Advances in Meteorology

Volume 2017, Article ID 4626585, 13 pages

https://doi.org/10.1155/2017/4626585

## The High Order Conservative Method for the Parameters Estimation in a PM_{2.5} Transport Adjoint Model

^{1}School of Science, Dalian Jiaotong University, Dalian 116028, China^{2}Physical Oceanography Laboratory/CIMST, Ocean University of China and Qingdao National Laboratory for Marine Science and Technology, Qingdao 266003, China^{3}Institute of Physical Oceanography, Ocean College, Zhejiang University, Zhoushan 316000, China^{4}School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Jicai Zhang; moc.361@gnahz_iacij and Kai Fu; nc.ude.cuo@ufk

Received 15 July 2017; Accepted 29 November 2017; Published 25 December 2017

Academic Editor: Yoshihiro Tomikawa

Copyright © 2017 Ning Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 well-posedness 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 well-behaved 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.

#### 1. Introduction

PM_{2.5} pollution, particulate matter with aerodynamic diameters less than 2.5 *μ*m, has gained widespread concern due to its adverse influence on human and ecosystem health. It has direct effect on climate by scattering and absorbing sunlight and has indirect effect by serving as cloud condensation nuclei and thereby affects the optical properties, lifetime of clouds, and precipitation. Therefore it is essential to estimate PM_{2.5} emissions to understand the changes to the atmospheric environment around the world. The research and prediction of PM_{2.5} pollution should be paid more attention, through which local air quality agencies can alert the public far from the unhealthy air in time.

For more in-depth 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 [1]. And the concentration of PM nitrate was found to be sensitive to reductions in VOC emissions. A Lagrangian air quality model was developed which represents the airborne particle complex as a source-oriented external mixture by Kleeman et al. [2]. Hudischewskyj and Seigneur [3] described the formulation and evaluation of a mathematical model that calculates the gas-phase and aerosol-phase pollutant concentrations in a plume that undergoes transport, dispersion, and dry deposition in the atmosphere. The performance of the Eta-Community Multiscale Air Quality (CMAQ) modeling system in forecasting PM_{2.5} and chemical species is assessed over the eastern United States by Yu et al. [4]. Seigneur [5] reviewed 3-dimensional (3D) Eulerian air quality models for PM in terms of their formulation and past applications. Basically, analysis of the PM_{2.5} on a continental scale depends on the sophisticated knowledge of the distribution. However, with incomplete observations in the spatial or temporal range, model estimates demonstrate significant uncertainties. Therefore, both further utilization of PM_{2.5} measurements and improvement of model estimates are identified as the important components for the continued analysis of PM_{2.5} sources.

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, four-dimensional variational (4D-Var) data assimilation is regarded as one of the most effective and powerful approaches developed over the past two decades (e.g., [6–9]). Efforts have been made with regard to the applications of the 4D-Var data assimilation in the study of PM_{2.5} emissions. The adjoint of GEOS-Chem, a four-dimensional variational data assimilation method, is used to map US air quality influences of inorganic PM_{2.5} precursor emissions by Henze et al. [10]. Zhang et al. [11] used the adjoint method for the examination of the source attribution of PM_{2.5} pollution over North China. Capps et al. [12] revealed the relative contributions of global emissions to PM_{2.5} air quality attainment in the US based on the GEOS-Chem adjoint model. Wang et al. [13] established a PM_{2.5} transport adjoint model which is applied for the estimation of the parameters successfully.

The ill-posedness of the inversion problem is the key part needed to be solved. The ill-posedness 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., [14, 15]). Many efforts have been made towards overcoming ill-posedness. And, it is justified that independent point scheme (IPS) is an effective approach for overcoming ill-posedness (e.g., [11–13]). In detail, we select several grids uniformly as independent points (IPs) in the space domain. Values of the control parameters at the IPs can be optimized, while those at other grids can be calculated through linear interpolating with the values at the IPs.

Referring to previous studies, Cressman interpolation [16] is used as the first-choice for the IPS owning to its simple operation. However, when applied to the datasets with large distances between grid points, it should be hard for Cressman interpolation to obtain satisfactory interpolation errors. As we know, mass conservation is an important issue for climate and atmospheric chemistry models. Thus, the Piecewise Parabolic Method (PPM) developed in [17], which has been proved to be a high order accurate interpolation in the past (e.g., [18–21]), is used in the same PM_{2.5} adjoint model shown in [13] to ensure the local mass conservation for the advection problems.

