The WRF model is a community mesoscale NWP system developed by the National Center for Atmospheric Research (NCAR) and partner institutions . The present study uses the Advanced Research WRF (ARW) core, configured as a single domain at 12 km horizontal grid spacing covering Luxembourg and the Greater Region (Fig. ). The domain comprises 120×120 horizontal grid points with 33 terrain-following vertical levels and a model top at 50 hPa, integrated with a 60 s time step. All simulations were performed with WRF version 4.5. The physics suite is summarised in Table .

The domain was defined to cover the Greater Region because the synoptic systems responsible for these floods develop at that scale, while the hydrological focus, and hence the density of verification stations, is Luxembourg. The parameterisations in Table  follow published evaluations of WRF over neighbouring domains, supported by exploratory comparisons against station observations. The two choices that most directly govern convective rainfall at this grid spacing are corroborated by an independent systematic study: in an ensemble of 30 parameterisation combinations applied to eight extreme precipitation events over Germany, identified the Grell–Freitas cumulus and YSU boundary-layer schemes as the best-performing members of their respective groups, both of which are adopted here. likewise employed the Noah land surface model and YSU boundary layer in convection-permitting simulations over Central Europe. The Thompson microphysics scheme was retained for its double-moment treatment of ice and rain, which is well suited to the mixed-phase and orographically enhanced precipitation characteristic of the Ardennes–Eifel uplands. A comprehensive sensitivity analysis of the full physics suite lies beyond the scope of the present comparison and is left for future work. The background error covariance was not taken from a default dataset but generated specifically for this domain using the NMC method over a one-month period, as described in Sect. .

The 3D-VAR assimilation mode within the WRFDA system was employed. The primary objective of 3D-VAR is to minimise a quadratic cost function representing the discrepancy between the model analysis and both the background field and observations : 1J(x)=(x-xb)TB-1(x-xb)+(H(x)-y)TO-1(H(x)-y), where xb is the background field, B is the background error covariance matrix, H is the observation operator, y represents the observations, and O is the observation error covariance matrix. The background error covariance was configured using the CV5 option with domain-specific statistics generated via the NMC method. The cost-function minimisation used two outer loops, and observations were accepted within a ± 1 h window centred on each analysis time.

Initial and lateral boundary conditions were taken from the ERA5 reanalysis at 0.25° resolution, sampled every 6 h across each 30 d window . Conventional observations were drawn from the NCEP PREPBUFR archive and comprise SYNOP and METAR surface reports, TEMP radiosonde profiles, TAMDAR aircraft measurements , and satellite atmospheric motion vectors (SATOB). In addition, GNSS ZTD observations were obtained from the Nevada Geodetic Laboratory . The composition of this observation set differs substantially between the three events, which proves important for interpreting the high-threshold results (Fig.  and Table ).

Observations were assimilated on a 6-hourly cycle following the Rapid Update Cycle (RUC) strategy . The model is first cold-started from ERA5 to produce a spin-up; thereafter, each analysis updates the short-range first guess with the available observations, and the corrected state initialises the next cycle (Fig. ).

Verification observations were assembled from four national and regional archives: the Luxembourg AgriMeteo network , the German Weather Service, Météo-France and the Walloon SPW network. Station lists, per-event completeness and the screening applied to each are given in the accompanying data files. Hourly composite weather-radar rainfall accumulations for the July 2021 event were obtained from the RADFLOOD21 product of the Royal Meteorological Institute of Belgium and aggregated to 6 h totals.

Meteorological variables were extracted from the WRF output using a custom Python workflow built on GDAL . Model values at station locations are taken from the nearest grid point, using a cKDTree query on the model grid ; the same nearest-neighbour rule is applied to all four systems so that they are sampled identically (Table ). No spatial averaging or weighting is applied.

Table  records the information required to reproduce the AI forecasts and their extraction to station locations.

