In this study, a four-dimensional variational (4D-Var) data assimilation system was developed based on the GRAPES–CUACE (Global/Regional Assimilation and PrEdiction System – CMA Unified Atmospheric Chemistry Environmental Forecasting System) atmospheric chemistry model, GRAPES–CUACE adjoint model and L-BFGS-B (extended limited-memory Broyden–Fletcher–Goldfarb–Shanno) algorithm (GRAPES–CUACE-4D-Var) and was applied to optimize black carbon (BC) daily emissions in northern China on 4 July 2016, when a pollution event occurred in Beijing. The results show that the newly constructed GRAPES–CUACE-4D-Var assimilation system is feasible and can be applied to perform BC emission inversion in northern China. The BC concentrations simulated with optimized emissions show improved agreement with the observations over northern China with lower root-mean-square errors and higher correlation coefficients. The model biases are reduced by 20 %–46 %. The validation with observations that were not utilized in the assimilation shows that assimilation makes notable improvements, with values of the model biases reduced by 1 %–36 %. Compared with the prior BC emissions, which are based on statistical data of anthropogenic emissions for 2007, the optimized emissions are considerably reduced. Especially for Beijing, Tianjin, Hebei, Shandong, Shanxi and Henan, the ratios of the optimized emissions to prior emissions are 0.4–0.8, indicating that the BC emissions in these highly industrialized regions have greatly reduced from 2007 to 2016. In the future, further studies on improving the performance of the GRAPES–CUACE-4D-Var assimilation system are still needed and are important for air pollution research in China.

Three-dimensional (3-D) atmospheric chemical transport models (CTMs) are important tools for air quality research, which are used not only for predicting spatial and temporal distributions of air pollutants but also for providing sensitivities of air pollutant concentrations with respect to various parameters (Hakami et al., 2007). Among several methods of sensitivity analysis, the adjoint method is known to be an efficient means of calculating the sensitivities of a cost function with respect to a large number of input parameters (Sandu et al., 2005; Hakami et al., 2007; Henze et al., 2007; Zhai et al., 2018). The sensitivity information provided by the adjoint approach can be applied to a variety of optimization problems, such as formulating optimized pollution-control strategies, inverse modelling and variational data assimilation (Liu, 2005; Hakami et al., 2007).

Four-dimensional variational (4D-Var) data assimilation, which is an
important application of adjoint models, provides insight into various model
inputs, such as initial conditions and emissions (Liu, 2005; Yumimoto
and Uno, 2006). In the past decades, many scholars have successively
developed adjoint models of various 3-D CTMs and the 4D-Var data
assimilation systems to optimize model parameters. Elbern and Schmidt (1999, 2001), Elbern et al. (2000, 2007) constructed the adjoint of the EURAD CTM and performed
inverse modelling of emissions and chemical data assimilation. Sandu et al. (2005) built the adjoint of the comprehensive chemical transport model
STEM-III and conducted the data assimilation in a twin-experiment framework
as well as the assimilation of a real data set, with the control variables
being O

Nowadays, several mega urban agglomerations in China, such as the Beijing–Tianjin–Hebei region, the Yangtze River Delta region and the Fenwei Plain, are still suffering from severe air pollution (Zhang et al., 2019; Xiang et al., 2020; Haque et al., 2020; Zhao et al., 2020). Previous studies have shown that emission-reduction strategies, which are mainly based on the results of atmospheric chemistry simulations, play an important role in reducing pollutant concentrations and improving air quality (Zhang et al., 2016; Zhai et al., 2016). The emission inventory represents important basic data for atmospheric chemistry simulation, and its uncertainty will affect the accuracy of air pollution simulation, which in turn will affect the accuracy of pollution-control measures based on the model results (Huang et al., 2018). In order to improve the accuracy of atmospheric chemistry simulation and the feasibility of the pollution-control strategies, the emission data obtained by the “bottom–up” method needs to be optimized, which can be done through the atmospheric chemical variational assimilation system, to reduce the impact of emission uncertainty.

