# Coupling Mage with Melissa to compute ubiquitous Sobol indices for river hydraulics

Terraz Théophile, Mendez-Rios Felipe

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## To cite this version:

Terraz Théophile, Mendez-Rios Felipe. Coupling Mage with Melissa to compute ubiquitous Sobol indices for river hydraulics. Proc. 7th SimHydro conference, Nov. 2023, Chatou, France, Nov 2023, Chatou, France. hal-04758745

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# COUPLING MAGE WITH MELISSA TO COMPUTE UBIQUITOUS SOBOL INDICES FOR RIVER HYDRAULICS

Théophile, Terraz<fnref n="1" />, Felipe, Mendez-Rios

INRAE, UR RiverLy, centre de Lyon-Villeurbanne, 5 Rue de la Doua, 69100 Villeurbanne, France

theophile.terraz@inrae.fr, felipe-alberto.mendez-rios@inrae.fr

## KEY WORDS

Global sensitivity analysis, 1D river modeling, ensemble runs, in-situ data analysis.

## ABSTRACT

In global sensitivity analysis, ubiquitous Sobol indices are used to quantify the effect of the variation of some input parameters of a model on the variance of the model outputs at each time-step and at each spatial point of the model. This paper introduces the coupling of the 1D numerical solver for transient open-channel flows Mage with Melissa, a framework for large scale in-transit sensitivity analysis. This framework is fault tolerant and can manage the specific cases in which some sets of parameters can lead to a divergence in the numerical solver that causes an early stop of the simulation. In addition, as the Sobol index estimators are computed iteratively and in-transit, the analysis is run in parallel to the simulations and results are available as soon as the last simulation ends. Moreover, as the simulation results are not stored on the file-system, it allows to run large scale sensitivity analysis without extra cost on disk usage and read/write overhead. The Lower Seine River, a tide river, is being used to demonstrate the quantification of the influence of a set of Strickler coefficients on the water elevation and the flow rate over a tidal cycle. The Lower Seine River model is segmented into 14 zones, each with independent constant Strickler coefficients. Tidal transient models are challenging to calibrate because the Strickler coefficients of a single zone influences the water elevation and the flow rate in different cross-sections at different time-steps of the model. In our study, the Strickler coefficient values are sampled from a uniform distribution using a Monte Carlo sampling method, the corresponding simulations are run on a desktop computer and the results are aggregated iteratively and in-transit by the Melissa server to generate ubiquitous Sobol index maps. Melissa provides first order and total order Sobol indices, as well as their 95% confidence interval. The Sobol indices are useful to identify the most influential Strickler coefficients for each zone at each time-step, leading to a better understanding of the model. The modeler can then adjust the model by either defining more relevant zones for the Strickler coefficients, or fine-tuning the most influential coefficients to fit the model results to field data, with setting the other ones once and for all. This approach can be applied to any other uncertain input parameter.

## 1. INTRODUCTION

When creating a digital river model, the modeler pays particular attention to the model calibration process. The modeler has to tune calibration variables to ensure that his model produces results that are consistent with the values measured in situ. However, these calibration variables are numerous and their effects are not known precisely in advance. This is particularly obvious when the calibration variables do not have a constant effect in time and space. In this case, the modeler can rely on the calculation of statistical indicators to quantify precisely the effect of each calibration parameter on the outputs of the model at each point in time and space. This is the role of ubiquitous Sobol indices [12]. Sobol indices quantify the impact of the variability of a model input on the model output. Their values are between 0 and 1: the higher the value, the greater the influence of model input variation on model output.

