The paper discusses a physical model, obtained with the aid of statistical analyses, of the relationships between the sun-induced chlorophyll a fluorescence quantum yield and marine environmental factors. The relationships are based on a large set of empirical data from various ocean regions with basins of different trophicity, at different depths and in different seasons. Underwater spectral radiance and irradiance in the PAR spectral range were used to determine the quantum yield of sun-induced chlorophyll a fluorescence. From a statistical analysis a preliminary mathematical expression was derived to describe the fluorescence quantum yield as a function of the scalar irradiance, basin trophicity and the water temperature in situ. These relationships may be useful for analysing the budget of the light energy absorbed by phytoplankton pigments utilized in chemical and non-chemical quenching.
© 2012 OSA
Sun-induced chlorophyll fluorescence – SICF (the main abbreviations and symbols used in the text are listed in Table 1 ) supplies information on the state and functioning of marine plant communities. Numerous attempts have therefore been made to determine, among other things, chlorophyll concentrations and levels of photosynthetic primary production in the sea on the basis of measured SICF values (e.g [1–3].). However, the use of SICF to determine these properties is severely limited: in spite of intensive research into the dependence of the SICF quantum yield on environmental conditions (see e.g [4–6].), no general mathematical description of SICF has yet been formulated. Our preliminary analyses of this dependence were presented at the Ocean Optics conference in 2010 and published in the conference materials (see ). The aim of the present work was to derive a more complex version of a mathematical model, based on a larger set of experimental data, of the dependence of the quantum yield of the natural fluorescence of phytoplankton on the three principal factors governing phytoplankton growth in the sea: the trophicity of the water body under scrutiny, the light conditions there, and the temperature at different depths in the water. It was based on a set of more than 1200 values of the above-mentioned principal factors governing phytoplankton growth, and on a set of values of the quantum yield of fluorescencecorresponding to these environmental conditions, determined from measurements of downward irradiance and upward radiance spectra, and of other spectra necessary for obtaining the relevant parameters. The details of these analyses will be discussed below.
2. Material and methods
In order to achieve the objectives of this work, empirical data gathered in various regions (mainly the southern part) of the Baltic Sea during serial cruises of r/v ‘Oceania’ (IO PAS Sopot) in 1999–2010 were employed. This bank of empirical data was extended by historical data gathered jointly by scientists from the Institute of Oceanology PAS and the Shirshov Institute of Oceanology RAS during the 23rd cruise of the r/v 'Vityaz' in the Atlantic Oceans: (1991) headed by Professor M.E Vinogradov from the Shirshov Institute of Oceanology RAS. These studies were carried out in seas of different trophicity, with chlorophyll concentrations from 0.02 mg m−3 (oligotrophic ocean centres) to ca 80 mg m−3 (supereutrophic waters in Baltic gulfs).
This database contains a large number of different marine environmental parameters and magnitudes characterizing the various properties of phytoplankton and marine photosynthesis. Of these parameters, the following, measured at different depths at 100 stations were used in the present analysis:
- - chlorophyll a concentration [mgchla m−3], measured at different depths in the sea using the traditional spectrophotometric method ;
- - the spectra of light absorption by phytoplankton [m−1], measured in vivo using non-extraction methods (see e.g [9–11].) in suitably prepared samples of water containing phytoplankton, drawn from different depths in the sea. The relevant spectral measurements were performed on a UNICAM UV4-100 spectrophotometer equipped with a LABSPHERE RSA-UC-40 integrating sphere. This is described in detail in ;
- - the downward irradiance in the PAR spectral range (400–700 nm) [μEin m−2s−1nm−1], the total (integrated from 400 to 700 nm) scalar downward irradiance in this range PAR(z) [μEin m−2s−2], and the spectra of upward radiance at the nadir in the PAR spectral range (400–700 nm) [μEin m−2 s−1 nm−1 srad−1] were measured:
- - in the Atlantic and Indian Oceans with underwater spectrophotometers constructed at IO PAN. With these, the downward irradiance Ed(λ,z) and upward radiance Lu(λ,z) were measured at nine wavelengths λ [nm]: 405, 450, 500, 525, 590, 600, 665, 683 and 710. The physical principles underlying these measurements are explained in e.g [13–15];
- - temperature temp  determined in situ with standard STD probes.
These parameters were measured in situ at different depths down to ca 60 m in the sea or in vitro in the laboratory in samples of sea water from different regions and depths but mostly (773 points at different stations and depths) in the Baltic Sea. On this basis 1224 values of the quantum yield of fluorescencewere obtained indirectly from an analysis of downward irradiance and upward radiance spectra, and of other spectra necessary for obtaining the relevant parameters; this was done in accordance with the two-stage scheme given in Ostrowska et al. (1997) and described below.
