Estimation of trophic states in warm tropical lakes and reservoirs of Latin America by using GPSS simulation/Estimacion de los estados troficos en lagos y embalses calidos tropicales en Latinoamerica usando simulacion GPSS/Estimacao dos Estados troficos em lagos e represas calidos tropicais na America Latina usando simulacao GPSS.
This paper proposes a stochastic simulation model to determine the boundaries of the trophic states of warm-water tropical lakes and reservoirs in Latin America based on statistical correlation and MonteCarlo techniques. The model was developed using GPSS as a discrete simulation language and calibrated by correlating a set of state variables of 27 Latin American lakes and reservoirs monitored by the Pan American Center of Sanitary Engineering and Environmental Sciences (CEPIS). In order to warrant a better stability in the resultant probabilistic behavior of the dependent variable, 10000 new virtual water bodies with different trophic states were generated to produce atrophic state index based on the total phosphorus concentration. Based on the obtained results, it is concluded that the applied methodology is appropriate to determine the boundaries of the trophic states of warm-water tropical lakes and reservoirs and is also able to generate results similar to those obtained using the existing applied estimation techniques.
KEY WORDS / Trophic State / Eutrophication / GPSS / Simulation Models / Lakes / Reservoirs /
En este trabajo se propone un modelo de simulacion estocastica para determinar los limites de los estados troficos en lagos y embalses calidos tropicales en Latinoamerica, basada en una correlacion estadistica y en tecnicas MonteCarlo. El modelo se desarrollo en el lenguaje de discreto GPSS y fue calibrado con un conjunto variables de estado de 27 lagos y presas de America Latina monitoreados por el Centro Panamericano de Ingenieria Sanitaria (CEPIS). A fin de garantizar una mejor estabilidad en el comportamiento probabilstico de la variable dependiente a partir de las distribuciones muestrales de las variables predictivas, se generaron 10000 cuerpos de agua de los diferentes estados troficos y se produjo un indice de estado trofico basado en la concentracion del fosforo total. Sobre la base de los resultados obtenidos se concluye que la metodologia es apropiada para estimar los limites entre los estados troficos de lagos y embalses y produce resultados similares a los obtenidos por otras metodologias.
Neste trabalho se propoe um modelo de simulacao estocastica para determinar os limites dos estados troficos em lagos e represas calidos tropicais em Latino America, baseada em uma correlacao estatstica e em tecnicas de MonteCarlo. O modelo se desenolveu na linguagem de simulacao discreta GPSS e foi calibrado com um conjunto de variaveis de estado de 27 lagos e represas de America Latina monitorizada pelo Centro Panamericano de Engenharia Sanitaria (CEPIS). Com o fim de garantir uma melhor estabilidade no comportamento probabilstico da variavel dependente a partir das distribuicoes amostrais das variaveis preditivas, foram gerados 10.000 corpos de agua dos diferentes estados troficos e se produziu um indice de estado trofico baseado na concentracao do fosforo total Sobre a base dos resultados obtidos se conclui que a metodologa e apropriada para estimar os limites entre os estados troficos de lagos e represas e produz resultados similares aos obtidos por outras metodologas.
In the last 50 years the extensive construction of dams and reservoirs in Latin America has produced a large number of artificial water bodies that have interfered with the hydrology and ecology of several basins, sub-basins and rivers. Most dams were built initially to generate electricity, but were later also used for other purposes, like fishery, irrigation, transportation, water supply source, sports, and recreation. Artificial and natural fresh water bodies have suffered under nutrient contamination, especially nitrogen and phosphorus, originated from point and diffuse sources, mainly municipal and industrial sewages and agriculture runoff. When an excessive amount of nutrients enter to the system, contaminating the water body, their elimination is technically difficult and costly (DVWK, 1988; UNEP, 1999). The quantity of organic matter in aquatic ecosystems defines their trophic state, which serves as an indicator of the contamination grade of the system. The trophic state depends on several biotic and abiotic factors. One of this factors is the in-lake total P concentration, which compared to other water quality parameters, could be considered as the most studied indicator to determine the trophic state in lakes and reservoirs (Konesky et al., 1999). In freshwater bodies the total P is widely accepted as the limiting nutrient, and controlling the total P inputs is regarded as essential in reversing freshwater eutrophication (OECD, 1982; Hecky and Kilham, 1988). There are numerous case studies of the reversal of eutrophication processes resulting from the reduction of P loadings (Willander and Personn, 2001).
