application of surface spline interpolation method in...

Research ArticleApplication of Surface Spline Interpolation Method inParameter Estimation of a PM25 Transport Adjoint Model

Ning Li12 Xianqing Lv 2 and Jicai Zhang 3

1School of Science Dalian Jiaotong University Dalian 116028 China2Physical Oceanography LaboratoryCIMST Ocean University of China and Qingdao National Laboratory for Marine Science andTechnology Qingdao 266003 China3Institute of Physical Oceanography Ocean College Zhejiang University Zhoushan 316000 China

Correspondence should be addressed to Jicai Zhang jicai zhang163com

Received 31 March 2018 Accepted 21 June 2018 Published 5 July 2018

Academic Editor Shuo Wang

Copyright copy 2018 Ning Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new method for the estimation of initial conditions (ICs) in a PM25 transport adjoint model is proposed in this paper In thismethod we construct the field of ICs by interpolating values at independent points using the surface spline interpolation Comparedto the traditionally used linear interpolation the surface spline interpolation has an advantage for reconstructing continuoussmooth surfaces The method is verified in twin experiments and the results indicate that this method can produce better invertedICs and less simulation errors In practical experiments simulation results showgood agreementwith the ground-level observationsduring the 22nd Asia-Pacific Economic Cooperation summit period demonstrating that the new method is effective in practicalapplication fields

1 Introduction

In recent years air pollution has escalated to hazardous levelsin Chinese cities Among numerous types of air pollutionPM25 (fine particles with diameter of 25120583m or less) isconsidered the most threatened kind to life expectancyand public health (eg [1ndash4]) Therefore simulation andprediction of PM25 pollution are always an area of interest toresearchers

An increasing number of atmospheric numerical modelshave been conducted and publicly available in various studiesfor more in-depth understanding of the physical chemicaland dynamical processes concerned with PM25 pollution(eg [5ndash12]) Initial condition (IC) plays an important rolein the model researches Providing the best possible ICsis essential for the success of the models Existing studiespresented several methods to obtain the IC The initialcondition was obtained from the National MeteorologicalCenter (NMC) global data and enhanced by rawinsonde andsurface observations by Liu et al [11] A background value(a small constant) was given to the model with the first few

days as the spin-up period tominimize the influence of initialconditions by Fu et al [12] Another efficient and accurateway to estimate the ICs is through data assimilation using theadjoint method (eg [13ndash17])

Data assimilation provides a configuration for combiningobservations and models to form an optimal initial modelstate In this method uncertain model parameters can beconstrained by minimizing the distance between the modelsimulations and observations In practice how to reducethe influence of ill-posedness caused by excessive controlparameters has been a key part of data assimilation andparameter estimation Several studies have proved that it canbe effectively solved by applying independent point scheme(IPS) (eg [13ndash17]) In detail several grids are selected asindependent points (IPs) in the space domain values of theICs at the IPs can be optimized while those at other grids canbe calculated through a certain interpolationmethodwith thevalues at the IPs Thus interpolation is another importantproblem that remains to be studied

Referring to previous studies Cressman interpolation(hereafter abbreviated as CI) is preferred for the adjoint

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6231271 11 pageshttpsdoiorg10115520186231271

2 Mathematical Problems in Engineering

model due to easy accessibility Although the adjoint modelwith the CI leads to satisfying results in general thereconstructed IC field is unsmooth around the indepen-dent points [18] Therefore a more reasonable interpolationmethod needs to be found to combine with the adjointmethod

The surface spline interpolation with a special type ofpiecewise polynomial called spline is widely used as onepopular technique for data interpolation The surface splineis a powerful tool for interpolating irregular continuous geo-logical or other surfaces and is often better than polynomialinterpolation because the interpolation error can be madevery small [19] Many real applications of the surface splinemethod indicate that the results are usually appropriate foroceanographic and meteorological contexts and it is particu-larly good for inferring smooth structure from scattered dataAs stated in the work of Harder et al [20] the surface splinewas proposed to interpolate wing deflections and computingslopes for aeroelastic calculations Based on the surface splineinterpolation a newmathematical model for tropical cyclonewind speeds is proposed in [21] Compared with the earlierwind model with linear interpolation the surface splinemodel could produce a more accurate wind estimate In [22]Yaghoobi et al described a scheme based on a cubic splineinterpolation which is applied to approximating the variable-order fractional integrals and is extended to solving a classof nonlinear variable-order fractional equations with timedelay Guo and Pan [18] validated this new IP scheme withtwin experiments and the results showed that the prescribednonlinear distribution of bottom friction coefficients arebetter inverted with the surface spline interpolation Due toits proved superior performance the surface spline interpola-tion (SSI) is applied to inversion of ICs with a PM25 transportadjoint model in this study

The paper is organized as follows Section 2 provides thedetailed descriptions of the numerical model and imple-mentation of SSI Twin experiments and practical experi-ments are carried out to evaluate the performance of SSI ininversion of ICs in Sections 3 and 4 respectively Finallysome key conclusions drawn from the work are presented inSection 5

2 Numerical Model and Settings

21 Forward Model The governing equations of the two-dimensional PM25 transport model are as follows

120597119862120597t + 119906120597119862120597119909 + V120597119862120597119910 = 120597120597119909 (119860119867120597119862120597119909 ) + 120597120597119910 (119860119867120597119862120597119910 )

+ 119878(1)

119862 (1199050 119909 119910) = 1198620 (119909 119910) (2)

where C denotes the ground-level PM25 concentration uand v are the horizontal wind velocity in x-coordinate andy-coordinate respectively 119860119867 is the horizontal diffusivitycoefficient S is the value of the source and sink and C0is theinitial condition (IC) to be estimated in this study

The boundary conditions are set as constant at the inflowboundary ΓIN and no gradient boundary conditions at theoutflow boundary ΓOUT

120597119862120597119905 = 0 on ΓIN (3)

120597119862120597997888rarr119899 = 0 on ΓOUT (4)

