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- 1. ARTICLE IN PRESS Water Research 39 (2005) 3686–3696 www.elsevier.com/locate/watres Reliability of parameter estimation in respirometric models Nicola Checchi, Stefano Marsili-LibelliÃ Department of Systems and Computers, University of Florence, Via Santa Marta, 3-50139 Firenze, Italy Received 18 September 2004; received in revised form 23 June 2005; accepted 25 June 2005 Available online 3 August 2005 Abstract When modelling a biochemical system, the fact that model parameters cannot be estimated exactly stimulates the deﬁnition of tests for checking unreliable estimates and design better experiments. The method applied in this paper is a further development from Marsili-Libelli et al. [2003. Conﬁdence regions of estimated parameters for ecological systems. Ecol. Model. 165, 127–146.] and is based on the conﬁdence regions computed with the Fisher or the Hessian matrix. It detects the inﬂuence of the curvature, representing the distortion of the model response due to its nonlinear structure. If the test is passed then the estimation can be considered reliable, in the sense that the optimisation search has reached a point on the error surface where the effect of nonlinearities is negligible. The test is used here for an assessment of respirometric model calibration, i.e. checking the experimental design and estimation reliability, with an application to real-life data in the ASM context. Only dissolved oxygen measurements have been considered, because this is a very popular experimental set-up in wastewater modelling. The estimation of a two-step nitriﬁcation model using batch respirometric data is considered, showing that the initial amount of ammonium-N and the number of data play a crucial role in obtaining reliable estimates. From this basic application other results are derived, such as the estimation of the combined yield factor and of the second step parameters, based on a modiﬁed kinetics and a speciﬁc nitrite experiment. Finally, guidelines for designing reliable experiments are provided. r 2005 Elsevier Ltd. All rights reserved. Keywords: Respirometry; Parameter estimation; Activated sludge models; Nitriﬁcation 1. Introduction et al., 1998; Petersen, 2000; Petersen et al., 2002), resulting in systematic estimation protocols (Petersen et The structural and practical identiﬁability of Monod- al., 2003b; Sin, 2004; De Pauw, 2005). This evolution based biochemical models has greatly progressed since was also stimulated by the introduction of the ASM the ﬁrst studies of Pohjanpalo (1978) and Holmberg models (Henze et al., 2000) whose parameter identiﬁca- (1982) and is still a viable research topic (Kesavan and tion is now considered such an important issue in Law, 2005). Important contributions now exist both on wastewater treatment that the ASM3 model has the principles of estimation (Dochain et al., 1995; replaced ASM1 not only for its better understanding Vanrolleghem and Keesman, 1996; Petersen et al., of the biochemical mechanisms, but also for its 2000; Brun et al., 2002; Petersen et al., 2003a) and on improved identiﬁability (Gernaey et al., 2004). In this its practical aspects (Vanrolleghem et al., 1995; Brouwer context, respirometry is a primary tool for model identiﬁcation and a great deal of research has been ÃCorresponding author. Tel./fax: +39 055 47 96 264. devoted to providing uncertainty limits to parameter E-mail address: marsili@dsi.uniﬁ.it (S. Marsili-Libelli). estimates and designing better experiments, especially 0043-1354/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2005.06.021
