...
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:
Qadj(t) = Q(t) + [Q̂(tref) − Q̂(t)]
where Q(t) is the original annual maximum, Q̂(t) is the value of the fitted linear trend for that year, and Q̂(tref) is the fitted value at the selected reference time.
...
Importantly, the adjustment does not replace individual annual maxima with values from the fitted trend. Instead, the original departure of each annual maximum from the fitted trend is retained:
Qadj(t) = Q̂(tref) + [Q(t) − Q̂(t)]
The term [Q(t) − Q̂(t)] represents the original year-to-year anomaly around the fitted linear trend. The adjustment therefore preserves the interannual variability and individual extreme events contained in the original AMAX series, while removing the systematic linear change through time and referencing all annual maxima to a common recent discharge level.
...
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. The resulting thresholds represent the historical range of interannual and extreme-event variability, but referenced to the selected recent discharge regime.
The adjustment is intended primarily to correct gradual, systematic temporal changes in the AMAX series. It does not imply that the underlying climate or hydrological system is strictly stationary, nor does it attempt to predict future changes. Instead, it provides a pragmatic way of producing a stationary-equivalent historical sample representative of a selected reference period while retaining as much information as possible from the available reanalysis.
The resulting thresholds represent the historical range of interannual and extreme-event variability, but referenced to the selected recent discharge regime.
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.
For each grid point, a linear trend is fitted to the annual maximum discharge series:
Q̂(t) = β₀ + β₁t
where β0\beta_0 is the intercept and β1\beta_1 is the estimated linear trend in discharge per year.
Qadj(t) = Q(t) + [Q̂(tref) − Q̂(t)] Qadj(t) = Q(t) + β₁(tref − t)
Q^(t)=β0+β1t\hat{Q}(t) = \beta_0 + \beta_1 t
where β0\beta_0 is the intercept and β1\beta_1 is the estimated linear trend in discharge per year.
The annual maximum for year tt is then adjusted to a common reference year treft_{\mathrm{ref}}:
Qadj(t)=Q(t)+[Q^(tref)−Q^(t)]\boxed{ Q_{\mathrm{adj}}(t) = Q(t) + \left[ \hat{Q}(t_{\mathrm{ref}}) - \hat{Q}(t) \right] }
Since the fitted trend is linear,
Q^(tref)−Q^(t)=β1(tref−t),\hat{Q}(t_{\mathrm{ref}})-\hat{Q}(t) = \beta_1(t_{\mathrm{ref}}-t),
and therefore
Qadj(t)=Q(t)+β1(tref−t)\boxed{ Q_{\mathrm{adj}}(t) = Q(t) + \beta_1(t_{\mathrm{ref}}-t) }
An alternative and perhaps more intuitive formulation is
Qadj(t)=Q^(tref)+[Q(t)−Q^(t)]\boxed{ Q_{\mathrm{adj}}(t) = \hat{Q}(t_{\mathrm{ref}}) + \left[ Q(t)-\hat{Q}(t) \right] }
which makes it explicit that the original anomaly around the fitted trend is preserved, but is transferred to the discharge level corresponding to the reference year.
So yes, for the final internal documentation I would definitely use equations formatted like this rather than the more text-like notation in the previous draft.
For each location, a trend function is fitted to the annual maximum discharge series. The fitted trend represents the gradual change in the expected magnitude of annual maxima through time. Each historical annual maximum is then shifted according to the difference between the fitted value for its year and the fitted value at a common reference time:
[
Q^{adj}y = Q_y + \left[f(t{ref})-f(t_y)\right],
]
where (Q_y) is the original annual maximum in year (y), (f(t_y)) is the fitted trend value for that year, (f(t_{ref})) is the fitted value at the selected reference time, and (Q^{adj}_y) is the adjusted annual maximum.
The adjustment therefore changes the location of individual annual maxima according to the estimated long-term trend but does not replace the observations with fitted values. Year-to-year deviations from the fitted trend are retained. In other words, the method aims to preserve the interannual variability and the individual extreme events contained in the original reanalysis while removing the systematic temporal change represented by the fitted trend.
The reference time is chosen to represent the recent discharge regime. Consequently, earlier annual maxima are shifted towards the level expected under the recent-period conditions. For an increasing trend, earlier maxima are generally adjusted upwards; for a decreasing trend, they are adjusted downwards. Annual maxima close to the reference period receive little or no adjustment.
A key advantage of this approach is that the complete historical AMAX sample remains available for the extreme-value analysis. No years are discarded and the effective sample size is therefore unchanged. This is particularly important for flood-frequency estimation, where the available record is relatively short and the largest events contain substantial information about the upper tail of the distribution.
Following the adjustment, the flood-frequency analysis is performed on the adjusted annual maxima. The resulting return-period thresholds can therefore be interpreted as thresholds based on the historical range of interannual and extreme-event variability, but referenced to the more recent discharge regime rather than to the average conditions over the entire reanalysis period.
The adjustment is intended primarily to correct gradual, systematic temporal changes in the AMAX series. It does not imply that the underlying climate or hydrological system is strictly stationary, nor does it attempt to predict future changes. Instead, it provides a pragmatic way of producing a stationary-equivalent historical sample representative of a selected reference period while retaining as much information as possible from the available reanalysis.