Research Article Generalized Likelihood Uncertainty Estimation Method in Uncertainty...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 701923 9 pageshttpdxdoiorg1011552013701923

Research ArticleGeneralized Likelihood Uncertainty Estimation Method inUncertainty Analysis of Numerical Eutrophication ModelsTake BLOOM as an Example

Zhijie Li1 Qiuwen Chen12 Qiang Xu1 and Koen Blanckaert1

1 RCEES Chinese Academy of Science Beijing 100085 China2 China Three Gorges University Yichang 443002 China

Correspondence should be addressed to Qiuwen Chen qchenrceesaccn

Received 17 February 2013 Accepted 7 May 2013

Academic Editor Yongping Li

Copyright copy 2013 Zhijie Li et alThis 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

Uncertainty analysis is of great importance to assess and quantify a modelrsquos reliability which can improve decision making basedon model results Eutrophication and algal bloom are nowadays serious problems occurring on a worldwide scale Numericalmodels offer an effective way to algal bloom prediction and management Due to the complex processes of aquatic ecosystem suchnumerical models usually contain a large number of parameters which may lead to important uncertainty in the model resultsThis research investigates the applicability of generalized likelihood uncertainty estimation (GLUE) to analyze the uncertainty ofnumerical eutrophication models that have a large number of intercorrelated parameters The 3-dimensional primary productionmodel BLOOM which has been broadly used in algal bloom simulations for both fresh and coastal waters is used

1 Introduction

Eutrophication and algal bloom are serious problems occur-ring on a worldwide scale which deteriorate the water qual-ities in many aspects including oxygen depletion bad smelland production of scums and toxins Accurate and reliablepredictions of algal blooms are essential for early warningand risk mitigating Numerical eutrophication models offeran effective way to algal blooms prediction andmanagementThere exist several well-developed eutrophication modelssuch as CE-QUAL-ICM [1 2] EUTRO5 [3ndash5] BLOOM [6ndash10] CAEDYM[11 12] and Pamolare [13 14]The choice of themost appropriate model may depend on the specific researchobjectives and data availability

Due to the complexity of algal bloom processes thesenumerical models usually have a large number of parame-ters which inevitably brings uncertainty to model resultsModeling practice typically includes model developmentcalibration validation and application while uncertaintyanalysis is often neglected Uncertainty analysis is essentialin the assessment and quantification of the reliability of

models Prior to the use of model results information aboutmodel accuracy and confidence levels should be provided toguarantee that results are in accordance with measurements[15] and that the model is appropriate for its prospectiveapplication [16]There are three major sources of uncertaintyin modeling systems parameter estimation input data andmodel structure [17ndash21] Understanding and evaluating thesevarious sources of uncertainty in eutrophication models areof importance for algal bloom management and aquaticecosystem restoration

Several methods for parameter uncertainty analysis areavailable for example probability theory method MonteCarlo analysis Bayesian method and generalized likelihooduncertainty estimation (GLUE) method Probability theorymethod employs probability theory of moments of linearcombinations of random variables to define means and vari-ances of random functions It is straightforward for simplelinear models while it does not apply to nonlinear systems[22] The Monte Carlo analysis computes output statistics byrepeating simulations with randomly sampled input variablescomplying with probability density functions It is easily

2 Mathematical Problems in Engineering

implemented and generally applicable but the results gainedfrom Monte Carlo analysis are not in an analytical formand the joint distributions of correlated variables are oftenunknown or difficult to derive [18 20 23] Bayesianmethodsquantify uncertainty by calculating probabilistic predictionsDetermining the prior probability distribution of modelparameters is the key step in Bayesian methods [24] GLUEis a statistical method for simultaneously calibrating theinput parameters and estimating the uncertainty of predictivemodels [25] GLUE is based on the concept of equifinalitywhich means that different sets of input parameter mayresult in equally good and acceptable model outputs fora chosen model [26] It searches for parameter sets thatwould give reliable simulations for a range of model inputsinstead of searching for an optimumparameter set that wouldgive the best simulation results [27] Furthermore modelperformance in GLUE is mainly dependent on parametersets rather than individual parameters whence interactionbetween parameters is implicitly accounted for

Beven and Freer [28] pointed out that in complexdynamic models that contain a large number of highlyintercorrelated parameters many different combinations ofparameters can give equivalently accurate predictions Inconsideration of equivalence of parameter sets the GLUEmethod is particularly appropriate for the uncertainty assess-ment of numerical eutrophication models which are anexample of complex dynamic models with highly intercorre-lated parameters [28ndash30]

The GLUE method has already been adopted for uncer-tainty assessments in a variety of environmental modelingapplications including rainfall-runoff models [25 31 32]soil carbon models of forest ecosystems [33] agriculturalnonpoint source (NPS) pollution models [34] groundwaterflow models [35] urban stormwater quality models [36 37]crop growthmodels [38] and wheat canopymodels [39]Thepopularity of the GLUE method can be attributed to its sim-plicity and wide applicability especially when dealing withnonlinear and nonmonotonic ecological dynamic models

The objectives of this paper are tomake use of the broadlyused eutrophication and algal bloommodel BLOOM[6 7] inorder to investigate the applicability of the GLUE method toanalyze and quantify the uncertainty in numerical eutroph-ication models that have a large number of intercorrelatedparameters and to provide a reference for method selectionwhen conducting uncertainty analysis for similar types ofmodels

2 Materials and Methodology

21 Study Area TheMeiliang Bay (31∘271015840N120∘101015840E) whichlocates at the north of Taihu Lake in China (Figure 1)is chosen as the study area Taihu Lake has high levelof eutrophication and algal blooms that cause enormousdamage to drinking water safety tourisms and fish farmingfrequently break out in summer and autumn

