Model Error-Induced Biases of Greenhouse Gas Contribution to Global Warming: A Piecewise Integration Approach

Many of the observed changes of the climate system since the 1950s are unprecedented, and there is a high level of condence in the conclusion that greenhouse gases (GHGs) caused a substantial part of the observed global warming. We need to consider the model errors, that usually accumulate in long-term integration as a result of imperfect physical and numerical representations, to attribute climate changes using model simulations. Here, we present a new method of the piecewise integration (PWI) with simulation corrected by the observation at each step, to identify model error-induced bias of global warming in the Community Earth System Model (CESM). To conrm the hypothesis of constant model-observation bias under different external forcing, we disturb the original CESM into a less low-cloud version and take its historical and GHGs-xed simulations as our “observations”. In the PWI historical and GHGs-xed runs of original CESM from 1958 to 2005, we use the difference between “historical observation” and PWI historical run to correct both PWI runs at the end of each 1-day step. The results show that the PWI can effectively reduce model’s cumulative error and presents a GHGs-induced global warming trend of 0.688 ℃ (48yr) -1 , which is very close to the “observational” trend of 0.683 ℃ (48yr) -1 , conrming the hypothesis of constant model bias under different external forcing. The continuous runs, as usually done by the Coupled Model Intercomparison Project (CMIP) models, present a much higher GHGs-induced global warming trend of 0.887 ℃ (48yr) -1 , which means that the model overestimates the GHGs’s role in global warming trend by 32.3% compared to our “observations”. Global distribution of this model bias is also discussed. The PWI method provides a new way to correct model biases in analyzing relative contribution of anthropogenic and natural forcing to global warming.


Introduction
The successive reports of the Intergovernmental Panel on Climate Change (IPCC) have offered increasingly con dent assessments of the dominant role of anthropogenic factors in causing current global warming (Parry et al. 2007). Despite the high level of con dence in the conclusion that greenhouse gases (GHG) caused a substantial part of the observed warming, however, there remain many uncertainties that have so far limited the potential to be more precise about attribution statements (Jones et al. 2013). These uncertainties include both observational uncertainty caused by measurement biases in datasets and uneven distribution of meteorological observation stations in the globe (Cowtan and Way 2014; Frei and Isotta 2019; Matthew et al. 2017), and simulation uncertainty of numerical models. The uncertainty of simulations is mainly due to the incomplete representation of the climate system and its dynamical and physical con gurations in numerical models (Jones et al. 2013; Morice et al. 2012). To reduce observational errors, data assimilation methods are often used to effectively absorb the observation information of different levels and spatial distribution; based on this, the quality of initial elds can be improved, and the errors caused by the lack of observations in high-latitude polar regions and vast ocean areas can be reduced to some extent, which can improve the quality of model simulations and reduce the growth of internal errors (Kalnay 2002). Moreover, improving physical module of the model is an important way to reduce simulation errors in the numerical model. Although current climate models have been greatly improved after decades of efforts, there still exist many shortcomings in describing some physical processes and the coupled ocean-atmosphere and land-atmosphere processes Improving the performance of the numerical model is an arduous and long-term task. Many researchers have improved simulation methods to reduce the uncertainty of simulation based on the existing models.
