Trade-off between number of constraints and primary-statement robustness in entropy models: the case of the open-channel velocity field.

In this research, the trade-off between the number of restrictions and the robustness of the primary formulation of entropy models was evaluated. The performance of six hydrodynamic models in open channels was assessed based on 1730 Laser-Doppler anemometry data. It was investigated whether it is better to use an entropy-based model with more restrictions and a weak primary formulation or a model with fewer restrictions, but with a strong formulation. In addition, it was also investigated whether the model performance improves with the insertion of restrictions. Three of the investigated models have a weak formulation (open-channel velocity field represented by Cartesian coordinates); while the other three models have a strong formulation, according to which isovels are represented by curvilinear coordinates. The results indicated that models with two restrictions performed better than those with one restriction, since the additional restriction includes information relevant to the system. Models with three restrictions perform worse than those with two restrictions, because the information lost due to the use of a numerical solution was more substantial than the information gained by the third restriction. In conclusion, a strong primary formulation brought more information to the system than the inclusion of a third constraint.


INTRODUCTION
Models are designed to explain physical processes, to which one may assign a probability of event occurrence, considering that each system state has a level of uncertainty. Welldesigned probabilistic models tend to enhance their capacity of representing reality because they consider the intrinsic uncertainties of the processes, which arise from several sources such as natural randomness, inaccuracy in data measurement, model structure imperfect parameterization, and others Gupta & Govindaraju (2019). Many water-related problems demand a probabilistic approach due to the considerable amount of uncertainty involved Mishra (2009), Cobo et al. (2017, e.g., rainfall occurrence, magnitude and intensity Mélèse et al. (2018), basin flow and sediment Shrestha et al. (2016), and hydraulics Tapoglou et al. (2019). Shannon (1948) investigated the information content and its relation to uncertainty measures while proposing uncertainty quantification, the so-called Shannon informational entropy, or simply Shannon entropy. Jaynes (1957a, b) physically formulated the informational principle of maximum entropy (PME) using Shannon entropy, which maximizes uncertainty under the given constraints and, thus, avoids the use of unproven assumptions. As a result, the probability density function associated to a researched process can be obtained by maximizing the constrained entropy function and using the variational calculus and the Lagrange multipliers method.
Therefore, the informational principle of maximum entropy Shannon (1948), Jaynes (1957a, b), Shore & Johnson (1980) provides an adequate approach for introducing probability into complex hydrodynamic problems such as modeling the velocity field in open channels. It has been successfully applied to several fields of Hydraulic and Environmental Engineering Harmancioglu & Singh (1998), Singh (2013), (2014), Ardiclioglu et al (2005). Other researches based on PME yielded encouraging results in areas such as water resources Cheng et al.  Moramarco et al. (2013), and open-channel hydrodynamics Chiu (1987Chiu ( ), (1988Chiu ( ), (1989Chiu ( ), (1991, Chiu et al. (2005), Barbé et al. (1991), de Araújo & Chaudhry (1998 Literature presents several robust, physically-based hydrodynamic models, such as Shiono-Knight Shiono & Knight (1991), Knight (2013, MIKE-11 DHI (1992), (2017), River2D Steffler & Blackburn (2002), Beakes et al. (2014), CE-QUAL-W2 Cole & Wells (2006), and Environmental Fluid Dynamics Code Tetra Tech (2007), among others. These models use complex spatially-distributed systems of equations that encompass the principles of mass, energy, and momentum conservation; as well as the effects of turbulence. Hydrodynamic models, such as the aforementioned ones, are able to solve complex problems Beakes et al. (2014), Knight (2013), Torres-Bejarano et al. (2015), Thanh et al. (2020), but demand a large number of parameters, which are often unavailable. When not based on measured data, the parameterization process may introduce uncertainty to such an extent that simple few-parameter models yield better results than the complex ones, especially in ungauged basins. Despite the fact that entropy equations tend to demand few parameters, they often out-perform equations based on different approaches, due to the robustness of the principle Chiu (1987), (1988) The entropy-equation optimization is subject to the given constraints, which represent information about the problem to be solved. Therefore, the greater the number of constraints, the more information there is about the system Jaynes (1957a). More information implies less uncertainty and more accurate models. Nevertheless, the density functions obtained by PME can be solved analytically only if a maximum of two constraints are used. If the optimization uses three or more constraints, it demands a numerical solution, which simultaneously lowers the model accuracy and increases the computational effort. This ambiguity -more than two constraints generate more information, but also weaken the computational solution -yields a non-trivial non-linear problem, which has not (to our best knowledge) been straightforward tackled in the Literature, especially by researches based on an accurate robust datasets. Therefore, the objective of this work was to assess the trade-off between number of constraints and strength of the primary statement on the performance of hydrodynamic entropy-based models using accurate laboratory data for validation. For this purpose, the following quests were analyzed: (i) is there an improvement in model performance when two constraints are used instead of one? (ii) is there an improvement in model performance when a third constraint is introduced, considering that its solution is not analytical? (iii) is it better to use a threeconstraint model with a weak primary statement, or a two-constraint model with a strong primary statement?
