A simple eutrophication model for the Bay of Keszthely,

G. Jolânkai and A. Szôllôsi-Nagy ,

Abstract. Lake Balaton and especially its Bay at Keszthely is facing the danger of increasing eutrophication. Utilizing former plant nutrient discharge studies and records and also the records of lake hydrology, hydrobiology and water chemistry, a simple algae-dissolved reactive phosphorus model has been constructed to describe the eutrophication process. The model is mainly based on the specific condition that in Lake Balaton the main direct source of phosphorus is the uptake from the the sediment with the indirect source being the input phosphorus load. The crucial point of model­ ling is the determination of the time varying (or rather varying as a function of meteorological factors) sediment phosphorus uptake coefficient. Consequently a special parameter estimation technique was required. The dynamics of the model are nonlinear. On the other hand the measure­ ment equation is linear with a white noise Gaussian measurement error sequence. To identify the unknown constants and the time varying parameters, the output error method was adopted. A recursive algorithm based on the maximum likelihood approach is developed for determining the unknown parameters and noise statistics. The algorithm utilizes the sensitivity matrix equation for the parameters.

Un modèle simple d'eutrophisation pour le baie de Keszthely, lac Balaton Résumé. Le lac Balaton et surtout sa baie à Keszthely est soumis à un danger d'eutrophisation. Un modèle simple algues-phosphore réactif dissous a été construit sur la base des données collectées sur l'hydrologie, l'hydrobiologie, la chimie des eaux et sur les études existantes sur les charges en nutrient du lac. Le modèle tient compte de la circonstance spéciale que le phosphore disponible pour les algues provient premièrement et directement des sédiments du fond et indirectement de la charge extérieure. Le point crucial de la modélisation est la détermination du coefficient de l'absorption du phosphore issu du fond, qui varie selon le temps (ou bien en fonction des paramètres météorologiques). On a eu besoin donc d'une technique spéciale pour l'estimation des paramètres. Le modèle est dynamique et non linéaire, tandis que l'équation de mesure (des variables d'état) est linéaire avec une série d'erreurs de bruit blanc dont la distribution est normale. L'identification des constantes inconnues et des paramètres variables dans le temps a été effectuée par la méthode de l'erreur 'output'. Un algorithme récursif a été élaboré sur la base de la méthode maximum-vraisemblance pour l'estimation des paramètres inconnus et de la statistique de bruit. L'algorithme utilise l'équation matricielle de sensibilité pour les paramètres.

INTRODUCTION In spite of the fact that according to the relevant literature the mathematical model­ ling of the eutrophication process is in a fairly well developed stage, at least as far as the theoretical approaches are concerned, the practical application of these models raises many additional problems. These can be divided into two main groups: (1) The desired level of model complexity is restricted by the available data base. Consequently where reliable records for some of the elements of nutrient trans­ port or the food chain are not available then at this point the modelling becomes only a mathematical game. (2) The next crucial point is the determination of model parameters. The adaptation of literature values or the estimation of the ranges of parameter values can be decided only with the possession of widespread knowledge on hydrological, hydro- biological and chemical processes taking place in the lake (or part of the lake) in question.

137 138 G. Jolankai and A. Szôllosi-Nagy Based on their former water quality modelling experiences the authors believe, contrary to the opinion of many others, that the main model parameters can only be determined with reasonable accuracy from the use of actual field data. There are two possible ways of model parameter determination: (1) To carry out objective oriented field experiments directly designed to measure the effects of the individual parameters, and (2) parameter estimation techniques with the utilization of available measure­ ment data. Using hydrological, nutrient input, chemical and hydrobiological data available for the Bay of Keszthely, which is the most endangered part of Lake Balaton, and also the results of earlier experimental and research work, a simple eutrophication model will be presented below. The basic principles of the model formulation were: The theoretical model selected to describe the main processes of nutrient transport and algae growth is calibrated by up-to-date parameter estimation techniques using the available records of about seven years in length, in such a manner that at a later model­ ling stage the dependence of model parameters on other parameters could also be taken into consideration.

