Randomly supported variation of deterministic models and its application to one-dimensional shallow water flows∗

Randomly supported variation of deterministic models and its application to one-dimensional shallow water flows

E. G. Birgin^† M. R. Correa^‡ V. A. Gonz´alez-L´opez^§ J. M. Mart´ınez^‡ D. S. Rodrigues^¶

February 21, 2022

Abstract

Mathematical laws that govern natural processes may consist in differential equations which, at each stage, are modified by a random effect that represents model inadequacies. In the presence of real data one is interested in the estimation of unknown physical parameters and unknown parameters that describe random perturbations. The case of one-dimensional artificial and natural channels usually represented by Saint-Venant equations is considered in this paper. A procedure for simultaneously estimating the main parameters of the under- lying physical model and the parameters that control the dispersion of the random effect is introduced. With this tool, we quantify the errors with respect to data in probabilistic terms.

Keywords: Decision-making, stochastic processes, partial differential modeling, Manning’s coefficient, hydraulic models.

1 Introduction

Decision-making requires predictions. Often, predictions involve the future although, more gen- erally, they are merely related to unobserved facts. Unfortunately, we can very rarely make predictions with absolute certainty. Therefore, decision-makers need to be informed by predic- tions of the type: “This event happens with this margin of error and with this probability”. On the other hand, in order to predict, it is common to make use of models which must be adjusted to best reproduce the data we already have about reality. In general, the higher the accuracy with which a model reproduces known data, the more efficient it will be in predicting unknown facts. Meanwhile, these considerations must be quantified in such a way that the performance

∗This work was supported by FAPESP (grants 2013/07375-0, 2016/01860-1, and 2018/24293-0) and CNPq (grants 304192/2019-8, 309517/2014-1 and 303750/2014-6).

†Dept. of Computer Science, Institute of Mathematics and Statistics, University of S˜ao Paulo, Rua do Mat˜ao, 1010, Cidade Universit´aria, 05508-090, S˜ao Paulo, SP, Brazil. e-mail: [email protected]

‡Dept. of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific Computing, University of Campinas, 13083-859, Campinas, SP, Brazil. e-mails: [email protected] and [email protected]

§Dept. of Statistics, Institute of Mathematics, Statistics, and Scientific Computing, University of Campinas, 13083-859, Campinas, SP, Brazil. e-mail: [email protected]

¶School of Technology, University of Campinas, 13484-332, Limeira, SP, Brazil. e-mail:

[email protected]

of a model in reproducing known data should yield both ranges of variation of the unobserved ones and probabilities for occurrences within well-determined intervals.

For example, suppose we need to make decisions in the presence of an epidemic. We have data on occurrences in a certain region up to the present, but we do not know about future occurrences, as well as occurrences in inaccessible regions. However, we need estimates of these unknown data to make appropriate decisions involving costs and benefits. Such estimates should be along the lines of “With a probability p there will be between n₁ and n₂ cases in a given region and a given period”.

To produce these results, we can assume that every event, observed or unobserved, is the result of a model based on reasonable physical considerations. Except in very special situations, such a model is far from deterministic, i.e. it fails to reproduce known data. Therefore, we can consider the known data as random variations of the ideal model. That said, the first task is to fit the model, with its random component, to the known data. Consequently, the random nature of the fitted model will make it possible the simulation of the unknown observations providing, by its very essence, confidence intervals and probabilities in a form that should be useful to decision-makers.

Reasonable mathematical models based on physical laws usually represent natural phenom- ena by a vector function u=u(x, t), which denotes the state u of a system at a spatial point x and a time instant t. Frequently, u is the solution of a system of partial differential equations (PDE) in which the derivative ofuwith respect to time must be equal to an expression involving x,t,u, spatial derivatives ofu, and unknown parameters. In real life, physical processes do not obey exactly the traditional deterministic laws with which they are usually associated. Fre- quently, the reasons for inadequacy rely on flaws in the models themselves. For example, many times, when dealing with the flow of natural channels, we use efficient models for rectilinear channels that, in nature, are practically nonexistent. Other reasons for inadequacy are related to the imperfect knowledge of physical and topographical features. Finally, since we are not able to solve differential equations analytically, we appeal to numerical schemes, which involve discretization and truncation errors.

Accordingly, starting fromu(x, t), the numerical scheme that allows us to obtain an approx- imation to u(x, t+ ∆t) for all x in the spatial domain must be complemented with a random term. This means that the knowledge we can obtain about state variables may not come from the rigorous solution of the differential equation, let alone from its discretization by reliable and stable methods, but of realizations on a random variable that acts as a perturbation of the result coming from the deterministic underlying law.

The problem that concerns us in this work is the prediction of flows by means of the dif- ferential equations that model natural channels. The standard procedure consists of using an optimization method applied to the minimization of the sum of squares of the differences be- tween real observations and observations predicted by the numerical resolution of differential equations, where the vector of unknowns of the optimization process is formed by constitutive coefficients, such as the Manning’s roughness coefficient, at different points of the spatial do- main. Least squares estimates are useful in many cases for this purpose but they do not give us clear meanings of the solutions obtained from the point of view of probability and statistics.

The questions that these estimates leave unanswered are:

• Which is the “Manning’s coefficient” that maximizes the probability that the process of solving the differential equations represents the measurements, when intrinsic perturba- tions are considered?

• What are the best parameters’ values to define random disturbances that can produce reliable bands of predicted trajectories?

The procedure suggested in this paper to solve the aforementioned problems consists of as- suming that both the Manning’s coefficientξand the parametersσfor the random perturbations are the independent variables of an optimization procedure that aims to maximize the likelihood of the observed data with respect toξandσ. The notion on which this procedure is supported is the maximum likelihood principle; see [14]. The maximum likelihood procedure, under certain conditions, can lead to the same results obtained by the least-squares method. For example, if the dispersion of the random term is known and under the usual linear regression model, both procedures lead to the same solution; see [5]. In addition, the maximization of the likelihood cor- responds to the minimization of the Kullback-Leibler divergence between the empirical law and the theoretical one. Maximum likelihood estimators have relevant properties, such as invariance, which allows one to easily estimate transformations of the parameters, if needed. Theorem 6.2.2 in [5] applies to these type of estimators and details the stochastic behavior of the maximum likelihood estimators when the sample size grows.

Suppose that we are interested in developing reasonable models to predict the evolution of fluid flows in natural channels. For this purpose, on the one hand, we have a set of measurements on which we know that there is a certain experimental error whose magnitude is approximately known. On the other hand, we have a preference for the Saint-Venant equations [28] in their explicitly discretized form with a numerical scheme. Finally, we do not know some essential constitutive parameters such as the Manning’s coefficient.