Abbreviations D : Flow depth at the channel F(u) Probability of the longitudinal velocity being less or equal to u H : Entropy function M : Entropy parameter. NSE : Nash& Sutcliffe p(u) : Probability density function PME : Principle of maximum entropy RMSE : Root mean square error SRP : Steffler, Rajaratnam and Peterson (1985) u : Longitudinal velocity u av : average velocity. U max : Maximum velocity in the cross section U1y : Model with one constraint and Cartesian coordinates. U2y : Model with two constraints and Cartesian coordinates. U3y : Model with three constraint and Cartesian coordinates. U1ξ : Model with one constraint and curvilinear coordinates. U2ξ : Model with two constraints and curvilinear coordinates. U3ξ : Model with three constraints and curvilinear coordinates. y : The vertical distance of a any point located in the flow from the channel bed z : The horizontal distance of a any point located in the flow from the nearest wall. β : Boussinesq coefficient. λ i : Lagrange parameters, i = 1,2,…,12. ξ : isovel δy, δi, βi, ε: shape parameters

MATERIALS AND METHODS
We investigated the performance of six entropy models designed to simulate open-channel velocity fields. The simulated velocities were compared with accurately-measured laboratory data. Three of the investigated models have a weak primary statement, i.e., they assume that isovels could be well represented by Cartesian coordinates; whereas the remaining three models have a strong statement according to which isovels are better represented by curvilinear coordinates. In this work, data were extracted from the experiments made by Steffler et al. (1985): run 1 (hereafter called SRP1), run 2 (SRP2), and run 3 (SRP3). The experiments (see the main characteristics in Table I) were performed at the Thomas Blench Laboratory flume located at the University of Alberta, Canada. The velocities were accurately measured using a Laser-Doppler anemometer. The models performance was assessed with the Nash-Sutcliffe coefficient (NSE) and the root mean square error (RMSE).

Models with weak primary statement
Two primary statements were assumed and models with one, two or three constraints used, respectively, in order to maximize the entropy function H (Equation 1), in which u means the longitudinal velocity; p(u) the respective probability density function; and U max the maximum velocity in the cross section. The six entropy models are divided into two groups: three models admit the Cartesian coordinate system (weak primary statement, Equation 2), whereas the three others admit the curvilinear coordinate system, as described in (Chiu 1988 Model U1y (one constraint and Cartesian coordinates) is based on Chiu (1987), who proposes the primary statement (Equation 2), according to which F(u) is the probability of the longitudinal velocity being less or equal to u at a point located at distance y from the channel bed. In Equation 2, D is the flow depth at the channel.
The first constraint is presented in Equation 3. It means that the integral of the probability density function p(u) over the whole dominium equals unity. We maximize the entropy function (H, Equation 1) Model U2y (two constraints and Cartesian coordinates) uses the same weak premise as model U1y (Equation 2). The entropy function (Equation 1) was maximized by Chiu (1987), using two constraints: Equations 3 and 6. The latter represents the mass conservation principle and indicates that the left-hand integral equals average velocity (u av ). As a result, the method yields the velocity-distribution Equation 7 with parameters λ 2 , λ 3 and M that can be estimated using the maximum and average velocities by equations 8 and 9: Chiu (1987).
