identification of dynamic models for horizontal subsurface constructed wetlands

Ecological Modelling 187 (2005) 201–218

Identification of dynamic models for horizontalsubsurface constructed wetlands

Stefano Marsili-Libelli∗, Nicola Checchi

Department of Systems and Computers, University of Florence, Via S. Marta 3, Firenze 50139, Italy

Received 5 February 2004; received in revised form 22 December 2004; accepted 3 January 2005Available online 26 February 2005

Abstract

The current trend in horizontal subsurface constructed wetlands (HSSCW) modelling advocates structures of increasingcomplexity, which however have produced a limited improvement in the understanding of their internal functioning or in thereliable estimation of their parameters. Following a different approach, this paper proposes the combination of a set of simplestructures and a robust identification method to approximate the dispersed flow and pollution reduction dynamics. The modelsare based on combinations of series/parallel CSTRs of unequal volumes in series with a plug-flow reactor. After introducing themodel structures, their identifiability is assessed with a method based on approximate parameter confidence regions based oneither the Fisher information matrix (FIM) or the Hessian matrix. Their relative agreement or disagreement yields informationabout model structural robustness and parameter identifiability. To demonstrate the method, the proposed models are calibratedwith data sets from several constructed wetlands, with widely differing hydraulics and pollution removal characteristics. Thei t and yets n and thep rder andM©

K

1

i

f

e oromtryuslvedthateticsber

0

dentification method can assist in the selection of the best combination of hydraulics and kinetics to obtain robusimple models for HSSCW. In particular, it is shown that the estimated volumes can be used for the wetland desigollution reduction kinetics are reassessed in terms of identifiability, providing a guideline to decide between first oonod kinetics.2005 Elsevier B.V. All rights reserved.

eywords: Constructed wetlands; Dispersed flow; Wastewater treatment; Parameter estimation; Mathematical modelling

. Introduction

Constructed wetlands represent an important low-mpact alternative to conventional wastewater treat-

∗ Corresponding author. Tel.: +39 055 47 96 264;ax: +39 055 47 96 264.

E-mail address: [email protected] (S. Marsili-Libelli).

ment processes for low strength domestic sewagfor tertiary treatment. Their design has evolved frearly empirical rules to advanced models, whichto explain the complexity of hydraulics in a poromedium combined with the many processes invoin pollution reduction. Still, recent studies suggestthe interdependence between hydraulics and kinis so strong and influenced by such a large num

304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2005.01.043

202 S. Marsili-Libelli, N. Checchi / Ecological Modelling 187 (2005) 201–218

Nomenclature

CDO dissolved oxygen concentration(mg DO L−1)

Cout model output concentration, either traceror pollutant (mg L−1)

Ci (0) initial tracer concentration in the firstmodel CSTR (mg L−1)

CW constructed wetlandsE(p) objective error function used in the esti-

mationF volumetric flow (m3 h−1)FSCW free surface constructed wetlandsH Hessian matrix of the error functional

E(p)HSSCW horizontal sub surface constructed wet-

landsJ Fisher information matrixkox organic carbon oxidation rate constant

(T−1)k1 first-order reaction rate (h−1)KDO Monod dissolved oxygen half-saturation

constant (mg DO L−1)KpH Monod pH half-saturation constantKs Monod organic carbon half-saturation

constant (mg COD L−1)Mo tracer mass injected into the system (g)np number of model parametersNexp number of experimental datapHo optimum pH for heterotrophic growthp vector of model parametersp estimated vector of model parametersrCorg total rate of organic carbon removal

(mg COD (L T)−1)rs settling rate of decaying organic carbon

(T−1)R Monod maximum reaction rate

(mg (L h)−1)RTD residence time distribution( )T transpose of a vector or matrixV volume (m3)VAF variance accounted for (%)XH heterotrophic biomass (mg COD L−1)

Greek lettersδ transport dead-time (h)δpj relative 95% confidence bound esti-

mated with the Jacobian approximationδpH relative 95% confidence bound esti-

mated with the Hessian approximationθ arrhenius temperature coefficientΓ equivalent parameter covariance matrix

for nonlinear systemsσ2

exp estimation error variance

σ2exp experimental data variance

of factors (Kadlec, 2000) that even the last genera-tion of models (Mashauri and Kayombo, 2000; Wynnand Liehr, 2001; Langergraber, 2003; Kincanon andMcAnally, 2004) is inadequate in fully explaining theobserved behaviours and providing a reliable estima-tion of their parameters. A different approach is pur-sued in this paper, which proposes a model with a verysimple structure to describe the hydraulics and carbonremoval of horizontal subsurface flow constructed wet-lands (HSSCW). Hoping that a simple model may besuccessful where complex ones have failed might ap-pear a futile and conceited claim. However, it will beshown that this approach makes sense if modelling isassisted by a robust identification technique to assess itsstructure and produce reliable parameter estimates. Themodel structure proposed here differs from the classicalplug-flow (PF) or tank-in-series approximations, beingderived from a previous laboratory study on diffusivereactors (Marsili-Libelli, 1997). The structural assess-ment is provided by a robust parameter calibration al-gorithm (Marsili-Libelli, 1992) recently enhanced tocompute the approximate confidence regions of the es-timated parameters (Marsili-Libelli et al., 2003). Thelatter theory is used as a structural identification toolby looking at the degree of coincidence of approximateconfidence regions obtained by two differing methods.The combination of modelling and identification toolsyields a simple but structurally robust model with reli-able parameters, which can be used for HSSCW mod-elling and design.

sig-n tedw the

The paper is organized as follows: first the mostificant papers marking the evolution of construcetlands modelling are reviewed in order to show

S. Marsili-Libelli, N. Checchi / Ecological Modelling 187 (2005) 201–218 203

current trend of increasing complexity. The proposedmodel structure is then introduced together with themain results of the confidence regions theory. In thelast section, these tools are applied to the identificationof several free-surface or horizontal subsurface con-structed wetlands.

