research article diffusion filters for variational data...

Research ArticleDiffusion Filters for Variational Data Assimilation ofSea Surface Temperature in an Intermediate Climate Model

Xuefeng Zhang12 Dong Li2 Peter C Chu3 Lianxin Zhang24 and Wei Li2

1College of Automation Harbin Engineering University Harbin 150001 China2Key Laboratory of Marine Environmental Information Technology National Marine Data and Information ServiceState Oceanic Administration Tianjin 300171 China3Naval Ocean Analysis and Prediction Laboratory Department of Oceanography Naval Postgraduate SchoolMonterey CA 93943 USA4College of Physical and Environmental Oceanography Ocean University of China Qingdao 266100 China

Correspondence should be addressed to Xuefeng Zhang xfz nmdis126com

Received 6 December 2014 Accepted 7 April 2015

Academic Editor Juan Jose Ruiz

Copyright copy 2015 Xuefeng Zhang et alThis 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

Sequential adaptive and gradient diffusion filters are implemented into spatial multiscale three-dimensional variational dataassimilation (3DVAR) as alternative schemes to model background error covariance matrix for the commonly used correctionscale method recursive filter method and sequential 3DVARThe gradient diffusion filter (GDF) is verified by a two-dimensionalsea surface temperature (SST) assimilation experiment Compared to the existing DF the new GDF scheme shows a superiorperformance in the assimilation experiment due to its success in extracting the spatial multiscale informationTheGDF can retrievesuccessfully the longwave information over the whole analysis domain and the shortwave information over data-dense regionsAfter that a perfect twin data assimilation experiment framework is designed to study the effect of the GDF on the state estimationbased on an intermediate coupled model In this framework the assimilation model is subject to ldquobiasedrdquo initial fields from theldquotruthrdquo model While the GDF reduces the model bias in general it can enhance the accuracy of the state estimation in the regionthat the observations are removed especially in the South Ocean In addition the higher forecast skill can be obtained through thebetter initial state fields produced by the GDF

1 Introduction

In general standard three-dimensional variational dataassimilation (3DVAR) can be formulated as theminimizationof the following cost function [1 2]

119869 (119909) =1

2(119909 minus 119909119887)

119879119861minus1(119909 minus 119909119887)

+1

2(119867119909 minus 119910)

119879119877minus1(119867119909 minus 119910) = 119869119887 + 119869119900

(1)

where 119909 is the analysis vector 119909119887 is the background vector 119910is the observation vector119867 is an interpolation operator frommodel space to observation space119877 is the observational errorcovariancematrix (sdot)119879 indicates transpose and (sdot)minus1 indicatesinversion 119861 is the background error covariance matrix

It is a challenge to determine 119861 in any data assimilationincluding 3DVar The spatial structure and the magnitudeof the correction for the state variables being estimated aredetermined completely by 119861

Two common approaches are used to prescribe 119861 Thefirst approach is the correlation scale method (CSM) [3] inwhich 119861 is represented by the Gaussian function

119861 = 119860 (119909 119910) exp(minus1199032

119909

1198712119909

minus1199032

119910

1198712119910

) (2)

where 119860 is an estimate of the magnitude of the backgrounderror 119903119909 and 119903119910 are the distances between two grid pointsand 119871119909 and 119871119910 are the characteristic length scales reflectingthe extent of spatial correction of the background error in the119909 and 119910 directions respectively A more general anisotropic

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2015 Article ID 751404 17 pageshttpdxdoiorg1011552015751404

2 Advances in Meteorology

shape with an ellipsoid about the spatial covariance canalso be found [4] It is noted that 119861 is explicitly generatedstatistically using the correlation scales [3 5ndash7] Howeverlimitations of the CSM are (1) positive value for each elementin119861 (which is not always true) (2) nonexistence of 119861minus1 unlessusing sufficiently small correction scales and (3) requirementof large computer memory to store 119861 since every elementin 119861 is calculated explicitly To avoid the inversion of 119861 andto speed up the convergence of descent algorithms such asthe steepest descent and conjugate-gradient methods a newvector 119908 is introduced by Lorenc [8] and Derber and Rosati[3] defined as

119908 = 119861minus1(119909 minus 119909119887) (3)

Then the cost function 119869 can now be rewritten as

119869 (119908) =1

2119908119879119861119879119908 +

1

2(119867119861119908 minus 119889)

119879119877minus1(119867119861119908 minus 119889) (4)

where 119889 = 119910 minus 119867119909119887 is the ldquoinnovationrdquo vector and forsimplification hereafter we will call it observation

Considering 119861119879 = 119861 (4) is equal to

119869 (119908) =1

2119908119879119861119908 +

1

2(119867119861119908 minus 119889)

119879119877minus1(119867119861119908 minus 119889) (5)

The effect of 119861119908 in (5) can be modeled by applying anequivalent spatial filter on 119908

The second approach to prescribe 119861 is the recursive filtermethod (RFM) [9]

119884119894 = 120572119884119894minus1 + (1 minus 120572)119883119894 for 119894 = 1 119899

119885119894 = 120572119885119894+1 + (1 minus 120572) 119884119894 for 119894 = 119899 1(6)

where119883119894 is the initial value at grid point 119894119884119894 is the value afterfiltering for 119894 = 1 to 119899 119885 is the initial value after one pass ofthe filter in each direction and120572 is the filter coefficient whichdetermines the extent of spreading of observational informa-tion over the analysis domain Multipass filter can be built upby repeated application of (6) Multidimensional filter can beconstructed by applying this one-dimensional filter in eachdirection It can be shown [10] that such multidimensionalfilter when applied with several passes can accurately modelisotropic Gaussian error correlations The implementationusing recursive filter to model 119861 has been widely used dueto its relatively computational inexpensiveness [10ndash13]

An outstanding issue of either CSM or RFM is itsinefficiency in capturing the spatial multiscale informationby observations A difficulty in practice is how to properlychoose the characteristic length scales 119871119909 and 119871119910 in theCSM or the filter coefficient 120572 in the RFM Observationalstudies show that (119871119909 119871119910) change with location depthand time [14ndash16] If they are too large the analysis is toosmooth and shortwave information is lost If they are toosmall the analysis lacks coherent structure in data sparseregions because the longwave information cannot be properlycorrected Thus in the past it has been thought that thecharacteristic length scales (119871119909 119871119910) in the CSM or the filtercoefficient 120572 in the RFM is responsible for the unsatisfactoryanalysis in the 3DVAR

To avoid empirically or statistically setting the character-istic length scales and to correctly minimize the longwaveand shortwave errors in turn a sequential 3DVAR (S3DVar)method was developed [17] to assimilate sea surface temper-ature (SST) in a global oceanmodel [18]The S3DVarmethodis simply composed of a series of 3DVars each of which usesrecursive filters with different filter coefficientsThese 3DVarssweep through all resolvable scales by observational networksfrom longwaves to shortwaves In addition a multigriddata assimilation scheme was also introduced to extract theresolvable information from longwave to shortwave in anobservational system [19] Recently a sequential variationalapproach based on the multigrid data assimilation methodwas proposed to accurately retrieve the multiscale informa-tion from available observation systems [20]

Since the matrix 119861 is treated as the Gaussian type inthe CSM and modeled as the diffusion process (or Gaussianfiltering process) in the RFM the spread of the informationfrom the analysis point to the entire region is interpretedas the diffusion phenomenon [21] The diffusion filter (DF)was developed on the base of the Gaussian diffusion processand therefore can be used directly to model 119861 Severalspatial multiscale variational analysis schemes based on themodification to the standard DF scheme are proposed in thisstudy As a pilot study one of the spatialmultiscale variationalanalysis schemes the gradientDF (GDF) is used to assimilateSST observations into an intermediate coupled model withina perfect ldquotwinrdquo experiment framework

The paper is organized as follows The methodology ofthe standard DF scheme is described in Section 2 Severalspatial multiscale DF schemes are presented in Section 3In Sections 4 and 5 simple observingassimilation systemsimulation experiments and global SST simulation withan intermediate coupled atmosphere-ocean-land model areconducted to evaluate one of the newDF schemes that is thegradient DF (GDF) on the model estimation and forecastThe conclusions are summarized in Section 6

2 Diffusion Filter

The DF is in fact a Gaussian filter Given the following initialvalue problem for one-dimensional diffusion equation

120597119906

120597119905= 119886

1205972119906

1205971199092

119906 = 119908 (119909)

119905 = 0

(7)

where 119886 gt 0 is the diffusion coefficient assumed to beconstant Its solution can be formulated by the convolutionof 119908(119909) with a Gaussian kernel 119866(119909 119905)

119906 (119909 119905) = 119908 (119909) lowast 119866 (119909 119905) (8)

where (lowast) indicates convolution 119866(119909 119905) = (1radic2120587120590)119890minus119909221205902

120590 = radic2119886119905 That is 119906(119909 119905) is equivalent to applying a Gaussianfilter on initial value 119908(119909) The second moment of the filterkernel is 1205902 = 2119886119905 which characterizes the intrinsic spatial

Advances in Meteorology 3

scale And 1205902 is only determined by diffusion coefficient 119886

when ldquotimerdquo duration 119905 is set to be constant which impliesthat the larger the value of 119886 is the lower the frequencyinformation of 119908(119909) would be acquired by 119906(119909 119905)

Generally in a two-dimensional finite domain the diffu-sion model can be written by