Our first attempt to tackle this problem would be fitting estimates of the Manning’s coef- ficient to the available data using an optimization algorithm. If, with such a fit, solving the equations yields values that agree with the observed ones, within the “experimental” error, it is sensible to consider that we are on the right track and that the equations reflect the physical phe- nomenon with the estimated Manning’s coefficient. If not, we wonder if any modification of the equations produces a forecast of the observations with error compatible with the measurement error. However, this question has been ambiguously formulated. What does “any modifica- tion” mean? Among the various possible answers, we choose one that is supported by plausible probabilistic considerations: We admit that the physical phenomenon can be produced by the Saint-Venant equations perturbed at selected steps with a random variable, with an adequate distribution, and we calculate the dispersion associated with the definition of such a variable by means of which the probability of the observations is maximal.

Many papers have been devoted to the problem of estimating the Manning coefficient. In [26]

the performance of the HEC-RAS software [7] for predicting inundation data was analyzed as a function of Manning’s roughness coefficient and weighting discretization parameters to pro- duce dynamic probability maps of flooding during the event. HEC-RAS was also used in [1], where different heuristic methods were used for the optimizing Manning’s coefficient. Ding and Wang [13] solved the Saint-Venant equations to simulate flows in channel networks and used the resulting deterministic model to compute the optimal Manning’s coefficient using standard quasi-Newton methods. Askar and Al-jumaily [2] estimated the Manning’s coefficient using plain Saint-Venant equations as predictors and sequential quadratic programming for optimiza- tion purposes. Ebissa and Prasad [18] used the GVF (Gradual Varied Flow) equations for simulations and genetic algorithms for deterministic optimization of the roughness parameter.

Birgin and Mart´ınez [6] developed a secant derivative-free optimization methods for determining the Manning’s coefficient in synthetic experiments. See, also [4, 15, 3, 20, 21, 25]. In all these

papers a mathematical model is assumed to be true and different optimization methods are employed to fit unknown parameters, most frequently using least squares. Therefore, the idea of using a stochastic model [23] reflecting imperfections of well-established deterministic ones seems to be new, at least for estimating hydraulic parameters with prediction purposes.

The rest of this paper is organized as follows. The proposed parameters estimation strat- egy based on a likelihood measure that rest upon a fuzzy favorability metric is introduced in Section 2. The Saint-Venant equations, selected as basic model to describe the flux of water in one-dimensional channels, are described in Section 3. Section 4 shows the numerical method that was chosen to solve the Saint-Venant equations. Implementation details and extensive nu- merical experiments are report in Section 5. The last section presents the conclusions and lines for future research.

2 Parameters estimation and fuzzy-favorability-based likelihood

Let Ω = (xI, xF) ⊂ R and I = (t0, T] ⊂ R be given spatial and time domains, respectively.

Assume thatT ={t₁, . . . , t_nt} ⊂I is such thatt₁<· · ·< t_nt =T and for alln= 0,1, . . . , n_t−1 and all x∈Ω, consider the stochastic process

u(x, tn+1) =F(x, tn, tn+1, u(x, tn), ξ) +δ(tn+1)V(x, t_n+1, u(x, tn), σ), (1) where ξ ∈ R^nξ is a vector of parameters with physical meaning, u ∈ R^nu, δ(t) is the indicator function of a given set T_γ ⊆ T, and V(x, t_n+1, u(x, tn), σ) ∈R^nu is a zero-mean random vector whose distribution depends on parameters σ ∈ R^nσ. In the typical case of (1), the equality u(x, t_n+1) = F(x, t_n, t_n+1, u(x, t_n), ξ) represents the discretization scheme of a partial differ- ential equation. Equation (1) indicates that, at each stage, the physical phenomenon under consideration is represented by a random variable whose expectation isF(x, t, t+ ∆t, u(x, t), ξ).

In the present work, we assume that the vectors of parameters ξ ∈ R^nξ and σ ∈ R^nσ are unknown and that nobs observations v_i^obs of the physical phenomenon represented by (1) at spatial-time coordinates (x^obs_i , t^obs_i )∈Ω×I fori= 1, . . . , n_obs are given. In addition, we assume that each observation v^obs_i is associated with a quantity ϑ_i > 0, that typically represents the measurement error of the observation. We wish to determinate ξ and σ from available data by means of the maximization of a likelihood measure based on a fuzzy favorability metric calculated through simulations. Let n_sim be the number of considered simulations and let v^sim_ij be the simulated value at spatial-time coordinate (x^obs_i , t^obs_i ) for i = 1, . . . , n_obs obtained at simulation j for j = 1, . . . , n_sim. The simulated values are calculated through a run of the process (1) using, for example, a uniform discretization of Ω given by x` = xI +`∆x for

`= 0, . . . , n_x, where ∆x= (x_F −x_I)/n_x. This means that the simulated values are calculated on a space-time grid. If any (x^obs_i , t^obs_i ) does not belong to the grid, the corresponding values of v_ij^sim may be calculated with interpolation. It is important to stress that simulated values v^sim_ij depend on ξ and σ.

The likelihood associated with a pair (ξ, σ) is intended to represent the probability that a given set of observations (subject to measurement errors) is generated by (1). Roughly speak- ing, the considered likelihood will be the ratio of the favorable cases to the total number of simulationsn_sim. However, for a simulation j, instead of a binary definition of favorability, we consider the fuzzy definition given by

exp(−d(v^obs₁ , . . . , v^obs_nobs, v^sim_1,j, . . . , v^sim_nobs,j)), (2)

where

d(v₁^obs, . . . , v_n^obs_obs, v^sim_1,j, . . . , v_n^sim_obs,j) = v u u t

1 n_obs

nobs

X

i=1

(v^obs_i −v_ij^sim)² ϑ²_i

!

(3) represents the root mean square deviation of simulation j with respect to observations. There- fore, the favorability of simulationjis 1 ifv_i^obscoincides withv_ij^simfori= 1, . . . , n_obs and is equal to 1/e≈0.37 if the distance betweenv_i^obs andv^sim_ij is equal toϑi fori= 1, . . . , nobs. So, ϑi must be chosen in practical cases as a representation of the measurement error of observation v_i^obs. Consequently, the likelihood associated with the pair (ξ, σ) is given by

L_ϑ(ξ, σ) = 1 nsim

nsim

X

j=1

exp(−d(v^obs₁ , . . . , v^obs_n

obs, v_1,j^sim, . . . , v^sim_n

obs,j)). (4)

Function L_ϑ(ξ, σ) as defined in (4) is stochastic and nondifferentiable. However, it is worth noticing that, whenσ ≡0, (4) reduces to (2) for any value ofnsim, thus being deterministic and cheaper to be evaluated. Moreover, maximizing (2) is equivalent to minimizing (3), i.e. solving a nonlinear weighted least-squares problem (recall thatv_ij^sim is a function of σ and ξ).