22 Adjoint Model The adjoint model is defined to calculatethe gradients of the cost function with respect to variousinput parameters which incorporate all physical processesincluded in the governing model to obtain the minimizationof the distance between the model output and observationsBased on the governing equations (1)ndash(4) its adjoint modelcan be constructed as follows

First the distance is expressed by the cost function whichis defined as

119869 = 12 intΣ119870(119862 minus 119862obs)2 d120590 (5)

where Σ denotes the set of the observations in time and spacedomain and C andCobs are the simulated and observed PM25concentrations respectively And K is the weighting matrixand should be the inverse of observation error covariancematrix theoretically K can be fixed simply assuming that theerrors in the data are uncorrelated and equally weighted Inthe present model K is 1 when the observations are availableand 0 otherwise

Then the Lagrangian function is constructed based on thetheory of Lagrangianmultipliermethod and can be expressedas

119871 = 119869 + int119901 times [120597119862120597t + 119906120597119862120597119909 + V120597119862120597119910 minus 120597120597119909 (119860119867120597119862120597119909 )

minus 120597120597119910 (119860119867120597119862120597119910 ) minus 119878] d120590(6)

where p is the adjoint variable of CAccording to the typical theory of Lagrangian multiplier

method we have the following first-order derivatives ofLagrangian function with respect to all the variables andparameters

120597119871120597119901 = 0 (7)

120597119871120597119860119867 = 0120597119871120597119878 = 0

(8)

120597119871120597119862 = 0 (9)

Equation (7) gives the governing equation (1) of theforward model The adjoint equation can be developed fromequation (9) which is given as follows

Mathematical Problems in Engineering 3

120597119901120597119905 + 119906120597119901120597119909 + V120597119901120597119910 + 120597120597119909 (119860119867 120597119901120597119909) + 120597120597119910 (119860119867120597119901120597119910)

minus 119870 (119862 minus 119862obs) = 0(10)

Equations of the adjoint model and the numerical scheme offorward and adjoint equations are similar to Wang et al [13]

23 Optimization of Independent ICs In this adjoint modelthe optimal parameters should be explored to minimize thecost function J Minimization of the cost function is imple-mented through parameter optimization This optimizationroutine is performed through iterative integrations of thegoverning and adjoint equations

According to the steepest decent (SD) method optimiza-tion of the control variable is conducted as follows

119896119894+1119897 = 119896119894119897 minus 120572119894 sdot 119901119894119897 (11)

In the above-mentioned equations i denotes the iterationstep kl is the l-th parameter 120572 is the step factor and N is thenumber of independent parameters It should be noted that pis the normalized gradient of J in respect to 1198620119897

119901119897 = 1205971198691205971198620119897radicsum119873119897=1 (1205971198691205971198620119897)2 (12)

The optimized gradients of the IC can be obtained bysolving the equation 1205971198711205971198620 = 0 We arrive at

1205971198691205971198620 = (120597119901120597119905 )0 + (nabla sdot 997888rarr119881119901)0 + (119860119867nabla2119901)0 (13)

If the ICs at some grid points are selected as independent ICswe can interpolate these independent ICs to obtain the wholeIC by the following equation

119896119894119895 = 119873sum119897=1

119882119897119894119895119896119897 (14)

where kij is the parameter at the grid point (ij) kl is the l-thindependent parameter and 119882119897119894119895 is the coefficient of inter-polation Then the optimized gradients of IC with respectto the l-th independent IC are

120597 and1198691205971198620119897 =1205971198691205971198620119897 +sum

119894119895

119882119897119894119895 1205971198691205971198620119894119895 (15)

where (ij) denotes grid points within the influence radius ofthe l-th independent IC

Referring to previous studies the parameters can beobtained by Cressman interpolation (hereafter abbreviated asCI) and the coefficient is traditionally calculated by Zhang etal [24]

119882119897119894119895 = 119908119897119894119895sum119873119897=1 119908119897119894119895 (16)

where119908119897119894119895 = (1198772minus(119903119897119894119895)2)(1198772+(119903119897119894119895)2) is the Cressmanweight119903119897119894119895 is the distance between the grid point (ij) and the l-th

independent point and R is the influence radius set based onexperience

In this study the surface spline interpolation is adoptedand the SSI coefficients are shown as follows

Assuming that ICs are known at N points the whole ICfiled can be constructed with a variation of the surface splineThe IC at grid points (ij) is expressed as [25]

1198620119894119895 = 119873sum119897=1

119860 119897(11990321198941198951198971198772 ln11990321198941198951198971198772 + 1 minus 11990321198941198951198971198772 ) (17)

where 119903119894119895119896 is the distance between the k-th independent pointand grid (ij)R is a prescribed radius and119860119896 is the coefficientto be determined Actually the relationship between theknown IC and the others can be expressed in matrix formas

119882119883 = 119884 (18)

The detailed forms of the three matrixes are given as

119883 = (119860 119897)119873times1119884 = (1198620119897)119873times1119882 = (119908119894119895)119873times119873

(19)

where 119908119894119895 = 1 119894=119895

(11990321198941198951198772)ln(1199032119894119895119877

2)+1minus11990321198941198951198772 119894 =119895 and 119903119894119895 is the distance

between the i-th and j-th independent points After calcula-tion we get

119860119896 = 119873sum119897=1

1198721198961198971198620119897 (20)

where119882minus1 = (119872119896119897)119873times119873 By Plugging (17) into (20) we arriveat

1198620119894119895 = 119873sum119897=1

1198821198971198941198951198620119897 (21)

where 119882119897119894119895 = sum119873119896=1119872119896119897((11990321198941198951198961198772)ln(11990321198941198951198961198772) + 1 minus 11990321198941198951198961198772) isthe coefficient corresponding to the l-th independent IC atthe grid (ij)