Calculating ubiquitous Sobol indices requires the model to be run a large number of times with different sets of parameters. Fortunately, tools have been developed to perform this task efficiently. This is the purpose of Melissa [13], an in-transit statistical analysis framework. In this paper, we present the coupling between Mage [5], a one-dimensional solver for river hydraulics, and Melissa. We illustrate this through the computation of ubiquitous Sobol indices for the Strickler friction coefficients in a model of the Lower Seine River. This example is particularly interesting because the Lower Seine River is strongly influenced by tidal

<footnote n="1">1 Corresponding author</footnote>
cycles [2][8]. This type of model is very difficult to calibrate because friction coefficients have a different effect on water level and flow rate depending on the phase of the tide. We show that computing the ubiquitous Sobol indices of the Strickler coefficients gives very rich and helpful information to the modeler for the calibration of his model with a relatively low effort.

## 2. MAGE AND MELISSA

### 2.1 Mage

Mage is a one dimensional solver for the Barré-de-Saint-Venant equations, which include the mass conservation equation (Eq. 1) and the momentum conservation equation (Eq. 2):

$$ \frac{\partial S}{\partial t} + \frac{\partial Q}{\partial x} = q_{lat} \qquad (1) $$

$$ \frac{\partial Q}{\partial t} + \frac{\partial}{\partial x} \left( \beta \frac{Q^2}{S} \right) + gS \frac{\partial Z}{\partial x} = -g \frac{Q |Q|}{K_s^2 S R_h^{4/3}} - gS J_s + k q_{lat} \frac{Q}{S} \qquad (2) $$

with S the wetted cross section, Q the water flow rate, $q_{lat}$ a lateral input or output, Z the water surface elevation, $R_h$ the hydraulic radius, $J_s$ a singular head loss and $k=1$ if $q_{lat}<0$, $k=0$ otherwise. Mage uses a Manning-Strickler law to account for the friction head loss, with $K_s$ a Strickler coefficient. These equations are solved using the Preissmann finite difference scheme, with an iterative Newton-Raphson method for solving the corresponding non-linear system [5]. Mage uses the Debord formulation to allow the flow calculation for compound channels, assuming the same energy slope for both the minor and the major bed [9]:

$$ Q_m = \varphi K_{sm} S_m R_{hm}^{2/3} J^{1/2} \qquad (3) $$

$$ Q_M = \left[ 1 + \frac{S_m}{S_M} (1 - \varphi^2) \right]^{1/2} K_{sm} S_M R_{hm}^{2/3} J^{1/2} \qquad (4) $$

with $\varphi \approx 0.9$ ($K_{sm}/K_{sM}$), subscript m and M correspond to the minor and major bed respectively. Mage is developed by INRAE (French Research Institute for Agriculture, Food and Environment), its source code is written in FORTRAN and can be downloaded here: https://gitlab.irstea.fr/jean-baptiste.faure/mage.

### 2.2 Melissa

Melissa is a framework developed by INRIA (French Research Institute for Informatics and Automatics) that enables large scale ensemble runs and, in particular, global sensitivity analysis with ubiquitous Sobol Indices [13]. Melissa is based on two key concepts: iterative algorithms to compute statistic estimators and in-transit asynchronous data transfer between the solvers that generate the data, and the analysis server that computes the statistic estimators. Melissa architecture is based on three main components: the server, the launcher and the clients.

Melissa clients: the numerical solver producing the data, Mage in our case. Clients are the members of the ensemble run. This part will be discussed in details in section 2.3.

Melissa launcher: a front-end Python script in charge of orchestrating the execution of the study. The launcher handles the interactions with the batch scheduler. These interactions include job submissions (clients and server), job status monitoring and fault tolerance.

Melissa server: a parallel executable that can receive data from the clients and update statistics on-the-fly. The server is also in charge of the design of experiment: it generates the members parameter sets and asks the launcher to submit the corresponding simulations to the batch scheduler. It runs from the beginning to the end of the study in its own job.

This architecture avoids the read/write bottleneck between the ensemble run and the analysis, and enables clients to be run in any order in an elastic asynchronous way. The server discards the data as soon as
the estimators are updated, saving storage on disc and I/O time. This architecture allows for a robust fault tolerance mechanism. Melissa is mostly written in Python and can be downloaded here: https://gitlab.inria.fr/melissa/melissa.

