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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 983090983088983089983091 Article ID 983095983088983089983097983090983091 983097 pageshttpdxdoiorg983089983088983089983089983093983093983090983088983089983091983095983088983089983097983090983091

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

983089 RCEES Chinese Academy o Science Beijing 983089983088983088983088983096983093 China983090 China Tree Gorges University Yichang 983092983092983091983088983088983090 China

Correspondence should be addressed to Qiuwen Chen qchenrceesaccn

Received 983089983095 February 983090983088983089983091 Accepted 983095 May 983090983088983089983091

Academic Editor Yongping Li

Copyright copy 983090983088983089983091 Zhijie Li et al Tis 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 o great importance to assess and quantiy 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 o aquatic ecosystem suchnumerical models usually contain a large number o parameters which may lead to important uncertainty in the model resultsTis research investigates the applicability o generalized likelihood uncertainty estimation (GLUE) to analyze the uncertainty o

numerical eutrophication models that have a large number o intercorrelated parameters Te 983091-dimensional primary productionmodel BLOOM which has been broadly used in algal bloom simulations or both resh 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 o scums and toxins Accurate and reliablepredictions o algal blooms are essential or early warningand risk mitigating Numerical eutrophication models offer

an effective way to algal blooms prediction and managementTere exist several well-developed eutrophication modelssuch as CE-QUAL-ICM [983089 983090] EURO983093 [983091ndash983093] BLOOM [983094ndash983089983088] CAEDYM[983089983089 983089983090] andPamolare [983089983091 983089983092] Te choice o themost appropriate model may depend on the speci1047297c researchobjectives and data availability

Due to the complexity o algal bloom processes thesenumerical models usually have a large number o parame-ters which inevitably brings uncertainty to model resultsModeling practice typically includes model developmentcalibration validation and application while uncertainty analysis is ofen neglected Uncertainty analysis is essentialin the assessment and quanti1047297cation o the reliability o

models Prior to the use o model results inormation aboutmodel accuracy and con1047297dence levels should be provided toguarantee that results are in accordance with measurements[983089983093] and that the model is appropriate or its prospectiveapplication [983089983094] Tere are three major sources o uncertainty in modeling systems parameter estimation input data andmodel structure [983089983095ndash983090983089] Understanding and evaluating these

various sources o uncertainty in eutrophication models are

o importance or algal bloom management and aquaticecosystem restorationSeveral methods or parameter uncertainty analysis are

available or example probability theory method MonteCarlo analysis Bayesian method and generalized likelihooduncertainty estimation (GLUE) method Probability theory method employs probability theory o moments o linearcombinations o random variables to de1047297ne means and vari-ances o random unctions It is straightorward or simplelinear models while it does not apply to nonlinear systems[983090983090] Te Monte Carlo analysis computes output statistics by repeating simulations with randomly sampled input variablescomplying with probability density unctions It is easily



983090 Mathematical Problems in Engineering

implemented and generally applicable but the results gainedrom Monte Carlo analysis are not in an analytical ormand the joint distributions o correlated variables are ofenunknown or difficult to derive [983089983096 983090983088 983090983091] Bayesian methodsquantiy uncertainty by calculating probabilistic predictionsDetermining the prior probability distribution o model

parameters is the key step in Bayesian methods [983090983092] GLUEis a statistical method or simultaneously calibrating theinput parameters and estimating the uncertainty o predictivemodels [983090983093] GLUE is based on the concept o equi1047297nalitywhich means that different sets o input parameter may result in equally good and acceptable model outputs ora chosen model [983090983094] It searches or parameter sets thatwould give reliable simulations or a range o model inputsinstead o searching oran optimum parameterset that wouldgive the best simulation results [983090983095] Furthermore modelperormance in GLUE is mainly dependent on parametersets rather than individual parameters whence interactionbetween parameters is implicitly accounted or

Beven and Freer [983090983096] pointed out that in complexdynamic models that contain a large number o highly intercorrelated parameters many different combinations o parameters can give equivalently accurate predictions Inconsideration o equivalence o parameter sets the GLUEmethod is particularly appropriate or the uncertainty assess-ment o numerical eutrophication models which are anexample o complex dynamic models with highly intercorre-lated parameters [983090983096ndash983091983088]

Te GLUE method has already been adopted or uncer-tainty assessments in a variety o environmental modelingapplications including rainall-runoff models [983090983093 983091983089 983091983090]soil carbon models o orest ecosystems [983091983091] agriculturalnonpoint source (NPS) pollution models [983091983092] groundwater1047298ow models [983091983093] urban stormwater quality models [983091983094 983091983095]crop growth models [983091983096] and wheat canopy models [983091983097] Tepopularity o the GLUE method can be attributed to its sim-plicity and wide applicability especially when dealing withnonlinear and nonmonotonic ecological dynamic models

Te objectives o this paperare to make use o the broadly used eutrophicationand algal bloom model BLOOM [983094 983095]inorder to investigate the applicability o the GLUE method toanalyze and quantiy the uncertainty in numerical eutroph-ication models that have a large number o intercorrelatedparameters and to provide a reerence or method selectionwhen conducting uncertainty analysis or similar types o models