The problem setting Lake Balaton, and especially its Bay at Keszthely, is facing the problem of eutro­ phication. To control this process first some basic questions should be answered: (1) What are the causes of this process, that is (a) what parameter controls this process, in other words what is the limiting factor in the eutrophication process, and (b) what are the sources of this pollution? (2) What are the main processes of eutrophication the modelling of which could be reasonably well performed using the available data base? With regard to the first question,most authors (see references) agree that the eutro­ phication process of Lake Balaton is phosphorus limited. The sources and extent of phosphorus mass flux into the lake have been discussed elsewhere (Jolankai, 1977). The main phosphorus source of the lake and especially of the Bay at Keszthely is its tributary the River (see Fig. 1). Since the lake receives about 40 per cent of the pollution via the Zala River that discharges into Keszthely Bay it is obvious that (1) Keszthely Bay is in the worst stage of degradation (highest eutrophic level) and (2) the eutrophication control of this Bay is a task of primary importance. Consequently the modelling of the eutrophication process in Keszthely Bay can be considered as the first necessary step. (Other parts of the lake still have water of fairly good quality.) Concerning the second question, a true description of the eutrophication process would include the modelling of the highly complex relationships of the aquatic ecosystem. For this purpose several models of different complexity levels are offered by the relevant literature starting with simple phosphorus-algae models and ending with multi parameter ecological models that include almost all aquatic life forms and water quality parameters (Park et al, 1974; Orlob and Chen, 1972; J^rgensen, 1975). However, from the practical point of view a reasonably applicable model should be based on the available data base and should at the same time focus on the main pro­ cesses. Though the results of several individual measurements and field experiments carried out in Keszthely Bay are available, sufficiently continuous and lengthy records A simple eutrophication model for the Bay of Keszthely 139

FIGURE 1. Lake Balaton and the Bay of Keszthely.

are fairly rare. However, records on algal biomass and various phosphorus forms are available. The characteristic features of the eutrophication process in Keszthely Bay are briefly described below, based on the very exhaustive work of Olâh et al. (1977) on the phosphorus transport processes of Lake Balaton. (1) Contrary to most of the other lakes cited in the relevant literature, dissolved reactive phosphorus (P04-P) in Lake Balaton has a 'bypassing' way of transport from its original source (inputs) towards the utilizing algae, namely : about 72 per cent of the received amount of P04-P load immediately precipitates in water that has a high CaCO3 content and gets adsorbed in the bottom sediment. Consequently the main direct source of phosphorus is the exchange process through the sediment—water interface. (2) The concentration of P04-P in the water has not changed significantly in the past and maintains an equilibrium level of about 0.003 mg/1. However, in the last few years higher concentrations have been observed in Keszthely Bay and the equilibrium concentration is much higher. In recent years, most likely due to the greater availability of phosphorus, algal growth has significantly increased and frequent algal blooms have been observed in Keszthely Bay. The only obvious cause of these apparent changes is the increasing phosphorus load. The main task is then to describe the interrelationships between phosphorus load and algal growth by an appropriate mathematical model, the use of which could enable planners to define the probable effects of eutrophication control measures.

THE NONLINEAR DYNAMIC LAKE MODEL Based on the foregoing considerations in respect of the phosphorus transport mechanism of the lake the simplest possible model that could describe the real situation with reasonable accuracy is a phosphorus-algae model that takes the sediment-water phosphorus exchange into consideration. However, contrary to the common practice of modelling the total phosphorus mechanism, the mechanism of the dissolved reactive phosphorus (P04-P) regime should be modelled, since according to experimental results in Hungary (Dobolyi and Ôrdôg, 1978), this is the only phosphorus form available for algae. At the present level of data availability the modelling procedure should end with the algae since only scattered zooplankton data are available. It also follows from the above statement that only a lumped algal loss coefficient can be used at this end of the modelling approach. 140 G. Jolànkai and A. Szôllosi-Nagy

FIGURE 2. The model relationships.