24 Independent Point Scheme Uniform selection of theindependent point over the survey region is quite commonlyused in the adjointmodel (eg [14ndash17]) Although this select-ing strategy can be realized easily but it is not a good choicedue to little consideration about the physical characteristics ofthe research object [18] As we know the spatial distributionof PM25 presents geographic features If the independent ICsare selected according to the spatial distribution of PM25the selection will be more reasonable At the area wherePM25 concentration is high more independent points will beneeded Moreover fewer independent points will be enoughat the area with low PM25 concentration

High concentration of PM25 was found in three areasof China (1) the North China Plain including two of


80 90 100 110 120 130 14070Longitude(E)

15

20

25

30

35

40

45

50

55

Latit

ude(

N)

Figure 1 Map of computing area and distribution of independentpoints (dots)

Chinarsquos four municipalities Beijing and Tianjin southernHebei Province western Shandong Province and northernHenan province (2) the Northeast China Plain and (3) theTaklimakan desert (eg [3 4]) Therefore grid points inthe above-mentioned high concentration area are selected asindependent points fromeach 5∘times5∘ areaGrid points in otherareas are selected as independent points from each 6∘times6∘ areaIn this study 67 independent points are selected and shownin Figure 1

25 Settings The area of modeling computation is Chinacovering 70∘-140∘E and 15∘-55∘N with a space resolutionof 05∘times05∘ There are 141 times 71 grids totally in the areaThe simulation period is 168 hours and the inverse integraltime step is 10 minutes The wind datasets used in ourexperiments are the NCEP Final Analysis (GFS-FNL) withspatial resolution of 25 degrees and temporal resolutionevery 24 hours We get the 05-degree data by using bilinearinterpolation for space and liner interpolation for time andkeep mass conservation of air The background value is fixedas 150120583gm3 Inflow boundary values are fixed equal to thebackground values The horizontal diffusion coefficient 119860119867is fixed as 1000 m2s

Seventy-four major Chinese cities are treated as theobservation positions Labeled as ldquoassimilated citiesrdquo theobservations at these cities are assimilated in the optimiza-tion procedure Meanwhile Jinan (JNa) Zhengzhou (ZhZ)Shenyang (SY) Quanzhou (QZ) Hangzhou (HaZ) Kunming(KM) Chengdu (CD) and Beijing (BeJ) are selected asldquochecked citiesrdquo to independently evaluate the ability of themodelrsquos simulation and inversion All of the 82 selected citiesare shown in Figure 2 PM25 concentrations at the 82 majorChinese cities are taken as the observations every two hoursin the present study

3 Numerical Experiments

Results obtained in previous studies indicated that the SSIproduced high efficiency to interpolate a smooth and exactlyfitting surface based on either scattered data or rectangular-grid data According to Guo et al [18] the inverted nonlinear

Figure 2 Map of computing area and the observation locations of82 citiesThe green dots denote the 82 cities used in the experimentsFull names of these cities can be found in [23]

parameter distribution was smoother than the surface con-structed by the traditionally used linear interpolation ina two-dimensional tidal adjoint model To evaluate theperformance of the SSI totally 12 twin experiments (TEs) arecarried out and classified in two groups for the comparisonwith the CI results in the PM25 adjoint model

For each experiment we run the forward model witha prescribed IC field The model-generated PM25 concen-trations at grid points of the observing positions shownin Figure 2 are regarded as the ldquoobservationsrdquo Consideringthe influence of the noises of in situ observations we addan estimated maximum error of 5 to the ldquoobservationsrdquoThe forward model is launched with an initial guess of ICsto obtain the simulations Thereafter the adjoint model isdriven by the discrepancy between simulations and ldquoobserva-tionsrdquo With variables obtained from the forward and adjointmodels gradients with respect to ICs are calculated ICs atthe independent points can be optimized according to (14)The ICs at other grids are determined by interpolation ofthe values at the independent points Repeat the procedurewith ICs being optimized until certain criteria are met Theoptimization algorithm used in this work is the steepestdecent method And the criterion is that the number ofiteration steps is equal to 300 at which the cost function canreach the minimum without large fluctuation The flowchartof the process is shown in Figure 3

Performance of the SSI or CI in combination of theadjoint model is evaluated by the mean absolute gross error(MAGE) [26] calculated as follows

MAGE = 1119873119873sum119894=1

1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816 (22)

where N is the number of observations andM and O are themodeled and observed results respectively

31 Results and Discussions of Group One In Group one twoIC fields are prescribed (see Figure 6) a paraboloid openingdownward (TE1) and a folding line distribution (TE2) TheTEs focus on evaluating feasibility of the SSI in inversion ofthe ICs


Table 1 Error statistics of the TEs in Group one before and after assimilation

Experiment J3001198691a K1a (120583gm3) K2a (120583gm3) K3a(120583gm3) K4aBefore After Before After Before After Before After

SSI (TE1) 950e-4 3281 097 2903 089 633 114 1334 609CI (TE1) 196e-3 3281 134 2903 113 633 143 1334 666SSI (TE2) 173e-4 5257 049 4922 036 2391 773 3925 1464CI (TE2) 760e-4 5257 104 4922 112 2391 787 3925 1517a119869300 is the final value of cost function and J1 is the initial value of cost function K1 is the MAGEs between simulated values and ldquoobservationsrdquo in ldquoassimilatedcitiesrdquo K2 is the MAGEs between simulated values and ldquoobservationsrdquo in ldquochecked cities K3 and K4 are the MAGEs and MNGEs between prescribed andinverted IC respectively

Observations

Calculate cost function

Run adjoint model

Adjust independent parameter

Interpolating the independent parameters to other girds

Judge if the criterion is satisfied

End

Begin

Initialize modelParameters

Run forward model

No

Yes

Figure 3 Flowchart of the experiments

Figure 4 presents the iteration histories of the normalizedcost function and MAGEs between the prescribed andinverted IC fields As can be seen the declining normalizedcost function indicates that the misfit between the observa-tions andmodel results is decreased and the observation datahas been assimilated efficiently Meanwhile the decreasedMAGEs suggest that the ICs are indeed optimized and gettingclose to the exact solution

