research article a fast independent component analysis algorithm for...

Research ArticleA Fast Independent Component Analysis Algorithm forGeochemical Anomaly Detection and Its Application to SoilGeochemistry Data Processing

Bin Liu12 Si Guo1 Youhua Wei12 and Zedong Zhan12

1 Geomathematics Key Laboratory of Sichuan Province Chengdu University of Technology Chengdu 610059 China2 College of Geophysics Chengdu University of Technology Chengdu 610059 China

Correspondence should be addressed to Bin Liu liubincim163com

Received 14 March 2014 Revised 21 June 2014 Accepted 7 July 2014 Published 24 July 2014

Academic Editor Chongbin Zhao

Copyright copy 2014 Bin Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fast independent component analysis algorithm (FICAA) is introduced to process geochemical data for anomaly detection Ingeochemical data processing the geological significance of separated geochemical elements must be explicit This requires thatcorrelation coefficients be used to overcome the limitation of indeterminacy for the sequences of decomposed signals by the FICAAso that the sequences of the decomposed signals can be correctly reflected Meanwhile the problem of indeterminacy in the scalingof the decomposed signals by the FICAA can be solved by the cumulative frequencymethod (CFM) To classify surface geochemicalsamples into true anomalies and false anomalies assays of the 1 10 000 soil geochemical data in the area ofDachaidan in theQinghaiprovince of China are processed The CFM and FICAA are used to detect the anomalies of Cu and Au The results of this researchdemonstrate that the FICAA can demultiplex themixed signals and achieve results similar to actual mineralization when 85 95and 98 are chosen as three levels of anomaly delineation However the traditional CFM failed to produce realistic results and hasno significant use for prospecting indication It is shown that application of the FICAA to geochemical data processing is effective

1 Introduction

An ore-forming system in the crust of the Earth is a highlynonlinear system which commonly involves coupled pro-cesses between material deformation pore-fluid flow heattransfer mass transport and chemical reactions [1ndash9]Because these processes can be described mathematically bya set of partial differential equations [1ndash3 10ndash13] it is import-ant to obtain theoretical and numerical solutions using bothmathematical and computational methods in the field ofapplied mathematics so that the distributions of mineralresources in the upper crust of the Earth can be better pre-dicted For this reason an emerging discipline known as com-putational geoscience [14 15] has been established in recentyears

For quantitative prediction of mineral resources geo-chemical data are an important information sourceResearchers need the data to understand and simulate thedynamic mechanisms of ore-forming systems Extracting

prospecting information from geochemical element data isa central purpose of metallogenic prediction [16] The geo-chemical element data are obtained from the field by sampl-ing and analysis During this process the correspondinginterferences are produced so collecting hidden anomalyinformation that may reflect the spatial distribution charact-eristics of geochemical elements is the key to suppressinginterference between geochemical elements [17ndash19] Com-mon data analysis methods are usually based on low-orderstatistical properties without considering the higher-orderstatistical characteristics of the data [20 21] Most research-ers believe that because samples disobey the normal distri-bution the existingmethods cannot be directly applied to rawdata analysis [22] Moreover due to the complexity of geo-logical information traditional data mining cannot reflectthe spatial distribution characteristics of mineral resourcesbecause the dynamic mechanisms that control the formationof these mineral resources are completely neglected [16ndash22]To better understand the geological information associated

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 319314 12 pageshttpdxdoiorg1011552014319314

2 Journal of Applied Mathematics

with geochemical data many researchers study the spatialdistribution characteristics of the elements and the physicaland chemical properties of the bare geologic body [23] Asa result several quantitative geochemical methods such asspatial statistics the fractal technique discriminant analysisand fuzzy clustering [24ndash28] have been developed overthe past few decades These methods are associated withthe density frequency of the element and are based onthe perceptions of mineralization which may lead to falseanomalies that have nothing to do with mineralization [29]In fact geochemical exploration data processing methodsshould only consider data cardinalities [30] For examplein the independent component analysis (ICA) methodhigher-order statistical features of data are taken intoaccount [31] The method isolates the independent sourcesignals that are implicit in the mixed signal under ldquoblindrdquoconditions In previous studies the ICA method was mainlyused for mineral predictions and is still at a fledging stage ingeochemical applications [32]

Although the mathematical analysis and computationalsimulation methods associated with applied mathematicshave been widely used to produce analytical solutions [1ndash333ndash36] and numerical results for understanding the dynamicmechanisms and processes of ore-forming systems [4ndash9] thisstudy attempts to utilize the FICAA for detecting anomaliesin geochemical data For example Zhao and his coworkershave successfully solved a set of partial differential equationsfor the convective pore-fluid problems that are closely relatedto hydrothermal ore-forming systems [1ndash3 33 34] Theyalso solved a different set of partial differential equationsfor the chemical dissolution front instability problems thatare present in many ore-forming systems within the uppercrust of the Earth [37ndash45] According to the FICAA andthe characteristics of the geochemical data processing rawgeochemical data with the FICAA can solve the sequenceindeterminacy problem of separated results caused by theFICAA by calculating the correlation coefficient matrix ofraw data and separating the independent components Inaddition the cumulative frequency method is used to solvethe indeterminacy problem in scaling by direct comparison tothe resultant anomalies When a late prospecting trench vali-dation is conducted the data (after FICAAprocessing) can beused to better reflect the spatial distribution characteristics ofthe elements This can provide useful geological informationfor understanding and simulating the dynamic mechanismsof ore-forming systems [14 15]

