research article analysis of the properties of adjoint equations...

Research ArticleAnalysis of the Properties of Adjoint Equations and AccuracyVerification of Adjoint Model Based on FVM

Yaodeng Chen12 Ruizhi Zhang13 Yufang Gao1 and Changna Lu1

1 Collaborative Innovation Center on Forecast and Evaluation of Meteorological DisastersKey Laboratory of Meteorological Disaster of Ministry of Education NUIST Nanjing 210044 China

2 College of Atmospheric Science Nanjing University of Information Science amp Technology Nanjing 210044 China3Meteorological Center of Huaihe River Basin Bengbu 23300 China

Correspondence should be addressed to Yaodeng Chen keyunuisteducn

Received 2 July 2014 Accepted 16 August 2014 Published 27 October 2014

Academic Editor Huanhe Dong

Copyright copy 2014 Yaodeng Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

There are two different approaches on how to formulate adjoint numerical model (ANM) Aiming at the disputes arising from theconstruction methods of ANM the differences between nonlinear shallow water equation and its adjoint equation are analyzedthe hyperbolicity and homogeneity of the adjoint equation are discussed Then based on unstructured meshes and finite volumemethod a new adjointmodel was advanced by getting numericalmodel of the adjoint equations directly Using a gradient check thecorrectness of the adjoint model was verified The results of twin experiments to invert the bottom friction coefficient (Manningrsquosroughness coefficient) indicate that the adjoint model can extract the observation information and produce good quality inversionThe reason of disputes about construction methods of ANM is also discussed in the paper

1 Introduction

Numerical simulation becomes an effective tool for the analy-sis and forecasting in hydrodynamics hydrology oceanic andatmospheric However the uncertainties in numericalmodelfor example initial conditions open boundary conditionsand physical parameters (wind drag coefficients bottomfriction coefficient and so on) make the underlying problemof calibration very challenging

Adjoint data assimilation method (shortened as ldquoadjointmethodrdquo hereafter) based on optimal control theory and 4-dimensional variational theory is one of the most effectivedata assimilation approaches with extensive applications inthe studies of meteorological and oceanic numerical simula-tion [1 2] It has the advantage of assimilating observationsthat are distributed in time and space in the numericalmodelwhile maintaining dynamical and physical consistency withthemodelThe adjointmethod is primarily focused on reduc-ingmodel errors induced by uncertainties in themodel [3ndash6]

The derivation of adjoint numerical model (ANM) isone of the essential issues for developing an adjoint dataassimilation system There are two different approaches on

how to get ANM [7] The first approach is ldquoadjoint of modelrdquothat is getting numerical solution of the forward continuousequations and then deriving ANM from this numericalmodel The second approach is ldquoadjoint of equationrdquo thatis deriving the adjoint equations directly from the forwardcontinuous equations and then determining the adjointnumerical model by modeling the adjoint equations

It is conventionally believed that the first method shouldbe adopted so most studies employed the ldquoadjoint of modelrdquoto construct ANM [1 7 8] It has to be stressed out that usingthe approach of ldquoadjoint of modelrdquo to get ANM from forwardnumerical model allows no selection of appropriate numeri-cal methods according to specific needs such as the problemof discontinuous solution In fact some studies suggest thatldquoadjoint of equationrdquo will have similar effectiveness and canalso get appropriate descend gradient [9 10] ldquoAdjoint ofequationrdquo avoids the limitations of forward numerical modeland is capable of designing numerical scheme according tomathematical properties of adjoint equation

Most adjoint numerical models are based on finitedifference method (FDM) FDM has good computationalperformance and coding efficiencyHowever the classic FDM

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 407468 7 pageshttpdxdoiorg1011552014407468

2 Abstract and Applied Analysis

based on rectangular meshes could not approximate bound-ary accurately in bays and estuaries which have complexgeometries [11 12] To improve the fitting ability of numericalmodel on the boundary in bays and estuaries finite elementmethod (FEM) based on adaptive meshes was introducedinto adjoint numerical model [13 14] However in FEM theestablished algebraic equations are nonlinear and have to besolved by iteration process [15]Moreover computation of theadjoint method is huge itself the forward model and adjointmodel would be integrated forward and integrated backwardonce respectively in each assimilation iteration So FEMwillcertainly increase the computational cost of the assimilationgreatly

In fact the finite volume method (FVM) which has themerits of FDM and FEM combines the best attributes ofFDM for simple discrete coding and computational efficiencyand FEM for geometric flexibility Furthermore the integralequations can be solved numerically by flux calculation theFVM is better suited to guarantee mass conservation [16]

To develop adjoint numerical model based on FVMthe properties of equation will be discussed first Althoughsome researches used ldquoadjoint of equationrdquo to develop adjointmodel [9 10 13] few studies deal with the properties ofadjoint equation such as hyperbolicity and homogeneity ofthe adjoint equation In addition What are the reasons thatFVM can be used to solve adjoint equations If the newadjoint model based on FVM viable In the study we willdiscuss these issues

This paper is organized as follows In Section 2 theadjoint equation of conservative shallow water equation ofCartesian coordinate was derived the formula for inversingbottom friction coefficient is given and the differencesbetween nonlinear shallow water equation and its adjointequation are analyzed In Section 3 properties of adjointequation such as characteristic structure hyperbolicity andhomogeneity of flux are discussed In Section 4 the forwardmodel and its adjoint model based on finite volume methodsand unstructured triangular meshes are described in briefIn Section 4 the correctness of the new adjoint model andcode is verified via the gradient check Then in Section 5the effects of the adjoint model by assimilating the twodifferent types of observation are discussed by a series of twinexperiments Concluding remarks are included in Section 6

2 Water Shallow Equations andTheir Adjoint Equations

Depth-averaged nonlinear conservative shallow water equa-tions with source terms are selected as the forward equationsin this study

120597119867

120597119905+120597 (119867119906)

120597119909+120597 (119867V)120597119910

= 0

120597 (119867119906)

120597119905+120597

120597119909(1198671199062+1

21198921198672) +

120597 (119867119906V)120597119910

= 119867119891V minus 119892119867120597ℎ

120597119909+ 119862119887119906119867

120597 (119867V)120597119905

+120597 (119906V119867)120597119909

+120597

120597119910(V2119867 +

1

21198921198672)

= minus119867119891119906 minus 119892119867120597ℎ

120597119910+ 119862119887V119867

(1)

where119867 = ℎ + 120577 is total water depth ℎ is static water depth120577 is water level 119906 V is the east and north component of waterflow respectively 119892 is gravitational acceleration119891 is Coriolisforce 119862