The application of the PPM in the PM_{2.5} adjoint model is presented in the following structure. Section 2 provides the detailed descriptions of the PPM and the PM_{2.5} transport adjoint model. Twin experiments are carried out, and the results are analyzed in Section 3. Based on the conclusion above, practical experiments are implemented in Section 4. Finally, some key conclusions drawn from the work are presented.

#### 2. The Piecewise Parabolic Method and the PM_{2.5} Transport Model

##### 2.1. The Piecewise Parabolic Method

For preserving mass, we define a particular parabolic interpolation distribution by the Piecewise Parabolic Method (PPM) [17]. The foundation of PPM is a piecewise parabolic interpolation scheme which generates a parabola to describe the internal structure of a computational cell, or zone, of the grid. A parabola is generated for any desired variable based upon knowledge of the zone-averaged values of that variable. In this section, the 2D PPM is established for solving the problems in two-dimensional PM_{2.5} transport adjoint model.

We divide the two-dimensional computing domain into regular Eulerian cells , where cells and cells centers are defined byLet be the independent point and be the value of parameters at the independent point. For the two-dimensional space, we first perform a computation along -direction and then compute along -direction. In this part, we just show the computation along -direction by using the conservative interpolation. The computation along -direction is similar. Then the values on at time can be obtained.

We apply a particular parabolic interpolation distribution [23], which can preserve mass conservation of the IPS. The interpolation distribution is written as

The interpolation distribution is defined on and is the value at the point of .

For obtaining second-order approximation scheme of PPM (PPM for short), is given as follows:

Then we can get all values at the point on by

For obtaining fourth-order approximation Scheme of the PPM (PPM for short), is given as follows:

Then we can get all values at the point on by

For the boundary region, we use Taylor expansion to get the lacked values

##### 2.2. PM_{2.5} Transport Model Description

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 gas-phase 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 [13]. Considering that the observations are ground-level, the vertical meteorology environment is indeterminate. A two-dimensional PM_{2.5} transport model is established in rectangular coordinates as follows:where denotes the PM_{2.5} concentration, and are the horizontal wind velocity in -coordinate and -coordinate, respectively, is the horizontal diffusivity coefficient, is the initial condition (IC), and is the value of SS.

The boundary conditions are set as constant at the inflow boundary and as no gradient boundary conditions at the outflow boundary .

The finite difference scheme of (2) is as follows:

Here, the upwind scheme is used in the advection term, that is,

And it is similar for the term in -coordinate.

With IC, SS, and the background value which is the constant in (12), the PM_{2.5} transport model could be solved by a sequence of time steps of length in the discrete schemes (i.e., (13)).

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 (10)–(13), its adjoint model can be constructed as follows.

First, the cost function is defined aswhere denotes the set of the observations and and are the simulated and observed PM_{2.5} concentrations, respectively. And is the weighting matrix and should be the inverse of observation error covariance matrix theoretically. can be fixed simply, assuming that the errors in the data are uncorrelated and equally weighted. In the present model, is 1 when the observations are available and otherwise.

Then the Lagrangian function is constructed based on the theory of Lagrangian multiplier method and can be expressed as where is the adjoint variable of .

According to the typical theory of Lagrangian multiplier method, we have the following first-order derivatives of Lagrangian function with respect to all the variables and parameters:

Equation (18) gives the governing equation (10). The adjoint equation can be developed from (20), which is given as follows:

The finite difference scheme of (21) is similar to (13).

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 (17) and can be described as follows:

#### 3. Twin Experiments and Analysis

We use the PM_{2.5} transport model shown in Section 3. Initial condition (IC) and “source and sink” (SS) are the major parameters in this model. As the key for the simulation of PM_{2.5}, inversions of SS and IC are the vital part of the whole model. For better model results, IC and SS will be inverted together in all the experiments.

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 1). There are 141 × 71 grids totally in the area. The simulation period is 168 hours and the inverse integral time step is 10 minutes. The background value is fixed as 35.0 *μ*g/m^{3} based on the actual PM_{2.5} concentration and the air quality standards in China. Inflow boundary values are fixed equal to the background values. The winds with spatial resolution 2.5° latitude by 2.5° longitude and 24-hour temporal resolution derived from the National Centers for Environmental Prediction (NCEP) are used in this work.