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# Benefits of upstream data for downstream streamflow forecasting: data assimilation in a semi-distributed flood forecasting model

Paul Royer-Gaspard, François Bourgin, Charles Perrin, Vazken Andréassian, Alban De Lavenne, Guillaume Thirel & François Tilmant

**To cite this article:** Paul Royer-Gaspard, François Bourgin, Charles Perrin, Vazken Andréassian, Alban De Lavenne, Guillaume Thirel & François Tilmant (2024) Benefits of upstream data for downstream streamflow forecasting: data assimilation in a semi-distributed flood forecasting model, LHB, 110:1, 2374081, DOI: 10.1080/27678490.2024.2374081

**To link to this article:** https://doi.org/10.1080/27678490.2024.2374081

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# Benefits of upstream data for downstream streamflow forecasting: data assimilation in a semi-distributed flood forecasting model

Paul Royer-Gaspard ID*, François Bourgin ID, Charles Perrin ID, Vazken Andréassian ID, Alban De Lavenne ID, Guillaume Thirel ID and François Tilmant ID

Université Paris-Saclay, INRAE, UR HYCAR, Antony, France

## ABSTRACT

An hourly hydrological forecasting model (GRP), used for flood forecasting in France, has been enhanced by developing a semi-distributed version (GRPS). This new model addresses some limitations of the original lumped approach by integrating flow observations from upstream stations to improve downstream flood predictions. A comparison of GRP and GRPS was conducted on a large set of flood events in nested catchments in France. Results indicate that GRPS slightly outperforms GRP up to the lead time of the last upstream flow observations reaching the downstream station, though performance gains vary widely across events. Event classification identified detrimental interactions between data assimilation methods and river routing model errors. A sensitivity test suggested that enhancing both the propagation model and the assimilation scheme could reduce poor performance at short lead times for some events. The study also demonstrated that upstream forecast quality significantly impacts downstream forecast accuracy. Overall, the findings confirm the benefits of incorporating multiple flow observations into a semi-distributed model and suggest several avenues for further improvement.

## Bénéfices des observations de débit à l'amont pour la prévision des crues à l'aval : assimilation de données dans un modèle de prévision hydrologique semi-distribué

## RÉSUMÉ

Un modèle de prévision hydrologique horaire (GRP), utilisé pour prédire les crues en France, a été adapté en une version semi-distribuée (GRPS), pour répondre à certaines limites de l'approche originale en assimilant les observations de débit des stations en amont pour améliorer les prévisions en aval. Une comparaison entre GRP et GRPS a été effectuée sur un grand échantillon d'événements de crue dans des bassins versants emboîtés français. Les résultats indiquent que GRPS surpasse en moyenne légèrement GRP jusqu'au moment où les dernières observations de débit en amont atteignent la station en aval, malgré une grande disparité d'un événement à l'autre. Une classification des événements a permis d'identifier des interactions néfastes entre les méthodes d'assimilation de données et les erreurs du modèle de routage. Un test de sensibilité a suggéré que l'amélioration du modèle de propagation et du schéma d'assimilation pourrait réduire les mauvaises performances à court terme pour certains événements. L'étude a également démontré que la qualité des prévisions en amont a un impact significatif sur la précision des prévisions en aval. Cette étude confirme l'intérêt de l'assimilation de multiples observations de débit dans un modèle semi-distribué et propose quelques pistes pour son amélioration.

## KEYWORDS

Flood forecasting;
hydrological model; spatial
discretisation; data
assimilation

## MOTS-CLES

prévision des crues ;
modélisation hydrologique ;
discrétisation spatiale ;
assimilation de données

# 1. Introduction

Since the creation of SCHAPI (i.e. the centralised French public flood forecasting service) in 2003, significant progress has been made in the field of flood risk forecasting. This has led to the operational deployment of the GRP model (Berthet, 2010) in most flood forecasting services in France. Successive improvements have made GRP a simple yet efficient

model (Berthet, 2010; Viatgé et al., 2019). Its use on many varied catchments has, nevertheless, demonstrated some limitations, which have been the starting point for several research projects (e.g. Astagneau, 2022; Peredo, 2021). In particular, the choice of a lumped catchment representation in GRP precludes the use of spatialised precipitation data and the

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assimilation into the model of flow observations avail-
able at internal gauge stations. The “one catchment –
one model” approach that lumped modelling implies
may also lead to inconsistencies in the simulation of
successive gauges from upstream to downstream on
the same river. Semi-distributed hydrological models –
dividing a catchment into nested sub-catchments – are
common alternatives for addressing these issues. In
the case of GR-like simulation models, the work of
Lobligeois et al. (2014) showed a clear benefit of semi-
distributed schemes in regions driven by highly het-
erogeneous rainfall.