3 Saint-Venant equations

Let us consider the one-dimensional shallow water flow in non-rectangular, non-prismatic chan- nel, with cross section varying with the coordinate x in a spatial domain Ω = (xI, xF) ⊂ R, as described in Figure 1. The governing equations, known as Saint-Venant equations [28], are

z_b h

z=z_b+h y= 0

y A

z = 0 B

b(y)

Figure 1: Geometry of a cross section of the channel at a pointx of the spatial domain Ω.

given by the mass balance of the fluid

∂A

∂t + ∂Q

∂x = 0 (5)

and the linear momentum balance equation which, according to [22], can be written as

∂Q

∂t + ∂

∂x Q²

A +gI1

=gI2−gA ∂zb

∂x +S_f

, (6)

where A = A(x, t) is the wetted cross-section area at position x and time t, Q = Q(x, t) is the flow rate, g is the gravitational acceleration (taken as 9.81 m/s²),I₁ and I₂ are terms that account for the hydrostatic pressure force and the wall pressure force, respectively. These terms are expressed by

I₁=

Z h(x,t) 0

(h−y)b(x, y)dy (7)

and

I₂ =

Z h(x,t) 0

(h−y)∂b(x, y)

∂x dy, (8)

whereb=b(x, y) is the channel width which, for a given pointx, is also a function of the depthy (as depicted in Figure 1) andh=h(x, t) is the water depth, which can be post-processed fromA and b. Finally, z_b(x) is the channel bed elevation at point x, measured from a datum, and S_f designates a friction slope term that can be written as

S_f = ξ²Q|Q|

R^4/3A², (9)

whereR is the hydraulic radius (ratio between the wetted area and the wetted perimeter of the channel) andξ (s/m^1/3) is the Manning’s roughness coefficient. The estimation of the Manning’s coefficient is a fundamental problem related to the simulation of floods in natural channels [12].

It is worth mentioning that the first theory for the derivation of the Manning’s equation from first principles was only proposed in 2001 [17].

Equations (5) and (6) can be obtained by integrating the Reynolds-averaged Navier-Stokes equations, under the shallow water hypothesis [29]. FromAandQ, we can univocally determine the mean velocityV =Q/A. The system of balance laws (5,6) is hyperbolic [24] and, under the assumption that

∂I₁

∂A ≈ A B,

whereB =B(x) is the width of the channel at the free surface at point x (see Figure 1), it has eigenvalues [22]

λ₁ =V −c and λ₂ =V +c, (10)

where

c= r

gA B

is the wave celerity. The general hydrostatic and wall pressure terms I1 and I2 can be difficult to evaluate in non-rectangular and non-prismatic channels [22]. Thus, under the simplification

g ∂I1

∂x −I₂

=gA∂h

∂x,

these terms can be combined in one single term, and then equation (6) can be written as

∂Q

∂t + ∂

∂x Q²

A

=−gA ∂z

∂x +S_f

, (11)

wherez(x, t) is the water level, defined as

z(x, t) =h(x, t) +z_b(x).

According to [29], the non-conservative form of the balance of linear momentum, given by (11), is adequate for the development of numerical schemes that preserve the “well-balanced”

property. This property states that initially still water with a horizontal surface must remain still regardless of the bed topography [29]. It is important to remark, though, that the system (5,11) is not strictly hyperbolic.

4 Numerical treatment of the Saint-Venant equations

Let Ω = (x_I, x_F) ⊂ R and I = (t₀, T] ⊂ R be given spatial and time domains, respectively.

The Saint-Venant equations (5,11) define the following model problem, written as a system of balance laws: find the vector of conserved variables u= [A , Q]^T such that

∂u

∂t +∂f(u)

∂x =s_z+s_f in Ω×I, (12)

satisfying the initial conditionu(x, t0) =u0(x) in Ω and appropriate boundary conditions. The fluxf, the source term due to the gradient of the water levelsz, and the source term due to the friction s_f are given by

f =

"

Q Q²/A

#

, s_z=



 0

−gA∂z

∂x



, and s_f =

"

0

−gAS_f

#

, (13)

respectively.

The development of accurate, efficient, and robust numerical schemes for such systems of balance laws is still a challenging issue that has been extensively investigated [8, 9, 22, 24, 29].

These schemes can be derived based either on (11), as in [29], or on (6), as in [24]. In both cases, the solutions admit non-smooth composite waves (shocks and rarefaction waves). A key element for the development of accurate numerical schemes is the treatment of the numerical fluxes, the source, and the bed slope term. A review of well-balanced finite-volume schemes for shallow-water equations can be found in [24], where the author describes some Godunov-type Riemann-problem-solver-free central schemes that incorporate upwinding information about the local speeds of propagation, also known as central-upwind schemes. These schemes have also been successfully applied in the context of complex multiphase flows in highly heterogeneous porous media; see [9, 10]. Another class of well-established methods for the solution of hyperbolic problems comprises the discontinuous Galerkin methods [8, 22], where piecewise polynomial (globally discontinuous) approximations are obtained from local weak forms of (12), connected by numerical fluxes.

In the present work, the Saint-Venant system is numerically solved, with the flux and the source written in the form (13), by using the upwind conservative finite volume scheme proposed in [29], which for the sake of completeness we briefly describe below. In this scheme, the intercell flux is computed by a one-sided upwind method and the water-surface gradient is evaluated by a weighted average of both upwind and downwind gradients.

Space and time discretization. LetX_h be a partition of the spatial domain Ω in n_x cells C_i= (x_i−¹

2, x_i+¹

2), i= 1, . . . , nx, (14)

of length ∆x_i=x_i+¹

2

−x_i−¹

2. For the time interval I, we define the partition

∆tn=tn+1−tn, (15)

with time steps ∆tn satisfying

X

n

∆t_n=T−t₀

and constrained by a Courant-Friedrichs-Lewy (CFL) stability condition to be specified below.

Upwind conservative scheme. Integrating (12) over a cellC_i and performing forward Euler time integration of all terms but sz (for which the approximation in time will be described below), we have the fully discrete scheme

uⁿ⁺¹_i =uⁿ_i −∆t_n

∆xi

f_i+^u 1

2

−f_i−^u 1 2

+ ∆t_ns^?_z,i+ ∆t_nsⁿ_f,i, (16) whereuⁿ_i denotes an approximation to the mean value of the vector of conserved variablesu in cellC_i attn, i.e.

uⁿ_i ≈ 1

∆xi

Z

Ci

u(x, tn)dx.