GRAPES–CUACE is an atmospheric chemistry model system developed by the Chinese Academy of Meteorological Sciences (CAMS) (Gong and Zhang, 2008; Zhou et al., 2008, 2012; Wang et al., 2010, 2015). GRAPES (Global/Regional Assimilation and PrEdiction System) is a numerical weather prediction system built by China Meteorological Administration (CMA), and it can be used as a global model (GRAPES-GFS) or as a regional mesoscale model (GRAPES-Meso) (Chen et al., 2008; Zhang and Shen, 2008). CUACE (CMA Unified Atmospheric Chemistry Environmental Forecasting System) is a unified atmospheric chemistry model constructed by CAMS to study both air quality forecasting and climate change (Gong and Zhang, 2008; Zhou et al., 2008, 2012). Using the meteorological fields provided by GRAPES-Meso, the GRAPES–CUACE model has realized the online coupling of meteorology and chemistry (Gong and Zhang, 2008; Zhou et al., 2008, 2012; Wang et al., 2010, 2015). The GRAPES–CUACE model not only plays an important role in the scientific research on air pollution in China (Gong and Zhang, 2008; Zhou et al., 2008, 2012; Wang et al., 2010, 2015) but has also been officially in operation since 2014 at the National Meteorological Center of CMA for providing guidance for air quality forecasting over China (Ke, 2019).

Recently, An et al. (2016) constructed the aerosol adjoint module of the GRAPES–CUACE model, which was subsequently applied in tracking
influential BC and PM

GRAPES-Meso is a real-time operational weather forecasting model used by China Meteorological Administration (Chen et al., 2008; Zhang and Shen, 2008). The GRAPES-Meso model uses fully compressible non-hydrostatic equations as its model core. The vertical coordinates adopt the height-based, terrain-following coordinates, and the horizontal coordinates use the spherical coordinates of equal longitude–latitude grid points. The horizontal discretization adopts an Arakawa-C staggered grid arrangement and a central finite-difference scheme with second-order accuracy, while the vertical discretization adopts the vertically staggered variable arrangement proposed by Charney-Phillips (Charney and Phillips, 1953). The time integration discretization uses a semi-implicit and semi-Lagrangian temporal advection scheme. The large-scale transport processes (both horizontal and vertical) for all gases and aerosols in GRAPES–CUACE are calculated by the dynamic framework of GRAPES-Meso, which implements the quasi-monotone semi-Lagrangian (QMSL) semi-implicit scheme on each grid (Wang et al., 2010). The physical processes principally involve microphysical precipitation, cumulus convection, radiative transfer, land surface and boundary layer processes. Each physical process incorporates several schemes and can also be tailored by the user (Xu et al., 2008). The major physical options that we selected include the WSM6 cloud microphysics scheme (Hong and Lim, 2006), the Betts–Miller–Janjic cumulus convection scheme (Betts and Miller, 1986; Janjić, 1994), the RRTM (Rapid Radiative Transfer Model; Mlawer et al., 1997) long-wave radiation scheme, the short-wave scheme based on Dudhia (1989), the Monin–Obukhov surface layer scheme (Monin and Obukhov, 1954), the MRF (medium-range forecast) planetary boundary layer scheme (Hong and Pan, 1996) and the Noah land surface scheme (Chen et al., 1996).

The atmospheric chemistry model CUACE mainly includes three modules: the aerosol module (module_ae_cam), the gaseous chemistry module (module_gas_radm) and the thermodynamic equilibrium module (module_isopia) (Gong and Zhang, 2008; Zhou et al., 2008, 2012; Wang et al., 2010, 2015). The interface program that connects CUACE and GRAPES-Meso is called aerosol_driver.F. This program transmits the meteorological fields calculated in GRAPES-Meso and the emission data processed as needed to each module of CUACE. The physical and chemical processes of 66 gas species and 7 aerosol species (sulfate, nitrate, sea salt, black carbon, organic carbon, soil dust and ammonium) in the atmosphere are comprehensively considered in the CUACE model (Wang et al., 2015).

CUACE adopts CAM (Canadian Aerosol Module; Gong et al., 2003) and RADM II
(the second-generation Regional Acid Deposition Model; Stockwell et al.,
1990) as its aerosol module and gaseous chemistry module, respectively. CAM
involves six types of aerosols: sulfate (SF), nitrate (NI), sea salt (SS),
BC, organic carbon (OC) and soil dust (SD), which are segregated
into 12 size bins with diameter ranging from 0.01 to 40.96

