Predicting discharge capacity of vegetated compound channels: uncertainty and identifiability of 1D process-based models

uncertainty and identifiability of 1D process-based models Adam Kiczko1, Kaisa Västilä2, 3, Adam Kozioł1, Janusz Kubrak1, Elżbieta Kubrak1, and Marcin Krukowski1 1Warsaw University of Life Sciences – SGGW, Institute of Environmental Engineering 2Department of Built Environment, Aalto University School of Engineering, Espoo, Finland 3Freshwater Centre, Finnish Environment Institute, Helsinki, Finland Correspondence: Adam Kiczko (adam_kiczko@sggw.pl)

. As shapes of parameter distributions are estimated through an inverse problem using the Bayes theorem, our results have the benefit that they are less affected by the initial assumption on the parameter variability compared to the "explicit" approaches.
The overall goal of this paper is to compare the uncertainty, parameter identifiability and physical interpretation of the parameters of discharge capacity methods characterized with different levels of parameterization. This work focuses on one-100 dimensional methods for compound channels with a significant share of the flow resistance generated by vegetation. The following methods were investigated: Manning based DCM, Pasche (Pasche and Rouvé, 1985) and Mertens (1989) methods designed for emergent rigid vegetation, and three versions of the two-layer model proposed by Luhar and Nepf (2013) as modified by Västilä and Järvelä (2018), designed for flexible submerged vegetation. All models were applied to vegetation conditions differing in relative submergence (covering both submerged and emergent conditions) and density, as motivated 105 by real cases where it is possible that e.g. a "rigid" vegetation model is applied for flexible vegetation because of lack of information on the vegetation properties. Parameter identification was conditioned on water depths instead of discharges to make the problem more similar to practical cases, like flood assessments, where a model outcome is usually the water level. It is out of the scope of the paper to provide a summary of all the available methods.