## 2.3 Coupling Mage with Melissa

Usually, a numerical solver produces results in the form of files written on a hard drive. In the scope of a Melissa study, this kind of output is not usable. The server needs the solver to send its data directly through the network: the solver has to become a client. To achieve that as seamlessly as possible, Melissa provides an API that can be plugged directly in the source code of the solver. It comes in the form of a dynamic library that provides three functions to call from the solver source code. This API is available in Python, C, FORTRAN 77 and FORTRAN 90. It is really close to the MPI API, except that the developer only needs to take care of the client side, with only 3 functions: melissa_init, melissa_send and melissa_finalize.

In Mage, we chose to define a dedicated output time step for the Melissa output. In that way, the user can do high frequency Melissa output and low frequency or no disc outputs. In addition to the melissa_send call at each output time step, Mage also sends the initial state and the final state of the simulation to the server. It needs to be taken into account because in the setup of the ensemble model, the user needs to tell the server how many times the function melissa_send will be called from each simulation. Coupling Mage with Melissa requires no additional work.

# 3. THE LOWER SEINE RIVER MODEL

## 3.1 Hydraulic Model

The case study is the Lower Seine River in Normandy, France (Figure 1). This river is characterized by strong tidal influence, leading to an amplitude variation between 3 m and 8 m downstream at the river mouth [1]. This tidal effect propagates approximately 160 km upstream until the Poses dam, where tidal propagation is dissipated [4]. The mean annual liquid flow entering Poses from upstream is estimated at 450 m³/s. Flood flows typically exceed 700 m³/s, while low-water flows remain under 200 m³/s [14]. Fluctuations in water level not only lead to rapid decline in flow rate but also introduce significant uncertainties into the measurements [3].

In tidal river, two periods are identified: one from low water (LW) to high water (HW), which has a shorter duration compared to the second period (from HW to LW) in this case. However, in terms of surface velocity, the period from LW to HW exhibits higher values (ranging from 1-2.5 m/s) compared to the period from HW to LW (ranging from 0.7-1.2 m/s) [7]. During the summer season, lower water levels are observed, leading to a reduction of the river flow's impact on tidal propagation from upstream to downstream. At the river mouth, the influence of low-water flow would be minimal. Conversely, a high water flow can push water masses out of the estuary [4].

This river is subject to some of the most significant anthropogenic pressures in France due to its industrial, agricultural, and maritime activities [6], which require continuous monitoring. In the same context, this estuary is well-instrumented with 17 tide gauges operated by Grand Port Maritime de Rouen (GPMR), providing water level data at a 5-minute interval. Additionally, gauging campaigns have been conducted to capture various tide states, reflecting the propagation towards the river mouth. Measurements taken at Rouen indeed illustrate the flow influenced by fluvial effects. At Heurteauville, previous gauging campaigns provide additional data to validate upcoming models. At Aizier, the boundary of brackish water up-welling is defined [10].
<figure><img src="image_3.png" /><figcaption>**Figure 1:** Lower Seine River estuary from its outlet at sea off Honfleur and Le Havre to the Poses dam</figcaption></figure>

To set up the model, some parameters are essential, including the number of reaches, cross-sectional geometry, Strickler friction coefficients ($K_s$), upstream discharge and downstream water elevation. Each reach is characterized by a sequence of cross-sections that provide information on the dimensions of the main channel and the floodplain. A model was built by SPC DREAL Normandie [2] using the Mascaret module, subsequently transferred to the Mage module by INRAE with modifications from original model.

The upstream boundary condition of the model is a discharge time series of the Seine at Poses. The downstream boundary condition is the stage time series of the Seine at Tancarville, reflecting the tidal signal (Figure 2). The structure of the Mage model has been split into two reaches, to plug the Eure tributary flow rate as a punctual intake between these two reaches. The upstream reach is composed of 34 cross-sections, while the downstream is composed of 568 cross-sections, each cross-section being represented by 24 data points. In each reach, a number of zones is also defined based on the number of friction coefficients to calibrate the model, assuming constant friction throughout the zone. The zones have been defined such as an hydrometric station installed on the river is present in the middle of each zone. This results in the upstream reach having only one zone, while the downstream has 13 zones.

