The archive (2009–2018) of radar rainfall composites in this study originates from two C-band weather radars operated by KNMI (Fig. ). Between September 2016 and January 2017, both radars were replaced by dual-polarization radars, and the radar in De Bilt (“DB” in Fig. ) was replaced by a new one in Herwijnen (“H” in Fig. ). The radar renewals and relocation have had a limited impact on the QPE product, mainly because the operational products are not yet (fully) using the additional information from the dual-polarization .

The radar product is Doppler-filtered for ground clutter. This product is then used to construct horizontal cross-sections at a nearly constant altitude of 1500 m, called pseudo-constant plan position indicators (pseudo-CAPPIs). Subsequently, range-weighted compositing is used to combine the reflectivities from both radars . Since 2013, non-meteorological echoes have been removed as an additional step with a cloud mask obtained from satellite imagery. As a final step, rainfall rates are estimated with a fixed Z–R relationship : 1Zh=200R1.6.

In this equation, Zh is the reflectivity at horizontal polarization (mm6m-3 but generally given in dBZ, according to 10×log⁡10[Zh]), and R is the rainfall rate (mmh-1). The final product is called the unadjusted radar QPE (RU) in this study.

KNMI also provides adjusted radar rainfall products, based on the aforementioned product, but adjusted with quality-controlled observations from both 32 automatic hourly and 319 manual daily rain gauges (; note that the number of rain gauges has slightly changed from 2009 until present). The same 32 automatic rain gauges are used for the MFB-adjustment method, which will be introduced in Sect. . In contrast to the spatially uniform hourly MFB adjustment, the observations from the manual rain gauges are used for daily spatial adjustments, based on distance-weighted interpolation of these observations . See Sect.  for a more detailed description of this method.

This product is considered as a reference rainfall product in the Netherlands, and it is therefore also regarded as the reference here (referred to as RA in this study). The RA data are not available in real time (available with a delay of 1 to 2 months because they only use quality-controlled and validated rain gauge observations), but they are archived and can therefore be used for “offline” methods. Both RA and RU have a 1 km2 spatial and 5 min temporal resolution.

The year 2008 is actually the first year in the KNMI archive of both datasets, but it was left out of the analysis here. RU for this year showed a significantly different behavior than the other years, especially during the first half year in which the product rarely underestimated and frequently even overestimated the rainfall sums. The reason for this behavior is not yet fully understood. reported that spring was exceptionally dry in the north of the country and that the months January and May were among the warmest on record. On some days with overestimations, clear bright band effects were visible in the radar mosaic, which may have contributed to the systematic differences.

Figure  indicates the need for correction of the real-time available radar rainfall product. RU systematically underestimates the true rainfall amounts, averaged for the land surface area of the Netherlands, by 55 %. This bias is not uniform in space, as will be highlighted in Sect. , and in time with higher underestimations during winter (on average 65 %) than during the other seasons (50 %–55 %). In the following two subsections, the operationally used MFB-adjustment method and the CARROTS method proposed in this study will be introduced.

Both bias adjustment methods were applied to the 10 years (2009–2018) of RU. In order to provide a hydrometeorological testbed, both the CARROTS and MFB-adjusted QPE products (from here on referred to as RC and RMFB, respectively) were validated against the reference rainfall. First, this was done at country level. The estimated daily rainfall sums for all grid cells within the land surface area of the Netherlands were compared to the reference in a similar way as the comparison in Fig. . To subdivide these results per year and season, an additional hourly rainfall sum validation was performed as well. The results of this analysis can be found in the Appendix, and the analysis was done as follows: for every rainy hour (when the sum of at least one grid cell was larger than 0.0 mm), we computed the root mean square error (RMSE) by squaring the differences between the three QPE products (RU, RC and RMFB) on the one hand and the reference on the other and taking the average of these squared differences over all grid cells within the land surface area of the Netherlands. Subsequently, the RMSE was averaged over all rainy hours in that season and year. Finally, the seasonal mean RMSE was divided by the average hourly rainfall rate for that season and year, resulting in the fractional standard error (FSE) score. The FSE score was calculated for every season in the 10 years to be able to compare the seasonal performance of the hourly rainfall estimates of RU, RC and RMFB.