Parameter identification and uncertainty analysis
There are two approaches for parameter identification (Figure 1), of which the conservative approach is typically used. In a typical engineering task, the resistance term is usually poorly recognized and in the DCM Manning roughness coefficients are identified as an inverse solution, ensuring the best possible fit of the modeled and observed water levels or inundation extents. For the more process-based methods, the conservative approach considers most of parameters, such as the vegetation 115 properties, as an input. The model identification applies then only to several minor values, like surface roughness, as illustrated in Figure 1a. Instead of the conservative approach, this study considers additional inputs in terms of parameter identification, which is beneficial for being able to apply the advanced methods despite limited data (Figure 1b). For instance, in real applications detailed information on e.g. floodplain cover is not readily available, while for the DCM approaches it is sufficient, at most, to know if dense vegetation is present or not. Thus, the concept of this study is to consider all additional inputs of 120 the process-based methods as parameters that have to be identified. Such an approach was previously presented by Kiczko et al. (2017) and revealed that process-based methods of Pasche and Mertens can be applied in the same manner as DCM. The technical difference between different the methods is just the number of parameters. Identifiability of model parameters and the quality of the obtained solution determine then the applicability of a method.
In the practical application usually there is no strong theoretical grounds for the assumption of a priori parameter distribution 130 and the shape of the likelihood function. In that case, as in the present study, a uniform/rectangular distribution is usually used.
The likelihood function, necessary to transform a priori into a posteriori distribution, in the original GLUE approach is used along with selecting so-called behavioral simulations, above the given level of fit measures. This allows to adjust the variation of the estimated uncertainty. In the present study the Gauss shaped function was used, where output uncertainty depends on the scaling factor κ (Romanowicz and Beven, 2006): with n standing for the number of observation points used in the parameter identification, σ 2 variation of model residuals,Ĥ and H observed and calculated water levels, respectively.
Determining the span of the model uncertainty variance is always an important part of the uncertainty estimation. A good uncertainty model ensures that a desired number of observations is enclosed within uncertainty intervals (Blasone et al., 2008). 140 This is particularly important in the present study, where different methods are compared in respect of their uncertainty.
Confidence intervals should be sufficiently wide to cover the required number of observations but not wider. This condition can be fulfilled with a sufficient variability range of model parameters, specified as a priori distribution P (θ) and appropriate shape of the likelihood function L (Y /θ) depending here on the κ coefficient.
Parameter ranges can be usually found by trials-and-errors while the shape of the likelihood function should be determined 145 in respect of observations. In the present study, the shape coefficient κ (Equation 2) was computed on the basis of minimization task: where H q L i , H q U i denote lower and upper quantile (q L , q U ) of the calculated water levels from the a posteriori distribution 150 (Equation 1), obtained with the likelihood function (Equation 2); p stands for confidence interval, defined as: p = q U − q L . In the present study 95% confidence intervals (p = 0.95) were used, with q L = 0.025 and q U = 0.975. is a small number as a penalty for too wide confidence intervals of water levels H. The minimum of the function 3 should be the smallest value of κ for which the last term in Eq. 3 equals zero: This is true when exactly n observations fall within the confidence intervals. For p = 0.95 and relatively small observation sets of n ∼ 10 in the present study, minimum is found when all observations are enclosed by intervals. In such a case, the sum term is equal to 1 and the difference becomes negative. It should be noted, that for a poor model and/or inappropriate variability ranges of its parameters, such a solution might not exist. Therefore it was necessary to control the solution of the minimization in respect of Equation 5. If the constraint was not fulfilled, it was necessary to revise assumptions on the a priori parameter 160 distribution P (θ). For a poor model, a solution fulfilling the constraints of Eq. 5 might not exist within parameter ranges that can be interpreted in terms of their physical characteristics. In such case, the model was considered as unidentifiable, i.e., inadequate for a given data set.
It is acknowledged that the parameter identification and associated uncertainty depend on the size of the observation data set. To address this issue, the parameter identification (Eq. 1) was performed for a varying number of observation points 165 n = 1, . . . , N , where N stands for the total size of a data set. The calculations include all possible combinations of observations with the given n i.e.
N ! n! (N −n)! . The total number of all combinations is then 2 N − 1, excluding the empty set. Such an approach allows eliminating the effect of non-representative observation samples. The method was discussed previously by Kiczko et al. Remaining observation points N −n act as a verification set. In this analysis, both the proportion of verification points that falls within estimated confidence intervals and the width of confidence intervals are used as measures of model performance. The more narrow the confidence bands and the less observation points falling outside them, the better a model is. On the opposite, a less adequate model requires larger spread of the solution, to enclose observations, as it wrongly explains their variability.
Because the different combinations of n points resulted in multiple uncertainty estimates, the results were presented in terms 175 of statistical moments, as a function of n. For a detailed description of results box-plots were used, where the median is given as a horizontal line within a box, that spans over 25% and 75% quantile, whiskers indicate the result extent, excluding extreme values given with cross marks. In the DCM approach (Posey, 1967), the channel cross section is divided in flow zones of similar hydraulic conditions, typically the main channel and floodplain. The interactions between the zones of significantly different mean velocities are reproduced with a rough imaginary wall, applied to the zone with the higher velocity, i.e. the main channel. In the present study, the roughness of the interface was assumed to equal the roughness of the channel banks next to the interface. Parameters of the method are the roughness coefficients for each flow zone. In the present study, DCM was based on the Manning formula, with 185 the common approach of having separate Manning coefficients for the main channel (n c ), and left (n L ) and right floodplain (n R ).

