1}(d) for drain flow in the winter period. Figure 4a supports the power law function, equation (7), previously assumed, although the result may also arise from dependency of a_{1} on the value of b_{0}, as explained below. The state dependence of a_{1} for drain flow is illustrated in Figure 4b, showing that the time constant T decreases as flow increases. This result may be interpreted in two ways: the state dependency of T illustrated by Figure 4b may be because a single drain flow store has been used to represent a dualpathway drain flow system (this is easily confirmed by applying the singlestore statedependent parameter analysis to the outputs of a parallel store model); alternatively, or additionally, the result may be interpreted as a kinematic routing effect.
3.4. Proposed Mechanistic Interpretations of the State Dependence
[29] The kinematic wave model [e.g., Beven, 2001, p. 177] requires that where v is celerity, q is flow, and and are parameters related to the physical properties of the flow path. Celerity may also be treated as equal to a representative length divided by a representative time of travel, which in this case, is the time constant T since there is no time delay ; hence, . Using the approximation to equation (3), parameter a_{1} can then itself be expressed in the form
where and have been combined into one parameter with units (mm/. The same form of a_{1}(d_{k}) is observed when fixing min; hence, (12) is valid for either or min. This theoretically derived expression is consistent with the form of the empirical relationship in Figure 4b, where < 1. Therefore, the nonlinear kinematic routing is accepted as a plausible model, and equation (11) becomes
[30] If the gain parameter G is forced to be constant, from equation (4), and the drain flow model would become
[31] An alternative explanation for the result in Figure 4a can now be seen: a_{1} may vary because of kinematic routing effects, and hence, b_{0} must vary in order to maintain a constant steady state gain between rainfall and flow. Therefore, the statedependent analysis results for the drain flow lead to three alternative propositions: (1) there are two parallel linear routing stores and nonlinear flow generation, (2) there is one nonlinear routing store as in equation (14) with linear flow generation, or (3) there is a mixture of the nonlinear flow generation and nonlinear routing which, assuming a single store, may be expressed as
3.5. Assessment of the Proposed Mechanistic Interpretations
[32] The first proposal, that there are two parallel linear routing stores and nonlinear flow generation, has already been assessed (the drain flow model in equation (10)) and was discounted. The second and third propositions are now explored by optimizing the parameters of the mixed model, equation (15). If the optimum value of is not significantly different from zero, then the second proposition is supported, while if it is significantly greater than zero, then the third proposition is supported. Prior to optimizing the parameters, the observed drain flow d_{k} in equation (15a) is replaced by simulated drain flow at the preceding time step . This prevents feedback effects that cause instability in the validation period results (described in section 3.8). Parameters G, , and are optimized to the drain flow observations using Matlab's nonlinear solver “fminsearch” with R_{T}^{2} as the criterion. The time delay parameter is also optimized by trial and error, and the c parameter is fixed, as in the previously optimized models, so that the volume of effective rainfall equals the volume of observed flow. The optimized parameter values are shown in Table 1 (model 9). The optimized value of (−0.06) implies that overall, there is less flow generated during wetter periods than in drier periods; however, this value is not considered to be significantly different from zero, and so our second proposition is supported: a linear drain flow generation model.
[33] Another feature of the optimized equation (15) is that G is close to unity. This is perhaps unsurprising, as the value of c has been fixed so that the volume of effective rainfall is equal to the value of observed flow, so that, in principle, there should be no need for a gain parameter. However, in practice, G ≈ 1 is a nontrivial result because it signifies that consistency between observed and simulated drain flow volumes coincides with the R_{T}^{2} optimal model (this consistency was not present when using linear routing, as indicated by the values of G in Table 1). Fixing and G = 1 and reoptimizing and yield the values in Table 1 (model 10) with associated R_{T}^{2} = 0.96. This is a remarkably good performance for a fourparameter (c, and ) model. Furthermore, unlike the majority of DBM rainfallflow models, this model avoids the use of observed flow as a surrogate measure of wetness and therefore may be used directly as a simulation model for prediction.
[34] However, model 10 is questionable for at least three reasons. First, recalling that the plot soil was not continually saturated, a linear flow generation model is contrary to the general experience that flow generation under unsaturated soil conditions is a nonlinear process. A second related issue is that the value is higher than expected a priori: calibrated values for catchmentscale models typically range from 0.2 to 0.7 [e.g., Wittenberg, 1993; Segond et al., 2007], and theoretical analysis for unconfined saturated subsurface flow gives a value of 0.5 [Moore and Bell, 2002]. These two issues point to the possibility of interaction between and within the optimization, so that is lower than expected and is higher than expected. However, plotting the response of R_{T}^{2} against the parameters of equation (15) shows unambiguously that value of is close to zero and that is close to 0.93. This plot is included in the auxiliary material in Figure S2.
[35] The remaining question about model 10 is whether a dualpathway drain flow model, where one or both pathways are nonlinear in the form of equation (15), might perform better. Indeed, when a statedependent analysis was applied to the disaggregated fast flow component of drain flow d_{f} (i.e., the observed drain flow minus the modeled slow drain flow from equation (10)), it did indicate nonlinearity of the form evident in Figure 4b. To answer the question, therefore, a nonlinear fast drain flow model was optimized conditional on the linear slow drain flow model specified within equation (10).