Second, the annual rainfall sums for 12 basins (a combination of catchments and polders) in the Netherlands (Fig. ) were compared with the reference. In addition, RC and RMFB were used as input for the rainfall-runoff models of the 12 basins. Most of the involved water authorities use these (lowland) rainfall-runoff models either operationally or for research purposes, often embedded in a Delft-FEWS system, which is a data-integration platform used worldwide by many hydrological forecasting agencies and water management organizations that brings data handling and model integration together for operational forecasting . For this reason, most models were already calibrated using interpolated rain gauge data or the RA product e.g.. The calibration period was based on the availability and quality of discharge observations for that basin, but it was generally 1 to 2 years within the period considered in this study (2009–2018). The WALRUS models for catchments Roggelsebeek and Dwarsdiep were not calibrated prior to this study and were therefore calibrated with the reference data (RA) for the periods 2013–2014 (Roggelsebeek) and 2016–2017 (Dwarsdiep). The choice for these periods was based on discharge observation availability and quality. The employed SOBEK RR(-CF) model is semi-distributed, and therefore we used sub-catchment-averaged rainfall sums from the gridded radar QPE. The four basins with a SOBEK model have the following number of sub-catchments: 7 for Gouwepolder, 1 for Beemster, 25 for Delfland and 23 for Linde. WALRUS is lumped, so the catchment-averaged radar QPE was used as input. A more detailed description of both rainfall-runoff models is outside the scope of this paper. All 12 model setups were run with a 5 min time step for the period 2009–2018.

The resulting discharge simulations were validated for the same period and 5 min time step using the Kling–Gupta efficiency (KGE) metric : 7KGE=1-(ρ-1)2+(α-1)2+(β-1)2, with 8α=σsσo,9β=μsμo.

Here, ρ is the Pearson correlation between observed and simulated discharge, α the flow variability error between observed and simulated discharge and β the bias between mean simulated (μs) and mean observed (μo) discharge. σs and σo are the standard deviations of the simulated and observed discharge. The KGE metric ranges from -∞ to 1.0, with 1.0 representing a perfect agreement between observations and simulations. In this study, the discharge simulated with RA as input was regarded as the observation.

Note that this validation method was not a leave-one-out or split-sample validation, as the full 10-year dataset was used for RA and the CARROTS- and MFB-adjustment derivation, and shorter periods in those 10 years were used for hydrological model calibration. However, the sensitivity of the CARROTS factor was tested by leaving individual years out of the derivation period (Sect. ).

As mentioned in Sect. , the purpose of the 31 d moving window in the factor derivation of CARROTS was to smooth the day-to-day variability of rainfall. To test the sensitivity of the method to the employed moving window size, the adjustment factors were re-derived for a range of moving window sizes (1 d, 1 week, 2 weeks, 6 weeks and 2 months). The derived factors were then compared to the original factor in this study, which was based on a moving window size of 31 d, and used to derive adjusted QPE products. Subsequently, these QPE products served as input for 1 of the 12 catchments, namely the WALRUS model for the Aa catchment (Fig. ), to test the effect on the simulated discharges (see Sect.  and Fig.  for the results). The Aa catchment was chosen because the unadjusted QPE product (RU) for this catchment has one of the highest biases of the 12 studied catchments (see Sect.  and Fig. ).

Besides the moving window choice, the length of the radar rainfall archive (10 years) was finite. To test whether or not this archive length was sufficient for reaching a stable factor derivation, individual years in the 10-year archive were left out of the CARROTS method. Hence, the adjustment factors were recalculated 10 times in a leave-one-year-out method, applied to RU and used as input for the WALRUS simulations for the Aa catchment. See Sect.  and Fig.  for the results.

The adjustment factors from CARROTS present the spatial variability in the radar QPE errors, with generally higher adjustment factors towards the edges of the radar domain (Fig. ). This difference is most pronounced from December through March, with factors in the south and east of the country more than 2 times higher than in the central and northwestern parts (Fig. a, b and l). Figure  demonstrates a clear annual cycle of the adjustment factors, with higher adjustment factors from December through March than in the other months. Figure a shows similar results for the catchment-averaged adjustment factors, with factors ranging from 2.1 for the Beemster polder to 3.2 for the Hupsel Brook catchment in January, whereas adjustment factors range from 1.3 for the Grote Waterleiding catchment to 1.6 for the Roggelsebeek catchment in June.