The source term due to the friction is explicitly evaluated attn as sⁿ_f,i=

0

−gAⁿ_iS_f(Aⁿ_i, Qⁿ_i)

(17) and the upwind flux vectorf_i+^u 1

2

is defined as

f_i+^u 1 2

=

"

Qⁿ_i+k Qⁿ_i+k2

/Aⁿ_i+k

#

(18) with

k=







0, if Qⁿ_i >0 andQⁿ_i+1>0, 1, if Qⁿ_i <0 andQⁿ_i+1<0,

1

2, otherwise,

(19) where the subscripti+¹₂ represents the average of the quantities at the neighbor cellsC_iandC_i+1. The approximation of the source term s^?_z,i requires special attention and is based on a two-step implementation of the scheme. In the first step, the first component of the vector equations (16–18) is used to evaluate the wetted area (and the water level) at the time levelt_n+1. Then, the following weighted combination of downwind and upwind gradients of the (updated) water surface, multiplied by the (also updated) wetted area, is proposed:

s^?_z,i =

0

−gAⁿ⁺¹_i [w_d(δz)_d+wu(δz)_u]

with

(δz)_d= zⁿ⁺¹_i+1−k−z_i−kⁿ⁺¹ xi+1−k−xi−k

and (δz)_u = z_i+kⁿ⁺¹−z_i−1+kⁿ⁺¹ x_i+k−xi−1+k

(20)

and weights defined based on local Courant numbers

w_d= 1−Cr^d and wu = Cr^u,

where Cr^d and Cr^u are evaluated by using the average velocity V_iⁿ = Qⁿ_i/Aⁿ_i, through the formulas

Cr^d= ∆tn

xi+1−k−xi−k

|V_i+1−kⁿ |+|V_i−kⁿ | 2

and Cr^u= ∆tn

x_i+k−xi−1+k

|V_i+kⁿ |+|V_i−1+kⁿ | 2

. (21) Note that, differently from the flux in (18) which is evaluated at the node x_i+1

2, the gradients in (20) are evaluated at the center of the cellC_i. Thus, in (20) and (23), the indexk is defined by

k=

0, if Qⁿ_i ≥0

1, if Qⁿ_i <0. (22)

Ifw_d=w_u= 1/2, then the classical second order two-point central difference approximation for the water surface gradient is recovered, which is known for producing nonphysical solutions, while the upwinding approximation (wd= 0 andwu= 1) results in an unstable scheme and the downwinding approximation (w_d= 1 and w_u= 0) may lead to over diffusive solutions; see [29].

As mentioned above, the non-uniform time steps ∆t_n are defined by ∆t_n ≤τ, where τ is a time scale parameter restricted by the CFL-type condition

τ

"

1≤i≤nmaxx

|V_iⁿ|+p

gAⁿ_i/Bi

∆xi

!#

= Cr, (23)

with Cr representing a prescribed Courant number.

5 Implementation and numerical experiments

In this section, we intend to embed the numerical solution of the Saint-Venant equations into the stochastic process (1) and to determine, from observed data, the Manning’s coefficientξ∈R and the standard deviation σ ∈R that are more likely to describe different open channel flow scenarios. In the embedding, function u(x, t_n+1) in (1) corresponds to a vector Uⁿ⁺¹ ∈ R^2nx, which is the composition of the vectors uⁿ⁺¹_i ∈ R² for i = 1, . . . , nx that appear on the left- hand side of (16). FunctionF(x, tn, tn+1, u(x, tn), ξ) in (1) corresponds to a vectorFⁿ⁺¹ ∈R^2nx, which is the composition of then_xtwo-dimensional terms on the right-hand side of (16). Finally, functionV(x, t_n, tn+1, u(x, tn), σ) in (1) corresponds to vectorVⁿ⁺¹ ∈R^2nx, which is the product of a diagonal matrix S = diag(s1, . . . , s2nx) by the vector Fⁿ⁺¹, where each si is a random variable that follows a normal distribution with zero mean and average deviationσ.

Given the observed data (heights and/or flow-rates for several values of the spatial variablex and the time variable t), finding optimal ξ^∗ and σ^∗ consists of maximizing function (4), that depends on the number of simulations n_sim. In numerical experiments, based on preliminary experiments, we considered,n_sim = 100. Function (4) is a function of two variables, stochastic, nonlinear, and non-differentiable. Considering that we know a priori intervals [ξmin, ξmax] and [σ_min, σ_max] within which lie the optimal values ξ^∗ and σ^∗, the simplest way to find these val- ues is to choose steps ξ_step and σ_step and do an exhaustive search inside this two-dimensional

box. Depending on the desired precision on the optimal values, iterative refinements may be performed.

The set of time instants T = {t₁, . . . , tnt} ⊂ I that constitutes the discretization of the time domain I = (t₀, T] is computed as the numerical method for solving the Saint-Venant equations proceeds. Each instant t_n+1 is chosen in such a way that the step ∆t_n =t_n+1−t_n satisfies ∆tn ≤ τ, where τ is given by (23). The set of instants T_γ = {t^γ₁, t^γ₂, . . .} ⊆ T that contains the instants t^γ₁ < t^γ₂ < . . . at which random perturbations occur is also calculated dynamically. Given a parameter γ,t^γ₁ is given by t^γ₁ = min{t∈ T |t−t₀≥γ}, while, for` >1, t<sup>γ</sup><sub>`</sub> = min{t∈ T |t−t^γ_`−1 ≥γ}. Roughly speaking, this means thatT_γis such that perturbations are made at time instants with intervals of size around γ.

The numerical resolution of the Saint-Venant equation described in Section 4, the objective function (4) described in Section 2, and the exhaustive search optimization procedure described in the paragraph above were implemented in Fortran 90. In order to allow reproducibility, the source code necessary to reproduce all numerical experiments described in the present section is available athttp://www.ime.usp.br/~egbirgin/. Tests were conducted on a computer with a 4.5 GHz Intel Core i7-9750H processor and 16GB 1600 DDR4 2666 MHz RAM memory, running Ubuntu 20.04. Code was compiled by the GFortran compiler of GCC (version 9.3.0) with the -O3 optimization directive enabled.

The numerical experiments are organized in three different sets, exploring steady-state and transient problems. In Section 5.1, we study the formation of a hydraulic jump in a horizontal flume. This example, although simple, is strategic to introduce the application of the proposed strategy in a problem with a steady-state solution. In Section 5.2, we apply the proposed methodology to the simulation of a partial dam-break. Finally, in Section 5.3, we study a problem with transient inflow boundary condition, simulating a flood in a rectangular open channel.

5.1 Hydraulic jump in a rectangular channel

In this first experiment, we study the formation of a hydraulic jump by taking as reference the experiment reported in [16]. There, the authors performed experimental investigations of the steady-state location of the hydraulic jump in a horizontal 14.0 m long and 0.46 m wide flume, for different Froude numbers Fr = V /√

gh, by starting from a supercritical flow in the entire channel and then controlling the tailwater depth by an adjustable downstream gate. The bottom of the flume is made up of metal and the walls are made up of glass for 0 m ≤x ≤3.05 m and of metal for 3.05 m< x ≤14 m. The Saint-Venant equations, subject to the same initial and boundary conditions of the experimental investigation, were then solved with the Upwind scheme of Section 4 in uniform meshes ofn_x = 50 cells with Cr = 0.1. Empirically, we consideredγ = 0.5 in the case Fr = 4.23 described in Section 5.1.1 and γ = 3.0 in the case Fr = 7.00 described in Section 5.1.2.