<figure><img src="image_4.png" /><figcaption>**Figure 2:** Stage time series of the Seine at Tancarville, the downstream boundary of the model</figcaption></figure>
A friction coefficient value must be given for each zone. In hydrodynamic modeling, these are the main calibration parameters, and the spatial variability of roughness combined with unsteady flow complicates their manual calibration. Moreover, in a tidal transient model the Strickler coefficient of a single zone can influences the water elevation and the flow rate in different cross-sections at different time-steps of the model. We focus in this example on the main channel Strickler coefficients $K_{sm}$. Before doing an automatic calibration of the $K_{sm}$ and to better understand the influence of these parameters, we can perform a global sensitivity analysis to assess the influence of these coefficients on the output fields of the model, namely the water level Z and the flow rate Q at each cross-section and at each timestep. We will simulate 71 hours to capture several tidal cycles.

## 3.2 Ensemble Model

To set up our sensitivity analysis we need to introduce variability to the main channel Strickler coefficients. We will start with an initial guess of $K_{sm} = 37$ for all the zones. The goal here is not to find the right $K_{sm}$ values to calibrate the model but to create some variance at the output to quantify the influence of each $K_{sm}$ on Z and Q at each cross-section and at each time step. Melissa needs a user-defined function to generate the parameter sets. Without prior knowledge, we can ask the Melissa server to randomly draw the 14 $K_{sm}$ independently between -10 and +10 around the initial guess with a continuous uniform probability distribution using the Python random module.

The Melissa server will draw independent sets of parameters and combine them to create the design of experiment described in [13]. The Melissa server will copy the original study and modify the friction coefficient file accordingly for each member, as we specified in the “preprocessing” section of the config.json file. If a simulation stops before completion, Melissa will try to re-launch it. At this point, we can not know in advance if some parameter sets generated by Melissa will lead to simulation failures. If the fault comes from the parameter set (a “bad” parameter set) then the simulation will fail again. As failures from other sources than a bad parameter set are unlikely to occur several times in a row for the same member, we set a limit of 2 failures before Melissa draws a new parameter set for this simulation.

In our example the Lower Seine model is very small and runs very fast (around 20 seconds). The more samples we have, the more accurate the statistic estimators will be. We can afford to ask Melissa for 10 000 iterations for this study. Melissa uses the pick-freeze Monte-Carlo method to compute Sobol indices estimators [11]. With this algorithm, each iteration is composed as a group of N+2 members, with N the number of varying parameters. In our case, 14 varying parameters lead to groups of 16 simulations, for a total of 160 000 simulation runs for the whole study. For our example, we request Melissa to compute the ubiquitous estimators of the mean, variance, first order and total order Sobol indices of the 14 $K_{sm}$ for both Z and Q outputs of the Seine model. As Melissa computes the statistic estimators through *in-transit* data transfer, we can ask for an output at every simulated hour of the study for each simulation without worrying about disk usage.

# 4. RESULTS

## 4.1 Raw Data

The ensemble study was run on a desktop computer. It is equipped with a Intel® Core™ i9-10900K CPU @ 3.70GHz. The server was running on one core and it launched 10 000 groups of 16 simulations, with one core per simulation. In our case a group is run in a single MPI MPMD call, so a total of 16 cores per group. The overall study took 63 hours and produced a total of 68 MB of data. If we did not analyze the data *in-transit* with Melissa, the outputs of all the 160 000 simulations in the Mage binary format would have weighted 63 GB. Melissa’s fault tolerance mechanism detected 1416 simulation crashes in 1411 different groups during the study. We can not know why they failed because simulation results are not stored on disk. After they were automatically re-launched by Melissa, 16 of these simulations failed a second time in 16 different groups: Melissa consider that these crashes are due to bad parameter sets. Melissa keeps record of these parameter sets if one wants to analyze the reason of these crashes, which we will not do here. Because all the
simulations in a group are in the same MPI communicator, when Melissa re-launch a simulation it has to relaunch the whole simulation group. In our case it led to a total of 22 832 additional simulations, which correspond to 12% of the computation time.

