# A Simple Approach to Account for Stage–Discharge Uncertainty in Hydrological Modelling

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}s

^{−1}for the outlet station and 1.1 m

^{3}s

^{−1}for the internal stations. In general, the consideration of the H-Q data uncertainty and the application of the MS-test resulted in remarkably better parameterisations of the model capable of simulating a particular peak event that otherwise was overestimated. The proposed model evaluation approach under discharge uncertainty is applicable to modelling conditions differing from the ones used in this study, as long as data uncertainty measures are available.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Study Site

^{2}. It is formed by the Kleine Gete (260 km

^{2}) and the Grote Gete (326 km

^{2}) sub-catchments. Elevation of the study area ranges from approximately 27 m in the north to 174 m in the south. Land use is mainly agricultural, with a predominance of pasture and cultivated fields while some local forested areas are also present. The catchment area is covered by nine soil units comprising loamy (predominant), sand-loamy, and clay soils, as well as soils with stony mixtures. Moderate humid conditions are observed in the catchment. The reader is referred for instance to [46] for additional details about the characteristics of the study site.

#### 2.2. Hydrometeorological Data

#### 2.3. The Hydrologic Code

_{act}; Kristensen and Jensen model), overland flow (two-dimensional, kinematic wave), channel flow (one dimensional, diffusive wave), flow in the unsaturated zone (one dimensional, Richards’ equation), flow in the saturated zone (two- or three-dimensional, Boussinesq equation), and exchange between aquifers and rivers. When using MIKE SHE in a catchment-scale framework, it is implicitly assumed that smaller scale equations (e.g., those that are embedded in the mathematical structure of MIKE SHE) are also valid on a larger (catchment) scale through the use of effective model parameters that implicitly perform an upscaling operation from field conditions. The MIKE SHE hydrological model uses a square-grid finite difference simulation scheme.

#### 2.4. Estimating the Uncertainty Attached to Stage–Discharge Data

^{3}T

^{−1}] and (H − H

_{0}) [L] are regression variables, a

_{i}is the ith regression coefficient [L

^{3−ei}T

^{−1}], e

_{i}is the ith regression exponent [––], i is an integer index [––], and l is an integer value defining the polynomial order [––]. H

_{0}is a lower stage benchmark [L], above which the rating curve is acceptably described by Equation (1). Q, (H − H

_{0}), and a

_{i}are positive real numbers. When e

_{i}is a positive integer number, then Equation (1) is the expression for the polynomial regression of order l. When e

_{i}adopts any positive real value, a

_{1}≠ 0.0 and a

_{i}= 0.0 for i ≠ 1, then Equation (1) is the expression for the power regression. Q was considered as the dependent variable, whilst (H − H

_{0}) was treated as the independent variable. Equation (1) was used in the current study by considering observational data originally utilised to derive the rating curve [47].

^{®}, Statistica

^{®}, StatGraphics

^{®}, and MS-Excel

^{®}. Furthermore, Practical Extraction and Report Language (PERL) subroutines were prepared for both processing information prior to the statistical analysis as well as for processing the results of the regression analyses and defining the time series of prediction bounds to be used in the hydrological modelling. Moreover, GNUPLOT

^{®}[53] was used in conjunction with PERL for automatic plotting purposes to display results.

#### 2.5. Initial Parameterisation of the Hydrological Model

_{p}) was estimated by combining crop coefficients (K

_{c}) and the crop reference (i.e., grass) potential evapotranspiration (ET

_{0}), which was in turn estimated by means of the modified Penman FAO-24 method [48]. Literature [48,49] values were adopted and assumed being constant for the parameters of the actual evapotranspiration (ET

_{act}) module of MIKE SHE [49].