1.1. A brief review of constructed wetlandsmodelling

Constructed wetlands represent an importantlow-impact complement to conventional wastewatertreatment processes if the amount of sewage is rel-atively small and energy requirements are moderate.Their popularity has been steadily increasing pro-vided the opportunities and limitations of this pro-cess are correctly understood (Kadlec and Knight,1996; Verhoeven and Meuleman, 1999). From themodelling viewpoint, constructed wetlands are defi-nitely more complex than conventional treatment pro-cesses because of the diffusive flow and the largenumber of processes involved in pollution reduction.For these reasons, many authors have pointed outthat the removal efficiency of constructed wetlandsis not easily predictable, being highly influenced bythe hydraulics or environmental conditions (Kadlec,2000; Wynn and Liehr, 2001). Early empirical designequations were proposed byUSEPA (1993), Cooperet al. (1996), IWA (1999), but soon after the lim-its of the first-order or plug-flow assumptions wered -eK ber,2 ro-c rveyo tedb a-t delc easei th-o

anta tialpa d ont ow( n( y pa-

rameter affecting efficiency since the work ofWernerand Kadlec (1996), who used standard RTD theory tocharacterise non-ideal flows in porous media. Later,Werner and Kadlec (2000)proposed an infinite numberof ideal CSTRs to model the wetland as a through-flowchannel flanked by side regions of limited flow. Theimportance of flow patterns was also considered byChazarenc et al. (2003), who analysed the RTD by atracer impulse method and proposed a hydraulic modelbased on ten cascaded CSTRs. Recently,Kincanon andMcAnally (2004)have reviewed the factors affectingthe removal efficiency by estimating an equivalentdetention time through site-specific laboratory analysisof soil structure and concluded that the most significantfeatures are those which determine the duration ofthe water–biota interactions. Working along similarguidelines,Persson and Wittgren (2003)point out theimportance of considering the actual mean residencetime in place of the nominal residence time and definethe effective volume relevant for removal. However,in spite of all these efforts, the fundamental problemposed byKadlec (2000)of the influence of hydraulicloading on all the primary wetland parameters and theneed for models with invariant parameters still remainswithout a definitive answer. This paper would like togive a contribution in this direction by proposing acombination of simple models and a new estimationtechnique to determine their parameters in a reliableway. First the proposed structures are analysed withrespect to hydraulics, then pollution removal termsa andr atedp t them

2

tillp lo atafi TRsib thet tived tingt qualv

emonstrated (Kadlec, 2000) and it is now acknowldged that even complex models (Mashauri andayombo, 2000; Wynn and Liehr, 2001; Langergra003) fail to describe the diffusion and removal pesses in a satisfactory manner. A definitive sun the modelling of HSSCW was recently preseny Rousseau et al. (2004)where many design equ

ions are critically assessed, confirming that moomplexity has not produced a corresponding incrn the reliability and accuracy of the design meds.

Flow characterisation in CW is extremely importnd hydraulic modelling has evolved from the inilug-flow and tank-in-series assumptions (Kadlecnd Knight, 1996) to more elaborate schemes base

he approximation of two-dimensional dispersed flLangergraber, 2003). Residence time distributioRTD) has been consistently considered as the ke

re added. Each combination of hydraulicseaction is structurally assessed using the estimarameters confidence regions, in order to selecost appropriate model/parameter combination.

. Model structure

The tank-in-series approximation, though sopular (Rousseau et al., 2004), is a very rigid modef diffusive flow and may produce unsatisfactory dtting even when the number of cascaded CSs large (Levenspiel, 1972; Marsili-Libelli, 1997)ecause a single structure must explain both

ransport dead-time and the diffusion. The alternaescribed here limits model complexity by separa

hese two aspects and relaxing the constraint of eolume typical of cascaded CSTRs models.


Fig. 1. Basic structure of the subsurface constructed wetland hy-draulic model (model A).

2.1. Dead-time modelling

In an ideal plug-flow, the dead-timeδ is equal tothe reactor travel time, hence the equivalent PF volumecan be computed asVPF= F × δ. In diffusive reactors,dead-time can still be defined as the time delay betweenpulse injection and the rising edge of the output concen-tration. In this sense, dead-time should not be confusedwith hydraulic residence time as it is normally defined(Levenspiel, 1972; Chazarenc et al., 2003; Persson andWittgren, 2003).

2.2. Diffuse flow modelling

Having relieved the cascaded CSTRs from mod-elling the transport dead-time, a CSTR approximationof diffusion can be obtained with very few elementsarranged in series/parallel combinations, relaxing therigid constraint of the tank-in-series model. Practicalexperience has shown that a single model structurecannot fit all situations, therefore two models are con-sidered here and the most appropriate one is selecteddepending on a structural assessment and data fitting.The starting model, originating from a previous paper(Marsili-Libelli, 1997), is shown inFig. 1. This struc-ture is composed of three cascaded CSTR, two of equalvolumesV1 and one of volumeV2 �= V1, in series with aPF element modelling the dead-timeδ. Assuming con-stant volume operation, the hydraulics of the systemcan be described by the equations inTable 1, which

TS

C

F

S

T

P

Fig. 2. An extension of the previous model A, splitting the flowbetween two parallel branches (model B).

are hereafter referred to as model A. Though attractivefor its simplicity, model A lacks generality, but can beextended to include series/parallel branches to accountfor parallel flow patterns. Several structures were testedand the most efficient was found to be the one shownin Fig. 2, with two parallel CSTR branches precededby a CSTR series and followed by a PF. In addition tousing three differing volumes, the partition factorb hasbeen introduced to split the flow between the two par-allel branches. This further model structure, hereafterreferred to as model B, is described by the equations inTable 2.

2.3. Hydraulic modelling based on tracer data

The residence time distribution is the system re-sponse to the pulse injection of a concentrated inerttracer. To reproduce this condition in the model, it isassumed that all the CSTRs have zero initial concen-tration except the first for whichC1(0) = Mo

V1, where

Mo is the tracer mass. The problem withMo is that itmay not be exactly known or that not all the injectedmaterial is recovered at the output. In either case a safechoice is to estimateMo along with the other parame-ters and ifMo is a priori known, its estimation may beused as an accuracy check.