Discretising a hydrological model from a lumped to
a semi-distributed scheme is not as simple as it may
seem. Although it should theoretically be beneficial to
the performance, its success strongly depends on the
interactions between spatial discretisation and available
data information content. The splitting of a catchment
into sub-catchments must be balanced between the –
often insufficient – quantity of available data (for cali-
bration or data assimilation) and the – often excessive –
quantity of parameters and states to be constrained. The
higher dimensionality of such models increases the
difficulty of identifying the appropriate parameters or
states, especially because observed data at downstream
gauges indiscriminately gather contributions from mul-
tiple upstream sub-catchments.

In the calibration field, regionalisation methods
now make it possible to achieve performances that
are often at least as good in semi-distributed set-
ups as in lumped ones (e.g. De Lavenne et al.,
2019). In the data assimilation field, however, deal-
ing with spatialised models increases the challenge
of state or parameter correction, which is not yet
fully addressed in hydrological modelling in general
(Wu et al., 2020). Along with the higher dimen-
sionality of the data assimilation problem, semi-
distributed models also add to the problem
encountered with lumped models of the time-
lagged correlation between most model states and
simulated streamflow. This problem alone requires
either introducing retrospective procedures in
streamflow data assimilation schemes (e.g. Li
et al., 2013; McMillan et al., 2013; Noh et al.,
2011; Rakovec et al., 2015) or – if no lag is con-
sidered – strongly reducing the list of candidate
states to those most correlated with streamflow
(Berthet, 2010; Piazzi et al., 2021). In the case of
semi-distributed models, an additional lag between
observations and states is introduced by the process
of river routing from upstream to downstream
gauges. Upward propagation of information pro-
vided by downstream streamflow observations is
challenging, and current schemes still struggle to
improve predictions at the outlet of ungauged
upstream sub-catchments (Li et al., 2015).
Moreover, even performing consistent assimilation

of multiple-gauge data from upstream and down-
stream stations is demanding due to timing errors
in downward river routing (Lee et al., 2011).

Ensemble data assimilation techniques such as the
ensemble Kalman filter or particle filter (or variants)
generally produce encouraging results for improve-
ments to flood forecasting systems. However, they do
not overcome the issues outlined in the previous para-
graph, since most of the referenced studies were based
on ensemble assimilation schemes. Moreover, they
often rely on a large ensemble size and thus come
with high computational costs. Proposed alternatives,
mixing data assimilation filters with sampling or opti-
misation techniques such as genetic algorithms,
Markov chain Monte Carlo, or variational assimila-
tion, show potential for improved numerical efficiency
(Abbaszadeh et al., 2019, 2018).

Still, there is considerable uncertainty about the
source of errors related to state correction in semi-
distributed models, and several questions seem to
remain without clear answers: To what extent do
these errors propagate from upstream simulations to
downstream simulations? Do they stem from an
incorrect representation of hydraulic transfers in the
hydrographic network? Do they stem from the obser-
vations themselves?

In this study, we sought to characterise the benefits
of assimilating flow observations at upstream stations
for downstream flood forecasting. To this end, we
built GRPS, a simple semi-distributed version of
GRP. Work was carried out to answer two questions:

* How does the forecast accuracy of GRPS compare with that of GRP?

* How sensitive is GRPS to errors in flow inputs from upstream, i.e. both forecast errors at upstream stations and routing errors?

## 2. Methodology

### 2.1. Description of the dataset

The study is conducted on a set of 136 nested catch-
ments (comprising a total of 285 gauging stations) in
continental France over the period 1998–2018
(Figure 1). The database comes from the work of
Thébault et al. (2023) and was built to gather
a sufficiently large sample to draw general conclusions
(Addor et al., 2020; Gupta et al., 2014).

Hourly streamflow time series were collected from
the national Hydroportail archive (Dufeu et al., 2022),
which provides open-access streamflow observation
data for around 5000 gauging stations.

Precipitation time series were extracted from the
Comephore gridded product (Tabary et al., 2012),
which provides hourly precipitation series at
1 × 1 km² resolution. The reanalysis is based on in situ
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Figure 1. Sample map of nested watersheds (red contours) and gauging stations (black dots).</figcaption></figure>

and radar measurements. Precipitation data were spatially averaged at catchment and sub-catchment scales.

Daily air temperature values from the SAFRAN reanalysis (Vidal et al., 2010), available at an 8 × 8 km² resolution, were used to estimate daily potential evaporation using the formula of Oudin et al. (2005). Evaporation data were then disaggregated to hourly time steps, assuming a parabolic shape from 6am to 7pm (UTC) and zero evaporation at night (Ficchi, 2017). The gridded hourly potential evaporation data were then spatially averaged over the catchments and sub-catchments.

