We start from the “classical” nudging-based run-time bias correction method described by . Following the notation used by , the original model increments for a state variable X (e.g. the meridional or zonal wind component) are noted M(X,t): 1∂tX=M(X,t).

When the model is nudged to a (re)analysis, these time-varying model increments become: 2∂tX=M(X,t)+Xa-Xτ≡N0(X,t), where Xa denotes the reanalysis state variable, τ the nudging timescale (here, 1 d), and N₀ the nudged version of the model. The influence and the choice of the nudging time constant is discussed in Sect. . The nudging strength is reduced to 0 near the surface (using a hyperbolic tangent factor transitioning from close to 1 to close to 0 over a range of about 500 m around 1500 m above the surface, and similarly at the upper limit of the atmosphere (beyond 5 hPa)).

The climatological nudging increments G0=(Xa-X)/τ‾ (the overbar denoting a cyclostationary time average, taking into account seasonal and daily cycles) are then used in a bias-corrected simulation denoted C₀: 3∂tX=M(X,t)+G0≡C0(X,t), where G0 is a function of space (latitude, longitude and vertical level) and time (day of the year). To reduce the high-frequency weather noise generated by averaging over only 20 years for any given day of the year, we apply a 20 d running-mean average to the final correction terms, but preserve the mean daily cycle.

Until this stage, this is, as stated, the “classical” run-time bias correction method described by .

We can now simply iterate this procedure for a first time, using the initial corrected model C₀: 4∂tX=C0(X,t)+Xa-Xτ≡N1(X,t).

Here, N₁ stands for the first iterated nudging. The cyclostationary climatological nudging increments G1=(Xa-X)/τ‾ of this first iteration are then added to the initial correction terms G0: G0+1=G0+G1. The first iterated corrected model run C₁ then uses these combined correction terms: 5∂tX=M(X,t)+G0+1≡C1(X,t).

This iteratively corrected model can then again be nudged to the (re)analysis. In general terms, the nth iteration of the bias-corrected model is then given by: 6∂tX=M(X,t)+G0+1+…+n≡Cn(X,t).

Because the nudging timescale τ is of the order of days (here, 1 d) and the model timestep of the order of minutes, the initial correction term G0 derived from Eq. () only partially offsets the model's systematic drift relative to the reanalysis. Similarly, the subsequent correction terms G0+1 and more generally G0+1+…+n only partially offset this systematic drift. The question of potential convergence is addressed in Sect. .

We use the LMDZ6 AGCM at low resolution (96 (longitude) × 95 (latitude) horizontal grid points and 79 vertical levels), coupled to the ORCHIDEE land surface model . The model is nudged towards the ERA5 reanalysis for the period 1981–2000 (plus the year 1980 for model spinup). Only the meridional and zonal wind components above about P=0.85Ps (where P and Ps are the atmospheric pressure at a given level and at the surface, respectively) are nudged towards ERA5 with a time constant of τ=1 d in the nudged simulations N_i (i=0…3). The corrected simulations C_i (i=0…3) and the uncorrected control simulations, denoted M, are run for the evaluation period 2001–2020 (plus the year 2000 for model spinup), distinct from the nudging period. These simulations are run twice, starting in 2000 with varying initial conditions. For these simulations we use atmospheric states obtained from other uncorrected LMDZ simulations for the years 1995 and 2000, respectively, and discard the first year of the integration as spinup, as just mentioned. The presented results for these simulations M and C_i are the average of these 2×20=40 model years. Table  provides a summary of the simulations described in this work.

In the Discussion, we briefly refer to one additional iteratively bias-corrected, supplementary test simulation that uses a nudging timescale of τ=3 d.

Evaluation is carried out against 2001–2020 ERA5 circulation characteristics such as climatological mean biases for meridional (v) and zonal (u) wind at selected standard pressure levels (850, 500 and 200 hPa), mean sea-level pressure biases, interannual patterns of sea-level pressure variability using empirical orthogonal functions, blocking frequencies based on 500 hPa geopotential indices, band-pass (between 2.5 and 7 d) sea-level pressure variability linked to storm track location and intensity , and frequencies of synoptic weather patterns using 5×4 self-organizing maps of hemispheric extratropical (polewards of 40° N and S) daily-mean geopotential heights. Furthermore, the amplitude of the u and v correction terms is analyzed to evaluate convergence of the correction procedure.

