A New Combined Stepwise-Based High-Order Decoupled Direct and

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A new combined stepwise-based high-order decoupled direct and reducedform method to improve uncertainty analysis in PM2.5 simulations Zhijiong Huang, Yongtao Hu, Junyu Zheng, Zibing Yuan, Armistead G. Russell, Jiamin Ou, and Zhuangmin Zhong Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.6b05479 • Publication Date (Web): 24 Feb 2017 Downloaded from http://pubs.acs.org on February 26, 2017

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Environmental Science & Technology

A new combined stepwise-based high-order decoupled direct and reduced-form method to improve uncertainty analysis in PM2.5 simulations Zhijiong Huang†, Yongtao Hu§, Junyu Zheng*,†, Zibing Yuan†, Armistead G. Russell*,§, Jiamin Ou†, Zhuangmin Zhong† †

College of Environment and Energy, South China University of Technology, Guangzhou, PR

China §

School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta,

GA 30332-0355, USA

Corresponding Author * Junyu Zheng. Phone: +86-20-39380021; fax: 86-20-39380021; e-mail: [email protected]; address: School of Environment and Energy, South China University of Technology, Guangzhou 510006, China * Armistead G. Russell. Phone: 404-374-7030; fax: 404-894-8266; e-mail: [email protected]; address: School of Civil & Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA

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ACS Paragon Plus Environment


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ABSTRACT

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The traditional reduced-form model (RFM) based on the high-order decoupled

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direct method (HDDM), is an efficient uncertainty analysis approach for air quality

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models, but it has large biases in uncertainty propagations due to the limitation of the

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HDDM in predicting nonlinear responses to large perturbations of model inputs. To

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overcome the limitation, a new stepwise-based RFM method that combines several

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sets of local sensitive coefficients under different conditions is proposed. Evaluations

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reveal that the new RFM improves the prediction of nonlinear responses. The new

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method is applied to quantify uncertainties in simulated PM2.5 concentrations in the

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Pearl River Delta (PRD) region of China as a case study. Results show that the

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average uncertainty range of hourly PM2.5 concentrations is -28% to 57%, which can

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cover approximately 70% of the observed PM2.5 concentrations, while the traditional

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RFM underestimates the upper bound of the uncertainty range by 1% to 6%. Using a

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variance-based method, the PM2.5 boundary conditions and primary PM2.5 emissions

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are found to be the two major uncertainty sources in PM2.5 simulations. The new RFM

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better quantifies the uncertainty range in model simulations and can be applied to

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improve applications that rely on uncertainty information.

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TOC/Abstract art

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1. Introduction

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Air quality models (AQMs) simulate the fate and transport of air pollutants by

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numerically solving a set of differential equations. They have been widely used for

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regulatory analysis, attainment demonstration, air quality forecasting, and the study of

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pollution formation mechanism. However, due to the highly varied uncertainties in

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input data (e.g., emissions, boundary conditions, and meteorological data) and

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simplified treatments of atmospheric chemistry and physics, AQMs generally contain

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significant uncertainty, which might confound the reliability of management decisions

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and air quality forecasting.1,2 Uncertainty analysis is an efficient means of

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understanding the potential errors in model simulations, which could provide

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scientific information to help improve decision making for policy makers.3,4 More

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importantly, uncertainty analysis could also provide insights into the causes of

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uncertainty and thus direct effective pathways for AQMs improvement.

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A simple yet popular method for uncertainty analysis is the Monte Carlo

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Method (MCM).5–7 Traditionally, MCM requires a sufficiently large number of model

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runs (at least 1,000) with randomly sampled model inputs for an accurate

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quantification of the probability distribution. However, this method is not practical for

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complex AQMs due to the large computational resources required for each simulation.

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Alternative methods based on the functional approximation of complex AQMs have

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been developed to reduce the number of model runs. These include Response Surface

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Modeling (RSM),8 Stochastic Response Surface Modeling (SRSM)9,10 and the

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Polynomial Chaos method (PC).11 However, their computational cost is still a major

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concern since these methods still require hundreds of model runs, especially for cases

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with a large number of uncertainty inputs.

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Recently, the reduced-form model (RFM) based on the high-order decoupled

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direct method (HDDM)12,13 proved to be an efficient method in uncertainty analysis

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for AQMs.3,4,14–16 The RFM combines sensitivity coefficients derived from HDDM to

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represent model responses to perturbations of inputs, such as emission rates, chemical

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reaction rates and boundary conditions (BCs). It requires a far less computational cost

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than MCM and SRSM, but the accuracy is highly subjected to the limitation of

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HDDM. As a local sensitivity analysis approach, HDDM (with a maximal order of 2)

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is not reliable for approximating highly nonlinear responses to large perturbations of

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inputs. The accurate perturbation range of HDDM for nonlinear responses is generally

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below 50%, as revealed by previous studies,14,17,18 and is far less than 50% in our case

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study in the Pearl River Delta (PRD) region (Figure S1). In comparison, uncertainties

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of some model inputs can be as high as 300%,19–21 indicating that the RFM cannot

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accurately predict nonlinear responses to large perturbations. Although higher-order

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derivatives calculated by the Brute-Force Method (BFM) can theoretically increase

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the accuracy of the RFM, it is an inefficient approach. As large perturbations

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generally relate to extreme values in distributions or high percentiles (e.g., 95% and

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99%),22 the limitation of HDDM inevitably bring large biases in uncertainty analysis

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and applications that rely on uncertainty information. Improving the accuracy of the

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RFM is thus essential in uncertainty analysis. 5



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In this study, we develop a new approach to precisely quantify the uncertainty of

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AQMs. This approach combines a stepwise-based HDDM that runs extra HDDM

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simulations under different conditions to overcome the limitation of HDDM. This

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approach was applied to a case study of the uncertainty analysis of PM2.5 simulations

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in PRD, one of the most developed and polluted areas in China. Emissions and BCs

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were considered as uncertainty inputs because of their large uncertainties.23 As

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revealed by this study, the new approach enabled a better quantification of uncertainty,

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particularly the upper bound of distributions, and can also be applied to improve the

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accuracy of, for example, probabilistic air quality forecasting.