120597119906

120597119905= 119886

1205972119906

1205971199092+ 119887

1205972119906

1205971199102(119909 119910) isin Ω 119905 isin (0 119878]

119906 = 119908 (119909 119910) (119909 119910) isin Ω 119905 = 0

120597119906

120597119899= 0 (119909 119910) isin Γ

(9)

whereΩ = [0119863]times[0119867] Ω = ΩΓ is the interior domain ofΩ Γ is the boundary ofΩ 119899 is the outer normal direction of Γand 119886 and 119887 are the diffusion coefficients in119909 and119910directionsrespectively

If 119906119878(119908) denotes 119906(119908)|119905=119878 the cost function (5) thenbecomes

119869 (119908) =1

2119908119879119906119878(119908)

+1

2(119867119906119878(119908) minus 119889)

119879

119877minus1(119867119906119878(119908) minus 119889)

(10)

Now the analysis is converted to the problem of optimizingthe initial value of the diffusion equation (9) To do sowe need the gradient of the cost function which can bederived by using adjoint methods just as four-dimensionalvariational (4DVAR) data assimilation usually does

For convenience of illustration a continuous adjointsystem is considered and 119869119887 is omitted It is also assumedthat the observations are located at analysis points and 119867 isthe identity matrix Then the adjoint of the tangential linearmodel of (9) takes the following form

minus120597119877

120597119905minus 119886

1205972119877

1205971199092minus 119887

1205972119877

1205971199102= 119891 (119909 119911) isin Ω 0 lt 119905 le 119878

119877 (119909 119910 119878) = 0 (119909 119911) isin Ω

120597119877

120597119899= 0 (119909 119910) isin Γ

(11)

where

119891 =

119889res (119908) = 119889 minus 119906119878(119908) 119905 = 119878

0 119905 lt 119878(12)

Note that 119889res(119908) is the observation residue which character-izes the remaining observational signals after the abstractedinformation at current solution 119908 119906119878(119908) has been removedform observations 119889 and 119889res(119908) is set to be zero at the gridpoints with no observations

The gradient of 119869 with respect to 119908 is 119892(119908) = minus1198770(119908)

where 1198770(119908) is the initial value of the adjoint variables Oncethe adjoint model is available the analysis can be performedin the following steps

(1) Choose an appropriate diffusion coefficient 119886 give theinitial guess of 119908 (119908 = 0 for instance)

(2) Integrate the diffusion model (9) from ldquotimerdquo 119905 = 0 to119878 to obtain 119906119878(119908)

(3) Calculate 119891 according to (12)(4) Integrate the adjoint model (11) from ldquotimerdquo 119905 = 119878 to

0 to obtain 1198770(119908) then the gradient 119892(119908) of the costfunction 119869 is minus1198770(119908)

(5) Use descent algorithms to adjust 119908(6) Loop from step (2) until the convergence criterion is

met

Use of DF for determining the matrix 119861 is called the DFmethod (DFM) which has the same computation loads asthe RFM if the ADI difference scheme (or the other operatorsplitting scheme see Appendix) is applied to calculate thediffusion equation (9) The diffusion filter scheme has thesame problem as the recursive filter scheme in extractingobservational information As the extent of spatial dispersionis only determined by diffusion coefficient 119886 when ldquotimerdquoduration 119905 is set to be constant if 119886 is large the shortwaveinformation will be lost Conversely if 119886 is small the long-wave information will not be properly captured Obviouslythe diffusion coefficient 119886 plays the same role as the filtercoefficient 120572 does in the recursive filter scheme

3 Spatial Multiscale Diffusion Filters

To retrieve longwave information over the whole domainand shortwave information over data-dense regions threespatial multiscale variational analysis schemes based on thediffusion filter are proposed

31 Sequential Diffusion Filter (SDF) The sequential diffu-sion filter (SDF) scheme is similar to the S3DVar methodderived by Xie et al [17] The SDF scheme uses a sequence of3DVars to obtain the final estimation to retrieve informationfrom all wavelengths from long- to shortwaves in turnThe matrix 119861 is modeled by applying the diffusion filtersequentially in 119909 and 119910 direction respectively SDF beginsits sequence with a big value of the diffusion coefficient119886 then an initial estimation is obtained through analyzingthe observed data After that a S3DVar is solved using thediffusion filter with a smaller 119886 than before For the S3DVarsobservations to be assimilated are produced by subtractingthe previously analyzed values from the observations assim-ilated by the previous 3DVar until the diffusion coefficient 119886is small enough The final estimation is the summation of allthe previous 3Dvar analyses based on the diffusion filter

From the above description it is noted that the SDFscheme is a simple extension of the DF in which informationis retrieved step by step from long- to shortwaves During theprocess of the SDF 119861 is changed gradually with the differentdiffusion coefficient 119886 and thus becomes flow dependentand anisotropic following the multiscale information of theobservation


32 Adaptive Diffusion Filter (ADF) Due to the introductionof the heat diffusion equation the gradient of the cost func-tion with respect to the state variables can be obtained usingthe adjoint method with 4DVar In general the diffusioncoefficients 119886(119909 119910) and 119887(119909 119910) are not constants but are spacedependent Therefore it is possible to optimize not only thestate variables but also the diffusion coefficients using 4DVarState variables and diffusion coefficients are used together ascontrol variables so values of 119886(119909 119910) and 119887(119909 119910) will changeadaptively according to the distribution of observations

Set 119868ℎ1

119868ℎ2

119868120591 as

119868ℎ1

= 119909119894 = 119894ℎ1 | 119894 = 0 1 119868 ℎ1 =119863

119868

119868ℎ2

= 119910119894 = 119895ℎ2 | 119895 = 0 1 119869 ℎ2 =119867

119869

119868120591 = 119905119899 = 119899120591 | 119899 = 0 1 119873 120591 =119878

119873

Ωℎ = 119868ℎ1

times 119868ℎ2

(13)

The cost function is transferred to the following form

119869 (119908 119886 119887)

=1

2

119868minus1

sum

119894=1

119869minus1

sum

119895=1

1199082

119894119895

+1

2

119872

sum

119898=1

119901(119898) [

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

2

(14)

where 119872 is the number of observations and 119902(119898)

119894119895is the

interpolation coefficient of the grid point (119894 119895)with respect tothe119898th observation 119901(119898) is the119898th element of the diagonalmatrix 119877minus1 For calculating the gradients of the cost function119869 with respect to 119908 119886 119887 in (12) the discrete adjoint modelsof (A1)ndash(A11) should be deduced firstly according to theLagrange multiplier method

119877119899

119894119895minus 119877119899minus12

119894119895

1205912+ Δ 119910 (119887119894119895Δ119910 (119877

119899

119894119895+ 119877119899minus12

119894119895)) = 119891

119899 isin [1119873] 119894 isin [1 119868 minus 1] 119895 isin [1 119869 minus 1]

119877119899+12

119894119895minus 119877119899

119894119895

1205912+ Δ 119909 (119886119894119895Δ119909 (119877

119899+12

119894119895+ 119877119899

119894119895)) = 0

119899 isin [0119873 minus 1] 119894 isin [1 119868 minus 1] 119895 isin [1 119869 minus 1]

119877119873

119894119895= 0

119894 isin [0 119868] 119895 isin [0 119869]

Δ 119909119877119899

0119895= 0

Δ119909119877119899

119868119895= 0

Δ 119910119877119899

1198940= 0

Δ119910119877119899

119894119869= 0

119899 isin [0119873 minus 1]

Δ 119909119877119899+12

0119895= 0

Δ119909119877119899+12

119868119895= 0

Δ 119910119877119899+12

1198940= 0

Δ119910119877119899+12

119894119869= 0

119899 isin [0119873 minus 1]

(15)

where119891

=

2

119872

sum

119898=1

119901(119898)119902(119898)

119894119895[

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

119899 = 119873

0 119899 lt 119873

(16)

The gradients of the cost function 119869with respect to119908 119886 119887 canbe expressed as follows

120597119869

120597119908119894119895= 119908119894119895 minus

1198770

119894119895

120591minus1

2Δ 119910 (119887119894119895Δ119910119877

0

119894119895)

120597119869

120597119886119894119895=1

2

119873minus1

sum

119899=1

[Δ119909119906119899+12

119894119895sdot Δ119909 (119877

119899

119894119895+ 119877119899+12

119894119895)]

120597119869

120597119887119894119895=1

2

119873minus1

sum

119899=1

(Δ119910119906119899

119894119895Δ119910119877119899

119894119895+ Δ119910119906

119899+1

119894119895Δ119910119877119899+12

119894119895)

(17)

Theprocess for the state estimationwith the 4DVar is outlinedas follows (a) Begin with the initial 119908 119886 119887 (b) Integratethe model equations (A1)ndash(A11) forward into a fixed timewindow and calculate the value of the cost function 119869(119908 119886 119887)using (14) (c) Integrate the adjoint model (15) backwardin time and calculate the values of the gradient of thecost function with respect to the control variables nabla119869 using(17) (d) With the values of the cost function 119869(119908 119886 119887) andthe gradient nabla119869 use the Broyden-Fletcher-Goldfarb-Shanno(BFGS) quasi-Newton minimization algorithm to obtain thenew values of the control variables namely the two diffusioncoefficients 119886 119887 and the state variables119908 (e)With the updatedcontrol variables from process (d) repeat processes (b) (c)and (d) until the convergence criterion for the minimizationis satisfied