Model U3y (three constraints and Cartesian coordinates), derived by Barbé et al. (1991), assumes the same premise as in the previous The symbols mean: n = number of measured points; n.vert = number of measured vertical profiles; Q = discharge; D = flow depth; B = flume width; A = wetted area; u av = average velocity, given by Q/A; U max = maximum measured velocity; ε = maximum-velocity dip. Data source : Steffler et al. (1983).

models (Equation 2) and three constraints:
Equations 3, 6, and 10. The third constraint represents the momentum conservation principle (Equation 10), in which β is the Boussinesq coefficient (Equation 11) and ρ is water density. The system generated by maximizing the entropy function (Equation 1) for the three constraints was solved using a numerical approach (MacLaurin series with the first two terms), which yields the approximate velocity field (Equations 13 and 14). The parameters λ 4 , λ 5 , and λ 6 can be obtained using the Boussinesq coefficient, maximum velocity and average flow velocity, as stated in Barbé et al. (1991). .
Models with a strong primary statement The strong statement admits that the longitudinal velocity is directly associated with the curvilinear, rather than with Cartesian coordinates Chiu & Chiou (1986); and that isovels can be represented by ξ coordinates (Equations 15-17), as proposed by Chiu (1986). The isovel (ξ) shape parameters (δy, ε) and variables (y, z) are defined in Figure 1. Parameter β i characterizes the velocity distribution of the primary flow.  Chiu and Chiou (1986).
Model U1ξ (one constraint and curvilinear coordinates) is based on Chiu (1988), who proposes the strong primary premise (Equation 18), according to which F(u) is directly associated with the isovel (ξ) spatial distribution, with the key parameters ξ max and ξ 0 , respectively, the In the present study, Model U3ξ (three constraints and curvilinear coordinates) used curvilinear coordinates (Equation 15), the strong statement (Equation 18) and three constraints, (Equations 3, 6 and 10 as in model U3y: Barbé et al. (1991), but substituting ratio y/D by The U3ξ model for the velocity field in open channels consists in solving Equations 22 and 23. The system parameters λ 10 , λ 11 , and λ 12 are estimated analogously as in model U3y, using the Boussinesq coefficient, maximum velocity and average flow velocity, as demonstrated by Barbé et al. (1991). .   (Figures 2a and 2b, respectively) show that the addition of the second constraint definitely improves the model predictability capacity. The term "n.vert" means the number of vertical profiles. Data source: Steffler et al. (1983). The black dots represent points located further than 3% of the channel width from the sidewalls and from the channel bed, whereas the plus (+) signs refer to the points located elsewhere (near wall and/or channel bed).

RESULTS AND DISCUSSION
However, although the three-constraint U3y model (Figure 2c) performs better than U1y, it is worse than U2y: the median NSE decreases to +0.32 and RMSE raises to 9%. This shows that the addition of extra information (third constraint) improves the model, i.e., U3y performs better than the one-constraint model U1y. The use of the numerical solution, as in Barbé et al. (1991), however, influences negatively this improvement and even limits its performance (U3y is worse than U2y). The same pattern can be observed for the three curvilinear-coordinate entropy models.
The one-constraint model U1ξ has negative Nash-Sutcliffe coefficients for all vertical profiles and errors ranging from 11% to 51%. According to the results, this is the worst model (Figure 3a) among the researched ones, with median NSE below -8. The inclusion of the second constraint notably improves model capacity: the median NSE is positive for all experiments (greater than +0.76) and the median error is as low as 4%. Comparison of Figures 3a and 3b also shows an improvement of the model when the second constraint is considered. The combination of two constraints and the curvilinear-coordinate system generates the best entropy model (U2ξ) among the investigated options. As in the weak primary-statement models, the inclusion of a third constraint yields a model (U3ξ) with a performance surpassing that of the one-constraint model (U1ξ) and raising the median Nash-Sutcliffe coefficient from -8.41 to +0.42; median error decreases from 26% to 8%, at the same time. In fact, U3ξ is the second best entropy model among the six tested formulations. Comparing Figures 3b and 3c reveals, however, that the U3ξ model does not represent the velocity-field data as well as the U2ξ model: in the balance between the advantage of having more information (third constraint) and the disadvantage of using a numerical solution, the negative aspect prevails. Besides, the numerical solution of the threeconstraint models generated instability during the parameterization process (when calculating the Lagrange multipliers) which augmented the computational effort. This was observed in all experiments.