5.1.1 Hydraulic jump for Fr = 4.23

The initial condition is a steady-state flow with water height h(x,0) = 0.043 m and velocity V(x,0) = 2.737 m/s for all x ∈ Ω. The respective Froude number for this initial condition is Fr = 4.23. The upstream boundary condition is given by these same values, while the

downstream boundary condition for the water depth changes according to

h(14, t) = min{0.222,0.043 + 0.00358t}. (24)

The observednobs = 10 values h^obs_i at the points x^obs_i ,i= 1, . . . , nobs, selected from the exper- imental measurements of [16], are given in Table 1. According to [16] these values correspond to the steady state of the system. However, it is not clear for which value of t^obs_i they are obtained. Therefore, in our experiments we consider two possibilities fort^obs_i : (i) t^obs_i = 60 s for i = 1, . . . , n_obs and (ii) t^obs_i = 180 s for i = 1, . . . , n_obs. From (24), the variation of the water depth at the downstream boundary ceases at 50 s. Thus, we may expect that in case (i) the solution is still transient, while the simulation for case (ii) is more likely to match the observed values of Table 1.

i x^obs_i (m) h^obs_i (m) i x^obs_i (m) h^obs_i (m)

1 0.30 0.043 6 3.05 0.226

2 0.61 0.043 7 3.35 0.223

3 0.91 0.046 8 3.66 0.223

4 1.22 0.049 9 3.96 0.223

5 2.74 0.229 10 4.27 0.223

Table 1: Observed data for the hydraulic jump problem for Fr = 4.23, taken from the experi- mental measurements of [16].

In this experiment, as in all the others that follow, we considered that the quantity ϑ_i >0 representing the measurement error of the ith observation, is equal for all i, i.e. that given ϑ > 0, ϑi = ϑ for i = 1, . . . , nobs. Table 2 shows the results for different values of ϑ ∈ {1.0,0.5,0.1,0.05,0.01,0.005,0.001}. In the table, column ξ^∗ shows the optimal value found for the Manning’s coefficientξ, columnσ^∗ shows the optimal value found for the deviationσ of the random effect, and L_ϑ(ξ^∗, σ^∗) corresponds to the optimal likelihood. As a reference, the table also includes (in the column named ξ|_σ=0) the optimal value of ξ that is obtained when the condition σ= 0 is imposed, as well as the corresponding likelihoodL_ϑ(ξ|_σ=0,0). As mentioned in Section 2, these values correspond to the least-squares approximation ofξ. Fortobs = 60 s the optimal Manning’s coefficient was ξ^∗ = 0.0139 forϑ≥0.010 and ξ^∗= 0.0142 for ϑ= 0.005 and ϑ= 0.001, with positive values ofσ^∗. Similarly, fort_obs = 180 s the optimal Manning’s coefficient and dispersion parameter were (ξ^∗, σ^∗) = (0.0122,0.0) for ϑ≥0.010, (ξ^∗, σ^∗) = (0.0127,0.017) forϑ= 0.005 and (ξ^∗, σ^∗) = (0.0128,0.022) forϑ= 0.001. The difference between the estimates forξ can be explained by the fact that the solution at t= 60 s is still transient, as expected.

There are seven possible scenarios with respect to the value of the precision-related parameter ϑ∈ {1.0,0.5,0.1,0.05,0.01,0.005,0.001}and two possibilities for the instantt^obs (identical for all observations) at which data could have been collected (60 s and 180 s). In the fourteen resulting cases, we estimated the Manning’s coefficient and the standard deviation using the proposed method. For the seven scenarios onϑ, the probability that the observed data was generated by the distribution defined byξ^∗ andσ^∗ turned out to be maximized whent^obs= 180 s. Therefore, it is sensible to conclude that the published data were obtained at this time instant or later.

Therefore, let us concentrate ourselves in the case defined by t^obs = 180 s. As we mentioned above, we made seven assumptions on the precision with which the observations were obtained.

Note that the estimated σ^∗ increases when ϑ decreases. This means that, as expected, if the

ϑ tobs = 60

σ^∗ ξ^∗ ξ|_σ=0 L_ϑ(ξ^∗, σ^∗) L_ϑ(ξ|_σ=0,0) 1.000 0.000 0.0139 0.0139 9.99849063E-01 – 0.500 0.000 0.0139 0.0139 9.99396389E-01 – 0.100 0.000 0.0139 0.0139 9.85018527E-01 – 0.050 0.000 0.0139 0.0139 9.41407375E-01 – 0.010 0.000 0.0139 0.0139 2.21024242E-01 –

0.005 0.037 0.0142 0.0139 3.31604786E-02 2.38649012E-03 0.001 0.059 0.0142 0.0139 1.15603418E-09 2.77965587E-66

ϑ tobs = 180

σ^∗ ξ^∗ ξ|_σ=0 L_ϑ(ξ^∗, σ^∗) L_ϑ(ξ|_σ=0,0) 1.000 0.000 0.0122 0.0122 9.99970118E-01 – 0.500 0.000 0.0122 0.0122 9.99880477E-01 – 0.100 0.000 0.0122 0.0122 9.97016216E-01 – 0.050 0.000 0.0122 0.0122 9.88118174E-01 – 0.010 0.000 0.0122 0.0122 7.41689584E-01 –

0.005 0.017 0.0127 0.0122 3.13619599E-01 3.02613794E-01 0.001 0.022 0.0128 0.0122 1.39467474E-04 1.05249005E-13

Table 2: Optimal Manning’s coefficient, standard deviation parameter, and likelihood obtained for varying values of the precision-related parameterϑ∈ {1.0,0.5,0.1,0.05,0.01,0.005,0.001}in the hydraulic jump problem for Fr = 4.23.

observations are made with maximal precision (ϑ = 0.001 in this case) their probability in the case of the deterministic model (σ = 0) is smaller than the probability in the case of the stochastic model with σ^∗= 0.022. On the other hand, the probabilityL_ϑ(ξ^∗, σ^∗) decreases very quickly withϑ. Again, this is the expected behaviour as far as the assumption of extremely good precision in observations decreases the probability that observations come from mathematical (obviously inexact) models. These results are illustrated in Figure 2, forϑ= 0.005 andϑ= 0.001, where we compare the results of the deterministic case (σ = 0) with the superposition of all the nsim= 100 simulations obtained for the optimal parameterσ^∗.

Let us interpret the results of this experiment in terms of the concepts introduced in the present work. The observed data are the heights at n_obs = 10 points between 0.23 and 4.27.

These data have been used to estimate the Manning’s coefficient and the standard deviation that defines the random effect of the deterministic model. Knowledge of the variance allows us to simulate the behaviour of the height at the unobserved points between 4.27 and 14. Moreover, this unobserved behaviour can be characterised in probabilistic terms. For example, we can establish that, say, with probability 0.99 the depth forx= 13 m is between two values produced by the simulation. In graphical terms, this result is shown in Figure 2. Note that the uncertainty forx= 13 m is lower than the uncertainty in the vicinity of the hydraulic jump.

