Determining Chlorophyll-a Concentrations in Aquatic Systems with New Statistical Methods and Models

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Determining Chlorophyll-a Concentrations in Aquatic Systems

with New Statistical Methods and Models

Peter Dimberg

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Abstract

Dimberg, Peter., 2011. Determining Chlorophyll-a Concentrations in Aquatic Systems with New Statistical Methods and Models. Institutionen för geovetenskaper. 25 pp. Uppsala. ISBN 978-91-506-2241-6

Chlorophyll-a (chl-a) concentration is an indicator of the trophic status and is extensively used as a measurement of the algal biomass which affects the level of eutrophication in aquatic systems. High concentration of chl-a may indicate high biomass of phytoplankton which can decrease the quality of water or eliminate important functional groups in the ecosystem.

Predicting chl-a concentrations is desirable to understand how great impact chl-a may have in aquatic systems for different scenarios during long-time periods and seasonal variation. Sev- eral models of predicting annual or summer chl-a concentration have been designed using total phosphorus, total nitrogen or both in combination as in-parameters. These models have high predictive power but are not constructed for evaluating the long-term change or predicting the seasonal variation in a system since the input parameters often are annual values or values from other specific periods. The models are in other words limited to the range where they were constructed. The aim with this thesis was to complement these models with other methods and models which gives a more appropriate image of how the chl-a concentration in an aquatic system acts, both in a short as well as a long-time perspective. The results showed that with a new method called Statistically meaningful trend the Baltic Proper have not had any change in chl-a concentrations for the period 1975 to 2007 which contradicts the old result observing the p-value from the trend line of the raw data. It is possible to predict seasonal variation of median chl-a concentration in lakes from a wide geographically range using summer total phosphorus and latitude as an in-parameter. It is also possible to predict the probability of reaching different monthly median chl-a concentrations using Markov chains or a direct relationship between two months. These results give a proper image of how the chl-a concentration in aquatic systems vary and can be used to validate how different actions may or may not reduce the problem of potentially harmful algal blooms.

Keywords: Chlorophyll-a, statistical models, aquatic systems, lakes

Peter Dimberg, Department of Earth Sciences, Uppsala University, Uppsala, Sweden

urn:nbn:se:uu:diva-160303 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-160303 ISBN 978-91-506-2241-6

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Sammanfattning

Koncentrationen av klorofyll-a (chl-a) är en indikator på vilken trofinivå ett akvatiskt system har och används som ett mått på algbiomassa som påverkar övergödningen i akvatiska system. Höga koncentrationer av chl-a i sjöar kan indikera hög biomassa av fytoplankton och försämra kvalitén i vattnet eller eliminera viktiga funktionella grupper i ett ekosystem. Det är önskvärt att kunna prediktera chl-a koncentrationer för att förstå hur stor påverkan chl-a kan ha för olika scenarier i akvatiska system under längre perioder samt under säsongsvariationer. Flera modeller har tagits fram som predikterar årsvärden eller sommarvärden av chl-a koncentrationer och i dessa ingår totalfosfor, totalkväve eller en kombination av båda som inparametrar. Dessa modeller har hög prediktiv kraft men är inte utvecklade för att kunna utvär- dera förändringar över längre perioder eller prediktera säsongsvariationer i ett system eftersom inparametrarna ofta är årsmedelvärden eller värden från andra specifika perioder. Modellerna är med andra ord begränsade till den domän som de togs fram för. Målet med denna avhandling var att komplette- ra dessa modeller med andra metoder och modeller vilket ger en bättre för- ståelse för hur chl-a koncentrationer i akvatiska system varierar, både i ett kortsiktigt och ett längre perspektiv. Resultaten visade att med en ny metod som kallas för Statistiskt meningsfull trend så har egentliga Östersjön inte haft någon förändring av chl-a koncentrationer under perioden 1975 till 2007 vilket motsäger tidigare resultat då p-värdet tas fram från en trendlinje av rådata. Det är möjligt att prediktera säsongsvariationer av median chl-a koncentrationer i sjöar från en bred geografisk domän med totalfosfor från sommar och latitud som inparametrar. Det är även möjligt att beräkna sanno- likheten av ett predikterat värde för olika månadsmedianer av chl-a koncentrationer med Markovkedjor eller ett direkt samband mellan två månader.