2.4. Modelling the carbon removal dynamics

selyl r(i od-e o(s botha an-

able 1tructure of model A

omponent Model equations

irst CSTR, volumeV1dC1

dt= F

V1Cin − F

V1C1

econd CSTR, volumeV1dC2

dt= F

V1C1 − F

V1C2

hird CSTR, volumeV2dC3

dt= F

V2C2 − F

V2C3

lug-flow dead-time Cout(t) = C3(t − δ)

Hydraulics and removal processes are cloinked, as shown byKadlec (2000), Wynn and Lieh2001)andRousseau et al. (2004). Therefore, if all thenteractions are taken into account very complex mls may result, as shown byMashauri and Kayomb2000)andLangergraber (2003). Vymazal (1999)hashown that the organic compounds are removederobically and anaerobically, but it is difficult to qu


Table 2Structure of model B

Component Model equations

Series branch, first CSTR, volumeV1dC1

dt= F

V1Cin − F

V1C1

Series branch, second CSTR, volumeV1dC2

dt= F

V1C1 − F

V1C2

Series branch, third CSTR, volumeV1dC3

dt= F

V1C2 − F

V1C3

First parallel branch, flow partitionb, volumeV2dC4

dt= bF

V2C3 − bF

V2C4

Second parallel branch, flow partition (1− b), volumeV3dC5

dt= (1 − b)F

V3C3 − (1 − b)F

V3C5

Output dilution C∗ = bC4 + (1 − b)C5

Plug-flow dead-time Cout (t) = C∗(t − δ)

tify the ratio between aerobic and anaerobic degrada-tion. The limited amount of dissolved oxygen avail-able reaches the porous media by diffusion through theplant to the root zones, so an oxygen limitation is oftenintroduced. Based on these considerations, a removalkinetics based on multiplicative Monod terms could beproposed.

rCorg = −XHµmax

Y

Ci

Ks + Ci

CDO

KDO + CDO

× KpH

KpH + yθ(T−20) − XArs

− koxCDO

KDO + CDOθ(T−20)Ci, (1)

where y = 10|pHo−pH| − 1 appears in the pH-modulating term. The three terms account respectivelyfor aerobic heterotrophic degradation, anaerobic degra-dation of settled material and oxidation. Model Eq.(1)is ecologically sound, but it is very difficult to computereliable estimates for all its parameters, though someof them could be obtained from the literature. Con-versely, a simple model can be based more on the datarather than on mechanistic assumptions and, followingthe approach ofPolprasert et al. (1998), a first-orderor a Monod term were added to either model A or B,assuming that no reaction occurs in the PF. In this way,there are four combinations of hydraulic and reactionterms, and the dynamics of each CSTR takes the form:

F

or

Monod kinetics :dCi

dt=Fi

Vi

Ci−1−Fi

Vi

Ci− RCi

Ks + Ci

,

(3)

whereCi is the pollutant concentration in the genericith CSTR,k1 the first-order kinetic rate andR,Ksare theMonod kinetic parameters. The temperature compen-sation was not included because temperature data werenot available. For identifiability reasons an unstruc-tured Monod growth rateR is used in Eq.(3) includingthe maximum growth rateµ, heterotrophic biomassXand yield factorY (Dochain and Vanrolleghem, 2001)because they cannot be estimated separately.R includesalso the dissolved oxygen limitation term appearing inEq.(1). However, its introduction requires more justifi-cation: in the beginning first-order kinetics was widelyemployed in CW design (Cooper et al., 1996; Reedet al., 1998; Vymazal et al., 1998), but Mitchell andMcNevin (2001)have later shown that CW do exhibit aMonod-like behaviour, which might in part answer thevariable kinetics question raised byKadlec (2000). Theinherent complexity of constructed wetland models isconfirmed byRousseau et al. (2004), who concludedthat the system parameters are context-dependent andnot even a complex model as that byWynn and Liehr(2001)can fully account for their variability. Further, ifthe organic loading is consistently much smaller thanKs, the Monod term may be approximated by a first-order kinetick1 ∼= R . So, Eq.(2)can be used for nearlyc -b ral,i ump-t h as

irst-order kinetics :dCi

dt= Fi

Vi

Ci−1 − Fi

Vi

Si − k1Ci,

(2)

Ksonstant loading, whereas Eq.(3) can give more flexiility in case a variable kinetics is required. In gene

t should be reminded that it is against the basic assion of this approach to use complex kinetics, suc


Table 3Constructed wetlands characteristics and tracer data calibrated models

Location Flow(m3 h−1)

Dead-timeδ (h)

Wetvolume(m3)

ModelvolumeVtot

a

(m3)

�Vb (%) Plug-flowvolumeVPF

c (m3)

Model ExperimentalHRT (h)

ModelHRT (h)

VAFd

Carville 70 21.60 2324 3301.0 42 1512.0 A 45.2674 45.4741 99.4971Castelnuovo

Bariano288 6.48 9000 7516.24 −16.5 1866.24 A 25.2865 25.9061 82.4683

CastelnuovoBariano

288 6.48 9000 8018.94 −10.9 1866.24 B 25.2865 24.5199 99.3506

Cerbaia 0.0708 5.98 2.81 2.5721 −8.5 0.4234 B 33.8486 33.9768 99.9256Sieci 0.0958 7.17 8.60 8.6637 0.7 0.6869 B 61.5202 60.6944 99.2277

a Vtot = 2× V1 + V2 + VPF for model A;Vtot = 3× V1 + V2 + V3 VPF for model B.b �V = 100Vtot−wet volume

wet volume .c VPF= δ × F.d VAF = variance accounted for.

Eq.(1), which could not be reliably calibrated given itslarge number of parameters.