Pasche and Mertens methods
A brief concept of the Pasche method is provided by Pasche and Rouvé (1985) and a detailed description of the algorithm used herein is provided in Kozioł et al. (2004). The model describes the discharge capacity of the compound cross section with rigid 190 vegetation, derived for steady flow conditions. Similarly to DCM, the model divides the compound cross-section into regions of the main channel and floodplains, dominated by bottom and vegetation roughness, respectively. It accounts additionally for the transition region between these two main zones. As in the DCM, the interactions between the main channel and floodplains are modeled using an imaginary rough wall. For the resistance the Darcy-Weisbach formula is used.
The Darcy-Weisbach friction coefficients are determined using a set of semi-empirical equations for each zone and the 195 imaginary wall, including transitional regions. The method explains the extent of the transition region within the vegetated region, affected by the higher flow velocity of the unvegetated main channel. The flow in the main channel depends on the apparent resistance of the imaginary wall. There is no general expression for the span of the transition region in the main channel, and it has to be established for each case.
Velocities in the flow zones and transitional regions are interrelated by the apparent resistance. Equations describing these 200 dependencies have an implicit form that requires iterative methods for solving, so that the Pasche method has a very complex numerical solution. Mertens (1989) attempted to improve the numerical efficiency of the Pasche concept by simplifying most of the demanding implicit formulas to less accurate but explicit ones, reducing the number of terms requiring iterative numerical solving.
In the Pasche and Mertens methods, a detailed parametrization of the channel, including plant properties, surface roughness 205 and the extent of the interaction zone in the main channel, is used. Assuming that the modeler has only knowledge on the geometry of the cross-section, the following parameters have to be identified: a x , a y , longitudinal and horizontal spacing of plant stems; d p average diameter of the stems; k f , k c roughness height of the floodplain and the main channel bed; b III /B c ratio of the interaction region width in the main channel (b III ) to the main channel width (B c ). Assuming that the channel is symmetric, the total number of parameters is six. Modeling different properties of vegetation on left (subscript L) and right 210 (subscript R) floodplains (a x,L :a x,R , a z,L :a z,R , d p,L :d P,R , k f,L :k f,R ) increases the number of parameters up to ten.

Generalized and Simplified Two-Layer Model
In the present study, the two layer model of Luhar and Nepf (2013), generalized by Västilä and Järvelä (2018) is considered as the state-of-art approach for submerged vegetation. This Generalized Two-Layer Model (GTLM) is based on the momentum balance with drag coefficients at the interfaces between vegetated and unvegetated areas of the channel cross section. General-215 ization proposed to the original model (Luhar and Nepf, 2013) by Västilä and Järvelä (2018) consists in replacing the channel width by the wetted perimeter (P ) and water depth by the hydraulic radius (R).
The channel discharge capacity is computed on the basis of equations for mean velocities in the unvegetated (u 0 ) and vegetated (u v ) parts of the cross section (Västilä and Järvelä, 2018): where g is the gravitational constant, S energy slope, u * 0 = u0 (gSR) 1/2 dimensionless velocity in unvegetated zone, C * the drag coefficient for shear stresses at the channel bed and at the interface between vegetated and unvegetated zones, L b and L v wetted lengths of the unvegetated channel margin and of the interface between vegetated and vegetated zones, respectively. B X denotes the vegetative blockage factor in the cross section, defined as the vegetated flow area divided by a total flow area.

