Flood return-period thresholds in GloFAS are derived from annual maximum discharge (AMAX) series extracted from the long-term river discharge reanalysis. The standard approach assumes that the annual maxima are stationary over the reanalysis period. However, long-term changes in the meteorological forcing or other components of the modelling system can introduce systematic trends in the simulated annual maxima. The existence of larger river discharge trends were documented, for example, in Zsótér et al. (2020), especially regarding the Congo basin or the southwestern USA. If such trends are sufficiently large, fitting a stationary extreme-value distribution directly to the complete historical sample may result in flood thresholds that are not representative of the more recent discharge regime, and thus not representative of the real time forecasts.
To address this, an adjustment is applied to the annual maxima before fitting the extreme-value distribution. The objective is to remove the estimated long-term change in the magnitude of the annual maxima, while retaining the complete historical sample and its interannual variability, and essentially transferring the AMAX time series to a common reference level representative of the recent discharge regime. For GloFAS v5, this reference level is the last reanalysis year, 2025, of the full period of 1979-2025.
For each location, a linear trend is fitted to the annual maximum discharge series using ordinary least-squares regression:
Q̂(t) = β₀ + β₁t
where t represents time (year), β₀ is the intercept and β₁ is the estimated linear trend (slope), representing the average change in annual maximum discharge per year over the historical period.
The strength of the linear relationship is quantified using the coefficient of determination, R², which measures the fraction of the variability in the annual maxima explained by the fitted linear trend. R² ranges from 0 to 1, with values close to 1 indicating that the temporal evolution of the AMAX series is well represented by the linear trend, while values close to 0 indicate that the fitted trend explains little of the year-to-year variability.
For each year t, the adjustment is calculated as the difference between the fitted trend value at the reference time and the fitted trend value for that year. But, rather than applying the full fitted trend correction in all locations, R² is used to scale the magnitude of the adjustment. The trend adjustment for each year is therefore:
ΔQ(t) = R² × [Q̂(tref) − Q̂(t)]
and the adjusted annual maximum is:
Qadj(t) = Q(t) + R² × [Q̂(tref) − Q̂(t)]
Since the fitted trend is linear, this can also be expressed as:
Qadj(t) = Q(t) + R² × β₁(tref − t)
This R² weighting reduces the adjustment where the estimated linear trend provides a poor representation of the temporal evolution of the AMAX series. When R² approaches 1, almost the full estimated linear trend is removed. When R² is small, only a small fraction of the fitted trend is used for the adjustment. In the limiting case of R² = 0, no adjustment is applied and the original annual maxima are retained.
For a positive trend, earlier annual maxima are generally shifted upwards, while for a negative trend they are shifted downwards. The magnitude of the adjustment progressively decreases towards the reference time. The actual correction depends on both the magnitude of the fitted slope and the strength of the linear relationship represented by R².
Importantly, the adjustment does not replace individual annual maxima with values from the fitted regression. The original annual maxima remain the basis of the adjusted series, with only the R²-weighted long-term component added:
Qadj(t) = Q(t) + R² × β₁(tref − t)
Consequently, the interannual variability and individual extreme events contained in the original AMAX series are retained. At the same time, systematic long-term changes are progressively corrected according to how strongly the fitted linear trend is supported by the AMAX series.
Following the annual maxima adjustment, the same stationary Gumbel extreme-value distribution is fitted to the adjusted annual maxima (as done for all GloFAS versions, see: GloFAS v4 and earlier versions) and the return-period discharge thresholds are calculated from this stationary-equivalent sample.
Zsótér, E., Cloke, H., Prudhomme, C., Harrigan, S., de Rosnay, P., Muñoz-Sabater, J. and Stephens, E. (2020): Trends in the GloFAS-ERA5 river discharge reanalysis. ECMWF Technical Memorandum No. 871.