- 2. ARTICLE IN PRESS N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3687 Nomenclature rend endogenous respiration rate (mg O2 LÀ1 minÀ1) ASM activated sludge model SNH ð0Þ initial concentration of ammonium-N C parameter covariance matrix, <np Ânp (mg N-NH4 LÀ1) CJ approximate covariance matrix based on the SNO ð0Þ initial concentration of nitrite-N (mg N- Fisher Information Matrix (FIM), <np Ânp NO2 LÀ1) CH approximate covariance matrix based on the SNH ammonium-N concentration (mg N- Hessian matrix, <np Ânp NH4 LÀ1) E(p) error functional for parameter estimation SNO nitrite-N concentration (mg N-NO2 LÀ1) y F ap ;NÀnp n F statistics at the 100(1Àa)% conﬁdence SPj output sensitivity to parameter pj level for np parameters and NÀnp degrees of s2 estimated variance of the residuals freedom (mg LÀ1)2 a=2 J Fisher Information Matrix (FIM), <np Ânp tNÀnp two-tails Student’s t distribution for the H Hessian (second derivative) matrix, <np Ânp given conﬁdence level 100(1Àa)% and N-np KNH ammonium oxidisers half velocity constant degrees of freedom (mg N-NH4 LÀ1) X A1 ammonium oxidisers biomass concentration KNO nitrite oxidisers half velocity constant (mg COD LÀ1) (mg N-NO2 LÀ1) X A2 nitrite oxidisers biomass concentration N number of experimental data (mg COD LÀ1) NOD nitrogen oxygen demand (mg O2 LÀ1) Y A1 ammonium oxidisers biomass yield np number of estimated parameters Y A2 nitrite oxidisers biomass yield RNH composite parameter related to the ﬁrst dpJ individual parameter conﬁdence interval oxidation stage (mg O2 LÀ1 minÀ1) estimated with the FIM approximation RNO composite parameter related to the second dPH individual parameter conﬁdence interval oxidation stage (mg O2 LÀ1 minÀ1) estimated with the Hessian approximation rNO synthesis oxygen uptake rate on nitrite-N mmaxA1 ammonium oxidisers maximum growth rate (mg O2 LÀ1 minÀ1) (minÀ1) rexp o experimental respiration rate mmaxA2 nitrite oxidisers maximum growth rate (mg O2 LÀ1 minÀ1) (minÀ1) ro model respiration rate (mg O2 LÀ1 minÀ1) t time constant of nitrite-N modiﬁed respira- tion model (min) when the model structure is a priori speciﬁed as in ASM estimation of the second step parameters is critical and models (Vanrolleghem et al., 1995; Vanrolleghem and assessing the inﬂuence of the number of data. Later, a Keesman, 1996; Brouwer et al., 1998; Petersen, 2000; speciﬁc nitrite experiment is performed, leading to a Petersen et al., 2000; Marsili-Libelli and Tabani, 2002; better estimation of those parameters. Eventually, an Sin, 2004; De Pauw, 2005). estimation protocol is deﬁned for the reliable estimation Based on the assumption that any given model of the whole parameter set. structure is necessarily a simpliﬁed approximation of a complex reality, the paper applies a new estimation 1.1. Theoretical identiﬁability of a two-step nitriﬁcation reliability test, derived from a previous method based on model the approximate conﬁdence regions (Marsili-Libelli et al., 2003), to a respirometric model based on ASM A two-step nitriﬁcation model derived from the ASM kinetics. Its aim is to determine under which conditions model suite (Henze et al., 2000) is considered, with two routine respirograms yield reliable estimates, without zero-growth autotrophic populations of ammonium performing complex experiments with inhibitors (e.g. oxidisers (XA1) and nitrite oxidisers (XA2). The zero- Nowak et al., 1995; Surmacz-Gorska et al., 1996) or growth assumption is justiﬁed by the short duration of combining respirometric and titrimetric measurements the experiments and the small amount of added (Petersen, 2000; Ficara et al., 2002; Gernaey et al., 2002). substrate. A similar model was used by Petersen (2000) After summarising the main results of Marsili-Libelli and Gernaey et al. (2001) for respirometric studies, et al. (2003), the theory is taken a step further, whereas a more recent paper (Insel et al., 2003) includes presenting a new reliability test which is applied to the hydrolysis. The endogenous respiration rend is directly estimation of a two-step nitriﬁcation kinetics. Initially, estimated by averaging the endogenous data. The the full two-step model is considered, showing that the identiﬁability of respirometric models without biomass
