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 8amin⁡fx,x∈Rn,8bsubject tol≤x≤u, where f is a nonlinear function, the vectors l and u represent lower and upper bounds on the variables, and the number of variables n is assumed to be large. The algorithm is also appropriate and efficient for solving unconstrained problems in which the variables have no bounds. With the supply of the objective function f and its gradient g, but with no requirement of knowledge about the Hessian matrix of the objective function f, the algorithm can be useful for solving large problems where the Hessian matrix is difficult to compute or is dense.

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 f. A search direction dk is computed by a two-stage approach. First, use the gradient projection method to identify a set of active variables, such as variables that will be held at their bounds. Then, the quadratic model is approximately minimized with respect to the free variables. The search direction is defined to be the vector leading from the current iterate to this approximate minimizer. Finally, a line search is performed along the search direction dk to compute a step length λk, and the variables are updated through xk+1=xk+λkdk. The L-BFGS-B algorithm has three termination criteria: the number of iterations reaches the set maximum value; the change of the objective function in consecutive iterations is relatively small; and the modulus of the projected gradient is small enough.

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: 9Jx=12γx-xbTB-1x-xb+12∑i=0pFix-yiTR-1Fix-yi, where x, which we sought to optimize, generally represents the state vector of emissions or their scaling factors, xb is the prior estimate of x, B is the error covariance estimate of xb, F is the forward model, y is the vector of measurements that are distributed during the time interval [t0, tp], R is the observation error covariance matrix, and γ is the regularization parameter.

In this study, we followed the method in Henze et al. (2009), and defined x as the state vector of scaling factors of BC emissions: 10x=ln⁡ssb, where s is the state vector of the daily gridded emissions of BC 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) and sb is the prior estimate of s. Thus, the prior estimate of x(xb) is equal to 0. According to Cao et al. (2011), the uncertainty of prior BC emissions used in this study is 76.2 %. Therefore, we assigned the prior error covariance matrix (B) to be diagonal and the uncertainty to be 76.2 % for BC emissions. Due to the lack of information to completely construct a physically representative B, the regularization parameter γ is introduced to balance the background and observation terms in the cost function. As described in Henze et al. (2009), an optimal value of γ can be found with the L-curve method (Hansen, 1998). Here, we followed this method and obtained γ=0.0001 through several emission inversions with a range of γ (10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001, 0.0000001).

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 PM2.5 concentrations were obtained from the public website of the China Ministry of Ecology and Environment (MEE) (http://www.mee.gov.cn/, last access: 14 January 2021). The network started to release real-time hourly concentrations of SO2, NO2, CO, ozone (O3), PM2.5 and PM10 in 74 major Chinese cities in January 2013, which further increased to 338 cities in 2016. The PM2.5 data were collected by the TEOM1405-F monitor, which draws ambient air through a sample filter at constant flow rate, continuously weighing the filter and calculates the near real-time mass concentration of the collected particulate matter. We used hourly surface PM2.5 concentrations for 1–31 July 2016 at 48 cities in northern China, including 12 cities in the Beijing–Tianjin–Hebei region (Fig. S1). Here, we have averaged PM2.5 concentrations at several monitoring sites in each city to represent a regional condition.

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 PM2.5 concentrations at 48 cities described above and the BC/PM2.5 ratio to obtain the hourly BC concentrations for 1–31 July 2016 at 48 cities in northern China. The detailed calculation process can be found in the Supplement.

The observation error covariance matrix (R), which is difficult to quantify, generally includes contributions from the measurement error, the representation error and the forward model error (Henze et al., 2009; Zhang et al., 2016; Cao et al., 2018). And there is also a certain error in calculating the BC concentration based on the BC/PM2.5 ratio. To reflect the possibly large uncertainties of the observation, we assumed R to be diagonal and with error of 100 %.

Minimization of the cost function Eq. (9) is performed through optimization. Starting from an initial guess (x equal to 0), the forward model simulates BC concentrations at each integration step during the time interval [t0, tp], and the adjoint model, which is driven by the discrepancy between simulated and observed BC concentrations, calculates the gradients of the cost function with respect to the scaling factors of BC emission (Fig. 2). Subsequently, the gradients are supplied to the L-BFGS-B optimization routine (Byrd et al., 1995; Zhu et al., 1997) to minimize the cost function iteratively (Fig. 2). At each iteration, the improved estimates of the scaling factors are implemented and the forward and adjoint models are integrated.

The simulation domain in this study is northern China (105–125∘ E, 32.25–43.25∘ N; Fig. S1), covering 41×23 horizontal grids with a resolution of 0.5∘×0.5∘ and vertically divided into 31 layers with an integration time step of 300 s. The National Centers for Environmental Prediction Final (FNL) Analysis dataset at a 6 h interval is used as meteorological input. The prior emission used here is the daily gridded BC emission 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) mentioned above. The results calculated by the BC/PM2.5 ratio show that the BC concentration in Beijing was high on 4 July 2016. So the assimilation window is from 20:00 CST, 3 July, to 19:00 CST, 4 July 2016. The hourly BC concentrations at 36 cities during this time interval are used for the emission inversion, and the BC concentrations of the remaining 12 cities are used for validation of the inversion effect (Fig. S1). The simulation is initialized at 20:00 CST, 30 June; the first 3 d are set as the spin-up time. The convergence criterion used in the optimization is that the objective function decreases by less than 1 % in consecutive iterations. According to the maximum estimation range of the prior emissions, here the upper and lower bounds of the scaling factors of BC emissions are ln(1.6) and ln(0.4), respectively.

The progression of the cost function at iteration k (Jk/J0) during the optimization procedure is shown in Fig. 3. The cost function quickly reduces and reaches the convergence criterion after eight iterations, with values of the converged cost function reduced by 37 %.

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 µg/m3), 26 % (from 6.6 to 4.9 µg/m3), 29 % (from 18.4 to 13.1 µg/m3), 20 % (from 6.6 to 5.3 µg/m3), 34 % (from 4.1 to 2.7 µg/m3) and 20 % (from 6.9 to 5.5 µg/m3), respectively (Fig. 4a, b). The results simulated with the optimized emission also show improved agreement with the observations over northern China with lower root-mean-square errors and higher correlation coefficients (Fig. 4a, b).

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 µg/m3) and 36 % (from 3.9 to 2.5 µg/m3), respectively. The improvements of the remaining 10 cities are also notable, with values of the model biases reduced by 1 %–20 %.

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.

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 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.

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 µg/m3) was about 2 times higher than the observations (5.7 µg/m3). With such a high initial concentration, the simulated BC concentrations with prior emissions were significantly higher than the observed concentrations at all times. The simulations with optimized emissions compensated for the over-predictions, but the BC concentrations were still higher than the observations in the first few hours (from 20:00, 3 July, to 02:00, 4 July 2016). And the model bias during this time period was 5.2 µg/m3. As the influence of the initial concentration on the simulation gradually weakened, the simulated BC concentrations with optimized emissions were largely reduced and much closer to the observations from 03:00 to 19:00, 4 July 2016, and the model biases during this time period also decreased, with the value of 1.9 µg/m3. This indicates that for short-term simulation, the influence of initial concentration on the simulation is not negligible. In further work, it is also of great significance to use the GRAPES–CUACE-4D-Var assimilation system to optimize the initial concentration to improve the simulation effect.

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.