^{1/3}s

^{−1}(i.e., about Manning’s n = 0.05 s m

^{−1/3}), which is a typical value for winding natural streams and channels with weeds and pools. For overland flow, Strickler values were defined in correspondence with vegetation cover and land use, ranging between 2 m

^{1/3}s

^{−1}(n = 0.5 s m

^{−1/3}, light underbrush) and 10 m

^{1/3}s

^{−1}(n = 0.1 s m

^{−1/3}, natural range) with most simulation cells having a value of 6 m

^{1/3}s

^{−1}(about n = 0.17 s m

^{−1/3}, dense grass/crops).

_{gr}= 1629) is larger than what has been used in some previous similar applications of MIKE SHE (i.e., [50]), where the number of modelled geological layers (6, after a simplification of the observed vertical succession of units [46]) created a complex arrangement of calculation grids leading to a large computational time for every model run. Consequently, a short six-month calibration period (1 March 1985–31 August 1985) was chosen preceded by a six-month spin-up period for attenuating the effects of the initial conditions. The evaluation period was fixed as (1 September 1985–1 March 1986). The simulations were made in a single run for every considered parameter set to ensure continuity of fluxes and internal state variables among the three periods. Furthermore, the initial conditions were the same for all simulations.

_{dr}) and is routed to streams with a velocity determined by the reciprocal time constant (T

_{dr}). This routing and time constant influences the peak of the hydrograph [46], while the drainage depth has more influence on its recession. Both drainage parameters were considered in model calibration.

_{x}and K

_{v}, the horizontal and vertical saturated hydraulic conductivities; and S

_{y}, the specific yield) of the Kw and Ln layers were also included in the calibration analysis since they likely influence the simulated hydrograph owing to river–aquifer interaction. Although the calibration of the hydrogeological parameters of all the geological units was considered through the piezometric information at different locations within the study catchment (Figure 1), this aspect is not covered in the current manuscript as we are exclusively focussing on river discharge simulation.

#### 2.6. Model Calibration, Validation and Sensitivity Analysis

_{O}(Ω

_{i}) is the prior likelihood distribution; L

_{O}(Ω

_{i}|O) is the likelihood measure, provided in the new observations (O) and computed in the newer period of observations; and L

_{p}(Ω

_{i}|O) is the posterior likelihood distribution. C

_{GL}is a scaling constant that enables the summation of the posterior likelihood measure of the behavioural simulations equal to one.

_{i}is the ith model prediction by the model; O

_{i}is the ith observation of interest (in the current study, derived from the rating curve); $\overline{O}$ is the mean value of the observations in the period of simulation; ${\sigma}_{obs}^{2}$ is the observed variance; and ${\sigma}_{i}^{2}$ is the error variance for the model. EF

_{2}varies between −∞ and 1.0; its optimal value is 1.0, whilst negative values indicate that the model performs worse than the mean value of the observations.

_{1}(the modified coefficient of efficiency) as a modification of the EF

_{2}index to reduce oversensitivity to the simulation of peak events by using the absolute value of the residual (i.e., res

_{i}= the distance between P

_{i}and O

_{i}), rather than its square. This second likelihood measure was formulated as:

_{2}or E

_{1}indexes is considered proportional to the linear fuzzy measure (Figure 3) and does not depend on residual values. For a given ith instantaneous T

_{i}(Figure 3a),

^{l}O

_{i}(lower than O

_{i}) and

^{u}O

_{i}(greater than O

_{i}) are the ith (data uncertainty) characteristic values associated with O

_{i}. Then, as shown in Figure 3b, any associated distribution density function may be “approximated” through a triangular “linear” fuzzy distribution (LFuzzy) on the basis of the parameters a

_{fz}and d

_{fz}, besides O

_{i},

^{l}O

_{i}, and

^{u}O

_{i}.

_{i}−

^{l}O

_{i}| and |O

_{i}−

^{u}O

_{i}| may be different, LFuzzy may approximate any skewness present in the original probability density function. Furthermore, since LFuzzy is used as an alternative likelihood measure, d

_{fz}adopts the value of 1.0 (while 0 < a

_{fz}< 1.0), which negates the concept of density function attached to LFuzzy, as the area under the curve is not equal to unity anymore. An arbitrary value of a

_{fz}= 0.1 was used throughout the study, implying a low likelihood value (i.e., a severe penalty) for P

_{i}departing from O

_{i}, even though not yet out of the data uncertainty band.