- 3. ARTICLE IN PRESS 3688 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 growth has been deﬁnitively assessed (Vanrolleghem et 1.3. Approximations of conﬁdence regions for the al., 1995; Petersen, 2000; Brun et al., 2002; Petersen et estimated parameters al., 2003a) and particularly by Brouwer et al. (1998), who estimated a model similar to Eq. (3) save for the Conﬁdence regions represent the set of parameter yield coefﬁcients which are supposed known. The values producing a model response within prescribed following parameter combinations can be estimated statistical boundaries. Being related to the error func- tional Eq. (2), any level EðpÞ4Eð^ Þ deﬁnes a region with p W1 ¼ ð3:43 À Y A1 ÞRNH ; W4 ¼ ð1:14 À Y A2 ÞRNO ; a given degree of conﬁdence. However, it is difﬁcult to W2 ¼ ð3:43 À Y A1 ÞS NH ð0Þ; W5 ¼ ð1:14 À Y A2 ÞSNO ð0Þ; specify statistically signiﬁcant levels of the increment W3 ¼ ð3:43 À Y A1 ÞK NH ; W6 ¼ ð1:14 À Y A2 ÞK NO ; DE ¼ EðpÞ À Eð^ Þ unless N is large, in which case DE p (1) has the required w2 asymptotic properties to apply the F statistics (Seber and Wild, 1989). The numerical where the two composite parameters RNH ¼ difﬁculty in estimating the exact conﬁdence regions ^ ^ mmax A1 X A1 =Y A1 and RNO ¼ mmax A2 X A2 =Y A2 are intro- has been examined by Vanrolleghem and Keesman duced for identiﬁability reasons (Dochain et al., 1995). (1996) and Dochain and Vanrolleghem (2001) who, on The ﬁrst-step parameters ðY A1 ; RNH ; K NH Þ are always the basis of a previous work by Lobry and Flandrois identiﬁable if S NH ð0Þa0, whereas if S NO ð0Þ ¼ 0 only (1991), proposed a successive contraction method to two of the three second-step parameters ﬁnd the value of E(p) corresponding to the prescribed ðY A2 ; RNO ; K NO Þ can be identiﬁed. In practice this F value. In this paper the approach of Marsili-Libelli condition implies a large estimation uncertainty of the et al. (2003) is used, based on linear or quadratic second step parameters, as will be shown later. The aim approximations, as suggested by Press et al. (1986) and of this paper is to assess the reliability of the estimates as Seber and Wild (1989). Conﬁdence regions in the np a function of SNH ð0Þ, showing that this parameter plays parameter space can be expressed as a the quadratic a crucial role in the model identiﬁability, and suggesting form appropriate values for this quantity. ðp À pÞC À1 ðp À pÞT pnp F ap ;NÀnp , ^ ^ n (4) 1.2. Calibration of model parameters where the matrix C is the equivalent of the parameter covariance matrix of the linear case (Ljung, 1999) and A numerical optimisation search is used to minimise 100(1Àa)% is the required conﬁdence level of the F the sum of squared errors between respiration data rexp o statistics with np parameters and NÀnp degrees of and model responses ro at the sampling times freedom. This matrix can be approximated either by i ¼ 1; 2; . . . ; N, extending the results of the linear theory or through a X N second-order expansion of the error functional Eq. (2). EðpÞ ¼ ðrexp ðiÞ À ro ðiÞÞ2 , o (2) The ﬁrst approach, used by Petersen (2000) and Dochain i¼1 and Vanrolleghem (2001), approximates C with the where p ¼ ½Y A1 RNH K NH Y A2 RNO K NO T is the para- inverse of the Fisher Information Matrix (FIM) J meter vector, N is the number of respirometric data (Ljung, 1999), deﬁned as a quadratic form of the output and the model output ro is computed from Eq. (3) sensitivity qy=qpjp ^ N exp 3:43 À Y A1 S NH 1 X qyðkÞ T qyðkÞ ro ¼ ^ mmaxA1 X A1 C J ¼ J À1 ; where J ¼ . (5) Y A1 K NH þ SNH s2 k¼1 qp qp 1:14 À Y A2 SNO þ ^ mmaxA2 X A2 þ rend . ð3Þ The measurement error variance is estimated as Y A2 K NO þ SNO s2 ¼ Eð^ Þ=N À np . As an alternative, C can be approxi- p The minimisation of the error functional Eq. (2) with mated by a second-order expansion of the objective the model constraint Eq. (3) is obtained with a combined error function in the neighbourhood of the minimum search algorithm starting with a modiﬁed genetic Eð^ Þ (Press et al., 1986; Seber and Wild, 1989) p algorithm (Marsili-Libelli and Alba, 2000) to determine
- 4. the initial search region containing the global minimum, q2 EðpÞ
- 5. . C H ð^ Þ ﬃ 2s2 H À1 ; where H ¼ p (6) which is then reﬁned with a modiﬁed simplex search qpqpT
- 6. p ^ (Marsili-Libelli, 1992). The calibration software was developed in the Matlab 6.5 platform (The Mathworks, Substituting either CJ or CH or in place of C in Eq. (4) Natick, MA, USA), using the stiff-Rosenbrock integra- two differing approximate conﬁdence ellipsoids are tion method with a relative accuracy of 10À6. The details obtained. Numerical algorithms for computing these of model implementation are described in Marsili-Libelli approximations are described in Press et al. (1986), and Tabani (2002). Marsili-Libelli et al. (2003) and De Pauw (2005).