The Meiliang Bay has a length of 166 km from south tonorth a width of 10 km from east to west and an averagedepth of 195m There are two main rivers the Zhihu Gangthat flows into Taihu Lake and the Liangxi River that flows out

N

Sample sitesRivers

Meiliang Bay

Liangxi RiverZhihu Gang 1 2

3

4

0 10 20(km)

China

Taihu

Figure 1 Location of the Taihu Lake and the Meiliang Bay(31∘271015840N120∘101015840E)

of the lake The exchange of substance between Meiliang Bayand the main body of Taihu Lake is taken into account in thestudyThemonthly observed data from four monitoring sitesin the Meiliang Bay were collected during 2009 to 2011 formodel calibration These data include river discharge waterlevel irradiance temperature concentrations of ammonianitrate nitrite phosphate and biomass concentration of blue-green algae green algae and diatom

In the composition of the algae blooms blue-green algaeis the dominant species andhas the highest percentage of totalbiomass Therefore it is selected to be the output variable ofBLOOM on which the uncertainty analysis is performed

22 BLOOMModel BLOOM is a generic hydroenvironmen-tal numerical model that can be applied to calculate primaryproduction chlorophyll-a concentration and phytoplanktonspecies composition [6ndash10] Fifteen algae species can bemod-eled including blue-green algae green algae and diatomsEach algae species has up to three types the N-type P-typeand E-type which correspond to nitrogen limiting condi-tions phosphorus limiting conditions and energy limitingconditions respectively Algae biomass in BLOOM mainlydepends on primary production and transport

The transport of dissolved or suspended matter in thewater body is modeled by solving the advection-diffusionequation numerically

120597119862

120597119905= 119863119909

1205972

119862

1205971199092minus V119909

120597119862

120597119909+ 119863119910

1205972

119862

1205971199102minus V119910

120597119862

120597119910+ 119878 + 119891119877 (119862 119905)

(1)

where 119862 concentration (kgsdotmminus3) 119863119909 119863119910 dispersion coef-

ficient in 119909- and 119910-direction respectively (m2sdotsminus1) S sourceterms 119891119877 (119862 119905) reaction terms 119905 time (s)

Mathematical Problems in Engineering 3

Primary production is mainly dependent on the specificrates of growth mortality and maintenance respirationwhich are modulated according to the temperature

119896119892119901119894

= Proalg0119894

times TcPalg119879119894

119896119898119903119905119894= Moralg0

119894

times TcMalg119879119894

119896119903119904119901119894

= Resalg0119894

times TcRalg119879119894

(2)

where 119896119892119901119894

potential specific growth rate of the fastestgrowing type of algae species (dminus1) Proalg0

119894

growth rate at0∘C (dminus1) TcPalg

119894

temperature coefficient for growth (ndash)119896119898119903119905119894 specific mortality rate (dminus1) Moralg0

119894

mortality rate at0∘C (dminus1) TcMalg

119894

temperature coefficient for mortality (ndash) 119896119903119904119901

119894

specific maintenance respiration rate (dminus1) Resalg0119894

maintenance respiration rate at 0∘C (dminus1) TcRalg

119894

tempera-ture coefficient for respiration (ndash) More details of BLOOMcan be found in Delft Hydraulics [6 7]

23 Generalized Likelihood Uncertainty Estimation TheGLUE methodology [25] is based upon a large number ofmodel runs performed with different sets of input param-eter sampled randomly from prior specified parameterdistributions The simulation result corresponding to eachparameter set is evaluated by means of its likelihood valuewhich quantifies how well the model output conforms tothe observed values The higher the likelihood value thebetter the correspondence between the model simulationand observations Simulations with a likelihood value largerthan a user-defined acceptability threshold will be retainedto determine the uncertainty bounds of the model outputs[33 40]Themajor procedures for performing GLUE includedetermining the ranges and prior distributions of inputparameters generating random parameter sets defining thegeneralized likelihood function defining threshold value forbehavioral parameter sets and calculating the model outputcumulative distribution function

BLOOM contains hundreds of parameters Ideally all theparameters should be regarded stochastically and includedin the uncertainty analysis However a more practical andtypical manner to conduct uncertainty analysis is to focuson a few key parameters [25 28 41] In eutrophicationmodels algal biomass is most closely related to the growthmortality and respiration processes resulting in the selectionof seven key parameters about blue-green algae accordingto (2) Table 1 summarizes the main characteristics of theseseven parameters The initial ranges of the parameters areobtained by model calibration and a uniform prior distri-bution reported in the literature [28] is considered for allparameters

Latin Hypercube Sampling (LHS) which is a type ofstratified Monte Carlo sampling is employed in this studyto generate random parameter sets from the prior parameterdistributions In total 60000 parameter combinations aregenerated for the model runs

The GLUE method requires the definition of a likelihoodfunction in order to quantify how well simulation results

conform to observed data The likelihood measure shouldincrease monotonically with increasing conformity betweensimulation results and observations [25] Various likelihoodfunctions have been proposed and evaluated in the literature[35 37 38 42] Keesman and van Straten [43] defined thelikelihood function based on the maximum absolute resid-ual Beven and Binley [25] defined the likelihood functionbased on the inverse error variance with a shape factor119873 Romanowicz et al [44] defined the likelihood functionbased on an autocorrelated Gaussian error model Freer et al[45] defined the likelihood function based on the Nash-Sutliffe efficiency criterion with shape factor 119873 as well asthe exponential transformation of the error variance withshaping factor 119873 Wang et al [41] defined the likelihoodfunction based on minimum mean square error In thisstudy the likelihood function 119871(120579

119894| 119874) of the model

run corresponding to the 119894th set of input parameters (120579119894)

and observations 119874 is defined based on the exponentialtransformation of the error variance 1205902

119890

and the observationvariance 1205902

0

with shape factor119873 [37 45]