33 Gradient Diffusion Filter (GDF) The algorithm is avariant of the spatial multiscale recursive filter [22] For smalldiffusion coefficients 119886 119887 the gradient contains not only allthe observational signals from longer to shorter wavelengthsbut also a lot of erroneous signals in data sparse regionswhich causes lack of coherent longwave structure in spaceIf this gradient is simply introduced into the minimizationalgorithmwithout careful considerations the analysis departsfar from reality Thus a prerequisite for the minimizationalgorithm used in 3DVAR is needed to extract the longwaveinformation from the gradient and at the same time topreserve the valuable shortwave signals

However the longwave information implied in the gra-dient cannot be made best use of to construct a reasonabledescent direction in general minimization algorithms Takethe steepest descent algorithm as an example in which thedescent direction is simply chosen as minus119892(119908) Suppose theinitial guess of119908 (ie1199080) is equal to zeroThen at the 119894th iter-ation the new solution119908119894 = 119908119894minus1+119897119894minus1lowast(minus119892(119908119894minus1)) is obtainedby using a line search algorithm to find an appropriate stepsize 119897119894minus1 According to what have been indicated the gradient119892(1199080) actually represents certain scales of observations119889 andthese scales will be extracted by the line search at the firstiteration and incorporated into a new solution 1199081 Howeverif the diffusion coefficients 119886 119887 are small the gradient 119892(1199080)will lack coherent structure in data sparse regions though itactually carries all observational signals And since the newsolution 1199081 is simply obtained along the descent directionminus119892(1199080) the same problem will also exist in 1199081 whichindicates that the longwave information of observations 119889is not effectively extracted from the gradient 119892(1199080) at thefirst iteration Similarly at the second iteration the longwaveinformation of the observation residue after the first iterationwill not be extracted from the gradient 119892(1199081) and incorpo-rated into the new solution 1199082 and so on As a consequencein data sparse regions the final analysis will also lose thelongwave structure of observations The same problem alsoexists for other minimization algorithms such as BFGS andthe conjugate gradient method for the same reason

The GDF scheme is designed to effectively retrieve thelongwave information over the whole domain and shortwaveinformation over data-dense regions Since the gradientcarries all observational information the main idea of thisnew scheme is to apply the diffusion filter on the gradient toextract the implied longwave signalWhile the diffusion coef-ficient decreases continuously with iteration the multiscaleinformation from long to short wavelengths can be extractedsuccessively The algorithm is designed as follows

(1) Give an initial guess of 119908 (ie 1199080) which equals zeroThen select diffusion coefficients 119886 119887 as small onesand give a large enough value to an extra diffusioncoefficient denoted as 120573

(2) Use the diffusion filter with coefficient 119886 119887 to calculate119861119908 in (5)

(3) Calculate the difference between observations 119889 and119867119861119908 namely the observation residue

(4) Calculate the gradient 119892 of the cost function 119869 withrespect to 119908 using the DF through the adjoint model

(5) Apply the diffusion filter with coefficient 120573 on minus119892

to calculate the descent direction 119864(minus119892) where 119864

represents a positive definite operator(6) Select 119864(minus119892) as the descent direction and use line

search algorithm to find the step size 119897 then 119908 isadjusted to 119908 = 119908 + 119897 lowast 119864(minus119892)

(7) The value of 120573 diminishes(8) Loop from step (2) until the convergence criterion is

metIf the background term 119869119887 is involved in the cost function

119869 the same procedure is performed except that 119892 calculatedin step (4) is the gradient of cost function 119869 which includesboth 119869119887 and 119869119900

4 ObservingAssimilation SystemSimulation Experiments

Observingassimilation system simulation experiments areperformed to evaluate the spatial multiscale variational anal-ysis The ldquotruthrdquo field in these experiments is represented byan analytic temperature field defined over the area of 100∘Endash110∘E and 30∘Nndash40∘N The ldquotruthrdquo field of the temperatureis plotted in Figure 1(b) whose high nonlinearity can beseen from Figure 1(a) The grid resolution is set to 18

∘times

18∘ and the total numbers of the grid are 80 times 80 The

observational dataset is generated using the analytic solutionObservational error is simulated by adding a sample of whitenoise with a standard deviation of 02 to the ldquotruthrdquo Threeexperiments are conducted in which different configurationsof numbers of observations are employed

41 Experiment 1 In this experiment the number of observa-tions is set to 2000 at first and the observations are randomlyand uniformly distributed in the whole domain which canbe seen from the black dots in Figure 1(b) In the experimentwith DF several values of the diffusion coefficient are usedto verify the impacts on the analyzed field In the experimentwith GDF the processes (1)ndash(8) described in Section 33 areconducted The diffusion coefficients 119886 119887 are set to a smallvalue of 01 which suggests almost all the observationalsignals from long to short wavelengths can be retrievedHowever a large enough value 10 is given to the extradiffusion coefficient 120573 of the gradient at the first step For thesubsequent steps 120573 is reduced by 01 from the previous stepAt the last step 120573 becomes 01 which is small enough for thecaseThe limitedmemory BFGS quasi-Newtonminimizationalgorithm [23] is used during the minimizing procedure

The major scales of the truth field are reconstructed by2000 observations almost fully using the GDF (Figure 2(a))but not well reconstructed using the DF with different diffu-sion coefficients (119886 119887) 10 (Figure 2(b)) 05 (Figure 2(c)) and01 (Figure 2(d)) The small scale features begin to dominatethe analyzed fieldswhen the diffusion coefficients are reducedgradually while the large scale signals are contaminateddramatically by an abundance of small scale features


30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)

Figure 1 The true temperature field to be analyzed (unit ∘C) (a) latitudinal variation along 100∘E and (b) ichnography image Black dots inthe panel (b) show the distribution of 2000 random observations

As He et al [18] indicated artificial signals can beproduced during the data assimilation if the chosen diffusioncoefficient cannot represent the actual scale In contrastthe GDF can handle spatial multiscale analysis pretty wellcompared to the simple DF with a fixed diffusion coefficientIn addition the GDF is easy to avoid in empirical selection ofthe diffusion coefficient

Figure 3 shows the performance of GDF and DF whenthe number of observations decreases from 2000 to 500 TheGDF (Figure 3(a)) can retrieve large scale information fromobservations and leave the unresolved scale as errors on top ofthe resolvable scalesThese errors are smaller than those gen-erated by DF with a fixed diffusion coefficient (Figure 3(b))in the condition of the sparseness of the observations and thelack of information

42 Experiment 2 The second experiment is conducted withremoval of observations in the area of 103∘Esim107∘E and 35∘Nsim40∘N (Figure 4) to further evaluate the GDF capability inretrieving the multiscale information from observationsTheanalyzed field of the GDF (Figure 5(a)) performsmuch betterin the data void region than that of the DF with (119886 119887) = 08(Figure 5(b)) and 05 (Figure 5(c)) The GDF can reconstructthe temperature field (Figure 5(a)) reasonably well despitethe absence of the observations in the region as shown inFigure 4 The spatial pattern of the whole temperature fieldcan be captured roughly according to the large scale informa-tion derived from all the observations in the whole analyzedregion However the DF fails to reconstruct the temperaturefield and produces false features especially in the data voidregion For example a strong cold tongue is produced for(119886 119887) = 08 (Figure 5(b)) and large scale temperature field isdistorted with displacement of the thermal front in the datavoid region for (119886 119887) = 05 (Figure 5(c)) Little information of

the observations can be extracted from data rich area to thedata void region using DF

Such capabilities make the GDF invaluable to get wellrepresented values for the data void (or insufficiently cov-ered) areas such as a typhoon-affected area during typhoonpassage or the Southern Hemisphere Oceans (compared toother ocean basins) The GDF can reconstruct the analyzedfield roughly according to the longwave information of theobservations beyond the data void area such as typhoon-affected region or the Southern Ocean On the other handboth the standard DF and the traditional RFmay lead to falseresults in the data void region as shown in Figures 5(b)-5(c)an improper analysis is also likely to be produced which willaffect the analysisforecast accuracy seriously

In addition several classical geostatistical tools such asinverse distance to a power triangulation with linear inter-polation and Kriging method are used to interpolate suchobservations (nowhite noise is imposed on the observations)Compared to the other two geostatistical tools the Krigingmethod is able to accurately fill in the hidden information(Figures 6(a) and 6(b) versus 6(c)) However compared withthe variational method the geostatistical tools have a limitedapplication and cannot handle corrections between differentanalysis variables or physical balances and other constraints[20]

5 Global SST Assimilation Using GDF

In this section we apply the GDF to assimilate the SSTinto an intermediate climate model to improve the climaterepresentation and forecast

51 Brief Description of an Intermediate Atmosphere-Ocean-Land Coupled Model An intermediate atmosphere-ocean-land coupled model [24] is employed as the first step to


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)