We used observations as forecasted forcing after state correction. This choice leads to idealised forecast conditions compared to the use of rainfall forecast archives but allows us to focus on hydrological model errors only.

## 2.2. Principle of spatial discretisation of the catchments

The catchment set used in this study was set up according to the hydrological situation of the catchments, data availability, and a specific rule to define their spatial structure. The gauging stations available in a French catchment database (Delaigue et al., 2020) were first filtered by removing those

significantly influenced by human activities. Snow-dominated catchments (here considered catchments with more than 10% solid precipitation) were also eliminated, as were those whose flow series contained more than 10% missing values per year.

The resulting set of stations was then processed to create a set of nested catchments. The term “nested catchments” refers to a group of gauging stations linked together by the hydrographic network, with a main output station downstream and a few stations upstream. The nesting process proposed by Thébault et al. (2023) enables the selection of simple nesting configurations: nested catchments have a single upstream discretisation level (i.e. the selected upstream catchments are not further discretised themselves) and homogeneous sub-catchment sizes to limit sensitivity issues. By dividing large, deeply nested catchments into numerous first-order nested catchments, the procedure has the advantage of creating a large, standardised dataset, with catchments sharing a similar spatial structure while covering a broad spectrum of physical properties (Table 1). Concentration times were computed as the time lag maximising correlation between catchment rainfall and streamflow at the outlet after spline-filtering of the lag correlation response curve to obtain more robust estimates. Propagation times were computed with similar
<figure><img src="image_16.png" /><figcaption>Table 1. Characteristics of the nested catchments.</figcaption></figure>

<table><thead><tr><th>Descriptor</th><th>Min.</th><th>Q<sub>25</sub></th><th>Med.</th><th>Q<sub>75</sub></th><th>Max.</th></tr></thead><tbody><tr><td>Catchment area (km<sup>2</sup>)</td><td>15</td><td>311</td><td>596</td><td>1135</td><td>10087</td></tr><tr><td>Sub-catchment area (km<sup>2</sup>)</td><td>3</td><td>94</td><td>232</td><td>459</td><td>7658</td></tr><tr><td>Upstream/downstream surface ratio (-)</td><td>0.05</td><td>0.23</td><td>0.42</td><td>0.57</td><td>0.86</td></tr><tr><td>Catchment average slope (°)</td><td>0.5</td><td>2.4</td><td>4.6</td><td>7.5</td><td>19.6</td></tr><tr><td>Hydraulic length of the river network (km)</td><td>0.2</td><td>14.9</td><td>26.1</td><td>41.0</td><td>125.1</td></tr><tr><td>Concentration time (h)</td><td>13</td><td>18</td><td>26</td><td>37</td><td>112</td></tr><tr><td>Propagation time in the river network (h)</td><td>0*</td><td>1</td><td>4</td><td>12</td><td>60</td></tr></tbody></table>

reasoning from lag correlation between streamflow at the outlet and streamflow at upstream gauges. In the case of multiple upstream gauges, a weighted-average lag was computed using the respective sub-catchment mean contribution to the outlet.

## 2.3. Selection of flood events

A selection of flood events was made using an automatic algorithm for (i) detection and (ii) spatio-temporal matching of flood events. The algorithm for detection of events, adapted from the work of Ficchì et al. (2016), was applied to each of the 285 gauging stations. The spatio-temporal matching procedure provided consistent semi-distributed events. It was set up for the purpose of this work. The process yielded 1360 events in a semi-distributed configuration, made up of the top 10 flood events for each nested catchment.

In the detection algorithm, peaks below the 95th flow quantile are ignored. A flood event is defined as occurring between the point at which the discharge exceeds 20% of the event peak discharge and the point at which the discharge falls below 25% of this value. No missing data are allowed during an event. A matching process was applied to identify events at upstream stations corresponding to an event at the main outlet, using a matching time window covering propagation time in the hydrographic network. Because of spatial heterogeneities in catchment responsiveness to precipitation, distinct events detected at the same upstream gauge may be associated with the same downstream events: such cases were addressed by merging upstream events. Leftover upstream (or downstream) events remaining without any associated downstream (or upstream) events were discarded.

## 2.4. Hydrological models

The GRP model (Berthet, 2010) is an hourly hydrological model with a simple three-parameter structure comprising a soil moisture accounting reservoir, a unit hydrograph, and a quadratic transfer reservoir whose output corresponds to discharge. GRP is designed to work with a companion flow assimilation scheme. The scheme is based on direct insertion of the observed flow to analytically correct the reservoir level. No post-processing for forecasted streamflow was used in this study in order to focus on the impact of the assimilation procedure.