2.5. Tracer data for the calibration of thehydraulic response

To calibrate models A and B with tracer data,four non-reactive tracer data sets were used fromwidely differing constructed wetlands, whose charac-teristics are listed inTable 3, together with some globalidentification results, more of which are reported inTable 4. The first data set was drawn from the literature

(USEPA, 1993) and refers to an HSSCW in Carville(Louisiana, USA) planted with arrowhead (Sagittariasp.). The physical volume was 2324 m3 with a depthof 76 cm, a porosity of 35% and an average flow of70 m3 h−1. The data were scanned and digitised di-rectly from the paper, which obviously limits theiraccuracy. Nevertheless, the experiment has been con-sidered representative enough to be used as a modeltest. These data were also used by the IWA special-ist group on macrophytes (p. 48,IWA, 1999) to fit amodel composed of three cascaded CSTRs and a PFreactor.

Table 4Estimated parameters and 95% confidence bounds for the tracer experiments

Location Parametersand confidencebounds (%)

CSTR volumes (m3) Flow split factorb ExperimentaltracerMo (g)

EstimatedtracerMo (g)

V1 V2 V3

Carville(model A)

p 240.4 1308.2 3743.4 3846.0δpJ ±6.385 ±6.417 ±2.743δpH ±6.635 ±6.482 ±2.665

CastelnuovoBariano (model A)

p 965 3720 1161.6 1093.6δpJ ±32.238 ±37.046 ±12.65δpH ±26.290 ±38.384 ±11.193

CastelnuovoBariano(model B)

p 1210.5 281.8 2239.4 0.2013 1161.6 1161.5δpJ ±17.546 ±69.429 ±125.817 ±75.123 ±12.742δpH ±17.646 ±64.886 ±141.381 ±75.012 ±13.485

Cerbaia(model B)

p 0.1226 0.2263 1.5545 0.6268 3.1011 3.1578δpJ ±2.246 ±23.514 ±4.313 ±7.881 ±1.789δpH ±2.271 ±25.049 ±4.313 ±8.891 ±1.820

S(

7±±

iecimodel B)

p 0.2341 0.404δpJ ±4.058 ±32.468δpH ±3.823 ±22.523

6.8698 0.7211 2.7974 3.263715.881 ±6.304 ±5.9539.972 ±4.996 ±4.169


The second data set comes from an experimentalfree-surface wetland in Castelnuovo Bariano, placedon the left embankment of the Po River near Rovigo,northern Italy. The tracer data were supplied by cour-tesy of the late Professor Giuseppe Bendoricchio, Uni-versity of Padua. This facultative pond was constructedas a pollution reduction facility in the framework ofthe Po river water quality management scheme. Asparse reed bed (Phragmites australis) colonisationwas present at the time of the experiment. This shal-low U-shaped area measures approximately 750 m inlength and 25 m in width. The flow and the water depthare controlled by a system of pumps and weirs. The ex-perimental conditions for the tracer experiments werean input flow of 80 L s−1 to obtain a maximum depthof 1.30 m, with an average water depth of 0.60 m andan average porosity of 80%.

The third data set shows the response to a lithiumcarbonate pulse of a small experimental HSCW at Cer-baia, Tuscany, central Italy, constructed for tertiarytreatment and discharging in a environmentally sen-sitive marshland area in central Tuscany. The HSCW,with a surface of 12 m2, was planted withPhragmitesaustralis 2 years before the experiment. The fillingmedium had an average diameter of 8 mm and a poros-ity of 35%. Similar to this is the fourth data set, from anHSCW located at Sieci, near Florence, Italy, treatingthe domestic effluent from the septic tanks of a smallrural community. The surface of 25 m2 was plantedwith Phragmites communis, whose roots were about3 lingc bled gri-c 5%.M ex-p(

2

et-i b-t herf al.,1b acerp ulicsa ted.

The Stretton-on-Fosse data were drawn from the WRcdatabase (Cooper et al., 1996) and refer to a plant withan effective volume of about 88 m3. A hydraulic per-turbation caused by a storm event entered the systemduring the observation interval, whereas the data fromPolprasert et al. (1998)refer to dry-weather operation.As in the Carville case, the data were scanned anddigitised directly from the paper drawings, so theiraccuracy is limited. Conversely, a direct COD removaltest was run at the Cerbaia plant (Bianco, 2002). Inthis case, the calibrated model B was used and onlythe COD kinetics was estmated.

3. Parameter estimation

The parameter estimation of HSCW is precededby a sensitivity analysis to check the suitability ofthe experiments and validated with the approximateconfidence regions (Marsili-Libelli et al., 2003). Firstthe hydraulic models A and B are identified and laterthe pollution kinetics is considered. In the case of theCerbaia and Sieci plants, the hydraulics identified withthe tracer experiment was retained and only the rate co-efficients in Eq.(2) or (3) were identified. Conversely,for the literature data (Cooper et al., 1996; Polprasertet al., 1998) both hydraulics and removal kinetics werejointly estimated, lacking specific tracer experiments.

3.1. Sensitivity analysis

s

g r oftL 96;D llie hert signa ofmc tendt aren t thet ulsee andt thep

0 cm deep at the time of the experiment. The filonsisted of a lower layer of gravel with a variaiameter of 10–30 mm and an upper layer of fine aultural soil. The estimated average porosity was 3ore information about these two plants and theeriments are reported in the thesis work ofBianco2002).

.6. Continuous operation data

For the calibration of the pollution reduction kincs Eqs.(2) and (3)more experimental data were oained, this time involving continuous operation, eitrom the literature (Cooper et al., 1996; Polprasert et998) or from the work ofBianco (2002)for the Cer-aia plant. In the case of literature data, no prior trulse data were available and, therefore, hydrand carbon removal dynamics were jointly estima

Parametric sensitivity (Perkins, 1972), defined a∂Cout∂pi

whereCout is the output concentration andpi is aeneric model parameter, is an important indicato

he parameter influence on the model output (Marsili-ibelli, 1992; Vanrolleghem and Keesman, 19ochain and Vanrolleghem, 2001; Marsili-Libet al., 2003). Sensitivity can be used to check whet

he given data are suitable for identification or to den efficient calibration experiment. The sensitivityodel B to a tracer experiment is shown inFig. 3. It

an be seen that the sensitivity trajectories nevero zero for the whole duration of the experiment,ot proportional to one another and are highest a

ime of the peak. All these features indicate that a pxperiment is suitable for parameter identificationhat the most informative data are those aroundeak.