No bias correction, calibration or downscaling was applied to any AI output. ERA5 fields were retrieved from the ECMWF MARS archive (class=ea, expver=1, stream=oper) at 0.25° on the pressure levels each model requires. The same reanalysis is distributed through the Copernicus Climate Data Store; we name MARS because that is the retrieval actually executed, and the exact requests are archived with the code. For GraphCast and FuXi, which run on that same regular grid, the only pre-processing was selection of the variables and levels each model's input schema specifies, plus the 6 h accumulation of total_precipitation_6hr described below. AIFS runs on the N320 reduced Gaussian grid, and its inputs were retrieved from MARS directly on that grid (grid=N320), together with the constant fields it requires. No regridding of the AIFS input was performed, either by us or inside anemoi-inference. WRF fields were extracted to station locations with the same nearest-neighbour approach on the model grid, so that all four systems are sampled identically. Precipitation is the 6 h accumulation valid at the verification time in every case.

One input requirement differs between the three AI systems and deserves explicit statement, because it is a common source of error. The GraphCast checkpoint used here takes total_precipitation_6hr as a model input as well as an output, and it expects a genuine 6 h accumulation: the six consecutive hourly ERA5 totals ending at the initialisation time, summed. Supplying instead the single hourly total ending at that time – the value returned by the common “hourly, sampled at 00/06/12/18” retrieval – supplies the wrong accumulation period and degrades the forecast substantially. It is not in general six times too small: for steady rain the 6 h total is about six times the final hour, but rainfall is intermittent, so the ratio varies from zero to much more than six from cycle to cycle. The same file also carries toa_incident_solar_radiation, which must not be summed in the same way; only the precipitation field is accumulated. Neither of the other two systems has this requirement. FuXi accepts a precipitation channel but its released checkpoint does not use it: forecasts produced with the correct 6 h accumulation, with zeros, and with random values are identical to within machine precision. AIFS treats total precipitation purely as a diagnostic output and takes no precipitation input at all. All GraphCast results reported here use the correctly accumulated 6 h input.

The two modelling systems were evaluated on three historic high-impact flood events over Luxembourg and the Greater Region: 2016, 2018, and 2021. Each event was simulated over a window of approximately one month (10 July–6 August 2016, 20 May–20 June 2018, and 20 June–20 July 2021), with forecasts initialised every 6 h, giving around 90 simulated days per configuration across the three events. The 2016 window is the shortest of the three, at 27 d, because the archived WRF output for that event ends on 6 August; the other two are full 30 d windows. Nominal 6-hourly sampling of a 30 d window gives about 120 valid times per event. Fewer are scored, because scores are computed only on the valid times common to all five products: 106, 122 and 117 6 h times enter the verification for 2016, 2018 and 2021, a total of 345 times or about 86 simulated days per configuration. The radar comparison of Sect.  uses 27 6 h periods and the resolution experiment of Sect.  22; neither is a subset of the station sample, since both are field-to-field comparisons. For WRF, two configurations were produced: a cold-start run without data assimilation (Before DA) and an observation-corrected run (After DA) using the 6-hourly RUC described in Sect. . WRF simulations were executed on 52 CPU cores of a high-performance computing cluster, distributed over dual-socket nodes with 28 Intel Xeon (Broadwell) cores each.

A 30 d window was chosen in preference to a window restricted to the flood days themselves. The first objective of the study is to establish whether 6-hourly assimilation of GNSS ZTD and conventional observations improves forecast skill, and a sample of two to four days does not contain enough independent precipitation events to separate that signal from sampling noise. The 30 d windows also include the pre-event synoptic evolution that conditions each flood, and they provide the model spin-up required by the Rapid Update Cycle. Scores restricted to the flood days alone are reported alongside the full-window scores in Sect. .

The three AI models were run on the same three events and 30 d windows. Forecasts were initialised from ERA5 analyses (at t-6 h and t) and advanced a single 6 h step to the verification time; each model was re-initialised from ERA5 every 6 h rather than rolled out freely. Inference was performed on a single NVIDIA A100 GPU with 40 GB of HBM2 memory. The 6 h cycling interval was chosen to align the products being compared. GraphCast, FuXi and AIFS all advance on a fixed 6 h native time step, which cannot be altered without retraining; that constrains the prediction interval, not necessarily the frequency at which forecasts could be launched, and off-synoptic initialisation is a separate question we have not tested. Cycling WRF more frequently would have given the physics-based system a shorter effective update interval than the products it is compared against. The sensitivity of the results to the cycling frequency is not examined here and we identify it as future work.