2 Materials and Methodology

983090983089 Study Area Te Meiliang Bay (983091983089∘983090983095N983089983090983088∘983089983088E) whichlocates at the north o aihu Lake in China (Figure 983089)is chosen as the study area aihu Lake has high levelo eutrophication and algal blooms that cause enormousdamage to drinking water saety tourisms and 1047297sh armingrequently break out in summer and autumn

Te Meiliang Bay has a length o 983089983094983094 km rom south tonorth a width o 983089983088 km rom east to west and an averagedepth o 983089983097983093 m Tere are two main rivers the Zhihu Gangthat 1047298ows into aihu Lakeand the Liangxi River that 1047298ows out

N

Sample sitesRivers

Meiliang Bay

Liangxi RiverZhihu Gang 1 2

3

4

0 10 20(km)

China

Taihu

F983145983143983157983154983141 983089 Location o the aihu Lake and the Meiliang Bay (983091983089∘983090983095N983089983090983088∘983089983088E)

o the lake Te exchange o substance between Meiliang Bay and the main body o aihu Lake is taken into account in thestudy Te monthly observed data rom our monitoring sitesin the Meiliang Bay were collected during 983090983088983088983097 to 983090983088983089983089 ormodel calibration Tese data include river discharge waterlevel irradiance temperature concentrations o ammonianitrate nitrite phosphate and biomass concentration o blue-green algae green algae and diatom

In the composition o the algae blooms blue-green algaeis the dominant species andhas the highest percentage o totalbiomass Tereore it is selected to be the output variable o BLOOM on which the uncertainty analysis is perormed

983090983090 BLOOM Model BLOOM is a generic hydroenvironmen-tal numerical model that can be applied to calculate primary production chlorophyll-a concentration and phytoplanktonspecies composition [983094ndash983089983088] Fifeen algae species can be mod-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 limiting

conditions respectively Algae biomass in BLOOM mainly depends on primary production and transport

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

10383891103925 = 907317

210383892

minus V

1038389 + 9073171038389

210383892

minus V 1038389

1038389 + + (1038389 1103925)

(983089)

where 1038389 concentration (kgsdotmminus3) 907317 9073171038389 dispersion coe-

1047297cient in - and -direction respectively (m2sdotsminus1) S sourceterms

(

1038389

1103925) reaction terms

1103925 time (s)



Mathematical Problems in Engineering 983091

Primary production is mainly dependent on the speci1047297crates o growth mortality and maintenance respirationwhich are modulated according to the temperature

1103925 = Proalg01103925 times cPalg

9073171103925

11039251103925 = Moralg

0

1103925 times cMalg

907317

1103925 1103925 = Resalg01103925 times cRalg

9073171103925

(983090)

where 1103925 potential speci1047297c growth rate o the astest

growing type o algae species (dminus1) Proalg01103925 growth rate at

983088∘C (dminus1) cPalg1103925 temperature coefficient or growth (ndash)

11039251103925 speci1047297c mortality rate (dminus1) Moralg01103925 mortality rate at

983088∘C (dminus1) cMalg1103925 temperature coefficient or mortality (ndash

) 1103925 speci1047297c maintenance respiration rate (dminus1) Resalg01103925

maintenance respiration rate at 983088∘C (dminus1) cRalg1103925 tempera-ture coefficient or respiration (ndash) More details o BLOOMcan be ound in Delf Hydraulics [983094 983095]

983090983091 Generalized Likelihood Uncertainty Estimation TeGLUE methodology [983090983093] is based upon a large number o model runs perormed with different sets o input param-eter sampled randomly rom prior speci1047297ed parameterdistributions Te simulation result corresponding to eachparameter set is evaluated by means o its likelihood valuewhich quanti1047297es how well the model output conorms tothe observed values Te higher the likelihood value thebetter the correspondence between the model simulationand observations Simulations with a likelihood value largerthan a user-de1047297ned acceptability threshold will be retainedto determine the uncertainty bounds o the model outputs[983091983091 983092983088] Te major procedures or perorming GLUE includedetermining the ranges and prior distributions o inputparameters generating random parameter sets de1047297ning thegeneralized likelihood unction de1047297ning threshold value orbehavioral parameter sets and calculating the model outputcumulative distribution unction

BLOOM contains hundreds o 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 ocuson a ew key parameters [983090983093 983090983096 983092983089] In eutrophicationmodels algal biomass is most closely related to the growthmortality and respiration processes resulting in the selection

o seven key parameters about blue-green algae accordingto (983090) able 983089 summarizes the main characteristics o theseseven parameters Te initial ranges o the parameters areobtained by model calibration and a uniorm prior distri-bution reported in the literature [983090983096] is considered or allparameters

Latin Hypercube Sampling (LHS) which is a type o strati1047297ed Monte Carlo sampling is employed in this study to generate random parameter sets rom the prior parameterdistributions In total 983094983088983088983088983088 parameter combinations aregenerated or the model runs