The GRPS model is a semi-distributed configuration of GRP (Figure 2). The contribution of each sub-catchment is modelled with GRP, and river routing from upstream to downstream is represented by a simple time-lag approach combined with a linear routing reservoir to account for wave diffusion in the network (Bentura & Michel, 1997).

In GRPS, flow data are assimilated to correct the model states of the sub-catchments at each gauging station. The correction of sub-catchment states depends solely on the flow observed at the gauging station and the flow simulated at this station. Consequently, at upstream stations, the state correction is identical to that of GRP. The flows predicted by GRPS upstream are identical to those of GRP at these stations. However, at the main downstream outlet, the simulated flow is the sum of the contribution from the intermediate sub-catchment and the flows propagated from the upstream stations. Adopting a similar assimilation scheme as for upstream gauges would lead to burdening the states of the downstream sub-catchment alone with the entire downstream model error and to neglecting errors in contributions from upstream sub-catchments.

<figure><img src="image_17.png" /><figcaption>**Figure 2.** Structure of GRPS. $Q_{upstr}$ is the contribution to streamflow from upstream sub-catchments 1 and 2, routed to the outlet of the nested catchment by the river routing process. $Q_{downstra}$ is the contribution from the downstream sub-catchment 3 at the outlet of the catchment. $Q_{tot}$ is the sum of all contributions at the outlet of the catchment.</figcaption></figure>

<footnote>*less than 1 h, i.e less than the model time step.</footnote>
<figure><img src="image_18.png" /></figure>

Considering the aforementioned issue, the assimilation process at the downstream station divides the correction proportionally to the flow contribution of the respective sub-catchments. It is therefore assumed that the simulated distribution of the contribution of each sub-catchment is representative of the true distribution, and that only the total simulated streamflow should be corrected in the model from the downstream observation. On the downstream side, the contribution of the intermediate sub-catchment is corrected by updating downstream model states with the same scheme as that of GRP. On the upstream side, the total contribution of the upstream sub-catchments is corrected by applying a multiplicative correction factor which persists in the forecast. The corrective factor is maintained at forecast times with an exponential decay tending towards 1 (the half-life time being equal to the catchment response time). This whole assimilation scheme ensures that the sum of corrected flows is equal to the flow measured at the station.

To strictly evaluate the benefit of upstream streamflow assimilation in downstream forecasting skills, a version of GRPS where only downstream gauge data are used for downstream state correction has also been tested. In this model, called GRPS "downstream", the assimilation of downstream streamflow is based on the same method as in GRPS, with a distribution of the error correction between downstream model states and upstream contributions.

## 2.5. Model evaluation

The models were evaluated using a split-sample approach (Klemeš, 1986), dividing the time series into two sub-periods (1999–2008 and 2009–2018). The GRPS calibration follows the sequential approach described by Lobligeois (2014), starting from upstream stations and then going to downstream. The models were calibrated with a fixed 24-h lead time for every catchment after sensitivity analysis showed that it had little impact on the results. The mean square error under the dimensionless form of the Nash and Sutcliffe criterion (or Nash-Sutcliffe efficiency, NSE; Nash & Sutcliffe, 1970) was used as the objective function. Streamflow forecasts were evaluated up to 5 days (120 h) ahead.

Model forecasts were evaluated as fixed-term forecasts for the rise time of each flood hydrograph, to characterise their performance from the start to the peak of the flood. In an operational context, the rising limb is decisive for adequately triggering warnings. Five forecast lead times were selected for the evaluation: three reference lead times were used uniformly in all the study catchments – 1 h, 12 h, and 120 h (short lead time, intermediate lead time, and long lead time,

respectively). Two other lead times differing between catchments were also chosen:

- Tp, which represents the average propagation time – as simulated by the model under the influence of assimilation – between the upstream stations and the downstream station, weighted by the contributions of each upstream sub-catchment to the downstream flow. On average, this is the time at which the simulated upstream hydrographs still under the influence of the last assimilation reach the downstream station.

- Tc, which represents the nested catchment time of concentration. Theoretically, this is the point at which the influence of the last assimilation has completely disappeared, i.e. when the forecasting model assumes the behaviour of a simulation model (without assimilation).

Two performance criteria were used to assess model performance:

- The NSE is used to estimate overall forecast quality. It varies between $-\infty$ and 1 (optimal value). The NSE reference "average" model corresponds to the average of observations over the duration of the event being evaluated, i.e. over the rise time of each flood.