_{i}values corresponding to the different ith instantaneous T

_{i}values that are included in the simulation. Nevertheless, it should be noted that any other type of fuzzy function besides triangular may be used as an alternative likelihood measure. Pragmatically, the linear type is preferred here because it is easier to program.

_{2}and E

_{1}), or any other index that is based on the explicit consideration of residuals, can be used to evaluate model performance, although with no explicit consideration of the discharge data uncertainty. There is then the need of a (simple) procedure to include data uncertainty considerations in the evaluation of model performance using residuals. This could be accomplished, for instance, by using a residual reduction fraction (w

_{lk}) since, given the data uncertainty associated with the “observations”, the residual should not be any longer entirely based on the distance between an “observation” and the respective predicted value. Otherwise, we might refuse combinations of model structures and parameters that should be retained and accept others that perhaps are not superior. Ultimately, the aim of the modelling should be to obtain predictions that are within the data uncertainty band rather than obtaining predictions that “match” perfectly “observations” whose values are uncertain.

_{lk}linearly varying within the data uncertainty band. Outside this uncertainty band, however, the traditional definition of the residual was accepted. That is, outside of the data uncertainty band, w

_{lk}= 1.0 (Figure 3c). In this context, the relaxed version of the residual was defined as the product of (w

_{lk})

_{i}and res

_{i}. Then, any model quality index based on residuals, such as EF

_{2}or E

_{1}, or any other index which the modeller is familiar with, may be evaluated using this relaxed value of the residual to account for data uncertainty.

_{O}(Ω

_{i}) = 0.5 (i.e., characterising a behavioural simulation in the calibration period) for a given ith model run and the prediction ability of the model is also the same for the validation period (i.e., L

_{O}(Ω

_{i}|O) = 0.5), then the numerator of Equation (2) would equal to 0.5 × 0.5 = 0.25, implying a drastic reduction in the resulting likelihood distribution (L

_{p}(Ω

_{i}|O)). For this condition, the respective parameter set would not be retained in the analysis (notwithstanding the model has the same predictive ability in the evaluation period as compared to the calibration period). This, therefore, encouraged adopting a likelihood threshold equal to 0.25 for the evaluation period.

_{k}is the local likelihood measure for the kth data station, and wst

_{k}is a weighting factor that explains the contribution of LL

_{k}to estimate GL

_{j}. The term wst

_{k}, defined subjectively so that the denominator of Equation (5) equals unity, accounts for data accuracy (uncertainty) and importance of the discharge station in the context of the modelling objectives. Combining these aspects, wst

_{k}for m = 3 was (subjectively) given the values 0.15 (Grote Gete), 0.25 (Kleine Gete), and 0.60 (Gete).

## 3. Results

#### 3.1. Uncertainty Attached to the Stage–Discharge Data

^{2}with r

^{2}= 0.97; for the Kleine Gete station, the respective regression curve is Q = 3.1873(H − 0.18)

^{1.11208}with r

^{2}= 0.97.

^{3}s

^{−1}. For the Kleine Gete station, this width is approximately 1 m

^{3}s

^{−1}. As already stated, a similar and constant discharge uncertainty interval width (i.e., 1.1 m

^{3}s

^{−1}) was assumed for the Grote Gete station. The use of the discharge data derived from rating curves for assessing the model performance implies a significant modelling uncertainty that must be addressed at the moment of evaluating model predictions. Figure 2 shows the time evolution of the discharge prediction band throughout the modelling period for these two gauging stations. These hydrographs are plotted along with the areal hyetograph for a visual comparison of the time evolutions of the “observed” discharge and the catchment-areal rainfall.

