Quantifying uncertainty in hydrological systems: A Bayesian point of view

Past decades have seen a flurry of activity in Bayesian applications in hydrology. These applications include those to quantify model and parameter uncertainty, develop alternatives to reduce structural uncertainty through sensible averaging procedures, to new alternatives of defining uncertainty by relaxing the framework for specifying model likelihoods. Additionally, hydrologists are starting to adopt data assimilation as a new way to both reduce predictive uncertainty, and also to assess where assumed model structures may not be fully adequate. This session invites contributions involving new and innovative ways of using Bayesian methods for the type of hydrological problems mentioned above, as well as other emerging problems that such techniques have been put for use in.

Convener: Ashish Sharma | Co-Conveners: Ilaria Prosdocimi, Dmitri Kavetski, Lucy Marshall, Mojtaba Sadegh, Alberto Viglione
| Thu, 02 Jun, 08:30–10:00|Room Auditorium Pasteur

Orals: Thu, 02 Jun | Room Auditorium Pasteur

Chairpersons: Alberto Viglione, Ashish Sharma, Ilaria Prosdocimi
Alexandre Devers, Claire Lauvernet, and Jean-Philippe Vidal

Daily climate projections have been recently created using several global climate models under different scenario emissions for their specific use as inputs for global hydrological models (Lange et Büchner., 2021). Despite the interest of these data, their coarse resolution prevent them to be used for hydrological models at the catchment scale and even more at the reach-scale.

This work therefore proposes a novel approach to make use of both global climate and global hydrology information and derive catchement-scale hydrological time series through a distributed hydrological model and a specifically designed data assimilation method. The final objective is to produce future streamflow series over 6 catchment case studies in Europe as part the DRYvER project (Datry et al., 2021). To that end, coarse climate forcings are downscaled through a spatial analogue procedure on precipitation, temperature, and potential evapotranspiration to derive ensemble time series of meteorological forcings. This ensemble then serves as an input to the distributed hydrological JAMS-J2K model (Krause et al., 2006). Finally, the hydrological distributed model JAMS-J2K is constrained by global hydrological time series available at the catchment outlet using a Particle Filter data assimilation procedure (van Leeuwen, 2009).

This setup is first applied to DRYvER catchments during the historical period 1995-2015, securing independent observations over the catchments to evaluate the impact of data assimilation. These experiments put forward the improvement of high-resolution hydrological simulations resulting from the assimilation of discharge from a global hydrological model, with respect to a simulation only driven by coarse meterological time series. Furthermore, by construction, the method allows deriving daily high-resolution hydrological time series coherent with global hydrological time series, paving the way for reach-scale hydrological projections under climate change scenarios.


Datry et al. (2021) Securing Biodiversity, Functional Integrity, and Ecosystem Services in Drying River Networks (DRYvER). Research Ideas and Outcomes. In press.

Krause et al. (2006) Multiscale investigations in a mesoscale catchment: hydrological modelling in the Gera catchment. Advances in Geosciences. doi:10.5194/adgeo-9-53-2006.

van Leeuwen (2009) Particle Filtering in Geophysical Systems. Monthly Weather Review. doi:10.1175/2009MWR2835.1.

Lange et Büchner (2021) ISIMIP3b bias-adjusted atmospheric climate input data (v1.1). ISIMIP Repository. doi:10.48364/ISIMIP.842396.1.

How to cite: Devers, A., Lauvernet, C., and Vidal, J.-P.: Data assimilation of global hydrological model discharge to constrain a distributed hydrological model, IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-439, 2022.

Hae Na Yoon, Lucy Marshall, Ashish Sharma, and Seokhyeon Kim

We launch a novel approach for hydrologic modeling in ungauged basins using satellite data. Due to universal availability, satellite data is an attractive option to fill the absence of in-situ data to calibrate a hydrologic model in ungauged or poorly gauged basins. The one specific satellite-derived calibration-measurement ratio (C/M ratio) is useful because of its physical property to indicate the water extent and its demonstrated correlation with observed streamflow. However, there are challenges in calibrating a hydrologic model using the C/M ratio because it has a different dimension to streamflow. Therefore, a new approach is required to use the modeled surrogate streamflow instead of the raw C/M ratio in place of streamflow. A new Bayesian approach is introduced here to identify parameters of surrogate streamflow in the absence of a time series of streamflow. This approach calibrates the conjugated probability of the parameter set of a hydrologic model and a surrogate streamflow model. Specifically, the proposed likelihood includes supplementary information, such as an estimated mean value of streamflow, to join the information of streamflow volume and dynamics of the modeled flow. Our new approach is assessed for multiple Australian Hydrologic Stations with distinct attributes. The strength in the new method is demonstrated with high Nash-Sutcliffe Efficiency values (0.535 ~ 0.781), and the uncertainties in the new model calibration are quantified via Markov Chain Monte Carlo sampling. The errors of the surrogate streamflow model and the hydrologic model are analyzed, and the predictive intervals are assessed with the benchmark model derived from the in-situ streamflow. Overall, our work improves previous studies on the hydrological predictions using the C/M ratio. Furthermore, it enables surrogate data to be highly correlated to the actual data, regardless of their dimensions.