Te GLUE method requires the de1047297nition o a likelihoodunction in order to quantiy how well simulation results

conorm to observed data Te likelihood measure shouldincrease monotonically with increasing conormity betweensimulation results and observations [983090983093] Various likelihoodunctions have been proposed and evaluated in the literature[983091983093 983091983095 983091983096 983092983090] Keesman and van Straten [983092983091] de1047297ned thelikelihood unction based on the maximum absolute resid-

ual Beven and Binley [983090983093] de1047297ned the likelihood unctionbased on the inverse error variance with a shape actor Romanowicz et al [983092983092] de1047297ned the likelihood unctionbased on an autocorrelated Gaussian error model Freer et al[983092983093] de1047297ned the likelihood unction based on the Nash-Sutliffe efficiency criterion with shape actor as well asthe exponential transormation o the error variance withshaping actor Wang et al [983092983089] de1047297ned the likelihoodunction based on minimum mean square error In thisstudy the likelihood unction (1103925 | ) o the modelrun corresponding to the th set o input parameters (1103925)and observations is de1047297ned based on the exponentialtransormation o the error variance 2

and the observation

variance 2

0 with shape actor [983091983095 983092983093]

10486161103925 | 1048617 = exp 983080minuslowast 220

983081 (983091)

where 2 = sum(sim minus obs)2 2

= sum(obs minus obs)2 sim isthe simulated blue-green algae biomass obs is the observedblue-green algae biomass obs is the average value o obs

Te sensitivity o the choice o the shape actor willbe analyzed and discussed I the likelihood value o asimulation result is larger than a user-de1047297ned threshold themodel simulation is considered ldquobehavioralrdquo and retained orthe subsequent analysis Otherwise the model simulationis considered ldquononbehavioralrdquo and removed rom urtheranalysis Tere are two main methods or de1047297ning thethreshold value or behavioral parameter sets one is to allow a certain deviation rom the highest likelihood value in thesample and the other is to use a 1047297xed percentage o the totalnumber o simulations [983092983094] Te latter is used in this studyand the acceptable sample rate (ASR) is de1047297ned as 983094983088 Tesensitivity o the choice o the threshold in the orm o theacceptable sample rate (ASR) will be analyzed and discussed

Te likelihood unction is then normalized such that thecumulative likelihood o all model runs equals 983089

104861611039251048617 = 10486161103925 | 1048617

sum1103925

10486161103925

| 1048617 (983092)

where (1103925) is the normalized likelihood or theth set o input parameters (1103925) Te uncertainty analysis is perormedby calculating the cumulative distribution unction (CDF) o the normalized likelihood together with prediction quantiles

Te GLUE-derived 983097983088 con1047297dence intervals or thebiomass o blue green are then obtained by reading 983093 and983097983093 percentiles o the cumulative distribution unctions

3 Results

983091983089 BLOOM Model Results Te calibration result or blue-green algae is shown in Figure 983090 and the calibration




983137983138983148983141 983089 Selected input parameters and their initial ranges

Parameter Category Equations (983090)

De1047297nition Unit Lower

boundUpperbound

Calibrated value

ProBlu 0Proalg0

1103925 Growth rate at 983088∘C or blue-green E-type 983089d 983088983088983089983091 983088983088983089983097 983088983088983089983094

cPBlu E cPalg1103925 emperature coefficient or growth or blue-green E-type mdash 983089983088983092983088 983089983089983088983088 983089983088983096

cPBlu N cPalg1103925 emperature coefficient or growth or blue-green N-type mdash 983089983088983092983088 983089983089983088983088 983089983088983096cPBlu P cPalg1103925

emperature coefficient or growth or blue-green P-type mdash 983089983088983092983088 983089983089983088983088 983089983088983096

MorBlu 0Moralg0

1103925 Mortality rate at 983088∘C or blue-green E-type 983089d 983088983088983090983096 983088983088983092983090 983088983088983091983093

cMBlu E cMalg1103925 emperature coefficient or mortality or blue-green E-type mdash 983089983088983088983088 983089983088983090983088 983089983088983089

cRBlu E cRalg1103925 emperature coefficient or maintenance respiration or blue-green E-type mdash 983089983088983092983088 983089983089983088983088 983089983088983095983090

983137983138983148983141 983090 Statistical characteristics o observed data rom 983090983088983088983097 to983090983088983089983089

Station Mean(gCm983091)

Standard deviation(gCm983091)

Maximum(gCm983091)

Minimum(gCm983091)

983089 983089983088983090983091 983089983093983088983092 983094983095983092983092 983088983088983089983093

983090 983089983089983092983092 983090983088983095983091 983097983096983096983093 983088983088983088983090

983091 983089983089983091983093 983089983091983094983097 983093983088983091983095 983088983088983088983092