- 7. ARTICLE IN PRESS N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3689 From Eqs. (5) or (6) the conﬁdence interval of the (Bates and Watts, 1988). Consider the maximum and individual parameter can be computed minimum curvature radii rmax ¼ ð1 À lmax ÞÀ1=2 and a=2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rmin ¼ ð1 À lmin ÞÀ1=2 , where lmax and lmin are the dpi ¼ ÆtNÀnp Cði; iÞ, (7) maximum and minimum eigenvalues of the matrix B a=2 where tNÀnp is the two-tails Student’s t distribution for related to the Hessian and FIM matrices by the the given conﬁdence level 100 (1Àa)% and NÀnp relationship (Seber and Wild, 1989) degrees of freedom. This statistics is consistent with B ¼ I np À 1K T HK, (8) 2 the multivariate F distribution for np ¼ 1, since qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ À1 T a=2 tNÀnp ¼ F a where K ¼ R and R R is the upper-triangular 1;NÀnp . Substituting CJ or CH in place of C decomposition of s2 J. Eq. (8) is the result of a in Eq. (7) yields the approximate conﬁdence bounds of coordinate transformation in the parameter space which ^ the estimated parameters pi . It is important to notice attempts at reducing the conﬁdence ellipsoid to a sphere that though dpi refers to a single parameter, it takes into (Seber and Wild, 1989). In the absence of curvature both account the full np-dimensional conﬁdence region rmax and rmin tend to 1, therefore the extent to which through the matrix C. their ratio exceeds unity yields an indication of the residual curvature in the neighbourhood of p. A ^ 1.4. A parameter estimation reliability test normalized threshold value for the ratio rmax =rmin can be obtained in the form In addition to yielding conﬁdence regions, the two rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ approximations Eqs. (5) and (6) provide a reliability test np r¼1þ ¯ Fa , (9) for the estimated parameters based on their inherent N À np np ;NÀnp conceptual difference. The FIM approximation CJ is where the square root represents the tolerance to keep based on the sensitivities, whereas the Hessian approx- into account the effective ellipsoids dimensions in the imation CH depends on the shape of the error surface. tangent plane (Bates and Watts, 1988; Seber and Wild, For nonlinear systems CH includes the effect of the 1989) normalized to be insensitive to scaling errors. curvature, reﬂecting the degree of nonlinearity induced Further, Quinn et al. (2005) recently demonstrated that by the model structure (Donaldson and Schnabel, 1987; the F statistics can be used to determine the conﬁdence Bates and Watts, 1988; Seber and Wild, 1989; Marsili- limits of a parameter function such as Eq. (8). Libelli et al., 2003, Appendix B). Conversely CJ, being a If rmax =rmin or the estimation can be considered ¯ linear approximation, does not contain this term. Since reliable at the 100(1Àa)% level, implying that the the curvature effect vanishes in the neighbourhood of search terminated at a point on the error surface the minimum of E(p), comparing the two conﬁdence where the effect of the curvature is negligible. This regions yields a measure of the estimation reliability, test is more practical than the mere visual compari- because if the two regions diverge, this implies that the son between ellipses because it does not involve search terminated at a point where the effect of the the subjectivity of the examiner and it takes into account curvature is still signiﬁcant and therefore this cannot be the full np dimension of the parameter space rather the real minimum of E(p). Conversely, if the two than the 2-dimensional subspace of the projected regions coincide the curvature effect is negligible and the ellipsoids. identiﬁcation can be considered reliable. In this case these regions coincide also with the exact conﬁdence region determined on the basis of the error surface 1.5. Optimal experiment design (OED) criteria (Lobry and Flandrois, 1991). It should be reminded, however, that the curvature, amplifying the estimation Other estimation accuracy indexes may be obtained as errors, is an indicator of a failure to reach the minimum, scalar functions of the covariance matrix C. Each of but is not inﬂuenced by model inadequacy and residual them emphasises differing aspects of the conﬁdence characteristics. Its vanishing merely indicates that ellipsoid, such as its volume or axes. Based on the residuals are orthogonal to the response surface in these indicators, a theory of optimal experiments has ^ the neighbourhood of p, but does not attempt at been developed (Fedorov, 1972) with the aim of characterizing them, e.g. whether they are gaussian minimising the estimates covariance. Table 1 illustrates and uncorrelated. This test is therefore an assessment of the main criteria used in this and similar studies (see e.g. the quality of the optimisation and not of the model Atkinson and Donev, 1992; Versyck et al., 1998; structure. Petersen, 2000; De Pauw, 2005). Of all the criteria Visual inspection of the agreement or divergence mentioned in Table 1, only D has the property of being between the conﬁdence regions involves a fair amount of scale invariant, whereas all the others are scale sensitive, subjectivity. To obtain an objective test the curvature particularly mod E. The least sensitive of all appears to radii are considered, which are related to the curvature be mod A.
