The aerosol adjoint module of the atmospheric chemical modeling system GRAPES–CUACE (Global–Regional Assimilation and Prediction System coupled with the CMA Unified Atmospheric Chemistry Environment) is constructed based on the adjoint theory. This includes the development and validation of the tangent linear and the adjoint models of the three parts involved in the GRAPES–CUACE aerosol module: CAM (Canadian Aerosol Module), interface programs that connect GRAPES and CUACE, and the aerosol transport processes that are embedded in GRAPES. Meanwhile, strict mathematical validation schemes for the tangent linear and the adjoint models are implemented for all input variables. After each part of the module and the assembled tangent linear and adjoint models is verified, the adjoint model of the GRAPES–CUACE aerosol is developed and used in a black carbon (BC) receptor–source sensitivity analysis to track influential haze source areas in north China.

The sensitivity of the average BC concentration over Beijing at the highest concentration time point (referred to as the Objective Function) is calculated with respect to the BC amount emitted over the Beijing–Tianjin–Hebei region. Four types of regions are selected based on the administrative division or the sensitivity coefficient distribution. The adjoint sensitivity results are then used to quantify the effect of reducing the emission sources at different time intervals over different regions. It is indicated that the more influential regions (with relatively larger sensitivity coefficients) do not necessarily correspond to the administrative regions. Instead, the influence per unit area of the sensitivity selected regions is greater. Therefore, controlling the most influential regions during critical time intervals based on the results of the adjoint sensitivity analysis is much more efficient than controlling administrative regions during an experimental time period.

In the large-scale scientific and engineering calculation fields, derivative calculation exists everywhere. Solving a nonlinear optimal problem requires calculating the gradient, as a Hessian matrix or in a higher-order reciprocal form (Cheng et al., 2009). The traditional finite difference method aims at some basic state, changing the concerned input variable values in a proper order, obtaining the difference between output variables, and determining the sensitivities of the output variables to each input variable (Cacuci, 1981a). This method usually creates truncation errors and is costly. Therefore, it is used only when there are few input variables. The decoupled direct method (DDM), which makes use of the tangent linear model (TLM), is an improvement of the finite difference method, but is still limited in cases of few input variables (Hakami et al., 2007). Comparatively, the adjoint method is an efficient sensitivity analysis approach, suitable for calculating the parametric sensitivities of complex numerical model systems and for solving various optimal problems based on sensitivity information. An adjoint model can be used to estimate the sensitivity of every variable in each time period and each simulation grid for the objective function in one simulation. Therefore, it is much more efficient than the finite difference method and the DDM. The adjoint method is used to calculate the derivatives of meromorphic functions based on machine precision; thus, it has higher calculation precision and it costs less, being propitious to large-scale nonlinear complex calculation and playing a significant role in meteorological and environmental fields. Based on the adjoint operator theory and the development of numerical models, the adjoint method is increasingly applied for the inversion of pollution sources and other calculations that involve many input parameters. Through this method, the TLM and the adjoint model of the original model can be obtained on the basis of the traditional Finite Difference Method combined with the adjoint equation theory. The principle is to build the objective function using the difference between modeled and the observed parameter values. Then, the gradient (sensitivity) of the objective function to the model input parameters is calculated using the adjoint model. This gradient can be used as a decreasing step length, correcting the input values, until the objective function reaches the minimum value through continuous iteration processes, therefore obtaining satisfactory input parameter values (Wang, 2000).

The adjoint method presents a unique advantage for complex multiparametric
systems. Only one simulation is required to estimate the sensitivity or
gradient of the objective function to all of the input parameters (Zhu and
Zeng, 2002; Zhu et al., 1999; Liu, 2005). Consequently, various types of
optimal control and inversion problems can be solved quickly using the
gradient information (Chen et al., 1998; Liu and Hu, 2003). Marchuk (1986)
and Marchuk and Skiba (1976) first applied the adjoint method to the
atmospheric environment field. They used the method in the optimal control
and reasonable site selection of pollution sources. They cleverly utilized
the conjugation property of the adjoint operator, thus avoiding the pollutant
transmission problems in repeated problem solving and greatly reducing the
calculation amount. Skiba and Parra-Guevara (2000a, b) and Skiba and
Davydova (2002, 2003) developed Marchuk's method and applied it to solving
atmospheric environment control problems. More recently, adjoint models were
developed for air quality models, and sensitivity analyses and assimilations
were conducted through them. Thus far, atmospheric chemistry adjoint models
include the adjoint of the European air pollution dispersion model (Elbern et
al., 2000), which is mainly used in the simulation of large areas; the
adjoint air quality model STEM-III (Sandu et al., 2005); the adjoint of the
atmospheric chemical transmission model CAMx (Liu et al., 2007); the adjoint
of the CMAQ model (Hakami et al., 2007; Turner, 2010); and the adjoint of the
GEOS-Chem model (Henze et al., 2007). The adjoint of the gaseous processes in
the CMAQ model was already developed, and it included the chemical conversion
and the transmission processes of 72 active species (Hakami et al., 2007). On
this basis, the adjoint of the aerosol processes in the CMAQ model is also
under development; this will be the first coupled gas–aerosol regional-scale
adjoint model to simulate specifically aerosol mass composition and size
distribution (Turner, 2010). Resler et al. (2010) presented a version of the
4D-Var (four-dimensional variation) method and successfully used the adjoint
of the CMAQ model to estimate the optimized diurnal profiles of NO

