We use the Max Planck Institute ESM MPI-ESM, which was also used in the Coupled Model Intercomparison Project Phase 6 (CMIP6) framework . We run the model MPI-ESM1-2-LR, the low-resolution configuration with 63 spherical harmonics in the atmosphere, a horizontal resolution of about 1.8∘ on land and about 1.5∘ in the ocean with daily coupling of the compartments. The time steps of the atmosphere–land and the ocean are 600 and 4320 s, respectively. We run the model with a prognostic atmospheric CO2 mixing ratio under preindustrial conditions (esm-piControl).

The marine biogeochemical cycle model HAMOCC is embedded in the ocean general circulation model MPIOM . HAMOCC includes carbonate chemistry and an extended NPZD-type cycle, including nutrient–light–temperature co-limitation and nitrogen-fixating cyanobacteria . The land carbon cycle model JSBACH includes dynamic vegetation, wildfires, and soil carbon decomposition and storage . The atmospheric general circulation model ECHAM6 transports the three-dimensional atmospheric prognostic atmospheric CO2 tracer with a flux-form semi-Lagrangian scheme .

Simulations in a perfect-model target reconstruction framework aim to reproduce the target climate evolution but are started from an independent initial state. Therefore, the initial conditions of the reconstruction simulation and the target do not match, but both the target and initial conditions share the same climatology. We choose a 50-year target period from model years 1850 to 1900 and an uncorrelated restart file from model year 2005 from the preindustrial control simulation (esm-piControl) submitted for the MPI-ESM1-2-LR model for C4MIP in CMIP6 .

In order to assess how many variables are needed to sufficiently reconstruct climate and biogeochemical cycles, we first perform reconstruction simulations only reconstructing physical state variables in atmosphere and/or ocean (Table ). In these simulations, the carbon cycle is only indirectly affected by the reconstruction of physical variables. In further simulations, we test how much carbon cycle states improve with respect to the target when carbon cycle state variables are reconstructed directly.

Newtonian or relaxation, which is often called “nudging”, is a simple four-dimensional assimilation technique that dynamically reconstructs variables in an ESM. A non-physical relaxation term with relaxation coefficient R (units  1s-1) is added to the prognostic equation to drag the model variable X, which is subject to model forcing Fm towards its target Xt: 1δXδt=Fm(X)+R(Xt-X).

For reconstruction of the dynamics of the ocean, we reconstruct three-dimensional temperature, salinity, and sea ice concentration and thickness (Table ). We label this reconstruction as indirect (Table ) from the carbon cycle's perspective, as the carbon cycle is not reconstructed directly but instead indirectly follows the reconstructed physical climate. Observational ocean data are often not available at each model time step. Therefore, we interpolate (without adjustments preserving the temporal mean) monthly model target output to daily frequency as has been done in previous studies . We choose a 60 d ocean relaxation time (converted to units 1s-1) like did in their perfect-model target reconstruction study. Reconstructions towards observations usually choose a stronger nudging strength .

We reconstruct the physics of the atmosphere by nudging temperature, vorticity, divergence and the logarithm of surface pressure . The high-frequency 6-hourly output serves as the target and is nudged into all 63 spherical harmonics. Temperature and the logarithm of surface pressure are nudged with a relaxation timescale of 24 h, vorticity is nudged with a relaxation timescale of 6 h, and divergence is nudged with a relaxation timescale of 48 h. Relaxation coefficients are converted to units of 1s-1 and are taken from previously used setups . Nudging the atmosphere with these quite short relaxation times is similar to the forced simulations, such as the Model Intercomparison Projects for ocean (OMIP) and land (LMIP) and Global Carbon Budget simulations, where (atmospheric) external boundary forcing drives the carbon cycle.

For reconstructions of oceanic carbon cycle, we use the same nudging approach and strength as for physical ocean reconstruction but on different variables. To reconstruct the components of the carbonate system, we nudge three-dimensional dissolved inorganic carbon (DIC) and total alkalinity (Table ).