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Physically, the drag coefficients for bed and the vegetation zone interface may take separate values. Following Luhar and Nepf (2013); Västilä and Järvelä (2018), it was herein assumed that the same value of C * can be used for both regions.
C d a is the vegetative drag per unit water volume, expressed conventionally as the product of a drag coefficient C d and the frontal projected plant area per unit water volume a, assuming that plants are rigid simple-shaped objects. To account for the presence of foliage and the flexibility of the plants inducing bending and streamlining, the vegetative drag per unit water volume can be parameterized as (Västilä and Järvelä, 2018) where u C is a characteristic approach velocity, taken here as equal to the velocity in a vegetation layer: u C ≈ u v . A S denotes total frontal projected areas of the plant stems and A L the total one sided leaf area per unit ground area A B . C D X ,S and C D X ,F represent constant coefficients for the drag of stems and foliage, respectively. The effect of streamlining and reconfiguration on 235 the drag is described using exponents χ S and χ F , for stems and foliage, respectively. u X,F and u X,S are reference velocities needed for determining the drag and reconfiguration coefficients.
Equations 6 and 8 implicitly depend on each other and require numerical solving. In the conservative approach vegetation parameters have to be known (Figure 1 a). The blockage factor B X requires knowledge on the vegetation distribution and/or height in the cross section. A S A B and A L A B ratios characterizing the plant structure can be measured or typical values for a certain 240 plant communities can be adopted. Drag coefficients C D X ,S , C D X ,F and reconfiguration exponents χ S and χ F , along with their reference velocities (u X,F and u X,S ), are plant species or plant type-specific factors and can be determined on the basis of laboratory measurements. Their values have been published for common plant species (Västilä and Järvelä, 2014;Västilä and Järvelä, 2018).
For channel flows with dense vegetation for which over 80 percent of the discharge is conveyed in the unvegetated regions, 245 the GTLM approach can be simplified by assuming that discharge in the vegetation layer is negligible with respect to the total discharge: u v ≈ 0 m/s (Luhar and Nepf, 2013;Västilä et al., 2016). The remaining Equation 6 does not require numerical solving. In the present study the above approach is referred as Simplified Two-Layer Model (STLM). By neglecting the Equation 7, the STLM requires five and GTLM nine parameters.
Parameters of GTLM and STLM, resulting from Equation 6 are the drag coefficient for shear stresses C * and Blockage It should be noted, that by parameterizing the Blockage Factor, the parameter identification task is much more complicated than in the conventional approaches. In the DCM the vegetation extent is equivalent to the division into main channel and 260 floodplains, which is known on the basis of the cross sectional geometry. Here, for GTLM and STLM it was considered as a part of the parameter identification problem.

Practical Two-Layer Model
Luhar and Nepf (2013) proposed a formula for the Manning coefficient n: where h stands for the vegetation height and K = 1 m 1/3 s −1 to ensure correct dimensions of the equation. The formula is derived for shallow channels, lined with vegetation, where the blockage factor was approximated as B X ≈ h H . In the presented form of the equation (9), following Västilä and Järvelä (2014), the water depth H was replaced with the hydraulic radius R.
The Luhar and Nepf (2013) formula 9 has a convenient form to be easily applied in practical cases, where usually the Manning equation is used. In the present study this approach is named Practical Two-Layer Model (PTLM) and applied as a 270 three parameter one, with the drag coefficient C * , average vegetation height h in the cross section and C D a.

Case studies
The analyses were conducted for a flume data set (Koziol, 2010) and a field data set (Västilä et al., 2016) collected from vegetated compound channels, interpreted herein as 5 distinct case studies, as detailed below. To our knowledge, the field cases are one of the most thorough characterizations on the dependency between vegetation properties and discharge capacity Experiments were performed for steady and uniform flow conditions. The water surface was kept parallel to the bed using a 290 weir localized at the flume outflow. Water discharge was measured using a circular weir and water levels were recorded in the middle of the channel.