## 4.1 Visualization and Interpretation

Melissa outputs its results in the form of a text file for each statistic estimator for each output field and for each time step. A file contains one value per line, in the order of the internal mesh numbering sent by the solver to the server. The Melissa server does not know about the geometry, so the values have to be mapped to the mesh afterward for visualization. In our case, with 1D fields, we can visualize the estimators along the section number axis. We plot various figures using a simple Python + PyQt script. The parameters, numbered from 1 to 14, correspond to the Strickler coefficients $K_{sm}$ in the 14 zones, from upstream to downstream. As mentioned previously, the river model divided into zones from left to right, upstream to downstream, and each zone comprises a different number of cross sections. Vertical dashed lines show the boundaries of the different zones in the x axis. This allows the visualization of Sobol indices either alongside other estimators with the $K_{sm}$ number in the y axis (Figure 3), for all parameters together at a given time step (Figure 4), or over all the time steps for a given parameter (Figures 5 and 6). Melissa also provides a 95% confidence interval for the Sobol indices estimators as described in [13] and shown in Figure 6. The smaller the confidence interval, the closer the estimator is to the real value.

The first case is particularly useful to compare the Sobol indices to the variance: a high Sobol' index value is only significant if it can explain some variance. For example, Figure 3 shows the first order Sobol' indices for the $K_{sm}$ in the third zone at t=64800 seconds (18 hours) on the water surface elevation. We can see that the Sobol indices are high around cross sections 100 to 200, so this parameter can be tuned to match water elevation measurements in that area. However the variance there is low, meaning that the $K_{sm}$ values need a wider range of variation to have a significant impact on the water elevation. In Figure 3 higher variance values are near the downstream end of the river. One can then attempt to find the parameters that have high Sobol index values around that zone using Figure 4. We can observe there that the Strickler coefficient of the zone number 14 has a significant influence in the downstream area: it means that the model is sensitive to the variations of the fourteenth $K_{sm}$ in that area at that time step. However, when quantifying this influence on the variance of water elevation while varying Strickler coefficients (as indicated by the order of magnitude of variance at bottom of the Figure 3), it remains minimal in contrast to their significant influence on flow rate, as seen at bottom of Figure 5. Influences of the $K_{sm}$ on the flow rate is not further discussed in this paper for the sake of conciseness. Nevertheless, it is essential to emphasize the strong relationship between these two output variables and their sensitivity to the variation of the friction coefficients.
<figure><img src="image_5.png" /><figcaption>**Figure 3:** First order Sobol indices of the Strickler coefficient of the third zone on the water elevation with mean and variance of the water elevation at t=18 hours</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4:** First order Sobol indices of all the Strickler coefficients on the water elevation at t=18 hours</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5:** First order Sobol indices of the Strickler coefficient of the third zone on the flow rate with mean and variance of the flow rate at t=18 hours</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6:** Confidence intervals of the first order Sobol indices of all the Strickler coefficients on the water elevation at t=18 hours</figcaption></figure>
We will now look at the evolution of the Sobol indices of the $K_{sm}$ on the water elevation Z trough time.
**Figure 7** shows that the $K_{sm}$ of the third zone does not have a constant influence on Z in space and time: it depends on the tidal cycles. Indeed, tide effects cause fluctuations in the downstream water level, which in turn lead to variations in the flow rate. These fluctuations translate into flow rate variations ranging from negative to positives values, signifying changes in flow direction. Additionally, these oscillations lead to instances where the flow rate crosses through a zero value. The bed friction do not have any effect on the flow when the velocity is zero, which explains why some time steps exhibit Sobol indices near zero.