Unfortunately, there is no nudging module available in the land surface model JSBACH. The current structure of JSBACH code is not flexible enough to allow frequent rewriting of physical variable fields, such as soil moisture or temperature, with external data. Here, we choose to manually reset the initial conditions every 1 January to the target values instead of the dynamic reconstruction at each time step. We thereby reconstruct land biogeochemistry and land surface physics such as soil moisture by resetting all restart variables every year. In Appendix Sect. , we provide several sensitivity analyses by resetting land only every 2 or 5 years and resetting the ocean every year in the same way.

We compare the target with reconstructions in the various metrics showing different attributes of tracking performance: bias, anomaly correlation coefficient and root-mean-square error. The non-physical relaxation terms in the prognostic equations can disturb the dynamics in the ESM and introduce biases defined as the differences in the reconstruction compared to the freely running target over time. The anomaly correlation coefficient skill score (ACC) shows the linear association between the reconstruction and the target over time and therefore measures synchronous evolution while ignoring bias. The root-mean-square error (RMSE) takes into account bias and measures the second-order Euclidian distance between reconstruction and target simulation over time. Under the assumption that persistent biases can be removed by post-processing, we also assess RMSE after having the mean monthly bias removed. For equations please consult Appendix . We calculate tracking performance over running 10-year chunks to capture the variability within tracking performance and reduce the influence of drifts over time.

How do we evaluate that a reconstruction is good enough? While good enough is a subjective judgment, we resample the target simulation along the time dimension with a block length of 10 years to check the metric of two randomly compared 10-year chunks. We consider the 95th quantile threshold for ACC and 5th quantile threshold for the remaining distance-based metrics as a baseline of internal variability to be a good-enough reconstruction , which we will refer to as “resampling threshold” in the following.

In the second part of this study, we perform initialized perfect-model experiments as in. The simulations in the perfect-model framework are started from the indirect and direct reconstructions as well the target representing perfect initial conditions. We take 19 initialization states chosen every second 1 January between 1860 and 1896, after allowing a 10-year adjustment phase after reconstructions were started. From each of those states from different reconstruction simulations, we fork five ensemble members and simulate 3 lead years. The perfectly initialized ensembles are started from the target initial conditions without any previous reconstruction simulation. We generate ensemble members by perturbing the stratospheric horizontal diffusion by a factor of 1.0000{member} in the first year, e.g., the factor is 1.00005 for the fifth ensemble member. This member-generating approach provokes only tiny initial perturbations to the climate system as the ocean and land initial conditions remain identical.

We compute predictive skill as the RMSE between the ensemble mean and the target as verification (Appendix ). Please find additional details about the predictive skill metrics and the uninitialized bootstrapping in . Acknowledging that our reconstruction simulation developed biases and that biases are commonly reduced by post-processing in predictability research, we also apply a simple lead-time-dependent mean bias reduction to the initialized ensembles to show whether skill improvements go beyond what a simple post-processing could deliver. For each initialization in turn, we first calculate the mean bias for all but that given initialization and then remove that mean bias from the given initialization. This implies using information about future initializations as in bias-reduced hindcasts . We also evaluate predictive skill from a perfectly initialized ensemble, which are started from the perfect initial conditions taken from target simulation, whereas the ensembles from reconstructed initial conditions are biased with respect to the target (Fig. ). This initialized predictive skill is also compared with uninitialized ensembles randomly generated from the target simulation representing ensembles without common initialization and hence no memory. This uninitialized reference skill is used in predictability research community to assign whether the skill increase stems from initialization.

Reconstructing the ocean and/or the atmosphere systematically disturbs the freely evolving model, which leads to annual mean biases with respect to the original target. We identify atmospheric circulation represented by winds and resulting precipitation and temperature to be descriptive for the impact of circulation on the carbon cycle. The gray stippling in Fig.  shows where this reconstruction bias is larger than the randomly resampling fifth-percentile mean absolute error threshold and therefore labels the reconstruction as being not significantly better than internal variability.