119871 (120579119894| 119874) = exp(minus119873 lowast

1205902

119890

12059020

) (3)

where 1205902119890

= sum (119910sim minus 119910obs)2 1205902119900

= sum(119910obs minus 119910obs)2 119910sim is

the simulated blue-green algae biomass 119910obs is the observedblue-green algae biomass 119910obs is the average value of 119910obs

The sensitivity of the choice of the shape factor 119873 willbe analyzed and discussed If the likelihood value of asimulation result is larger than a user-defined threshold themodel simulation is considered ldquobehavioralrdquo and retained forthe subsequent analysis Otherwise the model simulationis considered ldquononbehavioralrdquo and removed from furtheranalysis There are two main methods for defining thethreshold value for behavioral parameter sets one is to allowa certain deviation from the highest likelihood value in thesample and the other is to use a fixed percentage of the totalnumber of simulations [46] The latter is used in this studyand the acceptable sample rate (ASR) is defined as 60 Thesensitivity of the choice of the threshold in the form of theacceptable sample rate (ASR) will be analyzed and discussed

The likelihood function is then normalized such that thecumulative likelihood of all model runs equals 1

119871119908(120579119894) =

119871 (120579119894| 119874)

sum119894

119871 (120579119894| 119874) (4)

where 119871119908(120579119894) is the normalized likelihood for the119894th set of

input parameters (120579119894) The uncertainty analysis is performed

by calculating the cumulative distribution function (CDF) ofthe normalized likelihood together with prediction quantiles

The GLUE-derived 90 confidence intervals for thebiomass of blue green are then obtained by reading 5 and95 percentiles of the cumulative distribution functions

3 Results

31 BLOOM Model Results The calibration result for blue-green algae is shown in Figure 2 and the calibration


Table 1 Selected input parameters and their initial ranges

Parameter CategoryEquations (2) Definition Unit Lower

boundUpperbound

Calibratedvalue

ProBlu 1198640 Proalg0119894

Growth rate at 0∘C for blue-green E-type 1d 0013 0019 0016TcPBlu E TcPalg

119894

Temperature coefficient for growth for blue-green E-type mdash 1040 1100 108TcPBlu N TcPalg

119894

Temperature coefficient for growth for blue-green N-type mdash 1040 1100 108TcPBlu P TcPalg

119894

Temperature coefficient for growth for blue-green P-type mdash 1040 1100 108MorBlu 1198640 Moralg0

119894

Mortality rate at 0∘C for blue-green E-type 1d 0028 0042 0035TcMBlu E TcMalg

119894

Temperature coefficient for mortality for blue-green E-type mdash 1000 1020 101TcRBlu E TcRalg

119894

Temperature coefficient for maintenance respiration for blue-green E-type mdash 1040 1100 1072

Table 2 Statistical characteristics of observed data from 2009 to2011

Station Mean(gCm3)

Standard deviation(gCm3)

Maximum(gCm3)

Minimum(gCm3)

1 1023 1504 6744 00152 1144 2073 9885 00023 1135 1369 5037 00044 1179 1564 5534 0010

parameters are summarized in the last column of Table 1The statistical characteristics of the observed blue-green algaebiomass are shown in Table 2 The mean values of the blue-green algae biomass for the four sample sites are similarTherefore in order to reduce the sampling uncertaintiesthe average of the four sampling sites has been retained asdependent variable in the present study

The biomass of blue-green has a yearly cycle (Figure 2)with low values during spring followed by a rapid increasetowards peak values in summer or autumn The growthperiodicity of blue-green algae is mainly attributed to theperiodic variation of temperature and algae dormancy TaihuLake experiences a subtropical monsoon climate with fourdistinct seasons The lowest temperature is about 28∘C inaverage and appears in January and the highest temperatureis about 294∘C in average and usually appears in AugustThe suitable temperature range for growth of blue-green algaeis 25sim35∘C As a result the biomass of blue-green algae islow in spring When temperature increases in summer itis appropriate for blue-green algae breeding leading to thesharp increase in biomass and the occurrence of the peakvalue around August

The modes capture satisfactorily the observed evolutionof the blue-green algae biomass which indicates further anal-yses on model uncertainty are meaningful The coefficient ofdetermination (CoD) which is given by (5) is 085

CoD =sum119894

(119910119904119894minus 119910119900)2

sum119894

(119910119900119894minus 119910119900)2

(5)

where 119910119904119894 the simulated biomass of blue-green algae at time

step 119894 119910119900 the mean value of observed data 119910

119900119894 the observed

value of blue-green at time step 119894

0

1

2

3

4

5

6

SimulationObservation

2011-12010-12009-1

Biom

ass o

f blu

e gre

en (g

Cm

3)

Figure 2 Modeled results and observations of blue-green algae

32 Uncertainty Analysis Results The confidence interval(CI) is obtained by calculating the cumulative distributionfunctions of model outputs based on the normalized likeli-hood (4) with 119873 = 1 and ASR = 60 Figure 3 presents the90 confidence interval of blue-green algae biomass whichis estimated from the 5 and 95 quantiles of the cumulativedistribution functions and the corresponding observationsfrom January 2009 toDecember 2011 Table 3 summarizes thewidth of the 90 CI of each month and whether or not theobservations are located within the 90 CI

The 90 CI is narrow from January to May when thebiomass of blue-green algae is low The width of the 90 CIexpands as the biomass of blue-green algae increases duringsummer and autumn Among the total of 36 observations 13are located within the 90 CI of the simulations

The subjective choice of the shape factor119873 in (1) consid-erably influences the GLUE results whereas119873 is commonlytaken as 1 [35] Figure 4 displays the 90 CI when ASR =60 with shape factors119873 equal 50 and 100 respectivelyThesimulated 5 and 95 confidence quantiles and the weightedmean as well as the corresponding observations of blue-green algae biomass are shown