The model includes the following processes (Fig. 2): (1) Phosphorus precipitation and settling (since in this specific situation the precipitation of P04-P is a fairly constant and immediate process, the model considers this phenomenon with a preliminary subdivision of the phosphorus input load values). (2) Algal growth limited only by the available phosphorus concentration (tem­ perature and light limitation can be introduced at a later stage of modelling if required). (3) P04-P release from the sediment. (4) Algae loss (due to grazing, mortality, settling, etc.). (5) Outflow of phosphorus from the Bay (the average outflow value calculated from the water balance is considered). The model equations describing the above dynamic processes are dP(t) 1 T if) + -7^7 K it)P {t) - m(t)A(t) - Q{t)P(t) At w Hit) f s à Ait) = liA{t)-Ka(t)A(t) (1) at

dPs(t) &Ts(f)-Kf(t)Ps(S) at P{t) M(0 = Mn Kp + Pit) where 3 no = P04 -P concentration in the water [g/m ] ; = 03 Tit) [gm"3month-1]; Twit) 3 -1 Tit) = the total PO4-P input into the Bay [grrf month ] ; Hit) = depth of the Bay; Q(t) = specific outflow rate from the Bay [1 ./month]; Kfit) = exchangable phosphorus release coefficient from the sediment f 1 ./month] ; 2 Ps{t) = areal concentration of exchangeable phosphorus in the sediment [g/m ]; = phosphorus fraction of algae or chlorophyll-a phosphorus yield coefficient (dimensionless); Ait) = algal biomass or chlorophyll-fl concentration [g/m3] ; M(0 = algal growth coefficient [1 ./month] ; A simple eutrophication model for the Bay of Keszthely 141

Ka (t) = lumped algal loss coefficient [1 ./month] ; 2 _1 Ts(t) = 0.7 x Hit) Tit) [gm" month ] ; the portion of the total phosphorus input that is precipitated and settled; |3 = the portion of the total settled sediment phosphorus that will be exchangeable through the sediment water interface; Mmax = maximum specific growth rate of algae [ 1 ./month], assumed constant ; Kp = the concentration of P04-P at which half of the maximum algal growth rate is achieved, assumed constant = 0.001 g/m3; t = time [months]. Since it is proven that the direct process that governs eutrophication is phosphorus release from the sediment, the crucial point of modelling will be the determination of the phosphorus release coefficient Kf(t). This uptake value varies with time or (presumably) with meteorological factors, mainly the wind velocity and the resulting fluid motions. On the basis of the detailed work of Olâh et al. (1977) the possible minimum and maximum phosphorus release values can be determined reasonably well. The main objective of the modelling work is then to find, within this range, the actual values oîKf(t) as a function of time. Since the final objective of any water quality modelling is to provide a predictive model from which control measures can be designed, the next step is to define the variation of unknown model parameters, Kfit) andA"a(f), by finding experimental formulae that describe this variation as a function of other parameters that affect the values of the model parameters. Consequently when a best fit of time varying para­ meters^^) z.nàKait) is obtained using available past records of algae, P04-P concen­ trations in the water and P04-P input load, the next step is to search relationships between parameter values and past records of meteorological and hydraulic para­ meters. This step will be discussed in another paper while here only the model construction and verification will be presented. Available data and model input parameters are as follows:

Pit) P04-P concentration values in the water (about a seven year long record with a monthly sampling frequency); Ait) algal biomass concentration (five-year record with varying sampling frequency); chlorophyll-a concentration (seven-year record with about monthly sampling frequency); Tit) PO4-P input load records (ten-year record with varying bimonthly-daily frequency). Psif) areal exchangeable phosphorus concentration in the sediment (no records, only some approximate average values are available).