The emissions used in this study are based on statistical data of
anthropogenic emissions reported from government agencies for 2007 (Cao et
al., 2011). Emission source types included residences, industry, power
plants, transportation, biomass combustion, livestock and poultry breeding,
fertilizer use, waste disposal, solvent use, and light industrial product
manufacturing (Cao et al., 2011; Zhai et al., 2018). These emission data
were transformed through the Sparse Matrix Operator Kernel Emissions (SMOKE)
module into hourly gridded off-line data for 32 species, including BC, OC,
SF, NI, fugitive dust particles and 19 non-methane volatile organic
compounds (VOCs), CH

Assuming that

An atmospheric chemistry model can be viewed as a numerical operator

According to Eqs. (1) and (2), the adjoint model of the TLM can be expressed
as

The GRAPES–CUACE aerosol adjoint model was constructed by An et al. (2016) based on the adjoint theory (Ye and Shen, 2006; Liu, 2005) and the CUACE aerosol module, which mainly includes the adjoint of physical and chemical processes and flux calculation processes of six types of aerosols (SF, NI, SS, BC, OC and SD) in the CAM module, the adjoint of interface programs that connect GRAPES-Meso and CUACE, and the adjoint of aerosol transport processes.

As described in An et al. (2016), after the construction of the adjoint
model is completed, its accuracy must be verified to confirm its
reliability. Since the adjoint model is built on the basis of the TLM, the
validity of the TLM must be ensured before the accuracy of the adjoint model
is tested. The verification formula of tangent linear codes can be expressed
as

The adjoint codes can be validated on the basis of the correct tangent
linear codes. The adjoint codes and the tangent linear codes need to satisfy
Eq. (2) for all possible combinations of

After the TLM and the adjoint model were verified, the GRAPES–CUACE aerosol
adjoint model was constructed. The operation flowchart of the adjoint model
is shown in Fig. 1.

Running process of GRAPES–CUACE atmospheric chemistry model and its adjoint model.

The limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm (L-BFGS) is an optimization algorithm in the family of
quasi-Newton methods that approximates the BFGS using a limited amount
of computer memory (Liu and Nocedal, 1989). The L-BFGS-B algorithm extends
L-BFGS to solve large nonlinear optimization problems subject to simple
bounds on the variables (Byrd et al., 1995; Zhu et al., 1997), which can be
expressed as

The brief procedure of the L-BFGS-B algorithm is as follows. At each
iteration, a limited memory BFGS approximation to the Hessian is updated.
The limited memory BFGS matrix is used to define a quadratic model of the
objective function

The new 4D-Var data assimilation system, GRAPES–CUACE-4D-Var, was constructed on the basis of the GRAPES–CUACE atmospheric chemistry model, the GRAPES–CUACE aerosol adjoint model and the L-BFGS-B method. A schematic diagram of GRAPES–CUACE-4D-Var is shown in Fig. 2. The main parts of GRAPES–CUACE-4D-Var include GRAPES–CUACE atmospheric chemistry simulation, during which the basic-state values of the unequilibrated variables in checkpoint files are saved, observations and adjoint forcing term processing, GRAPES–CUACE aerosol adjoint model simulation, gradient extraction, cost function calculation, and optimization. The details of cost function, observations and optimization of emission inversion are as follows.

GRAPES–CUACE-4D-Var assimilation system.

Based on Bayesian theory and the assumption of Gaussian error distributions
(Rodgers, 2000) the cost function of the emission inversion is generally
defined as follows:

In this study, we followed the method in Henze et al. (2009), and defined

The surface measurements of BC were collected from the China Atmosphere Watch Network (CAWNET). The CAWNET was established by the China Meteorological Administration to monitor the BC surface mass concentration over China in 2004 and had 54 monitoring stations in the summer of 2016. The monitoring of BC was conducted by an aethalometer (Model AE 31, Magee Scientific Co., USA), which uses a continuous optical greyscale measurement method to produce real-time BC data (Gong et al., 2019). In this study, we used the recommended mass absorption coefficient for the instrument at an 880 nm wavelength with 24 h mean values of BC during 1–31 July 2016 at five representative stations of CAWNET in northern China (Fig. S1 in the Supplement).

The surface PM

To improve the performance of emission inversion, adequate observations are
needed for constraining the model. Due to the limited BC monitoring sites in
northern China, we used the surface PM

The observation error covariance matrix (

Minimization of the cost function Eq. (9) is performed through optimization.
Starting from an initial guess (

The simulation domain in this study is northern China (105–125

The progression of the cost function at iteration

Cost function reduction.