Figure 2 The analyzed temperature fields (unit ∘C) using (a) GDF (b) DF with 119886 119887 = 10 (c) DF with 119886 119887 = 05 and (d) DF with 119886 119887 = 01

examine the GDF Despite limitations in the representationsof some basic physical processes such as the absence ofENSO dynamical mechanism the model is of sufficientmathematical complexity for the purposes of this study Theintermediate coupledmodel has some successful applicationsin coupled data assimilation fields recently For example Wuet al [25] investigated the impact of the geographic depen-dence of observing system on parameter estimation andZhang et al [26] studied parameter optimization when theassimilation model contains biased physics within a biasedassimilation experiment frameworkThe configuration of themodel is presented here The atmosphere is represented by

a global barotropic spectral model based on the potentialvorticity conservation

120597119902

120597119905+ 119869 (120595 119902) =

120582 (119879119900 minus 120583120595) ocean sdot surface

120582 (119879119897 minus 120583120595) land sdot surface(18)

where 119902 = 120573119910 + nabla2120595 120573 = 119889119891119889119910 119891 is the Coriolis

parameter 119910 is the meridional distance from the equator(northward positive) and 120595 is the geostrophic atmospherestream function 120583 is a scale factor which converts streamfunction to temperature 120582 is the flux coefficient from theocean (land) to the atmosphere 119879119900 and 119879119897 denote SST


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)

Figure 3 The analyzed temperature fields (unit ∘C) with 500 observations analyzed fields using (a) GDF and (b) DF with 119886 119887 = 10

and land surface temperature (LST) respectively Wu et al[27] used the nonlinear atmospheric model to develop acompensatory approach of the fixed localization in EnKFanalysis to improve short-term weather forecasts

The ocean is composed of a 15-layer baroclinic oceanwith a slab mixed layer [28] as

120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601

= minus119870119879119879119900 + 119860119879nabla2119879119900 + 119904 (120591 119905) + 119862119900 (119879119900 minus 120583120595)

(19)

where 120601 is the oceanic stream function and 1198712

0= 1198921015840ℎ01198912

is the oceanic deformation radius with 1198921015840 and ℎ0 being

the reduced gravity and mean thermocline depth 120574 denotesmomentum coupling coefficient between the atmosphere andocean 119870119902 is the horizontal diffusive coefficient of 120601 119870119879and 119860119879 are the damping coefficient and horizontal diffusivecoefficient of 119879119900 119870ℎ = 119870119879 times 120581 times 119891119892

1015840 [29] where 120581 is theratio of upwelling to damping 119862119900 is the flux coefficient fromthe atmosphere to the ocean 119904(120591 119905) is the solar forcing whichintroduces the seasonal cycle

The evolution of land surface temperature (LST) is givenby

119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)

where 119898 represents the ratio of heat capacity between theland and the ocean mixed layer 119870119871 and 119860119871 are damping anddiffusive coefficients of 119879119897 respectively and 119862119897 denotes theflux coefficient from the atmosphere to the land

101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39

Figure 4 The distribution of 2000 random observations butremoval of the observations located in the range of 103∘Nsim107∘Nand35∘Esim40∘E

All the three model components adopt 64 times 54 Gaussiangrid and are forwarded by a leap frog time steppingwith a halfhour integration step sizeThere are 2287 and 1169 grid pointsover the ocean and land respectively AnAsselin-Robert timefilter [30 31] is introduced to damp spurious computationalmodes in the leap frog time integration Default values of allparameters are listed in Table 1 in Wu et al [24]

Starting from initial conditions 1198850 = (1205950 1206010 1198790

119900 1198790

119897)

where120595012060101198790119900 and1198790

119897are zonalmean values of correspond-

ing climatological fields the coupledmodel is run for 60 years


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)

Figure 5 The analyzed temperature fields (unit ∘C) using (a) GDF (b) DF with 119886 119887 = 08 (c) DF with 119886 119887 = 05

to generate the model states 1198851 = (1205951 1206011 1198791

119900 1198791

119897) The last 10

yearsrsquo model states (1198851) are used as the ldquotruthrdquo fields Figure 7shows the annual mean of 120595 (Figure 7(a)) 120601 (Figure 7(b))119879119900 and 119879119897 (Figure 7(c)) where the associated wave trains inthe 120595 field are observed For 120601 one can see the distinctpattern of the western boundary currents gyre systems andthe Antarctic Circumpolar Current (ACC) For 119879119900 and 119879119897reasonable temperature gradients are also produced Notethat the low temperature in tropical lands can be attributedto the linear damping of 119870119879 in the solar forcing The abovemodel configuration is called the ldquotruthrdquo model which hasreasonable but rough representation for the basic climatecharacteristics of the atmosphere land and ocean

52 Model ldquoBiasrdquo Arising from the Initial States Howeverstarting also from the same initial conditions 1198850 the Gaus-sian random numbers are added to 1205950 and 1206010 with standarddeviations of 107m2sminus1 (for 120595

0) and 105m2sminus1 (for 1206010)

respectively The coupled model is also run for 60 yearsto generate the model states 1198852 = (120595

2 1206012 1198792

119900 1198792

119897) The

last 10 yearsrsquo model states are used for analysis This modelconfiguration is called the biased model

Themodel ldquobiasesrdquo induced by perturbed initial fields areexamined Figure 8 shows time series of the spatial averagedroot mean square errors (RMSEs) of 120595 120601 119879119900 and 119879119897 for theassimilation model which are calculated according to thedifference in the assimilation model and the ldquotruthrdquo model


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)

Figure 6 Analyzed temperature results (unit ∘C) from classical geostatistical tools such as (a) inverse distance to a power (b) triangulationwith linear interpolation and (c) Kriging method

The obvious difference about all the four components canbe seen from Figure 8 The RMSE of 120595 reaches about 16 times107m2sminus1 with a high frequent oscillation (see Figure 8(a))In contrast the RMSE of 120601 performs smoothly and rapidlydecreases within the first year and gradually reaches a low andstable value about 104m2sminus1 (see Figure 8(b)) The RMSEs of119879119900 (Figure 8(c)) and 119879119897 (Figure 8(d)) increase rapidly in thefirst year which are generated by the initially perturbed 120595

0

and 1206010 through the coupling High frequency oscillation is

noted in the time series of the RMSE of 119879119897 which indicatesthat the land surface temperature 119879119897 is dominated by theatmospheric motion (120595) However time series of the RMSEof 119879119900 is much smoother than that of 119879119897 indicating that 119879119900 ismodulated by the oceanic motion (120601)

The spatial distribution of RMSE of 119879119900 for the biasedmodel (Figure 9) shows a notable bias in the ACC region ofthe SouthernOceanwith amaximumvalue over 15K near thesouthern tip of Africa Besides obvious biases also exist in thewest boundaries of the ocean in the subtropical regions

53 Twin Experiment Design In this section with theintermediate model and DFGDF data assimilation schemedescribed above a perfect twin experiment framework isdesigned with the assumption that the errors of initial modelstates are the only source of assimilation model biasesStarting from the model states 1198851 described in Section 51the ldquotruthrdquo model is run for 1 year to generate the timeseries of the ldquotruthrdquo states Only synthetic observations of 119879119900


0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)

Figure 7 Annual mean of (a) atmospheric stream function (unit m2sminus1) (b) oceanic stream function (unit m2sminus1) and (c) sea surfacetemperature and land surface temperature (unit K)

1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)

Figure 8 Time series of RMSEs of (a) atmospheric stream function (b) oceanic stream function (c) sea surface temperature and (d) landsurface temperature for the assimilation model


0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820

Figure 9 The spatial distribution of RMSE of sea surface tempera-ture for the biased model

are produced through sampling the ldquotruthrdquo states at specificobservational frequencies A Gaussian white noise is addedfor simulating observational errors The standard deviationsof observational errors are 05∘C for 119879119900 The sampling periodis 24 hours The ldquoobservationrdquo locations of 119879119900 are globalrandomly distributed with the same density of the oceanmodel grid points

The biased model uses the biased initial fields depictedin Section 52 Starting from the biased model states 1198852the experiment E GDF consists on assimilating observationsinto model states using the GDF scheme In comparisonthe experiment E DF is carried out where the standard DFscheme is used with the diffusion coefficients 119886 119887 = 05 Inaddition a control run without any observational constraintcalled CTRL serves as a reference for the evaluation ofassimilation experiments

54 Impact of the GDF on the Estimate of the States Theperformance of GDF is investigated Figures 10(a)ndash10(d)show time series of RMSEs of 120595 120601 119879119900 and 119879119897 for the CTRL(solid line) and the GDF (dash line) Compared to the CTRL(solid line in Figure 10(c)) 119879119900 of the GDF has significantimprovement (dash line in Figure 10(c)) in which the RMSEdecreases to approximately 05 K Figure 11 presents the spa-tial distributions of RMSEs of 119879119900 using GDF The RMSE of119879119900 over ocean is obviously reduced compared with that ofthe CTRL (see Figure 9) especially in the Southern Oceanthe subtropical and the subpolar regions In particular thereduction of RMSE is much significant in the ACC regionin which the RMSE decreases from above 15 K to below 3K

Unlike the RMSEs of 119879119900 there is no direct observa-tions constraint for 120595 120601 and 119879119897 therefore their RMSEsdecrease gradually owing to the effect of the coupling TheRMSEs of 120595 for the GDF are reduced significantly fromabout 16 times 107m2sminus1 to about 11 times 107m2sminus1 with a highfrequent oscillation (see Figure 10(a)) The 120601 in GDF is alsoimproved significantly comparing to CTRL (solid versus dashlines in Figure 10(b)) whose RMSE decreases gradually andsmoothly but it does not reach a stable value within theexperimental period indicating that the low frequency signalneeds a much longer time to reach equilibrium comparedto the high frequency signal For 119879119897 the GDF reduces theerror by approximately 60 Note that 119879119900 has no direct effecton 119879119897 which can be realized according to the framework

of the coupling model (see (18)ndash(20)) Instead 119879119900 affects119879119897 indirectly via 120595 The improved 119879119900 by the observationalconstraint increases the quality of 120595 over land through thedynamical constraint Then the improved 120595 ameliorates 119879119897through the process of the external forcing