Comparison of Figures 3 and 4 shows that the increase ofshape factor 119873 leads to a narrowing of the 90 CI Figure 5


Table 3 Width of 90 confidence interval (CI) for each month and indication whether (Y) or not (N) the observations is located within the90 CI band

Year Month 1 2 3 4 5 6 7 8 9 10 11 12

2009 90 CI (gCm3) 0004 0011 0010 0080 0349 1533 1646 0694 0989 1051 1538 0363within CI N Y N N Y N Y N Y N N N

2010 90 CI (gCm3) 0093 0029 0009 0013 0038 0147 1078 1362 1567 0768 1199 1272within CI N N N N N Y Y Y Y N Y Y

2011 90 CI (gCm3) 0504 0114 0039 0054 0182 0993 0865 1136 0872 1196 1500 0648within CI Y N N N Y N N N N N N Y

2009-1 2010-1 2011-1

005

115

225

335

445

555

Observation5

95Weighted mean

minus05

119873 = 1

Biom

ass o

f blu

e gre

en (g

Cm

3)

Figure 3 5 and 95 confidence quantiles and weigthed mean ofsimulated biomass of blue-green algae when ASR = 60 and 119873 =1 and corresponding observations from January 2009 to December2011

illustrates the effect of the shape factor 119873 which can beseen as a weight factor for the likelihood correspondingto each simulation When 119873 = 1 the magnitudes ofthe likelihood are similar for each simulation and thereis no clear division between acceptable and unacceptablesimulations As a result the cumulative distribution functionsincrease graduallyWith increasing119873 (eg119873 = 50) the highbehavioral simulations have a higher weight resulting in alarger gradient in the cumulative distribution function and anarrower CITheoretically when119873 = 0 every simulation hasequal likelihood and the widest CI will be obtained When119873 rarr infin the single best simulation will have a normalizedlikelihood of 1 while all other simulationswill get a likelihoodof zero resulting in the collapse of the 5 and 95 quantileson a single lineThis corresponds to the traditional calibrationmethod that omits uncertainty analysis

Previous studies have shown that the choice of thresholdvalues for the likelihood measures is particularly importantfor the GLUE method [34 36 47] In order to quantifythe effect of threshold values on the uncertainty analysesa series of acceptable sample rates (ASR) of 05 1 510 30 60 90 95 99 is investigated In this studyaverage relative interval length (ARIL) and percentage of

observations covered by the 90 confidence interval (11987590CI)

are adopted as metrics for the analysis These metrics aredefined as follows

ARIL = 1119899sum

Limitupper119905 minus Limitlower119905119861obs119905

(6)

where Limitupper119905 and Limitlower119905 are the upper and lowerboundary values of the 90 confidence interval 119899 is thenumber of time steps 119861obs119905 is the observed biomass of blue-green algae

11987590CI =

119873119876in119873obs

times 100 (7)

where 119873119876in is the number of observations located within90 CI119873obs is the total number of observations

Figures 6 and 7 present the influence of ASR onARIL and11987590CI for 119873 = 1 50 100 Figure 6 shows that for all ASR

values ARIL has the highest value for 119873 = 1 and decreaseswith increasing119873 which confirms the results of Figure 4 Fora given119873 value ARIL increases with ASRWhen ASRmovesfrom 05 to 99 the ARIL increases by 7393 4196and 524 for 119873 = 1 50 100 respectively An increasingASR which corresponds to a lower threshold of the acceptedlikelihood means that simulations with lower likelihood areconsidered ldquobehavioralrdquo which inevitably results in a largerARIL

From Figure 7 it is seen that 11987590CI becomes larger as

ASR increases for 119873 = 1 and 119873 = 50 while 11987590CI keeps

constant for 119873 = 100 This is because the increase of ASRresults in a larger ARIL which logically leads to an increasein observations located within the 90 CI When 119873 = 100the ARIL is low and119875

90CI does not increase with ASR becausethe 90 CI does not widen

The highest 11987590CI is obtained for ASR close to 100 and

119873 = 1 Its value of about 50 indicates that about half ofthe observed data remain outside the 90 CI for the greatestASR This can be attributed to other sources of uncertaintysuch as the input parameters or the observations

4 Discussion and Conclusion

The 90 confidence interval of the simulated results fails toenclose the peaks of the observed values in 2009 and 2011(Figure 3) Such a feature is not unusual and several reasonscan lead to this result Firstly there are inherent uncertaintiesfrom inputs boundaries and model structure which are not


2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)

Figure 4 5 and 95 confidence quantiles and weigthed mean of simulated biomass of blue-green algae when ASR = 60 and119873 = 50 (a)and 100 (b) and corresponding observations from January 2009 to December 2011

08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)

Figure 5 Dot plots of likelihood according to (4) when ASR = 100 and 119873 = 1 (a) 50 (b) and 100 (c) for TcPBlu E and TcRBlu E (cfTable 1)


0102030405060708090

0 20 40 60 80 100

ARI

L

Acceptable samples rate ()

119873 = 50

119873 = 100

119873 = 1

Figure 6 ARIL as function of ASR for119873 equals 1 50 and 100

0

10

20

30

40

50

0 20 40 60 80 100Acceptable samples rate ()

11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1

Figure 7 11987590CI as function of ASR for119873 equals 1 50 and 100

taken into account explicitly Secondly the observed valuesused for comparison are space averaged through arithmeticmean other than weighted mean which could also introducediscrepancy Finally the original observations from the fourstations containmeasurement uncertainties During summerand autumn when the algal blooms break out the biomassof blue-green algae is high and shows pronounced dailytemporal variations and spatial variations due to changesin irradiance transport by flow and wind drifting Themeasurements taken at a particular time and point cannotfully reflect these fine-scale spatial and temporal dynamicsThe model calibrated in this study is however capable ofsimulating blue-green algae dynamics at large spatial (spatialaverages) and temporal (seasonal) scales