The OECD (1982) determined the boundary values between oligotrophic and mesotrophic states for temperate lakes as 0.008mg x [l.sup.-1] P, and the boundary values between mesotrophic and eutrophic states was fixed at 0.267mg x [l.sup.-1] P. But these boundaries values are not applicable to warm tropical lakes and reservoirs in Latin America (Castagnino, 1983). Salas (2003) proposed a methodology to assess the eutrophication grade in warm tropical lakes, and a new alternative is proposed herein to improve the assessment process by using a simulation model.
Due to the requirement of a higher grade of mathematical complexity, few environmentalists use simulation techniques to model ecological problems. For example, Romeu (1995, 1997) briefly outlines a GPSS simulation model of an aquatic ecosystem. Some statisticians are already actively working in ecological problems using GPSS which are regularly publishing their work in refereed journals (Baltzer, 1996). GPSS is a consolidated discrete simulation language focused on the flow of transactions and is usually applied to simulate manufacturing processes (Chisman, 1992). A GPSS simulation consists of a network of blocks, which represents actions linked with a set of transactions, arranged sequentially in different blocks. Therefore the whole simulation is simply a sequence of one transaction entering to one of more blocks, then a new transaction starts, and so on. Modeling a real system is to piece together a set of blocks that cause transactions to behave in a manner resembling the real system (Minuteman Software, 2000).
A simulation model is a representation of the behavior of a set of variables in a system, performed with a computer and reflects, in a single way, the activities in which the variables are related. The simulated system in this work it is not the physical system itself, but it is a system which "generates" a great number of virtual lakes and reservoirs having similar characteristics to that encountered in the Latin American region with the objective of establishing the total P concentration limits between their different trophic states. To cover the goals of this work, the "transaction" is a virtual water body that flows through the system, while a block is defined as any operation that performs that transaction within the simulation model, such as its own generation (GENERATE), its transfer toward a specific block (TRANSFER), assignment of a value to one of its parameters (ASSIGN), evaluation of an arithmetic condition to modify its flow (TEST), and its own 'destruction' (TERMINATE). Since transactions take up memory space, all transactions that are no longer needed in the model should be removed in a TERMINATE block. Additionally, the system requires some control instructions such as definition of variables having floating point (FVARIABLE), definition of initiators to generate random numbers (RMULT), start up of the simulation processes (START), etc.
The data requirements of a GPSS simulation model include both parameters and variables. The parameters include particular values of the predictive variables: mean depth, P load and residence time of every water body and of every trophic state, in addition to the predicted total P value calculated from a regression model in function of the three previously mentioned variables. Variables are divided in input and output variables, and both sets of variables are classified as statistical ones.
Input variables include the probabilistic distributions of each one of the predictive variables corresponding to each trophic state, and are obtained from a set of 27 lakes and reservoirs located in the studied region. The output variables include the probabilistic distribution of the total P concentration and the trophic state of each one of the simulated water bodies.
Table I shows a database from 27 Latin American lakes and reservoirs collected by the Pan American Center of Sanitary Engineering and Environmental Sciences (CEPIS, 2001). The information in the database was obtained on the basis of the Clark method to quantify the nutrient load from the tributary streams (Sonzogni et al., 1978), the nutrient export coefficients developed by Rast and Lee (1978), the contribution of living beings considered by Castagnino (1982), and the laboratory APHA standards (13th to 16th eds.). For the trophic state classification of the lakes in Table I, the criteria applied by the Organization for Economic Cooperation and Development (OECD; Vollenweider and Kerekes, 1982, 1983; Salas, 2003) were used. However, the validity of these classification criteria is under discussion (Kietpawpan et al., 2003). According to these authors, out of the total lakes and reservoirs included in Table I, 55.88% would be classified as eutrophic, 29.41% as oligotrophic, and the remainder is mesotrophic.