55 Removal of Observational Data in the Southern Ocean Inthe real ocean the observations are scarce in the southernpolar region Therefore another set of data assimilationexperiment is carried out which is the same as the experi-ment in Section 53 but in which the observations south of50∘S and 50∘Esim300∘E are removed completely

Figures 12(a)ndash12(d) show the time series of RMSEs of 120595120601 119879119900 and 119879119897 with the GDF (black line) and the standardDF with 119886 119887 = 05 (red line) The RMSE of 119879119900 for the DFincreases persistently during the experimental period whilethe RMSE for the GDF begins to descend after 02 years andconverges after 06 years (Figure 12(c)) When the diffusioncoefficients are set to different values in the DF experiment(eg 119886 119887 = 02 08 10) similar results as the ones presentedin Figure 12 are obtained Results indicate that DF cannotcorrect the model bias in the data void region Howeverthe GDF is able to mitigate the model bias to some degreethrough extracting the spatial multiscale information fromthe available observations to the data void region Figure 13presents the spatial distributions of RMSEs of 119879119900 for the GDFand the standard DF with 119886 119887 = 05 Compared to the DFthe GDF produces a significant improvement within the datavoid region in the Southern Ocean (compare Figures 13(a)and 13(b))

The RMSE of 120595 in the GDF is not always smaller than theDF owing to the strong nonlinear nature of the high frequentatmosphere (red line versus black line in Figure 12(a)) Incontrast the evolution of 119879119897 in the model (see (20)) israther simple (ie linear) the RMSE in the GDF is almostalways smaller than that for the DF (red line versus blackline in Figure 12(d)) For the low frequent component 120601(see Figure 12(b)) the RMSEs of both the GDF and the DFdecrease gradually indicating that the effect of the data voidregion on the low frequent signal is small in the given timescale

56 Impact of the GDF on the Forecast From amore practicalpoint of view the role of the GDF should be judged fromthe model forecast In this section two forecast experimentswithout any observational constraint are integrated for 1 yearrespectively starting from the final analyzed states of theabove two assimilation experiments (the GDF and the DF)

Figures 14(a)ndash14(d) show the forecasted time series ofRMSEs of 120595 120601 119879119900 and 119879119897 for the GDF (black line) and theDF with 119886 119887 = 05 (red line) The GDF performs muchbetter than the DF in 1 yearrsquos forecast lead time of all the statevariables such as the high frequent component 120595 and the lowfrequent component 120601 (black versus red curves in Figures14(a) and 14(b)) It is interesting that the forecasted RMSEof 120601 still decreases inertially owing to the longer adjustmenttime of the low frequent signal but whose trend becomesmildly with the increase of the forecasted lead time For


01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)

Figure 10 Time series of RMSEs of (a) atmospheric stream function (b) oceanic stream function (c) sea surface temperature and (d) landsurface temperature for CTRL (solid curve) and GDF (dashed curve)

0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5

Figure 11 The spatial distribution of RMSE of sea surface tempera-ture using GDF

119879119900 because of the absence of the observational constraintthe forecasted RMSE has an obvious positive trend (seeFigure 14(c)) indicating that the forecasted state is graduallydrifting away from the truth Anyway the GDF retains itssuperiority relative to theDF during the entire forecasted leadtime The forecasted RMSEs of 119879119897 have similar patterns tothose of 119879119900 (see Figure 14(d))

6 Conclusions and Discussions

In this study the diffusion filter (DF) is introduced as aconcrete implementation of the 3DVAR scheme Similar tothe recursive filter (RF) the outstanding issue of DF is its

inefficiency in capturing the spatial multiscale informationresolved by observationsTherefore several spatial multiscalevariational analysis schemes based on the DF are proposed toretrieve the spatial multiscale information from longwaves toshortwaves As one of the spatial multiscale variational anal-ysis schemes the gradient diffusion filter (GDF) scheme isproposed and verified through a set of observingassimilationsystem simulation experiments where the ldquotruthrdquo field of thesea surface temperature is represented by a high nonlinearanalytic function in a given sea region and the observationsare sampled randomly and uniformly in the whole domainResults of the assimilation experiments indicate that theGDF has noticeable advantages over the standard RF and DFschemes especially in the data void region The GDF canretrieve the longwave information over the whole domainand the shortwave information over data-dense regions

After that a perfect twin experiment framework isdesigned to study the effect of theGDFon the state estimationbased on an intermediate atmosphere-ocean-land coupledmodel In this framework the assimilation model is subjectto ldquobiasedrdquo initial fields from the ldquotruthrdquo model The RMSEof the sea surface temperature can be reduced significantlythrough the observational constraint via the GDF At thesame time the RMSEs of the other model components suchas the land surface temperature and the atmospheric andoceanic stream functions can also bemitigated by the dynam-ical constraint and the external constraint through the ocean-atmosphere-land coupled process For simulating the real


01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)

Figure 12 Time series of RMSEs of (a) atmospheric stream function (b) oceanic stream function (c) sea surface temperature and (d) landsurface temperature for GDF (black curve) and DF with 119886 119887 = 05 (red curve)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(b)Figure 13 The spatial distributions of RMSEs of SST using (a) GDF and (b) DF with 119886 119887 = 05

observational networks in the world ocean roughly theobservations locating in the Southern Ocean are removedto investigate the role of the GDF in retrieving the multi-scale information from observations While the standard DFhardly removes the model bias in the data void region theGDF may mitigate the model bias to some degree throughextracting the multiscale information from the observationsbeyond the data void region In addition the higher forecastskill can also be obtained through the better initial state fieldsproduced by the GDF

It should be noted that the background term 119869119887 is omittedin the above assimilation experiments When high-densityaccurate resolvable information is available in observa-tional datasets it is much essential to extract the multiscaleinformation from the observations with deterministic dataassimilation approaches as this study does High-quality

background fields can be obtained firstly when determinis-tic data assimilation approaches are carried out Next thestatistical data assimilation approaches such as traditional3DVar and 4Dvar can be used to treat observations asrandom variables in which 119869119887 will be included to extractthe information that cannot be resolved by the observationnetworks

In spite of the promising results produced by the GDFin the intermediate climate model much work is neededto explore the impact of the multiscale variational analysisschemes on the state estimation and forecast in real applica-tions using general circulation models (GCMs) In additionother spatial multiscale variational analysis schemes based onthe DF such as the adaptive diffusion filter (ADF) schemeshould also be studied to further improve the convergencespeed and accuracy


1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)

Figure 14 Forecasted time series of RMSEs of (a) atmospheric stream function (b) oceanic stream function (c) sea surface temperature and(d) land surface temperature for GDF (black curve) and DF with 119886 119887 = 05 (red curve)

Appendix

Equivalence between the RF and DF Methods

Using ADI scheme (9) can be discretized as follows

119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)

where Δ 119909 Δ 119910 Δ119909 Δ119910 are forward and backward differenceoperator in 119909 and 119910 direction respectively 120591 is the time step119894 119895 and 119868 119869 are grid index and grid numbers in 119909 and 119910

direction respectively and 119899 and 119873 are the time index andthe total time step numbers The common tridiagonal matrixalgorithm (TDMA) can be used to solve both (A1) and (A2)For example the tridiagonal equation (A2) in the 119895th row canbe written as follows

119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =

1 minus 119897119894minus1 minus 119897119894 119894 isin [2 119868 minus 2]

1 minus 1198971 119894 = 1

1 minus 119897119868minus2 119894 = 119868 minus 1

120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)

It is easily testified that119860 is a positive definite and symmetri-cal matrix Therefore the Cholesky decomposition of 119860 canbe processed as follows

119860

= 119871119871119879

=

[[[[[[[[[[

[

1199011 sdot sdot sdot 0

1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)

Specially if 119886119894119895 is a constant which is equivalent to anisotropic filter we know that

119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)

Set 120572 = minus119902119901 then (18) and (19) can be formulated as

120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2

119906119894 = 120572119906119894+1 + (1 minus 120572) 120593119894 119894 = 119868 minus 2 119868 minus 1 1(A18)

Equation (A18) has the same form as (6) in the RFM

Conflict of Interests

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

Acknowledgments

The authors would like to express their gratitude to tworeviewers for their helpful comments and suggestions whichcontributed to greatly improve the original paper This workwas supported by the National Basic Research Programof China (no 2013CB430304) National High-Tech RampDProgram of China (no 2013AA09A505) and National Nat-ural Science Foundation of China (nos 41206178 4137601541376013 and 51379049)

References

[1] A C Lorenc ldquoAnalysis methods for numerical weather predic-tionrdquo Quarterly Journal of the Royal Meteorological Society vol112 no 474 pp 1177ndash1194 1986

[2] P Courtier ldquoVariational methodsrdquo Journal of the MeteorologicalSociety of Japan vol 75 no 1 pp 211ndash218 1997

[3] J Derber and A Rosati ldquoA global oceanic data assimilationsystemrdquo Journal of Physical Oceanography vol 19 no 9 pp1333ndash1347 1989