Fig. 3. Normalized sensitivity of model B to a tracer pulse. All theparameters exhibit high sensitivity for the entire pulse duration, con-firming that the pulse data are adequate for identification. The dottedlines represent the normalized outputCout.

3.2. Statement of the parameter estimationproblem

Before identifying the models A and B with eithertracer or continuous data, it was assumed that the valueof the transport dead-timeδ was a priori known be-cause of the influence that it has on all the parameters.Fig. 4 indicates that the sensitivity toδ extends wellbeyond the onset of the pulse response and influencesboth models for the duration of the whole experiment.On the other hand, includingδ in the parameter vec-tor would complicate the estimation process withoutsignificant benefits and it was preferred to estimate itdirectly, prior to the identification, by measuring thetime elapsed from the tracer injection to the onset ofthe output response.

The objective error functionE(p) to be minimized isdefined as the sum of the squared differences betweenobserved concentrationsCexp and model outputsCout.Dividing the squared sum by the difference between theexperimental dataNexp and the number of parametersnp yields a function with the statistical properties of theχ2 distribution withNexp− np degrees of freedom.

E(p) = 1

Nexp − np

Nexp∑k=1

[Cexp(k) − Cout(k)]2. (4)

Fig. 4. Normalized sensitivity of models A and B to the dead-timeδ, showing that its influence extends well beyond the onset of thepulse response, as demonstrated by the minimum halfway into thepulse response. The dotted lines represent the normalized outputCout.

Fig. 5A shows that the shape of this function for modelA is very regular, whereas for model B it presentsseveral numerical difficulties, as the local minima inthe (V1, V2) space (Fig. 5B) and (V3, b) subspace(Fig. 5D), or the long horizontal trough in the (V2,b) subspace (Fig. 5C). The latter is a consequence ofthe obvious fact that diverting the entire flow toV3(i.e. b = 1) makes the model insensitive toV2. Thisbehaviour, however, is not symmetrical betweenV2and V3, becauseV3 is the dominant branch. In thiscase, a local minimum is produced forb close to unity(Fig. 5D).

A modified simplex method (Marsili-Libelli, 1992)was selected to optimize Eq.(4) for its ability to copewith the irregularities and elongated valleys shownin Fig. 5, though it is defenceless against entrap-ments in local minima. To ensure proper termina-tion at the global minimum, an efficient initialization


Fig. 5. Contour representation of the objective error function(4) in the case of model A (A) and model B (B–D). For model A, the shape of theobjective function in the (V1, V2) space appears very regular and does not pose any numerical problem, whereas that of model B presents severalnumerical difficulties, as the local minimum in the parameter subspaces (V1, V2) (B) and (V2, b) (D), or the long horizontal trough parallel toV2

whenb = 1 (C).

was provided by a genetic algorithms (Marsili-Libelliand Alba, 2000). All the algorithms were coded inMatlab 6.5TM and the models A and B were imple-mented in Simulink, using a Rosenbrock stiff variable-step integration algorithm with an absolute toleranceof 10−6.

3.3. Approximations of confidence regions for theestimated parameters

Confidence regions of the estimated parametersp

indicate the spread of parameter values which are stillsafe for model functioning. This problem has beenextensively treated in the system-theoretic literature(Ljung, 1999) and in connection with environmentalmodels (Donaldson and Schnabel, 1987; Hakanson,

1996; Norton, 1996; Alewell and Manderscheid, 1998;Dochain and Vanrolleghem, 2001; Omlin et al., 2001).This problem is also at the basis of a previous paper(Marsili-Libelli et al., 2003) whose results are sum-marised here before being applied to the HSSCW iden-tification.

Generally, a confidence region is related to the con-tours ofE(p), in the sense that any levelE(p) > E(p)defines a region in the parameter space correspond-ing to a prescribed degree of confidence. This regionis exact because it is not based on any approxima-tion, but in practice it is difficult to specify statisti-cally significant levels of the incrementE(p) − E(p)unlessNexp is large, in which case theF statistics canbe used, having the requiredχ2 asymptotic properties(Seber and Wild, 1989). The numerical difficulty in


estimating the exact confidence region in the param-eter space has been examined byVanrolleghem andKeesman (1996), who suggested the use of extensiveMonte Carlo simulations. More recently,Dochain andVanrolleghem (2001)on the basis of a previous workby Lobry and Flandrois (1991), proposed a successivecontraction method to find the value ofE(p) corre-sponding to the prescribed value of theF statistics.This method, which has been used inMarsili-Libelliet al. (2003)and in this paper to construct the exactconfidence regions, is rather elaborate and computa-tionally intensive. On the other hand, in linear systemselliptical confidence region can be computed exactlyas a function of the estimated parameters covariancematrix (Ljung, 1999). Using the same approach, in thenonlinear case approximate confidence ellipsoids canstill be defined by a quadratic form similar to the linearcase (Press et al., 1986; Seber and Wild, 1989; Rooneyand Biegler, 1999)

(p − p)Γ −1(p − p)T = npF1−αnp,Nexp−np

, (5)

where the matrix� is the equivalent of the parametercovariance matrix. Eq.(5) is a quadratic function rep-resenting the approximate confidence ellipsoid in thenp-dimensional parameter space. This can be shownby diagonalizing the equivalent covariance matrix, i.e.�−1 = UUT, whereU is the matrix of eigenvectorsof �−1 andΛ = diag(λ1, λ2, . . . , λnp). Introducing thechange of coordinatesz = UT(p − p) and shifting theo le

∑

w en-v thel me-t

s toaL -i eart ob-j oui on

matrix J (Ljung, 1999) defined as a quadratic form of

the output sensitivity∂Cout∂p

∣∣∣p

computed in the neigh-

bourhood of the estimated parametersp.