All reconstructions yield identical results for winds and precipitation tracking performance. Reconstructing the ocean and/or the atmosphere introduces biases of up to 0.6 m/s in zonal and 0.9 m/s in meridional 10 m wind speed, depicting a southward shift of the Intertropical Convergence Zone (ITCZ). This bias results in a significant weakening of the Equator-ward latitudinal winds, whereas extratropical latitudinal winds intensify (Fig. a). The intensification and Equator-ward shift of the easterly trade winds and weakening of the Southern Hemisphere westerlies are both not significant (Fig. b). Precipitation is heavily impacted by these biases in atmospheric transport across many regions of the globe. Precipitation significantly shifts southward at the Equator with changes of more than 1 mmd-1 and increases in western Canada, western Russia and southern Australia (Fig. c). Unlike the previously described variables, the 2 m temperature bias depends on whether the ocean is reconstructed or not. Just reconstructing the ocean temperature and salinity (indirect_{OCEAN only}) leads to small, negative and significant biases in the tropical Atlantic and West Pacific. In addition, northern and southern Africa, as well as the Amazon and China, are subject to a small cold bias, whereas Saharan Africa and Southeast Asia get substantially warmer. The polar regions cool significantly (Fig. d). Only reconstructing the atmosphere (indirectATMonly) leads to a warm bias across nearly all oceans but less cold bias over northern and southern Africa and China (Fig. e). Combining atmosphere and ocean reconstruction (indirect) reduces the overall temperature bias, especially over the oceans (Fig. f).

While the biases explained above are liabilities of reconstructions, the linear association measured by the anomaly correlation coefficient (ACC) benefits from reconstruction. Reconstruction recreates climate variability of the target (Fig. g–l). The running 10-year correlation between the target and the reconstruction in atmospheric variables is in most grid cells above 0.4 and significantly better than the randomly resampling threshold. Reconstruction over the oceans is more successful in the tropics than in the extratropics, where the Northern Hemisphere and Southern Hemisphere midlatitude westerlies have low but still significant correlation. Generally, the atmosphere above the ocean is better reconstructed than above land, showing the stabilizing effect of an internally consistent ocean reconstruction on the atmosphere (Fig. g–l). The Southern Hemisphere tropical convergence of winds is well reconstructed, but the meridional winds in central Canada and tropical Africa are not significantly reconstructed (Fig. g). In addition, zonal winds across North America, southern Africa and Siberia have low correlation with the target, but the tropical zonal winds are very well reconstructed (Fig. h). Precipitation from the central Atlantic over central Africa is reconstructed worse than the resampling threshold, and the extratropical westerlies have low correlation with the target (Fig. i). Temperature is well reconstructed in the tropical oceans (Fig. j–l). Reconstructing both atmosphere and ocean (indirect) improves 2 m temperature correlation better than only reconstructing a single realm. The indirect carbon cycle reconstruction is significantly better than the resampling threshold except in central Africa, where the ITCZ shift changes the climate regime (Fig. l).

This physical bias due to reconstruction, especially in the tropics, can be explained by the sensitivity of atmosphere–ocean coupling to perturbation induced by nudging . Additionally nudging sea surface height might improve the El Niño–Southern Oscillation (ENSO) thermocline feedback . The reconstruction of ocean and atmospheric variables is perfectly aligned with the model climatology into that same model. Hence, the reconstruction error does not arise from inconsistent observations but from the perturbed interaction of atmospheric and oceanic dynamics. While reconstructing an increasing set of variables shows that nudging can be an efficient way to reconstruct variability , this reconstruction is biasing the climate state in the tropics at the same time also explained in.

Nudging atmospheric and ocean dynamics including sea ice all at once (indirect reconstruction), as is often done in state-of-the-art carbon cycle prediction systems, brings large-scale improvements over random resampling and atmosphere-only (indirectATM only) reconstruction, but strong regional biases remain (Fig. ).

