Determining Chlorophyll-a
Concentrations in Aquatic Systems
with New Statistical Methods
and Models
Peter Dimberg
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. Several 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
© Peter Dimberg 2011
urn:nbn:se:uu:diva-160303 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-160303
ISBN 978-91-506-2241-6
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 utvärdera 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 komplettera dessa modeller med andra metoder och modeller vilket ger en bättre förståelse 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 sannolikheten 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.
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
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
Dimberg, P.H., Hytteborn, J.K., Bryhn, A.C. (2011) Predicting
median monthly chlorophyll-a concentrations. Submitted for
publication to Water Science and Technology.
Dimberg, P.H., Bryhn, A.C., Hytteborn, J.K. (2011) Probabilities 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
Contents
1. Introduction ................................................................................................. 9
2. Methods .................................................................................................... 11
2.1 Analysis of long-term trends .............................................................. 11
2.2 Seasonal variations ............................................................................. 14
2.3 Probability of predicted concentrations .............................................. 15
3. Results and discussion .............................................................................. 18
3.1 Long-term trend of chlorophyll-a in the Baltic Proper ...................... 18
3.2 Seasonal variation of chlorophyll-a in lakes ...................................... 19
3.3 Probability of chlorophyll-a concentration in lakes ........................... 20
4. Concluding remarks .................................................................................. 22
5. Acknowledgements ................................................................................... 23
6. References ................................................................................................. 24
Abbreviations
Chl-a
n
r2
SMT
TN
TP
Chlorophyll-a
Number of data
Correlation of coefficient
Statistically meaningful trend
Total nitrogen
Total phosphorus
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 chla 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 Dillon 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 longterm changes of chl-a concentration since it is not included in the range of
9
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 userfriendly as the former mentioned and is limited for the lake Võrtsjärv in Estonia.
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.
10
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 longterm 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 examine 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 rejecting 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
11
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 pvalue 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 pvalue 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.
12
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.
13
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 weaknesses considering in-parameters and month.
14
Figure 2. Variation of empirical chl-a concentrations in lake Rotehogstjärnen (lat. 58.82 , long. 11.61 ). Empirical chl-a for January and December 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 stepby-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 formula and the equation to calculate the theoretically highest expected r2
(Håkanson, 1999).
(equation 2)
15
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.
16
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.
17
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.
18
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 p-value for full Number of intertrend
time trend
val divisions indicating SMT
SMT?
Chl-a
TP
TN
Negative
Positive
Positive
0.0004
0.006
0.003
No
No
No
0.010
<0.001
<0.001
0
0
0
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 explaining 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 geographical 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 Rotehogstjärnen. 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.
19
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
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
-0.26
-0.28
-0.43
-0.11
0.66
0.66
0.66
0.69
0.62
0.74
0.80
0.82
0.77
0.73
0.53
0.49
0.05
-0.07
-0.11
0.71
0.73
0.46
0.91
0.88
0.99
1.16
1.16
1.06
0.85
0.73
0.83
1.39
1.46
2.86
0.30
-0.40
-0.56
-0.72
-0.92
-0.48
0.16
0.56
-0.21
Figure 5. Modelled and empirical median chl-a concentration with one standard deviation for the lake Rotehogstjärnen (lat. 58.82 , long. 11.61 ). Empirical 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 dis20
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 concentrations in July. Three curves represent monthly median chl-a concentrations in the measured month February, 0.5, 5 and 50 µg/l.
21
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.
22
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 question. My assistant supervisor Gesa Weyhenmeyer for her support. Lars
Håkanson who was my assistant supervisor before his retirement. Julia Hytteborn for her contribution to the papers in this thesis. And everyone else.
23
6. References
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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) Modelling phytoplankton dynamics of the eutrophic Lake Võrtsjärv, Estonia. Hydrobiologia, 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.
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24
Phillips, P., Pietiläinen, O.P., Carvalho, L., Solimini, A., Lyche Solheim, A.,
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Yin, G., Zhang, Q. (2005) Discrete-time Markov chains. Two time-scale
methods and applications. New York Springer, 347p.
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