This paper is organized as follows The principles ofthe FICAA and the correlation coefficient algorithm areintroduced in Section 2 A brief introduction of the geologicalbackground of the area around Dachaidan in the Qinghaiprovince of China is given in Section 3The FICAA is appliedto the 1 10 000 soil geochemical data of this area in Section4The results are discussed and conclusions are stated in Sec-tions 5 and 6 respectively

2 Algorithm Principle

21The ProblemDescription of Independent Component Anal-ysis Independent component analysis (ICA) is a datamining

middot middot middot


S1 S2 S3 Sm

X1 X2 X3 Xm

Figure 1 Signal mixing process

technique that based on the hypothesis of statistical inde-pendence analyzes data from a perspective of higher-orderstatistical correlation [46] A set of random initial vectorsapproximates the independent signal source implicated in themixed signal by ldquodecomposingrdquo themixed signal Eachmixedsignal is a linear combination of original signals (Figure 1)The basic mathematical model of the ICA is as follows

119909 = 119860119904 (1)

where 119909 denotes some observed mixed signals matrix 119860

represents the mixing matrix of the system and vector 119904

represents unknown source signals that are assumed to bestatistically independent

When the source signals 119904 and the mixed matrix 119860 areboth unknown only 119909 can be obtainedThe ICA algorithm isdesigned to determine a matrix 119882 such that

119910 = 119882119909 (2)

where 119910 is an optimal estimate of the source signals 119904 Thelinear solution of the ICAmodel can be obtained with (1) and(2) Consider

119910 = 119882119860119904 = 119866119904 (3)

where 119866 = 119882119860 is an 119899 times 119899 identity matrixIn the late 1980s Jutten andHerault proposed the concept

of the ICA [47] In 1994 Comon extended the principalcomponent analysis based on data processing and com-pression as an independent component analysis algorithmand proposed independent component analysis based on theminimum of mutual information [48] In 2001 Hyvarinen etal proposed an algorithm for extracting the fixed-point fromablind signal also called the fast ICA (FICA) In 2012 Yu et alapplied the FICA to mineral prediction [32] However both

Journal of Applied Mathematics 3

the dynamic mechanisms and the processes of ore-formingsystems were completely neglected in their studies [32 4748]This is themain shortcoming of the existingmethod usedto treat geochemical data with statistical mathematics ratherthan simulating the dynamic mechanisms and the processesof ore-forming systems using applied mathematics [49 50]

22 The Data Requirements of the ICA Algorithm First theICA requires that each source signal be a random signalwith a zero mean value which is statistically independent atany moment although geochemical data are usually spatiallycorrelated Thus we should obtain various results due to thespatial correlation when the ICA is used Second it requiresan equal number of source signals andmixed signals (namelyit requires that the relationship among geochemical elementsis a linear combination) Third it requires that at most onesource signal obey the Gaussian distribution In practice theinfluence of the ldquonoiserdquo is usually not considered

23 The FICAA The rationale of the FICAA is to determinea target function by maximizing negentropy and to obtainthe optimal value of the target function by using Newtonrsquositerative method

Negentropy can be used to measure the non-GaussianityThe negentropy of a random variable 119910 is given by

119869 (119910) = 119867(119910gauss) minus 119867 (119910) (4)

where 119867(sdot) is the entropy function and 119910gauss and 119910 arethe Gaussian variable and the random variable respectivelywith the same mean and normalized variance If negentropyis zero then 119910 obeys the Gaussian distribution If 119910 is anon-Gaussian distribution negentropy must exceed zeroA non-Gaussian measurement reaches its maximum whennegentropy ismaximizedMeanwhile the optimal estimationof source signals is accomplished The approximate formulaof negentropy can be written as follows

119869 (119910) prop [119864 119866 (119910) minus 119864 119866 (119910gauss)]2

(5)

where 119866(sdot) is an arbitrary non-quadratic function Afteriterative trials it was discovered that the rate of convergence isfaster and the convergence effect is better when 119866(119910) = 119910

44

Tomaximize negentropy the optimal119864119866(119908119879119909)must be

achieved According to the Kuhn-Tucker conditions underthe constraint 119864119866(119908

119879119909)

2

= 1199082

= 1 the optima areobtained at points where the gradient of the Lagrangian iszero

119864 119909119892 (119908119879119909) minus 120573119909 = 0 (6)

where 120573 is a constant that can be easily evaluated as 120573 =

119864119908119879

0119909119892(119908119879

0119909)119908

0is the value of119908 at an optimum and119892(sdot) is

the first order derivative function of 119866(sdot) Assuming the dataare bounded 119864119909

119879119909 = 119868 the left part of (6) can be written

as 119865 and we can obtain the Jacobian matrix 119869119865(119908) as

119869119865 (119908) = 119864 1199091199091198791198921015840(119908119879119909) minus 120573119868 (7)

To simplify the matrix inversion we can reasonablyapproximate the first item of (7) as

119864 1199091199091198791198921015840(119908119879119909) asymp 119864 119909119909

119879 119864 119892

1015840(119908119879119909)

= 119864 1198921015840(119908119879119909) 119868

(8)

If1199080is approximated as119908 in120573 then 119869119865(119908) is transformed

into a diagonal matrix Thus we can obtain the approximateNewton iterative formula as

119908lowast

= 119908 minus

[119864 119909119892 (119908119879119909) minus 120573119908]

[119864 1198921015840(119908119879119909) minus 120573]

119908 =

119908lowast

119908lowast

(9)

where 119908lowast is the new value of 119908 and 120573 = 119864119908

119879119909119892(119908119879119909)

After simplification we can derive the iterative formula of theFICAA

119908lowast

= 119864 119909119892 (119908119879119909) minus 119864 119892

1015840(119908119879119909)119908

119908 =

119908lowast

119908lowast

(10)

