We propose to apply Piecewise Parabolic Method (PPM), a high order and conservative interpolation, for the parameters estimation in a PM_{2.5} transport adjoint model. Numerical experiments are taken to show the accuracy of PPM in space and its ability to increase the wellposedness of the inverse problem. Based on the obtained results, the PPM provides better interpolation quality by employing much fewer independent points. Meanwhile, this method is still wellbehaved in the case of insufficient observations. In twin experiments, two prescribed parameters, including the initial condition (IC) and the source and sink (SS), are successfully estimated by the PPM with lower interpolation errors than the Cressman interpolation. In practical experiments, simulation results show good agreement with the observations of the period when the 21th APEC summit took place.
PM_{2.5} pollution, particulate matter with aerodynamic diameters less than 2.5
For more indepth understanding of the physical, chemical, and dynamical processes concerned with PM_{2.5}, a lot of atmospheric numerical models have been conducted and are publicly available in various studies. A box model was used to simulate the atmospheric chemistry and gas/particle partition of inorganic compounds by Pun and Seigneur [
Data assimilation methods provide a configuration for combining observations and models to form an optimal estimate of the PM_{2.5} sources. In this method, observations are used to constrain estimates of model parameters that are both influential and uncertain. Among all data assimilation methods, fourdimensional variational (4DVar) data assimilation is regarded as one of the most effective and powerful approaches developed over the past two decades (e.g., [
The illposedness of the inversion problem is the key part needed to be solved. The illposedness is caused by the incompleteness of the observation data and excessive control parameters in practice, and it is generally characterized by the nonuniqueness and instability of the parameters in the identification process (e.g., [
Referring to previous studies, Cressman interpolation [
The application of the PPM in the PM_{2.5} adjoint model is presented in the following structure. Section
For preserving mass, we define a particular parabolic interpolation distribution by the Piecewise Parabolic Method (PPM) [
We divide the twodimensional computing domain
We apply a particular parabolic interpolation distribution
The interpolation distribution
For obtaining secondorder approximation scheme of PPM (PPM
Then we can get all values at the point on
For obtaining fourthorder approximation Scheme of the PPM (PPM
Then we can get all values at the point on
For the boundary region, we use Taylor expansion to get the lacked values
Generally speaking, the simulation and prediction of the PM_{2.5} are difficult due to the fact that PM_{2.5} is generally not directly emitted; instead, PM_{2.5} varies due to interactions among many processes including emissions, transport, photochemical transformation, and deposition, with meteorology playing an overarching role. The secondary PM_{2.5} is formed via chemical and thermodynamic transformations of gasphase precursors that may potentially emanate far from nonattainment regions. Just to investigate the estimation ability of adjoint method, the emitted and deposited primary PM_{2.5} and secondary PM_{2.5} are taken as a whole, called “source and sink” (SS), without considering the specific details.
We use the same adjoint tidal model as in [
The boundary conditions are set as constant at the inflow boundary
The finite difference scheme of (
Here, the upwind scheme is used in the advection term, that is,
And it is similar for the term in
With IC, SS, and the background value which is the constant in (
As a powerful tool for parameter estimation, the adjoint model is defined by an algorithm and its independent variables including initial conditions, boundary conditions, and empirical parameters. It allows for calculations of the gradients of the cost function with respect to various input parameters, which incorporate all physical processes included in the governing model, to obtain the minimization of the cost function. Based on the governing equations (
First, the cost function is defined as
Then the Lagrangian function is constructed based on the theory of Lagrangian multiplier method and can be expressed as
According to the typical theory of Lagrangian multiplier method, we have the following firstorder derivatives of Lagrangian function with respect to all the variables and parameters:
Equation (
The finite difference scheme of (
By running the adjoint model, the optimized gradients of the control variables and parameters including the horizontal diffusivity coefficient, SS, and IC can be obtained from (
We use the PM_{2.5} transport model shown in Section