Dessa resultat ger en reell förståelse för hur chl-a koncentrationer i akvatiska system varierar och kan användas till att validera hur olika åtgärder kan eller inte kan reducera problemet av de potentiellt skadliga algblomningarna.

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Bryhn, A.C., Dimberg, P.H. (2011) An Operational Definition of a Statistically Meaningful Trend. PLoS ONE, 6(4):e19241.

doi:10.1371/journal.pone.0019241

II Dimberg, P.H., Hytteborn, J.K., Bryhn, A.C. (2011) Predicting median monthly chlorophyll-a concentrations. Submitted for publication to Water Science and Technology.

III Dimberg, P.H., Bryhn, A.C., Hytteborn, J.K. (2011) Probabili- ties of monthly median chlorophyll-a concentrations in subarctic, temperate and subtropical lakes. Submitted for publication to Hydrobiologia.

In Paper I the author was accessory for collecting data, parts in developing theory, interpreting the results and contributed to the writing. In Paper II the author was responsible for collecting data, developing the theory and had the main responsibility for writing the paper. In Paper III the author was responsible for collecting data, developing the theory and program and had the main responsibility for writing the paper

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Abbreviations

Chl-a Chlorophyll-a

n Number of data

r² Correlation of coefficient

SMT Statistically meaningful trend

TN Total nitrogen

TP Total phosphorus

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1. Introduction

Chlorophyll-a (chl-a) is a green pigment which is found in aquatic systems and is extensive used as a trophic state indicator of eutrophication (Håkanson et al., 2007). Chl-a is involved in the photosynthesis and can be used to estimate the mass of phytoplankton in aquatic systems and also as a measurement of how great impact the activity of phytoplankton has in the ecosystem (Gregor and Marsálek, 2004). High load of nutrients, such as phosphorus and nitrogen, can promote the production of phytoplankton which increases the concentration of chl-a in the water mass. High concentrations of chl-a may indicate undesirable blooms of phytoplankton and in other words high content of cyanobacteria which can decrease the quality of water or eliminate important functional groups in aquatic systems (Håkanson and Bryhn, 2008). It is therefore important to be able to predict the concentration of chl- a and evaluate which variables that significantly influence the mass of phytoplankton. With such predictions it would be possible to understand and manage the trophic state in aquatic systems and these predictions would be decisive tools for overcoming the undesirable eutrophication in lakes. The reason for choosing chl-a concentrations when constructing models instead of phytoplankton biomass is due to there is often a scarcity of the data of phytoplankton biomass. The concentration of chl-a is used as an approximately variable for estimating the phytoplankton biomass.

Vollenweider (1968) derived a model from mass-balance of total phosphorus (TP) and showed that it is possible to predict water chemistry variables from simple equations in lakes assuming steady state. To understand what the chl-a concentration is related on in lakes, different statistical regression models have been developed, where Dillon and Rigler (1974) were among the first to show the relationship between TP and chl-a concentrations in lakes. These statistical models are often based on a relationship between concentrations of chl-a and phosphorus or nitrogen, or both in combination (Phillips et al., 2008). Several different regression models of chl-a after Dil- lon and Rigler (1974) have been developed, e.g. Jones and Bachmann (1976), Carlson (1977), Prepas and Trew (1983), Ostrofsky and Rigler (1987), Nürnberg (1996) and Phillips et al. (2008). These models predict the chl-a concentrations in the summer which uses nutrient concentration of the summer or the spring circulation as an in-parameter. It means that these regression models are not suited to predict seasonal variation or evaluate long- term changes of chl-a concentration since it is not included in the range of

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the models. It is therefore necessary to complement them with other statistical tools. Models for predicting phytoplankton have been developed e.g.

Frisk et al (1999). The disadvantage with this model is that it is not as user- friendly as the former mentioned and is limited for the lake Võrtsjärv in Es- tonia.

The aim with this thesis is to complement the regression models with other methods and models to give a more appropriate image and understand how the chl-a concentration acts in aquatic systems, both in a short as well as a long-time perspective. The new models and methods should be easy to use and include a wide range which means that they are not limited for a certain aquatic system and may be applied on the most systems there is.

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2. Methods

Three different methods of analysing chl-a concentrations in aquatic systems were developed in Paper I-III. One method was developed to analyse long- term trends, which is described in Paper I. A model was designed to predict seasonal variation of monthly median chl-a concentrations in lakes, Paper II.