How to cite: Yoon, H. N., Marshall, L., Sharma, A., and Kim, S.: Bayesian model calibration in ungauged catchments via satellite data, IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-313, 2022.

Igor Gejadze, Hind Oubanas, and Pierre-Olivier Malaterre

Models with spatially distributed parameters are commonly used in geophysical sciences. An important problem is to estimate the model's driving conditions using incomplete observations of the state variables,  obtained in-situ or remotely. This problem could be ill-posed when the information content of observations is not sufficient. The situation is further aggravated if the parameters of the model are not well known. A usual approach is to consider the extended formulation where the driving conditions and parameters are estimated simultaneously. The extended problem, however, often suffers from equifinality. For the high-dimensional distributed models (including those in hydraulics and hydrology applications) the variational approach is often used due to its computational feasibility. In this approach one looks for the mode of the posterior pdf using minimization methods. However, if the uniqueness of the solution is not guarantied, the posterior pdf could be multimodal or even infinite-modal, i.e. the mode may no longer be considered as a useful representation of the posterior pdf. In contrast, the mean of any (bounded) pdf is unique. The posterior mean is computed in full Bayesian methods, such as MCMC for example. However, these algorithms are not useful in high-dimensional estimation.

We present a hybrid semi-Bayesian method which combines the elements of Bayesian and variational estimation. Here, each distributed coefficient is represented as a product of its mean value and the 'shape' function. The latter are estimated using the variational estimation, whereas the means - using the direct Bayesian approach. The two steps constitute an estimation cycle, which is repeated after the information exchange. In a broad sense, the method can be considered as a variant of the variational Expectation-Maximization method. The method is illustrated in application to the convection-diffusion transport model (generalized Burgers equation). Next, the results of solving the river discharge estimation problem in the SWOT data processing context are presented. In the latter case the full Saint-Venant equations based hydraulic model SIC is used. 

How to cite: Gejadze, I., Oubanas, H., and Malaterre, P.-O.: A new semi-Bayesian robust estimation method for spatially distributed models, IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-115, 2022.

Léo Pujol, Pierre-André Garambois, Jérôme Monnier, Kevin Larnier, Lilian Villenave, and Robert Mosé
In a context of climate change and potential intensification of the hydrological cycle, improving representation of water fluxes within river basins is of paramount importance  for hydrological sciences and operational forecasts. To leverage multi-sourced observations (in situ, satellite, drones) of the critical zone, innovative approaches integrating hydraulic-hydrological modeling and multi-variate assimilation methods are needed. They should enable ingesting spatially distributed forcings, physiographic descriptors, hydrodynamic signatures from multi-source observables, and tackle calibration/correction problems in integrated models. Water surface observables are valuable to constrain hydraulic models of river reaches ([1] and references therein) and complex flow zones (confluences, floodplains), forced by spatially distributed inflows ([2], [3]). Since hydraulic large scale modeling can be computationally costly, a combination of effective 1D representations and 2D zooms, completed by hydrological modules, may be useful. This contribution presents the development of a complete multi-dimensional hydraulic-hydrological toolchain, based on the 2D hydraulic model and variational data assimilation platform DassFlow [4,5]. A new method for multi-dimensional hydraulic modeling, relying on a single 2D 2nd order solver applied to 1Dlike-2D meshes, is presented. Inferences of large composite control vectors, including hydrological and hydraulic controls, are carried out on academic and real cases in twin experiments. Accurate results are achieved given sufficient observability of parameters signatures, including information feedback from the river network to upstream hydrological catchments models.
[1] Larnier et al. "River discharge and bathymetry estimation from SWOT altimetry measurements", Inverse Problems in Science and Engineering 29, 6 (2021), pp. 759-789.
[2] Pujol et al. Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching Negro river. Journal of Hydrology, 591:125331, 2020.
[3] Malou et al. (2021). Generation and analysis of stage-fall-discharge laws from coupled hydrological-hydraulic river network model integrating sparse multi-satellite data. Journal of Hydrology, 603, 126993.
[4] Data assimilation for free surface flows. Technical report, Mathematics Institute of Toulouse-INSA group-CS corp. CNES-CNRS, 2019.
[5] Monnier et al. Inverse algorithms for 2D shallow 893 water equations in presence of wet dry fronts. application to flood plain dynamics. Advances in Water Resources, 894 97:11–24, 2016.

