Context: 

The ESA river discharge Climate Change Initiative (CCI) project is a precursor study. It aims to derive long term climate data records (at least over 20-years) of river discharge for some selected river basins (and some locations in the river network) using satellite remote sensing observations (altimetry and multispectral images) and ancillary data. It aims to provide a proof-of-concept for the feasibility for a potential River Discharge ECV product to meet the requirements for the Global Climate Observing System. This project covers precursor activities towards the production of data products that address the GCOS-defined requirements for the River Discharge ECV.

Data description :

Just as in-situ stage measurements can be used to gauge river discharge, altimetry-derived water surface elevation (WSE) can serve as an alternative means of estimating river discharge when discharge time series data is available. Several methodologies have been documented for deriving discharge time series from multimission altimetry observations and supplementary data (Biancamaria et al., 2024). At least two approaches will be used, depending on the available in situ discharge and altimetry water surface elevation (WSE) time series:

⋅ Method 1: The preferred approach relies on the altimetry water surface elevation time series and in situ discharge time series to create a rating curve (RC) characterized by a power relationship between these two variables following a Bayesian approach (Rantz et al., 1982). However, this method necessitates a significant overlap period between discharge data and radar altimetry measurements (e.g., Biancamaria et al., 2011; Papa et al., 2012), or it requires the assumption that the rating curve remains valid and consistent when discharge data is only available prior to the altimetry observation period.

⋅ Method 2: The final option, in cases where there is no temporal overlap between in-situ or simulated discharge and water surface elevation data, assumes that the validity and stability of the rating curve persist across the various time periods covered by the two datasets. Both of these time periods should be sufficiently long to encompass a wide range of events. With this assumption, Tourian et al. (2013, 2017) introduced a method for calculating the rating curve, not based on the time series of discharge and water surface elevation, but on the distribution of their quantiles. This method has been adopted by a limited number of recent studies (e.g., Belloni et al., 2021). However, it’s important to note that this methodology naturally introduces higher errors when compared to the preferred approach. For this reason, this methodology will be validated over some stations with various hydrological dynamics and satisfying previous methods (overlap period exists between WSE and Q).

Approaches to derive Rating Curve (RC) :

Bayesian Approach :

The Bayesian method is a robust statistical approach used for constructing a rating curve, frequently applied in the field of hydrology when the goal is to estimate unknown parameters from observed data, while taking into consideration the associated uncertainty in these estimates. 

According to this, the estimation of the rating curve using the Bayesian method involves several steps:

• The initial step entails defining a probabilistic model that describes the relationship between observed data and the parameters we aim to estimate. In many hydrological applications, the relationship between discharge data (Q) and water surface elevation data (WSE) is often expressed as a power function:

                                                                                      Q = a⋅(WSE-z0)^b

Here, a, z0 and b are the parameters of the rating curve. a, is a scaling coefficient governing the magnitude of the Q-WSE relationship, b, characterizes the nature of this relationship, and z0, represents the height of the free surface above the reference point, corresponding to the river bottom's altitude. The power relationship is especially pertinent due to its consistency with numerous hydrodynamic phenomena. The exponent b within the equation allows for the representation of distinctive flow characteristics, including factors like roughness and channel geometry. Moreover, it offers adaptability in modelling to accommodate variations in flow characteristics, whether they are turbulent or laminar. This relationship, despite its mathematical simplicity, facilitates the fine-tuning of model adjustments in accordance with observed data (Chow, 1959).

• The second step involves the use of prior normal distributions, reflecting our prior knowledge about these parameters. These distributions can either be informative or uninformative, depending on our level of knowledge. The limits and ranges for a, z0 and b can vary depending on the specific context of the study, the dataset used, and the characteristics of the river or channel being analysed.

- Coefficient “a”:  adjustment parameter for the rating curve representing the scaling factor for discharge. Its value can significantly fluctuate based on various factors such as the characteristics of the river or channel, hydraulic conditions, and other influencing factors. Consequently, "a" must be non-negative and constrained within a sensible range specific to the system under study. Following the Manning equation, “a” must be equal to W/n*S^1/2 (Chow et al., 1988) where W is the river’s width (m), n the Manning’s roughness coefficient and S the slope (m/m). Given the considerable variability in river width and slope across different stations, a feasible range for this coefficient can be considered as:

                                                                                      a ∈ [0; 3000]

- Coefficient “b”: adjustment parameter representing the exponent of the rating curve and indicating the hydraulic condition of the study site. Like "a," this value must comply with physical constraints and cannot be negative. Following the Manning equation, “b” must be equal to 5/3 for reference hydraulic condition (Rantz et al., 1982). To accommodate the variability in system characteristics across sites, the following range values can be considered for this coefficient:

                                                                                      b ∈ [0; 5]

- Coefficient “z0”: offset or the elevation at which discharge begins. It should be within the range of elevations relevant to your study. For this reason, the value cannot exceed the minimum value of water surface elevation (WSE) and the range value need to consider of the variability in term of water depth over the sites. A feasible range for this coefficient can be considered as:

                                                                                      z0 ∈ [min(WSE)-30; min(WSE)]

• The final step involves parameter estimation. The posterior distribution of the parameters yields probabilistic estimates of the rating curve parameters in the form of mean values (optimal values) and credibility intervals (95th percentiles). This accounts for the uncertainty associated with these parameters and is achieved through Markov Chain Monte Carlo (MCMC) sampling from the posterior distribution. Two commonly employed MCMC algorithms are "NUTS" (No-U-Turn Sampler) and "Metropolis-Hastings." The Metropolis-Hasting sampler "MH" algorithm, which is relatively simple and efficient where a balance between exploration and exploitation is desired. This algorithm can be adapted to sample from discrete state spaces.

Quantile approach : 

The Quantile approach employs statistical modelling using quantile functions to create a rating curve, eliminating the necessity for overlapping measurements. This algorithmic method enables the estimation of river discharge using satellite altimetry, even in instances where there are no in situ measurements within the altimeter's timeframe. This approach has undergone application and validation in diverse river basins spanning different climatic zones, such as the Amazon, Brahmaputra, Danube, Niger, and Ob (Tourian et al., 2013).

Assuming a stationary flow behaviour and no modification in the river bathymetry both at the altimetry virtual station and at the in-situ gage, this approach ensures the utilization of historical in situ data in current applications. This method computes the quantile functions of the altimetry water surface elevation on one hand and of the discharge time series on the other hand. Then a scatter plot of these in-situ discharge quantiles versus altimetry water surface elevation quantiles is computed to establish the rating curve using the bayesian approach described previously.

File description :

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