983092 983089983089983095983097 983089983093983094983092 983093983093983091983092 983088983088983089983088

parameters are summarized in the last column o able 983089Te statistical characteristics o the observed blue-green algaebiomass are shown in able 983090 Te mean values o the blue-green algae biomass or the our sample sites are similarTereore in order to reduce the sampling uncertaintiesthe average o the our sampling sites has been retained as

dependent variable in the present studyTe biomass o blue-green has a yearly cycle (Figure 983090)with low values during spring ollowed by a rapid increasetowards peak values in summer or autumn Te growthperiodicity o blue-green algae is mainly attributed to theperiodic variation o temperature and algae dormancy aihuLake experiences a subtropical monsoon climate with ourdistinct seasons Te lowest temperature is about 983090983096∘C inaverage and appears in January and the highest temperatureis about 983090983097983092∘C in average and usually appears in AugustTe suitable temperaturerange or growth o blue-green algaeis 983090983093sim983091983093∘C As a result the biomass o blue-green algae islow in spring When temperature increases in summer itis appropriate or blue-green algae breeding leading to thesharp increase in biomass and the occurrence o the peak

value around AugustTe modes capture satisactorily the observed evolution

o the blue-green algae biomass which indicates urther anal-yses on model uncertainty are meaningul Te coefficient o determination (CoD) which is given by (983093) is 983088983096983093

CoD = sum1103925 10486161103925 minus 10486172sum1103925 10486161103925 minus 10486172 (983093)

where 1103925 the simulated biomass o blue-green algae at timestep the mean value o observed data 1103925 the observed

value o blue-green at time step

0

1

2

3

4

5

6

Simulation

Observation

2011-12010-12009-1

B i o m a s s o

f b l u e g r

e e n

( g C m

3

)

F983145983143983157983154983141 983090 Modeled results and observations o blue-green algae

983091983090 Uncertainty Analysis Results Te con1047297dence interval(CI) is obtained by calculating the cumulative distributionunctions o model outputs based on the normalized likeli-hood (983092) with = 1 and ASR = 983094983088 Figure 983091 presents the983097983088 con1047297dence interval o blue-green algae biomass whichis estimated rom the 983093 and 983097983093 quantiles o the cumulativedistribution unctions and the corresponding observationsrom January 983090983088983088983097to December 983090983088983089983089 able 983091 summarizes thewidth o the 983097983088 CI o each month and whether or not theobservations are located within the 983097983088 CI

Te 983097983088 CI is narrow rom January to May when the

biomass o blue-green algae is low Te width o the 983097983088 CIexpands as the biomass o blue-green algae increases duringsummer and autumn Among the total o 983091983094 observations 983089983091are located within the 983097983088 CI o the simulations

Te subjective choice o the shape actor in (983089) consid-erably in1047298uences the GLUE results whereas is commonly taken as 983089 [983091983093] Figure 983092 displays the 983097983088 CI when ASR =983094983088 with shape actors equal 983093983088 and 983089983088983088 respectively Tesimulated 983093 and 983097983093 con1047297dence quantiles and the weightedmean as well as the corresponding observations o blue-green algae biomass are shown

Comparison o Figures 983091 and 983092 shows that the increase o shape actor

leads to a narrowing o the 983097983088 CI Figure 983093




983137983138983148983141 983091 Width o 983097983088 con1047297dence interval (CI) or each month and indication whether (Y) or not (N) the observations is located within the983097983088 CI band

Year Month 983089 983090 983091 983092 983093 983094 983095 983096 983097 983089983088 983089983089 983089983090

983090983088983088983097 983097983088 CI (gCm983091) 983088983088983088983092 983088983088983089983089 983088983088983089983088 983088983088983096983088 983088983091983092983097 983089983093983091983091 983089983094983092983094 983088983094983097983092 983088983097983096983097 983089983088983093983089 983089983093983091983096 983088983091983094983091

within CI N Y N N Y N Y N Y N N N

983090983088983089983088 983097983088 CI (gCm983091) 983088983088983097983091 983088983088983090983097 983088983088983088983097 983088983088983089983091 983088983088983091983096 983088983089983092983095 983089983088983095983096 983089983091983094983090 983089983093983094983095 983088983095983094983096 983089983089983097983097 983089983090983095983090

within CI N N N N N Y Y Y Y N Y Y

983090983088983089983089 983097983088 CI (gCm983091) 983088983093983088983092 983088983089983089983092 983088983088983091983097 983088983088983093983092 983088983089983096983090 983088983097983097983091 983088983096983094983093 983089983089983091983094 983088983096983095983090 983089983089983097983094 983089983093983088983088 983088983094983092983096

within CI Y N N N Y N N N N N N Y

2009-1 2010-1 2011-1

0

05

1

15

2

25

3

35

4

45

5

55

Observation

5

95

Weighted mean

minus05

= 1

B i o m a s s o f

b l u e g r e e n

( g C m

3

)

F983145983143983157983154983141 983091 983093 and 983097983093 con1047297dence quantiles and weigthed mean o simulated biomass o blue-green algae when ASR = 983094983088 and =1 and corresponding observations rom January 983090983088983088983097 to December