- The C2M criterion (Mathevet et al., 2006) is a bounded criterion between -1 and 1 that compares the mean square error of a model relative with another reference model. In our study, we compared GRPS with GRP: a C2M value greater than 0 means that GRPS performs better than GRP.

# 3. Results

## 3.1. Comparison of model performance

The distributions of NSE on the rising limb of the events are shown in Figure 3 for each forecast lead time of interest. In addition to the performance of GRP and of both versions of GRPS, the performance of the “naive” persistent model (constant forecast flow equal to the last observation assimilated) is presented, as a reminder of the value of hydrological models in streamflow forecasting. At short lead times, the benefits of the assimilation of the latest observations are maximised, and performance is therefore high. Overall quality then declines, stabilising around Tc. This is the lead time at which the simulated flow is mostly driven by post-assimilation rainfall: at this point, the last correction of model states has a diminished effect; the model hence behaves almost like a simulation model.
<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 3.** Performance distributions of the “naive” persistent model, GRP, GRPS (with assimilation of downstream data only) and GRPS on the rising limb of the 1 360 flood events for different forecasting lead times. Box plots represent 5%, 25%, 50%, 75%, and 95% quantiles.</figcaption></figure>

In general, the NSE distributions of the three models are close. However, the differences between the models per event can be quite significant, as shown by the C2M distributions. These distributions show an advantage for GRPS, particularly around Tp, where most values are positive. At subsequent lead times, the models show more balanced scores. Considering that Tp represents the last lead time at which a fraction of the flow simulated by GRPS is made up of upstream observed flows, the better score of GRPS compared to GRP was an expected result.

The performance of GRPS “downstream” does not exhibit any pattern across forecast lead times. The C2M score of GRPS “downstream” is evenly distributed around zero. Hence, the semi-distributed discretisation of the model can either benefit or impede its performance. Therefore, it is the assimilation of upstream streamflow data, rather than its spatialised structure, that stands out as the main cause for the gain of GRPS’s downstream forecast quality compared to GRP.

By applying Student's *t* test to the C2M event values for each catchment, we sought to quantify the number of catchments for which it is possible to make a significant decision regarding the choice of model between GRP and GRPS. The threshold *p* value was chosen as .01, given that the test was carried out 136 times. At lead time Tp, GRPS's and GRP's respective performance significantly differed in 22 catchments, including 21 where GRPS was significantly better than GRP. At lead time Tc, the number of such catchments drops to nine, including eight in which GRPS was significantly better than GRP. At both lead times, there were only two catchments left where GRPS was significantly better. This result confirms that the decisive argument to explain the superiority of GRPS over GRP is the assimilation of observed data at upstream

stations rather than the semi-distributed nature of the model. This result also shows that in most cases no model stands out, either because their performances are very similar or because of the variability between events. It should be noted, however, that the low number of events per catchment studied reduces the significance of Student's *t* test, and thus the number of catchments identified is likely to be a low estimate.

Flood hydrographs were visually analysed for the whole set of events, of which four are shown as examples in Figure 4. These hydrographs are selected to illustrate different situations which are typically encountered in the results or to stress rarer but decisive patterns. Overall, GRPS-simulated hydrographs are slightly more variable than GRP hydrographs.

Each sub-catchment model has specific simulated as well as observed streamflow patterns, which are all gathered as contributions to the downstream streamflow. Despite this, the hydrographs predicted by the two models exhibit similar shapes for most events. The example of the Meurthe flood illustrates this frequent case where the differences between the two models are visually small, but the values of the comparison criterion are non-zero, in this case to the advantage of GRPS at both lead times. The case of Le Lignon illustrates a rare situation where GRPS is better than GRP only at longer lead times, demonstrating an advantage strictly linked to the semi-distributed nature of the model and not to data assimilation. The Aveyron and Bourbre events demonstrate the limitations of GRPS when the assimilation of upstream observations leads to a general (Aveyron) or momentary (Bourbre) overestimation of downstream flood peak at short forecast lead times, whereas the differences are less clear at more distant lead times.
<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

Aveyron @ Montauban
O5882510 (05/02/2003 06h)

<figure><img src="image_23.png" /></figure>

Lignon @ Poncins
K0773220 (09/08/2013 01h)

<figure><img src="image_24.png" /></figure>

Bourbre @ Tignieu-Jameyzieu
V1774010 (15/11/2014 18h)

<figure><img src="image_25.png" /></figure>

**Figure 4.** Observed (circles) and predicted (fixed horizon) GRP (orange) and GRPS (blue) flood hydrographs at horizons Tp (light) and Tc (dark) for four selected events from different catchments. Scores are computed at the event scale.