<figure><img src="image_9.png" /><figcaption>**Figure 7:** First order Sobol indices of the third Strickler coefficient on the water elevation at every timesteps</figcaption></figure>

A similar effect can be observed in **Figure 8** for the $K_{sm}$ of the fourteenth zone. It is crucial to emphasize the propagation and dissipation of tide effects from downstream to upstream, particularly during time steps when flow rates are near zero, such as between 8 hours and 14 hours, as indicated by the increase in Sobol indices.

If the Sobol indices of a given parameter consistently remains low in space and time it means that this parameter has a minimal overall influence on the outputs of the model. In such cases, the modeler can choose to set a default fixed value for this parameter (keep the initial guess for example). Alternatively, some zones could be consolidated to reduce the number of parameters requiring calibration. In both cases, the objective is to simplify the model calibration process.
<figure><img src="image_10.png" /><figcaption>**Figure 8:** First order Sobol indices of the fourteenth Strickler coefficient on the water elevation at every time step</figcaption></figure>

The ubiquitous variance map is shown in **Figure 9**. It is useful to understand the overall effects of the parameters on the whole domain. We can remark that the variance is low around the boundaries and the initial time: this is due to all the members of the ensemble run having the same boundary conditions and initial conditions.

<figure><img src="image_11.png" /><figcaption>**Figure 9:** Ubiquitous variance of the water elevation</figcaption></figure>
To compare the relative influence of the Strickler coefficients in different times and locations we can multiply the Sobol indices by the normalized ubiquitous variance. This lead to indices between 0 and 1 where the values are high only if both the Sobol index and the corresponding variance are high (Figure 10).

<figure><img src="image_12.png" /><figcaption>**Figure 10:** Normalized first order Sobol indices of the third Strickler coefficient on the water elevation at every time step</figcaption></figure>

We can see in **Figure 10** that in fact the third zone $K_{sm}$ has a small overall effect on the water elevation at timestep number 18 as discussed in **Figure 3**, but we can also remark that it has more effect later, at time step 24. We can have a closer look at this time step in **Figure 11**. We see for that time step that the high Sobol indices values corresponds to the high variance area. The third $K_{sm}$ has a real influence on the water elevation at that time step between the 80th and the 160th section approximately.
<figure><img src="image_13.png" /><figcaption>**Figure 11:** First order Sobol indices of the Strickler coefficient of the third zone on the water elevation with mean and variance of the water elevation at t=23 hours (time step 24)</figcaption></figure>

All the examples given before can be extended to all the parameters and for all the output values of the model (in our case: the water elevation and the flow rate) and for the total order Sobol indices.

## 5. CONCLUSIONS

Coupling Mage with Melissa enable the computation of ubiquitous statistic estimators for river hydraulics with low efforts for the modeler. The light computational cost of 1D hydraulic models associated with Melissa *in-transit* and iterative data processing allows to run ensemble studies on a desktop computer without compromising on the quality of the results. In particular ubiquitous Sobol indices can produce very rich information on the behavior of a model. In this study we focused on the main channel Strickler coefficients but we can extend the sensibility analysis to the boundary conditions, the initial conditions, the tributary inputs, the flood plain friction coefficients or any other uncertain model input variable or parameter. Moreover, all the results proposed above can be visualized dynamically using the sliders or directly exported in the form of videos. All these results can lead to a better understanding of the model and can help for the next step in the modeler agenda: the calibration of the model.

## ACKNOWLEDGEMENTS

Thanks to Flavie Druine, Julien Deloffre, and the Lower Seine and Normandy Coastal Hydrometry Unit (DREAL Normandie) for the gauging campaigns, Stéphane Piney, and Marie Morin (SPC DREAL Normandie) for the 1D model of the Lower Seine.

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