A method to estimate a probability of exceeding or not exceeding a median chl-a concentration for a specified month in a lake is described in Paper III.

These three different methods are explained further in this section.

2.1 Analysis of long-term trends

The long-term variation of chl-a concentrations in aquatic systems can be evaluated by plotting raw data for the investigated period and by analysing the trend with additional significance level (p-value). To obtain the p-value equation 1 may be used to calculate a t-value which is used to obtain the p-value from a statistical software or table.

(equation 1)

where t is a statistical constant, r² is the coefficient of correlation and n is the number of data.

If the number of data (n) is large the p-value will often or always, depending on the quality (r²) of the data, show a significant trend. This has been a problem in aquatic sciences for example where the trend of chl-a concentrations in the Baltic Proper contradicts the trend of total nitrogen (TN) and TP, figure 1. The concentration of chl-a depends on the nutrients TP and/or TN which means that the trend of these variables should be the same. Another definition of a statistical trend is therefore needed. Paper I defines a new stricter method to determine whether a trend is significant or not, which is referred to as a Statistically meaningful trend (SMT), and was used to exam- ine whether the concentration of chl-a or the other nutrients had decreased or increased for the time period 1975 to 2007. The method of obtaining or re- jecting an SMT was to divide the raw data set into intervals and analyse the correlation of coefficients (r²) and the p-values. Prairie (1996) showed that

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an r² between 0.00 and 0.65 had approximately zero predictive power while the predictive power of an r² value above 0.65 increased exponentially. A p- value below 0.05 is in many scientific disciplines considered as a good threshold value to show a significant trend. If the raw data set was divided into several intervals the r² value would increase while the p-value also increased since less data were considered. Therefore, combining the r² and p- value for the divided intervals would give a stricter definition of when a trend should be considered as significant. If one interval had r²> 0.65 and a p-value < 0.05 the trend was considered as an SMT. An add-in to Excel (www.microsoft.com) was developed which calculates the r² and p-value for different intervals.

One other similar test of trends is the non-parametric Mann-Kendall test.

Since the Mann-Kendall test is non-parametric it is possible to use it on data without taking into account of what type of distribution the data has. The Mann-Kendall test does not take into account the magnitude of the investigated variable and is simply presenting the increase or decrease for an observed value as a binary result. It has been shown that when the variation of the data is increasing the power of the test decreases (Yue et al., 2002). The variation of the data is something that the SMT takes into account since the test is parametric and affected by the magnitude of the investigated data.

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13 Figure 1. The Baltic Proper, 1975-2007. A. TN concentrations, trend-slope

= positive, r² = 0.003, p-value < 0.001. B. TP concentrations, trend-slope = positive, r² = 0.006, p-value < 0.001. C. Chl-a concentrations, trend-slope = negative, r² = 0.0004, p-value = 0.010. From Paper I.

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2.2 Seasonal variations

Seasonal variation of chl-a concentration in lakes can be illustrated by making several measurements for every month in one year. By investigating the monthly chl-a concentrations, patterns or other specific behaviour in a lake may be detected. This can in several cases be related to the load of nutrients, TP and TN. One problem is that there is often a scarcity of data for the most lakes e.g. the lake Rotehogstjärnen (figure 2). It is not unusual that data are missing for some or several months, which mainly is due to insufficient time and money and in some cases harsh weather when the measurements are supposed to occur. Therefore a statistical model based on summer medians of TP including the latitude was designed to describe the seasonal variation of median chl-a concentrations, Paper II explains the procedure of developing the model using stepwise multiple regression. In this model 308 lakes were used, the latitude range of these lakes was 27-68.35 . Other previously published regression models of chl-a concentration from other studies (e.g.

Phillips et al., 2008) were tested against the data set and designed with a statistical constant to take account for the seasonal variation. The statistical constant was calculated by dividing the median empirical monthly chl-a concentration with the predicted chl-a for every lake and obtaining a median value from these constants which is referred to as a uniform constant. The uniform constant was used to calculate a predicted median chl-a concentration for a specific month and model. Wilcoxon’s test was used, p-level <

0.05, to detect whether the significance of the distribution could be rejected against the empirical data. This test included the output from the statistical model designed with summer TP and latitude and those which included the constant. Stability tests were made to evaluate the models’ different weak- nesses considering in-parameters and month.

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15 Figure 2. Variation of empirical chl-a concentrations in lake Rote- hogstjärnen (lat. 58.82 , long. 11.61 ). Empirical chl-a for January and De- cember are missing.