Results of the four TEs are summarized in Table 1 InTE1 it is apparent that the SSI shows an advantage overthe CI in the model results This is demonstrated by thenormalized cost function values and the MAGEs betweensimulated values and ldquoobservationsrdquo in ldquoassimilated citiesrdquoand ldquochecked citiesrdquo In addition the SSI enables the MAGEsbetween the prescribed and inverted IC reduced by 8199which is 458 better than the results obtained from the CI

To further quantify the discrepancy between the pre-scribed and inverted ICs the mean normalized gross error(MNGE) [26] is used and calculated as follows

MNGE = 1119873119873sum119894=1

[1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816119874119894 ] (23)

The calculated results are listed in Figure 5 The MNGEsbetween the prescribed and inverted paraboloid openingdownward at the initial iteration step are 1334 which areintroduced by the difference between the initial guess valuesand the true values of the adjusted model ICs After theassimilation the CI gets theMNGEs reduced to 666 whilethe SSI gets the MNGEs reduced to 609

In fact it is a statement that the SSI provides betterinterpolation quality Further evidence to support the con-clusion can be seen through the inversion IC showed inFigure 6 As can be seen the IC fields inverted by the adjointmodel combined with both methods are consistent with theprescribed field The SSI results display smoother patternconfirming that the SSI shows superiority in reconstructingIC fields in the PM25 adjoint model

The same conclusions can be drawn from analysis ofthe TE2 results The SSI results are superior to the CIresults manifested by the normalized cost function MAGEsbetween simulated values and ldquoobservationsrdquo and MAGEsbetween prescribed and inverted IC shown in Table 1 Forthe SSI simulations the MAGEs between simulated valuesand ldquoobservationsrdquo in the ldquochecked citiesrdquo before and afterthe assimilation are about two percent less than the CI sim-ulations The MNGEs between the prescribed and invertedfolding line distribution at the initial iteration step are 3925After the assimilation the CI gets the MNGEs reduced to1517 while the SSI gets the MNGEs reduced to 1463Although the SSI gets the MAGEs and MNGEs betweenprescribed and inverted ICs slightly better than the CI asmoother IC inversion also highlights the advantages of theSSI (Figure 6)

32 Results and Discussions of Group Two The ill-posednesscaused by incomplete number of the observation data is thekey part needed to be solved in inversion problem Feasibilityof the SSI in inversion of the ICs is investigated by anothergroup of twin experiments (Group two) under this assump-tion According to the geographic distribution of the original74 observation stations the number of assimilated stations is


50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)

Figure 4 Iteration histories of the normalized cost function and the MAGEs between the prescribed and inverted ICs for Group 1

005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)

50 100 150 200 250 3000Iterations Steps

015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)

Figure 5 Iteration histories of the mean normalized gross errors (MNGEs) between the prescribed and inverted ICs for Group 1


Table 2 Error statistics of TE3 and 4 and observations at 49 sites are assimilated in the TEs

Experiment 1198693001198691a K1a (120583gm3) K2a (120583gm3) K3a(120583gm3) K4aBefore After Before After Before After Before After


(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70

gmiddotＧ-3gmiddotＧ-3

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)

Figure 6 Two prescribed IC fields in twin experiments (a) a paraboloid opening downward (TE1) and (b) a folding line distribution (TE2)The IC fields inverted combined with (c) the SSI and (e) the CI in TE1 (d) the SSI and (f) the CI in TE2

artificially reduced by a third and a half respectivelyThe twokinds of prescribed IC fields in Group one are inverted in TE3and TE4 by assimilating the observations at 49 observationsites Meanwhile data at 37 sites are taken as the observationsin TE5 and TE6 Error statistics of the TEs in Group two aresummarized in Tables 2 and 3The assimilation procedures oftheMAGEs andMNGEs between the prescribed and invertedICs are shown in Figure 7

As shown in Tables 2 and 3 the cost function can atleast reach 10-2 of their initial value indicating the successfulICs inversion in all the TEs Our direct comparison of theerror statics shown in Tables 1ndash3 indicates that the SSI resultsremainmore accurate than the CI results in all the TEs of thisgroupTherefore the following analyses are carried out basedon the SSI results

When the observations at 49 observation sites are assim-ilated in the adjoint model the MAGEs between the pre-scribed and inverted folding line distribution in TE4 are atleast 343 better than that obtained when 37 observation

sites are adopted in TE6 Meanwhile the MNGEs in TE4get a 064 advantage than that in TE6 Compared with theinversion of the folding line distribution the inversion of theparaboloid opening downward is obviously affected by inad-equate observations The MAGEs between the prescribedand inverted paraboloid opening downward are 2733worsethan that obtained when more observations are used Andthe MNGE in TE3 is 148 worse than that in TE5 Thisis consistent with the conclusion obtained by Chertok andLardner in [27] that the ICs inversions can be more efficientif the provided observations from sufficient number of sitescan be used

Further comparison of the ICs inversion errors demon-strate that the SSI results get less MAGEs and MNGEsthan the CI results when fewer observations are assimi-lated in the model All the experiments indicate that theadjoint model combined with the SSI can much effectivelyovercome the ill-posedness caused by inadequate observa-tions


Table 3 Error statistics of TE5 and 6 observations at 37 sites are assimilated in the TEs



CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

Figure 7 Iteration histories of the mean normalized gross errors (MNGEs) between the prescribed and inverted IC for Group 2

4 Practical Experiments and Discussion

Performance of the adjoint model combined with thetwo methods is evaluated in a practical context Hourlyground-level PM25 observations (from 5 November to 11November 2014) obtained from the Data Center of Ministryof Environmental Protection of the Peoplersquos Republic ofChina (httpdatacentermepgovcn) are assimilated into themodel These observations were measured based on thetapered element oscillating microbalance (TEOM) method(eg [10 28]) Model-produced PM25 concentrations areused for evaluation of the simulated results