Ritobacken field experiment
The field data was obtained from an 11 m wide compound channel, Ritobacken Brook (Finland, Figure 3 The Manning coefficient of the narrow main channel as obtained from highest flows not inundating the floodplain was 305 n = 0.08 − 0.12 m −1/3 s due to irregular main channel geometry, woody debris and some aquatic vegetation. For Autumn 2011 Västilä and Järvelä (2014) estimated values of plant drag coefficients and reconfiguration exponents: C dx,F = 0.14, The discharge capacity at different flow conditions was obtained from water level data recorded at 5-15 min intervals with pressure transducers at the upstream and downstream ends of a 190 m long test reach. The discharge was obtained from a rating  The size of the Monte Carlo sample (n mc ) was determined in each case by trial and error, to satisfy the convergence of the 315 solution. In a similar way, the ranges of parameters for the a priori distributions P (θ) (Equation 1) were determined. It was done in respect of the parameter physical variability, by ensuring that observations will be enclosed by confidence intervals (Equation 5). Parameter bands with n mc Monte Carlo sample sizes are provided in Tables 1 and 2, separately for flume and Ritobacken field experiments. Parameter ranges were often defined outside bands expected in the nature, to allow fulfilling the constrain 5 by as many as possible models. For flume data sets calculations were performed for a symmetric channel, which 320 allowed to reduce the number of parameters, as the same values were used for the left and right floodplain.
The numerical results were analyzed from three perspectives: (1)  In this example, the parameters for discharge curves were identified at low flows, while the verification was conducted for high flows, which represents the common practical way of using hydraulic models to assess flood hazard at flows higher than the ones the models were calibrated with. In terms of parameter identification results are considered as successful, On the basis of the rating curves computed for each combination of n observation points, it is possible to analyze the estimated average widths of confidence intervals as a function of observation points used in the identification. In the present study, the confidence widths were provided in relative sizes as W , normalized by the median of the probabilistic solution H M and then averaged over computation points, corresponding to all n observation points: where H q L and H q U stands for the estimates of lower and upper confidence intervals for calculated water level.  Chosen results on the influence of the number of identification datapoints on the widths of the confidence intervals and the ratio of verification points included within the intervals are provided in Figures 5-7. In Figure 5 for GTLM applied for Ritobacken case study for Autumn 2011 and also the Pasche model used for the flume data set in case 1 it can be noticed 355 that: (1) the relative confidence interval widths (5a, 6a) are high for a small n as a result of the parameter equifinality; (2) with additional data points, the solution converges by reducing the span of intervals but also its variability due to different combination of observation points; (3)

Model identifiability
The model identifiability is understood here as the ability to determine the parameter a posteriori distribution that explains 365 the model uncertainty in relation to observations. This is satisfied by meeting the constraint given in Equation 5, as for cases presented in Figure 4.The criterion of Eq. 5 might be fulfilled even for a poor model by extending the parameter variability ranges (1 and 2), specified with a priori distribution P (θ). The only limitation could be the physical meaning of the parameters.  both flume cases. For flume experiments, the STLM likely did not work because the assumption that >80% of flow should be conveyed in the non-vegetated zones was not fulfill. The rest of the models, including DCM for all cases, were identifiable.

Widths of confidence intervals and quality of uncertainty estimation
To compare the performance of the applied identifiable discharge prediction methods, we show bar plots of average values observations from the verification set enclosed within these intervals informs how the estimated uncertainty is representative 400 for other data sets than these used for identification. A low ratio suggests that the probabilistic term incorrectly predicts the model uncertainty for the verification set. Therefore, narrow confidence intervals for small n numbers resulting in small ratios should be considered as unsuccessful, as the uncertainty analysis appears to be too optimistic. On the other hand, for larger n, good ratios might be obtained with very wide confidence intervals, indicating a poor model. The best solution is that one, which has the narrowest confidence intervals with satisfactory ratio of verification set enclosed within it. We interpret the results by 405 analyzing those both criteria together.

Flume data set, case 1
For the flume data in the case 1 (Figure 9), with rigid-high vegetation in floodplains and also channel banks, the best results were obtained with the Mertens method. It is characterized with the narrowest confidence intervals W , having a good predictive performance. Confidence intervals for n > 1 were below 5% and for n > 3 they already enclosed more than 50% of the 410 verification points. Almost similar performance was found for the DCM method, with slightly wider confidence intervals.   The PTLM had significantly higher ratios than DCM at n > 2, with GTLM falling between these. Among the two latter methods, GTLM had only slightly wider confidence intervals than DCM: 3% vs 2%. For PTLM, with similar share of verification points enclosed within confidence intervals, widths W were about 6%. At n = 1, GTLM had better performance than PTLM and DCM, with notably higher ratio of points enclosed. were about 5%, so just slightly higher than for PTLM.
For the Spring 2012 ( Figure 13), all methods have almost equal confidence widths and ratios of enclosed verification points.