24The Correlation CoefficientMatrix TheFICAA is limitedin the fact that separated signal sequences do not correspondone-to-onewith the order of the source signals In the absenceof any other prior knowledge this problem cannot be solvedIn this paper we use the correlation coefficientmatrix to solveit

Let (1198831 1198832 1198833 119883

119899) be one-dimensional random

variables If arbitrary 119883119894and 119883

119895have the correlation coef-

ficient 120588119894119895(119894 119895 = 1 2 119899) then an 119899 times 119899 matrix with

elements 120588119894119895

is the correlation coefficient matrix (119877) of(1198831 1198832 1198833 119883

119899) [51ndash53] Consider

119877 =

[

[

[

[

[

12058811

12058812

sdot sdot sdot 1205881119899

12058821

12058822


1205881198991

1205881198992

sdot sdot sdot 120588119899119899

]

]

]

]

]

(11)

where 120588119894119895

= cov(119883119894 119883119895)radic119863119883

119894radic119863119883119895 cov(119883

119894 119883119895) = 119864[119883

119894minus

119864119883119894] times [119883

119895minus 119864119883

119895]

To restore the source signals of independent componentsmixed signals and separated independent components aretaken as new observed data to calculate the correlationcoefficient matrix The two variables of maximum absolutevalues with the same correlation coefficient are deemed tohave a corresponding relation to each other

3 Geology of the Research Area

The research area is situated in the Qinghai province ofChina 700 km east of Xining and 35 km west of DachaidanThe 1 10 000 geographical coordinates cover the easternlongitude range from 95

∘4710158405110158401015840 to 95

∘5810158401810158401015840 and the north-

ern latitude range from 37∘4210158404410158401015840 to 37

∘4510158401410158401015840 The study

area is approximately 6 km2 The strata known in the area


Table 1 Parameters of soil geochemical elements in the given area of Qinghai

Element Sample number Min Max Mean StDev Skew KurtosisAu 3307 02 4100 7508 186540 9530 130097Pb 3307 23 13310 17398 335077 24929 813833Zn 3307 200 184 6686 18615 1137 2324Cu 3307 43 896 24959 98264 1481 4340As 3307 20 16230 40898 656605 8318 130047Sb 3307 02 13400 20290 326852 22701 826105

Table 2 Results of the Kolmogorov-Smirnov test on the variables

Element Kolmogorov-Smirnov 119885 Asymp Sig (2-tailed) DistributionPb 20782 lt001 NonnormalAu 20251 lt001 NonnormalCu 6078 lt001 NonnormalZn 5923 lt001 NonnormalAs 16895 lt001 NonnormalSb 17893 lt001 Nonnormal

are the Silurian (S) Permian (P) and Quaternary (Q4)

Rock types are conglomerate altered conglomerate sand-stone quartz schist altered quartz schist arkose quartzitelithic sandstone slate and phyllite The mineralizations aresilicification ferritization pyritization malachitization andchalcopyritization The study area mainly consists of an Aumineralization belt and a Cu mineralization belt The Au beltis mainly distributed in the Permian stratus and the Cu belt isprimarily found in the Silurian stratus (Figure 2) The majorminerals in this area are galena malachite and azurite

4 Application

The project involved setting up 34 soil geochemical profilelines and completing 6 km2 of soil geochemical surveys Thesampling network is 100m times 20m Soil geochemical samplesincluded 3307 samples that were analyzed for six elementsAu Cu Pb As Sb and Zn In this study area ten pros-pecting trenches were dug and 200 samples were collectedand notched Based on descriptive statistics analysis of soilgeochemical data characteristics indexes were determinedand are shown in Table 1 (Au is reported in ppb and theremaining metals are reported in ppm)

The existing geochemical data processing methods fre-quently rely on the assumption that geochemical data obeya normal distribution However the spatial distribution ofgeochemical element content is very complex so existingmethods have limitations Because the FICAA requires thatonly the contents of one element obey the normal distri-bution the first analysis of this framework examines theprobability plots reported in Figure 3The purpose is to verifywhether the raw data obey a normal distribution Figure 3shows that Au Pb As and Sb obviously disobey the normaldistribution Meanwhile Cu and Zn approximately obey anormal distribution

Graphical tools can support hypothesis tests for normal-ity The results of the tests for Cu and Zn are controversial as

Table 3 Correlation coefficient results

Variable Au Cu Pb Zn As Sb1199101 minus0213 minus0124 minus0982 0019 minus0067 00021199102 0001 minus0018 minus0006 minus0021 0009 minus09851199103 0040 0086 0020 0135 0949 00811199104 minus0967 0050 0099 0046 minus0169 minus00921199105 0047 minus0546 minus0092 0097 0031 minus00821199106 0128 minus0822 minus0130 minus0985 0256 minus0087

expected (Figure 3) Furthermore the Kolmogorov-Smirnov(K-S) normal distribution test is conducted on the soil sampledata Table 2 shows that the normality hypothesis cannot beaccepted (119875 lt 005) because the contents of the six elementsdisobey the normal distribution Thus these data meet thebasic requirement of the FICAA

Based on the FICAA and the statistical independence ofthe rawgeochemical data we consider the six element contentvalues as the mixed signals (the raw data) The oscillogramof the raw data is shown in Figure 4 and the oscillogram ofthe source signals after separation by the FICAA is shown inFigure 5 Because the FICAA is a copy or estimation of thesource signals the sequences and scaling of the source signalshave changedThe changes are displayed in the oscillogramofthe separated resultsWe introduce the correlation coefficientto solve this problem