The inverted ICs are shown in Figure 9 On the wholethey show similar patterns that ICs including the northeast-ern eastern central and northwestern regions are exposed to

hazardous levels of PM25 which reflects the observations at5 November 000 shown in Figure 8 However according tothe MAGEs between the observations and simulations in allthe observation cities at the initial time shown in Table 4 theadjoint model with SSI gets a 206 advantage over that withCI Meanwhile with a closer inspection it can be found thatthe IC surface obtained with SSI is much smoother while thatwith CI has bumps and depressions around several grids

Statistical errors are shown in Table 4 which demon-strates that the PM25 simulation is successful with bothinterpolation methods The normalized cost function of theSSI and CI is 0280 and 0304 respectively The SSI gets theMAGEs between simulated values and observations at theinitial time to be 206 better than the CI results MeanwhiletheMNGEs between simulated values and observations at the


Table 4 Error statistics of the practical experiment before and after assimilation


SSI 280e-1 3787 2082 4293 2215 4023 2427 4888 3727CI 304e-1 3787 2165 4293 2340 4023 2510 4888 3629a119869300 is the final value of cost function and J1 is the initial value of cost function K1 and K2 are the MAGEs between simulated values and observations inassimilated cities and checked cities respectively K3 and K4 are theMAGEs andMNGEs between the observations and simulated results in all the observationcities at the initial time

DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3

Figure 8 Observations at 5 November 000

initial time have a 098 advantage indicating superiority ofthe SSI This statement is further validated by the MAGEsbetween observations and simulations in ldquoassimilation citiesrdquoand ldquochecked citiesrdquo Simulations obtained with the SSI areused for further analysis of the model performance

The values of mean simulations normalized mean bias(NMB) normalized mean error (NME) and correlationcoefficients are calculated as the statistical measures (eg[26 28]) (domain wide averages) and plotted as an hourlytime series in Figure 9 A closer inspection of the resultsindicates that the domain mean simulations show goodagreement with the domain mean observations of the assim-ilation windows As can be seen from Figure 10 the cor-relation coefficients NMB and NME values at initial timeare 06289 -5486 and 5536 respectively indicating theunderestimations of PM25 Since tapered element oscillatingmicrobalance analyzers (TEOMs which are commonly usedin the EPA-Chinarsquos air quality monitoring network) measure-ments for PM25 should be considered as lower limits becauseof volatilization of soluble organic carbon species in thedrying stages of the measurement [26] the undersimulationis likely to be more severe than this evaluation suggests Thismay be caused by the lack of the observations in the firstfew hours In addition the NMB ranges from -2791 (200Beijing time 5 November) to 1193 (1600 Beijing time 5November) most of which is between -10 and 10 TheNME values are approximate to 2518 which is the NMEvalue of all the observations indicating the stability of theadjoint model during the selected time Meanwhile the cor-relation coefficients between the domain mean simulationsandmean observations aremost between 080 and 095Theseresults show that the scheme can effectively assimilate thePM25 observations

The spatial distribution of simulated averaged PM25concentrations during assimilation window across Chinais shown in Figure 11 Coal combustion leads to seasonalhigh level of PM25 concentrations in Northeast China inwinter which rose to 120-160 mgm3 during the assimilationwindow Meanwhile the concentrations are slightly largein middle and east coast of China which is in conformitywith industrial development Mean PM25 concentrations ofBeijing are about 60 mgm3 half values in the same periodof previous years [29] Air quality in Beijing and nearbyprovinces are better in the assimilation window a periodwhich coincided with the introduction of an emergencyemissions-reduction strategy during the Asia-Pacific Eco-nomic Cooperation (APEC) summit in November Abovespatial distribution were consistent with findings from pre-vious studies [30]

5 Summary

The independent point strategy serves as a satisfactorysolution to overcome the ill-posedness of the inverse prob-lem caused by excessive control variables Spatially varyingparameter is constructed by interpolating values at a groupof independent points under this scenario In this paperthe independent point strategy is described using the SSI asan alternative to the traditionally used linear interpolationThe adjoint model combined with the SSI is evaluated insimulation of PM25 pollution in China by optimizing theICs

In twin experiments the SSI results outperform theCI in the model simulations and inversion of nonlineardistribution of ICs The IC fields reconstructed by the SSIare smoother than that by the CI In addition the SSI isverified to be more effective in solving the ill-posedness ofthe inversion problem than the CI In practical experimentsICs are optimized and the PM25 concentrations in Chinaare simulated by assimilating the ground-level observationsduring the 22th APEC The comparatively accurate PM25distribution features demonstrate that the SSI plays a positiverole in ICs inversion and simulations in practical applica-tion

In conclusion the SSI inversion strategy is feasible andbetter than the CI and can improve numerical results tosome extentWe can obtainmuch smoother initial conditionswhich invertedwith the SSImethod and therefore aremore inline with reality And the improved IP scheme combined withthe SSI is showing promise in the numericalmodel parameterinversions


14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)

Figure 9 Inverted ICs obtained with the adjoint model combined with (a) SSI and (b) CI

Mean(model)Mean(obs)

r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

Time since assimilation start

minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB

Figure 10 Comparison of the modeled and observed PM25 con-centrations based on the results from the SSI temporal variation ofdomain mean NMB NME and correlation

Data Availability

All relevant data are available within the paper The winddatasets are available from the NCEP Final Analysis (GFS-FNL) with spatial resolution of 25 degrees and temporalresolution every 24 hours Hourly ground-level PM25 obser-vations are available from the Data Center of Ministry ofEnvironmental Protection of the Peoplersquos Republic of China(httpdatacentermepgovcn)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160

Figure 11 Spatial pattern of the average PM25 concentrations(120583gm3)

Acknowledgments

The authors highly appreciate the help and suggestionsof the manuscript given by Daosheng Wang And thanksare due to the surface spline numerical program providedby Haidong Pan This work was partly supported by theNational Key Research and Development Plan throughGrants 2017YFC1404000 and 2017YFA0604100 the NaturalScience Foundation of Zhejiang Province through GrantLY15D060001 and the Education Department Science Foun-dation of Liaoning Province of China (no JDL2017019)