Furthermore, scientists integrated population and mortality data into the
objective function, and apportioned source attribution to health impacts
through adjoint sensitivity analysis. For example, Pappin and Hakami (2013)
calculated health benefit influences separately from emissions of individual
source locations in Canada and the United States by estimating a certain
reduction in anthropogenic emissions of NO

GRAPES–CUACE is an online coupled model based on the atmospheric model GRAPES (Global–Regional Assimilation and Prediction System; Xue and Chen, 2008) and the air quality forecasting system CUACE (CMA Unified Atmospheric Chemistry Environmental Forecasting System; Zhou et al., 2012; Jiang et al., 2015). GRAPES is a numerical weather prediction system developed for the China Meteorological Administration (CMA). It can be used as a global model, GRAPES–GFS, as well as on a regional scale, as the GRAPES–MESO model. GRAPES–CUACE implements GRAPES–MESO. CUACE is an air quality forecasting and climate research system developed by the Chinese Academy of Meteorological Science (CAMS). In this research, the adjoint model of the GRAPES–CUACE aerosol module was developed and used in black carbon (BC) receptor–source sensitivity analysis.

The air quality forecasting system CUACE includes four major functional modules: emissions, gaseous chemistry, the size-segregated multicomponent aerosol algorithm, and data assimilation (Zhou et al., 2012). CUACE adopted CAM (Canadian Aerosol Module) as its aerosol module (Gong et al., 2003). The GRAPES–CUACE aerosol module has three parts: (1) CAM, (2) three interface programs that connect GRAPES–MESO and CUACE (in aerosol_driver.F, module_ae_cam.F, and aeroexe1.F), and (3) the aerosol transport processes that are embedded in GRAPES–MESO (see Fig. S1 in the Supplement).

CAM involves six types of particles – sulfate, organic carbon, black carbon,
nitrate, sea salt, and soil dust – which are divided into 12 sections using
the multiphase multicomponent aerosol particle size separation algorithm. The
mass conservation equation of the size-distributed multiphase, multicomponent
aerosols can be expressed as

CAM also involves the vertical diffusion processes of aerosols in the atmosphere (in chem_trvdiff2.F). By solving the vertical diffusion equation, the vertical diffusion trend of aerosol particles is calculated. The aerosol physical and chemical processes section (CAM_V5) is the core of this module, including some primary aerosol processes in the atmosphere: aerosol emission, moisture absorption increase, collision, coring, condensation, dry deposition, gravity setting, subcloud cleanup, aerosol activation, interaction between aerosols and clouds, and transmission of sulfate in the clouds and the clear sky (see Fig. S1 in the Supplement). The CAM_V5includes29 programs in total: 1 main program (cam1d.f), 4 auxiliary subroutines, and 24 subprograms related to the above-described aerosol physical and chemical processes.

In addition, the emission fluxes (both anthropogenic and natural emission sources) are calculated through the surface fluxes calculation module (SFFLUX). SFFLUX contains one master program and six subprograms. Each of the six subprograms calculates the emission fluxes of one component (see Fig. S1c in the Supplement). The three interface subroutines transfer meteorological parameters from GRAPES–MESO to CUACE, extend the spatial dimension from 1-D to 3-D, and read emissions for CAM. The transport processes (both horizontal and vertical) in GRAPES–CUACE are calculated by the dynamic framework of GRAPES–MESO, which implements the quasi-monotone semi-Lagrangian (QMSL) semi-implicit scheme on every grid (Wang et al., 2009). It includes an “upstream point” calculation subroutine (upstream_interp) and the QMSL scheme subroutine (BS_QMSL; Zhai, 2015).

In recent years, the GRAPES–CUACE modeling system was widely used in air pollutants simulation in China, and its performance is very well validated and improved (Zhou et al., 2012; Wang et al., 2015a, b; Jiang, 2015). These studies laid a good foundation for the development of the adjoint of GRAPES–CUACE aerosol model.