983090983088983089983089

illustrates the effect o the shape actor which can beseen as a weight actor or the likelihood correspondingto each simulation When = 1 the magnitudes o the likelihood are similar or each simulation and thereis no clear division between acceptable and unacceptablesimulations As a result the cumulative distribution unctionsincrease gradually With increasing (eg = 50) the highbehavioral simulations have a higher weight resulting in alarger gradient in the cumulative distribution unction and anarrower CI Teoretically when

= 0 every simulation has

equal likelihood and the widest CI will be obtained When rarr infin the single best simulation will have a normalizedlikelihood o 983089 while all other simulations willget a likelihoodo zero resulting in the collapse o the 983093 and 983097983093 quantileson a singleline Tiscorrespondsto thetraditionalcalibrationmethod that omits uncertainty analysis

Previous studies have shown that the choice o threshold values or the likelihood measures is particularly importantor the GLUE method [983091983092 983091983094 983092983095] In order to quantiy the effect o threshold values on the uncertainty analysesa series o acceptable sample rates (ASR) o 983088983093 983089 983093983089983088 983091983088 983094983088 983097983088 983097983093 983097983097 is investigated In this studyaverage relative interval length (ARIL) and percentage o

observations covered by the 983097983088 con1047297dence interval (90CI)are adopted as metrics or the analysis Tese metrics arede1047297ned as ollows

ARIL = 1 991761 Limitupper minus Limitlower

obs

(983094)

where Limitupper and Limitlower are the upper and lower

boundary values o the 983097983088 con1047297dence interval is thenumber o time steps obs is the observed biomass o blue-green algae

90CI = inobs

times100 (983095)

where in is the number o observations located within983097983088 CI obs is the total number o observations

Figures 983094 and 983095 present the in1047298uence o ASR on ARIL and90CI or = 1 983093983088 983089983088983088 Figure 983094 shows that or all ASR values ARIL has the highest value or = 1 and decreaseswith increasing which con1047297rms the results o Figure 983092 Fora given

value ARIL increases with ASR When ASR moves

rom 983088983093 to 983097983097 the ARIL increases by 983095983091983097983091 983092983089983097983094and 983093983090983092 or = 1 983093983088 983089983088983088 respectively An increasingASR which corresponds to a lower threshold o the acceptedlikelihood means that simulations with lower likelihood areconsidered ldquobehavioralrdquo which inevitably results in a largerARIL

From Figure 983095 it is seen that 90CI becomes larger asASR increases or = 1 and = 50 while 90CI keepsconstant or = 100 Tis is because the increase o ASR results in a larger ARIL which logically leads to an increasein observations located within the 983097983088 CI When = 100the ARILis low and 90CI does not increase with ASR becausethe 983097983088 CI does not widen

Te highest 90CI is obtained or ASR close to 983089983088983088 and = 1 Its value o about 983093983088 indicates that about hal o the observed data remain outside the 983097983088 CI or the greatestASR Tis can be attributed to other sources o uncertaintysuch as the input parameters or the observations

4 Discussion and Conclusion

Te 983097983088 con1047297dence interval o the simulated results ails toenclose the peaks o the observed values in 983090983088983088983097 and 983090983088983089983089(Figure 983091) Such a eature is not unusual and several reasonscan lead to this result Firstly there are inherent uncertaintiesrom inputs boundaries and model structure which are not




2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)

F983145983143983157983154983141 983092 983093 and 983097983093 con1047297dence quantiles and weigthed mean o simulated biomass o blue-green algae when ASR = 983094983088 and = 50 (a)and 983089983088983088 (b) and corresponding observations rom January 983090983088983088983097 to December 983090983088983089983089

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)

F983145983143983157983154983141 983093 Dot plots o likelihood according to (983092) when ASR = 983089983088983088 and = 1 (a) 983093983088 (b) and 983089983088983088 (c) or cPBlu E and cRBlu E (cable 983089)








(GLUE) methodrdquo Water Resources Research vol 983092983092 no 983089983090Article ID W983088983088B983088983094 983090983088983088983096

[983092983091] K Keesman and G van Straten ldquoIdenti1047297cation and predictionpropagation o uncertainty in models with bounded noiserdquoInternational Journal o Control vol 983092983097 no 983094 pp 983090983090983093983097ndash983090983090983094983097983089983097983096983097

[983092983092] R Romanowicz K J Beven and J awn ldquoEvaluation o predictive uncertainty in non-linear hydrological models usinga Bayesian approachrdquo in Statistics or the Environment Water Related Issues V Barnett and K F urkman Eds pp 983090983097983095ndash983091983089983095John Wiley amp Sons New York NY USA 983089983097983097983092

[983092983093] J Freer K Beven and B Ambroise ldquoBayesian estimation o uncertainty in runoff prediction and the value o data anapplication o the GLUE approachrdquo Water Resources Research vol 983091983090 no 983095 pp 983090983089983094983089ndash983090983089983095983091 983089983097983097983094