For example, the data assimilation method is used to introduce the observation as a constraint, which can improve the accuracy of the initial eld (Kalnay 2002 Similar to the re-initialization in the dynamic downscaling (Bennett and Leslie, 1981), the piecewise integration (PWI) approach was proposed for sensitivity experiment to mitigate accumulation systematic errors in the traditional continuous (CONT) long-term simulations (Zhang et al. 2008). The PWI splits the continuous long-term simulation into subintervals of sequential short-term simulations. The initial elds of the numerical model including the historical run and sensitivity run with certain external forcing xed are updated by re-initialization at the beginning of each subinterval, and the states of the simulation of historical run are constantly re-initialized with the observations or reanalysis data, while the states of sensitivity run are constantly updated by adding the difference between the state of historical run and reanalysis data. In this way, the drifts of the climate model can be effectively corrected, which will improve simulation accuracy of the climate model. This approach was validated with a simple sixparameter model, and the results indicated that it is able to reduce the error accumulation caused by longtime continuous integration and improve the credibility of numerical simulation (Zhang et al. 2008). A more comprehensive assessment of the applicability of the PWI was conducted under broader scenarios through experiments with the shallow-water model (Shao et al. 2015), and the in uences of analysis data error and subinterval length on the results were investigated (Shao et al. 2015). A series of idealized sensitivity experiments were carried out to study the contribution of GHGs emissions to climate change in the past 26 years of 1979-2004 by the Community Earth System Model (CESM), and the experiments showed that the PWI can effectively reduce the systematic error; however, the simulated time is not long enough to identify the global warming trend (Wang 2017), thus it is important to further analyze the performance of the PWI by using long-term ensembles.
In this study, we conducted a series of idealized experiments to investigate the in uence of external GHGs forcing change on the climate with the CESM, which include the historical simulations and GHGs-xed simulations. In addition, a more comprehensive assessment of the applicability of the PWI was assessed by the comparison of perturbed initial conditions ensembles of the CONT.

The PWI approach
To evaluate the contribution of the external forcing to past climate change, the climate model can be run with changing external forcing and constant external forcing. The rst simulation was performed with actual forcing including realistic changing of anthropogenic and natural forcing, which can be called historical simulation. The other simulation was performed with external forcing xed, for example, the GHGs concentration was xed at its initial value and kept constant throughout the run. We called this case a GHG-xed simulation. The averaged difference over a long period is then de ned as the effects of the GHGs on climate change; and it is also one of the most concerned quantity in the study of climate To improve the accuracy of the simulation, the state of historical simulation can be constrained with the observations or reanalysis data through nudging or re-initialization. The PWI splits the entire integration period into subintervals of sequential short-term simulations, and the state of historical simulation of the model is updated by observations or reanalysis data at the end of each subinterval. Due to the lack of observations or reanalysis data as constraints, the state of GHGs-xed simulation is also updated by superimposing the difference between re-initialization and the historical simulation state at the end of  resolution of the land-surface and land-ice models are 1.9° × 2.5°, and the sea-ice model has the horizontal resolution is 1°×1°. More detailed description of the CESM1.2.0 can be found at the website: https://www.cesm.ucar.edu/models/cesm1.2.

Experimental designs
To evaluate the advantages of the PWI compared to the CONT, a series of idealized sensitivity experiments were carried out with the model data, before using the observational data or reanalysis data for numerical simulations in the future works. Two different CESM versions denoted by CESM1.2.0 and CESM1.2.0_rl_0.9 were used. The CESM1.2.0_rl_0.9 is regarded as the "perfect model" without errors, and the simulation results are regarded as "observation" to test the accuracy of the other simulations and provide analysis elds for the PWI. The CESM1.2.0 represents a model with systematic errors, which is used for sensitivity experiments with the PWI and CONT (Fig. 1). The only difference between the two versions is that the minimum threshold for low stable clouds (denoted as rl): the rl in CESM1.2.0 is 0.8875, which is the default value of CESM1.2.0, while the rl value in CESM1.2.0_rl_0.9 is 0.9, resulting in less low clouds in CESM1.2.0_rl_0.9 than in CESM1.2.0.
To make it easier to distinguish the impact of GHGs emissions on climate change from that of the other forcing, we conduct two groups of simulation for climate sensitivity experiments as shown in Fig. 1. The rst group of simulations were performed with "actual forcing" including anthropogenic and natural forcing, which are historical simulations (called FULL simulations). The other group of simulations were performed with xed GHGs concentration throughout the integration, which have the same concentration as that in 1958 (called GHGs-xed simulations). The GHGs in the simulations include CO 2 , N 2 O, CH 4 , CFC 11 , and CFC 12 . The difference between the two groups could be regarded as the contribution of GHGs to climate change. The "observational" historical and GHG-xed runs were performed by using CESM1.2.0_rl_0.9 through CONT integration from 1958 to 2005. The PWI historical and GHG-xed runs were performed by using CESM1.2.0, which are both corrected by the same difference between the PWI and "observational" historical runs, based on the hypothesis of constant model error under different external forcing (Fig. 1).