#### 3.2. Model Calibration, Validation and Sensitivity Analysis

_{2}= 0.5) are shown in Figure 4 for the calibration period [1 March 1985–31 August 1985]. These were developed after conditioning analysis only on the stream discharges observed at the outlet of the study catchment, without including the analysis of the H-Q data uncertainty. In total, 4234 sets were considered behavioural upon the use of the EF

_{2}index. Out of the 15,000 parameter sets considered, only 47 produced models whose simulations failed due to instabilities.

_{2}index and without considering H-Q data uncertainty in the analysis (Figure 5a); using the EF

_{2}index and considering H-Q data uncertainty (Figure 5b); using the E

_{1}index and considering H-Q data uncertainty (Figure 5c); and using LFuzzy and considering H-Q data uncertainty (Figure 5d).

_{2}and E

_{1}model performance coefficients, suggesting that the E

_{1}coefficient is indeed less affected by the correct simulation of some peak flows; and (iii) the magnitudes of the values of the LFuzzy measure are similar to the ones obtained with the E

_{1}index (and the modified concept of modelling residual).

_{2}index (Figure 6a), using the E

_{1}index (Figure 6b), and using LFuzzy (Figure 6c).

_{2}and E

_{1}indices. Therefore, Figure 6 emphasises that the MS test is normally a more critical evaluation for distributed models. In addition, despite the significant low number of behavioural simulations, the parameter distributions after the MS validation (Figure 6) are practically the same as the ones obtained after traditional validation (Figure 5).

_{2}index (Figure 7c,d).

_{2}likelihood measure and the relaxation of the modelling residuals, shows a significant reduction in the width of the prediction band, after the application of Equation (2).

_{2}and the relaxation of the concept of modelling residuals illustrates that the few remaining behavioural parameterisations of the MIKE SHE structure (i.e., Figure 6a) simulated this peak event without losing that much simulation precision for the rest of events. This is true even though these remaining model parameterisations (26 when EF

_{2}was used or 14 when E

_{1}was used) have a tendency to over-predict low flows.

_{2}likelihood measure and (Figure 8c,d) using the E

_{1}likelihood measure.

_{2}or E

_{1}indices by relaxing the traditional concept of modelling residuals to account for H-Q data uncertainty contributed to a better identification of the MIKE SHE parameterisations that can simulate the peak discharge in January 1986 (Figure 8a,c). Namely, once Equation (2) was applied, the likelihood distributions (Figure 8b,d) were significantly modified on 25 January 1986, such that the range of the respective (discharge) prediction band was markedly reduced. For instance, Figure 8b, based on EF

_{2}, depicts that the prediction range for the prior likelihood distribution, i.e., [~7.0, ~21.0] m

^{3}s

^{−1}, was drastically reduced to [~7.0, ~12.2] m

^{3}s

^{−1}for the posterior distribution.

_{1}measure and the posterior likelihood distribution after conditioning based on observed streamflow at the three gauging stations and considering the H-Q data uncertainty for the Gete station (Figure 9a); the Grote Gete station (Figure 9b); and the Kleine Gete station (Figure 9c). The results depicted in Figure 9 are similar to those obtained on the basis of the EF

_{2}index. Figure 9b illustrates that the parameterised MIKE SHE structure had higher difficulties to model the discharge at the Grote Gete station, which is located in the mid-part of the study catchment (Figure 1). This difficulty was seen particularly for the peak recorded on 25 January 1986, but also the antecedent base flows. The peak flow was simulated much better for the Kleine Gete station and (as already discussed) for the Gete station. Since both stations are near to each other (Figure 1), the better simulation recorded for the Kleine Gete station compensated the overestimations recorded for the upper Grote Gete station.

## 4. Discussion

_{2}index (sometimes termed as NSE in related research). In addition, GLUE does not need to be strictly Bayesian-type in nature [54].

_{2}was used or 14 when E

_{1}was used, were left after the Bayesian-type update of the prior likelihood distribution.