How do these regional physical biases affect the reconstruction of oceanic carbon cycle? In order to assess the tracking performance in the indirect reconstruction of the oceanic carbon cycle, we focus on air–sea CO2 flux and surface oceanic pCO2 as the state variable of the ocean carbon sink, which is the oceanic driver of air–sea CO2 flux .

Reconstructing only the atmospheric dynamics (indirectATMonly) leads to strong positive biases across large parts of the global ocean, which can be reduced by also reconstructing oceanic temperature and salinity (indirect) (Fig. a, b, d, and e). The weakening of the Southern Hemisphere westerly winds decreases the magnitude of air–sea CO2 flux, but more importantly reduces the Southern Hemisphere overturning circulation and upwelling of carbon-rich waters, which leads to increased Southern Ocean carbon uptake (Fig. b and e). The intensification of easterly trade winds (Fig. b) strengthens upwelling and therefore higher pCO2 in the tropical Atlantic (Fig. b) . The bias pattern of air–sea CO2 flux is dominated by the bias of pCO2 (Fig. b and e).

The variations in the oceanic carbon cycle, described by the correlation coefficient, are better reconstructed than the resampling threshold. Indirect reconstruction of oceanic and atmospheric dynamics greatly improves tracking performance over atmosphere-only indirectATMonly reconstruction. The additional reconstruction of the physical ocean (Fig. e and f) largely enables a correlation above 0.7 (Fig. b and e). Only the carbon cycle in the tropical oceans remains difficult to reconstruct due to the strong biases in atmospheric circulation (Fig. a–c). Note that the land and atmospheric carbon bias due to indirect reconstruction are discussed in Fig. g–o in Sects.  and ).

Next, we compare the previously shown indirect carbon cycle reconstruction with direct carbon cycle reconstruction by nudging dissolved inorganic carbon (DIC) and alkalinity (ALK) towards the target.

While direct oceanic carbon cycle reconstruction reduces the magnitudes of the bias across the ocean, biases are still evident (Fig. c and f). These biases are caused by the physical biases, which the dynamical oceanic carbon cycle model is sensitive to. Hence, the biased ocean physics inhibits additional improvements in tracking performance from direct ocean carbon reconstruction.

Direct oceanic carbon cycle reconstruction improves the already high correlations across the oceans (Fig. c and f). The resampling threshold is surpassed nearly everywhere. Only coastal areas, especially those in the eastern tropical Atlantic with strong wind and precipitation biases, have a correlation below 0.7.

Section  shows how well indirect and direct reconstruction of the ocean carbon cycle work overall. While the direct reconstruction has slightly larger biases in air–sea CO2 flux, direct reconstruction also brings higher correlation. Note that the land and atmospheric carbon biases due to direct reconstruction are discussed in Fig. g–o in Sect.  and ).

How do these regional physical biases affect the reconstruction of the land carbon cycle? In order to assess the tracking performance in the best indirect reconstruction of the land carbon cycle, we focus on the state variable cVeg, which represents carbon storage in vegetation (leaves, stems, roots) and drives air–land CO2 flux and hence the land carbon sink.

For the land carbon cycle, the reconstruction of the ocean temperature and salinity did not matter, when atmospheric temperature was also reconstructed (Figs.  and ). Indirect reconstruction leads to biases compared to the target in carbon storage, and in particular cVeg (Fig. g and h), as the land carbon cycle is very sensitive to changes in atmospheric circulation, which are strongest in the tropics due to the ITCZ shift. In the Amazon and southern Africa, the air–land CO2 bias increases, most likely caused by the strong positive precipitation bias in these regions (Figs. c and j and k). Conversely, the carbon sink in Southeast Asia and central Africa has a carbon release bias due to less precipitation and a warm bias (Fig. j and k).