To comprehensively evaluate the reliability of the PWI, ensemble experiments of the CONT were also run with six ensemble members by using CESM1.2.0. The initial elds were unperturbed in the rst ensemble experiment, while in each of the remaining ve ensemble members, a small random disturbance was generated from the range (-0.01K, 0.01K) to the initial air temperature eld. It is worth pointing out that the temperature and wind elds of the PWI were updated after every 24 hours in all the numerical simulations, and the small initial disturbance has relatively small effects on the simulation results, so the PWI has only one ensemble. All these experiments are summarized in Table 1.

Results
To identify the model bias of GHG contribution to global warming, this section is organized in three parts. We rst show the advantage of the PWI is over the CONT by comparing the average 2-m temperature difference variation with latitude between the PWI simulation and the "observation", and that of between the CONT simulation and the "observation". We then present the temporal variations of global 2-m temperature and precipitation in the "observations", the CONT and the PWI simulations. Finally, we exhibit the spatial distributions of 2-m temperature trends and precipitation trends in the PWI and the CONT simulation, by compared to the "observation". Figure 2a shows the latitude distribution of average 2-m temperature difference in the historical simulation between the PWI and the "observation", and that between the CONT and the "observation". The globally averaged 2-m temperature variation with latitude in the CONT simulation is smaller than that of the "observation", and the uncertainty of the simulation in the mid-to-high latitudes of the Northern Hemisphere is larger than in tropics. The maximum bias of CONT ensemble mean reaches − 1.527℃ at 90°N, while the maximum bias of the PWI simulation is only 0.025℃ (). The 2-m temperature simulated by the PWI is closer to the "observation", and can better reproduce the "observational" change with altitude. Figure 2b shows the GHGs-xed simulation. Similar to historical simulation (Fig. 2a), the average 2-m temperature variation with latitude in the CONT simulation is also smaller than that of the "observation", and the uncertainty of simulation in the mid-to-high latitudes of the Northern Hemisphere is large. The maximum bias of CONT ensemble mean reaches − 0.876℃ at 90°N, while the maximum bias of the PWI is 0.32℃ at 64°S. The temperature simulated by the PWI is closer to the "observation", and can better describe the change with the latitude.  respectively. Thus, the error of the PWI relative to the "observation" is signi cantly smaller than that of CONT ensemble mean, which con rms that hypothesis of constant model-observation bias under different external forcing. Compared to the historical run, the PWI GHG-xed simulation is not that good as in historical run, since the state of the PWI GHGs-xed simulation is updated by adding the difference between "observational" and PWI historical runs. presents a much higher GHGs-induced global warming trend of 0.887℃ (48 year) −1 , which means that the model overestimates the GHGs role in global warming trend by 32.3% compared to the "observation." Therefore, the simulation accuracy and reliability of the PWI are higher than those of the CONT, which is usually used in our current CMIPs.