Model limitations Main assumptions:

(1) Algae growth is mainly governed by the available P04-P concentration in the water. (2) Available phosphorus concentration is governed by the uptake from the sediment and is proportional to the sediment phosphorus concentration. Main items neglected: (1) Light and temperature limitations for growth of algae have been neglected. However by obtaining actual values of Kait) a possible correlation of Kait) versus temperature and light intensity could result in a reasonable involvement of these limiting factors. 142 G. Jolankai and A. Szôllosi-Nagy (2) The effects of zooplankton grazing, respiration and mortality have been lumped into a single parameter. This limitation can be resolved only by obtaining more data. (3) It is believed that phosphorus uptake from the sediment is also a function of the existing P04-P concentration in the water and also the pH. It is assumed, however, that the effects of the above processes are negligible compared to the effect of the phosphorus available in the, sediment, and the uptake rate affected by the wind effect. (4) A portion of the phosphorus consumed by algae is continuously recycled into an available form. However, it is thought and calculated that this recycled amount is negligible compared to the other sources of phosphorus. In the opposite case this process can also be considered by introducing a new unknown parameter. It is believed that the modelling of the monthly variation of the parameters is sufficiently accurate, since the most frequent measurements (except some scattered studies) are also of monthly frequency.

RECURSIVE IDENTIFICATION OF THE MODEL PARAMETERS There are several approaches used in the literature to estimate the states and para­ meters of dynamic lake models. Many of these techniques use an adjustable parameter estimation scheme as illustrated in Fig. 3. For example Di Cola et al. (1976) use a conjugate gradient type algorithm to estimate the parameters of a compartmental aquatic ecosystem. Although their estimation technique is dynamic in character it is valid only for strict deterministic systems and excludes the possibility of using it in a recursive mode which is of particular importance from the point of view of controlling the trophic state of a lake. On the other hand Gnauck et al. (1976) recently demon­ strated the use of real-time estimation techniques based on statistical methods in limnological modelling. While their technique allows for considering the inherent un­ certainties and for directly using the results for control purposes it does not, however, consider the internal dynamics of the compartmental variables involved. The esti­ mation procedure presented here aims at unifying the advantages of the techniques cited above and at the same time tries to avoid their disadvantages. Namely, it is an on-line recursive algorithm for the simultaneous identification of the unknown constant and time varying parameters, as well as of the partially non-measurable states of a nonlinear compartment-type lake system. The algorithm does consider the uncertainties superimposed on the measured variables.

r~ I PARAMETERS © I NONLINEAR 1 DYNAMICAL LAKE PROCESSES 1 I l_ __l \' ADJUSTMENT ALGORITHM

ADJUSABLE LAKE MODEL / ADJUSTED PARAM E TERS ©( FIGURE 3. Block diagram of an adjustable model. A simple eutrophication model for the Bay of Keszthely 143 The identification problem The basic problem here is the recursive identification of the unknown parameters T 0(t) = [a, (3,Kf(t),Ka(t)] of the nonlinear lake system described by (1). Following Imboden and Gàchter (1975) equation (1) will be denoted by the nonlinear vector differential equation dx(f) -~=f[x(t),0(t),u(t)] (2) at T where x(?) = [P(t),A(t), Ps(t)] is a three-dimensional state vector of the lake system, representing the solution of the set of first-order differential equations (2) for given T initial conditions x(t0) = x° and input functions u(t) = [Tw(t),H(t), Q{t), Ts(t)] . Since there are no direct measurements on the phosphorus in the sediment, Ps(t), the set of observed data available for identification is given by z(f) = y(O+v(0 (3)

where y (t) is the two-dimensional vector of the real P04-P and algal biomass concen­ tration and v(/) is a two-dimensional vector representing the inevitable measurement noise. With the above definition of the states the measurement equation becomes z(t)=Hx(t) + \(t) (4) where the measurement matrix is 1 0 0" H 0 1 0 The measurement noise \{t) is assumed to be a white Gaussian noise (WGN) process, orthogonal to x(/), with zero mean E {v

Parameter estimation criterion and the batched-form identification In choosing an estimation criterion for the above problem one has to keep in mind that the criterion should allow for: (i) easy implementation of a recursive algorithm, (ii) obtaining linear dependence between the parameter variations and the output error vector, (iii) simultaneously generating the noise covariance matrix. To meet the second requirement the output error method was adopted in this study because, taking into account the existing measurement errors, the main objective is to develop unbiased estimates for the parameters. Equation error methods yield biased estimates in the same case. Details on this issue are given in Mehra (1970a). Looking at all the above requirements the maximum likelihood (ML) function seems to be the best choice of criterion because : (i) maximization of the likelihood function leads to unbiased, consistent and minimum variance estimates (Âstrôm, 144 G. Jolânkai and A. Szôllosi-Nagy

MEASUREMENI NOISE

MEASUREMENT DYNAMICAL LAKE SYSTEM (REAL) MATRIX u(t) 1 in-i[x(n)(t),§{n)(t) u(t)] —lA^y ^"

ADJUSTABLE LAKE MODEL *- f[x(t), e (t), u(t)] lr=5 |_| =^)0O=ij

i 7S A0 PARAMETER ADJUSTMENT E OUTPUT ERROR iz.