The spatial distribution of observed and simulated daily BC
concentrations on 4 July 2016. The observations (circles) are
overplotted over model simulations with the

Figure 4 shows the spatial distribution of observed and simulated daily BC
concentrations on 4 July 2016. In general, the results simulated with
the prior emission reflect the distribution characteristics of BC
concentration in northern China to a certain extent, with high values mainly located in Beijing and central and southern Hebei and low values mainly located in Inner Mongolia and eastern Shandong. However, the
differences between the simulated and the observed BC concentrations are
considerable, and almost all are over-predictions. The optimized
(posterior) emissions compensate for the over-predictions and largely reduce the model biases. For instance, the model biases for BC in Beijing, Tianjin,
Shijiazhuang, Jinan, Taiyuan and Zhengzhou are reduced by 46 % (from 5.4
to 2.9

It is crucial to validate the assimilation results by observations that were
not utilized in the assimilation. The BC concentrations at 12 cities were
used for validation (Fig. 4c, d). Assimilation compensates for over-predictions,
reduces the root-mean-square errors (from 5.2 to 4.4) and improves the
correlation coefficients (from 8.6 to 8.7). The model biases for BC in
Hengshui and Yizhou are reduced by 28 % (from 7.2 to 5.2

Figure 5 shows the spatial distributions of the prior and optimized daily BC emissions, which are at three vertical levels (non-point source on the ground, middle-elevation point source at 50 m and high-elevation point source at 120 m) as required by the GRAPES–CUACE model. The BC emissions on the ground are mostly non-point sources and at 50 m are concentrated middle-elevation sources; therefore they are distributed in a large area (Fig. 5a, c), while the BC emissions at 120 m are mainly from a few high-elevation point sources, so they are scattered in the area (Fig. 5e). It can be seen that the distributions of optimized BC emissions at three vertical levels are relatively consistent with those of prior emissions (Fig. 5b, d, f). However, the optimized emissions are considerably reduced (Fig. 5b, d, f). Especially for the regions where observation sites are located, such as southern Beijing, Tianjin, central and southern Hebei, northwest Shandong, central Shanxi, and northern Henan, BC emissions decrease significantly. As for the regions where observation sites are not located, such as Liaoning and Jiangsu, BC emissions are almost unchanged. The reason for this phenomenon is discussed in Sect. 4.3.

Spatial distributions of the

In order to analyse the differences between the prior and optimized BC emissions in Beijing–Tianjin–Hebei–Shanxi–Shandong–Henan region in more depth and detail, we calculated the ratio of the optimized emissions to prior emissions (optimized emissions divide by prior emissions), as shown in Fig. 6. In general, assimilation has the largest reduction in the non-point source on the ground (Fig. 6a), followed by the middle-elevation point source at 50 m (Fig. 6b), and the smallest reduction in the high-elevation point source at 120 m (Fig. 6c). This is related to the intensity of BC emissions at three vertical levels. The intensity of BC emissions on the ground (Fig. 5a) is about 1–2 orders of magnitude higher than that of the middle-elevation point source at 50 m (Fig. 5c) and the high-elevation point source at 120 m (Fig. 5e). In other words, the BC emissions on the ground have the most significant effect on the overall cost function and therefore reduced most during the progression of assimilation.

The ratio of optimized emissions to prior emissions (optimized
emissions divide by prior emissions).

The prior BC emissions used in this study are based on statistical data of anthropogenic emissions for 2007 (Cao et al., 2011), and the BC observations used for assimilation are from 2016, so the ratios of the optimized emissions to prior emissions can reflect the changes in BC emissions from 2007 to 2016 to a certain extent. From the perspective of each province, we can see that the ratios of the optimized emissions to prior emissions in Beijing, Tianjin, Hebei, Shanxi, Shandong and Henan are 0.4–0.8, 0.4–0.7, 0.4–0.8, 0.6–0.8, 0.4–0.8 and 0.5–0.8, respectively. This indicates that the BC emissions in these highly industrialized regions have greatly reduced from 2007 to 2016, which is consistent with previous studies (Zheng et al., 2018). Table 1 lists the anthropogenic BC emissions in China by province in 2006 and 2016 and the emission ratios of 2016 relative to 2006. According to previous research, the emission ratios of 2016 relative to 2006 are 0.41–0.82 in Beijing–Tianjin–Hebei–Shanxi–Shandong–Henan region and 0.73 over China. It can be seen that the ratios of the optimized emissions to prior emissions calculated in this study are within a reasonable range, which also shows that the newly constructed GRAPES–CUACE-4D-Var assimilation system can obtain reasonable BC emissions based on the observations.