[983092983094] R S Blasone J A Vrugt H Madsen D Rosbjerg B A Robin-son and G A Zyvoloski ldquoGeneralized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlosamplingrdquo Advances in Water Resources vol 983091983089 no 983092 pp 983094983091983088ndash983094983092983096 983090983088983088983096

[983092983095] L Li J Xia C Y Xu and V P Singh ldquoEvaluation o thesubjective actors o the GLUE method and comparison withthe ormal Bayesian method in uncertainty assessment o hydrological modelsrdquo Journal o Hydrology vol 983091983097983088 no 983091-983092 pp983090983089983088ndash983090983090983089 983090983088983089983088



Submit your manuscripts at

httpwwwhindawicom




implemented and generally applicable but the results gainedrom Monte Carlo analysis are not in an analytical ormand the joint distributions o correlated variables are ofenunknown or difficult to derive [983089983096 983090983088 983090983091] Bayesian methodsquantiy uncertainty by calculating probabilistic predictionsDetermining the prior probability distribution o model

parameters is the key step in Bayesian methods [983090983092] GLUEis a statistical method or simultaneously calibrating theinput parameters and estimating the uncertainty o predictivemodels [983090983093] GLUE is based on the concept o equi1047297nalitywhich means that different sets o input parameter may result in equally good and acceptable model outputs ora chosen model [983090983094] It searches or parameter sets thatwould give reliable simulations or a range o model inputsinstead o searching oran optimum parameterset that wouldgive the best simulation results [983090983095] Furthermore modelperormance in GLUE is mainly dependent on parametersets rather than individual parameters whence interactionbetween parameters is implicitly accounted or

Beven and Freer [983090983096] pointed out that in complexdynamic models that contain a large number o highly intercorrelated parameters many different combinations o parameters can give equivalently accurate predictions Inconsideration o equivalence o parameter sets the GLUEmethod is particularly appropriate or the uncertainty assess-ment o numerical eutrophication models which are anexample o complex dynamic models with highly intercorre-lated parameters [983090983096ndash983091983088]

Te GLUE method has already been adopted or uncer-tainty assessments in a variety o environmental modelingapplications including rainall-runoff models [983090983093 983091983089 983091983090]soil carbon models o orest ecosystems [983091983091] agriculturalnonpoint source (NPS) pollution models [983091983092] groundwater1047298ow models [983091983093] urban stormwater quality models [983091983094 983091983095]crop growth models [983091983096] and wheat canopy models [983091983097] Tepopularity o the GLUE method can be attributed to its sim-plicity and wide applicability especially when dealing withnonlinear and nonmonotonic ecological dynamic models

Te objectives o this paperare to make use o the broadly used eutrophicationand algal bloom model BLOOM [983094 983095]inorder to investigate the applicability o the GLUE method toanalyze and quantiy the uncertainty in numerical eutroph-ication models that have a large number o intercorrelatedparameters and to provide a reerence or method selectionwhen conducting uncertainty analysis or similar types o models

2 Materials and Methodology

983090983089 Study Area Te Meiliang Bay (983091983089∘983090983095N983089983090983088∘983089983088E) whichlocates at the north o aihu Lake in China (Figure 983089)is chosen as the study area aihu Lake has high levelo eutrophication and algal blooms that cause enormousdamage to drinking water saety tourisms and 1047297sh armingrequently break out in summer and autumn

Te Meiliang Bay has a length o 983089983094983094 km rom south tonorth a width o 983089983088 km rom east to west and an averagedepth o 983089983097983093 m Tere are two main rivers the Zhihu Gangthat 1047298ows into aihu Lakeand the Liangxi River that 1047298ows out

N

Sample sitesRivers

Meiliang Bay

Liangxi RiverZhihu Gang 1 2

3

4

0 10 20(km)

China

Taihu

F983145983143983157983154983141 983089 Location o the aihu Lake and the Meiliang Bay (983091983089∘983090983095N983089983090983088∘983089983088E)

o the lake Te exchange o substance between Meiliang Bay and the main body o aihu Lake is taken into account in thestudy Te monthly observed data rom our monitoring sitesin the Meiliang Bay were collected during 983090983088983088983097 to 983090983088983089983089 ormodel calibration Tese data include river discharge waterlevel irradiance temperature concentrations o ammonianitrate nitrite phosphate and biomass concentration o blue-green algae green algae and diatom

In the composition o the algae blooms blue-green algaeis the dominant species andhas the highest percentage o totalbiomass Tereore it is selected to be the output variable o BLOOM on which the uncertainty analysis is perormed

983090983090 BLOOM Model BLOOM is a generic hydroenvironmen-tal numerical model that can be applied to calculate primary production chlorophyll-a concentration and phytoplanktonspecies composition [983094ndash983089983088] Fifeen algae species can be mod-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 limiting

conditions respectively Algae biomass in BLOOM mainly depends on primary production and transport

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

10383891103925 = 907317

210383892

minus V

1038389 + 9073171038389

210383892

minus V 1038389

1038389 + + (1038389 1103925)