This work also considered P as the limiting nutrient in Latin American lakes and reservoirs (Salas, 2003). A Kolmogorov-Smirnov test was applied to the TP, Z, Lp, and Tw values showed in Table I. It was found that only their respective natural logarithms underwent a normal behavior (P>0.40). If TP', Z', Lp' and Tw' are the natural logarithms of TP, Z, Lp, and Tw, respectively, it is possible to adjust the next multiple linear regression model to the data showed in Table I, so that
TP' = [[beta].sub.0]+[[beta].sub.1] Z'+[[beta].sub.2]Lp'+[[beta].sub.3]Tw' (1)
where, [[beta].sub.0], [[beta].sub.1], [[beta].sub.2], [[beta].sub.3] are the regression coefficients corresponding to the ordinate at the origin, Z', Lp', and Tw', respectively. The model could be represented as a matrix form as follows:
TP' = Xb+e (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
and [TP'.sub.i], [Z'.sub.i], [Lp'.sub.i], [Tw'.sub.i] (i = 1, 2, ..., 39) are the values of the state variables shown in Table I and [[epsilon].sub.i], (i = 1, 2, ..., 39) is the error term of [TP'.sub.i]. For the data reported in Table I, it could be verified that MSE = 0.1222 and the variance-covariance matrix [(X'X).sup.-1] is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
And, therefore the vector of least squares estimates is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Finally, the multiple linear model in terms of natural logarithm of total phosphorus (TP') is
TP'=-1.237-0.934Z'+0.891Lp'+0.676Tw' (6) [R.sup.2]=0.902; SE=0.056)
If [Z'.sub.0], [Lp'.sub.0] and [Tw'.sub.0] are the particular values of Z', Lp' and Tw' respectively, the standard deviation of the mean value of TP' can be written (Younger, 1979) as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Table II shows the descriptive statistic values of the state variables of the water bodies reported in Table I for each trophic classification.
Because Volenweider and Kerkes (1981) assumed homogeneous standard deviations of TP' for the different trophic status, the standard deviation of TP' was calculated by applying Eq. 6 regardless of the trophic state. Table III shows the programming code of the simulation model carried out on GPSS World Version 4.3 (Minuteman Software, 2000). The goal of the simulation model is to simulate a number of water bodies in each trophic state where the state variables undergo random variations in relationship with their population means. In this way it is possible to warrant a better stability in the probabilistic behavior of the dependent variable from the distributions curves of the predictive variables obtained in the sample of 27 warm tropical lakes and reservoirs. According to the Kolmogorov-Smirnov test applied to the selected sample, if is assumed that such variations follow a normal distribution in relation to the means of the standard deviations reported in Table II. Additionally, it is also assumed that TP' follows a normal distribution in relationship with the mean and standard deviations given by Eqs. 6 and 7, respectively.
To add more realism to the simulation process, only positives values are used for Ks, which was obtained from the mass balance equation proposed by Vollenweider (1976),
TP = Lp/Z([1/Tw]+Ks) (8)
so that all the transactions (lakes and reservoirs) that did not satisfy the restriction
Lp/ZP[lambda] > 1/Tw (9)
Results and Discussion
Figure 1 shows the results (outputs) of the simulation model as a probability distribution function of natural logarithm of total phosphorus (TP') for each trophic level. The distribution curves are very close to the normal distribution pattern, and they have the same variance.
To validate the simulation model, the Z test is used to compare the mean values of TP' of the simulation model with the mean values of the real data. Non-significant differences between the compared means (P>0.32) were encountered. Table 1V indicates the estimated 95% of the confidence intervals for the TP' means of each trophic state based on the simulated data.
The limit between the oligotrophic and mesotrophic levels was obtained by averaging the upper limit of the 95% confidence interval for the TP' mean of the oligotrophic state and the lower limit of the 95% confidence interval for the TP' mean of the mesotrophic state. Similarly, the limit between the mesotrophic and eutrophic levels was obtained by averaging the upper limit of the 95% confidence interval for the TP' mean of the mesotrophic state and the lower limit of the 95% confidence interval for the TP' mean of the eutrophic state. According with Table V, the limit between the oligotrophic and mesotrophic states was established as the antilog (-3.22) = 0.04mg x [l.sup-1] P and the limit between the mesotrophic and eutrophic states was established as the antilog (-2.27) = 0.10mg x [l.sup-1] P. Vollenweider (1968) established the limits for temperate lakes as 0.01 and 0.03 mg x [l.sup-1] P, respectively.
The proposed GPSS simulation model seems to represent a useful technique to predict the trophic condition of warm tropical lakes and reservoirs. The results are not significantly different from those reported by CEPIS (2001). Utilizing the data reported on Table I, they fixed the limit between the oligotrophic and mesotrophic states as 0.03mg x [l.sup-1] and the limit between the mesotrophic and eutrophic states as 0.07 mg x [l.sup-1] P, respectively. In contrast, the USEPA (1974) fixed these limits as 0.01 and 0.02mg x [l.sup-1] P, and the OECD as 0.008 and 0.0267mg x [l.sup-1] P, respectively (Vollenweider and Kerekes, 1982).