2.3 Probability of predicted concentrations

Regression models to predict chl-a concentrations are not constructed to predict the probability for reaching a certain concentration, or the probability for reaching an interval between two concentrations. The probability is necessary when it is important to know how much the probability is of exceeding a certain level of a median chl-a concentration in a couple of months based on a previously measured value. Paper III discusses two different methods of calculating these probabilities where one method is built on discrete Markov Chains (MC; Yin and Zhang, 2005) and the other method (Reg) on a direct relationship between two months. MC is based on a step- by-step calculation for reaching different intervals of median chl-a concentrations from one month to the other etc. until reaching the target month.

Figure 3 illustrates the Reg method and figure 4 illustrates the MC method.

The difference between the MC and Reg method for the calculation of probabilities is that MC takes into account all the months between start and end month while the Reg method exclude these months and is only taking into account the start and the month. To calculate the uncertainty of data for these two methods an equation (equation 2) was derived from the Sampling for- mula and the equation to calculate the theoretically highest expected r² (Håkanson, 1999).

(equation 2)

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where L is the uncertainty, n is the number of data and r²r is the theoretically highest expected r².

In addition the two methods, MC and Reg were evaluated with equation 2 to detect which one could be preferable for different months. Since making these calculations manually is time-demanding a friendly use Java program was developed in NetBeans IDE 7.0 (www.netbeans.org) which can be used to calculate different probabilities for reaching different intervals of monthly median chl-a concentrations. A routine to suggest which method to use is included. The two methods, MC and Reg, were tested on 308 lakes with a latitude range of 27-68.35 . The outputs from PoCC were validated for the both methods against data which were calculated manually.

Figure 3. Illustration of the Reg method. Chl is different intervals of median chl-a concentrations and P is the probability of reaching a certain state. From Paper III.

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17 Figure 4. Illustration of the MC method. Chl is different intervals of median chl-a concentrations and P is the probability of reaching a certain state. From Paper III.

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3. Results and discussion

The results of long-term trend analysis in the Baltic Proper, seasonal variation of median chl-a concentrations in lakes and estimated probability of median chl-a concentrations for exceeding and not exceeding a predicted value in lakes are discussed in this section.

3.1 Long-term trend of chlorophyll-a in the Baltic Proper

The long-term trend of chl-a concentration in the Baltic Proper is not statistically meaningful according to the definition of an SMT, stated in Paper I.

This is a contradictory result compared to the other one (figure 1) analysing the trend line of the raw data with the p-value. With a null hypothesis of ‘No change of chl-a concentration in Baltic Proper’ the null hypothesis cannot be rejected using the SMT. The trend is however significant with a certainty of 99 % if only the p-value of the trend line is observed for the whole data set, and still the trend for TN and TP is even more significant despite the contradictory slope of the trend line. Using the definition of an SMT shows that the long-term trend is not significant for any of the variables chl-a, TN and TP in the Baltic Proper, table 1. These results coincide and give an indication that there has not been any significant change of the chl-a concentration in the Baltic Proper for the period 1975-2007. SMT would be a preferable method instead of only observing the p-value of the trend line from the data set since the p-value is, if the number of data is large, highly depended on the number of data and not the actual change or the quality (r²) of data used in the data set. This can be illustrated by using equation 1. Temporary fluctuations (or outliers) of chl-a concentrations in the raw data are eliminated by dividing the data set into intervals. However, the magnitude of the fluctuations or outliers will not be entire neglected. This increases the r² with an effect that the amount of n decreases. A balance between these two statistical parameters, r² and n, may be found and can therefore in some cases show an SMT.

Paper III has several examples of different variables which showed an SMT, for example economic growth, temperature deviations and population growth.

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19 Table 1. Test of an SMT for three different variables in the Baltic Proper. The test shows a positive result of SMT if p < 0.05 in combination with r²> 0.65 in any of the calculated intervals.

Time

series Significant

trend slope r² value for full time

trend p-value for full

time trend Number of inter- val divisions indi- cating SMT

SMT?