[36] With the R_{T}^{2} value remaining at 0.96, the addition of the slow flow store was not justified. Furthermore, a response surface analysis of the type shown in Figure S2 indicated poor identifiability, especially for the slow flow b_{0} parameter. On the other hand, an attraction of this dualpathway drain flow model is the more reasonable values (in terms of prior expectations) obtained for and (0.18 and 0.65, respectively). In other words, there seems to be a more reasonable split of nonlinearity between the flow generation component and the fast flowrouting component. While at this stage in the analysis a dualpathway model cannot be ruled out as an alternative description of the drain flow system, precedence is given to the more parsimonious singlestore model.
3.6. Surface Flow Model
[37] For the surface flow model, using surface flow as the state (rather than drain flow, aggregated flow, or soil water pressure) produced the clearest signals of state dependence in the parameters. Figure 4c shows a surprising reduction in b_{0} at the highest values of surface flow, and Figure 4d shows a_{1} increasing with increasing flow, particularly at the upper range of flows. This unexpected state dependence is largely because the surface runoff coefficient for the largest storm event (17–18 January 2007) was considerably less than those for the medium size events, and its time constant was larger. This is consistent with the field observation that the surface flow downpipe (see section 3.1) failed to collect and efficiently transmit the full surface flow on that occasion. If a nonlinear model of the form of equation (15) is pursued for the surface flow, despite the lack of clear evidence in Figure 4, then the optimized parameter values are as shown in Table 1 (model 5 when G is optimized and model 6 when G is fixed to 1.0).
[38] Model 6 has the same number of parameters as the linear singlestore model, while the R_{T}^{2} value has improved from 0.91 to 0.93. Also, the value is comparable with established values for overland flow; for example, the classic Chezy friction equation for surface flow would give . This supports an argument that the DBM results should be overruled in this case, given prior knowledge of process nonlinearity and field measurement problems. However, in keeping with the DBM ethos that adding complexity to the model on the basis of prior expectations should be avoided wherever possible, the linear surface flow routing in equation (10) is favored, although the nonlinear counterpart cannot be ruled out as an alternative plausible description.
3.7. Preferred Aggregated Flow Model
[39] Combining the preferred surface and subsurface models (models 3 and 10 in Table 1) leads to
[40] Equation (16) gives an R_{T}^{2} value for the aggregated flow of 0.97. To illustrate the performance in detail, four example events are included in Figure 3, comparing results to those obtained from the benchmark linear routing model, equation (10). The performance over the full winter time series is shown in the auxiliary material in Figure S3. These illustrations show that the overestimation of effective rainfall during the largest events, 12–13 December and 17–18 January, arising from the assumption of equation (10) that all the nonlinearity was in the flow generation has been solved. Figure S3 shows that the low flow bias has also been substantially solved, and the absence of a drain flow steady state gain (i.e., G = 1.0) signifies a more satisfactory mass balance than is implicit in equation (10). Adoption of the alternative more complex models (dualpathway nonlinear drain flow routing and/or nonlinear surface flow routing) produced the same or only marginally better performances.
[41] Some properties of the model residuals obtained from equation (16) are shown in Figure 5. Figures 5b and 5d show some bias toward negative residuals at the medium range of rainfall and flows. This indicates some systematic error in the drain flow recessions where, potentially, underestimation of medium flows is needed to achieve good peak flow and base flow performance. The bias persists if we use the dualpathway nonlinear drain flow routing, and so the bias cannot easily be interpreted as lack of flow pathways. Figures 5b, 5c, and 5d also show positive residuals during the large event of 17–18 January 2007, which was dominated by surface flow. These residuals are associated largely with timing errors (see Figure 3h); however, interpretation is complicated by the scope for observation error in surface flow during this event, as already noted. There is no evidence in Figures 3, 4, 5, or S3 that the swelling of the soil observed by Marshall et al. [2009] following the dry summer of 2006 has altered the flow response within this winter period.
3.8. A Summer Validation Period
[42] All the results presented so far are for the wet winter period 10 November 2006 to 25 January 2007. Insight into processes may be gained by closer analysis of the summer “validation” period of 9 May to 30 July. This period followed a dry April where observed flows fell to zero by the beginning of May (for this reason 1 May was not used as the start of the period; a zero initial condition is problematic because of the flow dependence of parameters). The dry spell continued until 15 June with only two significant flow events, and flows were sufficiently low that diurnal transpiration signals of around 0.001 L/s became visible, as previously noted. The subsequent wet period lasted until the end of July (Figure 6). Given the differences between the winter and summer conditions, this is a difficult validation period because some mechanisms operative over the summer period may not have been active over the winter, so the results reported in this section should be considered with this in mind.