The boundaries values predicted in this work for the trophic states in Latin American warm tropical lakes were 4 times higher than the corresponding values estimated in the temperate lakes located in the north hemisphere; nevertheless, they show the same trophic features. According with the OECD and USEPA criteria, a hypertrophic temperate lake has the same total P concentration than a warm tropical mesotrophic lake. This demonstrates that OECD criteria are largely inapplicable to warm tropical lakes. This is because the relative distributions of nutrient N and P in Latin American lakes and those of the OECD study are different in several ways. The main limnological feature to the OECD findings is that P is the nutrient limiting algal growth, because N usually is massively in excess of algal requirements. This is not true in Latin America, where nitrogenous material in fresh waters is much less abundant (Lewis, 2002). In many lakes of this region the relative N to P availability approximates that required for a balanced algal growth and, in some of them, shortage of N limits growth. Water managers aiming to control eutrophication in Latn America lakes are advised to use the OECD predictive equations with utmost caution.
Conclusions and Recommendations
Due to shortage of available trophic data in the Latin American region, a GPSS simulation technique is proposed for determining the trophic status of warm tropical lakes and reservoirs. This technique was applied to obtain a more realistic image of the trophic state of these water bodies and to predict it. The stochastic GPSS simulation models have some advantages when compared to the classical deterministic models: while the sampling of a water body permits to analyze just one picture of the film, simulation permits to have a wider vision of the reality. The simulation models are easier to understand and visualize than the pure analytical methods. They add more realism to the analysis because they introduce variability to the parameters describing the water body and they are cheaper to use because they requite less information, time and money to be developed. Simulation models like this can incorporate mathematical descriptions of physical, chemical and biological processes in lakes and reservoirs. If properly designed, these models can assist with management decisions that require considering alternative scenarios. The present study should encourage the practical uses of discrete event simulation in the area of ecological modeling and analysis.
Although this procedure provides a solution to the problem of determining the boundary values for trophic categories, its main disadvantage is its difficult application by managers and technicians having limited simulation modeling knowledge. The strength of this method is that it is strongly supported by the probabilistic distributions of the predictive variables of the water bodies in each trophic state and its effect in the variability of the dependent variable, which permits a better understanding of the trophic states of warm tropical lakes and reservoirs.
Abbreviations and Notations GPSS General Purpose Simulation System Z Average depth, m As Surface area, [km.sup.2] Tw Residence time, yr Lp Phosphorus load, g/[m.sup.2]yr Ks Overall loss rate of total phosphorus, [yr.sup.-1] TP Total phosphorus concentration, mg/l TP' Natural logarithm of TP Z' Natural logarithm of Z Lp' Natural logarithm of Lp Tw' Natural logarithm of Tw MSE Mean Square Error SD Standard Deviation y Column vector of the observed Tp' values (39x1) [beta] Column vector of the regression parameters (4x1) X Model specification matrix (39x4) [epsilon] Column vector o the random errors (39x1) [??] Vector of least squares estimates of [beta] (4x1)
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Received: 09/05/2005. Modified: 03/11/2005 Accepted: 04/11/2006.