0 0.05 0.1 0.15 0.2 0.25

0 2 4 6 8 10 12 14

h(m)

x (m)

σ^∗ = 0.017, ξ^∗ = 0.0127 σ = 0, ξ|_σ=0 = 0.0122 Observations

0 0.05 0.1 0.15 0.2 0.25

0 2 4 6 8 10 12 14

h(m)

x (m)

σ^∗ = 0.022, ξ^∗ = 0.0128 σ = 0, ξ|_σ=0 = 0.0122 Observations

Figure 2: Simulations of the hydraulic jump problem for Fr = 4.23 constructed with the optimal parameters that were obtained with ϑ = 0.005 (top) and ϑ = 0.001 (bottom), assuming that t^obs = 180 s. The graphics display the superposition of all then_sim= 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameterσ^∗. The pictures also show the least-squares solution, that corresponds to the caseσ = 0.

5.1.2 Hydraulic jump for Fr = 7.00

In this case, the initial condition is a steady-state flow with water height h(x,0) = 0.031 m and velocity V(x,0) = 3.831 m/s and the upstream boundary condition is given by these same values. The downstream boundary condition for the water depth changes according to

h(14, t) = min{0.265,0.031 + 0.00474t}.

The respective Froude number for this initial condition is Fr = 7.00. The observed n_obs = 30 values h^obs_i at the points x^obs_i , i = 1, . . . , n_obs were obtained from [16, Fig.5], and are given in Table 3. In this case, we kept the precision to the scale of centimeters in order to introduce an error in the measurements.

i x^obs_i (m) h^obs_i (m) i x^obs_i (m) h^obs_i (m) i x^obs_i (m) h^obs_i (m)

1 0.30 0.03 11 3.35 0.28 21 6.40 0.27

2 0.61 0.03 12 3.66 0.28 22 6.71 0.27

3 0.91 0.04 13 3.96 0.28 23 7.01 0.27

4 1.22 0.04 14 4.27 0.27 24 7.32 0.26

5 1.52 0.04 15 4.57 0.27 25 7.62 0.26

6 1.83 0.04 16 4.88 0.27 26 7.93 0.26

7 2.13 0.06 17 5.18 0.27 27 8.23 0.26

8 2.44 0.14 18 5.49 0.27 28 8.54 0.26

9 2.74 0.22 19 5.79 0.27 29 8.84 0.26

10 3.05 0.27 20 6.10 0.27 30 9.15 0.27

Table 3: Observed data for the hydraulic jump problem for Fr = 7.00, taken from the experi- mental measurements of [16].

The results considering that observations were all made at t^obs_i = t^obs = 240 s for i = 1, . . . , n_obs are shown in Table 4 and Figure 3. In this case, the optimal value for the Manning’s coefficient wasξ^∗= 0.0140 for all considered values ofϑ, independently of imposing the condition σ = 0 or not. Therefore, the fact of L_ϑ(ξ^∗, σ^∗) being larger than L_ϑ(ξ|_σ=0,0) (see the two last columns in the table) is due to the (positive) effect of the perturbation induced by the parameterσ^∗. The interpretation of Figure 3 in terms of the concepts introduced in the present work is similar to that of Figure 2. In this case, the observed data are between x= 0.30 m and x= 9.15 m and the “inaccessible zone” ranges from x= 9.15 m tox= 14 m. The results shown in Figure 3 show that, using the proposed methodology, with the observed data it is possible to estimate the behavior of the heights in the inaccessible zone with a high degree of certainty.

5.2 Partial dam-break

In this numerical experiment, we evaluate the performance of the proposed methodology in a scenario of partial dam-break. In this case, we take as reference the experimental investigation performed by the U.S. Army Corps of Engineers in 1960 [30], where it was studied the extent and magnitude of floods induced by the breaching of a 0.3048 m (1 ft) high dam, located in the middle of a 121.92 m (400 ft) long and 1.2192 m (4 ft) wide model flume with a bed slope of 0.005 and rectangular cross-section. From these investigation, we took the stage-time measurements

ϑ t^obs = 240

σ^∗ ξ^∗ ξ|_σ=0 L_ϑ(ξ^∗, σ^∗) L_ϑ(ξ|_σ=0,0) 1.000 0.0140 0.0140 0.0140 9.99928689E-01 9.99926964E-01 0.500 0.0140 0.0140 0.0140 9.99714786E-01 9.99707888E-01 0.100 0.0140 0.0140 0.0140 9.92894279E-01 9.92722747E-01 0.050 0.0145 0.0140 0.0140 9.71882340E-01 9.71207201E-01 0.010 0.0170 0.0140 0.0140 4.91903348E-01 4.81722953E-01 0.005 0.0225 0.0140 0.0140 6.29370475E-02 5.38504529E-02 0.001 0.0460 0.0140 0.0140 1.14903797E-19 1.90429401E-32

Table 4: Optimal Manning’s coefficient, standard deviation parameter, and likelihood obtained for varying values of the precision-related parameterϑ∈ {1.0,0.5,0.1,0.05,0.01,0.005,0.001}in the hydraulic jump problem for Fr = 7.00.

of the Test Condition 11.1, which is characterized by an initial state with the upstream side of the channel full of water and the downstream dry, and by the sudden opening of a 0.7315 m (2.4 ft) wide and 0.18288 m (0.6 ft) breach, from the top of the dam, att= 0. This test was also used in [29], in order to verify the robustness of the Upwind scheme described in Section 4.

Thus, from the experimental measurements given in Test Condition 11.1 of [30], we selected the set of observed values of the water height shown in Table 5, that represents a stage-time hydrograph placed at x = 68.58 m, consisting of nobs = 10 observations for t ≤ 20 s. As a reference, the center of the dam is located at x = 60.96 m. In all the simulations, we adopted an uniform mesh of n_x = 400 cells,γ = 4.9 s and Courant number Cr = 0.1. The treatment of the dry bed was done as described in [29], with hdry = 10⁻⁵m. Also, due to the sensibility of the numerical model to the dry bed treatment, in this numerical experiment we only consider perturbations due to the parameter σ on the flow rate Q (obviously, the water height h is indirectly affected by these perturbations).

i t^obs_i (m) h^obs_i (m) i t^obs_i (m) h^obs_i (m)

1 4.0 0.00 6 10.0 0.04

2 4.5 0.01 7 11.0 0.04

3 5.0 0.02 8 13.0 0.05

4 7.0 0.04 9 15.0 0.05

5 9.0 0.04 10 20.0 0.05

Table 5: Observed data for the partial dam-break problem, taken from the experimental mea- surements of the Test Condition 11.1 of [30]. Data consists of an hydrograph atx= 68.58 m for t^obs ≤20 s.