References

[1] L Han W Zhou andW Li ldquoCity as a major source area of fineparticulate (PM25) in Chinardquo Environmental Pollution vol 206pp 183ndash187 2015

[2] Y-RMaQ Ji andY Fan ldquoSpatial linkage analysis of the impactof regional economic activities on PM25 pollution in ChinardquoJournal of Cleaner Production vol 2139 pp 1157ndash1167 2016

[3] H Zhang Z Wang and W Zhang ldquoExploring spatiotemporalpatterns of PM25 in China based on ground-level observationsfor 190 citiesrdquo Environmental Pollution vol 216 pp 559ndash5672016

[4] L Han W Zhou and W Li ldquoCity as a major source area offine particulate (PM25) in Chinardquo Environmental Pollution vol2139 pp 1157ndash1167 2016


[5] B K Pun and C Seigneur ldquoSensitivity of particulate matternitrate formation to precursor emissions in the California SanJoaquinValleyrdquoEnvironmental ScienceampTechnology vol 35 no14 pp 2979ndash2987 2001

[6] S Yu R Mathur K Schere et al ldquoEvaluation of real-time PM25forecasts and process analysis for PM25 formation over theeastern United States using the Eta-CMAQ forecast model dur-ing the 2004 ICARTT studyrdquo Journal of Geophysical ResearchAtmospheres vol 113 no 6 Article ID D06204 pp 304ndash3122008

[7] C Seigneur ldquoCurrent status of air quality models for particulatematterrdquo Journal of the AirampWasteManagement Association vol51 no 11 pp 1508ndash1521 2001

[8] D K Henze J H Seinfeld and D T Shindell ldquoInversemodeling and mapping US air quality influences of inorganicPM25 precursor emissions using the adjoint of GEOS-ChemrdquoAtmospheric Chemistry and Physics vol 9 no 16 pp 5877ndash59032009

[9] L Zhang L Liu and Y Zhao ldquoSource attribution of PM25pollution over North China using the adjoint methodrdquo AGUFall Meeting Abstracts vol 1 2014

[10] S Capps D K Henze and A G Russell ldquoQuantifying relativecontributions of global emissions to PM25 air quality attain-ment in the USrdquo AGU Fall Meeting Abstracts vol 1 2011

[11] S C Liu S A McKeen E-Y Hsie et al ldquoModel study oftropospheric trace species distributions during PEM-West ArdquoJournal of Geophysical Research Atmospheres vol 101 no 1 pp2073ndash2085 1996

[12] K Fu and D Liang ldquoThe conservative characteristic FDmethods for atmospheric aerosol transport problemsrdquo Journalof Computational Physics vol 305 pp 494ndash520 2016

[13] D Wang N Li Y Shen and X Lv ldquoThe parameters estimationfor a PM25 transportmodel with theAdjointmethodrdquoAdvancesin Meteorology vol 9 Article ID 9873815 pp 1ndash13 2016

[14] D Wang A Cao J Zhang D Fan Y Liu and Y ZhangldquoA three-dimensional cohesive sediment transport model withdata assimilation Model development sensitivity analysis andparameter estimationrdquo Estuarine Coastal and Shelf Science2017

[15] D Wang J Zhang Y P Wang et al ldquoA methodology forestimating the parameters in three-dimensional cohesive sed-iment transport models by assimilating in situ observationswith the adjoint methodrdquo Journal of Atmospheric and OceanicTechnology vol 34 no 7 pp 1469ndash1482 2017

[16] D Wang J Zhang X He et al ldquoParameter estimation fora cohesive sediment transport model by assimilating satelliteobservations in the Hangzhou Bay Temporal variations andspatial distributionsrdquoOceanModelling vol 121 pp 34ndash48 2018

[17] N Li Y Liu X Lv J Zhang and K Fu ldquoThe High OrderConservative Method for the Parameters Estimation in a PM25transport adjointmodelrdquoAdvances inMeteorology vol 2017 pp1ndash13 2017

[18] Z Guo H Pan W Fan and X Lv ldquoApplication of surfacespline interpolation in inversion of bottom friction coefficientsrdquoJournal of Atmospheric and Oceanic Technology vol 34 no 9pp 2021ndash2028 2017

[19] G D Knott ldquoInterpolating cubic splinesrdquo Progress in ComputerScience and Applied vol 11 pp 319ndash321 1999

[20] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972

[21] J S Wijnands G Qian and Y Kuleshov ldquoSpline-basedmodelling of near-surface wind speeds in tropical cyclonesrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 19-20pp 8685ndash8707 2016

[22] S Yaghoobi B P Moghaddam and K Ivaz ldquoAn efficient cubicspline approximation for variable-order fractional differentialequations with time delayrdquo Nonlinear Dynamics vol 87 no 2pp 815ndash826 2017

[23] W J Qu R Arimoto X Y Zhang et al ldquoSpatial distributionand interannual variation of surface PM10 concentrations overeighty-six Chinese citiesrdquo Atmospheric Chemistry and Physicsvol 10 no 12 pp 5641ndash5662 2010

[24] J Zhang and Y P Wang ldquoA method for inversion of periodicopen boundary conditions in two-dimensional tidal modelsrdquoComputerMethods AppliedMechanics and Engineering vol 275pp 20ndash38 2014

[25] D Liu J Zhang and S Wang ldquoConstrained fitting of faultedbedding planes for three-dimensional geological modelingrdquoAdvances in Engineering Software vol 33 no 11-12 pp 817ndash8242003

[26] Y Zhang M Bocquet V Mallet C Seigneur and A BaklanovldquoReal-time air quality forecasting part I history techniquesand current statusrdquo Atmospheric Environment vol 60 pp 632ndash655 2012

[27] D L Chertok and R W Lardner ldquoVariational data assimilationfor a nonlinear hydraulic modelrdquo Applied Mathematical Mod-elling vol 20 no 9 pp 675ndash682 1996

