Introduction

To address non-stationarity, an adjustment is applied globally on all 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, essentially transferring the AMAX time series to a common reference level representative of the recent discharge regime. For areas/pixels, where no long-term change (i.e. trend) is present, there will be no change in the annual maxima. While for area/pixels with well-defined changes/trends, the adjustment of the annual maxima values will be potentially large. For GloFAS v5, this reference level is the last reanalysis year, 2025, of the full 1979–2025 period, which is the closest year in the reanalysis to the real-time forecasts.

Annual maxima adjustment methodology

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 potential adjustment is calculated as the difference between the fitted trend value at the reference time and the fitted trend value for that year. Rather than applying the full fitted trend correction at all locations, the square root of R², √R², is used to scale the magnitude of the adjustment. The adjustment weight is therefore:

w = √R²

and the trend adjustment for each year is:

ΔQ(t) = √R² × [Q̂(tref) − Q̂(t)]

The adjusted annual maximum is then:

Q_adj(t) = Q(t) + √R² × [Q̂(tref) − Q̂(t)]

Since the fitted trend is linear, this can also be expressed as:

Q_adj(t) = Q(t) + √R² × β₁(tref − t)

Adjusted annual maxima are constrained to be non-negative. Therefore, values that would become negative following the adjustment are set to zero.

The √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, √R² also approaches 1 and almost the full estimated linear trend correction is applied. When R² is small, only a fraction of the fitted trend difference is used for the adjustment. In the limiting case of R² = 0, the adjustment weight is zero and the original annual maxima are retained.

For a positive trend, earlier annual maxima are shifted upwards, while for a negative trend they are shifted downwards. The magnitude of the adjustment progressively decreases towards the reference year and is zero at the reference year itself. The actual correction therefore 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 trend difference added:

Q_adj(t) = max[0, Q(t) + √R² × β₁(tref − t)]

Consequently, the interannual variability and individual extreme events contained in the original AMAX series are retained, while the estimated systematic long-term change is 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 used for previous GloFAS versions (see: GloFAS v4 and earlier versions). The Gumbel distribution parameters are estimated using the same sample L-moment method, and the return-period discharge thresholds are calculated from the resulting adjusted, stationary-equivalent AMAX sample.

Example catchment demonstrating the impact of the annual maxima adjustment

The plot below shows an example of the method described above and demonstrates the impact of removing the estimated long-term trend from the annual maxima on the resulting flood thresholds. The example represents a river in the Congo basin, where a large decreasing trend is present in the GloFAS v5 reanalysis.

The original annual maxima are shown by dark red stars and exhibit a strong decreasing trend, from around 20,000 m³/s in the early part of the reanalysis to less than 10,000 m³/s towards 2025. Accordingly, the fitted linear trend has a steep negative slope and the R² value is high (0.829), indicating that a large proportion of the variability in the AMAX series is associated with the fitted linear temporal trend.

The annual maxima adjustment applies the difference between the fitted trend value at each year and the fitted trend value at the 2025 reference year to the corresponding annual maximum, scaled by √R². In this example, √R² = √0.829 = 0.9105, meaning that approximately 91% of the estimated linear trend difference is applied. Because the fitted trend is negative, the earlier annual maxima are shifted downwards towards the discharge regime represented by the end of the reanalysis period. The size of this adjustment progressively decreases towards 2025, where no adjustment is applied.

The resulting adjusted annual maxima no longer exhibit the strong underlying long-term trend, while retaining the interannual variability and individual extreme events of the original series. The impact on the derived flood thresholds is substantial. Using exactly the same Gumbel distribution and L-moment fitting methodology for both series, the thresholds fitted to the adjusted annual maxima are considerably lower than those fitted to the original annual maxima, approaching approximately half of the original values for some return periods in this example.

Example catchment in the Congo basin to demonstrate the behaviour and impact of the annual maxima adjustment on the related flood thresholds.

Impact on real-time GloFAS forecasts

The use of the adjusted flood thresholds in place of the original thresholds is expected to have a substantial positive impact on the reliability of the real-time GloFAS ensemble forecasts in areas affected by large trends in the reanalysis. In such locations, the original thresholds can be poorly representative of the more recent discharge regime and therefore inconsistent with the discharge levels of the real-time forecasts. The adjustment brings the flood thresholds into better alignment with the recent modelled discharge regime, while retaining the information provided by the complete 1979–2025 reanalysis period.

This impact can be particularly large in regions such as the Congo basin example shown above, where the original thresholds were substantially higher than those derived from the trend-adjusted annual maxima.

References

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.