The results, shown in Table 6, lead to an optimal Manning’s coefficient ξ^∗ = 0.0094 with small values of the standard deviation σ^∗ for ϑ ≥ 0.005, indicating a behavior close to the deterministic ones. For ϑ= 0.001, however, the methodology returned σ^∗ = 0.230, limited to this value in order to avoid the occurrence of unphysical (numerical) instabilities. The plots of the solutions obtained for ϑ= 0.001, compared with the observed data of Table 5, are shown in Figure 4. Finally, in Figure 5, we show the simulations fort= 30 s and t= 60 s, compared with

0 0.05 0.1 0.15 0.2 0.25 0.3

0 2 4 6 8 10 12 14

h(m)

x (m)

σ^∗= 0.0225, ξ^∗ = 0.0140 σ = 0, ξ|_σ=0 = 0.0140 Observations

0 0.05 0.1 0.15 0.2 0.25 0.3

0 2 4 6 8 10 12 14

h(m)

x (m)

σ^∗= 0.0460, ξ^∗ = 0.0140 σ = 0, ξ|_σ=0 = 0.0140 Observations

Figure 3: Simulations of the hydraulic jump problem for Fr = 7.00 constructed with the optimal parameters that were obtained with ϑ = 0.005 (top) and ϑ = 0.001 (bottom), assuming that t^obs = 240 s. The graphics display the superposition of all then_sim= 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameterσ^∗. The pictures also show the least-squares solution, that corresponds to the caseσ = 0.

the measured data also taken from the Test Condition 11.1 of [30]. These results show that the simulations performed with the parameters calculated with the time-stage information obtained at a single spatial point, fort≤20 s, can provide a good prediction of the flood induced by the dam break, at future time instants.

ϑ σ^∗ ξ^∗ ξ|_σ=0 L_ϑ(ξ^∗, σ^∗) L_ϑ(ξ|_σ=0,0) 1.000 0.002 0.0094 0.0094 9.99985221E-01 9.99985220E-01 0.500 0.002 0.0094 0.0094 9.99940884E-01 9.99940880E-01 0.100 0.002 0.0094 0.0094 9.98523148E-01 9.98523060E-01 0.050 0.002 0.0094 0.0094 9.94105668E-01 9.94105316E-01 0.010 0.002 0.0094 0.0094 8.62608850E-01 8.62600897E-01 0.005 0.003 0.0094 0.0094 5.53679297E-01 5.53655504E-01 0.001 0.230 0.0094 0.0094 1.02462106E-04 3.81057525E-07

Table 6: Optimal Manning’s coefficient, standard deviation parameter, and likelihood obtained for varying values of the precision-related parameterϑ∈ {1.0,0.5,0.1,0.05,0.01,0.005,0.001}in the partial dam-break problem.

0 0.01 0.02 0.03 0.04 0.05 0.06

0 5 10 15 20

h(m)

t (s)

σ^∗ = 0.230, ξ^∗ = 0.0094 σ = 0, ξ|_σ=0 = 0.0094 Observations

Figure 4: Simulations of the partial dam-break constructed with the optimal parameters that were obtained withϑ= 0.001. The graphic displays the superposition of all thensim = 100 sim- ulations associated with the optimal Manning’s roughness coeffcient ξ^∗ and standard deviation parameter σ^∗. The pictures also show the least-squares solution, that corresponds to the case σ= 0. In this case the graphic represents an hydrograph at x= 68.58 m for t^obs≤20 s.

It must be emphasized that in this experiment the estimates of Manning’s coefficient and variance were obtained using data from a single hydrograph at x = 68.58 m. Despite this, the forecasts (including uncertainty) shown in Figure 5 were very good with respect to the actual data which, in the estimation process, were considered to be unknown.

5.3 Open channel with growing flow-rate in the boundary

In this experiment, we consider an example employed in [19, 27, 6] for modeling open channels and estimating Manning’s coefficients, that simulates a flood in a rectangular open channel. The channel under consideration extends from x_I = 0 m to x_F = 3000 m. The transversal area is rectangular with a width of 5 m and the bed slope is 0.001. The initial conditions represent a subcritical flow with Q= 8.245 m³/s and height h = 1.2 m. The upstream boundary condition (x = 0) establishes that the flow rate increases linearly from its initial value to the maximum value of 200 m³/s in 20 minutes. After that, it decreases during 40 minutes, also linearly, reaching the initial value of 8.245 m³/s and remaining stationary thereafter.

x (m)

600 1200 1800 2400 2950 t(s)

720 5.7 4.5 3.3 2.2 1.4

1500 10.7 10.0 9.2 8.4 7.8 2220 10.7 10.7 10.6 10.6 10.5

Table 7: Observed data for open channel problem with growing flow-rate in the boundary. Data correspond to depths of the channel at different times and points.

In Table 7, we display measured depths at five different points and times 720 s, 1500 s, and 2220 s. We used these (training) data to predict depths at the same spatial positions and future times 3000 s, 3240 s, and 3480 s. Considering a uniform mesh of nx = 100 elements, Cr = 0.01, and γ = 16 s, we obtained a Manning’s coefficient ξ^∗ = 0.0248. The optimal deviations σ^∗ and the respective likelihoods are show in Table 8.

ϑ σ^∗ ξ^∗ ξ|_σ=0 L_ϑ(ξ^∗, σ^∗) L_ϑ(ξ|_σ=0,0) 2.000 0.00000 0.0248 0.0248 9.80454076E-01 – 1.000 0.00000 0.0248 0.0248 9.24078838E-01 – 0.500 0.00000 0.0248 0.0248 7.29182269E-01 –

0.100 0.00095 0.0248 0.0248 3.18653777E-03 3.72308107E-04 0.050 0.00095 0.0248 0.0248 2.00908073E-08 1.92136542E-14

Table 8: Optimal Manning’s coefficient, standard deviation parameter, and likelihood obtained for varying values of the precision-related parameter ϑ ∈ {2.0,1.0,0.5,0.1,0.05} in the open channel problem with growing flow-rate in the boundary.

Let us now focus on the case ϑ = 0.1, whose results for t^obs = 720 s, t^obs = 1500 s, and t^obs = 2200 s, are shown in Figure 6. Since, in this case, the estimated variance is strictly positive, the randomized Saint-Venant model produces a bundle of predictions for the future. Therefore, using simulation, predictions can be expressed in probabilistic terms. Table 9 displays the

predictions in the form of confidence intervals with their corresponding probabilities, obtained from n_sim= 100 simulations with standard deviationσ^∗.

Table 9 provides the type of information that is useful for a decision maker that only knows the data given in Table 7. For example, this table says that, at t = 3480 s, one may expect heights between 7.4 and 13.2 m with probability 0.9. However, if this risk is considered to be excessive and more conservative estimations are needed, the table shows that, with probability 0.99 the maximal height would not exceed 27.8 m. Finally, in order to illustrate the solution at a further time instant, in Figure 7 we show the simulation for t^obs = 3000 s and respective the observed data.