J = Γ J−1 = s2

Nexp∑k=1

(∂Cout(k)

∂p

)T (∂Cout(k)

∂p

),

(7)

In Eq. (7), s2 = E(p) is an estimate of the mea-surement error variance. As an alternative, the errorfunction can be approximated with a second order ex-pansion of the objective error function in the neigh-bourhood of the minimumE(p) (Press et al., 1986)

Γ H (p) = s2H(p)−1 where H(p) = 1

2

∂2E(p)

∂p∂pT

∣∣∣∣p

.

(8)

Substituting either�J or �H in place of� in Eq. (5)yields the approximations of the confidence ellipsoids

(p − p)TΓ −1J (p − p)T = npF

1−αnp,Nexp−np

, (9)

(p − p)TΓ −1H (p − p)T = npF

1−αnp,Nexp−np

. (10)

For direct visual inspection, projections onto a two-dimensional subspace is required. Let (pi, pj) be theparameter couple of interest, then Eq.(5) becomes

[

[Γ (i, i) Γ (i, j)

]−1[p −p

]

stillt -m the2 -l en-vRy

λ

-m n-t

rigin in p, Eq.(5) can be written as anp-dimensionallipsoid.

np

k=1

λk z2k = npF

1−αnp,Nexp−np

, (6)

ith axis lengths inversely proportional to the eigaluesλk. Therefore, the smaller is the eigenvalue,arger is the uncertainty of the corresponding paraer.

For nonlinear models there are several waypproximate the matrix� as described inMarsili-ibelli et al. (2003). In summary,� can be approx

mated either by extending the results for the linheory or through a second order expansion of theective error function Eq.(4). The first approach, alssed byDochain and Vanrolleghem (2001), approx-

mates� with the inverse of the Fisher Informati

pi − pi pj−pj ]Γ (j, i) Γ (j, j)

i i

pj − pj

= 2F1−α2,Nexp−np

. (11)

This reduced-order representation, however,akes into account that the parameters (pi, pj) were estiated together with all the others. In fact, although× 2 sub-matrix in Eq.(11) is formed by the four se

ected elements of the full covariance matrix, its eigalues differ from the correspondingλi andλj of �−1.ewriting Eq.(11) in the form of Eq.(6) with np = 2ields:

1z21 + λ2z

22 = 2F1−α

2,Nexp−2, (12)

In polar coordinates, Eq.(12) provides the increentsξi and ξj to trace the confidence ellipse ce

red in (pi, pj) using the eigenvaluesu andv obtained


from the diagonalization of

[Γ (i, i) Γ (i, j)

Γ (j, i) Γ (j, j)

]−1

=

[ u v ]diag(λ1λ2)

[u

v

], namely:

ξi =√

2F1−α2,Nexp−2

λ1u1 cos(ϕ) −

√2F1−α

2,Nexp−2

λ2u2 sin(ϕ) + pi,

ξj =√

2F1−α2,Nexp−2

λ1v1 cos(ϕ) −

√2F1−α

2,Nexp−2

λ2v2 sin(ϕ) + pj,

for 0 ≤ ϕ ≤ 2π. (13)

3.4. Confidence intervals of individual parameters

The percentage relative confidence interval of theindividual parameter can be computed as a function ofthe covariance matrix:

δpi = ±t1− α

2Nexp−np

√Γ (i, i)

pi

× 100 (14)

wheret1− α

2Nexp−np

is the two-tail Student’st distribution forthe given confidence levelα andNexp− np degrees offreedom. Substituting�J or�H in place of� in Eq.(14)yields the relative confidence bounds of the estimatedparameters ˆpi.

δpJ (i) = ±t1− α

2Nexp−np

√Γ J (i, i)

pi

× 100 and

δ

I fer toa thate i-m trix� s oftL

3

thet nb etecti tion� the

Hessian approximation�H depends on the shape of theerror surface. For nonlinear systems, this may be heav-ily influenced by thecurvature, reflecting the degreeof nonlinearity in the parametrization (Donaldson andSchnabel, 1987; Seber and Wild, 1989), so neglectingthis term may have a significant effect. Comparing the

regions obtained with�J or �H represents a way to as-sess the curvature effect and hence the extremal natureof the estimated parametersp. It has been shown inMarsili-Libelli et al. (2003)that when the two regionscoincide the curvature effect is negligible and the iden-tification can be considered reliable. In this case, theseregions coincide also with the exact confidence regiondetermined on the basis of the error surface (Lobry andFlandrois, 1991). Conversely, if the search terminatesat a non extremal point, where the curvature term maycause the two approximations to diverge, warns that theestimation problem becomes critical. In this sense, thispaper follows the suggestions ofvan Tongeren (1995),who advocates structural validation combined with ajoint correlation analysis.

4

ioni Bwet npd dt ausen cali-b el Ap rst-o esti-m rderd iler

pH (i) = ±t1− α

2Nexp−np

√Γ H (i, i)

pi

× 100. (15)

t should be reminded that though these bounds resingle parameter, they take into account the factach parameter is part of the jointnp-dimensional estation through the diagonal elements of the full ma. The algorithms for the numerical approximation

he FIM and Hessian matrices are described inMarsili-ibelli et al. (2003).

.5. A parameter estimation validity test

There is an important conceptual difference inwo approximations�J and �H and the comparisoetween their confidence regions can be used to d

naccurate estimation results. The FIM approximaJ is based on the sensitivity trajectories, whereas

. Discussion of model calibration results

The calibration algorithm of the previous sects now applied to the identification of models A andith the data sets described in Sections2.5 and 2.6. Thestimation results are summarised inTables 3 and 4for

he hydraulics and inTable 5for the pollution reductioart. In the case of thePolprasert et al. (1998)and WRcata (Cooper et al., 1996), hydraulics and kinetics ha

o be jointly estimated from the same data set, beco separate tracer experiment was available for theration of the hydraulic model. In these cases, modrovided the hydraulic part and either a Monod or firder kinetics were used, whichever gave the bestation results. In the Cerbaia plant, a COD first-oynamics Eq.(2) was independently identified, whetaining the already identified hydraulic model B.