119887= (120588119892119899

2radic1199062 + V2)11986743 120588 is water body densityand 119899 is Manning coefficient

The purpose of adjoint method is to select appropriatemodel parameters or initial fields to minimize the distancebetween the calculated values and observed values Thus theobjective functional is constructed as follows

119869 (119866) =1

2∭119909119910119905

(1205791

1003816100381610038161003816120577 minus 120577obs10038161003816100381610038162

+ 1205792

1003816100381610038161003816119906 minus 119906obs10038161003816100381610038162

+ 1205793

1003816100381610038161003816V minus Vobs10038161003816100381610038162

) 119889119909 119889119910119889119905

(2)

where 1205791 1205792 1205793are weight coefficients 120577 119906 V are numerical

values and 120577obs 119906obs Vobs are observed values in the assimilatewindow where there are observed data otherwise they areequal to numerical value By means of Lagrange multipliermethod 120582 120572 120573 are introduced into (1) respectively asLagrange multiplier The objective functional is rewritten as

119871 =∭119909119910119905

120582(120597119867

120597119905+120597 (119867119906)

120597119909+120597 (119867V)120597119910

)

+ 120572 [120597 (119867119906)

120597119905+120597

120597119909(1198671199062+1

21198921198672)

+120597 (119867119906V)120597119910

minus119867119891V + 119892119867120597ℎ

120597119909minus 119862119887119906119867]

+ 120573[120597 (V119867)120597119905

+120597 (119906V119867)120597119909

+120597

120597119910(V2119867 +

1

21198921198672)

+ 119863119891119906 + 119892119867120597ℎ

120597119910minus 119862119887V119867]119889119909119889119910119889119905 + 119869

(3)

There must be nabla119880119871 = 0 while the function has extreme

value Utilize (120575120577 120575119906 120575V) = 0 on the boundary and initialfields of forward process 120577|

1198790= 119906|1198790

= V|1198790

= 0 and let120582|119879= 120572|119879= 120573|119879= 0 be the initial field Thus the adjoint

equation is obtained as

120597120582

120597119905+120597 (120582119906 + 119892119867120572)

120597119909+120597 (120582V + 119892119867120573)

120597119910

= 120582(120597119906

120597119909+120597V120597119910) + 1205791

1003816100381610038161003816120577 minus 120577obs1003816100381610038161003816

Abstract and Applied Analysis 3

120597119867120572

120597119905+120597 (119867120582 + 119867120572119906)

120597119909+120597 (119867120572V)120597119910

= 120582120597120577

120597119909+ 119867120572

120597119906

120597119909+ 119867120573

120597V120597119909

+ 119867120572119862119887

119867

+119867120573119891 + 1205792

1003816100381610038161003816119906 minus 119906obs1003816100381610038161003816

120597119867120573

120597119905+120597 (119867120573119906)

120597119909+120597 (119867120582 + 119867120573V)

120597119910

= 120582120597120577

120597119910+ 119867120572

120597119906

120597119910+ 119867120573

120597V120597119910

+ 119867120573119862119887

119867

minus119867120572119891 + 1205793

1003816100381610038161003816V minus Vobs1003816100381610038161003816

(4)

And the formula for inversing bottom friction coefficient is

119863119896+1

= 119863119896minus 120596

1

119879∭119909119910119905

(120572119867119906 + 120573119867V)120588119892radic1199062 + V2

11986743119889119905

(5)

where 119863 = 1198992 is the bottom friction coefficient 120596 isin [0 1]

and 119896 is the times of assimilationThe most significant difference between shallow water

equation and its adjoint equation is that shallow waterequations are forward integration while adjoint equationsare backward integration so shallow water equations areoften called forward equations and adjoint equations areoften called backward equations In addition shallow waterequations are forced by gravity term bottom friction termand other external forces the variables and terms in shallowwater equations have physical meanings However adjointequations are derived from forward equations and driven bythe difference between observed values and numerical values

3 Properties of Adjoint Equation

31 Characteristic Structure of Adjoint Equation The adjointequations can be rewritten as

120597119880lowast

120597119905+120597119891lowast

120597119909+120597119892lowast

120597119910= 119878lowast (6)

where

119880lowast= [120582119867120572119867120573]

119879

119891lowast= [120582119906 + 119892119867120572119867120582 + 119867120572119906119867120573119906]

119879

119892lowast= [120582V + 119892119867120573119867120572V 119867120582 + 119867120573V]119879

119878lowast= [120582(

120597119906

120597119909+120597V120597119910) +

1003816100381610038161003816120577 minus 120577obs1003816100381610038161003816

120582120597120577

120597119909+ 119867120572(

120597119906

120597119909+119862119887

119867) +119867120573(

120597V120597119909

+ 119891)

120582120597120577

120597119910+ 119867120572(

120597119906

120597119910minus 119891) + 119867120573(

120597V120597119910

+119862119887

119867)]

119879

(7)

Jacobian matrices of 119891lowast 119892lowast are

119860lowast=120597 (119891lowast)

120597119880lowast= [

[

119906 119892 0

119867 119906 0

0 0 119906

]

]

119861lowast=120597 (119866lowast)

120597119880lowast= [

[

V 0 119892

0 V 0

119867 0 V]

]

(8)

Thus if 119888 = radic119892119867 then the eigenvalues of119860lowastare 120582119860lowast1= 119906minus119888

120582119860lowast2= 119906 and 120582

119860lowast3= 119906 + 119888

Then corresponding eigenvectors are119882119860lowast1= 1205831199031[1 minus119888

119892 0]119879 119882119860lowast2= 1205831199032[0 0 1]

119879 and 119882119860lowast3= 1205831199033[1 119888119892 0]

119879where 120583

1199031 1205831199032 1205831199033

are constants And left eigenvectors are119882119897

119860lowast1= 1205831198971[1 minus119888119867 0] 119882119897

119860lowast2= 1205831198972[0 0 1] and 119882

119897

119860lowast3=

1205831198973[1 119888119867 0] where 120583

1198971 1205831198972 1205831198973are constants

Eigenvalues of 119861lowast are 120582119861lowast1= V minus 119888 120582

119861lowast2= V and 120582

119861lowast3=

V + 119888 Thus the eigenvalues (right) of 119861lowast can be obtained as119882119861lowast1= ]1199031[1 0 minus119888119892]

119879 119882119861lowast2= ]1199032[0 0 1]

119879 and 119882119861lowast3=

]1199033[1 0 (119888119892)]

119879 where ]1199031 ]1199032 ]1199033are constants Left eigen-

vectors of 119861lowast are119882119897119861lowast1= ]1198971[1 0 minus119888119867]119882119897