The computational contrast between the two approaches is large. On 52 CPU cores, one 6 h WRF cycle takes approximately 34 min without assimilation and 37 min with the 3D-VAR analysis included, so a complete 30 d window requires roughly 70 and 75 h of wall-clock time respectively. On a single NVIDIA A100 GPU, one 6 h AI forecast takes about 1.5 min, so the 120 initialisations of a 30 d window take about 30 min of inference. That is roughly 140 times faster than the physics-based chain for the same product, albeit after a training phase whose cost is borne once by the model developers. The timings are wall-clock for the forecast step alone and exclude ERA5 retrieval, checkpoint loading and output writing.

Three distinct evaluations are reported, each covering the largest period for which the necessary observations exist. They are summarised here so that the domain of validity of each result is unambiguous.

The model intercomparison (Sect. –) covers all three flood events over their full evaluation windows, verified against surface stations. This is the primary evaluation and provides the categorical and continuous scores.

The verification against weather radar (Sect.  and ) is restricted to the July 2021 event. The RADFLOOD21 composite was produced specifically for that flood and is extensively documented ; no equivalent open radar product covers the 2016 and 2018 events. It has two parts. Section  compares the fields visually at the three 6 h periods spanning the peak, 14 July 18:00 to 15 July 06:00 UTC. Section  scores them with the Fractions Skill Score over the 27 6 h periods from 11 July 06:00 to 17 July 18:00 UTC for which the radar total is complete and every system has a field.

The resolution experiment (Sect. ) is restricted further, to the 22 6 h periods from 12 July 12:00 to 17 July 18:00 UTC, because the 4 and 1.33 km nests were not integrated before that time. It is therefore a targeted test of one question, whether horizontal resolution explains the behaviour seen at 12 km, and its results are not carried into the pooled statistics.

A point of uniformity should be stated explicitly, because the three events differ in the observations available for assimilation (Table ). That heterogeneity affects the input to a single configuration, WRF (After DA). In every event all systems draw on the same ERA5 fields, are advanced on the same 6-hourly cycle, and are verified against the same station network with the same metrics and the same matching procedure. The verification is therefore uniform across the three events. The initial states themselves are not identical, and we do not claim they are: the AI models are re-initialised directly from ERA5 at each cycle, whereas the WRF first guess is a 6 h forecast from the previous cycle and, in the assimilated configuration, is corrected by observations. This study is best read as a comparison of ERA5-driven 6 h hindcast systems that includes an explicit WRF assimilation sensitivity, rather than as an isolation of physics from data-driven formulation under identical initial states. We regard this variation as a property of the observing system rather than of the experimental design, since aircraft reporting depends on commercial air traffic and GNSS coverage depends on which regional archives extend back to a given event.

Verification uses surface stations that are independent of the observations assimilated into WRF. Two criteria were applied. First, any station lying within 12 km of a surface observation assimilated in that event's own analyses was excluded; the assimilated positions were taken from the observation files actually ingested by the Rapid Update Cycle rather than assumed, and the screen was applied separately for each event. Second, stations whose records cannot support a 6 h accumulation, several of which report only 12 or 24 h totals, were excluded, retaining only those supplying at least 95 % of the 6-hourly windows. The retained networks comprise 259, 280 and 264 stations for precipitation in 2016, 2018 and 2021. Of these, 119, 134 and 148 stations also carry a 2 m temperature channel and pass the same two criteria, and the temperature scores reported below are computed on those; the smallest station-to-observation separation anywhere is 12.04 km. The counts differ between events because the available station networks themselves differ – gauges are commissioned and decommissioned, records begin and end, and the assimilated observing system was denser in 2021 – not because a different rule was applied. For the 2021 event the retained network comprises 105 German, 92 French, 34 Luxembourgish and 33 Belgian stations occupying 234 distinct 12 km model grid cells. All 264 report precipitation; 148 of them report 2 m temperature as well, leaving 116 that report precipitation only (Fig. ). The two variables use different networks because not every station reports both: the Belgian SPW gauges are precipitation-only, only a subset of the German stations carries an hourly temperature record, and fewer still report it without long gaps.