[4] P Tandeo E Autret B Chapron R Fablet and R Garello ldquoSSTspatial anisotropic covariances from METOP-AVHRR datardquoRemote Sensing of Environment vol 141 pp 144ndash148 2014

[5] D W Behringer M Ji and A Leetmaa ldquoAn improved coupledmodel for ENSO prediction and implications for ocean initial-ization Part I the ocean data assimilation systemrdquo MonthlyWeather Review vol 126 no 4 pp 1013ndash1021 1998

[6] S Masina and N Pinardi ldquoA global ocean temperature andaltimeter data assimilation system for studies of climate vari-abilityrdquo Climate Dynamics vol 17 no 9 pp 687ndash700 2001


[7] B Huang J L Kinter III and P S Schopf ldquoOcean dataassimilation using intermittent analyses and continuous modelerror correctionrdquo Advances in Atmospheric Sciences vol 19 no6 pp 965ndash992 2002

[8] A C Lorenc ldquoOptimal nonlinear objective analysisrdquo QuarterlyJournalmdashRoyalMeteorological Society vol 114 no 479 pp 205ndash240 1988

[9] C M Hayden and R J Purser ldquoRecursive filter objective analy-sis of meteorological fields applications to NESDIS operationalprocessingrdquo Journal of Applied Meteorology vol 34 no 1 pp3ndash15 1995

[10] R J Purser W-S Wu D F Parrish and N M RobertsldquoNumerical aspects of the application of recursive filters tovariational statistical analysis Part I spatially homogeneousand isotropic Gaussian covariancesrdquo Monthly Weather Reviewvol 131 no 8 pp 1524ndash1535 2003

[11] A Lorenc ldquoIterative analysis using covariance functions andfiltersrdquoQuarterly Journal of the RoyalMeteorological Society vol118 no 505 pp 569ndash591 1992

[12] X-Y Huang ldquoVariational analysis using spatial filtersrdquoMonthlyWeather Review vol 128 no 7 pp 2588ndash2600 2000

[13] J Gao M Xue K Brewster and K K Droegemeier ldquoA three-dimensional variational data analysis method with recursivefilter for Doppler radarsrdquo Journal of Atmospheric and OceanicTechnology vol 21 no 3 pp 457ndash469 2004

[14] P C Chu S K Wells S D Haeger C Szczechowski and MCarron ldquoTemporal and spatial scales of the Yellow Sea thermalvariabilityrdquo Journal of Geophysical Research C Oceans vol 102no 3 pp 5655ndash5667 1997

[15] P C Chu W Guihua and Y Chen ldquoJapan Sea thermohalinestructure and circulation Part III autocorrelation functionsrdquoJournal of Physical Oceanography vol 32 no 12 pp 3596ndash36152002

[16] K-A Park and J Y Chung ldquoSpatial and temporal scalevariations of sea surface temperature in the East Sea usingNOAAAVHRR datardquo Journal of Oceanography vol 55 no 2pp 271ndash288 1999

[17] Y Xie S E Koch J A McGinley S Albers and N WangldquoA sequential variational analysis approach for mesoscale dataassimilationrdquo in Proceedings of the 21st Conference on WeatherAnalysis and Forecasting17th Conference on Numerical WeatherPrediction 15B7 American Meteorological Society Washing-ton DC USA 2005 httpamsconfexcomamspdfpapers93468pdf

[18] Z He Y Xie W Li et al ldquoApplication of the sequential three-dimensional variational method to assimilating SST in a globalocean modelrdquo Journal of Atmospheric and Oceanic Technologyvol 25 no 6 pp 1018ndash1033 2008

[19] W Li Y Xie Z He et al ldquoApplication of the multigrid dataassimilation scheme to the China seasrsquo temperature forecastrdquoJournal of Atmospheric and Oceanic Technology vol 25 no 11pp 2106ndash2116 2008

[20] Y Xie S E Koch J AMcGinley et al ldquoA spacendashtimemultiscaleanalysis system a sequential variational analysis approachrdquoMonthly Weather Review vol 139 no 4 pp 1224ndash1240 2011

[21] AWeaver and P Courtier ldquoCorrelationmodeling on the sphereusing a generalized diffusion equationrdquoQuarterly Journal of theRoyal Meteorological Society vol 122 pp 535ndash561 2001

[22] D Li X Zhang H L Fu L Zhang X Wu and G HanldquoA spatial multi-scale three-dimensional variational analysisbased on recursive filter algorithmrdquo Journal of Atmospheric andOceanic Technology In press

[23] DC Liu and JNocedal ldquoOn the limitedmemoryBFGSmethodfor large scale optimizationrdquo Mathematical Programming vol45 no 1ndash3 pp 503ndash528 1989

[24] X Wu S Zhang Z Liu A Rosati T L Delworth and YLiu ldquoImpact of geographic-dependent parameter optimizationon climate estimation and prediction simulation with anintermediate coupledmodelrdquoMonthlyWeather Review vol 140no 12 pp 3956ndash3971 2012

[25] X Wu S Zhang Z Liu A Rosati and T L Delworth ldquoA studyof impact of the geographic dependence of observing systemon parameter estimation with an intermediate coupled modelrdquoClimate Dynamics vol 40 no 7-8 pp 1789ndash1798 2013

[26] X Zhang S Zhang Z Liu X Wu and G Han ldquoParameteroptimization in an intermediate coupled climate model withbiased physicsrdquo Journal of Climate vol 28 no 3 pp 1227ndash12472015

[27] XWuW Li G Han S Zhang and XWang ldquoA compensatoryapproach of the fixed localization in EnKFrdquo Monthly WeatherReview vol 142 no 10 pp 3713ndash3733 2014

[28] Z Liu ldquoInterannual positive feedbacks in a simple extratropicalair-sea coupling systemrdquo Journal of the Atmospheric Sciencesvol 50 pp 3022ndash3028 1993

[29] S G H Philander T Yamagata and R C Pacanowski ldquoUnsta-ble air-sea interactions in the tropicsrdquo Journal of theAtmosphericSciences vol 41 no 4 pp 604ndash613 1984

[30] R Asselin ldquoFrequency filter for time integrationsrdquo MonthlyWeather Review vol 100 no 6 pp 487ndash490 1972

[31] A Robert ldquoThe integration of a spectral model of theatmosphere by the implicit methodrdquo in Proceedings of theWMOIUGG Symposium on NWP pp 19ndash24 Japan Meteoro-logical Society Tokyo Japan 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of


Geology Advances in


shape with an ellipsoid about the spatial covariance canalso be found [4] It is noted that 119861 is explicitly generatedstatistically using the correlation scales [3 5ndash7] Howeverlimitations of the CSM are (1) positive value for each elementin119861 (which is not always true) (2) nonexistence of 119861minus1 unlessusing sufficiently small correction scales and (3) requirementof large computer memory to store 119861 since every elementin 119861 is calculated explicitly To avoid the inversion of 119861 andto speed up the convergence of descent algorithms such asthe steepest descent and conjugate-gradient methods a newvector 119908 is introduced by Lorenc [8] and Derber and Rosati[3] defined as

119908 = 119861minus1(119909 minus 119909119887) (3)

Then the cost function 119869 can now be rewritten as

119869 (119908) =1

2119908119879119861119879119908 +

1

2(119867119861119908 minus 119889)

119879119877minus1(119867119861119908 minus 119889) (4)

where 119889 = 119910 minus 119867119909119887 is the ldquoinnovationrdquo vector and forsimplification hereafter we will call it observation

Considering 119861119879 = 119861 (4) is equal to

119869 (119908) =1

2119908119879119861119908 +

1

2(119867119861119908 minus 119889)

119879119877minus1(119867119861119908 minus 119889) (5)

The effect of 119861119908 in (5) can be modeled by applying anequivalent spatial filter on 119908

The second approach to prescribe 119861 is the recursive filtermethod (RFM) [9]

119884119894 = 120572119884119894minus1 + (1 minus 120572)119883119894 for 119894 = 1 119899

119885119894 = 120572119885119894+1 + (1 minus 120572) 119884119894 for 119894 = 119899 1(6)

where119883119894 is the initial value at grid point 119894119884119894 is the value afterfiltering for 119894 = 1 to 119899 119885 is the initial value after one pass ofthe filter in each direction and120572 is the filter coefficient whichdetermines the extent of spreading of observational informa-tion over the analysis domain Multipass filter can be built upby repeated application of (6) Multidimensional filter can beconstructed by applying this one-dimensional filter in eachdirection It can be shown [10] that such multidimensionalfilter when applied with several passes can accurately modelisotropic Gaussian error correlations The implementationusing recursive filter to model 119861 has been widely used dueto its relatively computational inexpensiveness [10ndash13]

An outstanding issue of either CSM or RFM is itsinefficiency in capturing the spatial multiscale informationby observations A difficulty in practice is how to properlychoose the characteristic length scales 119871119909 and 119871119910 in theCSM or the filter coefficient 120572 in the RFM Observationalstudies show that (119871119909 119871119910) change with location depthand time [14ndash16] If they are too large the analysis is toosmooth and shortwave information is lost If they are toosmall the analysis lacks coherent structure in data sparseregions because the longwave information cannot be properlycorrected Thus in the past it has been thought that thecharacteristic length scales (119871119909 119871119910) in the CSM or the filtercoefficient 120572 in the RFM is responsible for the unsatisfactoryanalysis in the 3DVAR