[43] The model of equation (16), estimated using the winter period, performed well on the wet summer period (Figure 6, from 15 June), with an R_{T}^{2} value for the aggregated flow of 0.96, and if the parameters are reoptimized (as before, for drain and surface flow models separately), their values are close to those estimated for the winter period. However, equation (16) performs less well if the whole 2 month summer period is included (the entire period in Figure 6), with an R_{T}^{2} value for the aggregated flow of 0.83. The loss in performance lies in both surface and drain flow components. The error structure over the summer period is dominated by overestimation of flow events within and immediately after the dry weather and underestimation of similarly sized events in the subsequent wetter period. If the summer period is extended to include the dry weather in August, this pattern repeats.
[44] However, reoptimizing the parameters within the model structure of equation (16) (including in the drain flow model), giving an optimal R_{T}^{2} value of 0.91 for the aggregated flow for the period shown in Figure 6, does not recover the level of performance achieved in the wet periods. Plausible reasons for the apparent structural change in the model during and after the dry periods are the increased significance of the diurnal transpiration cycle, the presence of the strong nonlinearity and hysteresis usually found in dryingwetting cycles of clay rich soils [e.g., Topp, 1971], the presence of infiltration excess flow (although there is no observed evidence of this), changes in the properties of the grass, and/or changes in the soil structure, for example, a stronger role of macropores after dry periods. As noted, Marshall et al. [2009] reported large cracks appeared in the plot's soil during the dry summer of 2006, and these appeared to affect the groundwater dynamics. However, there was no visible change to the soil structure or unusual groundwater dynamics during the 2 month validation period. Furthermore, there was no evidence that the change in response was related to the available potential evaporation estimates. A statedependent parameter estimation exercise on the 2 month summer period produced results similar to those for the winter period, except with a very noisy relationship between b_{0} and flow for the drain flow model. This implies that influences other than rainfall and integrated antecedent wetness affected the variability of drain flow generation in the summer, potentially relating to the factors suggested above. Examination of runoff and soil moisture data over multiple experimental plots is currently underway to help resolve this.
3.9. Wetness Index Simulation
[45] One motivation for the DBM analysis is to assist in identification of a model that is capable of predicting flow response to scenarios of future rainfall inputs. The routing models identified here may be applied directly to prediction and so can the wet period drain flow generation model. The optimal surface flow and dry period drain flow generation models, however, used observed flow as a surrogate wetness index. A possible solution is to use the flow estimated at the previous time step; however, this led to unsatisfactory results for both drain and surface flows. Therefore, the value of the DBM analysis for making predictions partly rests in its potential for use in multistepahead forecasting when the data exhibits time delays and to inform hypotheses about models that can simulate catchment wetness.
[46] A range of simple models were tested, and the discussion here focuses on the one which gave the most satisfactory results. The DBM results in Figure 4 are closely associated with models that replace the observed flow in equation (7) with a simulated catchment wetness index [Young, 2003]. This includes the models described by Jakeman et al. [1990] and the subsequent versions described by Ye et al. [1995] and McIntyre and Al Qurashi [2009]. For the winter period, attempts to optimize these models led to the same conclusion as the DBM analysis: the generated flow is not significantly related to variability of wetness, and the linear effective rainfall model of equation (16b) is preferred. For the summer period, a threeparameter version of the catchment wetness index model (which assumes a firstorder loss from the soil store and that effective rainfall is proportional to wetness raised to a power ) combined with the nonlinear routing store in equation (16a) achieved a satisfactory performance over the full range of summer flows, with R_{T}^{2} = 0.95. Arguably, such simple drain flow generation models would not have been presumed to work well prior to a DBM analysis, and parameter values would have been significantly biased if the routing nonlinearity had not been separated out. However, given the difficulty of addressing the hysteretic and nonstationary behavior proposed in section 3.8, identifying a single predictive model to simulate flows over both the winter and summer periods is more challenging and has not been addressed in this work.
[47] Identifying a predictive simulation model of surface flow also remains a challenge The best performance for surface flow in the winter period was R_{T}^{2} = 0.79 using a fiveparameter catchment wetness index model, which accounted for losses using daily estimates of potential evaporation and included an infiltration excess parameter, along with a linear routing store structure (as in model 3 in Table 1). Using a nonlinear surfacerouting store (as in model 5) improved the result only slightly, with R_{T}^{2} = 0.81. In the summer period, the best model (the fiveparameter catchment wetness index model with a single linear routing store) achieved only R_{T}^{2} = 0.55 after optimization. It seems that the surface flow generation processes are not controlled solely by a lumped soil wetness index: the observed flow was by far the best wetness index, as suggested in the previously cited DBM rainfallflow modeling studies.
[48] It may be conceptually attractive to use the same wetness index for the generation of flow for all pathways, as is normal practice in conceptual modeling at both plot and catchment scales [e.g., Lee et al., 2005; Krueger et al., 2010]. The previous DBM results (Figure S1) indicate that this may not be effective for the case study plot, and this is confirmed by testing wetness index models and explicit soil moisture accounting models, including those proposed for this same plot prior to DBM analysis [Wheater et al., 2008]. While it would not be reasonable to suggest that the surface and subsurface flow processes are independent, the DBM results illustrate that the separation is stronger than presumed a priori.