## 5. Conclusions

_{2}, and its alternative E

_{1}, proved to be more effective than the use of a linear fuzzy measure of likelihood. Of course, this has significance specific to coping with data uncertainties when selecting the most appropriate model parameterisations under the current discharge data uncertainty constraints.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spatial distribution in the study site of the stream gauging stations and wells used in model calibration and evaluation (after [48]).

**Figure 2.**Rating curves and associated 90% prediction intervals and estimated hydrograph and associated data uncertainty band for (

**a**) the Gete station; and (

**b**) the Kleine Gete station.

**Figure 3.**Schematic description of the linear approach that was used for defining (

**a**,

**b**) the linear fuzzy (likelihood) measure (LFuzzy) and (

**c**) the residual reduction fraction (w

_{lk}).

**Figure 5.**Scatter plots of behavioural parameter sets (likelihood target was 0.25), for three of the inspected parameters, in the validation period (after the Bayesian-type update of likelihood measures) and after conditioning based on observed streamflow at the outlet of the catchment: (

**a**) using the EF

_{2}index and without considering H-Q data uncertainty in the analysis; (

**b**) using the EF

_{2}index and considering H-Q data uncertainty in the analysis; (

**c**) using the E

_{1}index and considering H-Q data uncertainty in the analysis; and (

**d**) using the linear fuzzy likelihood measure (LFuzzy) and considering H-Q data uncertainty in the analysis.

**Figure 6.**Scatter plots of behavioural parameter sets (likelihood target was 0.25), for three of the inspected parameters, in the validation period [1 September 1985–1 March 1986], after conditioning based on observed streamflow at the outlet of the catchment and at two internal locations (i.e., a multi-site (MS) validation test) and considering the H-Q data uncertainty: (

**a**) using the EF

_{2}index; (

**b**) using the E

_{1}index; and (

**c**) using the linear fuzzy likelihood measure (LFuzzy). “G” denotes that the likelihood measure is “global” (MS test).

**Figure 7.**Ninety percent streamflow prediction limits (the Gete station) in the validation period [1 September 1985–1 March 1986], using both prior (likelihood target was 0.50) and posterior (likelihood target was 0.25) likelihood distributions, after conditioning based on observed streamflow at the three study gauging stations (i.e., the multi-site test) and considering the H-Q data uncertainty: (

**a**,

**b**) using the LFuzzy likelihood measure and (

**c**,

**d**) using the EF

_{2}index.

**Figure 8.**Ninety percent streamflow prediction limits for the Gete gauging station (posterior likelihood distribution), in the validation period [1 September 1985–1 March 1986], after conditioning based on observed stream flow at the three study gauging stations (i.e., the multi-site test) and the H-Q data uncertainty; and respective likelihood cumulative prior and posterior distributions on 25 January 1986, (

**a**,

**b**) using the EF

_{2}likelihood measure and (

**c**,

**d**) the E

_{1}likelihood measure.

**Figure 9.**Ninety percent streamflow prediction limits in the validation period [1 September 1985–1 March 1986] using the E

_{1}likelihood measure and the posterior likelihood distribution, after conditioning based on observed streamflow at the three study gauging stations (i.e., the multi-site test) and the H-Q data uncertainty for: (

**a**) the Gete station; (

**b**) G the rote Gete station; and (

**c**) the Kleine Gete station.

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**MDPI and ACS Style**

Vázquez, R.F.; Hampel, H.
A Simple Approach to Account for Stage–Discharge Uncertainty in Hydrological Modelling. *Water* **2022**, *14*, 1045.
https://doi.org/10.3390/w14071045

**AMA Style**

Vázquez RF, Hampel H.
A Simple Approach to Account for Stage–Discharge Uncertainty in Hydrological Modelling. *Water*. 2022; 14(7):1045.
https://doi.org/10.3390/w14071045

**Chicago/Turabian Style**

Vázquez, Raúl F., and Henrietta Hampel.
2022. "A Simple Approach to Account for Stage–Discharge Uncertainty in Hydrological Modelling" *Water* 14, no. 7: 1045.
https://doi.org/10.3390/w14071045