RECURSIVE IDENTIFICATION ALGORITHM

FIGURE 4. The identification scheme of the adjustable nonlinear dynamic lake model.

1970) which are easy to obtain in a recursive way, (ii) the change in the parameter vector is linearly related to the output error vector and (iii) maximization of the likelihood function leads to the evaluation of the measurement noise covariance matrix. The identification scheme is of the adjustable type and in principle very much like the one depicted in Fig. 3. Figure 4 shows the conceptual setting of the system identification with the following notational modifications: (i) For the real (nominal) dynamic lake system, whose parameters are to be identified, the equations X(„)(0= f[X(„)(0, 0<„)(O,U(O] (6)

z(n)(0 = #x(„)(f) + v(0 (7) are used, where #(„)(/) are the nominal values of the (unknown) parameters, instead of (2) and (4). (ii) The adjustable model for a particular parameter vector 0 (t) is described by x(O=f[x(O,0(O,u(O] (8) z(t) = Hx(t) (9) where x(t) and z(t) are the best available model state and model output, respectively. The output error is defined by

«(0 = z(n)(r)-z(0 (10) = Hx(n)(t) + v(t)-z(t) For using the likelihood principle one has to find the best estimate of 6, which will be denoted by 0*, based on the measurement sequence Zk. The ML estimate of 6 is obtained by maximizing the conditional probability of Zk given 0, i.e.

0*= arc maxp(Zfc|0) (11) A simple eutrophication model for the Bay of Keszthely 145

With successive applications of Bayes rule an expression forp(Zfc (0) can be derived as

p(Zk\0)=p(z(l),...,z(k)\0)

= p(z(k)\Zk-i,0)p(Zk_l\e)

= p(.z(k)\Zk_1,o)p(z(k-l)\Zk_2,o)p(Zk_2\0)

= hp[zmZi-uo] (i2)

Since the logarithm is a monotonie function, the ML estimate can also be written as

r k 9* = arc max [In p(Zk |

^{z(OI^-_1,0)}=z(/|/-l) (14) and the covariance matrix

E{[z{i)-z{i\i-l)] [z(0—z(i|/-l)]r}=/î(i|/-l) (15) where the quantity v (z) = z(z') — z(i\i - 1) is called the innovation being the 'new piece of information' not available beforehand. Thus, denoting the determinant of the covariance matrix by \R (i | i — 1 ) |,

T p(z(j)\Z,_u 0) = —-j-2 exp [- V4 v (i)R-\i\i - 1) v (i)](16) therefore (13) becomes 0*=arc m?LxL(0,R) (17) where

L( 0,R) = constants — £ {vT(i)R~\i\i-1) v (i) + \n\R(i\i - 1)1} (18) 2 i=i and maxL(0,R)=L(0*,R)

The problem of determining the ML estimate has now become one of finding the conditional mean and error covariance. These quantities however, are precisely the output of a Kalman filter for given parameter vector 0 (t). [The reader may find details on the Kalman-filtering together with hydrological applications in Szôllosi-Nagy (1976) and Szôllosi-Nagy et al. (1977).] Unfortunately L(0, R) is not explicit in 0 or R and hence it is not possible to determine the vector 9 unless a proper optimization technique is used. Even in the case when R were known exactly the problem still remains involved since the ML estimate of 0 is obtained by maximizing (18) with respect to 0 subject to the constraints given by the Kalman filter. And this is indeed a very difficult optimization problem as was demonstrated by Mehra (1970b). 146 G. Jolénkai and A. Szôllosi-Nagy Instead of solving this problem, an iterative maximization procedure is suggested here for considering the dependence on 0 and R. Assume that there is given a trajec­ tory x^) corresponding to a certain value of the parameter vector 0^), considered as the best value at this stage, and the control u(f). In this case one can conclude that e (J) and v (/) are equivalent