Mario A. Ortiz-Jimenez. Chemical Engineer, Instituto Tecnologico de Tepic. Mexico. M.Sc. in Statistics, Colegio de Posgraduados. Mexico. Ph.D. Candidate at the Postgraduate Program in Science and Technology, Consejo Nacional de Ciencia y Tecnologia, Mexico. Professor, Instituto Tecnologico de Tepic. Mexico. email: email@example.com
Jose de Anda. Chemical Engineer, Universidad de Guadalajara, Mexico. M.Sc. in Chemical Engineering, Universidad Autonoma Metropolitana-Iztapalapa, Mexico. Ph. D. in Earth Sciences, Universidad Autonoma de Mexico. Researcher, Centro de Investigacion y Asistencia en Tecnologia y Diseno del Estado de Jalisco, A.C. Address: Normalistas 800, CP 44270 Guadalajara, Jalisco, Mexico. e-mail: firstname.lastname@example.org
Ulrich Maniak. Doctor-Engineer, Leichtweiss-Institut fur Wasserbau, Abteilung Hydrologie und Wasserwirtschaft, Braunschweig, Germany. Emeritus Professor, Leichtweiss-Institut fur Wasserbau, Germany. e-mail: U.Maniak@tu-bs.de
TABLE I MONITORED PARAMETERS OF SOME LATIN AMERICAN LAKES AND RESERVOIRS Lake/Reservoir Year Country Z Tw Salto Grande 1982 Argentina 7.80 0.097 Salto Grande 1981-83 Argentina 8.40 0.032 Descoberto 1980 Brazil 6.90 0.280 Paranoa 1980 Brazil 14.30 0.731 Funil 1978-79 Brazil 22.80 0.151 Funil 1987 Brazil 22.80 0.131 Funil 1988-89 Brazil 21.00 0.081 Vigario 1988-89 Brazil 9.80 0.008 Lajes 1988-89 Brazil 13.60 0.760 Americana 1982 Brazil 9.20 0.047 Americana 1986 Brazil 7.80 0.084 Atibainha 1986 Brazil 12.50 0.388 Barra Bonita 1978-79-80 Brazil 8.30 0.269 Barra Bonita 1983 Brazil 9.20 0.073 Barra Bonita 1982-84 Brazil 9.30 0.211 Barra Bonita 1986 Brazil 8.60 0.221 Cachoeira 1986 Brazil 10.70 0.130 Guarapiranga 1982-83 Brazil 4.90 0.238 Guarapiranga 1986 Brazil 4.90 0.331 Itupararanga 1986 Brazil 7.80 0.660 Jaguari 1986 Brazil 16.80 1.225 Paiva Castro 1982-83 Brazil 5.70 0.059 Paiva Castro 1986 Brazil 5.40 0.028 Paraibuna 1986 Brazil 26.40 1.923 Ponte Nova 1982-83 Brazil 8.90 0.615 Ponte Nova 1986 Brazil 8.30 0.806 Taiacupeba 1986 Brazil 2.20 0.135 Laguna de Sonso 1988 Columbia 1.00 0.041 Pozo Honda 1981 Ecuador 20.00 4.845 Pozo Honda 1982 Ecuador 20.14 1.574 Livingston 1975 USA 6.30 0.243 Chapala 1983-84 Mexico 4.20 11.050 Chapala 1986-87 Mexico 4.43 15.940 Tequesquitengo 1986 Mexico 16.00 98.500 Requena 1986-87 Mexico 5.00 0.260 La Plata 1981-82 Puerto Rico 10.00 0.090 Loiza 1973-74 Puerto Rico 6.14 0.054 Tortuguero 1974-75 Puerto Rico 1.20 0.133 Laguna Grande 1980 Venezuela 3.50 0.120 Lake/Reservoir Lp TP Classification Salto Grande 12.6000 0.082 E Salto Grande 14.5000 0.046 M Descoberto 0.6500 0.016 O Paranoa 2.9300 0.040 E Funil 9.1900 0.041 M Funil 18.6000 0.048 E Funil 29.3000 0.066 E Vigario 142.9000 0.084 E Lajes 0.7950 0.018 O Americana 37.7000 0.081 E Americana 31.6200 0.098 E Atibainha 1.2900 0.023 O Barra Bonita 3.4300 0.058 M Barra Bonita 23.0000 0.115 E Barra Bonita 7.4500 0.094 E Barra Bonita 6.9600 0.059 M Cachoeira 4.1400 0.032 O Guarapiranga 1.4000 0.052 E Guarapiranga 1.7000 0.044 M Itupararanga 2.1100 0.029 M Jaguari 3.2800 0.036 O Paiva Castro 7.3000 0.040 M Paiva Castro 5.4800 0.023 O Paraibuna 0.7900 0.016 O Ponte Nova 0.5500 0.027 O Ponte Nova 0.4700 0.025 O Taiaqupeba 0.6400 0.031 M Laguna de Sonso 6.5800 0.210 HE Pozo Honda 5.0600 0.400 HE Pozo Honda 15.1000 0.200 HE Livingston 14.6000 0.200 E Chapala 0.9320 0.426 M Chapala 1.5080 0.680 E Tequesquitengo 0.0460 0.023 -- Requena 13.2290 