Following the suggestions ofvan Tongeren (1995),in assessing each model, a balance between the sum ofsquared errors and the confidence regions agreementwas sought. The latter is a very strong indicator of areliable model structure, but in some cases (e.g. withscanned, low-accuracy data) the minimisation of thesum of squared errors has been given more importance.A further performance index, the “variance-accounted-for” (VAF), was also considered. It is defined as thepercentage of experimental data variance explained bythe model, i.e.

VAF = 100×(

1 − σ2err

σ2exp

), (16)

whereσ2err is the estimation error variance andσ2

exp isthe experimental error variance.

Table 3 summarizes the plant characteristics inthe first four columns (location, flow, dead-time andwet volume, considering porosity). Dead-time (col-umn 3) was independently estimated and introducedas a known quantity in the subsequent calibration.Column 5 shows the total model volume, com-puted asVtot = 2× V1 + V2 + VPF for model A andVtot = 3× V1 + V2 + V3 VPF for model B. The percent-age estimation error on total volumes is shown incolumn 6, whereas column 7 reports the plug-flow vol-umes obtained asVPF= δ × F. Columns 9 and 10 com-pare the hydraulic retention times (HRT) computedfrom the experimental data and the model by numer-i ac-c tm thet odelB iecip nedf faceC

oft s thee nds,iC uatem reast ables ta.N stills ment

between data and model response, indicated by VAF, isabove 95%. The last two columns ofTable 4comparethe experimental amount of tracerMo, computed byintegrating the tracer concentrations, with the esti-matedMo, computed by integrating the correspondingmodel response.Table 5summarizes the identificationof the pollution removal dynamics. Three cases wereanalysed, two of which taken from the literature and thethird from a specific experiment at the Cerbaia plant.In this last case, the hydraulic parameters previouslyidentified were retained and only the removal kineticswas calibrated. There are two rows inTable 5for thisplant because both first-order and Monod kineticswere tested, but the first-order kinetics was preferredfor the higher VAF value and the smaller confidenceregions. The identification results for each data set arenow analysed, first the tracer experiments and then thepollution data.

4.1. Carville tracer hydraulic model

Given the inherent low accuracy of the data, modelA was selected and a good fit was obtained, as shownin Fig. 6, though Table 3 shows that the volumeis overestimated by more than 40%. A high VAFscore indicates an adequate the model structure, con-firmed by the coincidence of the confidence intervals(Table 4).

4.2. Castelnuovo Bariano hydraulic model

oorr l-u acyo eentw themi be av ionsi oft se.O pro-v n-t theg eat ith

cal integration. In the last column, the varianceounted for Eq.(16) is evaluated.Table 3shows thaodel A is appropriate for the Carville data, though

otal volume is largely overestimated, whereas mperforms better with the small Cerbaia and S

lants. A comparatively less satisfactory fit is obtaior Castelnuovo, possibly because it is a free-surW.Table 4shows more detailed estimation results

he tracer experiments, giving for each parameterstimated values and the relative confidence bou

ndicated byδpJ and δpH, computed from Eq.(15).oincidence of these intervals indicate an adeqodel structure and a reliable identification, whe

heir divergence denotes a model with a less relitructure, critical fitting or poorly informative daevertheless, such a low-accuracy model mayerve a practical purpose provided that the agree

Model A was tested with these data with pesults, shown inTable 3 (underestimated total vome, HRT disagreement, low VAF). The inadequf model A is confirmed by the disagreement betw

he approximate confidence ellipses shown inFig. 7,hich also shows the exact regions, computed withethod ofLobry and Flandrois (1991). This figure is

mportant because it shows that the ellipses mayery poor approximation of the real confidence regn the case of misalignment, confirming the limitshe linear approximations in a structurally critical can the other hand, the exact regions alone would

ide no information about the criticality of the ideification. Model B, on the other hand, producedood data fit shown inFig. 8 and confirmed by thgreement between confidence intervals (Table 4). The

otal volume underestimation is 11%, compared w


Tabl

e5

Est

imat

edpa

ram

eter

san

d95

%co

nfide

nce

boun

dsfo

rth

eca

rbon

rem

oval

kine

tics

Dat

aso

urce

Par

amet

ers

and

confi

denc

ebo

unds

(%)

V1

(m3)

V2

(m3)

k 1(d

−1)

R(d

−1)

Ks

(mg

L−1)

VAF

Str

etto

n-on

-Fos

seW

Rc

data

base

(Coo

per

etal

.,19

96)m

odel

A+

Mon

odE

q.(3

)

p1.

8524

18.8

214

–3.7

011

16.2

232

94.3

75δp

J±1

95.0

92±2

57.5

61–

±522

.303

±20.

794

–δp

H±7

3.82

7±2

9.10

4–

±24.

906

±43.

850

–

Pol

pras

erte

tal.

(199

8)mod

elA

+fir

st-o

rder

kine

tics

Eq.(

2)

p0.

4068

2.009

20.5

030

––

16.3

59δp

J±8

21.7

92±1

018.

655

±174

.404

––

–δp

H±7

38.2

35±5

2.98

5±2

6.68

8–

––

Cer

baia

mod

elB

+fir

st-o

rder

kine

tics

Eq.

(2)

pS

eeTa

ble

4for

the

hydr

aulic

para

met

ers

0.02

65–

–74

.568

δpJ

±48.

792

––

–δp

H±5

1.09

4–

––

Cer

baia

mod

elB

+M

onod

Eq.

(3)

pS

eeTa

ble

4for

the

hydr

aulic

para

met

ers

204.

977

37.7

64.4

92–

δpJ

±452

1.17

9±4

605.

091

––

δpH

±152

5.34

2±1

536.

262

––

Fig. 6. Fitting model A to a tracer experiment from a horizontalsubsurface flow constructed wetland in Carville (Louisiana, USA).The data were scanned from US EPA (1993).

16.5% of model A (Table 3), which can be explainedwith the presence of lateral impoundments which donot contribute to the active volume. The estimation ofthe recovered tracerMo = 1161.5 g is in excellent agree-ment with the same quantity obtained from the data,1161.6 g.