The modeling domain covers China, from 70°E to 140°E and from 15°N to 55°N, with a grid resolution of 0.5° latitude by 0.5° longitude (see Figure
Map of computing area and the observation locations of 74 cities. The green circles denote the 74 cities used in the experiments. Full names of these cities can be found in [
For each experiment, the iteration process is designed as follows.
Run the governing model with the prescribed parameters. The modelgenerated PM_{2.5} concentrations at grid points of the observation positions are regarded as the “observations.”
An initial guess value of control parameters, SS (or IC), is taken as zero in this work (or the background value for IC). Run the governing model with the initial values; the simulations are obtained.
The adjoint model is driven by the discrepancy between simulations and observations. The gradients with respect to control parameters are calculated by the adjoint variables obtained by backward integration of the adjoint equations. The control parameters at the independent grids should be updated with a certain optimization algorithm until the convergence criterion is met. The parameters at other points are determined by the interpolation of the values at the independent points.
With the procedure repeated, the parameters will be optimized increasingly. The differences between simulated results and “observations” will decrease. Meanwhile, differences between the prescribed parameters and the inverted ones will be decreased as well.
During the iterative minimization of the cost function, the parameters are optimized with the steepest descent method. The iteration will be terminated once a convergence criterion is reached. In this work, the criterion is that the number of iteration steps is equal to 300 exactly in the twin experiments and the practice experiments.
We pick out seventyfour major cities in China (see Figure
In this section, three versions of the IPS are considered and they are different in the method of interpolation. Cressman interpolation, PPM
The important factors that affect the result of IPS are the selection of independent points and interpolation format. Studies, such as [
We choose one from each 2 points as the independent one; the total number of the independent points is 71 × 36 and each is from a 1° × 1° area.
We choose one from each 4 points as the independent one; the total number of the independent points is 36 × 19 and each is from a 2° × 2° area.
We choose one from each 6 points as the independent one; the total number of the independent points is 25 × 13 and each is from a 3° × 3° area.
We choose one from each 8 points as the independent one; the total number of the independent points is 19 × 10 and each is from a 4° × 4° area.
We choose one from each 10 points as the independent one; the total number of the independent points is 15 × 8 and each is from a 5° × 5° area.
With different IPs distribution schemes, the MAEs of the inversion parameters and the values of the normalized cost function obtained from different interpolations are presented in Figures
(a) The MAE of the IC versus the IPs distribution schemes in Group One. (b) The MAE of SS versus the IPs distribution schemes in Group One. The abscissa indicates distribution scheme category of IPs, while the ordinate indicates the MAE of inversion results. The solid lines are values of the given interpolations and the dotted lines indicate the minimum values of the solid lines, respectively.
(a) The values of normalized cost functions versus IPs distribution schemes as after the assimilation. (b) The iteration histories of the cost function for three version IPS matched the optimal IPs distribution schemes. The abscissa indicates distribution scheme category of IPs. The solid lines are values of the given interpolations and the dotted lines indicate the minimum values of the solid lines, respectively.
For Cressman interpolation, when too many (Scheme (A)) independent points are used, the simulation errors after assimilation are relatively large due to illposedness caused by excessive control parameters. When too few independent points (Schemes (D), (E)) are used, it leads to poor interpolation effect, which can fail to give satisfactory results. Just in this case, in Schemes (D) and (E), PPM
We take the distribution scheme, in which the experiment has the smallest normalized cost function and the minimum MAEs of the inversion parameters, as the optimum scheme for the interpolations. It can be readily seen that the optimum scheme is Scheme (C) for Cressman interpolation, while those for PPM
(a) The prescribed IC, (b) the prescribed SS, (c) the inverted IC obtained from Cressman interpolation, (d) the inverted SS obtained from Cressman interpolation, (e) the inverted IC obtained from PPM
As shown in Figure
Error statistics of different interpolation matched optimal IPs distribution scheme.
Experiment 