119861lowast2= ]1198972[0 1 0]

and119882119897119861lowast3= ]1198973[1 0 119888119867] where ]

1198971 ]1198972 ]1198973are constants

32 Hyperbolicity of Adjoint Equation Let 119862lowast matrix bethe linear combination of Jacobian matrices 119860lowast and 119861

lowast119862lowast= [

1199061205961+V1205962 1198921205961 1198921205962

1198671205961 1199061205961+V1205962 0

1198671205962 0 1199061205961+V1205962] 1205961 1205962are real parameters

andradic12059621+ 12059622gt 0 Thus

120597119880lowast

120597119905+ 119862lowast(nabla sdot 119880

lowast) = 119878lowast (9)

Let 119888 = radic119892119867 and the eigenvalues of 119862lowast are

1205821= 1199061205961+ V1205962minus 119888 |120596| 120582

2= 1199061205961+ V1205962

1205823= 1199061205961+ V1205962+ 119888 |120596|

(10)

According to the definition of hyperbolic equation wehave the following

(1) For 120596 = 0 all eigenvalues of 119862lowast are real numbersthat is there are 119899 real eigenvalues Thus the adjointequations are hyperbolic equations

(2) When 119867 gt 0 that is when 119888 = 0 the eigenvaluesof 119862lowast do not equal each other Then the adjointequations are strict hyperbolic equations

33Homogeneity of Flux According to the expressions of119860lowast119861lowast 119880lowast 119891lowast and 119892lowast we have

119860lowast119880lowast= 119891lowast 119861

lowast119880lowast= 119892lowast (11)

This means that 119891lowast(119880lowast) 119892lowast(119880lowast) in adjoint equationsmeet the requirements of homogeneity property so that theadjoint equation can be used to construct flux vector splittingdirectly


4 Finite Volume Scheme of Adjoint Equation

In the section the FVM is used to solve the forward equationsand their adjoint equations and the computational controlvolumes are based on the same unstructured triangularmeshes

As regards the forward equations a lot of discontinuitycapture methods are used to solve the shallow water equa-tions including the cases with strong discontinuity such astidal bore [15] In this study the numerical flux is evaluated byRoersquos scheme [17 18] and the cell variables are reconstructedby the piecewise linear model [19] For the sake of brevity wedo not describe them again

It has been proved that adjoint equations are hyperbolicequations in Section 32 A significant feature of hyperbolicequations is discontinuity of their solution Therefore weused the discontinuity capture methods which are used tosolve the shallow water equations to solve adjoint equations

Using FVM based on unstructured meshes computa-tional domain is performed on triangular element and thevariable is set at the center of each element Let997888119865

lowast

= (119891lowast 119892lowast)

be the numerical flux at interfaces of control volumes thefinite volume scheme of adjoint equations can be formulatedas follows

119880lowast119899+1

119894= 119880lowast119899

119894minusΔ119905

Ω119894

3

sum

119895=1

997888119865lowast

119894119895Δ119897119895+ 119878lowast

119894Δ119905 (12)

whereΩ119894is the area of 119894th control volume Δ119897

119895is the length of

the 119895th side of the 119894th control volume and 119880lowast119894 119878lowast119894is the mean

value of 119880lowast119894 119878lowast119894on the control volume Ω

119894

The adjoint variables at the two sides of control volumeboundary are considered as different values (119880lowast

119871 119880lowast

119877) The

variables are set at the center of grid element and119880lowast119871 119880lowast119877are

reconstructed with piecewise constant approximation Thenumerical flux of adjoint equations based on Roersquos schemecan be written as

119865lowast=1

2[119865 (119880

lowast

119894119871) + 119865 (119880

lowast

119894119877) minus |119860| (119880

lowast

119894119877minus 119880lowast

119894119871)]

where |119860| = 119877 |Λ| 119871

(13)

According to the derivation in Section 31 in aboveequation

119860 =[

[

119906119899119909+ V119899119910

119892119899119909

119892119899119910

119867119899119909

119906119899119909+ V119899119910

0

119867119899119910

0 119906119899119909+ V119899119910

]

]

119877 =

[[[[[

[

1 0 1

minus119888

119892

119899119909

997888119899

119899119910

119888

119892

119899119909

997888119899

minus119888

119892

119899119910

997888119899

119899119909

119888

119892

119899119910

997888119899

]]]]]

]

119871 =

[[[[[

[

1 minus119888

119867

119899119909

997888119899

minus119888

119867

119899119910

997888119899

0 2119899119910

2119899119909

1119888

119867

119899119909

997888119899

119888

119867

119899119910

997888119899

]]]]]

]

|Λ| =[[

[

10038161003816100381610038161003816119906119899119909+ V119899119910minus 119888119899

100381610038161003816100381610038160 0

010038161003816100381610038161003816119906119899119909+ V119899119910

100381610038161003816100381610038160

0 010038161003816100381610038161003816119906119899119909+ V119899119910+ 119888119899

10038161003816100381610038161003816

]]

]

(14)

o

c a

b

Figure 1 Triangular element

5

3210

minus1minus2minus3minus4minus5

0 5 10 15 20

4

X (km)

Y(k

m)

Figure 2 Mesh setup of generalized estuary

Using the Green formula the gradient terms in theadjoint equation source terms (119878lowast) can be approximatelycomputed as

120597119906

120597119909cong

1

119878119886119887119888

[(119906119886+ 119906119887) (119910119887minus 119910119886)

2

+(119906119887+ 119906119888) (119910119888minus 119910119887)

2+(119906119888+ 119906119886) (119910119886minus 119910119888)

2]

(15)

where 119886 119887 119888 are the centers of the neighbor cells respectively(Figure 1) and 119878

119886119887119888is the area of triangle 119886119887119888 For 120597119906120597119910

120597V120597119909 120597V120597119910 120597120577120597119909 120597120577120597119910 they can be approximatelycomputed similarly to 120597119906120597119909

5 Verification of the Adjoint ModelBased on FVM

In this section we perform data assimilation experiments toverify the correctness and to discuss the calculation efficiencyof the adjoint model in tidal bore estuary The domain isa generalized estuary the length is 20 km the width atupstream is 2 km the width of estuary is 10 km and thewater depth is 10 meters The flow is calculated on a mesh ofunstructured triangle total of 588 nodes and 1064 cells Theminimum side of the grid is about 100 meters (Figure 2)