How to cite: Pujol, L., Garambois, P.-A., Monnier, J., Larnier, K., Villenave, L., and Mosé, R.: Variational data assimilation on a multi-dimensional hydraulic-hydrological model of hydrographic network: inference of large composite control vectors on the DassFlow paltform, IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-126, 2022.

Rafael Navas, Willem Vervoort, and Pablo Gamazo

Building Rating Curves is complex since it depends on the availability of direct instantaneous stage-velocity/area measurements (gaugings) and variable hydraulic conditions. This work proposes a methodology to estimate synthetic (non-gauged) daily rating curves. A regional rainfall-runoff model (GR4J) is coupled with an instantaneous/stage - daily/discharge relationship based on third-order Chebyshev polynomials. Bayesian inference with Delayed Rejection Adaptive Metropolis algorithm is used to optimize the parameters and to quantify the uncertainty in the joint daily rating curve and the regional rainfall-runoff model. The likelihood function based on the model residuals is assumed to be gaussian, homoscedastic, and independent. As a study area, a region with 4 gauging sites located in New South Wales, Australia was chosen, and periods with no changes in the stage-discharge relationship were selected. Then, the method is implemented 4 times across the gauging sites, where 3 sites are supposed to be gauged and 1 site is supposed to be partially gauged (with only instantaneous water level). The proposed approach can increase the spatial representation of hydrological data when used in combination with satellite river altimetry. Another extension would be to identify the error if fewer gaugings are available.  Future works could address how to incorporate heteroscedastic, autocorrelation, and zero-inflation of model residuals; as well as how to account for the variation in the range of model residuals across the gauging sites.


  • u: gauged site
  • v: partially gauged site
  • q: streamflow (mm/day)
  • h: water level (m)
  • h^: Transformed h
  • Ø: climatological forcing (mm/day, P & PET)
  • ε: model residuals
  • x1...x4: GR4J parameters
  • x5...x7:Chebyshev parameters
  • σ: variance of residuals


Gauging sites (Figure 1) are located in Lachlan River at Reids Flat   (3742 km2), Lachlan River at Narrawa  (2256 km2), Abercrombie River at Abercrombie (2636 km2), and Abercrombie River at Hadley  (1635 km2).


The best fit of the prediction is obtained in the central limb of the rating curves (Figure 2).



As a result, the misfit residuals show a pattern of under and over estimations for high/low
flows (Figure 3).


How to cite: Navas, R., Vervoort, W., and Gamazo, P.: Bayesian inference of synthetic daily rating curves by coupling Chebyshev polynomials and the GR4J model, IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-2, 2022.

Cousquer Yohann, Sivelle Vianney, and Jourde Hervé

Uncertainty quantification in hydrological modeling, and noticeably in the lumped modelling of karst systems, generally attributes source of uncertainty only to parameters or to input data.  Uncertainty in model conceptualization, which often leads to one unique model structure, is frequently neglected. This issue is particularly important for karst hydrology, where hydrological systems are highly heterogenous and information about the structure is difficult to obtain. In this work, we present a Bayesian Model Averaging (BMA) approach to assess predictive uncertainty with errors due to either model structure and model parameters. A set of plausible model structures is first selected, and a prior parameter space is sampled. For each model structure, the ability of the model to reproduce the observed behavior (discharge and water level) of the karst system is evaluated and a likelihood measures of acceptable structure is assigned to the model.  Then, the likelihood measures of each acceptable structure are integrated over the parameter space to obtain total likelihood of model structures. The latter is used to weight the predictions obtained with each model structure in the ensemble predictions. This method is illustrated with the KarstMod modeling platform that allows to run large range of lumped parameter model structures dedicated to karst hydrology. Then, the importance of structure error to predict spring discharge and water table levels of a well known karst system, the Lez karst system (Montpellier, France), is discussed.

How to cite: Yohann, C., Vianney, S., and Hervé, J.: Estimating the Structural Uncertainty of Lumped Parameter Models in Karst Hydrology: a Bayesian Model Averaging (BMA) Approach  , IAHS-AISH Scientific Assembly 2022, Montpellier, France, 29 May–3 Jun 2022, IAHS2022-522, 2022.