Contrary to previous studies , we refrained from bias-correcting atmospheric temperatures because we have realized that run-time temperature bias corrections in the LMDZ AGCM destructively interfere with the physical parameterizations in the tropics. In particular, the temperature corrections significantly reduce a large-scale tropospheric cold bias in the model, leading to reduced convective instability and thus to reduced simulated convective activity, accumulation of water vapour in the atmospheric boundary layer, and a substantial surface energy flux imbalance linked to reduced surface evaporation and increased downwelling longwave radiation.

The amplitude of the nudging increment decreases for higher iterations, as shown in Fig.  for the zonal mean absolute zonal wind nudging increments. While the spatial patterns are broadly similar for all simulations N_i, the amplitude of the nudging increments decreases by approximately a factor of 2–3 from N₀ to N₃. Contrary to the nudging increments, which decrease, the amplitude of the bias correction increments applied in simulations C_i increases for higher iterations, as the correction increments G0+…+i for simulation C_i are the sum of the average nudging increments of N₀ to N_i (see Eq. ). The spatial structure of the bias correction increments G0+…+i also remains similar for the successive iterations C_i, but for the higher iterations it is not a simple spatially constant multiple of the initial correction increments G0 used for C₀, as shown by the ratio of the zonal mean absolute zonal wind correction increments between G0+…+3 and G0, displayed in Fig. . Instead, this ratio has distinctive spatial structures varying in space and in time through the annual cycle (not shown), indicating that the iterative procedure is not identical to a simple uniform amplification of the initial correction. We interpret this as being linked to the non-linear and non-local nature of the atmospheric system, where the bias corrections implemented during a correction step can increase tendency errors elsewhere, which are then corrected during the next iteration, and where spatially unequal bias reduction at one iteration can also lead to modified tendency errors, and thus subsequent bias reduction, in the following iteration.

Do these successive correction terms converge towards some “final” correction term? Figure d shows that the nudging increments in the third iterated nudging step N₃ do not vanish, although they are substantially weaker than in N₀ (Fig. a). The global mean of the absolute zonal wind nudging tendencies (in January, to be consistent with Fig. ) is 0.50 ms-1d-1 for N₀, 0.35 ms-1d-1 for N₁, 0.29 ms-1d-1 for N₂, and 0.26 ms-1d-1 for N₃. This means that the intensity of the remaining nudging tendencies decreases at higher iterations, but convergence towards potentially vanishing final nudging tendencies is still far away after 3 iterations. The combined correction terms arising from the sum of these absolute zonal wind nudging tendencies have global mean values of 0.50 ms-1d-1 for G0 (because G0 is identical to the mean nudging tendencies of N₀), 0.83 ms-1d-1 for G0+1, 1.09 ms-1d-1 for G0+1+2, and 1.30 ms-1d-1 for G0+…+3, and are thus somewhat lower than the corresponding sums of the global mean of the absolute zonal wind nudging tendencies (which would be 0.5, 0.85, 1.14 and 1.40 ms-1d-1, respectively), indicating that some local-scale compensation occurs between different iterations, as already shown by Fig. .

Plots for wind bias reductions obtained with the standard bias correction method and its successive iterations are provided in Fig.  (maps for u at 500 hPa) and in Fig.  (zonal means for u and v). Figure  displays the 1981–2000 annual mean bias of the zonal wind component at 500 hPa for the different simulations. Figure  displays, in each of its two panels, the ratio of the 1981–2000 RMSE in each of the corrected simulations C_i (i=0,…,3) and the RMSE of the uncorrected control simulation M: RMSE(C_i)/RMSE(M), for the annual mean of the zonal (a) and meridional (b) wind speed, where the RMSE is calculated with respect to ERA5.

By construction, mean wind biases at various levels in the atmosphere are decreased in the corrected simulation C₀ compared to the uncorrected control simulation M. Relative to M, the RMSE of the zonal and meridional wind components u and v is decreased by ≈20 % near the atmospheric boundary layer (where no correction is applied) to ≈60 % (for u) at 200 hPa. When the correction procedure is iterated once (simulation C₁), a further substantial RMSE reduction is obtained. In most cases except close to the atmospheric boundary layer, no substantial further improvement is obtained in subsequent iterations of the correction procedure.