The reconstruction correlations in the land carbon cycle are much lower than for the oceanic carbon cycle. cVeg is well reconstructed in the extratropics, but the biases in the tropics result in correlations with the target lower than the resampling threshold (Fig. g and h). Air–land CO2 shows the same patterns with lower correlations, which are below the resampling threshold in the tropics (Fig. j and k).

Direct reconstruction of the land carbon cycle, which is here performed by resetting all restart files of the land carbon sub-model to the target every 1 January, greatly enhances tracking performance of cVeg by simulation design. A sensitivity analysis for less frequent resetting can be found in the supplementary information (Sect. ).

This direct resetting reconstructs cVeg much better than the resampling threshold in the extratropics. However, the physical climate biases during the course of a year even introduce cVeg biases stronger than the resampling threshold in the tropics (Fig. i). In addition, the biases in the air–land CO2 flux are not improved (Fig. l), which indicates that this hard reset of restart files introduces a shock to the dynamical land model.

On the other hand, correlations in cVeg and air–land CO2 flux increased to above 0.5 everywhere except in the tropics, where the ITCZ shift changes the climate regime (Fig. i and l).

Section  shows the direct land carbon cycle reconstruction yields stronger correlation improvements than ocean direct carbon cycle reconstruction because the indirect reconstruction of the ocean was already quite good. Direct reconstruction reduces biases in land carbon cycle state variables, but the resulting air–land CO2 flux biases becomes worse.

Tracking performance for prognostic atmospheric CO2 integrates the air–sea and air–land CO2 fluxes over time . As atmospheric CO2 mixes quickly across the globe, we first examine globally aggregated quantities driving globally averaged atmospheric CO2 (Fig. ).

We first examine the indirect reconstruction represented by the green error bars in Figs.  and . The indirect reconstruction has a negative bias in global pCO2 in the annual mean (Figs. a and a). This bias is slightly higher than the magnitude of the resampling mean absolute error threshold, which resembles the temporal standard deviation (Fig. a). The global oceanic CO2 flux is biased low but within the resampling threshold magnitude range (Figs. b and d).

On the other hand, the variations of the global oceanic carbon cycle measured by ACC are well reconstructed, surpassing the resampling threshold (Fig. b and e).

When biases are persistent, they can be reduced by a bias reduction procedure, which is often done when applying climate model output to a real-world application. After applying a simple mean bias reduction, RMSE is well below the resampling threshold (Fig. c and f).

The indirect reconstruction also leads to biases in the land carbon cycle (Fig. c and d). Vegetation carbon pools (cVeg) have a strong positive bias that is much larger than the resampling threshold (Fig. c). The bias of global air–land CO2 flux is very small in the annual mean.

Global annual cVeg has a 0.5 correlation with the target, which is lower than the resampling threshold. Global air–land CO2 variations are well reconstructed, surpassing the resampling threshold (Fig. h and k).

Without bias reduction, accuracy measured by RMSE is worse than the resampling cVeg threshold. After bias reduction, cVeg accuracy is still slightly worse than the threshold, but accuracy improves from 5 PgC to below 1 PgC, which is the magnitude of the resampling threshold (Fig. i). Global air–land CO2 flux accuracy is below the resampling threshold (Fig. l).

Global atmospheric CO2 has larger variations in reconstruction skill depending on which 10-year chunk is used to calculate the metric. And the skill has a nearly constant level throughout the year (Fig. m–o). The mean bias is close to zero (Figs. e and m). Correlation with the target is above 0.7 and in the range of the resampling threshold (Fig. n). accuracy is at 0.7 ppm in the range of the resampling threshold. Mean bias reduction improves accuracy to below 0.5 ppm (Fig. o).