a (0 = z(n) (0 - z (v)(i) = v (i) (19) z^) being the best current estimate of Z(„). The likelihood function then is

k I(0(„),*(„)) = const-14 X {«T(0*fo)(0«(0 + ln|tf(,,)0')l} (20) i = i where for brevity the error covariance matrix is written as R(/). Suppose now that, = being at the stage (17), an improved value of the parameter vector 0(n +1) 0(n) + A0(n), (Afl^) is a small perturbation) is determined to maximize the likelihood function. Accordingly, a new state trajectory X(n + j) is generated corresponding to the improved parameter vector 0(v + i)(t)- Define

Ax(r) = x(T) + 1)(f)-x(T))(0 which, by the definition of sensitivity (Tomovic, 1963)

5(Tj)(r) = Ax(f)/A0(T)), can be written as

Ax

V(TJ + 1)0') = Z(n)(0 - Z(n + 1)0') = Z(n)(0 ~ [Z(TJ)(0 + Az(z)]

= Z(„)(i) - z(rt)(i) - H Ax(t)

= Z(«)0) - Z(T,)0") - HS(n)(i) A0(r))

= e(i)-HS(n)(i)A0(v) (22) The 3x4 matrix S^) is the systems sensitivity matrix whose computation will be given in a later section. At the new stage the value of the likelihood function is a function of A0 and R, i.e.

T L( 6 + A0,R) = const - fc £ {[«(0 -HS^i) A0{ri)) R-\ï) [.] 1 = 1 + In |/? 0)1} (23) To maximize this function with respect to A0 and R the partial derivatives dL[8 + A0,R(i)] dL [0+ A0,R(i)] and 3A0 bR(i) are computed. Setting these derivatives equal to zero one obtains the following neces­ sary conditions for the maximization of the likelihood function:

T T T T 1 X iS (i) H R-'ii) HS(i)} A0*(t!) = X iS (i) H iT (i) 6 (i)] (24) i=i 1=1 A simple eutrophication model for the Bay of Keszthely 147 and k k T £ R(i) = £ {[«(0 - HS(i) A0fo] [s(i) -HS(f) A0(*}] (25) /=i i=i These equations allow one to establish a batch-iterative procedure to compute AOfa) andR(i). For real-time usage,however, these formulae are reformulated in a recursive fashion.

The recursive identification algorithm Defining A0% = A0^ and rewriting (24) for the interval [0, k] one obtains

£{.}A6J!+{ST(k)HTk-l(,kJHS(.k)}A0£= "t* [ST(f)HTA~l(f)*(J)] i=i /=l + [ST(k)HTR-1(k)e(k)] (26) Since (24) is valid for all k, it holds as well for the interval [0, k - 1 ], i.e.

T T k l T T î {S {i)H k-'{i)HS{i)}A0t_ï = ± [S (i)H R-\i)B(i)] (27) « = i ,- = i To establish a recursive algorithm the following condition is imposed .„* i^-i for(0,*-l) k ~\ <, (28) 'Stfjf for(*-l,Jfc) These conditions and (27) imply that (26) has the form

T 1 X {.)A0^-1+{S (k)HR- HS(k)}Ô0^= Ï UA0jf_i i=\ i=1 + [ST(k)HT R-\k)s(k)] from which the change ô0% in 0k for the interval {k - \,k) needed to identify the parameter vector is 801 = ^(^HR-'HSik)}-1 [ST(k)HTR-1(k)]e(k) (29) The final recursive ML parameter identification algorithm is then 0{k) = 0(k - 1) +A~\k)B(k) s (k) (30) where B(k) = ST(k)HTR-1(k) (31) is a 4 x 2 matrix and A(k) = ST(k)HTR-1(k)HS(k) (32) is a 4 x 4 matrix which is equivalent to the incremental Fisher information matrix for the unknown parameter vector. It is noted that if the Fisher matrix is singular then the parameters are not identifiable. Since ^{E(Z')} = 0, for i > 1, one obtains from (29) that E{S0t}= 0, i.e. the parameter vector identified by (30) is unbiased. A recursive algorithm for the noise covariance matrix can be obtained from (25) along the same lines. Since from the point of view of parameter identification the inverse error covariance matrix is of interest [c/(31), (32)] an algorithm for com- 148 G. Jolànkaî and A. Szôllosi-Nagy puting R~l is given below