[28] Y Zhang M Bocquet V Mallet C Seigneur and A BaklanovldquoReal-time air quality forecasting part II state of the sciencecurrent research needs and future prospectsrdquo AtmosphericEnvironment vol 60 pp 656ndash676 2012

[29] K B He F Yang Y Ma et al ldquoThe characteristics of PM25 inBeijing Chinardquo Atmospheric Environment vol 35 no 29 pp4959ndash4970 2001

[30] J L Hu YWang Q Ying and H Zhang ldquoSpatial and temporalvariability of PM25 and PM10 over the North China Plain andthe Yangtze River Delta Chinardquo Atmospheric Environment vol95 pp 598ndash609 2014

Hindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


model due to easy accessibility Although the adjoint modelwith the CI leads to satisfying results in general thereconstructed IC field is unsmooth around the indepen-dent points [18] Therefore a more reasonable interpolationmethod needs to be found to combine with the adjointmethod

The surface spline interpolation with a special type ofpiecewise polynomial called spline is widely used as onepopular technique for data interpolation The surface splineis a powerful tool for interpolating irregular continuous geo-logical or other surfaces and is often better than polynomialinterpolation because the interpolation error can be madevery small [19] Many real applications of the surface splinemethod indicate that the results are usually appropriate foroceanographic and meteorological contexts and it is particu-larly good for inferring smooth structure from scattered dataAs stated in the work of Harder et al [20] the surface splinewas proposed to interpolate wing deflections and computingslopes for aeroelastic calculations Based on the surface splineinterpolation a newmathematical model for tropical cyclonewind speeds is proposed in [21] Compared with the earlierwind model with linear interpolation the surface splinemodel could produce a more accurate wind estimate In [22]Yaghoobi et al described a scheme based on a cubic splineinterpolation which is applied to approximating the variable-order fractional integrals and is extended to solving a classof nonlinear variable-order fractional equations with timedelay Guo and Pan [18] validated this new IP scheme withtwin experiments and the results showed that the prescribednonlinear distribution of bottom friction coefficients arebetter inverted with the surface spline interpolation Due toits proved superior performance the surface spline interpola-tion (SSI) is applied to inversion of ICs with a PM25 transportadjoint model in this study

The paper is organized as follows Section 2 provides thedetailed descriptions of the numerical model and imple-mentation of SSI Twin experiments and practical experi-ments are carried out to evaluate the performance of SSI ininversion of ICs in Sections 3 and 4 respectively Finallysome key conclusions drawn from the work are presented inSection 5

2 Numerical Model and Settings

21 Forward Model The governing equations of the two-dimensional PM25 transport model are as follows

120597119862120597t + 119906120597119862120597119909 + V120597119862120597119910 = 120597120597119909 (119860119867120597119862120597119909 ) + 120597120597119910 (119860119867120597119862120597119910 )

+ 119878(1)

119862 (1199050 119909 119910) = 1198620 (119909 119910) (2)

where C denotes the ground-level PM25 concentration uand v are the horizontal wind velocity in x-coordinate andy-coordinate respectively 119860119867 is the horizontal diffusivitycoefficient S is the value of the source and sink and C0is theinitial condition (IC) to be estimated in this study

The boundary conditions are set as constant at the inflowboundary ΓIN and no gradient boundary conditions at theoutflow boundary ΓOUT

120597119862120597119905 = 0 on ΓIN (3)

120597119862120597997888rarr119899 = 0 on ΓOUT (4)

22 Adjoint Model The adjoint model is defined to calculatethe gradients of the cost function with respect to variousinput parameters which incorporate all physical processesincluded in the governing model to obtain the minimizationof the distance between the model output and observationsBased on the governing equations (1)ndash(4) its adjoint modelcan be constructed as follows

First the distance is expressed by the cost function whichis defined as

119869 = 12 intΣ119870(119862 minus 119862obs)2 d120590 (5)

where Σ denotes the set of the observations in time and spacedomain and C andCobs are the simulated and observed PM25concentrations respectively And K is the weighting matrixand should be the inverse of observation error covariancematrix theoretically K can be fixed simply assuming that theerrors in the data are uncorrelated and equally weighted Inthe present model K is 1 when the observations are availableand 0 otherwise

Then the Lagrangian function is constructed based on thetheory of Lagrangianmultipliermethod and can be expressedas

119871 = 119869 + int119901 times [120597119862120597t + 119906120597119862120597119909 + V120597119862120597119910 minus 120597120597119909 (119860119867120597119862120597119909 )

minus 120597120597119910 (119860119867120597119862120597119910 ) minus 119878] d120590(6)

where p is the adjoint variable of CAccording to the typical theory of Lagrangian multiplier

method we have the following first-order derivatives ofLagrangian function with respect to all the variables andparameters

120597119871120597119901 = 0 (7)

120597119871120597119860119867 = 0120597119871120597119878 = 0

(8)

120597119871120597119862 = 0 (9)

Equation (7) gives the governing equation (1) of theforward model The adjoint equation can be developed fromequation (9) which is given as follows


120597119901120597119905 + 119906120597119901120597119909 + V120597119901120597119910 + 120597120597119909 (119860119867 120597119901120597119909) + 120597120597119910 (119860119867120597119901120597119910)

minus 119870 (119862 minus 119862obs) = 0(10)




119896119894+1119897 = 119896119894119897 minus 120572119894 sdot 119901119894119897 (11)


119901119897 = 1205971198691205971198620119897radicsum119873119897=1 (1205971198691205971198620119897)2 (12)




119896119894119895 = 119873sum119897=1

119882119897119894119895119896119897 (14)


120597 and1198691205971198620119897 =1205971198691205971198620119897 +sum

119894119895

119882119897119894119895 1205971198691205971198620119894119895 (15)



119882119897119894119895 = 119908119897119894119895sum119873119897=1 119908119897119894119895 (16)





1198620119894119895 = 119873sum119897=1

119860 119897(11990321198941198951198971198772 ln11990321198941198951198971198772 + 1 minus 11990321198941198951198971198772 ) (17)