Because adjoint operators in Hilbert spaces are more convenient to deal with
than adjoint operators are in Banach spaces, we take advantage of the
simplified geometrical properties of Hilbert spaces in developing the adjoint
model (Cacuci, 1981b). In a Hilbert space, the inner product is denoted by

An atmospheric chemical transport model (CTM) solves the mass conservation
equations and can be expressed as

In constructing the adjoint of the GRAPES–CUACE aerosol model, we developed the TLM and the adjoint of the three parts (CAM, interface subroutines, and aerosol transport processes) involved in the GRAPES–CUACE aerosol module.

First, the TLM of CAM_V5 (CAM_V5–TLM) was constructed and validated (validation details in Sect. 2.2.3). Then, the adjoint of CAM_ V5 (CAM_V5–ADJ) was developed and verified based on CAM_V5–TLM (verification details in Sect. 2.2.4). CAM_V5–ADJ comprises 58 programs in total: all 29 original source codes of CAM_V5, 25 corresponding adjoint codes (except the 4 auxiliary subroutines), 1 stack manipulation function definition program for saving the basic state in the inner structure (in adBuffer.f), and 3 zero-assignment subroutines (in putzeroint.f, initial0.f, and initial0all.f).

CAM_V5, CAM_V5-TLM, and CAM_V5–ADJ are box modules with spatially fixed coordinates. To update the spatial 1-D CAM–ADJ to the spatial 3-D CUACE–ADJ aerosol module, the adjoints of the interface subroutines (in aerosol_driver.F, module_ae_cam_ad.F, and aeroexe1_ad.F) and the transport processes (in ad_uptream_interp.F and ad_bs_qmsl.F) were developed to transfer the 3-D parameters from GRAPES to CUACE. Then, the adjoints of SFFLUX (in cam_sfflux_ad.F, cam_sfbc_ad.F, cam_sfnt_ad.F, cam_ sfoc_ad.F, cam_sfrd_ad.F, cam_sfss_ad.F, and cam_sfsf_ad.F) were integrated in CUACE–ADJ. The CUACE–ADJ aerosol module is capable of extending sensitivity values from the time series, at a horizontal grid cell, to the 3-D variations in a reverse chronological order, displaying inverse aerosol transport processes.

The physical processes (aerosol processes included) were calculated at the model's vertical half levels. However, the aerosol transport processes, which are embedded in the dynamic framework of GRAPES–MESO, were calculated at the model's full vertical levels. Therefore, the interpolation routines (in phy_post_back.F, phy_prep.F) and their corresponding adjoints (in ad_phy_post_back.F, ad_phy_prep.F) were additionally integrated in the CUACE–ADJ aerosol model. In addition, basic states in the outer structure correspond to the output and input (O/I) of the binary file (read_initialdata.F).

Building an adjoint model for a forward model is a very complex task. To
speed up the process and reduce mistakes, the entire model is divided into
many small subprograms. In this study, the adjoint model was developed both
manually and automatically. The automatic differentiation engine TAPENADE
(Tangent and Adjoint PENultimate Automatic Differentiation Engine;

After the adjoint model is built, its accuracy must be verified to confirm its reliability. The adjoint model is a concomitant of the TLM. Thus, the validity of the TLM must be ensured before the accuracy of the adjoint model is tested. If all of the codes are tested together, then it is difficult to isolate error locations. To overcome this problem, both the TLM and the adjoint model are divided into smaller sections, which are then tested separately. After these sections are confirmed, the assembled TLM and the adjoint model are tested.

Supposing that the code of every small section is regarded as

All input variables in the model should pass the TLM validation. There are
many input variables in the model, but as the space of this paper is limited,
we only choose two representative variables and provide the validation
results here. For instance, the concentration value of pollutants
(

Validation results of the tangent linear model.

After all tangent linear codes have passed the testing, the adjoint codes can
be tested on the basis of the TLM. The adjoint codes and the tangent linear
codes need to satisfy Eq. (5) for all possible combinations of

Validation results of the adjoint model.

Considering pollutant concentration variable

Flow chart of GRAPES–CUACE, aerosol adjoint, and the flowchart of the parameters transmission process.

As observed from the results in Table 2, both sides of the equation produce
values with 14 identical significant digits or more. This result is within
the range of computer errors, so the values of the left and the right sides
are considered equal. Thus, the pollutant concentration variable

After each part of the assembled TLM and the adjoint model were verified, the
GRAPES–CUACE aerosol adjoint model was constructed. The structures and
parameters flowchart is shown in Fig. 1. ADJ is short for adjoint;

When operating, the forward GRAPES–CUACE simulation should be run first to save the basic-state values of the unequilibrated variables in checkpoint files. Intermediate values are recalculated or saved in stack during the adjoint integration. Then, the saved basic-state values during the forward integration and the forcing terms are used as inputs for the adjoint backward simulation.