In the absence of run-time temperature corrections in this study, substantial large-scale air temperature biases remain in our wind-only corrected simulations C_i (see Fig. ). While the wind corrections lead to improvements in the zonal-mean temperature simulation in the upper atmosphere, above ≈300 hPa, the bias reduction is only modest below this level. In addition, further bias reduction between 200 and 300 hPa in polar regions during higher iterations (>2) of the bias correction procedure come at the expense of a degradation of the simulated temperature patterns at these latitudes at about 150 hPa. Because the wind corrections act through the model dynamics, its influence on temperature is expected to arise primarily through dynamical balance (e.g. thermal wind, geostrophic and hydrostatic adjustment), which constrains horizontal temperature gradients rather than the global-mean temperature at a given pressure level. As a result, wind-only bias correction can reasonably be expected to improve regional temperature anomalies about the level-wise global mean while leaving the global-mean temperature largely unaffected, since the latter is controlled by the column-integrated energy balance and model physics. Therefore, when for a given pressure level the global mean bias is accounted for (i.e. subtracted, yielding a centered temperature T̃=T-ΔT, where ΔT is the global mean temperature bias at a given atmospheric level), the remaining regional bias patterns are reduced in the corrected simulations except very close to the surface, as can be seen in Fig.  that displays the RMSE of this centered temperature T̃. In this case the iterative run-time bias correction procedure leads to further improvement in most parts of atmosphere above about 900 hPa (with the exception of a small degradation of the biases at about 850 hPa, which are already not well corrected in C₀, and an already-mentioned degradation at about 150 hPa). This means that, while the global mean temperature biases – predominantly cold below about 150 hPa and very likely caused by deficiencies in the model physics – are not corrected, regional-scale temperature bias patterns are smoothed because of the corrected model dynamics.

Sea-level pressure biases are only weakly reduced in our simulations that only bias-correct wind patterns. In C₀, the global RMSE of the annual mean sea-level pressure (with respect to ERA5) is reduced by 10 % compared to the uncorrected control simulation M, and iterations of the bias correction method only lead to insignificant overall improvements (the reduction compared to M is 13 % for simulation C₃). The weak correction of sea-level pressure errors is in contrast to earlier results for the Southern Hemisphere obtained using run-time temperature corrections .

As stated, building on cyclostationary correction terms, the run-time bias correction procedure used here is, by construction, designed to better represent climatological mean values of the corrected variables. However, it is certainly of equal importance to assess whether these run-time bias corrections also improve the way the model represents circulation variability on different timescales.

The dominant patterns of interannual variability of monthly circulation structures are indeed more realistically depicted in the corrected simulations C₀ than in the uncorrected control simulations M, and even more so in simulations with iterated run-time bias corrections (C₁₋₃). This can be seen in Table  that displays the seasonal squared spatial Pearson correlation coefficient r2 between simulated (LMDZ) and “observed” (ERA5) dominant modes of monthly 500 hPa geopotential anomalies, as identified by principal component analysis for the different corrected model runs C_i, relative to the corresponding squared spatial correlation coefficient for the uncorrected control simulations M: R=rCi2/rM2.

Only seasons for which the first principal component explains more than 40 % of the variance in the ERA5 dataset are retained here in order to restrict the analysis to the most stable and physically meaningful patterns. Although the 20-year observational period might be a bit short and could be the reason for some remaining noise in the results reported in Table  – the tendencies for a given season and hemisphere from C₀ to C₃ are not always steady –, it seems that an iteration of the run-time bias correction leads in most cases to a better representation of the interannual circulation variability patterns. The results reported here are obtained for the total 40 years of the two 20-year runs of each experiment C_i, but the results for individual 20-year runs of these experiments are very similar, increasing the confidence in the robustness of the results. Furthermore, the squared correlation coefficients overall increase between C₁ and C₂.

Diagnostics of short-term atmospheric circulation variability indicate an overall positive impact of the wind-based bias corrections for spatial means of bandpass-filtered extratropical winter sea-level pressure variability and average blocking frequencies (Table ), but there is no clear improvement over multiple iterations of the correction procedure.

The winter short-term (2.5–8 d) mid-latitude (40–60° N and 40–60° S) sea-level pressure variability, diagnosed from the temporal standard deviation of sea-level pressure from its seasonal mean (σ2.5–8 d, using a bandpass filtering in the frequency domain, and limited to grid points with surface height below 1000 m) shows no consistent improvement, and even slight degradation, for the iterated corrections C_1…3 compared to the simple non-iterated bias correction applied in C₀. However, the short-term pressure variability is overall better depicted in the corrected than in the uncorrected control simulations.