It should be emphasised that precipitation, the primary verification variable, is not assimilated by any observation type in this configuration. The WRFDA observation files carry only pressure, wind speed and direction, height, temperature, dew point and humidity, together with sea-level pressure and precipitable water; radar reflectivity and rain-rate assimilation are disabled. No precipitation observation therefore enters the regional analysis, and the station-selection criteria extend the same separation to the 2 m temperature verification. Two limits on that statement should be kept in view. The assimilated temperature, humidity and wind observations can inform the model's rainfall indirectly, so the verification is independent of direct reuse rather than statistically independent of the analysis. And ERA5, which supplies the initial and boundary conditions for every system here, has itself assimilated many of the same observing networks.

The retained network spans four countries and samples the model domain rather than concentrating on the hydrological area of interest. For the 2021 event it comprises 105 German, 92 French, 34 Luxembourgish and 33 Belgian stations occupying 234 distinct model grid cells, and the 2016 and 2018 networks are similarly distributed. Two features are worth noting. The Luxembourg AgriMeteo gauges provide continuous 6-hourly records at high spatial density over the small, fast-responding catchments described in Sect. , and because they are agro-meteorological stations that do not report on the Global Telecommunication System they cannot enter an operational analysis: they are independent by construction rather than by selection. The German, French and Belgian stations, by contrast, sample the wider domain over which the responsible synoptic systems develop. The scores reported here therefore describe domain-scale performance and are not dominated by conditions in a single catchment.

Because precipitation is intermittent and highly skewed, categorical scores derived from a 2×2 contingency table of forecast/observed rainfall occurrence at or above a threshold are used, with hits a, false alarms b, misses c, correct negatives d, and total n=a+b+c+d. The comparison is inclusive, ≥ rather than >. This is worth stating because 692 of the 92 119 observations in each model's sample are reported at exactly 1.0 mm, and would change category under the strict inequality: 2POD=aa+c,3FAR=ba+b,4CSI=aa+b+c,5ETS=a-ara+b+c-ar,ar=(a+b)(a+c)n, where POD is the probability of detection, FAR the false-alarm ratio, CSI the critical success index, and ETS the equitable threat score. Scores are reported at a 1.0 mm(6h)-1 threshold throughout, and additionally at 5, 10 and 20 mm(6h)-1 in Sect.  to assess whether the conclusions depend on the rainfall rate considered. Near-surface temperature and precipitation amount were verified with continuous statistics over N forecast–observation pairs (Fi,Oi): 6Bias=1N∑i=1N(Fi-Oi),7MAE=1N∑i=1NFi-Oi,8RMSE=1N∑i=1N(Fi-Oi)2,9r=∑i(Fi-F‾)(Oi-O‾)∑i(Fi-F‾)2∑i(Oi-O‾)2. For POD, CSI, ETS, and r, higher is better; for FAR, MAE, and RMSE, lower is better; the bias is best near zero. Bias, MAE and RMSE are used for the station verification of 2 m temperature; r is used only for the field-to-field comparison against radar (Sects. –), where Fi and Oi are the model and radar values at the points of the common mesh rather than at stations.

The aggregated scores are complemented by a spatial evaluation against the Belgian weather-radar composite for the peak of the July 2021 flood. Hourly RADFLOOD21 radar accumulations were aggregated to 6 h windows. This evaluation is restricted to 2021 because RADFLOOD21 was released specifically for this event, whereas no equivalent open radar product covers 2016 and 2018. Figure  compares the 6 h precipitation fields at three consecutive timestamps spanning the event peak (14 July 18:00 UTC to 15 July 06:00 UTC).