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2. Materials and methods

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2.1 Modeling system and input data.

Model simulations were conducted

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using the WRF/SMOKE-PRD/CMAQ modeling system.24,25 Detailed model

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configurations of CMAQ and WRF are shown in Table S1 and Figure S2 of the

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supporting information (SI). Three one-way nested domains were applied for CMAQ.

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Here, we focus on the third domain (D3), which covers the PRD region. The second

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domain (D2) provided BCs for D3. Considering that the stepwise-based HDDM was

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applied to both emissions and BCs, two periods in April and December 2013 that had

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different PM2.5 formation patterns (higher local emission contributions in April and

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higher regional contributions in December) were selected as simulation periods to

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validate the new approach.26

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The most up-to-date emissions inventory for Guangdong province in 2010

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was

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adopted for the Guangdong area.27 Annual emissions in Hong Kong provided by the

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Hong Kong Environment Protection Department, Multi-resolution Emission

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Inventory for China (MEIC, http://www.meicmodel.org) and the Regional Emission

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inventory in Asia (REAS, http://web.nies.go.jp/REAS/) were adopted for areas

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outside Guangdong province.

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Hourly observations of PM2.5 concentrations at 12 monitoring stations of the

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PRD Regional Air Quality Monitoring Network (PRDRAQM) and hourly

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observations of PM2.5 chemical composition were used to evaluate the PM2.5

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simulations.28 Detailed results of the evaluation are shown in the S2 of SI. Briefly,

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CMAQ generally captured the trends and chemical compositions of ambient PM2.5 in

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PRD and the errors were within the recommended “criteria” range.29

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2.2 A combined stepwise-based HDDM and RFM method.

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Eq 1 shows the

formulas for the traditional RFM (hereafter referred to as the RFM). 30

= + ∆ ×

, +

+ ∆ ∆ × ,,

∆ × ,

(1)

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Where is the predicted concentration with j sensitivity inputs perturbed simultaneously, ∆ is the fractional perturbation of input j, is the

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concentration in the base case, and , , , and ,, are the first- and

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second-order sensitivities of input j and the cross-sensitivity between inputs j and k.

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Limited by local sensitivities, eq 1 cannot precisely predict highly nonlinear

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PM2.5 responses to large perturbations of inputs. To overcome this limitation, we 7



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devised a new stepwise-based HDDM method (SB-HDDM), building on the 3-step

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approach proposed by Simon et al.31 The 3-step approach reruns two extra HDDM

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simulations to adjust the prediction but ignores the fact that responses to fix-length

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perturbations with different starting and ending points are different, such as nonlinear

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responses to a 50% reduction and to a 50% increase are different although they have

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the same length perturbation (refer to S3 of SI for more details). The new SB-HDDM

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considers the response difference and encompasses negative and positive

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perturbations. This is necessary because both negative and positive perturbations are

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involved in uncertainty analysis.

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Figure 1. The conceptual figure of the stepwise-based HDDM (SB-HDDM). Case1 and Case2 are HDDM simulations for positive perturbations, and Case3 and Case4 are HDDM simulations for negative perturbations. The figure shows positive perturbations as an example. The red line denotes the response approximated by the SB-HDDM. The black, blue, and green dashed line represents the responses approximated by the sensitivities in the base case (black dot), Case1 (blue dot) and Case2 (green dot), respectively. The gray, blue and green shadow bins indicate the perturbation ranges in which the responses are estimated by the base case, Case1 and Case2. X1 and Y1 are the switching points that determine these perturbation ranges (X2 and Y2 are the switching points for negative perturbations).

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negative, the SB-HDDM uses the base case, Case3 and Case4 to predict the model

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response; otherwise, the base case, Case1 and Case2 are used. For illustration, a

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Figure 1 describes the conceptual figure of the SB-HDDM. If the perturbation is

prediction of the response to positive perturbation ! is shown as an example (eq 2). 8


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Sensitivities at the base level are used to predict the concentration change between the

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base case and the switching point X1. For the change between X1 and Y1 (denoted as

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∆C# $ ) and the additional change beyond Y1 (denoted as ∆C# % ), sensitivities in

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Case1 and Case2 are used, respectively. Considering that the response might be

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nonlinear, ∆C# $ is the sum of the absolute response to negative perturbation b1

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and the response to positive perturbation b2. Similarly, ∆C# % is the sum of the

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absolute response to negative perturbation c1 and the response to positive perturbation

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c2. Note that if the predicted point lies between X1 and Case1 (Figure S6), the change

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between X1 and the predicted point should be the difference between the response to

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negative perturbation b2 and the response to negative perturbation b1. The

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perturbations b1, b2, c1 and c2 are determined according to the values of X1, Y1 and

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the predicted point and should be normalized to the coordinate system of Case1 or

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Case2 by dividing by 1 + '$ or 1 + '% . ∆()*+,, = × +

/ 0 + . × / +

/ 2 + 1/ × / +

/

× − . × / +

× / 0 − 1/ × / +

× / 2

! 567 |!| ≤ |:1| a=4 :1 567 |!| > |:1|

0 567 |!| ≤ |:1|

A New Combined Stepwise-Based High-Order Decoupled Direct and

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