Chl-a Negative 0.0004 0.010 0 No

TP Positive 0.006 <0.001 0 No

TN Positive 0.003 <0.001 0 No

3.2 Seasonal variation of chlorophyll-a in lakes

The seasonal variation of median chl-a concentration in lakes can be illustrated with 12 different statistical regression models, one for each month (table 2). When the p-value of latitude did not exceed 0.05 in the stepwise multiple regression for the different months it was considered as an explain- ing variable in the models. The results showed that it is important to consider the latitude in models if they are being used for lakes with an extensive geo- graphical range, especially for colder periods. Such models can have summer median TP as an input variable which means that the overall picture of chl-a variation during a year can be obtained for lakes even if data for other periods are missing. It is important to stress that the modelled seasonal variation of median chl-a concentrations will be an approximate result compared to the empirical concentrations. The modelled median chl-a concentrations for the months far from summer may differ more with the empirical concentrations compared to the summer months. Nevertheless, the model was only rejected for April when the outputs from the model were tested against the empirical values with Wilcoxon’s test. Figure 5 illustrates how the regression models can be used and are exemplified on data from the lake Rote- hogstjärnen. For the summer period the concentration of chl-a has a greater variation than during winter, which is indicated by the standard deviations.

The modelled concentration of chl-a had the same seasonal variation shape as the empirical concentration. However, even though the modelled median chl-a concentrations are lower than the empirical chl-a concentrations during summer they are still within the range of plus and minus the standard deviations. Paper II discusses the results of seasonal variation of median chl-a concentration in more detail. Since empirical concentrations of chl-a were missing for January and December for the lake Rotehogstjärnen the model can be used to investigate the overall seasonal variation with and without including the empirical chl-a concentrations for the other months.

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Table 2. Monthly regressions of log(chl-a) for different months. Lat is the latitude and TPs is TP for the summer.

Month r² sqrt(Lat) log(TPs) Intercept

Jan 0.66 -0.26 0.71 1.39

Feb 0.66 -0.28 0.73 1.46

Mar 0.66 -0.43 0.46 2.86

Apr 0.69 -0.11 0.91 0.30

May 0.62 0.88 -0.40

Jun 0.74 0.99 -0.56

Jul 0.80 1.16 -0.72

Aug 0.82 0.05 1.16 -0.92

Sep 0.77 1.06 -0.48

Oct 0.73 -0.07 0.85 0.16

Nov 0.53 -0.11 0.73 0.56

Dec 0.49 0.83 -0.21

Figure 5. Modelled and empirical median chl-a concentration with one stan- dard deviation for the lake Rotehogstjärnen (lat. 58.82 , long. 11.61 ). Em- pirical chl-a for January and December are missing, November had one value. The r² between modelled and empirical were 0.63.

3.3 Probability of chlorophyll-a concentration in lakes

The Probability of exceeding a certain median chl-a concentration for one month assuming a measured chl-a concentration a previous month is dis-

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21 cussed in Paper III. Figure 6 gives an example of a result between the months February to July for the methods MC and Reg, where the measured values are 0.5, 5 and 50 µg/l. The results from these two methods differ where a measured chl-a concentration of 5 µg/l in February gives a 25 % probability of exceeding 11.8 µg/l in July using a curve fit for the MC method and a 25 % probability of exceeding 28.2 µg/l in July using a curve fit for the Reg method. The advantage by using the MC method is that one month often has a high r² with the next month and by using MC a high r² is considered for every calculation step. The Reg method has the disadvantage that one month often has a low r² with another month several months away, but should be used if the target month is the month after or if a significant high r² value can be considered between the measured and target month. In this example, using the MC method, the r² value has the lowest value between April and May with an r² = 0.55 and for the entire data set r² = 0.61.

The r² for the Reg method between February and July is 0.56. However, since the MC method takes account for more data, and the r² is approximately the same for the both methods, MC should be considered as more reliable. Equation 2 confirms this conclusion with a calculated uncertainty for MC with 0.05 compared to the uncertainty for Reg with 0.22. The same conclusion was made for every tested case when there was one or more months between the start and end month.

Figure 6. Cumulative probabilities of exceeding monthly median chl-a con- centrations in July. Three curves represent monthly median chl-a concentrations in the measured month February, 0.5, 5 and 50 µg/l.

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4. Concluding remarks

The chl-a concentration has not changed in the Baltic Proper under the period 1975 to 2007, and the same conclusion can be made for the nutrients total nitrogen and total phosphorus. With a Statistically meaningful test analysis it is possible to determine whether different aquatic variables have decreased or increased in a long-term period in the Baltic Proper. However, it can also be used to determine the long-term trends in other areas such as lakes, entire Baltic Sea or in other scientific disciplines (Paper I).