To perform the sensitivity analysis and solve environmental optimization
problems, we usually take into account various factors, including air quality
standards, economic losses, health benefits, the emissions reduction
enforceable ratio range, and suitable locations for factories. Hence, a
reasonable evaluation function

The principle application of the adjoint model is sensitivity analysis, and
all its other applications may be considered to derive from it (Errico,
1997). In this research,

Because of its high efficiency in calculating sensitivity (or gradient), the
adjoint model plays an important role in optimization problems. For example,
in emission inventory optimization problems,

In this study, the GFS reanalysis data, which are collected six times a day
with 1

Left: model domain settings; right: the locations of Nanjiao (NJ) and Shangdianzi (SDZ) observation sites and the Tongzhou (TZ) and Daxing (DX) districts.

The data used in this paper were obtained from the Beijing Meteorological
Observatory Nanjiao station and Shangdianzi station. The Nanjiao station (NJ;
39.8

BC is an important component of atmospheric aerosols. It is emitted directly into the atmosphere predominantly during combustion (Seinfeld and Pandis, 2006). Its sources include anthropogenic and natural emission sources. Natural sources (e.g., volcanic eruption and forest fires) are occasional and regional, contributing little to the long-term background BC concentration in the atmosphere (Parungo et al., 1994). Comparatively, many human activities increase the concentration of BC aerosols; therefore, anthropogenic sources are the primary sources of BC. Streets et al. (2001) and Cao et al. (2006) noted that the vast majority of BC emissions in China are produced by the untreated raw coal, honeycomb briquettes, and biomass fuels that people use in their daily lives.

BC is the main light-absorbing aerosol species; it alters the radiative properties of other aerosols with which it is mixed. In addition, it may also affect cloud formation and precipitation (Hakami et al., 2005), reduce crop production, decrease visibility, and harm human health. In one word, BC plays an essential role in atmospheric radiative forcing, climate change, and air quality evaluation.

The simulated ground BC concentration distributions from 20:00 BT 3 July to
11:00 BT 4 July are shown in Fig. 3. These six graphs illustrate the
formation and transportation processes of this high BC concentration episode
over Beijing. At 20:00 BT 3 July, two small spots of high BC concentrations
appeared around Shijiazhuang (SJZ; 114.48

BC concentration distribution at ground level (Unit:

Figure 4 shows the hourly variation of ground-level BC concentration in
Beijing. It is easy to notice that during the first 2 simulated days, the
BC concentration reached its peak at approximately 02:00 BT on 2 and 3 July,
and its lowest value at approximately 15:00 BT on the same days. Thus, the
absolute BC concentration in this case appears to be affected by the diurnal
height variation of the boundary layer, atmospheric stability, and diffusion
conditions. On the contrary, the highest BC concentration on 4 July
(15.7

The model results are compared with the above observation data in Fig. 5. The
correlation coefficients of the simulated and the observed BC concentrations
at Shangdianzi and Nanjiao stations are 0.65 and 0.54, respectively.
Therefore, the general variation trends of the simulated and observed BC
concentrations are consistent. However, the simulated BC concentration values
are greater than the corresponding observed values at both stations, with
MR

Hourly variation of simulated ground BC concentration over the Beijing municipality.

As mentioned above, the adjoint method can provide information about the
influences of location-specific sources on the objective function. To
determine the area and the time period when the most important emission
sources fed the highest BC concentration over Beijing as recorded at
11:00 BT 4 July 2008 (Fig. 4), we define the objective function

The adjoint input, also regarded as a forcing term, is

Comparisons of observed and modeled hourly BC concentrations at
Nanjiao and Shangdianzi stations from 20:00 1 July 2008 to 19:00 4 July 2008. Statistical parameters used are mean functional bias (MFB), mean
functional error (MFE), mean value of the simulated (

In this way, the emission sensitivity coefficient

Cumulative sensitivity coefficient distribution.
Panels

To control air quality, usually emissions are cut over a certain period,
e.g., 1–3 days ahead of the predicted severe
pollution day. Based on this practical concept, sensitivity coefficients at
every model's backward integral time step are added from the objective time
point (highest BC concentration: 11:00 BT 4 July 2008) to a certain
preceding time point, as illustrated in Fig. 6. Figure 6 shows a
spatial–temporal cumulative effect from BC emissions to the objective
function

As shown in Fig. 6, sensitivity coefficients accumulate along an inverse time
series. When sensitivity coefficients from the previous hour until the
objective time point are added, only the Tongzhou (TZ) and Daxing (DX)
districts (locations of TZ and DX districts were shown in Fig. 2) in Beijing
have sensitivity coefficients of 0.05–0.1

Adjoint sensitivity analysis is a powerful complement to forward methods. While forward techniques are source-based, backward methods provide receptor-based sensitivity information. Under this conception, we use the adjoint method to locate the most influential emission sources area and the most influential emission time period.