**3.2. Link between model performance and event characteristics**

We studied the correlations between model performance and the characteristics of flood events and catchments.

It appeared that the most obvious explanatory factor for the NSE values of GRP and GRPS is the time of concentration of the catchments (also highly correlated with their size and the autocorrelation of flows). This result (not shown) can be explained both by the greater difficulty of hydrological models in simulating highly reactive catchments and very short events in general (Astagneau et al., 2021; Merz et al., 2009), and by the lower persistence of assimilation benefits, as flows in reactive catchments are often less autocorrelated.

Two other major explanatory factors for the comparative performance of GRPS versus GRP were identified. The distributions of C2M values are shown in Figure 5 for different event classes based on the ranges of these factors.

The first event classes are determined according to the observed contribution of upstream catchments to

streamflow during the event. Differences in the respective performance of GRPS and GRP are clearly more visible for events where the upstream contribution is greater, although this is not necessarily to the advantage of one model or the other. Conversely, GRP and GRPS show very similar performances for events where the contribution of upstream sub-catchments is low. This was expected since the behaviour of GRPS is then very close to a lumped model, since the downstream catchment both produces most of the flow and does not depend on the river routing in the hydrographic network.

A second explanatory factor is the spatial correlations between the hydrographs observed at downstream and upstream stations – with consideration of the time lag. The higher this correlation (i.e. the more similar the upstream hydrographs are to the downstream hydrograph – reduced by a factor of proportionality and shifted in time) – the more homogeneous the sub-catchments are in hydrological terms. As a result, while we would expect GRPS to outperform GRP in nested catchments with highly heterogeneous hydrographs where spatialised forcings and processes
<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /><figcaption>**Figure 5.** Distribution of GRPS versus GRP performance by event class for different forecasting lead times based on the 1960 flood events. In the top row, event classes are based on the relative contributions of upstream sub-catchments. In the bottom row, event classes are based on the spatial correlation of hydrographs observed at different stations.</figcaption></figure>

should yield better performances, the opposite is observed at lead times less than or equal to Tp. The situation returns to equilibrium at later lead times, and there are no longer significant differences. This result indicates that the assimilation scheme struggles when dealing with spatially variable contributions, highlighting potential issues in the way the correction is distributed to the downstream sub-catchment states or to the upcoming hydrographs from upstream sub-catchments. At this point, it is difficult to conclude whether this relates to erroneous upstream simulated contributions before analysis or to an inappropriate procedure for the correction of upstream forecasted contributions. Obviously, the hypothesis made about respecting the simulated distribution of correction between upstream and downstream contributions before the analysis lacks robustness, since it is very sensitive to errors in peak arrival, volume, or shape in any of these contributions.

# 4. Discussion

To take this a step further, we sought to identify ways of improving the semi-distributed model, by studying issues related to the river routing in the network and to the quality of upstream forecasts.

## 4.1. Sensitivity to the river routing scheme

To understand the effect of river routing errors on forecasts, we chose to compare the results obtained with a simpler hydraulic propagation module than

GRPS, removing the distributed correction between downstream model states and upstream contributions. The C2M criteria obtained by the models constructed are shown in Figure 6. The simplest model is GRPS with a river routing module consisting of a simple time lag (GRPS-SL for "simple lag"). The intermediate model is GRPS with a diffusive river routing where the streamflow observed at the downstream station is used to correct downstream states only (GRPS-DL for "diffusive lag"). The final model is GRPS as used previously, i.e. GRPS-DL with corrections distributed between upstream and downstream contributions.

It is quite clear that improvements in the river routing and correction scheme only bring about marginal gains in performance distributions, concentrated at short lead times. At a 1-h lead time, the respective performances of the models follow the order of their complexity rather well. Although the median performance is only slightly improved as the river routing becomes more complex, the occurrence of poor scores is strongly reduced. The corresponding events are those where the upstream contributions exceed the observed flow – which is why the distributed correction was implemented. Apart from the negative extremes, the benefits are mixed and inconclusive on the overall effect of improved river routing and error distribution methods.