4.3. Cerbaia and Sieci hydraulic model

Model B produced a good fit for both plants, asshown inFigs. 9 and 10. Table 3indicates a good agree-ment between the estimated and experimental globalvariables (volume and HRT) and the VAF is also high.The model adequacy is confirmed by the agreement ofthe confidence ellipses between themselves and withthe exact confidence regions, as shown inFig. 11 forthe Cerbaia case. This figure should be compared withFig. 7 to underline that ellipses alignment implies thecoincidence with the exact confidence regions, con-firming the validity of the linear approximation in thecase of a correct identification, in spite of the numericaldifficulties shown inFig. 5. Similar results have beenfound for the Sieci plant.

4.4. Pollution reduction models

The chemical models Eq.(2) or (3) were calibratedwith the three data sets already described in Section2.6 and the results are summarized inTable 5. In the


Fig. 7. Disagreement between approximate confidence regions obtained by fitting model A to the Castenuovo Bariano tracer data, indicatingpoor identifiability. The exact regions, represented by the dots, were computed for comparison with the method ofLobry and Flandrois (1991).

Stretton-on-Fosse data, drawn from the WRc database(Cooper et al., 1996), a combination of model A andMonod kinetics provided the best performance. A hy-draulic perturbation entered the system (the flow is dis-played inFig. 12 ) and allowed a good identificationof the hydraulics, as confirmed by the high VAF andthe visual agreement ofFig. 12, though the divergence

Fig. 8. Fitting model B to the tracer data from Castelnuovo Barianofree-surface wetland.

between confidence bounds denotes a critical identifi-cation, largely due to the low quality of the data.

The experiment reported byPolprasert et al. (1998)had no such hydraulic variations but the combinationof model A and first-order kinetics could still be iden-tified, albeit with a lower accuracy and a very low VAF.The visual agreement, shown inFig. 13, indicates that

Fig. 9. Fitting model B to the tracer data from the experimentalHSCW in Cerbaia.


Fig. 10. Fitting model B to the tracer data from the experimentalHSCW in Sieci.

the model response tends to act as a moving average ofthe output, smoothing out the observed abrupt changes,therefore the limited output variations suggested theadoption of a first-order kinetics. The borderline iden-

tification results are confirmed by the huge confidencebounds forV1 andV2.

At the Cerbaia plant first-hand data were available,which allowed the re-use of the previously calibratedhydraulic part of model B, and both first-order andMonod kinetics were tried. The calibration results areshown inFig. 14. The Monod kinetics resulted in largeand diverging confidence regions, indicating poor iden-tifiability, whereas the first-order kinetics produced amore reliable result. It is interesting to notice that theratio of the Monod parametersR

Ks= 204.9

7737.7 = 0.02648is in excellent agreement with the calibrated value ofk1 = 0.0265. This confirms the assumption of Section2.4 that the first-order kinetics can be used to approx-imate the Monod kinetics when the substrate is con-sistently much less than the half-saturation constant.Further, the inadequacy of the Monod kinetics is con-firmed by the very high correlation betweenR andKs,as shown by the very narrow and elongated shape ofthe confidence ellipses, shown inFig. 15. This confirms

F obtain te ellipsoidsb compa tem

ig. 11. Two-dimensional projections of the confidence regionsetween themselves and with the exact regions, computed forodel structure and a robust identification.

ed for the Cerbaia traced data. The coincidence of approximarison with the method ofLobry and Flandrois (1991), indicates an adequa


Fig. 12. Fitting the BOD data from the Stretton-on-Fosse plant(Cooper et al., 1996) to the combination of model A and Monodkinetics.

that the only quantity which can be reliably estimatedis their ratio, equal tok1, rather than their separate val-ues. The low value of the kinetic constantsk1 is alsoin agreement with the conclusion ofWynn and Liehr(2001), who report a heterotrophic maximum growthrates three orders of magnitude smaller than the valuesused for conventional treatments.

As a concluding remark, model B proved the mostversatile and it can predict the HSSCW wet volumeVtot = 3× V1 + V2 + V3 + VPF with sufficient accuracyfor simulation and design. As to the pollution reduc-

Fig. 13. Fitting the COD data fromPolprasert et al. (1998)to thecombination of model A and first-order kinetics.

Fig. 14. Fitting the COD data from the Cerbaia plant to the combi-nation of model B and first-order kinetics.

tion, the identification of Monod kinetics may be diffi-cult and a first-order model can represent a robust alter-native. Whether such kinetics may be used for designshould be decided after careful assessment of the avail-able information regarding the kind of effluent, typeof plants and other operational circumstances, whichmay heavily influence the actual behaviour, as alreadypointed out by many authors (Kadlec, 2000; Wynn andLiehr, 2001; Rousseau et al., 2004).

Fig. 15. Comparison of approximate ellipses in the (R, Ks) spacefor the Cerbaia identification reveals an extremely high parameterc s.
orrelation, indicating a poor identifiability of the Monod kinetic


5. Conclusion

This paper has proposed a simple method to ap-proximate the transport, diffusion and reaction dynam-ics in subsurface horizontal constructed wetlands witha lumped-parameter model composed of a combina-tion of series/parallel CSTR elements in series with aplug-flow reactor. The motivation for proposing sucha simple model comes from the growing awarenessthat the current level of model complexity is not pro-ducing a proportional insight into the factors affectingpollution removal vis-a-vis the difficulty of estimat-ing a large number of parameters. The approach pur-sued here, instead, is based on a combination of simplemodels assisted by a comparatively more sophisticatedestimation technique, which computes the parametersconfidence regions based on two differing approxima-tions. Since they produce coincident regions only inthe case of a consistent identification, this approachallows structural discrimination on the basis of theagreement of these regions. This test is very sensitiveto structural and parametric perturbations and can de-tect model criticality not revealed by other performanceindexes.

The proposed models were applied to several datasets drawn either from the literature of from field ex-periments and the confidence regions method was usedto select the most appropriate model combination foreach situation. In particular, the hydraulic model B ap-proximated the actual wet volume with an accuracya sur-f tionr be-t cultt vedm e or-g theM thatl am-p osec toryr

R

A for. Mod.

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identification of dynamic models for horizontal subsurface constructed wetlands

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