The seasonal variation of median chl-a concentration in lakes can be predicted from the summer median total phosphorus concentration along with the latitude. The model described in Paper II can be used for lakes which empirical chl-a concentrations are scarce for different months to predict the overall seasonal variation of median chl-a concentrations.

The probability of reaching a chl-a concentration in one month given a certain empirical median concentration of chl-a from a previous month can be predicted using Markov chains or a direct relationship between two months.

The methods described in Paper III may be used to estimate the probability of exceeding or not exceeding a specific chl-a concentration in a single lake.

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5. Acknowledgements

I would like to thank the following people: my main supervisor Andreas Bryhn for always encouraging and giving me ideas whenever I had a ques- tion. My assistant supervisor Gesa Weyhenmeyer for her support. Lars Håkanson who was my assistant supervisor before his retirement. Julia Hyt- teborn for her contribution to the papers in this thesis. And everyone else.

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6. References

Carlson, R.E. (1977) A trophic State Index for Lakes. Limnology and Oceanography, 2(2):361-369.

Dillon, P.J., Rigler, F.H. (1974) The Phosphorus-Chlorophyll Relationship in Lakes. Limnology and Oceanography, 19(5):767-773.

Frisk, T., Bilaletdin, Ä., Kaipainen, H., Malve, O., Möls, M. (1999) Model- ling phytoplankton dynamics of the eutrophic Lake Võrtsjärv, Estonia. Hy- drobiologia, 414:59-69.

Gregor, J., Marsálek, B. (2004) Freshwater phytoplankton quantification by chlorophyll a: a comparative study of in vitro, in vivo and in situ methods.

Water Res, 38:517-522.

Håkanson, L. (1999) On the principles and factors determining the Predic- tive success of ecosystem models, with a focus on lake Eutrophication mod- els. Ecological modelling, 121:139-160.

Håkanson, L., Bryhn, A.C., Hytteborn, J. (2007) On the issue of limiting nutrient and predictions of cyanobacteria in aquatic systems. Science of the Total Environment, 379:89-108.

Håkanson, L., Bryhn, A.C. (2008) Eutrophication in the Baltic Sea. Present Situation, Nutrient Transport Processes, Remedial Strategies. Springer- Verlag Berlin Heidelberg, 310p.

Jones, J.R., Bachmann, R.W. (1976) Prediction of phosphorus and chloro- phyll levels in lakes. J. Water Pollut. Cont. Fed., 48:2176-2182.

Nürnberg, G.K. (1996) Trophic State of Clear and Colored, Soft- and Hard- water Lakes with Special Consideration of Nutrients, Anoxia, Phytoplankton and Fish. Lake Reserv. Manage., 12:432-447.

Ostrofsky, M.L., Rigler, F.H. (1987) Chlorophyll-Phosphorus Relationships for subarctic lakes in Western Canada. Can. J. Fish. Aquat. Sci., 44:775-781

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25 Phillips, P., Pietiläinen, O.P., Carvalho, L., Solimini, A., Lyche Solheim, A., Cardoso, A.C. (2008) Chlorophyll-nutrient relationships of different lake types using a large European dataset. Aquat Ecol., 42:213-226. DOI 10.1007/s10452-008-9180-0.

Prairie, Y.T. (1996) Evaluating the predictive power of regression models.

Can J Fish Aquat Sci., 53:490-492.

Prepas, E.E., Trew, D.O. (1983) Evaluation of the Phosphorus-Chlorophyll Relationship for Lakes Off the Precambrian Shield in Western Canada. Can.

J. Fish. Aquat. Sci., 40:27-35.

Vollenweider, R.A. (1968) Scientific Fundamentals of the Eutrophication of Lakes and Flowing Waters, with Particular Reference to Phosphorus and Nitrogen as Factors in Eutrophication. OECD Technical Report, DAS/CSI/68.27, 159p.

Yin, G., Zhang, Q. (2005) Discrete-time Markov chains. Two time-scale methods and applications. New York Springer, 347p.

Yue, S., Pilon, P., Cavadias, G. (2002) Power of the Mann-Kendall and Spearman’s rho tests for detecting monotonic trends in hydrological series.

Journal of Hydrology, 259:254-271.

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Determining Chlorophyll-a Concentrations in Aquatic Systems with New Statistical Methods and Models