Four types of regions are defined according to administrative division and
the sensitivity coefficients distribution (Table 3 and Fig. 7). BTH refers to
the administrative Beijing–Tianjin–Hebei region, which covers 105 grid cells
and is approximately 318 000 km

Different influential regions. BTH: red dashed frame; InR-1: blue dashed frame; InR-2: pinkish red solid frame; object region: yellow shadow.

Information on the four emission reduction regions.

18 h (17:00 BT 3 July–11:00 BT 4 July) cumulative SC and SC/Grid over the four emission reduction regions.

SC: sensitivity coefficient. SC/Grid: sensitivity coefficient per simulation grid.

To compare the effect of emission sources reduction at different time points
in the four regions, we add the BC emission sensitivity coefficients
vertically and extract their inverse time series values (Fig. 8). Figure 8a
shows the inverse time series of the sensitivity coefficients at every 5 min
integration time steps. It reflects the influence of BC emissions on the
objective function

In Fig. 8a, the sensitivity coefficients of BTH, InR-1, and BJ reach their
peak values at 18:00 BT 3 July, whereas that InR-2 is maximized at 17:00 BT
3 July. Afterward, they all decrease sharply along a backward time sequence.
This phenomenon indicates that the impact of emissions on

Then, we compare the preceding 18 h cumulative sensitivity coefficients,
from 17:00 BT 3 July to 11:00 BT 4 July, for the above four regions
(Table 4), given that the sensitivity coefficient on 17:00 BT 3 July is
still relatively high (for BTH, InR-1, and BJ). From Table 4, the simulated
SC (sensitivity coefficient) of BTH is 7.3

In this study, based on the adjoint theory and methods, we constructed and tested an adjoint model for an aerosol module of the atmospheric chemical model GRAPES–CUACE. Developing the GRAPES–CUACE aerosol adjoint model included constructing and validating the tangent linear and the adjoint models of the three parts involved in the GRAPES–CUACE aerosol module: CAM, interface programs, and the aerosol transport processes. Meanwhile, strict mathematical validation schemes for the tangent linear and the adjoint models were carried out for all input variables. After the assembled tangent linear and the adjoint models for each part were verified, the adjoint model of the GRAPES–CUACE aerosol was constructed. At the same time, the GRAPES–CUACE model and its aerosol adjoint were adopted to perform a numerical simulation and a receptor–source sensitivity test. Compared with the BC aerosol observations from the Nanjiao and Shangdianzi stations, the hourly trends of BC concentration estimated through the present model were similar, with correlation coefficients 0.65 and 0.54, respectively.

The GRAPES–CUACE adjoint model simulated the sensitivity of the concentration on emission, and it was adopted to track the most influential emission sources regions and the most influential time intervals for the high BC concentrations. Four types of regions were selected and compared based on the administrative divisions and the adjoint-sensitivity coefficient distribution. The result of the aerosol adjoint model suggested that the regions divided based on the sensitivity values could be correlated to the influence emission sources regions better than the administratively divided regions could. In particular, in the example used here, the BC emissions at 18:00 BT on 3 July to the objective time point (about 17–18 h) had a much greater influence than emissions emitted earlier than that.

The BC adjoint sensitivity results presented here could help design efficient haze control schemes using the adjoint method. It is found that to increase the emission reduction efficiency, influential regions should be located scientifically (e.g., according to the adjoint sensitivity coefficients distribution) rather than based on administrative divisions.

We used the GRAPES–CUACE as distributed by the Numerical Weather Prediction
Center of Chinese Meteorology Administration (

This study was supported by the National Natural Science Foundation of China (41575151), the National Science-Technology Support Program (2014BAC16B03), and the CMA Innovation Team for Haze-fog Observation and Forecasts. We appreciate Lin Zhang, Feng Liu, Qiang Cheng, Hongliang Zhang, and Min Xue for providing technical support in adjoint model construction. Thanks are also owed to the developers of the GRAPES–CUACE aerosol model. The authors are indebted to the anonymous referees for their valuable comments. Edited by: J. Annan