As clearly demonstrated in Figure 5 the waveforms of thesix elements after separation correspond to the waveformsof the raw data Treating the survey data and indepen-dent components as new observation variables we calculatethe correlation coefficient between them The correlationcoefficient results in Table 3 show that themaximum absolutevalues of each column are 0967 0546 0982 0985 0949and 0985 respectivelyThus it can be concluded that 1199101 11991021199103 1199104 1199105 and 1199106 correspond to Pb Sb As Au Cu and Zn


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

TC01

TC02TC03

TC04

TC05

TC06

TC07

TC08

TC09TC10

P1a1

P1a1

S

S

SQ

N

The type of stratum

Q Quaternary

Permian

SilurianThe type of alteration

Ore-bearing alteration

Copper-bearing alteration

General symbols

Fault

Line of geologicallimitationUnconformity

Water system

Prospecting trench

1100000 100 200 400 600 800

QinghaiChina

Study area

0 500 1000

(km)

1000

(m)

Figure 2 Simplified geological map and location map of the study area with alterations


Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Figure 3 Normal probability plots of Au Cu Pb Zn As and Sb


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200

Figure 4 Oscillogram of the raw data

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0

Figure 5 Oscillogram of the data separated by the FICAA

FICAA processing is only a copy or estimation of theraw data and only reflects the general trend of the dataThe cumulative frequency method is employed to determinethe distribution characteristics of the elements and to solvethe problem of sequence indeterminacy Selecting 85 95and 98 as the intrazone mesozone and external zone ofthe anomaly the isograms of the raw data and separateddata are depicted using the cumulative frequency methodTheir three-level cumulative frequencies are given in Table 4Because this studymainly analyzes themetallogenic elementsAu andCu Figures 6 through 9 show theAu andCu isogramsof the raw data and data separated by the FICAA respectively

5 Discussion

The anomaly analysis of Au and Cu and the prospectingtrench work are carried out based on the anomaly isogramsThe anomaly isograms show that Au is mainly distributed inthe Permian while Cumostly spreads over the SilurianThereare obvious differences between the isograms of Au and Cuthat are delineated by the raw data and the FICAA processeddata Test and analysis results are shown in Table 5 In trenchTC04 the element content of Au is 01 where the raw data ofAu has an obvious anomaly but the FICAA processed datadoes not The FICAA processed result is consistent with the


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52

Figure 6 The isogram of Au

748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288

Figure 7 The isogram of Au after processing by the FICAA

Table 4 Zoning sequences delineated by the cumulative frequency

Zoning Au Cu Pb Zn As Sb 1199101 1199102 1199103 1199104 1199105 1199106

Exozone 98 34 195 854 592 275 02379 04269 16729 08691 minus0151 44311Mesozone 251 429 279 102 135 449 04289 0889 26092 15673 06145 52717Intrazone 52 509 415 115 227 798 0871 1935 40833 2883 17436 59494


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51

Figure 8 The isogram of Cu

748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015

Figure 9 The isogram of Cu after processing by the FICAA

actual situation In trenches TC01 and TC05 the elementcontents of Cu are 015 and 199 The raw data show normaldistributions while the FICAA processed data reveal appar-ent anomalies that match the true conditions Meanwhileother trenches reasonably show the distribution abnormal-ities of Au and Cu Thus employing the FICAA to processthe geochemical data can better reflect the spatial distributioncharacteristics of elements This is because when the existingstatistical data processing method is used we must assumethat the data obey the normal distribution or lognormaldistribution However the actual geochemical data may not

satisfy this assumption Therefore these existing methodshave limitationsThe FICAA can preprocess the geochemicaldata under unknown circumstances This can eliminate themutual interference between elements so that the data mayprovide a clearer direction for geochemical data processing

The analysis results show that the anomaly isogramsprocessed by the FICAA are more in line with the actualelement distributions Although we solve the problem ofscaling indeterminacy the anomalies still cannot be displayeddirectly by the processed data Moreover the FICAA shouldbe verified in other geological areas


Table 5 The analysis results of chemical groove samples

Serial number of trench Au (max) Cu (max)TC01 lt010 015TC02 lt010 105TC03 lt010 019TC04 lt010 214TC05 lt010 199TC06 lt010 05TC07 037 lt001TC08 045 lt001TC09 169 lt001TC10 161 lt001

More importantly the existing geochemical data process-ing methods with the statistical mathematics characteristic(such as the FICAA used in this study) cannot be usedto simulate the dynamic mechanisms and processes of ore-forming systems so they are invalid for predicting concealedore deposits within the upper crust of the Earth To solvethis problem applied mathematics methods have been usedto establish an emerging discipline known as computationalgeoscience during most of the past two decades As a resultcomputational simulation methods have become importanttools not only for simulating the dynamic mechanisms andprocesses of ore-forming systems but also for predicting thepotential locations of ore deposits within the upper crust ofthe Earth [14 15 49]

6 Conclusions

This study identifies soil geochemistry anomaly areas basedon the presence or absence of mineralization in trenchsamples Use of the correlation coefficient and the cumu-lative frequency can solve indeterminacy problems in bothsequences and scaling Comparing FICAA processed rawdata to the results of the CFM beforehand can better reflectthe distribution characteristics of the geochemical elementsBecause geological backgrounds are different when the studyareas are different the geochemical exploration methodshould be applied on the basis of the actual situation

The existing geochemical data processing methods withthe statistical mathematics characteristic (such as the FICAAused in this study) cannot be used to simulate the dynamicmechanisms and processes of ore-forming systems thereforethey are invalid for predicting concealed ore deposits withinthe upper crust of the Earth Because computational geo-science methods with the applied mathematics characteristiccan effectively simulate the dynamic mechanisms and pro-cesses of ore-forming systems they should be used in futureresearch to predict potential locations of ore deposits withinthe upper crust of the Earth