t x true Confidence intervals

99% 98% 95% 90%

3000 600 9.000 8.484 – 11.241 8.484 – 10.366 8.484 – 10.020 8.529 – 9.707 3000 1200 9.321 8.663 – 11.902 8.663 – 11.043 8.663 – 10.629 8.711 – 10.206 3000 1800 9.588 8.755 – 12.145 8.755 – 11.983 8.755 – 11.117 8.912 – 10.686 3000 2400 9.779 8.655 – 12.803 8.655 – 12.537 8.655 – 11.665 8.748 – 11.236 3000 2950 9.919 8.395 – 13.545 8.395 – 13.118 8.395 – 12.257 8.757 – 11.763 3240 600 8.157 7.558 – 11.956 7.558 – 11.633 7.558 – 10.141 7.558 – 9.397 3240 1200 8.868 7.838 – 12.662 7.838 – 12.477 7.838 – 10.773 7.838 – 10.189 3240 1800 8.903 7.973 – 14.099 7.973 – 11.977 7.973 – 11.514 7.973 – 10.854 3240 2400 9.164 8.168 – 15.377 8.168 – 13.729 8.168 – 12.343 8.337 – 11.417 3240 2950 9.349 8.381 – 16.131 8.381 – 15.825 8.381 – 13.364 8.381 – 11.911 3480 600 7.174 6.523 – 15.255 6.523 – 15.042 6.523 – 11.048 6.523 – 9.690 3480 1200 7.679 7.168 – 16.489 7.168 – 15.460 7.168 – 12.539 7.227 – 10.276 3480 1800 8.091 7.569 – 19.465 7.569 – 17.534 7.569 – 13.771 7.569 – 10.709 3480 2400 8.414 7.859 – 23.285 7.859 – 20.035 7.859 – 15.459 7.859 – 12.247 3480 2950 8.644 7.888 – 22.972 7.888 – 22.809 7.888 – 16.928 7.888 – 11.957 Table 9: Predictions of the open channel problem with growing flow-rate in the boundary at t ∈ {3000 s,3240 s,3480 s}, constructed with the optimal parameters that were obtained with ϑ= 0.1. The table shows predictions in terms of intervals with different degrees of confidence that are based on thensim = 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameter σ^∗. Column “true” displays synthetic data that were generated along with the training data shown in Table 7, but that were not used in the process of calculating the optimal model parameters.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 20 40 60 80 100 120

zb+h(m)

x (m) z_b σ^∗ = 0.230, ξ^∗ = 0.0094 σ = 0, ξ|_σ=0 = 0.0094 Experimental data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 20 40 60 80 100 120

zb+h(m)

x (m) zb

σ^∗ = 0.230, ξ^∗ = 0.0094 σ = 0, ξ|_σ=0 = 0.0094 Experimental data

Figure 5: Simulations of the partial dam-break at t = 30 s (top) and t = 60 s (bottom), con- structed with the optimal parameters that were obtained withϑ= 0.001. The graphics display the superposition of the water level z = h+z_b of all the n_sim = 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameterσ^∗. The pictures also show the least-squares solution, that corresponds to the case σ = 0. In this case, simulations representpredictions, since observed data correspond to t≤20 s and simulations correspond tot= 30 s (top) andt= 60 s (bottom). The experimental data shown in the graphs were taken from [30] to assess the quality of the prediction but, as shown in Table 5, were not

0 2 4 6 8 10 12

0 500 1000 1500 2000 2500 3000

h(m)

x (m)

σ^∗= 0.00095, ξ^∗ = 0.0248 σ = 0, ξ|_σ=0 = 0.0248 Training data

0 2 4 6 8 10 12

0 500 1000 1500 2000 2500 3000

h(m)

x (m) σ^∗ = 0.00095, ξ^∗= 0.0248

σ= 0, ξ|_σ=0= 0.0248 Training data

0 2 4 6 8 10 12

0 500 1000 1500 2000 2500 3000

h(m)

x (m) σ^∗ = 0.00095, ξ^∗= 0.0248

σ= 0, ξ|_σ=0= 0.0248 Training data

Figure 6: Simulations of the open channel problem with growing flow-rate in the boundary at t^obs = 720 s (top), t^obs = 1500 s (middle), and t^obs = 2220 s (bottom), constructed with the optimal parameters that were obtained with ϑ = 0.1. The graphics display the superposition of all thensim= 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameter σ^∗. The pictures also show the least-squares solution, that corresponds to the case σ= 0.

2 4 6 8 10 12 14 16

0 500 1000 1500 2000 2500 3000

h(m)

x (m) σ^∗ = 0.00095, ξ^∗= 0.0248

Test data

Figure 7: Simulation of the open channel problem with growing flow-rate in the boundary at t = 3000 s, constructed with the optimal parameters that were obtained with ϑ = 0.1. The graphic displays the superposition of all the nsim = 100 simulations associated with the optimal Manning’s roughness coeffcientξ^∗ and standard deviation parameterσ^∗. The picture also shows the least-squares solution, that corresponds to the case σ = 0. In this case, the simulation represents a prediction, since observed data correspond to t^obs ∈ {720 s,1500 s,2220 s} while the simulation corresponds to t= 3000 s. The test data shown in the graph are synthetic data (shown in column “true” of Table 9) that were generated along with the training data, but were not used in the process of calculating the optimal model parameters.

6 Conclusions

One of the main reasons why one makes use of mathematical models is to obtain information that is not explicitly present in the data. Except in very special cases, mathematical models cannot provide such information with absolute certainty. Relying on model forecasts over and above their actual limitations can lead to disastrous decisions. For this reason, models that suggest different possibilities for the predicted variables together with the corresponding probabilities are preferable to purely deterministic ones.

Deterministic models are well known for their successful predictions in the fundamental branches of Physics. Often, these models consist of systems of partial differential equations whose numerical solution is the subject of in-depth studies reported in the specialized literature.

It is therefore natural to rely on these models to produce stochastic counterparts that allow us to obtain sensible forecasts with their variations and uncertainties. In this paper, the physical problem under consideration was the flux of water in channels and the deterministic model over which we developed the stochastic counterpart was given by the Saint-Venant equations.

Examples taken from the Hydraulic literature were analyzed in order to corroborate the reliability or the new approach. These examples corroborate that Saint-Venant equations were very effective for defining a reliable underlying deterministic model. The simulations obtained by the stochastic approach were able to produce sensible bundles of possibilities for unknown variables, including useful confidence intervals and the corresponding probabilities. We believe that these are useful tools for decision-makers.

We are aware that the proposed approach does not contemplate explicitly correlations be- tween different state variables. However, we believe that the trick of considering all the states as a single observation in a high-dimensional space (which opposes to the computation of like- lihood as a product of individual probabilities) alleviate this drawback. Needless to say, our procedure makes it possible to obtain results with a moderate amount of computer work. A natural extension would be the estimation of a full covariance matrix related to successive stages of the system. This would involve very hard computer work even using the analytic tools of Gaussian processes. Computing full covariance matrices would be probably impossible using our optimization-simulation approach. Nevertheless, intermediate alternatives are possible and will be the subject of future study. One of these alternatives consists in assuming that non-null correlations appear only when spatial states are close in spatial and temporal terms. Such an assumption would alleviate computer work, as the number of correlations that we would need to estimate could be drastically reduced.

References

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[2] M. Kh. Askar and K. K. Al-jumaily, A nonlinear optimization model for estimating Man- ning’s roughness coefficient, in Proceedings of the Twelfth International Water Technology Conference, IWTC12, Alexandria, Egypt, 2008, pp. 1299–1306.