To avoid empirically or statistically setting the character-istic length scales and to correctly minimize the longwaveand shortwave errors in turn a sequential 3DVAR (S3DVar)method was developed [17] to assimilate sea surface temper-ature (SST) in a global oceanmodel [18]The S3DVarmethodis simply composed of a series of 3DVars each of which usesrecursive filters with different filter coefficientsThese 3DVarssweep through all resolvable scales by observational networksfrom longwaves to shortwaves In addition a multigriddata assimilation scheme was also introduced to extract theresolvable information from longwave to shortwave in anobservational system [19] Recently a sequential variationalapproach based on the multigrid data assimilation methodwas proposed to accurately retrieve the multiscale informa-tion from available observation systems [20]

Since the matrix 119861 is treated as the Gaussian type inthe CSM and modeled as the diffusion process (or Gaussianfiltering process) in the RFM the spread of the informationfrom the analysis point to the entire region is interpretedas the diffusion phenomenon [21] The diffusion filter (DF)was developed on the base of the Gaussian diffusion processand therefore can be used directly to model 119861 Severalspatial multiscale variational analysis schemes based on themodification to the standard DF scheme are proposed in thisstudy As a pilot study one of the spatialmultiscale variationalanalysis schemes the gradientDF (GDF) is used to assimilateSST observations into an intermediate coupled model withina perfect ldquotwinrdquo experiment framework

The paper is organized as follows The methodology ofthe standard DF scheme is described in Section 2 Severalspatial multiscale DF schemes are presented in Section 3In Sections 4 and 5 simple observingassimilation systemsimulation experiments and global SST simulation withan intermediate coupled atmosphere-ocean-land model areconducted to evaluate one of the newDF schemes that is thegradient DF (GDF) on the model estimation and forecastThe conclusions are summarized in Section 6

2 Diffusion Filter

The DF is in fact a Gaussian filter Given the following initialvalue problem for one-dimensional diffusion equation

120597119906

120597119905= 119886

1205972119906

1205971199092

119906 = 119908 (119909)

119905 = 0

(7)

where 119886 gt 0 is the diffusion coefficient assumed to beconstant Its solution can be formulated by the convolutionof 119908(119909) with a Gaussian kernel 119866(119909 119905)

119906 (119909 119905) = 119908 (119909) lowast 119866 (119909 119905) (8)

where (lowast) indicates convolution 119866(119909 119905) = (1radic2120587120590)119890minus119909221205902

120590 = radic2119886119905 That is 119906(119909 119905) is equivalent to applying a Gaussianfilter on initial value 119908(119909) The second moment of the filterkernel is 1205902 = 2119886119905 which characterizes the intrinsic spatial





120597119906

120597119905= 119886

1205972119906

1205971199092+ 119887

1205972119906

1205971199102(119909 119910) isin Ω 119905 isin (0 119878]

119906 = 119908 (119909 119910) (119909 119910) isin Ω 119905 = 0

120597119906

120597119899= 0 (119909 119910) isin Γ

(9)



119869 (119908) =1

2119908119879119906119878(119908)

+1

2(119867119906119878(119908) minus 119889)

119879

119877minus1(119867119906119878(119908) minus 119889)

(10)



minus120597119877

120597119905minus 119886

1205972119877

1205971199092minus 119887

1205972119877

1205971199102= 119891 (119909 119911) isin Ω 0 lt 119905 le 119878

119877 (119909 119910 119878) = 0 (119909 119911) isin Ω

120597119877

120597119899= 0 (119909 119910) isin Γ

(11)

where

119891 =

119889res (119908) = 119889 minus 119906119878(119908) 119905 = 119878

0 119905 lt 119878(12)









met








Set 119868ℎ1

119868ℎ2

119868120591 as

119868ℎ1

= 119909119894 = 119894ℎ1 | 119894 = 0 1 119868 ℎ1 =119863

119868

119868ℎ2

= 119910119894 = 119895ℎ2 | 119895 = 0 1 119869 ℎ2 =119867

119869

119868120591 = 119905119899 = 119899120591 | 119899 = 0 1 119873 120591 =119878

119873

Ωℎ = 119868ℎ1

times 119868ℎ2

(13)


119869 (119908 119886 119887)

=1

2

119868minus1

sum

119894=1

119869minus1

sum

119895=1

1199082

119894119895

+1

2

119872

sum

119898=1

119901(119898) [

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

2

(14)


119894119895is the


119877119899

119894119895minus 119877119899minus12

119894119895

1205912+ Δ 119910 (119887119894119895Δ119910 (119877

119899

119894119895+ 119877119899minus12

119894119895)) = 119891


119877119899+12

119894119895minus 119877119899

119894119895

1205912+ Δ 119909 (119886119894119895Δ119909 (119877

119899+12

119894119895+ 119877119899

119894119895)) = 0


119877119873

119894119895= 0

119894 isin [0 119868] 119895 isin [0 119869]

Δ 119909119877119899

0119895= 0

Δ119909119877119899

119868119895= 0

Δ 119910119877119899

1198940= 0

Δ119910119877119899

119894119869= 0

119899 isin [0119873 minus 1]

Δ 119909119877119899+12

0119895= 0

Δ119909119877119899+12

119868119895= 0

Δ 119910119877119899+12

1198940= 0

Δ119910119877119899+12

119894119869= 0

119899 isin [0119873 minus 1]

(15)

where119891

=

2

119872

sum

119898=1

119901(119898)119902(119898)

119894119895[

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

119899 = 119873

0 119899 lt 119873

(16)


120597119869

120597119908119894119895= 119908119894119895 minus

1198770

119894119895

120591minus1

2Δ 119910 (119887119894119895Δ119910119877

0

119894119895)

120597119869

120597119886119894119895=1

2

119873minus1

sum

119899=1

[Δ119909119906119899+12

119894119895sdot Δ119909 (119877

119899

119894119895+ 119877119899+12

119894119895)]

120597119869

120597119887119894119895=1

2

119873minus1

sum

119899=1

(Δ119910119906119899

119894119895Δ119910119877119899

119894119895+ Δ119910119906

119899+1

119894119895Δ119910119877119899+12

119894119895)

(17)



















∘times






30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)












100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





120597119906

120597119905= 119886

1205972119906

1205971199092+ 119887

1205972119906

1205971199102(119909 119910) isin Ω 119905 isin (0 119878]

119906 = 119908 (119909 119910) (119909 119910) isin Ω 119905 = 0

120597119906

120597119899= 0 (119909 119910) isin Γ

(9)



119869 (119908) =1

2119908119879119906119878(119908)

+1

2(119867119906119878(119908) minus 119889)

119879

119877minus1(119867119906119878(119908) minus 119889)

(10)



minus120597119877

120597119905minus 119886

1205972119877

1205971199092minus 119887

1205972119877

1205971199102= 119891 (119909 119911) isin Ω 0 lt 119905 le 119878

119877 (119909 119910 119878) = 0 (119909 119911) isin Ω

120597119877

120597119899= 0 (119909 119910) isin Γ

(11)

where

119891 =

119889res (119908) = 119889 minus 119906119878(119908) 119905 = 119878

0 119905 lt 119878(12)









met








Set 119868ℎ1

119868ℎ2

119868120591 as

119868ℎ1

= 119909119894 = 119894ℎ1 | 119894 = 0 1 119868 ℎ1 =119863

119868

119868ℎ2

= 119910119894 = 119895ℎ2 | 119895 = 0 1 119869 ℎ2 =119867

119869

119868120591 = 119905119899 = 119899120591 | 119899 = 0 1 119873 120591 =119878

119873

Ωℎ = 119868ℎ1

times 119868ℎ2

(13)


119869 (119908 119886 119887)

=1

2

119868minus1

sum

119894=1

119869minus1

sum

119895=1

1199082

119894119895

+1

2

119872

sum

119898=1

119901(119898) [

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

2

(14)


119894119895is the


119877119899

119894119895minus 119877119899minus12

119894119895

1205912+ Δ 119910 (119887119894119895Δ119910 (119877

119899

119894119895+ 119877119899minus12

119894119895)) = 119891


119877119899+12

119894119895minus 119877119899

119894119895

1205912+ Δ 119909 (119886119894119895Δ119909 (119877

119899+12

119894119895+ 119877119899

119894119895)) = 0


119877119873

119894119895= 0

119894 isin [0 119868] 119895 isin [0 119869]

Δ 119909119877119899

0119895= 0

Δ119909119877119899

119868119895= 0

Δ 119910119877119899

1198940= 0

Δ119910119877119899

119894119869= 0

119899 isin [0119873 minus 1]

Δ 119909119877119899+12

0119895= 0

Δ119909119877119899+12

119868119895= 0

Δ 119910119877119899+12

1198940= 0

Δ119910119877119899+12

119894119869= 0

119899 isin [0119873 minus 1]

(15)

where119891

=

2

119872

sum

119898=1

119901(119898)119902(119898)

119894119895[

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

119899 = 119873

0 119899 lt 119873

(16)


120597119869

120597119908119894119895= 119908119894119895 minus

1198770

119894119895

120591minus1

2Δ 119910 (119887119894119895Δ119910119877

0

119894119895)

120597119869

120597119886119894119895=1

2

119873minus1

sum

119899=1

[Δ119909119906119899+12

119894119895sdot Δ119909 (119877

119899

119894119895+ 119877119899+12

119894119895)]