(983089)

where 1038389 concentration (kgsdotmminus3) 907317 9073171038389 dispersion coe-

1047297cient in - and -direction respectively (m2sdotsminus1) S sourceterms

(

1038389

1103925) reaction terms

1103925 time (s)






9073171103925

11039251103925 = Moralg

0

1103925 times cMalg

907317


9073171103925

(983090)















and the observation

variance 2

0 with shape actor [983091983095 983092983093]

10486161103925 | 1048617 = exp 983080minuslowast 220

983081 (983091)





104861611039251048617 = 10486161103925 | 1048617

sum1103925

10486161103925

| 1048617 (983092)



3 Results








boundUpperbound

Calibrated value

ProBlu 0Proalg0





MorBlu 0Moralg0







Maximum(gCm983091)

Minimum(gCm983091)

983089 983089983088983090983091 983089983093983088983092 983094983095983092983092 983088983088983089983093

983090 983089983089983092983092 983090983088983095983091 983097983096983096983093 983088983088983088983090

983091 983089983089983091983093 983089983091983094983097 983093983088983091983095 983088983088983088983092

983092 983089983089983095983097 983089983093983094983092 983093983093983091983092 983088983088983089983088








0

1

2

3

4

5

6

Simulation

Observation

2011-12010-12009-1

B i o m a s s o

f b l u e g r

e e n

( g C m

3

)












Year Month 983089 983090 983091 983092 983093 983094 983095 983096 983097 983089983088 983089983089 983089983090

983090983088983088983097 983097983088 CI (gCm983091) 983088983088983088983092 983088983088983089983089 983088983088983089983088 983088983088983096983088 983088983091983092983097 983089983093983091983091 983089983094983092983094 983088983094983097983092 983088983097983096983097 983089983088983093983089 983089983093983091983096 983088983091983094983091


983090983088983089983088 983097983088 CI (gCm983091) 983088983088983097983091 983088983088983090983097 983088983088983088983097 983088983088983089983091 983088983088983091983096 983088983089983092983095 983089983088983095983096 983089983091983094983090 983089983093983094983095 983088983095983094983096 983089983089983097983097 983089983090983095983090


983090983088983089983089 983097983088 CI (gCm983091) 983088983093983088983092 983088983089983089983092 983088983088983091983097 983088983088983093983092 983088983089983096983090 983088983097983097983091 983088983096983094983093 983089983089983091983094 983088983096983095983090 983089983089983097983094 983089983093983088983088 983088983094983092983096


2009-1 2010-1 2011-1

0

05

1

15

2

25

3

35

4

45

5

55

Observation

5

95

Weighted mean

minus05

= 1

B i o m a s s o f

b l u e g r e e n

( g C m

3

)


983090983088983089983089







obs

(983094)



90CI = inobs

times100 (983095)












2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)


08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)


















httpwwwhindawicom






9073171103925

11039251103925 = Moralg

0

1103925 times cMalg

907317


9073171103925

(983090)















and the observation

variance 2

0 with shape actor [983091983095 983092983093]

10486161103925 | 1048617 = exp 983080minuslowast 220

983081 (983091)





104861611039251048617 = 10486161103925 | 1048617

sum1103925

10486161103925

| 1048617 (983092)



3 Results








boundUpperbound

Calibrated value

ProBlu 0Proalg0





MorBlu 0Moralg0







Maximum(gCm983091)

Minimum(gCm983091)

983089 983089983088983090983091 983089983093983088983092 983094983095983092983092 983088983088983089983093

983090 983089983089983092983092 983090983088983095983091 983097983096983096983093 983088983088983088983090

983091 983089983089983091983093 983089983091983094983097 983093983088983091983095 983088983088983088983092

983092 983089983089983095983097 983089983093983094983092 983093983093983091983092 983088983088983089983088








0

1

2

3

4

5

6

Simulation

Observation

2011-12010-12009-1

B i o m a s s o

f b l u e g r

e e n

( g C m

3

)












Year Month 983089 983090 983091 983092 983093 983094 983095 983096 983097 983089983088 983089983089 983089983090

983090983088983088983097 983097983088 CI (gCm983091) 983088983088983088983092 983088983088983089983089 983088983088983089983088 983088983088983096983088 983088983091983092983097 983089983093983091983091 983089983094983092983094 983088983094983097983092 983088983097983096983097 983089983088983093983089 983089983093983091983096 983088983091983094983091


983090983088983089983088 983097983088 CI (gCm983091) 983088983088983097983091 983088983088983090983097 983088983088983088983097 983088983088983089983091 983088983088983091983096 983088983089983092983095 983089983088983095983096 983089983091983094983090 983089983093983094983095 983088983095983094983096 983089983089983097983097 983089983090983095983090


983090983088983089983089 983097983088 CI (gCm983091) 983088983093983088983092 983088983089983089983092 983088983088983091983097 983088983088983093983092 983088983089983096983090 983088983097983097983091 983088983096983094983093 983089983089983091983094 983088983096983095983090 983089983089983097983094 983089983093983088983088 983088983094983092983096