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The overall measures are similar to those from Autumn 2011. At n > 5, STLM had a slightly higher ratio of verification data enclosed compared to the other methods. The confidence widths are about 3% and for n > 5 for all methods more than 70% of points fall within confidence intervals.

Physical interpretation of identified parameters
The obtained parameter values were compared with the measured ones for the two most complex models of Pasche and  and stem, which fell close to the values observed for willows and other woody species (e.g. Västilä and Järvelä, 2018). Wide ranges for the vegetation heights h results from interactions with l/L and from the model insensitivity, when vegetation exceed 480 the water level and there is no free flow zone above. The values in the distribution of the identified C * were notably larger than the experimentally derived C * value (∼ 0.034 − 0.08, Västilä et al. (2016)), which is compensated by the notably lower identified A L /A B and C D a compared to the measured vegetation densities at Ritobacken (a ≈ 10−25 for the grassy vegetation, Västilä et al. (2016)). This is another example of the parameter equifinality that can result if all the vegetation properties have to be identified because of lack of available measurement data.

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The comparison of real and measured values on vegetation extent for the Ritobacken case study was possible for the blockage factor B X . In the presented approach it probabilistic estimates can be calculated using values of h L , h R , l L /L L and l R /L R .
In Figure 16, the measured and the identified blockage factor B X of GTLM is given as a function of the water depth. It can be noted, that confidence intervals for the B X are wide. The observed values are shifted from the median of a probabilistic solution towards 0.6 quantile. A large spread of values for B X , with very small variation of water levels for that solution 490 (Figure 12), suggest a moderate model sensitivity on B X .
The present study is according to our knowledge the first one, where different discharge capacity methods were compared in the respect of their uncertainty, estimated along with model parameters, using probabilistic formulation of the problem of the parameter identification. It should be noted that noticeable focus was made to ensure that the uncertainty analysis was objective 495 and repeatable, which can be seen in the proposed technique for scaling the likelihood function. The novelty of the proposed approach includes the analysis of obtained confidence widths, together with the ratio of independent observations explained by them, with respect of the number of observations used in the model identification.
Our results show that the number of parameters is not a factor precluding the use of a given method for predicting the channel discharge. It was possible to identify a model with more than ten parameters (i.e. GTLM), almost as well as three parameter Both studies demonstrate however a characteristic stabilization of uncertainty estimates for larger sets of observation points 510 (Figure 7). Her and Chaubey (2015) investigated also effect of additional data points, referring to other model derivatives than the main output, such as information on the flow from sub-basins. For the present study, the analysis could be amended using e.g. observations of velocities in the channel and floodplains. However, such data were not available for all cases and were thus not included here.
The results confirm previous findings of ( Kiczko and Mirosław-Światek, 2018;Romanowicz and Kiczko, 515 2016), that for discharge formulas the probabilistic solution differs form the deterministic one. This is evident from Figure 4 for calculated rating curves or parameter distributions in Figure 14. This obvious behavior of nonlinear models highlights the needs for such uncertainty analyses.
The more complex, process-based methods were usually better than the classical DCM, having more narrow confidence intervals, enclosing larger ratios of observations, when applied to vegetative conditions they were developed for. This important indicate that the choice of the resistance formula is important for cases where vegetative resistance dominates. On the other hand, one-dimensional models may be more sensitive to uncertainty related to the identification of the resistance parameters 530 than are two-dimensional models.
The performance of a model depends on its adequacy for the given vegetative and flow conditions. For unsubmerged sparse rigid vegetation, the most reliable method was the Mertens model with mostly explicit formulas. Because of a simpler numerical form than in the Pasche method, the Mertens method was less vulnerable to numerical instabilities, which probably affected the outcomes of the Pasche uncertainty estimation. In the case of dense flexible vegetation typically observed on natural floodplains 535 (Figure 3), the most reliable performance with respect to uncertainty estimates was obtained with the two-layer approaches GTLM and PTLM that were performed well for both dense submerged and emergent vegetation (Figures 11-13).
The GTLM was in this paper amended with a vegetation parameterization (Eq. 8) that describes the influence of the plant streamlining and reconfiguration on flow resistance. The GTLM with (Eq. 8) performed particularly well when the vegetation was high (Figure 12), appearing to be the most reliable method for predicting the discharge capacity during the most critical 540 conditions when the vegetative flow resistance is high. The GTLM parameterized at low flows reliably predicted the water levels during high discharges, including the more rapid increase in discharge at water levels exceeding vegetation height (Figure4a).
Although Eq. 8 has been developed for woody vegetation, it was applicable to the predominantly grassed vegetation at the field site. Field surveys indicated that much of the plants consisted of a main stem and more flexible leaves, conceptionally similar in structure to foliated woody vegetation. Eq. 8 describes the drag from stem and leaves and allows to set different values for 545 the flexibility-induced reconfiguration for the stem and foliage.
It should be noted that the results for the DCM with constant values of the Manning coefficient were quite good except for flume case 2. In all cases it had worse performance than the process-based methods, but was applicable in all these cases. Based on the results, the process-based methods are expected to perform better than DCM when several important sources of flow resistance, such as rough floodplain surface and vegetative drag, are present.