Conflict of Interests

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

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 41272363)

References

[1] C Zhao H B Muhlhaus and B E Hobbs ldquoFinite elementanalysis of steady-state natural convection problems in fluid-saturated porous media heated from belowrdquo International Jour-nal for Numerical and Analytical Methods in Geomechanics vol21 no 12 pp 863ndash881 1997

[2] C Zhao B E Hobbs and H B Muhlhaus ldquoFinite elementmodelling of temperature gradient driven rock alteration andmineralization in porous rock massesrdquo Computer Methods inApplied Mechanics and Engineering vol 165 no 1ndash4 pp 175ndash187 1998

[3] C Zhao B E Hobbs and H B Muhlhaus ldquoTheoretical andnumerical analyses of convective instability in porous mediawith upward throughflowrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 23 no 7 pp 629ndash646 1999

[4] B E Hobbs Y Zhang A Ord and C Zhao ldquoApplication ofcoupled deformation fluid flow thermal and chemical mod-elling to predictivemineral explorationrdquo Journal of GeochemicalExploration vol 69-70 pp 505ndash509 2000

[5] A Ord B E Hobbs Y Zhang et al ldquoGeodynamic modelling ofthe Century deposit Mt Isa Province Queenslandrdquo AustralianJournal of Earth Sciences vol 49 no 6 pp 1011ndash1039 2002

[6] P Sorjonen-Ward Y Zhang and C Zhao ldquoNumerical mod-elling of orogenic processes and gold mineralisation in thesoutheastern part of the Yilgarn Craton Western AustraliardquoAustralian Journal of Earth Sciences vol 49 no 6 pp 935ndash9642002

[7] P A Gow P Upton C Zhao and K C Hill ldquoCopper-goldmineralisation in New Guinea numerical modelling of colli-sion fluid flow and intrusion-related hydrothermal systemsrdquoAustralian Journal of Earth Sciences vol 49 no 4 pp 753ndash7712002

[8] P M Schaubs and C Zhao ldquoNumerical models of gold-depositformation in the Bendigo-Ballarat Zone Victoriardquo AustralianJournal of Earth Sciences vol 49 no 6 pp 1077ndash1096 2002

[9] Y Zhang B E Hobbs A Ord et al ldquoThe influence offaulting on host-rock permeability fluid flow and ore genesisof gold deposits a theoretical 2D numerical modelrdquo Journal ofGeochemical Exploration vol 78-79 pp 279ndash284 2003

[10] C Zhao B E Hobbs A Ord S Peng H B Muhlhaus andL Liu ldquoTheoretical investigation of convective instability ininclined and fluid-saturated three-dimensional fault zonesrdquoTectonophysics vol 387 no 1ndash4 pp 47ndash64 2004

[11] C Zhao B E Hobbs A Ord P Hornby S Peng and L LiuldquoMineral precipitation associated with vertical fault zones theinteraction of solute advection diffusion and chemical kineticsrdquoGeofluids vol 7 no 1 pp 3ndash18 2007

[12] C Zhao B EHobbs andAOrdConvective andAdvectiveHeatTransfer in Geological Systems Springer Berlin Germany 2008

[13] C Zhao Dynamic and Transient Infinite Elements Theory andGeophysical Geotechnical and Geoenvironmental ApplicationsAdvances in Geophysical and Environmental Mechanics andMathematics Springer Berlin Germany 2009

[14] C Zhao B E Hobbs and A Ord ldquoInvestigating dynamicmechanisms of geological phenomena using methodology of


computational geosciences an example of equal-distant min-eralization in a faultrdquo Science in China D Earth Sciences vol 51no 7 pp 947ndash954 2008

[15] C Zhao B E Hobbs and A Ord Fundamentals of Computa-tional Geoscience Numerical Methods and Algorithms SpringerBerlin Germany 2009

[16] S CWangTheory andMethods of Mineral Resources PredictionBased on Synthetic Information Science Press Beijing China2002

[17] Q Cheng F P Agterberg and G F Bonham-Carter ldquoA spatialanalysis method for geochemical anomaly separationrdquo Journalof Geochemical Exploration vol 56 no 3 pp 183ndash195 1996

[18] Q Cheng ldquoSpatial and scaling modelling for geochemicalanomaly separationrdquo Journal of Geochemical Exploration vol65 no 3 pp 175ndash194 1999

[19] Q Cheng Y Xu and E Grunsky ldquoIntegrated spatial and spec-trum method for geochemical anomaly separationrdquo NaturalResources Research vol 9 no 1 pp 43ndash52 2000

[20] Q Cheng ldquoMapping singularities with stream sediment geo-chemical data for prediction of undiscovered mineral depositsin Gejiu Yunnan Province Chinardquo Ore Geology Reviews vol32 no 1-2 pp 314ndash324 2007

[21] QM Cheng C Lu and C Ko ldquoGIS spatial temporal modelingof water systems in greaterrdquo Earth Science Journal of ChinaUniversity of Geosciences vol 15 no 13 pp 1ndash8 2004

[22] P Filzmoser K Hron and C Reimann ldquoInterpretation ofmultivariate outliers for compositional datardquo Computers andGeosciences vol 39 pp 77ndash85 2012

[23] W Wang J Zhao and Q Cheng ldquoAnalysis and integration ofgeo-information to identify granitic intrusions as explorationtargets in southeastern Yunnan District Chinardquo Computers ampGeosciences vol 37 no 12 pp 1946ndash1957 2011