The station gauges overlaid on each panel provide a partial check on the radar composite: within Luxembourg the two agree closely on both the location and the magnitude of the heaviest 6 h accumulations. That check is weaker than it may appear. RADFLOOD21 is a radar–gauge merged analysis whose inputs include the Belgian SPW network, so the SPW gauges in our verification set are not independent of it; the Luxembourg AgriMeteo gauges are, and it is those that support the agreement quoted here. Agreement at a few stations also does not validate the composite across the whole scoring domain – at Mirecourt, near the edge of the composite, the radar gives 42.3 mm against 62.0 mm at the gauge. We therefore treat the radar as a useful spatial reference rather than as ground truth.

Including the unassimilated WRF run in the comparison isolates the effect of the assimilation on the spatial field, and here the visual impression and the quantitative score disagree, so we report both. Both configurations place the rain band in approximately the correct position and the assimilated field looks the sharper of the two. Scored against the radar over these three panels alone, however, the unassimilated run is the better of the two at every threshold and every neighbourhood width, and its spatial correlation with the radar accumulation is higher (0.671 against 0.578); at the 12 km neighbourhood and the 1 mm(6h)-1 threshold it attains an FSS of 0.849 against 0.801, using the same kilometre-sized neighbourhoods and native-to-common-grid treatment as Sect. . Over the full week (Sect. ) the ordering is the opposite at the lower thresholds, where assimilation raises the panel-mean FSS from 0.502 to 0.519 at 1 mm and from 0.366 to 0.385 at 5 mm at 12 km, and raises correlation from 0.290 to 0.306. These are descriptive differences; the pooled FSS can rank the two WRF configurations differently, and no dependence-aware uncertainty analysis of radar FSS is presented here. The peak and week summaries therefore need not support identical statements about assimilation. We draw attention to this because it also illustrates why the visual comparison of Fig.  cannot stand on its own, and why the quantitative scores of Sect.  are reported alongside it.

The radar shows an intense, elongated rain band stretching from Belgium across Luxembourg into western Germany. Both WRF configurations reproduce this banded mesoscale structure and its sharp gradients, and the assimilated field looks the sharper of the two, although as reported above the cold-start run scores higher over these three panels. GraphCast captures the correct location of the rainfall envelope but is strongly smoothed and underestimates peak intensities. FuXi produces almost no precipitation signal during the peak, consistent with its poor categorical scores. AIFS produces the largest and most coherent precipitation envelope of the three AI systems, which explains its high detection rates, but its fields remain smooth and broad relative to the radar. Thus, even where the aggregated skill of AIFS or GraphCast matches or exceeds WRF, the physics-based model retains a clear advantage in representing the fine-scale structure and amplitude of extreme convective rainfall.

Figure  shows 6-hourly precipitation time series from 10 to 17 July 2021 at four illustrative stations near the verification network described in Sect. . Three span the physiographic gradient of the study area described in Sect. : Briedfeld in the northern Oesling, Ettelbruck near the Sauer–Alzette confluence, and Oberkorn in the southern Gutland. Mirecourt lies some 150 km to the south in France and tests behaviour away from the core catchments. Briedfeld passes the 12 km separation screen but fails the 95 % completeness rule, so it is not included in the pooled verification sample.

The time series corroborate both the aggregated metrics and the spatial analysis. At the three Luxembourgish stations the gauge and radar records agree closely on the timing and magnitude of the episode (130.6, 121.8 and 131.8 mm at the gauges against 129.3, 116.2 and 116.3 mm from the radar over the 8 d window), providing a well-constrained target against which the forecasts can be judged. Of the systems plotted, the unassimilated WRF comes closest at Briedfeld (134.5 mm against 130.6 mm observed) while AIFS is closest at the other two (112.0 and 113.0 mm; 121.4 mm at Briedfeld), and AIFS captures the timing of the heaviest 6 h accumulations; GraphCast is close behind at all three (113.7, 106.5 and 100.4 mm); the assimilated WRF follows with somewhat weaker peaks (97.9, 87.5 and 94.5 mm); and FuXi remains far too low throughout the episode (56.0, 48.8 and 42.5 mm), less than half the observed total at every station, consistent with its poor categorical scores. At Mirecourt, where the event was much weaker, the ordering differs: WRF tracks the observed total most closely (60.5 mm against 62.0 mm observed) whereas AIFS overestimates it (89.6 mm), indicating that the tendency of AIFS to produce broad, coherent precipitation envelopes helps it during widespread heavy rainfall but costs accuracy in more marginal situations.