- 8. ARTICLE IN PRESS 3690 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 Table 1 Optimal experiment design criteria based on FIM Criteria Method Effect A minðtrðJ À1 ÞÞ Minimization of the arithmetic mean of parameter errors. mod A maxðtrðJÞÞ Same as A, but insensitive to FIM ill-conditioning D maxðdetðJÞÞ Minimizes the volume of the conﬁdence ellipsoid E maxðlmin ðJÞÞ Minimizes the length of the largest axis of the conﬁdence ellipsoid mod E minðlmax =lmin Þ Minimizes the ill-condition number ro (mgO2 L-1 min-1) 2. Calibration of the respirometric model 1 SNH (0) = 2.5 mg L-1 0.8 model The purpose of this section is to investigate the data inﬂuence of the initial amount of substrate SNH(0) on 0.6 the identiﬁability of the model parameters 0.4 p ¼ ½Y A1 RNH K NH Y A2 RNO K NO T , using three data sets 0.2 obtained with the respirometer described in Marsili- 0 5 10 15 20 25 30 Libelli and Tabani (2002). The validity of the identiﬁed time (min) parameters is assessed through several tests: sensitivity, curvature radii and OED criteria. ro (mgO2 L-1 min-1) 1.2 1 SNH (0) = 5 mg L-1 model 2.1. Assessment of estimated parameters 0.8 data 0.6 Though the estimation results, shown in Fig. 1, 0.4 appear visually acceptable in all cases, the inspection 0.2 of the parameter values and conﬁdence intervals of 0 5 10 15 20 25 30 35 40 45 Table 2 reveals that under certain experimental condi- time (min) tions the results may be seriously ﬂawed. As a ro (mgO2 L-1 min-1) preliminary test, parameter values should always check 1.2 1 SNH (0) = 10 mg L-1 for physical constraints; in this sense the negative values model YA2 for low and medium SNH(0) indicates an unreliable 0.8 data estimation, though its wide conﬁdence interval encom- 0.6 passes reasonable values. On the other hand, imposing 0.4 positivity constraints to the simplex algorithm may 0.2 disrupt the search and produce awkward numerical 0 10 20 30 40 50 60 70 80 90 results, as demonstrated by the results of Alfonso and da time (min) -˜ Conceic ao Cunha (2002), later criticized by De Pauw Fig. 1. Fitting the model Eq. (3) to the batch respirometric et al. (2004). For this reason it was preferred to use an data. unconstrained method, but rather use the conﬁdence regions approach to design reliable experiments. The most serious ﬂaws appearing in Table 2 are the negative whereas the full data set failed. This results is in values of the yield coefﬁcients for low and medium agreement with De Pauw (2005), who found that initial SNH(0) and the abnormally large values RNO and under-sampling may results in an improved accuracy KNO for SNH(0) ¼ 5 mg LÀ1. Further, since the duration and reduced parameter correlation. of the three experiments differs, the number of available data is not the same and this may affect the estimation 2.2. Sensitivity analysis accuracy. To obtain a fair comparison among the three experiments, the data sets for SNH(0) ¼ 5 and 10 mg LÀ1 The sensitivity trajectories for the three initial were under-sampled selecting every other data, in order substrate conditions SNH(0) are shown in Fig. 2. They to have approximately the same experimental basis for were computed with an incremental method similar to all of them. The results were rather counterintuitive: not that described by De Pauw and Vanrolleghem (2003) only the estimation did not get worse, but rather it and De Pauw (2005). The sensitivity to YA1 and YA2 is improved considerably for SNH(0) ¼ 5 mg LÀ1, particu- very small for all initial SNH(0). This is in agreement larly for the yield coefﬁcients and the critical second-step with the results of Table 2 and suggest that the best way parameters. The radii test Eq. (9) was also passed, to identify the yield coefﬁcients is to estimate their sum