In the numerical experiments the gravitation constantis taken as 98ms the initial velocity and the free surfaceelevation above the still water level are set to be zero as


minus6

minus5

minus4

minus3

minus2

minus1

0

075

08

085

09

095

1

105

1E minus 011E minus 061E minus 11 1E minus 011E minus 061E minus 11

120593(120572)

Log 10

|120593(120572)minus1|

120572 120572

Figure 3 The curve of Φ(120572) and log10|Φ(120572) minus 1| varying with disturbance 120572

Table 1 Φ(120572) and Log10|Φ(120572) minus 1| corresponding to different disturbance 120572

120572 10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

Φ(120572) 07981 09776 09977 09997 09999 09999 09999 09999 09999 09997 09968 09913Log10|Φ(120572) minus 1| minus0695 minus1650 minus2645 minus3634 minus4546 minus5218 minus5238 minus5189 minus4701 minus3575 minus2504 minus2061

the initial conditions The time steps of the forward modeland the adjoint model are both 40 seconds Tidal component1198722at the east open boundary of estuary is given with

amplitude of 2metersThe absorbing extrapolation boundarycondition for upstream is used and the reflection boundarycondition is used for solid wall boundary

The forward model is integrated five period When flowpropagates to the straight channel the tidal bore is formedAs discussed in [20] the tidal height reaches 3 meters withina short period (tidal bore) 9 kilometers away from the estuary

For adjoint data assimilation descent gradient of costfunction is obtained by the calculation of ANM Thus thecorrectness of the adjoint model and code should be verifiedwhich can be performed via a gradient check [14 21] that isthe quantity

Φ (120572) =119871 (119883 + 120572ℎ) minus 119871 (119883)

120572ℎ119879nabla119871 (119883)(16)

is considered and it is verified that it is of order 1+119874(120572) 119871(119883)is the cost function at a general point 119883 (in the study it isbottom friction coefficient) nabla119871(119883) is the computed gradient120572 is a disturbance quantity and ℎ is an arbitrary unit vector(such as ℎ = nabla119871nabla119871minus1)

Table 1 shows typical values of Φ(120572) and Log10|Φ(120572) minus 1|

for a sequence of decreasing values of 120572 Clearly Φ(120572) is1 + 119874(120572) Figure 3 is the curve of Φ(120572) and Log

10|Φ(120572) minus 1|

varying with disturbance 120572 It can be seen that Φ(120572) has 1-order accuracy and the curve of Log

10|Φ(120572)minus1| varying with

120572 also shows an obviousV-shape also as expected in literature[14] Thus it can thus be concluded at least for this model

problem setup that the new ANM based on FVM and Roersquosscheme is correct and feasible

6 Assimilation Effect in Generalized TidalBore Estuary

In this section In order to discuss and display the perfor-mance of assimilation of the adjoint model in generalizedtidal bore estuary we perform ldquotwin experimentsrdquo to invertManningrsquos roughness coefficient by assimilating the observa-tion of water level or velocity the observed values are fourkinds water level or velocity at all the cells at cells with oddnumbers (half of the grids) at cells with the number endingwith 3 (110 of the grids) and at the four cells (3 km 0) (8 km0) (13 km 0) and (18 km 0) The domain and experimentsetup are the same as in Section 5

Process for the twin experiments is as follows

(1) Given true bottom friction coefficient (002) calculateforwardmodel 5 periods and then savewater level andcurrent velocity of last periods as ldquoobserved valuerdquo

(2) Given the initial or calibrated bottom friction coeffi-cient calculate forward model 5 periodic and calcu-late cost function based on the calculated value of thelast periodic and ldquoobserved valuerdquo If the cost functionsatisfies the specified termination conditions then theassimilation process ends otherwise proceed to thenext procedure

(3) Calculate the ANM and calibrate bottom frictioncoefficientThen go back to procedure (2) to continueassimilation


Table 2 Manningrsquos roughness coefficient (119899) inverse results ofassimilating different quantities of water level data

Test Data Initialvalues

Inversedvalue

Iterationtimes

1 All cells 0010000 0020001 212 One-half of all cells 0010000 0019998 323 One-tenth of all cells 0010000 0019995 394 Four cells 0015000 0019991 51

Table 3 Different case of Manningrsquos roughness coefficient inver-sion

Case Data Initialvalues

Inversedvalue

Iterationtimes


61 Assimilate Different Quantities of Water Level Data Inthis part Manningrsquos roughness coefficient is inverted byassimilating different quantities of water level data

The initial values and inverted results for the four casesare given in Table 2 It can be seen from Table 2 that theassimilation algorithm converges to the true value (002)when initial values are given as 001 in the four testsIt also can be found that when more observations wereassimilated the assimilation algorithm conversed faster Theclosest result to the true value with minimum assimilationtimes (21) is obtained when assimilating water level at allcells Satisfactory results are also obtained when assimilatingwater level data at only four cells though the assimilationtimes (51) are greater When the initial values of 002 005and 015 are given similar results are also obtained

It should be pointed out that the adjoint assimilationalgorithm is quite sensitive to initial values when assimilatingwater level data at only four cells In this case the initialvalues should near to the true value or the algorithm doesnot converge

62 Assimilate Different Quantities of Current VelocityObserved Data Assimilating the observation of currentvelocity if we let the initial guesses drag coefficient equal tothe value which used in assimilating the observation of waterlevel the assimilation process cannot converse When we letthe initial guesses drag coefficient near to the true value itworks By assimilating the observation of current velocity atall cells we let the initial guess coefficient be 0015 after 29assimilationswe get the inverted result to be 0019977At one-half of all cells the initial guess is 0015 after 37 assimilationsthe inverted result is 000045 At one-tenth of all cells theinitial guess is 00015 after 52 assimilations the invertedresult is 0019964 At four cells we let it be 0017 after 73assimilations we get the inverted result to be 0019959 Theinitial values and inverted results for the four cases are givenin Table 3

05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations

Water levelCurrent velocity

Figure 4 The first fifteen iterations of the cost function

Figure 4 shows the first fifteen iterations of the cost func-tion with assimilating water level observed value at all cells(real line) and with assimilating current velocity observedvalue at all cells (dashed) respectively It can be seen thatthe performance with assimilating the two types of observedvalue behaves differently When assimilating current velocitythe cost function decline fast but its convergence speed isslow When water level was assimilated the cost functionconvergence to minimum value smoothly The phenomenamay result in the fact that the observation of current velocityis more sensitive than the water level for the coefficientSo when current velocity was assimilated the cost functionresponded more quickly also because of this it is hard toconvergeThe results suggest that the two types of observationshould be assimilated by a suitable balance