Ideally accurate predictions require that the resultsare consistent with the observations while the uncertaintyspread of the results quantified by the 90CI is as narrow aspossible [46] From Figures 6 and 7 it can be seen that whilekeeping ASR fixed the 90 CI is narrowed by increasing theshape factor 119873 at the expense of decreasing the percentageof observations that it covers (119875

90CI) Similarly while keeping

119873 fixed the 90 CI is narrowed by reducing ASR but atthe same time also 119875

90CI decreases As a consequence it isessential to optimally choose119873 and ASR in order to find theoptimal compromise between the uncertainty spread and itscoverage of observations

As illustrated by the application to the BLOOMmodel foralgal bloom GLUE is an appropriate method for uncertaintyanalysis that can cope with equifinality between differentparameter sets incurred by high level of model complexityIn conclusion the study demonstrates that GLUE is aneffectivemethod for uncertainty analysis of complex dynamicecosystem models which provides a solid foundation for theuse of the model predictions in decision making

Acknowledgments

The authors are grateful for the financial support of theNationalNature Science Foundation ofChina (50920105907)National Basic Research Program 973 (2010CB429004) ldquo100Talent Program of Chinese Academy of Sciences (A1049)rdquoand the Chutian Scholarship (KJ2010B002) Koen Blanckaertwas partially funded by the Chinese Academy of SciencesVisiting Professorship for Senior International Scientists(2011T2Z24)

References

[1] C F Cerco and T Cole ldquoThree-dimensional eutrophicationmodel of Chesapeake Bayrdquo Journal of Environmental Engineer-ing vol 119 no 6 pp 1006ndash1025 1993

[2] C F Cerco D Tillman and J D Hagy ldquoCoupling andcomparing a spatially- and temporally-detailed eutrophicationmodel with an ecosystem network model an initial applicationto Chesapeake Bayrdquo Environmental Modelling and Software vol25 no 4 pp 562ndash572 2010

[3] V J Bierman S C Hinz D ZhuW JWiseman N N RabalaisandR E Turner ldquoA preliminarymass balancemodel of primaryproductivity and dissolved oxygen in the Mississippi RiverPlumeInner Gulf Shelf Regionrdquo Estuaries vol 17 no 4 pp886ndash899 1994

[4] W S Lung and C E Larson ldquoWater quality modeling of upperMississippi River and Lake Pepinrdquo Journal of EnvironmentalEngineering vol 121 no 10 pp 691ndash699 1995

[5] P Hernandez R B Ambrose Jr D Prats E Ferrandis andJ C Asensi ldquoModeling eutrophication kinetics in reservoirmicrocosmsrdquoWater Research vol 31 no 10 pp 2511ndash2519 1997

[6] Delft Hydraulics Technical Reference Manual Delft3D-WAQWL Delft Hydraulics Delft The Netherlands 2005

[7] Delft Hydraulics Delft3D-WAQ Users Manual WL DelftHydraulics Delft The Netherlands 2009

[8] F J Los and J W M Wijsman ldquoApplication of a validatedprimary production model (BLOOM) as a screening tool formarine coastal and transitional watersrdquo Journal of MarineSystems vol 64 no 1ndash4 pp 201ndash215 2007

[9] F J Los M T Villars and M W M van der Tol ldquoA 3-dimensional primary production model (BLOOMGEM) andits applications to the (southern) North Sea (coupled physical-chemical-ecological model)rdquo Journal of Marine Systems vol 74no 1-2 pp 259ndash294 2008


[10] K Salacinska G Y El Serafy F J Los and A Blauw ldquoSensitivityanalysis of the two dimensional application of the genericecological model (GEM) to algal bloom prediction in the NorthSeardquo Ecological Modelling vol 221 no 2 pp 178ndash190 2010

[11] D P Hamilton and S G Schladow ldquoPrediction of water qualityin lakes and reservoirsmdashpart I model descriptionrdquo EcologicalModelling vol 96 no 1ndash3 pp 91ndash110 1997

[12] D Trolle H Skovgaard and E Jeppesen ldquoThewater frameworkdirective setting the phosphorus loading target for a deep lakein Denmark using the 1D lake ecosystem model DYRESM-CAEDYMrdquo Ecological Modelling vol 219 no 1-2 pp 138ndash1522008

[13] S E Jrgensen H T Tsuno H Mahler and V SantiagoldquoPAMOLARE Training Package Planning and Management ofLakes andReservoirsModels for EutrophicationManagementrdquoUNEP DTIE IETC and ILEC Shiga Japan 2003

[14] Z Gurkan J Zhang and S E Joslashrgensen ldquoDevelopment ofa structurally dynamic model for forecasting the effects ofrestoration of Lake Fure Denmarkrdquo Ecological Modelling vol197 no 1-2 pp 89ndash102 2006

[15] N Oreskes K Shrader-Frechette and K Belitz ldquoVerificationvalidation and confirmation of numerical models in the earthsciencesrdquo Science vol 263 no 5147 pp 641ndash646 1994

[16] E J Rykiel ldquoTesting ecological models the meaning of valida-tionrdquo Ecological Modelling vol 90 no 3 pp 229ndash244 1996

[17] M B Beck ldquoWater quality modeling a review of the analysis ofuncertaintyrdquoWater Resources Research vol 23 no 8 pp 1393ndash1442 1987

[18] R W Katz ldquoTechniques for estimating uncertainty in climatechange scenarios and impact studiesrdquo Climate Research vol 20no 2 pp 167ndash185 2002

[19] M Radwan P Willems and J Berlamont ldquoSensitivity anduncertainty analysis for river quality modelingrdquo Journal ofHydroinformatics vol 6 no 2 pp 83ndash99 2004