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Despite the larger number of parameters, the process-based methods were less flexible than the Manning based DCM approach. Pasche and Mertens methods were only suitable for rigid unsubmerged vegetation, for which they were derived. The two-layer approaches GTML and PTML, although it was possible to identify them, had a very poor performance when applied to sparse emergent vegetation ( Figure 2). Further, our findings confirmed that the STLM is strict about the assumption of neg-  (Figure 14). In the case of two-layer approaches, the fit measures reveal a low sensitivity of GTLM to the blockage factor: B X has large variability, while variation of computed water levels was very small. The application of process-based methods with numerous parameters seems to be inseparably connected with the problem of the equifinality. Similar behavior was reported for the Shiono-Knight model by Knight et al. (2007). The parameter equifinality, as over-parametrization is a basic assumption of the probabilistic approach in the parameter identification Binley, 565 1992, 2014). Overall, the parameter identification is expected to result in more physically realistic values if at least some of the required vegetation properties or the channel bed roughness can be directly measured and used as the input (see Figure 1a).
Discharge formulas analyzed in the study are usually only a part of the one-dimensional model. The uncertainty of such models depends also on additional elements, like spatial variability of resistance and simplification of the channel geometry. It should be also noted, that the investigated cases had fairly regular cross-section and homogeneous vegetation. Therefore, care 570 should be taken when attempting to generalize the presented findings to all one-dimensional approaches. In complex real-world cases, it might be beneficial to include several discharge formulas through an ensemble approach, which is also used in other fields, such as climate modeling.

Conclusions
In this study, six methods for estimating the channel discharge capacity were analyzed in terms of their uncertainty, for two 575 experiments: a flume experiment with rigid submerged vegetation and a field experiment with flexible vegetation under both emergent and submerged conditions. The outcomes of the study are summarized as follows: 1. The numerical experiments showed that it is possible to identify parameters of process-based methods including a large number of parameters on the basis of the inverse problem with narrow uncertainty bands.
2. The number of parameters is not a factor determining the applicability of the method. It was possible to obtain similar 580 uncertainty estimates for models with both a low and a high number of parameters.
3. The uncertainty related to the parameter equifinality is noticeable only when a small number of observations is used in parameter identification.
4. The parameters obtained through the identification differ from their measured physical values, which results from the parameter equifinality. The equifinality does not, however, affect the uncertainty of a model.