7 Conclusions and Outlook

This study gives a detailed analysis and summary of theproperties of adjoint equation and analyzes the characteristicstructure of adjoint equation hyperbolicity of equationequation homogeneity and the splitting characteristic ofJacobian matrix It has been proven that the adjoint equationbelongs to hyperbolic equation and demonstrated that theattention should be paid to the solution discontinuity ofhyperbolic equations when constructing adjoint numericalmodel We also demonstrate the homogeneity of the adjointequation and demonstrate that ANM in flux vector splittingcan be constructed directly Based on the theoretical analysiswe verify the accuracy and assimilation effect of ANMconstructed by finite volume scheme Regarding the disputesarising from two different construction methods of ANMswe believe that it is associated with the design and selectionof numerical algorithm which significantly influence theaccuracy and validity of ANM Thus theoretical analysis onadjoint equation should be carried out before developingANMs However this study is only a preliminary researchon adjoint equation theory Further researches on issuesrelated to adjoint equation are required including the basicproperties of adjoint equation solutions relation of adjointequation and certainmathematical and physical equations aswell as physicalmeaning of adjoint equationThe clarificationof these problems may provide necessary guidance for theconstruction of reasonable adjoint numerical model


The ldquotwin experimentsrdquo of generalized tidal bore estuaryto invert Manningrsquos roughness coefficient show that thespatial distribution of the bottom friction coefficient willinfluence the precision of the inversion plenty of observationdata can improve the accuracy of the inversion The resultsalso indicate that the two types of observation should beassimilated in a suitable balance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was jointly sponsored by the 973 Program(2013CB430102) the National Natural Science Foundationof China (41205082 41105077) the Meteorological OpenFund of Huaihe River Basin and the Priority Academic Pro-gram Development of Jiangsu Higher Education Institutions(PAPD)

References

[1] W CThacker and R B Long ldquoFitting dynamic to datardquo Journalof Geophysical Research vol 93 pp 1227ndash1240 1988

[2] F-X Le Dimet and O Talagrand ldquoVariational algorithmsfor analysis and assimilation of meteorological observationstheoretical aspectsrdquo Tellus A vol 38 no 2 pp 97ndash110 1986

[3] A F Bennett Inverse Methods in Physical Oceanography Cam-bridge University Press New York NY USA 1992

[4] A W Heemink E E A Mouthaan M R T Roest E AH Vollebregt K B Robaczewska and M Verlaan ldquoInverse3D shallow water flow modelling of the continental shelfrdquoContinental Shelf Research vol 22 no 3 pp 465ndash484 2002

[5] S-Q Peng L Xie and L J Pietrafesa ldquoCorrecting the errors inthe initial conditions and wind stress in storm surge simulationusing an adjoint optimal techniquerdquo Ocean Modelling vol 18no 3-4 pp 175ndash193 2007

[6] A Zhang E Wei and B B Parker ldquoOptimal estimation of tidalopen boundary conditions using predicted tides and adjointdata assimilation techniquerdquoContinental Shelf Research vol 23no 11ndash13 pp 1055ndash1070 2003

[7] O Talagrand and P Courtier ldquoVariational assimilation ofmeteorological observations with the adjoint vorticity equationI theoryrdquo Quarterly Journal of the Royal Meteorological Societyvol 113 no 478 pp 1311ndash1328 1987

[8] J Schroter U Seiler and M Wenzel ldquoVariational assimilationof geosat data into an eddy-resolving model of the gulf streamextension areardquo Journal of Physical Oceanography vol 23 no 5pp 925ndash953 1993

[9] Z Sirkes and E Tziperman ldquoFinite difference of adjoint oradjoint of finite differencerdquo Monthly Weather Review vol 125no 12 pp 3373ndash3378 1997

[10] X-Q Lu Z-K Wu Y Gu and J-W Tian ldquoStudy on the adjointmethod in data assimilation and the related problemsrdquo AppliedMathematics and Mechanics vol 25 no 6 pp 636ndash646 2004

[11] C Chen J Zhu E Ralph S A Green J Budd Wells and F YZhang ldquoPrognostic modeling studies of the Keweenaw Current

in Lake Superior Part I formation and evolutionrdquo Journal ofPhysical Oceanography vol 31 no 2 pp 379ndash395 2001

[12] C Chen J Zhu L Zheng E Ralph and J W Budd ldquoA non-orthogonal primitive equation coastal ocean circulation modelapplication to Lake Superiorrdquo Journal of Great Lakes Researchvol 30 no 1 pp 41ndash54 2004

[13] F Fang C C Pain M D Piggott G J Gorman and AJ H Goddard ldquoAn adaptive mesh adjoint data assimilationmethod applied to free surface flowsrdquo International Journal forNumericalMethods in Fluids vol 47 no 8-9 pp 995ndash1001 2005

[14] F Fang M D Piggott C C Pain G J Gorman and A J HGoddard ldquoAn adaptivemesh adjoint data assimilationmethodrdquoOcean Modelling vol 15 no 1-2 pp 39ndash55 2006

[15] C Lu J Qiu and R Wang ldquoWeighted essential non-oscillatoryschemes for tidal bore on unstructured meshesrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 6 pp 611ndash630 2009

[16] C Chen R C Beardsley andG Cowles AnUnstructuredGridFinite-Volume Coastal Ocean Model FVCOM User Manual2006

[17] P L Roe ldquoApproximate Riemann solvers parameter vectorsand difference schemesrdquo Journal of Computational Physics vol43 no 2 pp 357ndash372 1981

[18] D Ambrosi ldquoApproximation of shallow water equations byRoersquos approximate Riemann solverrdquo International Journal forNumerical Methods in Fluids vol 20 pp 157ndash168 1995

[19] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics Springer New York NY USA 1997

[20] Y-D Chen R-Y Wang W Li and C-N Lu ldquoInversing openboundary conditions of tidal bore estuary using adjoint dataassimilation methodrdquo Advances in Water Science vol 18 no 6pp 829ndash833 2007

[21] I M Navon X Zou J Derber and J Sela ldquoVariational dataassimilation with an adiabatic version of the NMC spectralmodelrdquoMonthly Weather Review vol 120 no 7 pp 1433ndash14461992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


based on rectangular meshes could not approximate bound-ary accurately in bays and estuaries which have complexgeometries [11 12] To improve the fitting ability of numericalmodel on the boundary in bays and estuaries finite elementmethod (FEM) based on adaptive meshes was introducedinto adjoint numerical model [13 14] However in FEM theestablished algebraic equations are nonlinear and have to besolved by iteration process [15]Moreover computation of theadjoint method is huge itself the forward model and adjointmodel would be integrated forward and integrated backwardonce respectively in each assimilation iteration So FEMwillcertainly increase the computational cost of the assimilationgreatly