0.383 -- La Plata 34.9000 0.220 E Loiza 52.9100 0.330 E Tortuguero 0.3000 0.010 O Laguna Grande 13.7000 0.290 E O: oligotrophic, M: mesotrophic, E: eutrophic, HE: hypertrophic. TABLE II DESCRIPTIVE STATISTICS OF EACH TROPHIC CLASSIFICATION State Oligotrophic Mesotrophic Eutrophic variables Mean SD Mean SD Mean SD Z' 2.17 0.83 1.90 0.64 2.15 0.52 Tw' -1.00 1.27 -1.35 1.67 -1.95 1.64 Lp' 0.11 1.00 1.20 1.08 2.71 1.30 TABLE III PROGRAMMING CODE OF THE SIMULATION MODEL FOR 10000 WARM TROPICAL LAKES AND RESERVOIRS 100 ;MYLAKES.GPS--By Mario A. Ortiz-Jim6nez 110 ;MEAN AND STANDARD DEVIATION OF EACH TROPHIC STATE 120 PLO FVARIABLE (-1.2370-0.9335#P$Z+0.891#P$LP+0.67634#P$TW) 130 SUMA1 FVARIABLE 0.3106-0.1299#P$Z+0.0199#P$LP+0.0345#P$TW 140 SUMA2 FVARIABLE P$Z#(-0.1299+0.0640#P$Z-0.0159#P$LP-0.0159#P$TW) 150 SUMA3 FVARIABLE P$LP#(0.0199-0.0159#P$Z+0.0209#P$LP+0.0143#P$TW) 160 SUMA4 FVARIABLE P$TW#(0.0345-0.0159#P$Z+0.0143#P$LP+0.0179#P$TW) 170 DEPLO FVARIABLE 0.3495762#SQR(V$SUMAI+V$SUMA2+V$SUMA3+V$SUMA4) 180 ;CALCULO DE Ks 190 TERMI FVARIABLE EXP(P$LP)/EXP(P$PL)#EXP(P$Z)) 200 TERM2 FVARIABLE 1/EXP(P$TW) 210 KS FVARIABLE LOG(V$TERM1-V$TERM2) 220 ;LAKES AND RESERVOIRS GENERATION 230 RMULT 9999,6666,7777,3333 240 GENERATE 0 250 OPEN (<<MISDATOS.TXT>>)" 260 TRANSFER .5588,EUTRO,INTER 270 ;EUTROPHIC LAKES AND RESERVOIRS 280 EUTRO ASSIGN Z,(NORMAL(1,2.1453,0.5247)) 290 ASSIGN LP,(NORMAL(2,2.711,1.2973)) 300 ASSIGN TW,(NORMAL(3,-1.95,1.6374)) 310 ASSIGN PL,(NORMAL(4,V$PLO,V$DEPLO)) 320 ASSIGN 1,3 330 TEST G V$TERMI,V$TERM2,SALE 340 TRANSFER AEGIS 350 INTER TRANSFER .5263,MESO,OLIGO 360 ;MESOTROPHIC LAKES AND RESERVOIRS 370 MESO ASSIGN Z,(NORMAL(5,1.9033,0.6388)) 380 ASSIGN LP,(NORMAL(6,1.2015,1.0815)) 390 ASSIGN TW,(NORMAL(7,-1.3452,1.6706)) 400 ASSIGN PL,(NORMAL(8,V$PLO,V$DEPLO)) 410 ASSIGN 1,2 420 TEST G V$TERMI,V$TERM2,SALM 430 TRANSFER REGIS 440 ;OLIGOTROPHIC LAKES AND RESERVOIRS 450 OLIGO ASSIGN Z,(NORMAL(9,2.1703,0.8331)) 460 ASSIGN LP,(NORMAL(10,0.1112,0.9964)) 470 ASSIGN TW,(NORMAL(I1,-0.9972,1.2656)) 480 ASSIGN PL,(NORMAL(12,V$PLO,V$DEPLO)) 490 ASSIGN 1,1 500 TEST G V$TERMI,V$TERM2,SALO 510 REGIS WRITE (POLYCATENATE(P$PL,>> <<,P1)),,SALE 520 CLOSE 530 TERMINATE 1 540 SALE CLOSE 550 TERMINATE 1 560 SALM CLOSE 570 TERMINATE 1 580 SALO CLOSE 590 TERMINATE 1 600 START 10000 TABLE IV UPPER AND LOWER LIMITS OF THE POPULATION PARAMETERS OF TP' BY TROPHIC CLASSIFICATION Classification N Mean SD 95% Conf. LCL UCL Oligotrophic 2804 -3.75 1.41 [+ or -] 0.05 -3.80 -3.70 Mesotrophic 2455 -2.67 1.46 [+ or -] 0.06 -2.73 -2.61 Eutrophic 4055 -1.87 1.54 [+ or -] 0.05 -1.92 -1.82 TABLE V TROPHIC CLASSIFICATION LIMITS Trophic state Average (TP') Trophic limit (TP) Oligotrophic/mesotrophic [(-3.70)+(-2.73)]/2=-3.22 0.04 Mesotrophic/eutrophic [(-2.61)+(-1.92)]/2=-2.27 0.10 The main lines of the programming code given in Table III are: 120 Definition of the mean of TV according to Eq. 6. 170 Definition of the standard deviation of TV according to Eq. 7. The expression V$SUMAI+ ... +V$SUMA4 represents the result of the following matrix product: II [Z'.sub.0] [Lp'.sub.0] [Tw.sub.0]]x[(X'X).sup.-1]x[1 [Z'.sub.0] [Lp'.sub.0] [Tw'.sub.0] where [Z'.sub.0], [Lp'.sub.0] and [Tw'.sub.0] are random values normally and independently distributed with the mean and standard deviation of every trophic state. 210 Definition of the natural logarithm of the variable Ks starting from the mass balance equation proposed by Vollenweider (1976) (Eq. 8). 