[20] H Li and J Wu ldquoUncertainty analysis in ecological studiesrdquoin Scaling and Uncertainty Analysis in Ecology Methods andApplications J Wu K B Jones H Li and O L Loucks Edspp 45ndash66 Springer Dordrecht The Netherlands 2006

[21] K E Lindenschmidt K Fleischbein and M BaborowskildquoStructural uncertainty in a river water quality modellingsystemrdquo Ecological Modeling vol 204 no 3-4 pp 289ndash3002007

[22] P Wiwatanadate and H G Claycamp ldquoExact propagation ofuncertainties in multiplicative modelsrdquo Human and EcologicalRisk Assessment vol 6 no 2 pp 355ndash368 2000

[23] M J W Jansen ldquoPrediction error through modelling conceptsand uncertainty from basic datardquo Nutrient Cycling in Agroe-cosystems vol 50 no 1ndash3 pp 247ndash253 1998

[24] G Freni and G Mannina ldquoBayesian approach for uncertaintyquantification in water quality modelling the influence of priordistributionrdquo Journal of Hydrology vol 392 no 1-2 pp 31ndash392010

[25] K Beven and A Binley ldquoThe future of distributed modelsmodel calibration and uncertainty predictionrdquo HydrologicalProcesses vol 6 no 3 pp 279ndash298 1992

[26] R E Brazier K J Beven J Freer and J S Rowan ldquoEquifinalityand uncertainty in physically based soil erosion models appli-cation of the GLUE methodology to WEPPmdashthe water erosionprediction projectmdashfor sites in the UK and USArdquo Earth SurfaceProcesses and Landforms vol 25 no 8 pp 825ndash845 2000

[27] A Candela L V Noto and G Aronica ldquoInfluence of surfaceroughness in hydrological response of semiarid catchmentsrdquoJournal of Hydrology vol 313 no 3-4 pp 119ndash131 2005

[28] K Beven and J Freer ldquoEquifinality data assimilation anduncertainty estimation in mechanistic modelling of complexenvironmental systems using the GLUE methodologyrdquo Journalof Hydrology vol 249 no 1ndash4 pp 11ndash29 2001

[29] J Yang P Reichert K C Abbaspour J Xia andH Yang ldquoCom-paring uncertainty analysis techniques for a SWAT applicationto the Chaohe Basin in Chinardquo Journal of Hydrology vol 358no 1-2 pp 1ndash23 2008

[30] R F Vazquez K Beven and J Feyen ldquoGLUE based assessmenton the overall predictions of a MIKE SHE applicationrdquo WaterResources Management vol 23 no 7 pp 1325ndash1349 2009

[31] R Lamb K Beven and S Myraboslash ldquoUse of spatially distributedwater table observations to constrain uncertainty in a rainfall-runoff modelrdquo Advances in Water Resources vol 22 no 4 pp305ndash317 1998

[32] H T Choi and K Beven ldquoMulti-period and multi-criteriamodel conditioning to reduce prediction uncertainty in anapplication of TOPMODEL within the GLUE frameworkrdquoJournal of Hydrology vol 332 no 3-4 pp 316ndash336 2007

[33] C Ortiza E Karltuna J Stendahl A I Gardenasa and G IAgren ldquoModelling soil carbon development in Swedish conif-erous forest soilsmdashan uncertainty analysis of parameters andmodel estimates using theGLUEmethodrdquo EcologicalModellingvol 222 no 17 pp 3020ndash3032 2011

[34] Y Gong Z Shen Q Hong R Liu and Q Liao ldquoParameteruncertainty analysis in watershed total phosphorus modelingusing the GLUE methodologyrdquo Agriculture Ecosystems andEnvironment vol 142 no 3-4 pp 246ndash24255 2011

[35] A E Hassan H M Bekhit and J B Chapman ldquoUncertaintyassessment of a stochastic groundwater flowmodel usingGLUEanalysisrdquo Journal of Hydrology vol 362 no 1-2 pp 89ndash1092008

[36] G Freni G Mannina and G Viviani ldquoUncertainty in urbanstormwater quality modelling the effect of acceptability thresh-old in the GLUEmethodologyrdquoWater Research vol 42 no 8-9pp 2061ndash2072 2008

[37] G Freni G Mannina and G Viviani ldquoUncertainty in urbanstormwater quality modelling the influence of likelihood mea-sure formulation in theGLUEmethodologyrdquo Science of the TotalEnvironment vol 408 no 1 pp 138ndash145 2009

[38] J He J W Jones W D Graham and M D Dukes ldquoInfluenceof likelihood function choice for estimating cropmodel param-eters using the generalized likelihood uncertainty estimationmethodrdquoAgricultural Systems vol 103 no 5 pp 256ndash264 2010

[39] X Mo and K Beven ldquoMulti-objective parameter conditioningof a three-source wheat canopy modelrdquo Agricultural and ForestMeteorology vol 122 no 1-2 pp 39ndash63 2004

[40] P Smith K J Beven and J A Tawn ldquoInformal likelihoodmeasures in model assessment theoretic development andinvestigationrdquo Advances in Water Resources vol 31 no 8 pp1087ndash1100 2008

[41] X Wang X He J R Williams R C Izaurralde and J DAtwood ldquoSensitivity and uncertainty analyses of crop yields andsoil organic carbon simulated with EPICrdquo Transactions of theAmerican Society of Agricultural Engineers vol 48 no 3 pp1041ndash1054 2005

[42] J R Stedinger R M Vogel S U Lee and R BatchelderldquoAppraisal of the generalized likelihood uncertainty estimation


(GLUE) methodrdquo Water Resources Research vol 44 no 12Article IDW00B06 2008

[43] K Keesman and G van Straten ldquoIdentification and predictionpropagation of uncertainty in models with bounded noiserdquoInternational Journal of Control vol 49 no 6 pp 2259ndash22691989