[24] Q Cheng F P Agterberg and S B Ballantyne ldquoThe separationof geochemical anomalies from background by fractal meth-odsrdquo Journal of Geochemical Exploration vol 51 no 2 pp 109ndash130 1994

[25] R Ghavami-Riabi M M Seyedrahimi-Niaraq R KhalokakaieandM R Hazareh ldquoU-spatial statistic data modeled on a prob-ability diagram for investigation of mineralization phases andexploration of shear zone gold depositsrdquo Journal of GeochemicalExploration vol 104 no 1-2 pp 27ndash33 2010

[26] M A Goncalves A Mateus and V Oliveira ldquoGeochemicalanomaly separation by multifractal modellingrdquo Journal of Geo-chemical Exploration vol 72 no 2 pp 91ndash114 2001

[27] U Kramar ldquoApplication of limited fuzzy clusters to anomalyrecognition in complex geological environmentsrdquo Journal ofGeochemical Exploration vol 55 no 1ndash3 pp 81ndash92 1995

[28] C Li T Ma and J Shi ldquoApplication of a fractal method relatingconcentrations and distances for separation of geochemicalanomalies from backgroundrdquo Journal of Geochemical Explo-ration vol 77 no 2-3 pp 167ndash175 2003

[29] P Roshani A R Mokhtari and S H Tabatabaei ldquoObjectivebased geochemical anomaly detectionmdashapplication of discrim-inant function analysis in anomaly delineation in the Kuh Panjporphyry Cu mineralization (Iran)rdquo Journal of GeochemicalExploration vol 130 pp 65ndash73 2013

[30] P Lenca P Meyer B Vaillant and S Lallich ldquoOn selectinginterestingness measures for association rules user orienteddescription andmultiple criteria decision aidrdquoEuropean Journalof Operational Research vol 184 no 2 pp 610ndash626 2008

[31] T Lee M Girolami A J Bell and T Sejnowski ldquoA unifyinginformation-theoretic framework for independent componentanalysisrdquo Computers amp Mathematics with Applications vol 39no 11 pp 1ndash21 2000

[32] X C Yu L W Liu D Hu and Z N Wang ldquoRobust ordinalindependent component analysis (ROICA) applied to mineralresources predictionrdquo Journal of Jilin University (Earth ScienceEdition) vol 42 no 3 pp 872ndash880 2012

[33] C Zhao B E Hobbs H B Muhlhaus A Ord and G LinldquoConvective instability of 3-D fluid-saturated geological faultzones heated from belowrdquo Geophysical Journal Internationalvol 155 no 1 pp 213ndash220 2003

[34] C Zhao B E Hobbs A Ord S Peng H B Muhlhaus andL Liu ldquoDouble diffusion-driven convective instability of three-dimensional fluid-saturated geological fault zones heated frombelowrdquoMathematical Geology vol 37 no 4 pp 373ndash391 2005

[35] C Zhao B E Hobbs and A Ord ldquoTheoretical analyses ofnonaqueous phase liquid dissolution-induced instability intwo-dimensional fluid-saturated porous mediardquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 34 no 17 pp 1767ndash1796 2010

[36] C Zhao B E Hobbs A Ord P Hornby and S Peng ldquoMor-phological evolution of three-dimensional chemical dissolutionfront in fluid-saturated porous media a numerical simulationapproachrdquo Geofluids vol 8 no 2 pp 113ndash127 2008

[37] C Zhao B E Hobbs P Hornby A Ord S Peng and L LiuldquoTheoretical and numerical analyses of chemical-dissolutionfront instability in fluid-saturated porous rocksrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 32 no 9 pp 1107ndash1130 2008

[38] C Zhao B E Hobbs A Ord P Hornby and S Peng ldquoEffect ofreactive surface areas associated with different particle shapeson chemical-dissolution front instability in fluid-saturatedporous rocksrdquo Transport in Porous Media vol 73 no 1 pp 75ndash94 2008

[39] C Zhao B E Hobbs A Ord and S Peng ldquoEffects of mineraldissolution ratios on chemical-dissolution front instability influid-saturated porous mediardquo Transport in Porous Media vol82 no 2 pp 317ndash335 2010

[40] C Zhao B E Hobbs and A Ord ldquoTheoretical analyses of theeffects of solute dispersion on chemical-dissolution front insta-bility in fluid-saturated porous mediardquo Transport in PorousMedia vol 84 no 3 pp 629ndash653 2010

[41] C Zhao B E Hobbs and A Ord ldquoEffects of medium and pore-fluid compressibility on chemical-dissolution front instability influid-saturated porous mediardquo International Journal for Num-erical and Analytical Methods in Geomechanics vol 36 no 8pp 1077ndash1100 2012

[42] C Zhao Physical and Chemical Dissolution Front Instability inPorous Media Theoretical Analyses and Computational Simula-tions Springer Berlin Germany 2014

[43] C Zhao L B Reid K Regenauer-Lieb and T Poulet ldquoA poro-sity-gradient replacement approach for computational simula-tion of chemical-dissolution front propagation in fluid-satu-rated porous media including pore-fluid compressibilityrdquo Com-putational Geosciences vol 16 no 3 pp 735ndash755 2012

[44] C Zhao B E Hobbs and A Ord ldquoTheoretical analyses ofacidization dissolution front instability in fluid-saturated car-bonate rocksrdquo International Journal for Numerical and Analyti-calMethods inGeomechanics vol 37 no 13 pp 2084ndash2105 2013

[45] C Zhao B E Hobbs and A Ord ldquoEffects of medium perme-ability anisotropy on chemical-dissolution front instability in


fluid-saturated porous mediardquo Transport in Porous Media vol99 no 1 pp 119ndash143 2013