Anthropogenic BC emissions in China by province in
2006

Note: numbers in parentheses represent emission ratios relative to 2006.

Although Sect. 4.1 shows that assimilation reduces the model biases and
improves all statistical values at each site, there are still
over-predictions to a certain degree. In addition to the emission, the
initial concentration is also an important factor that affects the BC
simulation. Here, we take Beijing as an example to analyse the influence of
the initial concentration on the BC simulation. Figure 7 shows the comparison
of observed and simulated (with the prior and optimized BC emissions,
respectively) BC concentrations in Beijing from 20:00, 3 July, to 19:00, 4 July 2016. At the initial moment (20:00, 3 July 2016), the
initial concentration for simulation (11.5

Comparison of observed and simulated (with the prior and
optimized BC emissions, respectively) BC concentrations in Beijing from
20:00, 3 July, to 19:00, 4 July 2016. The observed BC
concentrations were calculated by the

As described in Sect. 4.2, after assimilation, the BC emissions in the regions where observation sites are located decrease significantly, while in the regions where observation sites are not located, such as Liaoning and Jiangsu, BC emissions are almost unchanged. This may be due to the short period of the observation data used for assimilation. In this study, since the GRAPES–CUACE-4D-Var assimilation system has not yet implemented parallel computing, the assimilation window was set for 24 h. In such a short period of time, the pollutants emitted from Liaoning and Jiangsu may not have been transported to the regions where observation sites are located, thus having little impact on the BC concentrations in these regions. In other words, the emissions from Liaoning and Jiangsu have little effect on the overall cost function, so there is little change during the progression of assimilation. Therefore, implementing parallel computing of the GRAPES–CUACE-4D-Var assimilation system and performing emission inversion for a longer period (i.e. 1 month) is another important task in the future.

In this study, we developed a new 4D-Var data assimilation system for the
GRAPES–CUACE atmospheric chemistry model (GRAPES–CUACE-4D-Var) and applied
it for assimilating surface BC concentrations and optimizing its daily
emissions in northern China on 4 July 2016, when a pollution
event occurred in Beijing. The main conclusions are as follows.

The newly constructed GRAPES–CUACE-4D-Var assimilation system is feasible and can be applied to perform BC emission inversion in northern China.

The BC concentrations simulated with optimized emissions show improved agreement with the observations over northern China with lower root-mean-square errors and higher correlation coefficients. The model biases are reduced by 20 %–46 %. The validation of assimilation results with observations that were not utilized in the assimilation shows that assimilation compensates for over-predictions, reduces the root-mean-square errors and improves the correlation coefficients. The improvements are also notable, with values of the model biases reduced by 1 %–36 %.

Compared with the prior BC emissions, the optimized emissions are considerably reduced. Especially for Beijing, Tianjin, Hebei, Shandong, Shanxi and Henan, the ratios of the optimized emissions to prior emissions are 0.4–0.8, indicating that the BC emissions in these highly industrialized regions have greatly reduced from 2007 to 2016, which is consistent with previous studies.

Meanwhile, several mega urban agglomerations in China are facing atmospheric
compound pollution with high PM

The GRAPES–CUACE atmospheric chemistry model
used in this study was distributed by the National Meteorological Center of the Chinese Meteorology Administration (2021,

The supplement related to this article is available online at:

XA envisioned and oversaw the project. XA and CW designed and developed the GRAPES–CUACE-4D-Var assimilation system and prepared the paper. CW designed the experiments and carried out the simulations with contributions from all other co-authors. QH and ZS provided the observation data used in the study. YL and JL processed the data and prepared the data visualization. All authors reviewed the paper.

The authors declare that they have no conflict of interest.

We thank Lin Zhang from the Numerical Forecast Center of the China Meteorological Administration for providing technical support with the optimization algorithm.

This research has been supported by the National Key Research and Development Program of China (grant no. 2017YFC0210006) and the National Natural Science Foundation of China (grant nos. 41975173 and 91644223).

This paper was edited by Augustin Colette and reviewed by two anonymous referees.