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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.

SUMMARY

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 /

RESUMEN

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.

RESUMO

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.

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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.

Methodology

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)

where

[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)

were excluded.

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)


REFERENCES

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Baltzer (1996) Environmental Modeling and Assessment. Vol.1. Baltzer Science Publ. Amsterdam, Holland. XXX pp.

Castagnino WA (1982) Investigacion de modelos simplificados de eutroficacion en lagos tropicales. Pan American Center for Sanitary Engineering and Environmental Sciences (CEPIS). Rev. Vers. PAHO. Washington DC, USA. XXX pp.

CEPIS (2001) Metodologas simplificadas para la evaluacion de eutroficacion en lagos calidos tropicales. In: Henry J. Salas y Paloma Martino (Eds.). Programa Regional CEPIS/ HEP/OPS 1981-1990. Version actualizada 2001. Lima. Peru. 63 pp.

Chisman JA (1992) Introduction to simulation modelling using GPSS/PC. Prentice Hall. Englewood Cliffs, NJ, USA. 242 pp.

DVWK (1988) Sanierung un Restaurierung ron Seen. Merkblatter zur Wasserwirtschaft 213/ 1988. Kommissionsvertrieb. Deutscher Verband fur Wasserwirtschaft und Kulturbau e.V. Paul Parey. Hamburg/Berlin, Germany. 33 pp.

Hecky RE, Kilham P (1988) Nutrient limitation of phytoplankton in freshwater and marine environments: A review of recent evidence on the effects of enrichment. Limnol. Oceanogr: 33: 796-822.

Kietpawpan M, Visuthismajarn P Ratanachai C (2003) Statistical assessment of trophic conditions: squared Euclidean distance approach. Songklanakarin J. Sei. Technol. 25: 359-365.

Koneski Z, Davcev C, Mitresky K, Gorgoski J (1999) Trophic state estimation for lakes using fuzzy logic. Proc. LASTED Int. Conf. Applied Modelling and Simulation. Cairns, Australia.

Lewis WM (2002) Causes for the high frequency of nitrogen limitation in tropical lakes. Verh. Int. Verein. Limnol. 28: 210-213.

Minuteman Software (2000). GPSS World Tutorial Manual. Holly Springs, NC, USA. http:// www.minutemansoftware.com.

OECD (1982) Eutrophication of waters. Monitoring assessment and control. OECD. Paris, France. 154 pp.

Rast W, Lee GF (1978) Summary analysis of the North American (U.S. portion) OECD Eutrophication Project: Nutrient loading-lake response relationships and trophic state indices.

Ecological Research Series, No EPA-600/ 3-78-008. US Environmental Protection Agency. Corvallis. OR, USA. 454 pp.

Romeu JL (1977) More on Simulation and Statistical Education. Am. J. Math. Manag. Sci. 17(3,4).

Romeu JL (1995) Simulation and Statistical Education. Winter Simulation Conference Proceedings of the 27th conference on winter simulation. Arlington, Virginia. USA. pp. 1371-1375.

Salas HJ (2003) A Simplified Phosphorus Trophic State Model for Warm-Water Tropical Lakes. 2003 AWWA Source Water Protection Symposium. January 19-22. Albuquerque, NM, USA.

Sonzogni WC, Monteith TJ, Bach WN, Hughes VG (1978) United States Great Lakes tributary loadings. Great Lakes Pollution From Land Use Activities Reference Group Report. International Joint Commission, Great Lakes Regional Office. Windsor. ON, Canada. 187 pp.

UNEP (1999) Planning and Management of Lakes and reservoirs: An Integrated Approach to Eutrophication. Technical Publication Series 11. UNEP. Nairobi. Kenya.

USEPA (1974) An approach to a relative trophic index system for classifying lakes and reservoirs. Working paper No 24. United States Environmental Protection Agency. National Eutrophication Survey. Pacific Northwest Environmental Research Laboratory, Colvallis, OR. USA.

Vollenweider RA (1968) Scientific fundamentals of the eutrophication of lakes and ,flowing waters. whit particular reference to nitrogen and phosphorus as factors in eutrophication. Tech. Report No DAS/CS1/68.27. OECD. Paris, France.

Vollenweider RA (1976) Advances in defining critical loading levels for phosphorus in lake eutrophication. Mem. Inst. Ital. Idrobiol. Bott Marco de Marchi: 33: 53-83.

Vollenweider RA, Kerekes JJ (1981) Background and summary results of the OECD cooperative program on eutrophication. In Restoration of Lakes and Inland Waters. EPA/440/5-81-010. pp. 25-36.

Vollenweider RA. Kerekes J (1982) Eutrophication of Waters. Monitoring, Assessment and Control. Organization for Economic Co-Operation and Development (OECD), Paris. 156p.

Willander A, Personn G (2001) Recovery from eutrophication: experiences of reduced phosphorus input to the four largest lakes of Sweden. Ambio 30: 475-485.

Younger MS (1979) Handbook for Linear Regression. Duxbury. Belmont, CA. USA. 570 pp,

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: ojimez@nayar.uan.mx

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: janda@ciatej.net.mx

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.
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Author:Ortiz-Jimenez, Mario A.; de Anda Jose; Maniak, Ulrich
Publication:Interciencia
Date:May 1, 2006
Words:5634
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