Updated climatology by PWI
Temporal variation of global annual precipitation is provided in Fig. 4a. The globally averaged error of the PWI relative to the "observation" is signi cantly smaller than that of CONT ensemble mean, and the PWI can better reproduce the trend of global annual precipitation of the "observation" than the CONT. The globally averaged precipitation trends of the "observation", the PWI, and the CONT ensemble mean are 0.017mm day − 1 (48 year) −1 , 0.013mm day − 1 (48 year) −1 , and − 0.02mm day − 1 (48 year) −1 , respectively. It is obvious that the accuracy of the PWI simulation is higher than that of the CONT ensemble mean. The trend of global precipitation simulated by the PWI and the CONT ensemble mean are lower than the "observation". The simulation of precipitation by the PWI is not as good as that of 2-m temperature, and the reason may be due to the fact that the speci c humidity has not been updated, which should be considered in the future work. Figure 4b shows the precipitation change in the GHG-xed simulations. The temporal variation of globally averaged precipitation trend of the PWI simulation is slightly closer to the "observation" than that of CONT ensemble mean; and the globally averaged precipitation trends of the "observation," PWI, and CONT ensemble mean in the GHGs-xed simulation are 0.003mm day − 1 (48 year) −1 , -0.008mm day − 1 (48 year) −1 , and − 0.037mm day − 1 (48 year) −1 , respectively. Since the state of the GHGs-xed simulation by the PWI is updated by superimposing the difference between "observation" and the historical simulation at the end of each subinterval, and the speci c humidity closely related to precipitation is also not updated, the high accuracy of PWI simulation cannot be guaranteed. However, the error of the PWI simulation is still much smaller than that of the CONT ensemble mean. Compared to temperature, the trend of precipitation change is much weaker due to more stable troposphere under the anthropogenic GHG forcing-induced global warming (Held and Soden 2006). This may be one reason that the PWI method didn't perform well in diagnosing the small precipitation trend. Implement of the moisture in PWI may also improve our PWI method isolating the model bias.
3.3 Spatial distributions of 2-m temperature and precipitation trends  60°N) and Russia. However, none of the CONT ensemble members' simulation, including the ensemble mean, is consistent with the "observation" in these cooling areas, while the PWI can simulate some cooling areas, say the Paci c Ocean near South America as in the "observation". Moreover, Fig. 6 shows that the difference of GHGs-induced 2-m temperature trend between the PWI simulation and the "observation" is smaller than that between the CONT simulation and the "observation". The difference of 2-m temperature trend between the simulation of each member of the CONT ensemble and the "observation" range from 0.08°C to 0.407°C (48 year) −1 , and that between the CONT ensemble mean and the "observation" is 0.204°C (48 year) −1 . While the difference of 2-m temperature trend between the PWI simulation and the "observation" is only 0.006°C (48 year) −1 . Each member of the CONT ensemble, as well as the CONT ensemble mean and the PWI simulation, are quite different from the "observation" in high latitudes of the Northern Hemisphere.
The degree of resemblance between the "observation" and simulated trend maps can be quanti ed in terms of centered pattern correlation (e.g., the area-averaged 2-m temperature trend is removed before computing the pattern correlation) and root-mean-square error (RMSE). Pattern correlations between each member of the CONT ensemble and the "observation" range from 0.192 to 0.360, and RMSE range from 0.429°C to 1.087°C (48 year) −1 . Pattern correlation between the CONT ensemble mean and the "observation" is 0.356, and RMSE is 0.410°C (48 year) −1 . While the pattern correlation of the PWI and "observation" is 0.533, and the RMSE is 0.299°C (48 year) −1 (Fig. 5). The results show that the ensemblemean of the CONT simulation still cannot eliminate possible systematic errors, but the PWI can effectively reduce the cumulative error and improve the simulation accuracy and reliability of sensitivity experiments. Precipitation trend varies obviously across individual ensemble members, despite the fact that each simulation was conducted using the same model and subjected to the identical radiative forcing. Further, the patterns of precipitation trend can be nearly opposite between individual runs. For example, CONT2 exhibits signi cantly decreased precipitation over the eastern equatorial Paci c Ocean and the Indian Ocean near Australia, while signi cantly increased precipitation over the western equatorial Paci c Ocean and western Indian Ocean, while CONT4 shows opposite trends.