230 Initial number given to start up the process of random numbers generation. 240 Creates virtual water bodies, which are used as future inputs in the simulation model. 250 A file named MISDATOS.TXT is opened. 260 55.88% of the generated virtual water bodies are delivered to a block identified as INTER (line 350). The remaining 44.12% are delivered to a block identified as EUTRO (line 280), which corresponds to the percentage of the eutrophic water bodies in the Table I. 280 Based on the values on Table II, the random numbers generator I assigns to the eutrophic water bodies a normal distributed mean depth of 2.15 m and a standard deviation of 0.52m. 290 Based on the values on Table 2, the random numbers generator 2 assigns to the eutrophic water bodies a normal distributed mean depth of 2.71m and a standard deviation of 1.30m. 300 Based on the values in Table 2, the random numbers generator 3 assigns to the eutrophic water bodies, a normal distributed mean depth of 1.95m and a standard deviation of 1.64m. 310 The random numbers generator 4 assigns to the eutrophic water bodies, a normal distributed mean total phosphorus concentration with mean and standard deviation defined on lines 120 and 170, respectively. 320 The code 3 is assigned to all simulated eutrophic water bodies. 330 If Ks>0 (that is, the inequality (9) is satisfied), the eutrophic water body will go to next block (line 340), otherwise will be deliver to the SALE block (line 540). 340 The water body is unconditionally transferred directly to the block labeled REGIS (line 510). 350 All transactions arriving at INTER (55.88% of the generated transactions), randomly sends 52.63% of the leveled transactions to the OLIGO location. This fraction corresponds to 55.88x0.5263 = 29.41% of all oligotrophic water bodies described on Table 1. The remainder generated water bodies (55.88%-29.41% = 26.47%) are transferred to the MESO location, which corresponds to mesotrophic lakes described on Table 1. 360-430 Instructions are repeated in similar way to the lines 280 to 340, but the data now correspond to mesotrophic water bodies. 450-500 Instructions are repeated in similar way to the lines 280 to 330, but the data now correspond to oligotrophic water bodies. 510 A chain of text is written on the file with the generated values of TV, as is the code of the water body classification (1, 2 or 3). If an error occurs, the active water body is sent to the block leveled as SALE. 520 The active water body closes the text file. 530 As Ks>0, the active oligotrophic lake is 'removed' from the simulation and its statistic is not taken into account. 540 As Ks[less than or equal to]0, the eutrophic lake closes the text file and its statistic is taken into account. 550 As Ks[less than or equal to]0, the active eutrophic lake is 'removed' from the simulation. 560 As Ks[less than or equal to]0 the mesotrophic lake closes the text file and its statistic is taken into account. 570 As Ks[less than or equal to]0, the active mesotrophic lake is 'removed' from the simulation. 580 As Ks[less than or equal to]0, the oligotrophic lake closes the text file and its statistic is taken into account. 590 As Ks[less than or equal to]0, the active oligotrophic lake is 'removed' from the simulation and its statistics is discarded. 600 The logic sequence of the previous blocks is repeated ten thousand times.