k R~\k)- ÈT^k-l)- R-1(k^l)s(k)eT(k)R-1(k~l) (33) k-1 ~ ' (k-l)2 where the matrix inversion lemma was also utilized.

Computation of the sensitivity matrix As is known (Tomovic, 1963) it is possible to assign a so-called sensitivity system to any dynamic system. What is more interesting, however, is that this sensitivity system is linear even in the case of nonlinear dynamics. Defining the sensitivity function as

dxj(k) sif(k) = (34) dO;(k) which indicates the sensitivity of the rth state variable with regard to the/th para­ meter, one can show that the sensitivity system obeys a linear matrix difference equation (for continuous systems: differential equation)

3f(*-l) ôf(Jfc-l) (35) dx'ik-l) dO'ik-l) where 9f/9xr and df/907* are the Jacobians of the nonlinear lake model (2). The initial condition for (35) is 5(0) = 0. The Jacobians, for a given k, are as follows

Kn p 1 -Mn A~Q Mmax -Kr (Kp+py Kp+P 1 K„ P (36) ax 0 (.Kp+Pf Kp+P "

0 0 -Kf

Kn 1 K +P — P* o p H (37) d0T

0 0 and the sensitivity matrix is a 3 x 4 rectangular matrix. Summarizing the computational steps in the recursive identification process: (1) specification of the initial conditions, (2) solution of the sensitivity equation (35), (3) solution of the system equation (2), (4) calculation of the output error (10), (5) computation oîR'1 by (33), A and B by (32), (31), respectively, (6) computation of the new parameters (30), (7) repeat the computation for the next time interval starting at Step (2). A simple eutrophication model for the Bay of Keszthely 149

1971 ~T~ 1972 "• 1973 ' T97Â "" 1975 '~1 "'" l976~ "T" 1977 FIGURE 5. Measured and simulated values of the model components and parameters.

RESULTS AND DISCUSSION From Fig. 5 which shows the measured data of model variables and also the simulated changes off, A, mdPs concentrations, the following conclusions can be drawn:

(1) The measured equilibrium level of past P04-P concentrations together with the also insignificant changes of algae growth in the first part of the measurement period (1971—1973) could only be maintained with a continuous accumulation of exchangeable phosphorus in the sediment. Though there were no measurement data of Ps to prove this modelling conclusion it seems to be obvious and this conclusion meets the opinion of other authors in this field. (2) After some years of continuous phosphorus accumulation a sudden change in algae concentration has started and then with somewhat more moderate but still higher algae activity the phosphorus content in the water also began to rise. (3) Simultaneously with the increased algae activity and phosphorus content rise, the accumulation of exchangeable phosphorus in the sediment slowed down. (4) Though the simulated values follow the measured data fairly well, the model cannot simulate the sudden algae growth without its preceding higher phosphorus concentration in the water. It is probable, however, that these simulated changes follow the real 'but unobserved' situation, since algae growth can only result from rising phosphorus levels in the water. 150 G. Jolankai and A. Szôllosi-Nagy (5) Reviewing the real and simulated past trends one reaches the conclusion that algae growth is likely to increase periodically also in the future, probably even more drastically, unless other aquatic life forms take over the situation or management measures are introduced. (6) The simulation of future changes will be attempted in a future study on the

careful calibration of model parameters Kf and Ka.

Acknowledgement. The authors wish to express their sincere thanks to Mr Làszlô Tôth senior research engineer at VITUKI for his kind advice and for making measurement data available, as well as to Miss Veronika Major research engineer for preparing the computer programs.

REFERENCES

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