- 9. ARTICLE IN PRESS N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3691 Table 2 Estimated parameters for the respirometric experiments, conﬁdence intervals and reliability indicators SNH(0) No. of data Y A1 RNH K NH Y A2 RNO K NO Y A ¼ Y A1 þ Y A2 rend rmax =rmin r ¯ 2.5 18 ^ p 2.5288 0.1909 0.2124 -2.2039 2.4849 1.8595 0.3249 0.2725 36.4446 2.3900 dpJ 75.9618 70.0110 70.0498 75.7584 713.9352 712.1508 70.2115 dpH 75.2857 70.0077 70.0280 75.1212 71.7371 71.3545 70.1721 5.0 15 ^ p 0.1973 0.2063 0.2006 -0.0258 0.2036 0.1371 0.1714 0.2710 1.1565 2.6387 dpj 70.7820 70.0078 70.0351 70.7577 70.0513 70.2092 70.0492 dpH 70.7023 70.0072 70.0352 70.6809 70.0473 70.1893 70.0478 5.0 29 ^ p 0.7340 0.2032 0.2088 -0.4982 13.4362 24.443 0.2358 0.2710 202.6126 1.9416 dpj 70.7867 70.0037 70.0273 70.7582 7584.9534 71081.8783 70.0451 dpH 70.5556 70.0022 70.0207 70.5379 73.8382 76.3424 70.0393 10.0 21 ^ p 0.0657 0.2061 0.2440 0.1557 0.1777 0.0819 0.2214 0.2657 1.6819 1.8830 dpj 70.0998 70.0015 70.0214 70.0986 70.0097 70.0513 70.0232 dpH 70.0946 70.0014 70.0216 70.0939 70.0069 70.0340 70.0232 10.0 42 ^ p 0.0388 0.2058 0.2416 0.1936 0.1824 0.1103 0.2324 0.2657 1.3155 1.6205 dpj 70.0870 70.0013 70.0196 70.0875 70.0079 70.0467 70.0205 dpH 70.0820 70.0013 70.0208 70.0829 70.0066 70.0389 70.0204 Y A ¼ Y A1 þ Y A2 and perform a separate experiment amount SNH(0) and the consumed oxygen expressed as for calibrating YA2, as will be shown later. For NOD, where YA is the yield factor of the two steps SNH ð0Þ ¼ 2:5 mg LÀ1 , the proportional sensitivities of combined and a fourth experiment with SNH ð0Þ ¼ the second step parameters RNO and KNO denote poor 1 mg LÀ1 was added for improved accuracy. The identiﬁability. This effect decreases as SNH(0) increases, resulting value Y A ¼ 0:245 is very close to 0.24, which but still underlines the critical identiﬁability of the is generally indicated as the typical yield factor for second step parameters. It should also be noticed that autotrophs (Orhon and Artan, 1994; Henze et al., 2000; the shape of the RNO trajectory is largely inﬂuenced by Petersen, 2000; Sin, 2004). SNH(0), exhibiting a large peak at the beginning of the second step for SNH(0) ¼ 10 mg LÀ1. This indicates that 2.5. Respiration on nitrite the transition between the ﬁrst and the second step is a critical event for parameter estimation and it will be The estimation of the second step has always posed investigated further with a speciﬁc nitrite experiment. special problems and in the past the estimation of both steps was achieved by introducing selective inhibitors 2.3. Evaluation of design criteria (Nowak et al., 1995; Surmacz-Gorska et al., 1996) such as Allylthiourea (ATU) or sodium chloride. However, The OED indicators of Table 1 were evaluated for the the use of inhibitors, in addition to requiring a sample of calibrated model of Fig. 1 and the results of Table 3 were fresh biomass for each experiment, is not advisable for obtained. In agreement with the conﬁdence intervals of their lack of selectivity. On another front, Ficara et al. Table 2, the uncertainty affecting the estimates of RNO (2002) characterize the two steps by measuring pH or and KNO is not monotonic and this reﬂects into the OED performing two separate experiments, if only DO criteria of Table 3, save for mod A, which is not affected measurements are considered. To investigate further by FIM ill-conditioning (Petersen, 2000) and shows a the dynamics of the second step, a nitrite respirogram monotonic increase with SNH(0). In fact it was found that with SNO ð0Þ ¼ 2:7 mg LÀ1 was performed. According to if RNO and KNO are not estimated all the OED indexes Eq. (3) the pertinent model should be exhibit a monotonic behaviour. Further, all the OED SNO criteria were only slightly affected by data decimation. rNO ¼ ð1:14 À Y A2 ÞRNO þ rend . (10) K NO þ SNO 2.4. Estimation of the yield factors This, however, does not take into account the gradual response of the nitrite oxidisers observed by Vanrolle- Fig. 3 shows the ﬁtted regression line nitrogen oxygen ghem et al. (1998) and Vanrolleghem et al. (2004). To demand (NOD) ¼ (4.57ÀYA) SNH(0) between the initial account for this biochemical fact, an exponential term is