[46] A Hyvarinen P O Hoyer andM Inki ldquoTopographic indepen-dent component analysisrdquoNeural Computation vol 13 no 7 pp1527ndash1558 2001

[47] C Jutten and J Herault ldquoBlind separation of sources part I anadaptive algorithm based on neuromimetic architecturerdquo SignalProcessing vol 24 no 1 pp 1ndash10 1991

[48] P Comon ldquoIndependent component analysis A new conceptrdquoSignal Processing vol 36 no 3 pp 287ndash314 1994

[49] C Zhao L B Reid and K Regenauer-Lieb ldquoSome fundamentalissues in computational hydrodynamics of mineralization areviewrdquo Journal of Geochemical Exploration vol 112 pp 21ndash342012

[50] C Zhao B E Hobbs and A Ord ldquoTheoretical and numericalinvestigation into roles of geofluid flow in ore forming systemsintegrated mass conservation and generic model approachrdquoJournal of Geochemical Exploration vol 106 no 1ndash3 pp 251ndash260 2010

[51] TW AndersonAn Introduction toMultivariate Statistical Ana-lysis John Wiley amp Sons New York NY USA 1984

[52] R J Serfling Approximation Theorems of Mathematical Statis-tics John Wiley amp Sons New York NY USA 1980

[53] EMasry ldquoThe estimation of the correlation coefficient of bivari-ate data under dependence convergence analysisrdquo Statistics ampProbability Letters vol 81 no 8 pp 1039ndash1045 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with geochemical data many researchers study the spatialdistribution characteristics of the elements and the physicaland chemical properties of the bare geologic body [23] Asa result several quantitative geochemical methods such asspatial statistics the fractal technique discriminant analysisand fuzzy clustering [24ndash28] have been developed overthe past few decades These methods are associated withthe density frequency of the element and are based onthe perceptions of mineralization which may lead to falseanomalies that have nothing to do with mineralization [29]In fact geochemical exploration data processing methodsshould only consider data cardinalities [30] For examplein the independent component analysis (ICA) methodhigher-order statistical features of data are taken intoaccount [31] The method isolates the independent sourcesignals that are implicit in the mixed signal under ldquoblindrdquoconditions In previous studies the ICA method was mainlyused for mineral predictions and is still at a fledging stage ingeochemical applications [32]

Although the mathematical analysis and computationalsimulation methods associated with applied mathematicshave been widely used to produce analytical solutions [1ndash333ndash36] and numerical results for understanding the dynamicmechanisms and processes of ore-forming systems [4ndash9] thisstudy attempts to utilize the FICAA for detecting anomaliesin geochemical data For example Zhao and his coworkershave successfully solved a set of partial differential equationsfor the convective pore-fluid problems that are closely relatedto hydrothermal ore-forming systems [1ndash3 33 34] Theyalso solved a different set of partial differential equationsfor the chemical dissolution front instability problems thatare present in many ore-forming systems within the uppercrust of the Earth [37ndash45] According to the FICAA andthe characteristics of the geochemical data processing rawgeochemical data with the FICAA can solve the sequenceindeterminacy problem of separated results caused by theFICAA by calculating the correlation coefficient matrix ofraw data and separating the independent components Inaddition the cumulative frequency method is used to solvethe indeterminacy problem in scaling by direct comparison tothe resultant anomalies When a late prospecting trench vali-dation is conducted the data (after FICAAprocessing) can beused to better reflect the spatial distribution characteristics ofthe elements This can provide useful geological informationfor understanding and simulating the dynamic mechanismsof ore-forming systems [14 15]

This paper is organized as follows The principles ofthe FICAA and the correlation coefficient algorithm areintroduced in Section 2 A brief introduction of the geologicalbackground of the area around Dachaidan in the Qinghaiprovince of China is given in Section 3The FICAA is appliedto the 1 10 000 soil geochemical data of this area in Section4The results are discussed and conclusions are stated in Sec-tions 5 and 6 respectively

2 Algorithm Principle

21The ProblemDescription of Independent Component Anal-ysis Independent component analysis (ICA) is a datamining



S1 S2 S3 Sm

X1 X2 X3 Xm

Figure 1 Signal mixing process

technique that based on the hypothesis of statistical inde-pendence analyzes data from a perspective of higher-orderstatistical correlation [46] A set of random initial vectorsapproximates the independent signal source implicated in themixed signal by ldquodecomposingrdquo themixed signal Eachmixedsignal is a linear combination of original signals (Figure 1)The basic mathematical model of the ICA is as follows

119909 = 119860119904 (1)

where 119909 denotes some observed mixed signals matrix 119860

represents the mixing matrix of the system and vector 119904

represents unknown source signals that are assumed to bestatistically independent

When the source signals 119904 and the mixed matrix 119860 areboth unknown only 119909 can be obtainedThe ICA algorithm isdesigned to determine a matrix 119882 such that

119910 = 119882119909 (2)

where 119910 is an optimal estimate of the source signals 119904 Thelinear solution of the ICAmodel can be obtained with (1) and(2) Consider

119910 = 119882119860119904 = 119866119904 (3)

where 119866 = 119882119860 is an 119899 times 119899 identity matrixIn the late 1980s Jutten andHerault proposed the concept

of the ICA [47] In 1994 Comon extended the principalcomponent analysis based on data processing and com-pression as an independent component analysis algorithmand proposed independent component analysis based on theminimum of mutual information [48] In 2001 Hyvarinen etal proposed an algorithm for extracting the fixed-point fromablind signal also called the fast ICA (FICA) In 2012 Yu et alapplied the FICA to mineral prediction [32] However both






119869 (119910) = 119867(119910gauss) minus 119867 (119910) (4)