120597119869

120597119887119894119895=1

2

119873minus1

sum

119899=1

(Δ119910119906119899

119894119895Δ119910119877119899

119894119895+ Δ119910119906

119899+1

119894119895Δ119910119877119899+12

119894119895)

(17)



















∘times






30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)












100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



Set 119868ℎ1

119868ℎ2

119868120591 as

119868ℎ1

= 119909119894 = 119894ℎ1 | 119894 = 0 1 119868 ℎ1 =119863

119868

119868ℎ2

= 119910119894 = 119895ℎ2 | 119895 = 0 1 119869 ℎ2 =119867

119869

119868120591 = 119905119899 = 119899120591 | 119899 = 0 1 119873 120591 =119878

119873

Ωℎ = 119868ℎ1

times 119868ℎ2

(13)


119869 (119908 119886 119887)

=1

2

119868minus1

sum

119894=1

119869minus1

sum

119895=1

1199082

119894119895

+1

2

119872

sum

119898=1

119901(119898) [

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

2

(14)


119894119895is the


119877119899

119894119895minus 119877119899minus12

119894119895

1205912+ Δ 119910 (119887119894119895Δ119910 (119877

119899

119894119895+ 119877119899minus12

119894119895)) = 119891


119877119899+12

119894119895minus 119877119899

119894119895

1205912+ Δ 119909 (119886119894119895Δ119909 (119877

119899+12

119894119895+ 119877119899

119894119895)) = 0


119877119873

119894119895= 0

119894 isin [0 119868] 119895 isin [0 119869]

Δ 119909119877119899

0119895= 0

Δ119909119877119899

119868119895= 0

Δ 119910119877119899

1198940= 0

Δ119910119877119899

119894119869= 0

119899 isin [0119873 minus 1]

Δ 119909119877119899+12

0119895= 0

Δ119909119877119899+12

119868119895= 0

Δ 119910119877119899+12

1198940= 0

Δ119910119877119899+12

119894119869= 0

119899 isin [0119873 minus 1]

(15)

where119891

=

2

119872

sum

119898=1

119901(119898)119902(119898)

119894119895[

[

119868minus1

sum

119894=1

119869minus1

sum

119895=1

(119902(119898)

119894119895119906119873

119894119895) minus 119889(119898)]

]

119899 = 119873

0 119899 lt 119873

(16)


120597119869

120597119908119894119895= 119908119894119895 minus

1198770

119894119895

120591minus1

2Δ 119910 (119887119894119895Δ119910119877

0

119894119895)

120597119869

120597119886119894119895=1

2

119873minus1

sum

119899=1

[Δ119909119906119899+12

119894119895sdot Δ119909 (119877

119899

119894119895+ 119877119899+12

119894119895)]

120597119869

120597119887119894119895=1

2

119873minus1

sum

119899=1

(Δ119910119906119899

119894119895Δ119910119877119899

119894119895+ Δ119910119906

119899+1

119894119895Δ119910119877119899+12

119894119895)

(17)



















∘times






30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)












100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


















∘times






30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)












100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


30 31 32 33 34 35 36 37 38 39 40Distance

1

2

3

4

5

6

7

Value

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

(b)












100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(c)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(d)




120597119902

120597119905+ 119869 (120595 119902) =






100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

minus05

03

11

19

27

35

43

51

59

(b)




120597

120597119905(minus

120601

11987120

) + 120573120597

120597119909120601 = 120574nabla

2120595 minus 119870119902nabla

2120601

120597

120597119905119879119900 + 119906

120597119879119900

120597119909+ V

120597119879119900

120597119910minus 119870ℎ120601


(19)


0= 1198921015840ℎ01198912





119898120597

120597119905119879119897 = minus119870119871119879119897 + 119860119871nabla

2119879119897 + 119904 (120591 119905) + 119862119897 (119879119897 minus 120583120595) (20)


101 102 103 104 105 106 107 108 109

31

32

33

34

35

36

37

38

39




119900 1198790

119897)

where120595012060101198790119900 and1198790




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



119900 1198791

119897) The last 10



0) and 105m2sminus1 (for 1206010)


2 1206012 1198792

119900 1198792

119897) The




100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(a)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(b)

100 101 102 103 104 105 106 107 108 109 11030

31

32

33

34

35

36

37

38

39

40

0

08

16

24

32

4

48

56

64

(c)



0






0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 50 100 150 200 250 300 350

minus50

0

50

minus1E + 008

minus7E + 007

minus3E + 007

1E + 007

5E + 007

9E + 007

(a)

0 50 100 150 200 250 300 350

minus50

0

50

minus35000

minus15000

5000

25000

45000

(b)

0 50 100 150 200 250 300 350

minus50

0

50

250258266274282290298306

(c)


1 2 3 4 5 6 7 8 9 10

10

12

14

16

18

times106

(a)

1 2 3 4 5 6 7 8 9 10

2

4

6

8

times104

(b)

1 2 3 4 5 6 7 8 9 10

2

4

6

(c)

1 2 3 4 5 6 7 8 9 10

01

02

03

04

05

(d)



0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 50 100 150 200 250 300 350

minus50

0

50

02468101214161820













01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


01 02 03 04 05 06 07 08 09 1

8

10

12

14

16

18times10

6

(a)

01 02 03 04 05 06 07 08 09 16500

7000

7500

8000

8500

9000

(b)

01 02 03 04 05 06 07 08 09 10

2

4

6

8

(c)

01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

1

2

3

4

5








01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


01 02 03 04 05 06 07 08 09 1

10

12

14

16times10

6

(a)

01 02 03 04 05 06 07 08 09 16000

6500

7000

7500

8000

(b)

01 02 03 04 05 06 07 08 09 1

2

25

3

35

4

45

(c)

01 02 03 04 05 06 07 08 09 1

01

015

02

025

03

(d)


0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20

(a)

0 50 100 150 200 250 300 350

minus50

0

50

0

4

8

12

16

20







1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


1 11 12 13 14 15 16 17 18 19 2

10

12

14

16times10

6

(a)

1 11 12 13 14 15 16 17 18 19 2

6000

6200

6400

6600

6800

7000

(b)

1 11 12 13 14 15 16 17 18 19 21

2

3

4

5

(c)

1 11 12 13 14 15 16 17 18 19 2

01

015

02

025

03

035

(d)


Appendix



119906119899+12

119894119895minus 119906119899

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899

119894119895) = 0


(A1)

119906119899+1

119894119895minus 119906119899+12

119894119895

1205912minus Δ 119909 (119886119894119895Δ119909119906

119899+12

119894119895)

minus Δ 119910 (119887119894119895Δ119910119906119899+1

119894119895) = 0


(A2)

1199060

119894119895= 119908119894119895 119894 isin [0 119868] 119895 isin [0 119869] (A3)

Δ 119909119906119899

0119895= 0 (A4)

Δ119909119906119899

119868119895= 0 (A5)

Δ 119910119906119899

1198940= 0 (A6)

Δ119910119906119899

119894119869= 0 119899 isin [0119873] (A7)

Δ 119909119906119899+12

0119895= 0 (A8)

Δ119909119906119899+12

119868119895= 0 (A9)

Δ 119910119906119899+12

1198940= 0 (A10)

Δ119910119906119899+12

119894119869= 0 119899 isin [0119873 minus 1] (A11)



119860119906 = 120574 (A12)

where

119860 =

[[[[[[[[[[

[

1198981 1198971 sdot sdot sdot 0

1198971 1198982 1198972

1198972 1198983

d d 119897119868minus2

0 119897119868minus2 119898119868minus1

]]]]]]]]]]

]


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


119906 =

[[[[[[[[[[[

[

119906119899+12

1119895

119906119899+12

2119895

119906119899+12

3119895

119906119899+12

119868minus1119895

]]]]]]]]]]]

]

120574 =

[[[[[[[[[

[

1205741

1205742

1205743

120574119868minus1

]]]]]]]]]

]

119897119894 = minus120591

2ℎ21

119886119894+1119895 119894 isin [1 119868 minus 2]

119898119894 =


1 minus 1198971 119894 = 1


120574119894 = 119906119899

119894119895+120591

2Δ 119910 (119887119894119895Δ119910119906

119899

119894119895) 119894 isin [1 119868 minus 1]

(A13)


119860

= 119871119871119879

=

[[[[[[[[[[

[


1199021 1199012

1199022 1199013

d d

0 119902119868minus2 119901119868minus1

]]]]]]]]]]

]

[[[[[[[[[[

[

1199011 1199021 sdot sdot sdot 0

1199012 1199022

1199013 d d 119902119868minus2

0 119901119868minus1

]]]]]]]]]]

]

(A14)

which leads to

119871120593 = 120574

119871119879119906 = 120593

(A15)

where

120593119894+1 = minus119902119894

119901119894+1120593119894 +

1

119901119894+1120574119894+1 119894 = 1 2 119868 minus 2

1205931 =1

11990111205741

119906119894 = minus119902119894

119901119894119906119894+1 +

1

119901119894120593119894 119894 = 119868 minus 2 119868 minus 1 1

119906119868minus1 =1

119901119868minus1120593119868minus1

(A16)


119901119894 = 119901 119894 isin [1 119868 minus 2]

119902119894 = 119902 119894 isin [1 119868 minus 2]

119901 + 119902 = 1

(A17)


120593119894+1 = 120572120593119894 + (1 minus 120572) 120574119894+1 119894 = 1 2 119868 minus 2





Acknowledgments


References










































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

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