- 10. ARTICLE IN PRESS 3692 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 SNH (0) = 2.5 mg L-1 SNH (0) = 5 mg L-1 1°step 2°step 1°step 2°step 4 4 0.05 0.15 KNH 2 RNO KNO 2 0.1 KNH 0 0 RNO YA1 0.05 KNO -2 YA1 0 -4 -0.05 0 -2 -6 -0.05 -8 -0.1 -4 -0.1 -10 RNH YA2 RNH YA2 -12 -0.15 -6 -0.15 -14 -8 -0.2 -16 -0.2 0 10 20 30 0 10 20 30 10 20 40 60 10 20 40 60 time (min) time (min) time (min) time (min) SNH (0) = 10 mg L-1 1°step 2°step 5 2 kNH KNO 0 0 YA1 YA2 -2 -5 -4 -10 -6 -15 -8 RNH RNO -20 -10 -25 -12 0 50 100 0 50 100 time (min) time (min) Fig. 2. Sensitivity trajectories of the respirometric model Eq. (3) for the three values of SNH(0) over the two steps. The solid lines refer to Y’s, the dotted lines to R’s and the dashed lines to K’s. introduced to model the initial time lag t. From the identiﬁcation viewpoint, the initial rising data are incompatible with the Monod term, which SNO cannot ﬁt the OUR initial increase. It is therefore fair to rNO ¼ ð1 À eÀt=t Þð1:14 À Y A2 ÞRNO þ rend . K NO þ SNO omit them and calibrate the model Eq. (10) only with the (11) data which it can explain. The technique of eliminating
- 11. ARTICLE IN PRESS N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3693 Table 3 Optimal experiment design criteria evaluated for the three respirograms. The FIM results are in italic, the Hessian in bold SNH(0) A mod A D E mod E 2.5 8.6480 Â 101 1.9961 Â 106 7.1748e Â 1012 1.3325 Â 10À2 1.4888 Â 108 1.2432 Â 101 1.9859 Â 106 7.9761 Â 1014 8.5083 Â 10À2 2.3194 Â 107 5 3.5347 Â 105 4.7588 Â 106 7.4034 Â 109 2.8290 Â 10À6 1.6758 Â 1012 1.2983 Â 101 4.6728 Â 106 3.5911 Â 1014 9.7351 Â 10À2 4.7818 Â 107 10 4.4101 Â 10À3 1.5971 Â 107 2.9120 Â 1027 2.6742 Â 102 5.3342 Â 104 3.8515 Â 10À3 1.6782 Â 107 4.9547 Â 1027 3.0053 Â 102 4.9163 Â 104 50 High High NOD= 4.325SNH (0) sensitivity to sensitivity to YA = 4.57−4.325=0.245 τ KNO , RNO 40 R2 = 0.9992 NOD (mgO2 L-1) rNO 30 0 20 YA2 -0.5 10 -1 0 0 5 10 15 20 25 0 2 4 6 8 10 12 1 SNH(0) (mg NH4+− N L-1 ) RNO 0 Fig. 3. Computation of the combined yield factor YA from respirograms. For improved accuracy a fourth experiment with -1 0 5 10 15 20 25 SNH ð0Þ ¼ 1 mg LÀ1 was considered. 1 KNO 0 the initial respiration data has been frequently used in -1 the literature (Vanrolleghem et al., 1995; Vanrolleghem 0 5 10 15 20 25 and Coen, 1995; Vanrolleghem and Keesman, 1996; 0 - Ossenbruggen et al., 1996; Ubay C okgor et al., 1998; ¨ Mathieu and Etienne, 2000) in estimating a Monod -0.5 τ kinetics. Conversely, Eq. (11), including both start-up -1 and nitriﬁcation, can explain all the experimental data 0 5 10 15 20 25 and the introduction of the exponential term in Eq. (11) time (min) is fully supported by experimental evidence and bio- Fig. 4. Sensitivity analysis of the nitrite respirogram. The chemical theory (Vanrolleghem et al., 1998; Petersen, dotted line represents the reference rNO trajectory. 2000; Baeza et al., 2002; Vanrolleghem et al., 2004). Further, this additional term does not pose special estimation problems given the sensitivity separation with correlation matrix R the other parameters YA2, RNO or KNO. In fact the trajectories of Fig. 4 indicate two high-sensitivity zones: Y A2 RNO K NO t at the beginning of the experiment, both the time lag t 2 3 1:0000 À0:1859 À0:2803 À0:4453 Y A2 and the yield factor YA2 are the most sensitive R ¼ 6 À0:1859 6 1:0000 0:8912 0:3143 77 RNO parameters, whereas in the last part of the respirogram 6 7 4 À0:2803 0:8912 1:0000 0:2863 5 K NO the two most critical parameters are RNO and KNO. The introduction of the time lag allows the preservation À0:4453 0:3143 0:2863 1:000 t of the biological meaning of YA2, RNO or KNO. The (12) relative ﬁt of the two models is shown in Fig. 5 and the estimated parameters of the two models are that apart from the couple (RNO, KNO), all other compared in Table 4. It is evident from the parameter correlations are moderate.