119882119883 = 119884 (18)



(19)

where 119908119894119895 = 1 119894=119895

(11990321198941198951198772)ln(1199032119894119895119877



119860119896 = 119873sum119897=1

1198721198961198971198620119897 (20)


1198620119894119895 = 119873sum119897=1

1198821198971198941198951198620119897 (21)





80 90 100 110 120 130 14070Longitude(E)

15

20

25

30

35

40

45

50

55

Latit

ude(

N)











MAGE = 1119873119873sum119894=1

1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816 (22)







Observations


Run adjoint model




End

Begin


Run forward model

No

Yes





MNGE = 1119873119873sum119894=1

[1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816119874119894 ] (23)






50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)


005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)


015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)






(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









120597119901120597119905 + 119906120597119901120597119909 + V120597119901120597119910 + 120597120597119909 (119860119867 120597119901120597119909) + 120597120597119910 (119860119867120597119901120597119910)

minus 119870 (119862 minus 119862obs) = 0(10)




119896119894+1119897 = 119896119894119897 minus 120572119894 sdot 119901119894119897 (11)


119901119897 = 1205971198691205971198620119897radicsum119873119897=1 (1205971198691205971198620119897)2 (12)




119896119894119895 = 119873sum119897=1

119882119897119894119895119896119897 (14)


120597 and1198691205971198620119897 =1205971198691205971198620119897 +sum

119894119895

119882119897119894119895 1205971198691205971198620119894119895 (15)



119882119897119894119895 = 119908119897119894119895sum119873119897=1 119908119897119894119895 (16)





1198620119894119895 = 119873sum119897=1

119860 119897(11990321198941198951198971198772 ln11990321198941198951198971198772 + 1 minus 11990321198941198951198971198772 ) (17)


119882119883 = 119884 (18)



(19)

where 119908119894119895 = 1 119894=119895

(11990321198941198951198772)ln(1199032119894119895119877



119860119896 = 119873sum119897=1

1198721198961198971198620119897 (20)


1198620119894119895 = 119873sum119897=1

1198821198971198941198951198620119897 (21)





80 90 100 110 120 130 14070Longitude(E)

15

20

25

30

35

40

45

50

55

Latit

ude(

N)











MAGE = 1119873119873sum119894=1

1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816 (22)







Observations


Run adjoint model




End

Begin


Run forward model

No

Yes





MNGE = 1119873119873sum119894=1

[1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816119874119894 ] (23)






50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)


005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)


015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)






(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









80 90 100 110 120 130 14070Longitude(E)

15

20

25

30

35

40

45

50

55

Latit

ude(

N)











MAGE = 1119873119873sum119894=1

1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816 (22)







Observations


Run adjoint model




End

Begin


Run forward model

No

Yes





MNGE = 1119873119873sum119894=1

[1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816119874119894 ] (23)






50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)


005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)


015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)






(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018












Observations


Run adjoint model




End

Begin


Run forward model

No

Yes





MNGE = 1119873119873sum119894=1

[1003816100381610038161003816119872119894 minus 1198741198941003816100381610038161003816119874119894 ] (23)






50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)


005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)


015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)






(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

CI(TE1)SSI(TE1)

(a)

50 100 150 200 250 3000Iteration Steps

0

2

4

6

MAG

E

CI(TE1)SSI(TE1)

(b)

CI(TE2)SSI(TE2)

50 100 150 200 250 3000Iteration Steps

10minus4

10minus2

100

J 300J 1

(c)

50 100 150 200 250 3000Iteration Steps

5

10

15

20

25

MAG

E

CI(TE2)SSI(TE2)

(d)


005

006

007

008

009

01

011

012

013

014

MN

GE

50 100 150 200 250 3000Iteration Steps

CI(TE1)SSI(TE1)


015

02

025

03

035

04

MN

GE

CI(TE2)SSI(TE2)






(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018












(e)

(b)

(d)

(a)

(c)

(f)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

20

30

40

50

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

80 90 100 110 120 130 14070Longitude(E)

20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

80

90

100

20

30

40

50

60

70


20

30

40

50

Latit

ude(

N)

20

30

40

50La

titud

e(N

)

20

30

40

50

Latit

ude(

N)

20

30

40

50

60

70

(b)

(d)

(f)











CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018












CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

2

4

6

MAG

E

50 100 150 200 250 3000Iteration Steps

006

008

01

012

014

MN

GE

CI(TE3)SSI(TE3)

CI(TE5)SSI(TE5)

50 100 150 200 250 3000Iteration Steps

025

03

035

04

MN

GE

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)

50 100 150 200 250 3000Iteration Steps

10

15

20

25

MAG

E

CI(TE4)SSI(TE4)

CI(TE6)SSI(TE6)











DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018












DTTY

QHRHRB

MDJ

BeJYT

WHuNJ

ZJJZG

QJKMYXGY

QZFZ

20

40

60

80

100

120

140

80 90 100 110 120 130 14070Longitude(E)

152025303540455055

Latit

ude(

N)

gmiddotＧ-3





5 Summary





14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









14080 90 100 110 120 13070Longitude(E)

152025303540455055

Latit

ude(

N)

2030405060708090100110120

(a)

152025303540455055

Latit

ude(

N)

80 90 100 110 120 130 14070Longitude(E)

2030405060708090100110

(b)



r

116 000 117 000 118 000 119 000 1110 000 1111 000115 000


minus100minus50

050

100

NM

B an

d N

ME

()

070809

1

Cor

rela

tion

116 000 117 000 118 000 119 000 1110 000 1111 000115 000

116 000 117 000 118 000 119 000 1110 000 1111 000115 000405060708090

PM2

5 conc

entr

atio

n (

gm

3 )

NMENMB


Data Availability




20

25

30

35

40

45

50

55

80 90 100 110 120 130 140700

20

40

60

80

100

120

140

160


Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018










































Journal of












Journal of







Volume 2018






Volume 2018








application of surface spline interpolation method in...

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