It should be remembered that the river routing schemes tested are conceptual and thus simple regarding the true wave propagation conditions. Nevertheless, the “simple lag” method is widely used in hydrological modelling because it is often sufficient to produce satisfactory
<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /><figcaption>**Figure 6.** Performance distribution relative to GRP for alternative models to GRPS for different forecast horizons on the 1 360 flood events. GRPS-SL, GRPS-DL and GRPS are described in 4.1.</figcaption></figure>

simulation performance (Lerat et al., 2012; Lobligeois, 2014; Pechlivanidis et al., 2010; Saulnier & Le Lay, 2009). It is possible that the hydrological forecasting exercise stresses the drawbacks of the simple river routing scheme due to interactions with data assimilation. In our study, this hypothesis is supported by the performance gains at 1-h lead time between “simple lag” and “diffusive lag”. To discuss this issue further, however, we would need to consider solutions derived from the shallow water equations in order to closely simulate wave deformation processes that lag methods cannot reproduce. With a more detailed routing scheme, it would then be possible to concentrate research on the data assimilation method with less doubt about detrimental interactions with propagation errors.

Still, even with a perfect routing scheme, other sources of error remain in the assimilation process, such as inconsistent streamflow data due to measurement errors or due to human influence between upstream and downstream stations. As outlined in the introduction, the issue of uncertain streamflow data alone has driven research on adequate assimilation methods for hydrological forecasting. The simple assimilation scheme adopted in our study does not manage such uncertainties, although it could be easily adapted in an ensemble context by adding noise to streamflow observations as typically performed in the ensemble Kalman filter. Nonetheless, this simple scheme allowed us to estimate the raw benefit of adding upstream streamflow data for the quality of flood forecasting.

In addition, it is worth noting that the spatial configuration of the semi-distributed model probably hinders the added value of the spatialisation of meteorological

forcings. A part of the spatial information carried by the fine resolution of the gridded precipitation product (1 km²) is lost by the spatial averaging at the scale of the sub-catchments of our study (generally more than 100 km²). The spatial configuration of GRPS was made from existing gauging stations, was limited to a single upstream discretisation level, and focused on maintaining consistency between the respective size and contribution of the sub-catchments within a given nested catchment. Therefore, our study lacks a consistent spatial discretisation across the catchment set to assess the relationship between model resolution and input resolution – or, even further, between model resolution and river routing scheme complexity.

## 4.2. Sensitivity to upstream forecast quality

Comparisons between GRP and GRPS show that the lead times where improvements are mostly observed merely correspond to the time of arrival of quasi-observed hydrographs from upstream (potentially imperfect because of river routing errors). To further characterise the contribution of upstream observations, we set up an idealised model in which upstream models would be perfect, i.e. in which contributions from upstream sub-catchments would only be observed hydrographs routed downstream. The results are shown in Figure 7.

The results show that the idealised GRPS model performs much better than GRPS as soon as lead times exceed the propagation time of upstream hydrographs. This supports the hypothesis that the gains disappear after Tp due to the quality of upstream forecasts. The excellent performance of the idealised GRPS model shows that the quality of upstream
<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /><figcaption>**Figure 7.** Distribution of performance relative to GRP from GRPS (blue) and idealised GRPS (with upstream observations) (green) at different forecast horizons on the 1 360 flood events. GRPS (idealized) is described in 4.2.</figcaption></figure>

hydrographs is essential for modelling downstream flows in a semi-distributed model, and that the simplicity of the river routing scheme is not the main cause for the limited benefits of GRPS compared to GRP.

Our results also show that the C2M of the idealised GRPS is still negative for almost one-quarter of the events. These are events where GRP already performs well (correlation of C2M with GRP NSE at Tc = -0.26), where GRPS already demonstrated difficulty against GRP (correlation with GRPS C2M vs. GRP = +0.46), and where upstream contributions are significant (correlation with upstream contribution percentage = +0.35). These characteristics show that the quality of upstream forecasts is not sufficient to compensate for errors elsewhere in the semi-distributed model (localised precipitation errors, parameter robustness, river routing scheme). As stressed previously, further work is required to disentangle the sources of these errors.

Possible pathways for upstream hydrological forecast improvements could, for example, be a breakthrough in understanding the modelled processes (e.g. Astagneau et al., 2022) or multi-model averaging procedures (Thébault et al., 2023). Astagneau et al. (2024) showed, for example, that a modified version of GRP that better accounts for high-intensity events in dry conditions can significantly improve forecasting efficiency. Another idea could be going a step further in the spatial discretisation of the catchments by dividing each gauged sub-catchment into – not necessarily gauged – multiple sub-catchments, and thus better accounting for the spatial variability of hourly precipitation. Astagneau (2022) demonstrated that intense and/or spatially heterogeneous rainfall can strongly undermine the performance of GR5H, a lumped hourly hydrological model whose structure is close to GRP's though slightly more complex. A comparison performed by Astagneau (2022) between an improved lumped model and a highly discretised semi-distributed

model showed that both approaches could reduce simulation errors. However, transposed in a forecasting framework, the set-up of a further distributed model would require a semi-distributed state-updating procedure for ungauged sub-catchments, which is outside the scope of this study. We may simply comment that the direct state-updating method adopted in this study is not really suited for such a purpose. As highlighted in the introduction, retrospective procedures would be a necessary yet probably not sufficient requirement. Uncertainty managing frameworks in data and models such as filtering or variational methods should also help in mitigating inadequate state corrections.