- 12. ARTICLE IN PRESS 3694 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 The primary aim of this experiment is the direct Y A1 þ Y A2 with the area method of Section 2.4 and then estimation of YA2, and in this sense model Eq. (11) is perform a separate nitrite respirogram. deﬁnitely superior to Eq. (10), in which the large estimated value for YA2 can be explained observing that the oxygen consumption is given by NOD ¼ (1.14ÀYA2) 3. Conclusion SNO(0). If the respirogram area decreases, as a consequence of discarding the initial four data, NOD This paper has advanced the results of a previous decreases and therefore the right-hand-side must de- study to obtain an estimation reliability test, which was crease. Since SNO(0) is constant, this can only be applied to a respirometric model to assess its estimates accomplished by increasing YA2, which leads to the and design efﬁcient experiments. A test (Eq. (9)) is observed error. This value is clearly unfeasible, being presented, based on the computation of the approximate larger than the sum of YA1 and YA2 of Table 2 for conﬁdence regions, from which a discriminating thresh- SNH ð0Þ ¼ 10 mg LÀ1 whereas its conﬁdence interval is old is derived: if the ratio of the curvature radii is below unrealistically narrow, indicating a great conﬁdence in a a given value, then the estimation can be considered wrong estimate. On the other hand, the parameter reliable in the sense that the estimation procedure ended values of Eq. (11) are in good agreement with those of in a well-behaved region of the error functional, Table 2 for SNH ð0Þ ¼ 10 mg LÀ1 and are consistent with corresponding to acceptable parameters. This test is the literature values (Orhon and Artan, 1994; Petersen, based on the effect of curvature on parameter estimation 2000). They conﬁrm, however, the large inherent and detects possible obstacles preventing the successful uncertainty, about 36%, in the estimation of YA2, in termination of the search, but is not inﬂuenced by model line with the results found by Chandran and Smets inadequacy and residual characteristics, therefore it (2001) by an indirect method based on electron balance. should not be directly used to discriminate among Respirometric experiments based on titrimetric and off- model structures. In the paper this test has been used to gas analysis (Gapes et al., 2003) report a similar range assess the role played by the initial amount of (0.02–0.07). As expected, both models satisfy the ammonium-N SNH(0) in identifying a two-step respiro- curvature criterion Eq. (9), conﬁrming that this test is metric model and the effect of decimating the data, capable of detecting only error in the optimisation and which may produce more accurate estimates. Increasing should not be used to discriminate among model the amount of SNH(0) produces a general decrease in the structures. From these experiments it appears that the uncertainty of the estimated parameters. However, it is best way to estimate YA2 is to compute the sum Y A ¼ necessary to take into account other factors which limit this quantity: (1) the model does not consider biomass growth (as in Nowak et al., 1995; Brun et al., 2002) and therefore the experiment must have a rather short Model Eq. (11) Model Eq. (10) duration; (2) the nitriﬁcation process is very sensitive Data not used in calibrating Model Eq. (10) to several factors: not only pH and temperature, but also Data used in calibrating both models Eqs. (10) and (11) a high ammonium-N concentration as an inhibiting 0.45 agent. This conﬂicts with the requirement of a high rNO (mg L-1 min-1) SNO (O)= 2.7 mg L1 SNH(0) for good identiﬁability, though a high SNH(0) 0.4 produces more data, which increase the estimation 0.35 accuracy. Therefore one of the aims of the paper has been that of determining the best compromise for 0.3 producing a well-identiﬁable two-step respirogram with- rend 0.25 out sacriﬁcing either the accuracy of the ﬁrst step 0 5 10 15 20 25 30 with an insufﬁcient SNH(0) or inhibiting the process time (min) with a too large SNH(0). It was also shown that a Fig. 5. Calibration of the second step with nitrite data. The nitrite experiment could be used to estimate the second initial transient data (hollow circles) were not used in the step alone, adding an exponential term to take into calibration of model Eq. (10). account the gradual nitrite uptake by the biomass. The Table 4 Estimated parameters of the nitrite respiration models Y A2 RNO K NO rend t rmax =rmin r ¯ Model Eq. (10) 0.361570.0155 0.23670.007 0.14570.0138 0.263 — 1.1330 1.9891 Model Eq. (11) 0.063770.0233 0.166570.066 0.092170.0428 0.263 1.069470.3425 1.2358 1.9430
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