The comparison in Sect.  is visual, and visual comparison of precipitation fields is subjective. The week covered by the radar composite was therefore scored quantitatively using the Fractions Skill Score FSS;, which compares the fraction of grid points exceeding a threshold within a neighbourhood of increasing size and so credits a forecast that places rainfall approximately, rather than exactly, in the right place.

Four points of sampling and method are stated explicitly, because each of them can move the scores. First, a 6 h radar total is formed only where all six hourly composites exist; a partial window is discarded rather than treated as containing dry hours. Of the 28 6 h periods of 11–17 July 2021 this removes one, 11 July 00:00 UTC, for which only the final hour is archived. Second, the systems are scored on the intersection of the valid times at which the radar and every system has a field, which leaves 27 common periods, 11 July 06:00 to 17 July 18:00 UTC. Third, all fields reach the common mesh by the same nearest-neighbour step from their own native grids: the 0.02° (∼ 2.2 km) mesh spanning 47.5–51.5° N and 4.0–8.5° E. Inside that box the native grids provide 1764 points for WRF, 323 each for GraphCast and FuXi, and 159 for AIFS on its N320 reduced Gaussian grid, so AIFS is the most coarsely sampled of the systems compared here, not the least. Fourth, a period in which neither the forecast nor the radar exceeds the threshold carries no information and cannot be scored; such periods are excluded, so the number of scored periods falls as the threshold rises, from 25–27 at 1 mm to 18 at 20 mm(6h)-1. Those counts, and a pooled aggregate FSS formed from summed numerator and denominator terms over the common periods, are given in the data file accompanying this paper; the aggregate weights panels by their FSS denominator and can change the ordering. At 1 mm and 12 km, for example, GraphCast leads the aggregate (0.748), ahead of WRF Before DA (0.738), WRF After DA (0.732) and AIFS (0.730), whereas AIFS leads the panel mean. At 100 km the aggregate leader is WRF Before DA (0.874). At 20 mm, both estimators place WRF ahead of the AI models, although their preferred WRF configuration differs at 100 km. Rankings must therefore name the estimator. Neighbourhood filtering uses zero padding outside the rectangular scoring domain.

Results are given in Fig. , together with the spatial correlation of the 6 h accumulation fields.

The quantitative scores show the same threshold dependence as the station scores. At 1 mm(6h)-1 AIFS leads the panel mean at every neighbourhood width (0.574 at 12 km), ahead of GraphCast (0.530), the assimilated WRF (0.519) and the cold-start run (0.502); at 5 mm AIFS leads at 12–50 km, while WRF After DA leads at 100 km (0.573 versus 0.537). The ordering reverses as intensity increases. At 20 mm(6h)-1 a WRF configuration is the most skilful system at every width: the assimilated run leads at 12, 25 and 50 km (0.178 at 12 km) and the cold-start run at 100 km (0.324 against 0.310). At 12 km GraphCast and AIFS attain 0.126 and 0.122, and at 100 km 0.197 and 0.198, while FuXi produces no exceedance anywhere in the scored periods. FSS is non-decreasing with neighbourhood width in these results. This alone does not distinguish amplitude from displacement errors; the amplitude statistics provide separate evidence of peak attenuation. The spatial correlation of the 6 h accumulation fields runs the other way at all thresholds – 0.414 (AIFS), 0.410 (GraphCast), 0.306 (WRF After DA), 0.290 (WRF Before DA), 0.081 (FuXi) – so the AI fields track the broad pattern of the accumulation more closely even where they fail to reproduce its extremes.

Two caveats apply. All three global models are coarser than the 12 km neighbourhood – GraphCast and FuXi at 0.25° and AIFS on N320, giving 323, 323 and 159 native points inside the scoring box against 1764 for WRF – so that column lies below the scale any of them resolves and none of the three is exempt from the limitation. Whether a sub-grid neighbourhood score is a strict lower bound on what the model could achieve is not established, and we do not claim it. And although the sample spans 27 6 h periods, they all fall within a single week of one event, so these values characterise this flood episode rather than model behaviour in general.