Understanding the tracking performance of the ocean and land carbon cycle, we can now evaluate the spatial distribution of globally averaged atmospheric CO2. Reconstructing only the atmosphere warmed the globe and also increased atmospheric CO2 globally (Figs. k and m). Additionally reconstructing the ocean keeps the temperature stable but introduces a less than 1 ppm low bias across the Southern Hemisphere, reflecting the higher uptake of the Southern Ocean carbon sink and the Southern Hemisphere land carbon sink (Fig. e, k, and n). The variations in atmospheric CO2 are well reconstructed with correlation coefficients above 0.6 in the Southern Hemisphere, but across the Northern Hemisphere extratropics and land regions with strong physics biases correlation is at 0.5 below the resampling threshold (Fig. m and n).

Now, we assess the potential improvements in the global carbon cycle due to direct reconstruction of the global carbon cycle variables shown in orange in Figs.  and .

The global ocean carbon cycle improves after direct DIC and alkalinity reconstruction (Fig. a). Monthly biases remain but are now within the resampling threshold (Fig. a). Correlation improves from 0.8 to above 0.9 in surface pCO2. Air–sea CO2 correlation does not improve but only because correlations above 0.9 for the indirect reconstruction were already very high (Fig. b). Correlation for boreal winter is above 0.95, indicating that initial conditions in winter are well reconstructible for initializing forecasts of oceanic carbon sink (Fig. b). Direct reconstruction improves pCO2 accuracy to 0.2 ppm. Mean bias reduction can hardly improve accuracy after direct reconstruction (Fig. c). Air–sea CO2 flux accuracy degrades in comparison to indirect reconstruction. This degradation is removed by the mean bias reduction (Fig. f).

All results for the direct reconstruction of the land carbon cycle must be understood in the context of the method chosen for the direct reconstruction. Because we reset the restart files in 1 January to the target, the metrics are near perfect in January by design. However, then the biogeochemistry is not modified directly for 12 months and only follows the physical climate reconstruction indirectly, and thus biases triggered by physical biases unaligned with the reset land biogeochemistry pools quickly build up and may approach the metric of the indirect reconstruction. Likewise, there is no bias in global cVeg in January by design. The bias increases with the physical biases until it surpasses the resampling threshold in August and continues increasing until the end of the year (Fig. g). Annual cVeg bias is strongly improved by direct reconstruction (Fig. g). Global air–land CO2 flux has a stronger bias than the indirect reconstruction (Fig. j). Correlation in the global cVeg is near perfect in January by design and slowly decreases to 0.8 in December while still being better than the resampling threshold (Fig. h). Annual cVeg variations are reconstructed much better by the direct method compared to the indirect method (Fig. h). Global air–land CO2 flux variations increase by 0.2 (Fig. k). Direct reconstruction improves global cVeg accuracy. Accuracy is better than the resampling threshold after mean bias reduction. Direct reconstruction slightly improves CO2 flux accuracy. Furthermore, a mean bias reduction slightly improves accuracy (Fig. i and l).

The global CO2 bias in the direct reconstruction increases to +1.8 ppm (Fig. m), but correlation increases from 0.7 to 0.9 (Fig. n). The direct reconstruction has worse accuracy than the indirect due to established bias, but after mean bias reduction the accuracy is below 0.3 ppm (Fig. o).

How does direct carbon cycle reconstruction affect tracking performance in prognostic atmospheric CO2? The time series already indicate that there is a 1–2 ppm atmospheric CO2 positive bias in the direct reconstruction (Fig. e). This bias is very homogeneous over the oceans (Fig. o). However, correlation strongly increased to 0.9 above the oceans and above 0.7 on land, except for central Africa with its persistent biases, where the reconstruction is not better than the resampling threshold.

Section  shows that atmospheric CO2 follows the reconstructed land and ocean carbon cycle, integrating their respective fluxes over time. The direct carbon cycle reconstruction introduces a large bias in the atmospheric CO2 distribution that the indirect reconstruction did not suffer from even after mean bias reduction (Fig. ). Globally averaged atmospheric CO2 after direct reconstruction had a better accuracy tracking performance after the mean bias reduction, showing how global aggregation can balance regional biases. The direct land and ocean carbon cycle reconstructions track targets much better than the indirect reconstruction when measured by correlation.