The "observation" trend of GHGs-induced precipitation (labeled 'OBS') (Fig. 7, bottom- (Fig. 7). Moreover, Fig. 8 shows the difference of precipitation trend  caused by GHGs between the PWI simulation and the "observation" is smaller than that between the CONT simulation and the "observation". The difference of precipitation trend between the simulation of each member of the CONT ensemble and the "observation" range from 0.012mm day − 1 (48 year) −1 to 0.031mm day − 1 (48 year) −1 , and that between the CONT ensemble mean and the "observation" is 0.022mm day − 1 (48 year) −1 . While the difference of precipitation trend between the PWI simulation and the "observation" is only 0.007mm day − 1 (48 year) −1 .

Conclusions And Discussion
In the long-term integration, the model systematic errors may accumulate during long-term integration as a result of incomplete physical and numerical representation, which may cause large uncertainty when evaluating climate response to changes in external forcing. To reduce these errors, we present a new PWI method to attribute climate change. The PWI splits the continuous long-term simulation into subintervals of sequential short-term simulations, and the simulation is updated by re-initialization at the end of each subinterval for the experiment.
We rst performed a sets of sensitivity experiments based on the "perfect" CESM1.2.0_rl_0.9 with higher rl of 0.9 and original CESM1.2.0 with default rl of 0.8875 to test the hypothesis that the model error, interpreting the long-term trend, keeps the same under different external forcing.
The rl in CESM1.2.0 is lower than that in CESM1.2.0_rl_0.9, which leads to the increase of low-cloud coverage, the decrease of solar shortwave radiation to the Earth surface, and the decrease of surface air temperature. Therefore, in the historical simulation, the 2-m temperature by the CONT are lower than the "observation," and the model-"observation" biases are large. The simulation by the PWI is able to reproduce the changing of the "observation." The PWI can effectively reduce the cumulative error and improve the simulation accuracy and reliability. The same results are also obtained in GHG-xed simulations, but its simulation accuracy is not as good as the historical run, because the state of GHGsxed simulation by PWI is updated by adding the difference between "observational" and PWI historical runs.
Our PWI simulations present a GHGs-induced global warming trend of 0.688℃ (48 year) −1 , which is very close to the "observational" trend of 0.683℃ (48 year) −1 . This result con rms our hypothesis of constant model bias under different external forcing and shows that the PWI method can effectively reduce model's cumulative error. The continuous runs, as usually done by the CMIP models, present a much higher GHGs-induced global warming trend of 0.887℃ (48 year) −1 , which means that the model overestimates the GHGs's role in global warming trend by 32.3% compared to our "observations". Further analyses of 48-yr GHGs-induced 2-m temperature trend show that the model bias mainly occur in highlatitude region.
The PWI simulations presents a GHGs-induced global precipitation trend of 0.020mm day − 1 (48 year) −1 , which means that the PWI overestimates the GHGs role in global precipitation trend by 53.8% compared to the "observation" which is 0.013mm day − 1 (48 year) −1 . While that the CONT ensemble mean of 0.035mm day − 1 (48 year) −1 overestimates the GHGs role in global precipitation trend by 169.2% compared to the "observation". Further analyses maps of 48-yr GHGs-induced precipitation trend, the results show that the error of PWI simulation is less than that of CONT simulation. The accuracy of precipitation simulated by the PWI is not as good as that of 2-m temperature, the main reason may be that the speci c humidity is not updated.
The above results show that the ensemble mean of the CONT still cannot eliminate possible systematic errors, while the PWI can effectively reduce the cumulative errors and improve the simulation accuracy and reliability of the sensitivity experiments. The PWI has provided good results in the test and application of complex climate model of the CESM. The realistic observation or reanalysis data instead of our mimicking "observation" can be used to the attribution analysis in the future works.

Declarations
Con ict of interest: The authors declare that they have no con ict of interest.      Two-meter temperature (°C (48yr)-1) trends  caused by GHGs for each member of the CONT ensemble (CONT0: unperturbed initial eld; CONT1-CONT5: perturbed initial eld), the CONT ensemble mean (labeled EM), PWI (labeled PWI), and "observation" (labeled 'OBS'). Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.   Same as in Fig. 6 except for difference of precipitation (mm day-1 (48yr)-1). Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.