[44] R Romanowicz K J Beven and J Tawn ldquoEvaluation ofpredictive uncertainty in non-linear hydrological models usinga Bayesian approachrdquo in Statistics for the Environment WaterRelated Issues V Barnett and K F Turkman Eds pp 297ndash317John Wiley amp Sons New York NY USA 1994

[45] J Freer K Beven and B Ambroise ldquoBayesian estimation ofuncertainty in runoff prediction and the value of data anapplication of the GLUE approachrdquo Water Resources Researchvol 32 no 7 pp 2161ndash2173 1996

[46] R S Blasone J A Vrugt H Madsen D Rosbjerg B A Robin-son and G A Zyvoloski ldquoGeneralized likelihood uncertaintyestimation (GLUE) using adaptive Markov Chain Monte Carlosamplingrdquo Advances in Water Resources vol 31 no 4 pp 630ndash648 2008

[47] L Li J Xia C Y Xu and V P Singh ldquoEvaluation of thesubjective factors of the GLUE method and comparison withthe formal Bayesian method in uncertainty assessment ofhydrological modelsrdquo Journal of Hydrology vol 390 no 3-4 pp210ndash221 2010

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


implemented and generally applicable but the results gainedfrom Monte Carlo analysis are not in an analytical formand the joint distributions of correlated variables are oftenunknown or difficult to derive [18 20 23] Bayesianmethodsquantify uncertainty by calculating probabilistic predictionsDetermining the prior probability distribution of modelparameters is the key step in Bayesian methods [24] GLUEis a statistical method for simultaneously calibrating theinput parameters and estimating the uncertainty of predictivemodels [25] GLUE is based on the concept of equifinalitywhich means that different sets of input parameter mayresult in equally good and acceptable model outputs fora chosen model [26] It searches for parameter sets thatwould give reliable simulations for a range of model inputsinstead of searching for an optimumparameter set that wouldgive the best simulation results [27] Furthermore modelperformance in GLUE is mainly dependent on parametersets rather than individual parameters whence interactionbetween parameters is implicitly accounted for

Beven and Freer [28] pointed out that in complexdynamic models that contain a large number of highlyintercorrelated parameters many different combinations ofparameters can give equivalently accurate predictions Inconsideration of equivalence of parameter sets the GLUEmethod is particularly appropriate for the uncertainty assess-ment of numerical eutrophication models which are anexample of complex dynamic models with highly intercorre-lated parameters [28ndash30]

The GLUE method has already been adopted for uncer-tainty assessments in a variety of environmental modelingapplications including rainfall-runoff models [25 31 32]soil carbon models of forest ecosystems [33] agriculturalnonpoint source (NPS) pollution models [34] groundwaterflow models [35] urban stormwater quality models [36 37]crop growthmodels [38] and wheat canopymodels [39]Thepopularity of the GLUE method can be attributed to its sim-plicity and wide applicability especially when dealing withnonlinear and nonmonotonic ecological dynamic models

The objectives of this paper are tomake use of the broadlyused eutrophication and algal bloommodel BLOOM[6 7] inorder to investigate the applicability of the GLUE method toanalyze and quantify the uncertainty in numerical eutroph-ication models that have a large number of intercorrelatedparameters and to provide a reference for method selectionwhen conducting uncertainty analysis for similar types ofmodels

2 Materials and Methodology

21 Study Area TheMeiliang Bay (31∘271015840N120∘101015840E) whichlocates at the north of Taihu Lake in China (Figure 1)is chosen as the study area Taihu Lake has high levelof eutrophication and algal blooms that cause enormousdamage to drinking water safety tourisms and fish farmingfrequently break out in summer and autumn

The Meiliang Bay has a length of 166 km from south tonorth a width of 10 km from east to west and an averagedepth of 195m There are two main rivers the Zhihu Gangthat flows into Taihu Lake and the Liangxi River that flows out

N

Sample sitesRivers

Meiliang Bay

Liangxi RiverZhihu Gang 1 2

3

4

0 10 20(km)

China

Taihu

Figure 1 Location of the Taihu Lake and the Meiliang Bay(31∘271015840N120∘101015840E)

of the lake The exchange of substance between Meiliang Bayand the main body of Taihu Lake is taken into account in thestudyThemonthly observed data from four monitoring sitesin the Meiliang Bay were collected during 2009 to 2011 formodel calibration These data include river discharge waterlevel irradiance temperature concentrations of ammonianitrate nitrite phosphate and biomass concentration of blue-green algae green algae and diatom

In the composition of the algae blooms blue-green algaeis the dominant species andhas the highest percentage of totalbiomass Therefore it is selected to be the output variable ofBLOOM on which the uncertainty analysis is performed

22 BLOOMModel BLOOM is a generic hydroenvironmen-tal numerical model that can be applied to calculate primaryproduction chlorophyll-a concentration and phytoplanktonspecies composition [6ndash10] Fifteen algae species can bemod-eled including blue-green algae green algae and diatomsEach algae species has up to three types the N-type P-typeand E-type which correspond to nitrogen limiting condi-tions phosphorus limiting conditions and energy limitingconditions respectively Algae biomass in BLOOM mainlydepends on primary production and transport

The transport of dissolved or suspended matter in thewater body is modeled by solving the advection-diffusionequation numerically

120597119862

120597119905= 119863119909

1205972

119862

1205971199092minus V119909

120597119862

120597119909+ 119863119910

1205972

119862

1205971199102minus V119910

120597119862

120597119910+ 119878 + 119891119877 (119862 119905)

(1)

where 119862 concentration (kgsdotmminus3) 119863119909 119863119910 dispersion coef-

ficient in 119909- and 119910-direction respectively (m2sdotsminus1) S sourceterms 119891119877 (119862 119905) reaction terms 119905 time (s)