An explanation for these higher adjustment factors from December through March is that radar QPE often severely underestimates the rainfall amounts for stratiform systems, which regularly occur during the Dutch winter. This especially holds when the QPE is constructed from reflectivities sampled above the melting layer . This seems to be the case here as well. A simple first-order estimation of the 0 ∘C isotherm level, using a constant wet adiabatic lapse rate of 5.5 Kkm-1 with ground temperature data for all rainy hours in the 10 years (Fig. b), indicates that the 1500 m pseudo-CAPPI is generally above the 0 ∘C isotherm level from December through March. This coincides with the months with higher adjustment factors (Fig. c) and could thus explain the winter effect on the adjustment factors. This effect is presumably even stronger further away from the radars because the QPE product consists of samples at even higher altitudes than 1500 m for locations more than 120 km from the radars. Besides, an additional dependency of the monthly factor on the time of year that cannot be explained by temperature seems to be present, with lower adjustment factors during spring and early summer and higher factors for the subsequent period (Fig. c).

The MFB-adjusted QPE (RMFB) significantly reduces the systematic bias of RU (Fig. 2), from a 55 % underestimation on average for the Netherlands to 10 % (Fig. a and Table ). However, the remaining bias in RMFB is generally caused by a systematic underestimation of the reference rainfall. The overall underestimation is less for RC (8 %, Fig. b) but results from estimation errors that are associated with either under- or overestimates of the reference rainfall. The spread in Fig. b is significantly wider than in Fig. a, indicating that the country-wide QPE error of RC is often higher than for RMFB. The yearly FSE in Table  clearly indicates this too, with a systematically higher FSE for RC than for RMFB.

An advantage of the MFB adjustment is that it corrects for the circumstances during that specific day and thus also for instances with overestimations (Fig. a). On a country-wide level, this is clearly advantageous, also compared to CARROTS (Fig. ). The negative effect of the spatial uniformity of the factor, however, becomes apparent in Fig. , which compares the annual precipitation sums of the two adjusted radar rainfall products with the reference and RU for the 12 basins. For all basins, both adjusted products manage to significantly increase the QPE towards the reference. However, for 9 out of 12 basins, RC outperforms RMFB (Fig. e). Exceptions are Beemster, Luntersebeek and Dwarsdiep, where the performance of both products is similar. Differences between the performance of RC and RMFB become most apparent for catchments that are located closer to the edges of the radar domain. For instance, RMFB for the Aa and Regge catchments, which are located in the far south and east of the country, still underestimates the annual reference rainfall sums, with on average 20 % for the Aa (mean annual RMFB is 610 mm, and mean annual RA=761 mm) and 13 % for the Regge (mean annual RMFB is 673 mm, and mean annual RA=776 mm), while this is on average only 5 % (both under- and overestimations occur) for RC (Fig. 6b and c).

The MFB-adjusted QPE performs better for the Beemster polder, Dwarsdiep polder (Fig. d) and Luntersebeek catchment (Fig. a) due to their location in the radar mosaic. The Luntersebeek catchment (central Netherlands, Fig. ) is located closer to both radars. There, RMFB generally performs better and sometimes even overestimates the true rainfall, which is consistent with Holleman (2007). The performance of RMFB for the Dwarsdiep catchment is similar to its performance for the Linde catchment (both in the north of the country), but RC shows more variability in the error from year to year for the Dwarsdiep catchment (Fig. d), leading to a better relative performance of RMFB. The CARROTS QPE tends to overestimate the rainfall amount of the three aforementioned basins (Beemster, Dwarsdiep and Luntersebeek) for some years (e.g., by 16 % for the Luntersebeek in 2016). Overall, the performance of RC and RMFB is not that different for these three basins, with on average just a lower MAE for RMFB than for RC for the Luntersebeek catchment and Dwarsdiep polder (Fig. e).