Blocking anticyclones are a significant atmospheric phenomenon, although there is not one standard accepted definition . Here we follow the widely used definition by : Blocking events are diagnosed at a given point in time and space based on the reversal of the meridional gradient of geopotential height measured at 500 hPa, extending over at least 15° of continuous longitude, and persisting within a 5° latitude × 10° longitude box centered on a given grid point for at least 5 d. We then calculate the annual mean frequency of these events for each grid point, and calculate the spatial RMSE E of these frequencies compared to ERA5. For each corrected simulation, the ratio between ECi for that simulation and EM of the uncorrected control simulation is then computed, yielding a score fB=ECi/EM. While the iterated procedure seems to improve the model score for the 30–75° N annual and spatial mean blocking frequency, (i.e., fB (30–75° N) <1), with best results obtained for 2 and 3 iterations, the bias-corrected simulations show degraded performance compared to the uncorrected control simulations (i.e., fB (30–75° S) >1) in the Southern Hemisphere (30–75° S). However, in the Southern Hemisphere, the iterated bias correction procedure leads to less degraded performance compared to a single bias correction in C₀.

In the Southern Hemisphere, the mean RMSE of the frequencies of typical hemispheric-scale daily circulation patterns, as identified using 20 (5×4) self-organizing maps for each season (DJF and JJA) and denoted as fS in Table , is slightly degraded in the corrected simulations compared to the uncorrected control simulation (see fS for 40–90° S in the table). Frequency errors are increased by about 15 % on average compared to the uncorrected control simulation. Iterations of the bias correction procedure (C_1…3) do not show any consistent change (neither improvement nor degradation) compared to the non-iterated bias correction applied in C₀, even if C₂ and C₃ show the least degradation in JJA and DJF, respectively, compared to C₀. In the Northern Hemisphere, the non-iterated bias correction C₀ procedure produces the best results of all simulations (including the uncorrected control simulation M). In other words, while the bias correction method appears to improve the model performance in this particular respect, iterating the bias correction procedure does not lead to further improvement of the frequency of the simulated main synoptic weather patterns (see fS for 40–90° S in the table).

Overall, taking these diagnostics together, the wind bias correction does lead to a modestly improved representation of short-term atmospheric circulation variability, but iterating the bias correction procedure does not necessarily lead to clear further improvement. We note that in the cases where the model performance is degraded by applying the bias correction once (that is, C₀ shows degraded performance relative to M), iterating the bias correction can lead to improved (less degraded) results.

In studies using the nudging technique, the optimal choice of the nudging time constant τ, and more generally of technical choices in the implementation of the nudging procedure, has repeatedly been the object of attention e.g.,. Tests of a range of different nudging time constants τ within the rather usual range between 1 and 3 d, both for the non-iterated and the iterated bias correction procedure, have shown that τ=1 d overall yields the best results in our modeling framework consisting of nudging wind in the LMDZ AGCM , at least for the range of time constants tested here. show that under certain conditions small nudging time constants (below τ=1 d) can lead to strongly degraded ERBC results. However, in a study of the effect of “classical” ERBCs on the simulation of the Antarctic climate , a nudging timescale of τ=6 h was used, substantially less than that used here (τ=1 d), and that in that study, improvements of the simulated climate were shown across all timescales, including high-frequency variability. We note in this respect that the iterative procedure used here can be seen as a method that leads to a stronger effective nudging, as shown by the fact that the amplitude of the combined correction terms G0+…+i for higher numbers i of iterations increases (see Fig. ). Additional systematic tests of the effects of smaller time constants τ<1 d in LMDZ are planned for future studies.

Related to this question, it is interesting to ask whether an iterated bias correction with a larger time constant τ could actually be a simple equivalent of a “classical” non-iterated bias-correction with a correspondingly smaller time constant. Figure  shows that this is not the case: it displays the ratio between the bias correction increments in a “classical” non-iterated simulation with τ=1 d (as in the rest of the simulations discussed in detail in this paper), relative to the increments used in a supplementary, twice iterated bias-corrected simulation with τ=3 d (which uses three nudging simulations instead of only one). This clearly shows that the iteration procedure is not equivalent to a non-iterated bias correction procedure with a correspondingly smaller nudging time constant. A more detailed comparison of these simulations with different nudging time constants and numbers of iterations is provided in .