In fact the finite volume method (FVM) which has themerits of FDM and FEM combines the best attributes ofFDM for simple discrete coding and computational efficiencyand FEM for geometric flexibility Furthermore the integralequations can be solved numerically by flux calculation theFVM is better suited to guarantee mass conservation [16]

To develop adjoint numerical model based on FVMthe properties of equation will be discussed first Althoughsome researches used ldquoadjoint of equationrdquo to develop adjointmodel [9 10 13] few studies deal with the properties ofadjoint equation such as hyperbolicity and homogeneity ofthe adjoint equation In addition What are the reasons thatFVM can be used to solve adjoint equations If the newadjoint model based on FVM viable In the study we willdiscuss these issues

This paper is organized as follows In Section 2 theadjoint equation of conservative shallow water equation ofCartesian coordinate was derived the formula for inversingbottom friction coefficient is given and the differencesbetween nonlinear shallow water equation and its adjointequation are analyzed In Section 3 properties of adjointequation such as characteristic structure hyperbolicity andhomogeneity of flux are discussed In Section 4 the forwardmodel and its adjoint model based on finite volume methodsand unstructured triangular meshes are described in briefIn Section 4 the correctness of the new adjoint model andcode is verified via the gradient check Then in Section 5the effects of the adjoint model by assimilating the twodifferent types of observation are discussed by a series of twinexperiments Concluding remarks are included in Section 6

2 Water Shallow Equations andTheir Adjoint Equations

Depth-averaged nonlinear conservative shallow water equa-tions with source terms are selected as the forward equationsin this study

120597119867

120597119905+120597 (119867119906)

120597119909+120597 (119867V)120597119910

= 0

120597 (119867119906)

120597119905+120597

120597119909(1198671199062+1

21198921198672) +

120597 (119867119906V)120597119910

= 119867119891V minus 119892119867120597ℎ

120597119909+ 119862119887119906119867

120597 (119867V)120597119905

+120597 (119906V119867)120597119909

+120597

120597119910(V2119867 +

1

21198921198672)

= minus119867119891119906 minus 119892119867120597ℎ

120597119910+ 119862119887V119867

(1)

where119867 = ℎ + 120577 is total water depth ℎ is static water depth120577 is water level 119906 V is the east and north component of waterflow respectively 119892 is gravitational acceleration119891 is Coriolisforce 119862

119887= (120588119892119899

2radic1199062 + V2)11986743 120588 is water body densityand 119899 is Manning coefficient

The purpose of adjoint method is to select appropriatemodel parameters or initial fields to minimize the distancebetween the calculated values and observed values Thus theobjective functional is constructed as follows

119869 (119866) =1

2∭119909119910119905

(1205791

1003816100381610038161003816120577 minus 120577obs10038161003816100381610038162

+ 1205792

1003816100381610038161003816119906 minus 119906obs10038161003816100381610038162

+ 1205793

1003816100381610038161003816V minus Vobs10038161003816100381610038162

) 119889119909 119889119910119889119905

(2)

where 1205791 1205792 1205793are weight coefficients 120577 119906 V are numerical

values and 120577obs 119906obs Vobs are observed values in the assimilatewindow where there are observed data otherwise they areequal to numerical value By means of Lagrange multipliermethod 120582 120572 120573 are introduced into (1) respectively asLagrange multiplier The objective functional is rewritten as

119871 =∭119909119910119905

120582(120597119867

120597119905+120597 (119867119906)

120597119909+120597 (119867V)120597119910

)

+ 120572 [120597 (119867119906)

120597119905+120597

120597119909(1198671199062+1

21198921198672)

+120597 (119867119906V)120597119910

minus119867119891V + 119892119867120597ℎ

120597119909minus 119862119887119906119867]

+ 120573[120597 (V119867)120597119905

+120597 (119906V119867)120597119909

+120597

120597119910(V2119867 +

1

21198921198672)

+ 119863119891119906 + 119892119867120597ℎ

120597119910minus 119862119887V119867]119889119909119889119910119889119905 + 119869

(3)

There must be nabla119880119871 = 0 while the function has extreme

value Utilize (120575120577 120575119906 120575V) = 0 on the boundary and initialfields of forward process 120577|

1198790= 119906|1198790

= V|1198790

= 0 and let120582|119879= 120572|119879= 120573|119879= 0 be the initial field Thus the adjoint

equation is obtained as

120597120582

120597119905+120597 (120582119906 + 119892119867120572)

120597119909+120597 (120582V + 119892119867120573)

120597119910

= 120582(120597119906

120597119909+120597V120597119910) + 1205791

1003816100381610038161003816120577 minus 120577obs1003816100381610038161003816


120597119867120572

120597119905+120597 (119867120582 + 119867120572119906)

120597119909+120597 (119867120572V)120597119910

= 120582120597120577

120597119909+ 119867120572

120597119906

120597119909+ 119867120573

120597V120597119909

+ 119867120572119862119887

119867

+119867120573119891 + 1205792

1003816100381610038161003816119906 minus 119906obs1003816100381610038161003816

120597119867120573

120597119905+120597 (119867120573119906)

120597119909+120597 (119867120582 + 119867120573V)

120597119910

= 120582120597120577

120597119910+ 119867120572

120597119906

120597119910+ 119867120573

120597V120597119910

+ 119867120573119862119887

119867

minus119867120572119891 + 1205793

1003816100381610038161003816V minus Vobs1003816100381610038161003816

(4)


119863119896+1

= 119863119896minus 120596

1

119879∭119909119910119905

(120572119867119906 + 120573119867V)120588119892radic1199062 + V2

11986743119889119905

(5)






120597119880lowast

120597119905+120597119891lowast

120597119909+120597119892lowast

120597119910= 119878lowast (6)

where

119880lowast= [120582119867120572119867120573]

119879

119891lowast= [120582119906 + 119892119867120572119867120582 + 119867120572119906119867120573119906]

119879

119892lowast= [120582V + 119892119867120573119867120572V 119867120582 + 119867120573V]119879

119878lowast= [120582(

120597119906

120597119909+120597V120597119910) +

1003816100381610038161003816120577 minus 120577obs1003816100381610038161003816

120582120597120577

120597119909+ 119867120572(

120597119906

120597119909+119862119887

119867) +119867120573(

120597V120597119909

+ 119891)

120582120597120577

120597119910+ 119867120572(

120597119906

120597119910minus 119891) + 119867120573(

120597V120597119910

+119862119887

119867)]