 

Before  After  Before  After  Before  After  
Cressman 

32.81  0.74  6.33  1.35  3.16  0.93 
PPM 

32.81  0.28  6.33  1.13  3.16  0.86 
PPM 

32.81  0.19  6.33  1.10  3.16  0.81 
As we know, insufficient observations may cause the illposedness of inversion problem. For further investigation of the performance of PPM
The MAEs statistics before and after assimilation, with 55 observation points.
Experiment 



 

Before  After  Before  After  Before  After  
Cressman 

30.49  0.79  6.33  1.58  3.16  1.07 
PPM 

30.49  0.43  6.33  1.42  3.16  1.01 
PPM 

30.49  0.35  6.33  1.19  3.16  0.87 
The MAEs statistics before and after assimilation, with 49 observation points.
Experiment 



 

Before  After  Before  After  Before  After  
Cressman 

29.76  0.81  6.33  1.66  3.16  1.12 
PPM 

29.76  0.60  6.33  1.52  3.16  1.06 
PPM 

29.76  0.41  6.33  1.22  3.16  0.89 
The MAEs statistics before and after assimilation, with 37 observation points.
Experiment 



 

Before  After  Before  After  Before  After  
Cressman 

28.30  0.85  6.33  1.80  3.16  1.21 
PPM 

28.30  0.58  6.33  1.70  3.16  1.15 
PPM 

28.30  0.51  6.33  1.27  3.16  0.93 
We compare the error statistics from Tables
PPM
In this part, practical experiments are carried out during the 21th APEC summit taking place in Beijing. During the assimilation procedure, both the position and the values of observations are used in the practical experiments. The simulation and data assimilation were conducted for the period of one week from 5 November to 11 November 2014. SS in the practical experiments has temporal variation.
The seventyfour major Chinese cities in Figure
Error statistics of the practical experiments are listed in Table
Error statistics before and after assimilation.
Experiment 



 

Before  After  Before  After  Before  After  
Cressman 

37.87  11.70  42.93  19.31  40.23  19.67 
PPM 

37.87  10.02  42.93  19.15  40.23  17.15 
PPM 

37.87  9.47  42.93  18.61  40.23  17.05 
Measurements at four representative stations, Fuzhou (FZ), Quanzhou (QZ), Jinan (JNa), and Beijing (BeJ), are selected to evaluate the forward model and inverse modeling, as well as the assimilated results. In the practical experiment, Fuzhou is the city with the smallest MAEs between simulated results and observations in all the “assimilated cities.” Among all the “checked cities,” MAEs between simulated results and observations in Quanzhou are the smallest, while in Jinan it is the largest. Beijing is a major city of the “checked cities.” We will focus on the comparison and analysis of the simulations from PPM
In detail, from Table
The MAE (
Experiment  Fuzhou  Quanzhou  Jinan  Beijing 

Cressman  2.27  8.43  27.79  12.21 
PPM 
2.04  8.18  24.76  8.22 
The timevarying PM_{2.5} concentrations of four representative cities during the assimilation window.
The scatterplot of Figure
Comparison of the modeled and observed PM_{2.5} concentrations based on the results from the PPM
By way of illustration, Figure
The mean field of PM_{2.5} concentrations (
We have presented a general high order, conservative method for the parameter inversions of a PM_{2.5} transport adjoint model. It is verified that the application of the PPM into the independent point scheme can solve the illposedness of the inversion problem much effectively. A series of test cases have demonstrated that the PPM is superior to Cressman interpolation for the simulations and the inversion parameters in this model.
We believe the evidence presented here clearly indicates that PPM produces better approximations than can be obtained by using Cressman interpolation. With the availability of the PPM, the illposedness caused by insufficient observations is effectively improved. Regarding the interpolation technique, PPM is favourite in this work. This is mostly due to its ability to achieve high order accuracy in space. In the practical experiments, with the PPM
The numerical results shown here are performed in twodimensional adjoint model only. However, the PPM has a full threedimensional capability. Additional future work will extend to the verification of the PPM in threedimensional adjoint model simulations.
The last two authors made equal contributions to this work and are equally considered to be corresponding authors.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was partly supported by the National Key Research and Development Plan of China (Grants nos. 2017YFC1404000, 2017YFA0604100, and 2016YFC1402304), the Fundamental Research Funds for the Central Universities of China (Grant no. 201513059), the Natural Science Foundation of Zhejiang Province (no. LY15D060001), the National Natural Science Foundation of China (no. 41606006 and no. 41371496), the Education Department Science Foundation of Liaoning Province of China (no. JDL2016029), and the Education Department Science Foundation of Liaoning Province of China (no. JDL2017019).