An obvious risk of iterative bias correction procedures is over-correction of model biases, leading to degraded or physically less meaningful results when the model is applied outside of the calibration period. To detect possible signs of over-correction, the difference of the global mean RMSE reduction in corrected simulations between the calibration period (1981–2000, “in-sample”) and the normal evaluation period (2001–2020, “out of sample”) is displayed in Fig.  for mean annual zonal and meridional wind speed at different levels in the atmosphere. The idea is that if, as the number of iterations increases, the performance during the out-of-sample period (2001–2020) is systematically degrading compared to the performance during the in-sample period (1981–2000), then this means that the real added value of additional iterations is decreasing. As Fig.  suggests, this seems to be the case at least after three iterations. In that case, while for several variables the in-sample score still continues to increase, the out-of-sample score starts to decrease. This is particularly the case for variables which have low overall bias reduction scores, for example the 700 hPa meridional wind. The difference between the evolution of the in-sample and the out-of-sample scores is weaker for variables which have strong bias correction scores, such as the 200 hPa zonal wind.

It therefore seems that after about three iterations of the bias-correction method, the performance during the out-of-sample period is degrading compared to the performance during the in-sample period, in particular for variables for which still some potential for correction is left. As a precaution, we interpret this as a sign of over-correction, suggesting that two iterations of the bias correction method should be the maximum number in practical applications.

It is possible, however, that there is some interference between possible over-correction and the fact that the study period, 1981–2020, is a period of strong climatic change. Although it has been shown before that nudging-based ERBC remains valid under strong climate change , part of the performance over the 2001–2020 period might therefore be influenced by the climate change signal. One could, as an additional test, use 2001–2020 as the ERBC calibration period and 1981–2000 for validation, or use pair years between 1981 and 2020 for the ERBC calibration and odd years of the same period for out-of-sample testing. However, the main motivation for our use of various ERBC approaches is to eventually use it in climate change simulations. In that sense, evaluating the effect of the bias corrections in a climate that is warmer than the calibration period, but still known, is a relevant test for the intended use of the ERBC approach.

We have shown that globally the mean biases are progressively reduced during the iteration of the bias correction, at least until the point when over-correction starts to occur. However, it is clear that the overall structure of the bias patterns is robust, as can be seen very clearly for example in Fig. . This is not surprising, as climate model bias patterns are quite persistent even under strong climate change .

What is noteworthy nevertheless is that the strength of attenuation of mean biases is to some degree spatially variable. For example, while Fig.  shows a steady decrease of the annual mean 500 hPa zonal wind mean bias in the Southern Indian Ocean with increasing number of iterations of the bias correction, the corresponding bias pattern over Western Europe (positive over the Mediterranean and negative over North-western Europe) quickly stabilises, and is even amplified in simulations C₂ and C₃ relative to C₀ and C₁. Similarly, while biases continue to decrease over the North Pacific, they already start to re-increase in simulation C₂ over the South Pacific. The reasons for regional persistence of biases are not clear, and they probably are region-specific. It is for example possible that these regionally varying behaviours are linked to region-specific factors such as topography, or that they are caused by teleconnection patterns in interaction with error compensation (in the sense that in two regions linked by teleconnection mechanisms, well correcting an error in one region might unmask errors in the other region).

As a consequence, iterating the bias correction method might allow regional climate modelers to dispose of much improved atmospheric boundary conditions for the RCM in one specific region, but not in another. A detailed analysis for the reasons of bias persistence in specific regions is beyond the scope of this paper, and these reasons can depend on various factors, possibly including resolution, and are probably model-dependent. Regional bias persistence in different variations of ERBC is analysed in more detail in .

As we have shown here (see for example Fig. ), the geographical patterns of the mean model biases are somewhat independent of the number of iterations, and the mean absolute bias correction increments increase with the number of iterations (see Fig. ). This might suggest that as a first simple approach one could try to use a multiple of the bias correction terms obtained during the first nudging (that is, those applied in C₀) as a surrogate for the more costly iteration procedure. We have carried out a simple test in which the correction terms of C₀ are amplified by 50 %; otherwise this simulation (referred to as C_0,×1.5 in the following) is identical to C₀. The analysis of C_0,×1.5 shows that the bias correction is less efficient than in C₀. For example, with respect to ERA5, the annual mean global RMSE of the 500 hPa zonal and meridional wind components is 7.5 % (u500) and 15.6 % (v500) higher in C_0,×1.5 than in C₀. However we note that C_0,×1.5 still shows a substantial improvement compared to the uncorrected model run M.

It is fairly easy to understand why the model performance degrades between C₀ and C_0,×1.5. As one can see in Fig. , although the amplitude of the correction terms does increase essentially everywhere for higher iterations of the correction procedure, the ratio of the correction terms between different phases of the iteration procedure is not uniform – it clearly has a spatial structure. A simple overall increase of the initial correction terms, although modest in our test, does not reproduce this spatial structure.