119896119892119901119894

= Proalg0119894


119896119898119903119905119894= Moralg0

119894


119896119903119904119901119894

= Resalg0119894


(2)

where 119896119892119901119894


119894


119894


119894


119894


119894



119894







119894| 119874) of the model



119890


0


119871 (120579119894| 119874) = exp(minus119873 lowast

1205902

119890

12059020

) (3)

where 1205902119890

= sum (119910sim minus 119910obs)2 1205902119900





119871119908(120579119894) =

119871 (120579119894| 119874)

sum119894

119871 (120579119894| 119874) (4)





3 Results





boundUpperbound

Calibratedvalue



119894


119894


119894


119894


119894


119894



Station Mean(gCm3)


Maximum(gCm3)

Minimum(gCm3)

1 1023 1504 6744 00152 1144 2073 9885 00023 1135 1369 5037 00044 1179 1564 5534 0010




CoD =sum119894

(119910119904119894minus 119910119900)2

sum119894

(119910119900119894minus 119910119900)2

(5)





0

1

2

3

4

5

6


2011-12010-12009-1

Biom

ass o

f blu

e gre

en (g

Cm

3)








Year Month 1 2 3 4 5 6 7 8 9 10 11 12




2009-1 2010-1 2011-1

005

115

225

335

445

555

Observation5

95Weighted mean

minus05

119873 = 1

Biom

ass o

f blu

e gre

en (g

Cm

3)






ARIL = 1119899sum


(6)


11987590CI =

119873119876in119873obs

times 100 (7)













2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)


08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)



0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119892119901119894

= Proalg0119894


119896119898119903119905119894= Moralg0

119894


119896119903119904119901119894

= Resalg0119894


(2)

where 119896119892119901119894


119894


119894


119894


119894


119894



119894







119894| 119874) of the model



119890


0


119871 (120579119894| 119874) = exp(minus119873 lowast

1205902

119890

12059020

) (3)

where 1205902119890

= sum (119910sim minus 119910obs)2 1205902119900





119871119908(120579119894) =

119871 (120579119894| 119874)

sum119894

119871 (120579119894| 119874) (4)





3 Results





boundUpperbound

Calibratedvalue



119894


119894


119894


119894


119894


119894



Station Mean(gCm3)


Maximum(gCm3)

Minimum(gCm3)

1 1023 1504 6744 00152 1144 2073 9885 00023 1135 1369 5037 00044 1179 1564 5534 0010




CoD =sum119894

(119910119904119894minus 119910119900)2

sum119894

(119910119900119894minus 119910119900)2

(5)





0

1

2

3

4

5

6


2011-12010-12009-1

Biom

ass o

f blu

e gre

en (g

Cm

3)








Year Month 1 2 3 4 5 6 7 8 9 10 11 12




2009-1 2010-1 2011-1

005

115

225

335

445

555

Observation5

95Weighted mean

minus05

119873 = 1

Biom

ass o

f blu

e gre

en (g

Cm

3)






ARIL = 1119899sum


(6)


11987590CI =

119873119876in119873obs

times 100 (7)













2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)


08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)



0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












boundUpperbound

Calibratedvalue



119894


119894


119894


119894


119894


119894



Station Mean(gCm3)


Maximum(gCm3)

Minimum(gCm3)

1 1023 1504 6744 00152 1144 2073 9885 00023 1135 1369 5037 00044 1179 1564 5534 0010




CoD =sum119894

(119910119904119894minus 119910119900)2

sum119894

(119910119900119894minus 119910119900)2

(5)





0

1

2

3

4

5

6


2011-12010-12009-1

Biom

ass o

f blu

e gre

en (g

Cm

3)








Year Month 1 2 3 4 5 6 7 8 9 10 11 12




2009-1 2010-1 2011-1

005

115

225

335

445

555

Observation5

95Weighted mean

minus05

119873 = 1

Biom

ass o

f blu

e gre

en (g

Cm

3)






ARIL = 1119899sum


(6)


11987590CI =

119873119876in119873obs

times 100 (7)













2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)


08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)



0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Year Month 1 2 3 4 5 6 7 8 9 10 11 12




2009-1 2010-1 2011-1

005

115

225

335

445

555

Observation5

95Weighted mean

minus05

119873 = 1

Biom

ass o

f blu

e gre

en (g

Cm

3)






ARIL = 1119899sum


(6)


11987590CI =

119873119876in119873obs

times 100 (7)













2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)


08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)



0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2009-1 2010-1 2011-1

005

115

225

335

445

555

minus05

119873 = 50

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(a)

2009-1 2010-1 2011-1

119873 = 100

005

115

225

335

445

555

minus05

Observation5

95Weighted mean

Biom

ass o

f blu

e gre

en (g

Cm

3)

(b)


08

06

04

02

0104 106 108 11

Like

lihoo

d

08

06

04

02

0104 106 108 11

Like

lihoo

d

119873 = 1

TcPBlu E

TcRBlu E

(a)

1

08

06

04

02

0104 106 108 11

times10minus8

1

08

06

04

02

0104 106 108 11

times10minus8

Like

lihoo

dLi

kelih

ood

119873 = 50

TcPBlu E

TcRBlu E

(b)

119873 = 100

8

6

4

2

0

times10minus17

104 106 108 11

Like

lihoo

d

104 106 108 11

8

6

4

2

0

times10minus17

Like

lihoo

d

TcPBlu E

TcRBlu E

(c)



0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0102030405060708090

0 20 40 60 80 100

ARI

L


119873 = 50

119873 = 100

119873 = 1


0

10

20

30

40

50


11987590

CI(

)

119873 = 50

119873 = 100

119873 = 1








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Generalized Likelihood Uncertainty Estimation Method in Uncertainty...

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Transcript of Research Article Generalized Likelihood Uncertainty Estimation Method in Uncertainty...