119879

(7)


119860lowast=120597 (119891lowast)

120597119880lowast= [

[

119906 119892 0

119867 119906 0

0 0 119906

]

]

119861lowast=120597 (119866lowast)

120597119880lowast= [

[

V 0 119892

0 V 0

119867 0 V]

]

(8)


120582119860lowast2= 119906 and 120582

119860lowast3= 119906 + 119888


119892 0]119879 119882119860lowast2= 1205831199032[0 0 1]

119879 and 119882119860lowast3= 1205831199033[1 119888119892 0]

119879where 120583

1199031 1205831199032 1205831199033


119860lowast1= 1205831198971[1 minus119888119867 0] 119882119897

119860lowast2= 1205831198972[0 0 1] and 119882

119897

119860lowast3=

1205831198973[1 119888119867 0] where 120583

1198971 1205831198972 1205831198973are constants


119861lowast2= V and 120582

119861lowast3=


119879 119882119861lowast2= ]1199032[0 0 1]

119879 and 119882119861lowast3=

]1199033[1 0 (119888119892)]



119861lowast2= ]1198972[0 1 0]

and119882119897119861lowast3= ]1198973[1 0 119888119867] where ]

1198971 ]1198972 ]1198973are constants



1199061205961+V1205962 1198921205961 1198921205962

1198671205961 1199061205961+V1205962 0



120597119880lowast




1205821= 1199061205961+ V1205962minus 119888 |120596| 120582

2= 1199061205961+ V1205962

1205823= 1199061205961+ V1205962+ 119888 |120596|

(10)














lowast



119880lowast119899+1

119894= 119880lowast119899

119894minusΔ119905

Ω119894

3

sum

119895=1

997888119865lowast

119894119895Δ119897119895+ 119878lowast

119894Δ119905 (12)





119894


119871 119880lowast

119877) The



119865lowast=1

2[119865 (119880

lowast

119894119871) + 119865 (119880

lowast

119894119877) minus |119860| (119880

lowast

119894119877minus 119880lowast

119894119871)]

where |119860| = 119877 |Λ| 119871

(13)


119860 =[

[

119906119899119909+ V119899119910

119892119899119909

119892119899119910

119867119899119909

119906119899119909+ V119899119910

0

119867119899119910

0 119906119899119909+ V119899119910

]

]

119877 =

[[[[[

[

1 0 1

minus119888

119892

119899119909

997888119899

119899119910

119888

119892

119899119909

997888119899

minus119888

119892

119899119910

997888119899

119899119909

119888

119892

119899119910

997888119899

]]]]]

]

119871 =

[[[[[

[

1 minus119888

119867

119899119909

997888119899

minus119888

119867

119899119910

997888119899

0 2119899119910

2119899119909

1119888

119867

119899119909

997888119899

119888

119867

119899119910

997888119899

]]]]]

]

|Λ| =[[

[

10038161003816100381610038161003816119906119899119909+ V119899119910minus 119888119899

100381610038161003816100381610038160 0

010038161003816100381610038161003816119906119899119909+ V119899119910

100381610038161003816100381610038160

0 010038161003816100381610038161003816119906119899119909+ V119899119910+ 119888119899

10038161003816100381610038161003816

]]

]

(14)

o

c a

b


5

3210


0 5 10 15 20

4

X (km)

Y(k

m)



120597119906

120597119909cong

1

119878119886119887119888

[(119906119886+ 119906119887) (119910119887minus 119910119886)

2

+(119906119887+ 119906119888) (119910119888minus 119910119887)

2+(119906119888+ 119906119886) (119910119886minus 119910119888)

2]

(15)








minus6

minus5

minus4

minus3

minus2

minus1

0

075

08

085

09

095

1

105


120593(120572)

Log 10

|120593(120572)minus1|

120572 120572



120572 10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12






Φ (120572) =119871 (119883 + 120572ℎ) minus 119871 (119883)

120572ℎ119879nabla119871 (119883)(16)




10|Φ(120572) minus 1|














Inversedvalue

Iterationtimes




Inversedvalue

Iterationtimes






05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations










Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120597119867120572

120597119905+120597 (119867120582 + 119867120572119906)

120597119909+120597 (119867120572V)120597119910

= 120582120597120577

120597119909+ 119867120572

120597119906

120597119909+ 119867120573

120597V120597119909

+ 119867120572119862119887

119867

+119867120573119891 + 1205792

1003816100381610038161003816119906 minus 119906obs1003816100381610038161003816

120597119867120573

120597119905+120597 (119867120573119906)

120597119909+120597 (119867120582 + 119867120573V)

120597119910

= 120582120597120577

120597119910+ 119867120572

120597119906

120597119910+ 119867120573

120597V120597119910

+ 119867120573119862119887

119867

minus119867120572119891 + 1205793

1003816100381610038161003816V minus Vobs1003816100381610038161003816

(4)


119863119896+1

= 119863119896minus 120596

1

119879∭119909119910119905

(120572119867119906 + 120573119867V)120588119892radic1199062 + V2

11986743119889119905

(5)






120597119880lowast

120597119905+120597119891lowast

120597119909+120597119892lowast

120597119910= 119878lowast (6)

where

119880lowast= [120582119867120572119867120573]

119879

119891lowast= [120582119906 + 119892119867120572119867120582 + 119867120572119906119867120573119906]

119879

119892lowast= [120582V + 119892119867120573119867120572V 119867120582 + 119867120573V]119879

119878lowast= [120582(

120597119906

120597119909+120597V120597119910) +

1003816100381610038161003816120577 minus 120577obs1003816100381610038161003816

120582120597120577

120597119909+ 119867120572(

120597119906

120597119909+119862119887

119867) +119867120573(

120597V120597119909

+ 119891)

120582120597120577

120597119910+ 119867120572(

120597119906

120597119910minus 119891) + 119867120573(

120597V120597119910

+119862119887

119867)]

119879

(7)


119860lowast=120597 (119891lowast)

120597119880lowast= [

[

119906 119892 0

119867 119906 0

0 0 119906

]

]

119861lowast=120597 (119866lowast)

120597119880lowast= [

[

V 0 119892

0 V 0

119867 0 V]

]

(8)


120582119860lowast2= 119906 and 120582

119860lowast3= 119906 + 119888


119892 0]119879 119882119860lowast2= 1205831199032[0 0 1]

119879 and 119882119860lowast3= 1205831199033[1 119888119892 0]