Summarizing, the CARROTS factors have a clear annual cycle, with generally higher adjustment factors further away from the radars (Sect. ). On average for the Netherlands, the MFB-adjusted QPE outperforms the CARROTS-corrected QPE. However, the spatial variability in the CARROTS factors, in contrast to the uniform MFB adjustment, results in estimated annual rainfall sums for the 12 hydrological basins that are generally closer to the reference (for 9 out of 12 basins) than with the MFB-adjusted QPE, especially for the east and south of the country. This effect is expected to become more pronounced when the adjusted QPE products are used for discharge simulations.

The severe underestimations of RU have a considerable effect on the discharge simulations for the 12 basins (Fig. ). This leads to hardly any discharge response and thus negative KGE values for most basins as compared to discharge simulations with the reference rainfall data. The effect is most pronounced for the freely draining catchments in the east and south of the country. These catchments are more driven by groundwater flow than the polders in the west of the country. Groundwater flow gets hardly replenished because of similar estimated annual evapotranspiration and RU sums, resulting in baseflows that are too low. The polders, especially Delfland and Beemster, are an exception to this because they are less driven by groundwater-fed baseflow and more by direct runoff from greenhouses or upward seepage flows, which makes them more responsive to individual rainfall events, leading to higher KGE values (with RU as input) compared to the other basins.

The model runs using RMFB as input significantly improve the simulated discharges, compared to the runs with RU. Nevertheless, the model runs still strongly underestimate the simulated discharges compared to those from the reference runs for the catchments in the south and east of the country (Fig. a–f). This is particularly noticeable for the catchments Reusel (KGE = 0.26) and Roggelsebeek (KGE = 0.04). The spatial uniformity of the MFB factors is identified as the cause of these effects because the MFB method can not correct for the sources of errors leading to the biased QPE in space. This already led to clear underestimations in the annual rainfall sums for these regions (Fig. ).

The CARROTS QPE outperforms RMFB when this product is used as input for the 12 rainfall-runoff models. This is not exclusively the case for the six catchments in the east and south of the country (Fig. a–f), but also for the other polder and catchment areas. The exception to this is the Beemster polder. The Beemster is mostly fed by upward seepage, leading to a more predictable baseflow for all models runs. In addition, the catchment is located close to an automatic weather station and is located between both operational radars, which makes the MFB adjustment more beneficial for this region. The difference in performance between the hydrological model simulations is small, with a KGE of 0.92 (using RC) versus 0.96 for RMFB, as compared to the reference run.

The use of a different moving window size hardly influences the CARROTS factors for moving window sizes of 2 weeks or longer, but this does not hold for moving window sizes of a day or, to a lesser extent, 1 week (Fig. a). The factor derived with a moving window size of 1 d fluctuates heavily from day to day. This suggests that the adjustment factor is still quite sensitive to individual events in the 10-year period, when a moving window size of 7 d or shorter is used. Moving window sizes of more than a month (6 weeks and 2 months were tested here) lead to similar CARROTS factors as with a 1-month (31 d) moving window size but somewhat more smoothed. A similar effect likely takes place for a seasonal (3-month) moving window. For larger moving window sizes (half a year to a year, for instance), we expect that the seasonality in the factor is lost and that an average correction factor remains.

In contrast to this, the differences between these six sets of CARROTS factors (Fig. a) lead to minimal variations in the simulated discharges for the Aa catchment when these factors are used to adjust the input QPE (Fig. b). Differences in timing and magnitude (0.2–0.3 mmd-1) are visible during peaks and recessions, for instance in early April. However, these are small compared to the differences between the model runs with RC and RMFB (Fig. ). However, the use of a window size of 1 d or, to a lesser extent, of a week clearly leads to more fluctuations in the CARROTS factor (Fig. a) and can therefore influence the rainfall estimation for individual events (and the factor will also be influenced by these individual events). For quickly responding catchments and urban catchments, this could still lead to different results. In conclusion, a 31 d smoothing of the climatological adjustment factor is warranted.

In addition, leaving individual years out of the 10-year archive has a limited impact on the CARROTS factors (see also the vertical bars in Fig. c). Similar to the aforementioned results for the moving window size analysis, it leads to hardly any variations in the simulated discharges for the Aa catchment (not shown here). This suggests that the 10-year archive length was sufficiently long for the factor derivation.