The comparison above sets a 12 km limited-area model against global systems at 25–28 km, and it is reasonable to ask whether the advantage of WRF follows from its finer grid rather than from its formulation. The 12 km domain used throughout this study is the outermost member of a 12–4–1.33 km nest configured following the recommended 1:3 ratio, the innermost member of which is intended to drive hydraulic simulation of the Luxembourgish catchments at approximately 1 km. That nest was integrated from 12 July 12:00 to 17 July 18:00 UTC, which allows the question to be addressed directly, although only for those 22 6 h periods and only over the footprint of the innermost domain (49.2–50.1° N, 5.6–7.0° E). The three nest members were scored against the radar composite exactly as in Sect.  (Table ).

The nest is run with two-way feedback, so the inner domains are not passive receivers: the solution on each nest is communicated back to its parent at every coupling step, and the analysis increment introduced on the 12 km domain is therefore propagated through all three members. Assimilating the same reports separately on the finer grids would reuse observations that have already been applied on the parent domain, which would count the same information twice rather than add new constraint. The comparison in this section is consequently a test of horizontal resolution at fixed analysis, which is the quantity in question. Assimilation performed directly at convection-permitting scale is a different experiment, requiring observation error statistics and a background error covariance appropriate to that scale, and we identify it as future work.

Refining the grid does not improve the position-based scores. FSS at 10 mm(6h)-1 falls from 0.307 at 12 km to 0.278 at 4 km and 0.274 at 1.33 km, and the spatial correlation declines in the same order (0.145, 0.135, 0.106). This is consistent with the double-penalty behaviour familiar from convection-permitting verification, in which a finer model generates sharper and more numerous small-scale features and any displacement of them is penalised twice, once as a miss and once as a false alarm. We offer that as the likely explanation rather than a demonstrated one: FSS responds to coverage and amplitude as well as to displacement, and separating them would require object-based displacement diagnostics or matched-smoothing experiments that we have not performed.

The amplitude statistics move in the opposite direction, but only in the tail. The maximum 6 h accumulation rises from 39.97 mm at 12 km to 53.40 mm at 4 km and 58.12 mm at 1.33 km, against the 53.97 mm recorded by the radar over the same periods: the 12 km domain falls well short of the observed peak, the 4 km domain reaches it almost exactly, and the 1.33 km domain exceeds it by some 8 %. The bulk of the distribution does not follow. All three members carry a domain-mean 6 h total of 3.6–3.8 mm against 4.72 mm from the radar, so every member is 20 %–24 % low in the mean and refinement does not close that gap; the 99th percentile is likewise flat across the three at 33.2–33.5 mm against 39.62 mm observed. The finer nests therefore do what convection-permitting simulation is expected to do for the heaviest cells – to the point of slight overshoot at 1.33 km – while leaving a persistent deficit in the bulk of the accumulation and being scored more harshly by neighbourhood metrics.

What this experiment can and cannot settle should be stated carefully. It shows that within WRF, refining the grid by an order of magnitude raises peak accumulation but does not raise the neighbourhood scores: from 12 to 1.33 km the 10 mm score at the 12 km neighbourhood changes by -0.033 (0.307 to 0.274). It does not test WRF against the AI systems at a matched 25–28 km grid, which is the comparison that would isolate resolution in the cross-model difference, and a non-monotonic response within one model does not by itself exclude a role for grid spacing between models. The two experiments also use different geography and different samples: the wide domain over 27 periods in Sect. , the inner footprint over 22 here. For scale, at 10 mm and the 12 km neighbourhood the spread between WRF and the two stronger AI systems in Fig.  is 0.020 (0.254–0.274), smaller than the 0.033 produced by refinement within WRF; the much larger spread across all systems depends on including FuXi. What can be said is that the AI fields do not reach the observed intensities at any scale, whereas the WRF nest attains and slightly exceeds the observed peak once the grid is refined to 4 km and below. Resolution contributes to the sharpness of the WRF fields, as acknowledged in Sect. ; whether it is also the origin of the contrast between the two modelling paradigms is not resolved by this experiment.