119879where 120583

1199031 1205831199032 1205831199033


119860lowast1= 1205831198971[1 minus119888119867 0] 119882119897

119860lowast2= 1205831198972[0 0 1] and 119882

119897

119860lowast3=

1205831198973[1 119888119867 0] where 120583

1198971 1205831198972 1205831198973are constants


119861lowast2= V and 120582

119861lowast3=


119879 119882119861lowast2= ]1199032[0 0 1]

119879 and 119882119861lowast3=

]1199033[1 0 (119888119892)]



119861lowast2= ]1198972[0 1 0]

and119882119897119861lowast3= ]1198973[1 0 119888119867] where ]

1198971 ]1198972 ]1198973are constants



1199061205961+V1205962 1198921205961 1198921205962

1198671205961 1199061205961+V1205962 0



120597119880lowast




1205821= 1199061205961+ V1205962minus 119888 |120596| 120582

2= 1199061205961+ V1205962

1205823= 1199061205961+ V1205962+ 119888 |120596|

(10)














lowast



119880lowast119899+1

119894= 119880lowast119899

119894minusΔ119905

Ω119894

3

sum

119895=1

997888119865lowast

119894119895Δ119897119895+ 119878lowast

119894Δ119905 (12)





119894


119871 119880lowast

119877) The



119865lowast=1

2[119865 (119880

lowast

119894119871) + 119865 (119880

lowast

119894119877) minus |119860| (119880

lowast

119894119877minus 119880lowast

119894119871)]

where |119860| = 119877 |Λ| 119871

(13)


119860 =[

[

119906119899119909+ V119899119910

119892119899119909

119892119899119910

119867119899119909

119906119899119909+ V119899119910

0

119867119899119910

0 119906119899119909+ V119899119910

]

]

119877 =

[[[[[

[

1 0 1

minus119888

119892

119899119909

997888119899

119899119910

119888

119892

119899119909

997888119899

minus119888

119892

119899119910

997888119899

119899119909

119888

119892

119899119910

997888119899

]]]]]

]

119871 =

[[[[[

[

1 minus119888

119867

119899119909

997888119899

minus119888

119867

119899119910

997888119899

0 2119899119910

2119899119909

1119888

119867

119899119909

997888119899

119888

119867

119899119910

997888119899

]]]]]

]

|Λ| =[[

[

10038161003816100381610038161003816119906119899119909+ V119899119910minus 119888119899

100381610038161003816100381610038160 0

010038161003816100381610038161003816119906119899119909+ V119899119910

100381610038161003816100381610038160

0 010038161003816100381610038161003816119906119899119909+ V119899119910+ 119888119899

10038161003816100381610038161003816

]]

]

(14)

o

c a

b


5

3210


0 5 10 15 20

4

X (km)

Y(k

m)



120597119906

120597119909cong

1

119878119886119887119888

[(119906119886+ 119906119887) (119910119887minus 119910119886)

2

+(119906119887+ 119906119888) (119910119888minus 119910119887)

2+(119906119888+ 119906119886) (119910119886minus 119910119888)

2]

(15)








minus6

minus5

minus4

minus3

minus2

minus1

0

075

08

085

09

095

1

105


120593(120572)

Log 10

|120593(120572)minus1|

120572 120572



120572 10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12






Φ (120572) =119871 (119883 + 120572ℎ) minus 119871 (119883)

120572ℎ119879nabla119871 (119883)(16)




10|Φ(120572) minus 1|














Inversedvalue

Iterationtimes




Inversedvalue

Iterationtimes






05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations










Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















lowast



119880lowast119899+1

119894= 119880lowast119899

119894minusΔ119905

Ω119894

3

sum

119895=1

997888119865lowast

119894119895Δ119897119895+ 119878lowast

119894Δ119905 (12)





119894


119871 119880lowast

119877) The



119865lowast=1

2[119865 (119880

lowast

119894119871) + 119865 (119880

lowast

119894119877) minus |119860| (119880

lowast

119894119877minus 119880lowast

119894119871)]

where |119860| = 119877 |Λ| 119871

(13)


119860 =[

[

119906119899119909+ V119899119910

119892119899119909

119892119899119910

119867119899119909

119906119899119909+ V119899119910

0

119867119899119910

0 119906119899119909+ V119899119910

]

]

119877 =

[[[[[

[

1 0 1

minus119888

119892

119899119909

997888119899

119899119910

119888

119892

119899119909

997888119899

minus119888

119892

119899119910

997888119899

119899119909

119888

119892

119899119910

997888119899

]]]]]

]

119871 =

[[[[[

[

1 minus119888

119867

119899119909

997888119899

minus119888

119867

119899119910

997888119899

0 2119899119910

2119899119909

1119888

119867

119899119909

997888119899

119888

119867

119899119910

997888119899

]]]]]

]

|Λ| =[[

[

10038161003816100381610038161003816119906119899119909+ V119899119910minus 119888119899

100381610038161003816100381610038160 0

010038161003816100381610038161003816119906119899119909+ V119899119910

100381610038161003816100381610038160

0 010038161003816100381610038161003816119906119899119909+ V119899119910+ 119888119899

10038161003816100381610038161003816

]]

]

(14)

o

c a

b


5

3210


0 5 10 15 20

4

X (km)

Y(k

m)



120597119906

120597119909cong

1

119878119886119887119888

[(119906119886+ 119906119887) (119910119887minus 119910119886)

2

+(119906119887+ 119906119888) (119910119888minus 119910119887)

2+(119906119888+ 119906119886) (119910119886minus 119910119888)

2]

(15)








minus6

minus5

minus4

minus3

minus2

minus1

0

075

08

085

09

095

1

105


120593(120572)

Log 10

|120593(120572)minus1|

120572 120572



120572 10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12






Φ (120572) =119871 (119883 + 120572ℎ) minus 119871 (119883)

120572ℎ119879nabla119871 (119883)(16)




10|Φ(120572) minus 1|














Inversedvalue

Iterationtimes




Inversedvalue

Iterationtimes






05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations










Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus6

minus5

minus4

minus3

minus2

minus1

0

075

08

085

09

095

1

105


120593(120572)

Log 10

|120593(120572)minus1|

120572 120572



120572 10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12






Φ (120572) =119871 (119883 + 120572ℎ) minus 119871 (119883)

120572ℎ119879nabla119871 (119883)(16)




10|Φ(120572) minus 1|














Inversedvalue

Iterationtimes




Inversedvalue

Iterationtimes






05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations










Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Inversedvalue

Iterationtimes




Inversedvalue

Iterationtimes






05000

1000015000200002500030000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cos

t fun

ctio

n

Iterations










Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article analysis of the properties of adjoint equations...

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