119869 (119910) prop [119864 119866 (119910) minus 119864 119866 (119910gauss)]2

(5)


44



119879119909)

2

= 1199082


119864 119909119892 (119908119879119909) minus 120573119909 = 0 (6)


119864119908119879

0119909119892(119908119879

0119909)119908





119869119865 (119908) = 119864 1199091199091198791198921015840(119908119879119909) minus 120573119868 (7)


119864 1199091199091198791198921015840(119908119879119909) asymp 119864 119909119909

119879 119864 119892

1015840(119908119879119909)

= 119864 1198921015840(119908119879119909) 119868

(8)



119908lowast

= 119908 minus

[119864 119909119892 (119908119879119909) minus 120573119908]

[119864 1198921015840(119908119879119909) minus 120573]

119908 =

119908lowast

119908lowast

(9)


119879119909119892(119908119879119909)


119908lowast

= 119864 119909119892 (119908119879119909) minus 119864 119892

1015840(119908119879119909)119908

119908 =

119908lowast

119908lowast

(10)


Let (1198831 1198832 1198833 119883





elements 120588119894119895



119877 =

[

[

[

[

[

12058811

12058812


12058821

12058822


1205881198991

1205881198992

sdot sdot sdot 120588119899119899

]

]

]

]

]

(11)

where 120588119894119895

= cov(119883119894 119883119895)radic119863119883

119894radic119863119883119895 cov(119883

119894 119883119895) = 119864[119883

119894minus

119864119883119894] times [119883

119895minus 119864119883

119895]




∘4710158405110158401015840 to 95

∘5810158401810158401015840 and the north-


∘4510158401410158401015840 The study









4 Application










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

TC01

TC02TC03

TC04

TC05

TC06

TC07

TC08

TC09TC10

P1a1

P1a1

S

S

SQ

N

The type of stratum

Q Quaternary

Permian




General symbols

Fault


Water system

Prospecting trench

1100000 100 200 400 600 800

QinghaiChina

Study area

0 500 1000

(km)

1000

(m)



Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200



0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119869 (119910) = 119867(119910gauss) minus 119867 (119910) (4)


119869 (119910) prop [119864 119866 (119910) minus 119864 119866 (119910gauss)]2

(5)


44



119879119909)

2

= 1199082


119864 119909119892 (119908119879119909) minus 120573119909 = 0 (6)


119864119908119879

0119909119892(119908119879

0119909)119908





119869119865 (119908) = 119864 1199091199091198791198921015840(119908119879119909) minus 120573119868 (7)


119864 1199091199091198791198921015840(119908119879119909) asymp 119864 119909119909

119879 119864 119892

1015840(119908119879119909)

= 119864 1198921015840(119908119879119909) 119868

(8)



119908lowast

= 119908 minus

[119864 119909119892 (119908119879119909) minus 120573119908]

[119864 1198921015840(119908119879119909) minus 120573]

119908 =

119908lowast

119908lowast

(9)


119879119909119892(119908119879119909)


119908lowast

= 119864 119909119892 (119908119879119909) minus 119864 119892

1015840(119908119879119909)119908

119908 =

119908lowast

119908lowast

(10)


Let (1198831 1198832 1198833 119883





elements 120588119894119895



119877 =

[

[

[

[

[

12058811

12058812


12058821

12058822


1205881198991

1205881198992

sdot sdot sdot 120588119899119899

]

]

]

]

]

(11)

where 120588119894119895

= cov(119883119894 119883119895)radic119863119883

119894radic119863119883119895 cov(119883

119894 119883119895) = 119864[119883

119894minus

119864119883119894] times [119883

119895minus 119864119883

119895]




∘4710158405110158401015840 to 95

∘5810158401810158401015840 and the north-


∘4510158401410158401015840 The study









4 Application










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

TC01

TC02TC03

TC04

TC05

TC06

TC07

TC08

TC09TC10

P1a1

P1a1

S

S

SQ

N

The type of stratum

Q Quaternary

Permian




General symbols

Fault


Water system

Prospecting trench

1100000 100 200 400 600 800

QinghaiChina

Study area

0 500 1000

(km)

1000

(m)



Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200



0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















4 Application










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

TC01

TC02TC03

TC04

TC05

TC06

TC07

TC08

TC09TC10

P1a1

P1a1

S

S

SQ

N

The type of stratum

Q Quaternary

Permian




General symbols

Fault


Water system

Prospecting trench

1100000 100 200 400 600 800

QinghaiChina

Study area

0 500 1000

(km)

1000

(m)



Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200



0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

TC01

TC02TC03

TC04

TC05

TC06

TC07

TC08

TC09TC10

P1a1

P1a1

S

S

SQ

N

The type of stratum

Q Quaternary

Permian




General symbols

Fault


Water system

Prospecting trench

1100000 100 200 400 600 800

QinghaiChina

Study area

0 500 1000

(km)

1000

(m)



Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200



0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Au Cu

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Pb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Zn

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

As

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200

Sb

Observed cum prob

10

08

06

04

02

00

Prob

abili

ty

100806040200



0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0

500

0

50

100

0

1000

2000

0

100

200

0

1000

2000

0

100

200


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

minus50

0

50

minus50

0

50

minus50

0

50

minus50

0

50

minus10

0

10

minus10

minus5

0



5 Discussion



748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

10

25

52


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleAu

087

157

288






748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

34

43

51


748473 749000 750000 751000 751773

4182000

4183000

4182000

4183000

748473 749000 750000 751000 751773

Soil sampleCu

061

174

minus015









6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













6 Conclusions





Acknowledgment


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a fast independent component analysis algorithm for...

Documents