In order to investigate whether it is better to use a three-constraint model with a weak primary statement, or a two-constraint model with a strong statement, the marginal improvement of model U2y was compared with models U3y and U2ξ, respectively. The results show that the third constraint has the drawback of demanding a numerical solution, which increases computer time demand and worsens result accuracy; whereas the combination of an analyticalsolution system (two constraints) with a strong statement yields a high-performance model. In fact, model U2ξ has a median NSE of +0.77 against +0.32 of U3y, whereas the median error of U3y (9%) is more than twice that of U2ξ (4%). When using the Student t-test (5% significance) to compare the NSE between U2y and U3y, it shows that both models are statistically equal, i.e., the simple addition of the third constraint does not upgrade the model capability because the threeconstraint model demands a numerical solution of its equations. Contrastingly, when we apply the t-test to compare U2y and U2ξ, the results indicate that they are statistically different, with clear superiority of the latter: NSE improves from +0.51 to +0.77 and the average NSE raises from -2.31 to +0.66. Figure 4 provides a synthesis of the performance of the models, considering only the number of constraints. It is clear that models with only one constraint perform much worse than those with two or three constraints (negative NSE and high RMSE). It is also visible that, despite the similarity of the results of the U2 and U3 models, the performance of U2 models is higher. Besides, if one compares the Nash-Sutcliffe coefficient for the best-fit U2 model (U2ξ) with that for the best-fit U3 model (U3ξ) using the Student t-test with a 5% significance, it shows that the models are statistically different and that U2ξ is clearly superior.
From Figures 2, 3, and 4 it is noteworthy that all models have flaws in representing some vertical profiles. Figure 5 indicates that mal-represented profiles are those near the sidewall, which can be confirmed by Figures  2 and 3. This flaw occurs even when the best models (U2ξ, U3ξ, and U2y) are used. For the near-wall verticals, for example, model U2y exhibits an NSE coefficient as low as -24 and a corresponding error RMSE as high as 32%. It still mimics accurately the measured data for more centralized verticals with a maximum NSE of 0.98 and minimum RMSE of 1% (Table II). This fact is directly connected to sidewall proximity, as shown in de Araújo & Chaudhry (1998) and in Greco (2015). The analyzed entropy models perform well for profiles further than 3% of the channel width, however, as can be depicted from Figure 5. Nonetheless, the use of curvilinear coordinates improves model performance for The black dots represent points located further than 3% of the channel width from the sidewalls and from the channel bed, whereas the plus (+) signs refer to the points located elsewhere (near wall and/or channel bed). one, two, or three constraints, particularly in the vicinity of walls. This is most emphasized in the routine U2ξ, the best-performance model in the context of this research. In fact, Chen & Chiew (2004) experimentally observed the significant velocity gradient near the channel bed; Patel et al. (2016) showed that near-wall gradients influence turbulence and quasi-streamwise vortices in channel flow; whereas Ninto & Garcia (2006) emphasized the influence of the nearwall flow on sediment re-suspension, which was confirmed by Mohan et al. (2019). These features have been experimentally observed, among others, by Steffler et al. (1985), de Araújo constraints performs better than those with one constraint, and that the second constraint includes relevant information for the system. Contrastingly, models with three constraints perform worse than those with two constraints, showing that the loss of information due to the use of numerical solutions may surpass the gain of information due to the third constraint. The best-performance entropy model (that with two constraints and curvilinear coordinates -U2ξ) was able to mimic well the accurately-measured laboratory data for vertical profiles further than 3% of channel width. For vertical profiles closer than 3% of the width, the studied models do not perform well because the specific information concerning the prevailing processes is neither provided in the primary statement, nor in the constraints. In the present case study, the replacement of a weak primary-statement (use of Cartesian coordinates, Equation 2) by a strong one (use of curvilinear coordinates, Equation 18) brings more information to the system than the inclusion of a third constraint.