2009-1 2010-1 2011-1

0

05

1

15

2

25

3

35

4

45

5

55

Observation

5

95

Weighted mean

minus05

= 1

B i o m a s s o f

b l u e g r e e n

( g C m

3

)


983090983088983089983089







obs

(983094)



90CI = inobs

times100 (983095)












2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)


08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)


















httpwwwhindawicom







boundUpperbound

Calibrated value

ProBlu 0Proalg0





MorBlu 0Moralg0







Maximum(gCm983091)

Minimum(gCm983091)

983089 983089983088983090983091 983089983093983088983092 983094983095983092983092 983088983088983089983093

983090 983089983089983092983092 983090983088983095983091 983097983096983096983093 983088983088983088983090

983091 983089983089983091983093 983089983091983094983097 983093983088983091983095 983088983088983088983092

983092 983089983089983095983097 983089983093983094983092 983093983093983091983092 983088983088983089983088








0

1

2

3

4

5

6

Simulation

Observation

2011-12010-12009-1

B i o m a s s o

f b l u e g r

e e n

( g C m

3

)












Year Month 983089 983090 983091 983092 983093 983094 983095 983096 983097 983089983088 983089983089 983089983090

983090983088983088983097 983097983088 CI (gCm983091) 983088983088983088983092 983088983088983089983089 983088983088983089983088 983088983088983096983088 983088983091983092983097 983089983093983091983091 983089983094983092983094 983088983094983097983092 983088983097983096983097 983089983088983093983089 983089983093983091983096 983088983091983094983091


983090983088983089983088 983097983088 CI (gCm983091) 983088983088983097983091 983088983088983090983097 983088983088983088983097 983088983088983089983091 983088983088983091983096 983088983089983092983095 983089983088983095983096 983089983091983094983090 983089983093983094983095 983088983095983094983096 983089983089983097983097 983089983090983095983090


983090983088983089983089 983097983088 CI (gCm983091) 983088983093983088983092 983088983089983089983092 983088983088983091983097 983088983088983093983092 983088983089983096983090 983088983097983097983091 983088983096983094983093 983089983089983091983094 983088983096983095983090 983089983089983097983094 983089983093983088983088 983088983094983092983096


2009-1 2010-1 2011-1

0

05

1

15

2

25

3

35

4

45

5

55

Observation

5

95

Weighted mean

minus05

= 1

B i o m a s s o f

b l u e g r e e n

( g C m

3

)


983090983088983089983089







obs

(983094)



90CI = inobs

times100 (983095)












2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)


08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)


















httpwwwhindawicom





Year Month 983089 983090 983091 983092 983093 983094 983095 983096 983097 983089983088 983089983089 983089983090

983090983088983088983097 983097983088 CI (gCm983091) 983088983088983088983092 983088983088983089983089 983088983088983089983088 983088983088983096983088 983088983091983092983097 983089983093983091983091 983089983094983092983094 983088983094983097983092 983088983097983096983097 983089983088983093983089 983089983093983091983096 983088983091983094983091


983090983088983089983088 983097983088 CI (gCm983091) 983088983088983097983091 983088983088983090983097 983088983088983088983097 983088983088983089983091 983088983088983091983096 983088983089983092983095 983089983088983095983096 983089983091983094983090 983089983093983094983095 983088983095983094983096 983089983089983097983097 983089983090983095983090


983090983088983089983089 983097983088 CI (gCm983091) 983088983093983088983092 983088983089983089983092 983088983088983091983097 983088983088983093983092 983088983089983096983090 983088983097983097983091 983088983096983094983093 983089983089983091983094 983088983096983095983090 983089983089983097983094 983089983093983088983088 983088983094983092983096


2009-1 2010-1 2011-1

0

05

1

15

2

25

3

35

4

45

5

55

Observation

5

95

Weighted mean

minus05

= 1

B i o m a s s o f

b l u e g r e e n

( g C m

3

)


983090983088983089983089







obs

(983094)



90CI = inobs

times100 (983095)












2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)


08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)


















httpwwwhindawicom




2009-1 2010-1 2011-1

0

05

1

15

2

253

35

4

45

5

55

minus05

= 50

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(a)

2009-1 2010-1 2011-1

= 100

0

05

1

15

2

253

35

4

45

5

55

minus05

Observation

595Weighted mean

B i o m a s s o

f b l u e g r e e n

( g C m

3

)

(b)


08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

08

06

04

02

0104 106 108 11

L i k e

l i h o o

d

= 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

L i k e l i h o o

d

L

i k e

l i h o o

d

= 50

TcPBlu E

TcRBlu E

(b)

= 100

8

6

4

2

0

times10minus17

104 106 108 11

L

i k e

l i h o o

d

104 106 108 11

8

6

4

2

0

times10minus17

L i k e l i h o o

d

TcPBlu E

TcRBlu E

(c)


















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the ash prediction problem

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