It should also be noted that, under the conditions of our study, an improvement in upstream forecast quality would also become an improvement of the lumped model that is taken as reference for the performance computations, so that it would have a limited impact on our results. It should also be kept in mind that precipitation forecast data are probably the more challenging limitation to the quality of hydrological forecasts at long lead times in operational conditions, and that this issue was not addressed here.

## 5. Conclusion

This study presented the results of the spatial discretisation of the GRP hourly hydrological forecasting model into a semi-distributed model, called “GRPS”. The aim was to characterise the benefits of the assimilation of flow observations at upstream gauge stations for downstream flood forecasting.

The models were compared using a database of 1 360 events selected from 136 nested catchments in continental France over the period 1998–2018. The models were evaluated using a split-sample procedure and two performance criteria (NSE and C2M) for forecast lead times up to 120 h.

The results show that, on average, GRPS slightly outperforms GRP up to the lead times corresponding
<figure><img src="image_32.png" /></figure>

to the downstream arrival times of the last upstream flow observations. A comparison with a version of GRPS where only downstream streamflow data would be used in the assimilation scheme clearly established the positive impact of upstream data assimilation on downstream performance. Still, a wide disparity was observed in average performance gains from one event to the other. The events where GRPS and GRP performances differ the most are those where the contribution of the upstream sub-catchments represents a significant proportion of the total flow, although this is not necessarily to the systematic advantage of GRPS. Events characterised by very similar observed hydrographs at upstream and downstream gauging stations are generally better simulated by GRPS at short forecast lead times. This result highlights the detrimental interactions between errors in the propagation of upstream streamflow in the river network and the data assimilation scheme. A sensitivity test for the river routing scheme and the downstream state correction procedure shows that it is possible to limit the occurrence of events where GRPS strongly underperforms. Finally, an experiment with perfect upstream forecasts shows that the quality of upstream forecasts is driving the quality of downstream forecasts once the benefit of streamflow assimilation has passed.

This study takes us a step closer towards improving real-time forecasting tools, by showing that a simple semi-distributed hydrological model can be beneficial in some contexts. It also invites us to investigate further the interactions between assimilation schemes and hydrological model errors. The added value of assimilation schemes representing uncertainty in observed streamflow or model parameters for operational forecasting – such as ensemble or variational techniques – should be addressed in future work.

## Acknowledgment

This work was made possible thanks to the provision of climatic data by Météo-France and hydrological data by SCHAPI. We also thank Cyril Thébault for providing us with the catchment set he compiled for his thesis, and for his meticulous critique of the data. We thank Maria-Helena Ramos, Olivier Delaigue, Paul Astagneau, and Daniela Peredo for their suggestions for this work. We thank Jérémy Verrier for his help in using a computer cluster to run our model. For the time given to the first author to finalise the writing of this article, we thank Dennis Fox, professor at the Université Côte d'Azur, and the Hydroclimat company executives Adrien Lambert and Magali Troin. Finally, we would like to thank the Associate Editor, Michel Lang, and two anonymous reviewers for their constructive comments that helped us improve the paper.

## Disclosure statement

No potential conflict of interest was reported by the author(s).

## Funding

This work was supported by the French Ministry of Environment (DGPR/SNRH/SCHAPI) and the RenovRisk-Transfert project (OSU-Réunion, FEDER).

## ORCID

Paul Royer-Gaspard http://orcid.org/0000-0002-5136-6585

François Bourgin http://orcid.org/0000-0002-2820-7260

Charles Perrin http://orcid.org/0000-0001-8552-1881

Vazken Andréassian http://orcid.org/0000-0001-7124-9303

Alban De Lavenne http://orcid.org/0000-0002-9448-3490

Guillaume Thirel http://orcid.org/0000-0002-1444-1830

François Tilmant http://orcid.org/0000-0003-3115-9535

## Data availability

The meteorological forcing data are produced and provided by Météo France (Tabary et al., 2012; Vidal et al., 2010). The hydrometric data are collected and provided by the French Ministry of Environment. To request access to the data, please contact the data owners: Météo France (https://meteo.data.gouv.fr/; Météo France, 2024) for the meteorological data; Banque Hydro (https://hydro.eaufrance.fr/edito/a-propos-de-lhydroportail, French Ministry of Environment, 2024) for the hydrometric data.

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