Generally speaking, this first part showed how direct carbon cycle reconstruction improves linear association between reconstruction and target (measured by ACC) but often increases biases degrading accuracy (measured by RMSE). Only after bias reduction does accuracy improve with respect to the indirect carbon cycle reconstruction.

The RMSE between the initialized ensembles and the target simulations in annual globally averaged pCO2 continuously increases from lead year 1 to lead year 3 as expected. While perfectly and indirectly initialized ensembles stay below the resampling uninitialized threshold for the first 2 lead years, indicating that global pCO2 is predictable due to initialization (Fig. a), the direct initialization has a larger error due to the offsets in global atmospheric CO2, which pCO2 tries to equilibrate to (Fig. e). Therefore, this persistent bias causes lead year 3 to be not predictable. A simple mean bias reduction resolves this issue, making all 3 lead years predictable. Direct initialization only beats indirect initialization for lead year 1 with RMSE of 0.35 ± 0.05 versus 0.45 ± 0.05 ppm (Fig. f).

Global air–sea CO2 flux is predictable for 3 years in all initialization methods, which is 1 year longer than in , possibly because here we use more and more equally distributed initialization dates. Direct initialization is advantageous over the indirect initialization because the initial lead offset is smaller (0.14 ± 0.01 versus 0.18 ± 0.02 PgCyr-1) (Fig. b). The simple mean bias reduction improves the skill of the non-perfect initializations to identical magnitudes (Fig. g).

Indirect initialization makes cVeg not predictable. The physical reconstruction biases drive larger errors in lead year 1 than in later lead years and also to a lesser extent for the direct reconstruction, where some biases are corrected. But both reconstructed initialized ensembles show decreasing distances towards the target, whereas increasing distances are expected for vanishing predictive skill as in the perfectly initialized ensembles (Fig. c). Mean bias reduction eliminates the differences between direct and perfect reconstruction, making both predictable unlike the indirect reconstruction (Fig. h).

Global air–land CO2 flux is predictable for 3 years, again 1 year longer than found in . Both reconstructed initializations start with a higher error of 1.1 ± 0.2 PgCyr-1 in lead year 1 compared to perfectly initialized 0.7 ± 0.1 PgCyr-1 (Fig. d). Mean bias reduction brings non-perfect initializations within the error bars of the perfect initialization after lead year 1 (Fig. i). A recent analysis focused on process-based understanding of land carbon predictability using JSBACH indicates that soil moisture and soil carbon storage, both reconstructed by the direct method, influence the air–land CO2 flux the most .

Perfect and indirect initialization atmospheric CO2 predict the target for 3 years, as found in . While the perfect initialization error grows continuously from zero, the indirect initialization error stays nearly constant at 0.7 ± 0.1 ppm, but the error stays below the direct initialization error, which suffers from the bias in the direct reconstruction simulation (Fig. e). Mean bias reduction improves RMSE, making direct initialization better but still within the margins of the indirect initialization. After lead year 1, indirect and direct initializations are similar to perfect-initialization predictive skill at 0.7 ppm (Fig. j).

The anomaly correlation coefficient measures how predictable variations are and is independent of the mean bias (Fig. k–o) . Measuring predictive skill with ACC shows very similar behavior across all variables. While perfect initialization is the most predictable, indirect and direct carbon cycle initialization are fairly similar. Predictive ACC skill seems to saturate after lead year 2.

These initialized predictive skill results show that indirectly initialized ensembles predict the target quite reasonably. Direct initialization suffers strong shocks in some variables, when reconstruction is started and stopped, but these shocks can be partly reduced by a mean bias reduction. The improvements of direct reconstruction over indirect reconstruction in the global carbon cycle predictive skill after bias reduction are not significant, except for vegetation carbon pools (cVeg) (Fig. f–j).