2.2 Used computation methods
(Stage 1) The total upward radiance Lint,fl at the nadir in the chlorophyll a fluorescence spectral band was determined from an analysis of spectral upward radiance values at the nadir (containing effects due to elastic – see Rayleigh and Mie theory – and non-elastic Raman scattering, apart from fluorescence) for a number of wavelengths λ in the spectral region, coinciding with the above-mentioned chlorophyll a fluorescence band centred on 683 nm. To this end we used the algorithm for determining this fluorescence based on measurements of the water-leaving radiance at the nadir for three wavelengths in the region of this chlorophyll a fluorescence band peaking at around 683 nm (see ). Similar methods for determining the total radiance at the nadir due to chlorophyll fluorescence have often been used by other authors employing the Fluorescence Line Height (FLH) algorithm [2,3,18–20]. As already mentioned, we realize that the FLH algorithm takes into account the reduced influence of elastic and non-elastic scattering in the total radiance spectrum Lu(λ) to only an approximate extent. It is well-known that the Raman scattering spectrum is shifted in relation to the radiation eliciting it towards long wavelengths and is selective. A more detailed analysis of the influence of this effect on radiation will be found in . In addition, we would like to draw attention to the fact that CDOM and SPM can also affect the estimated fluorescence yield. When the concentrations of these constituents are high, the effect due to elastic scattering may increase strongly, especially on large suspended particles and phytoplankton; seen against this background, the fluorescence spectrum may be relatively small. We assume, however, that these effects are largely eliminated by the FLH method. In view of the lack of suitable measurements, we have not taken these subtelties into consideration in this discussion. One has to be mindful of the fact that they are of an approximate nature and are simplified, whereas the model descriptions presented here may be a first step to further, more detailed and comprehensive analyses.
(Stage 2) The SICF quantum yield was calculated on the basis of the above-mentioned values of the radiance Lint,fl. For this we used the approximate method we derived for determining the SICF quantum yield by solving the inverse optical problem (see below).
2.3. The solution to the inverse optical problem
The inherent quantum yield of SICF in the sea at depth z defines the ratio of the number of quanta (in the spectral band around 685 nm) emitted by phytoplankton in unit volume to the number of quanta from the entire spectral range 400 – 700 nm absorbed by its pigments.
As a result of fluorescence, phytoplankton at depth z’ emits radiation from a layer of thickness dz’ vertically upwards in an amount described by the equation (for a vertically stratified sea):
- Lint,fl(z') [Ein m−2s−1srad−1] – the part of the total upwelling radiance in the chlorophyll a fluorescence band at the nadir due to fluorescence;
- [m−1] - the mean phytoplankton light absorption coefficient weighted by the irradiance spectrum;
- [m−1] - spectrum of the phytoplankton light absorption coefficient;
- ≈1.2PAR(z') [Ein m−2s−1] - scalar irradiance in the 400-700 nm spectral range;
- E0(λ,z') [Ein m−2s−1nm−1] - spectrum of scalar spectral irradiance;
- [Ein m−2s−1] - downwelling irradiance in the 400-700 nm spectral range.
- Ed(λ,z') [Ein m−2s−1nm−1] - spectrum of downwelling vector irradiance;
- – a factor resulting from the isotropicity of fluorescence.
The contribution to the upward radiance, Lint,fl,z(z'), from the nadir at depth z, derived from the fluorescence at depth z', Lint,fl(z = z'), differs approximately by a factor linked to the coefficient of light absorption by the water body a683nm(z') or the coefficient of attenuation of radiance KL,u,683nm(z'):s Lu,683nm(z). Analysis of empirical data acquired by our research group back in the 1970s, mainly in the Baltic Sea and in some parts of the Atlantic Ocean [22, 23], shows that in the red region the ratio a(λ,z)/KL,u(λ,z) does not diverge from unity by more than 20%, the standard deviation being 0.05. In our assessment this does not, in the majority of cases, introduce errors greater than 5%.
The total radiance Lint,fl,z is obtained by integrating Eq. (3) with respect to depth z’ within the limits (∞ - z). If, further, we assume a simplified, homogeneous model of the medium (K0(z') = K0(z)≡K0,z, KL,u,683nm(z') = KL,u,683nm(z)≡KL,u,683nm,z, are constant and depth-independent) and bearing in mind the relationship , the expression for Lint,fl,z can be reduced to the approximate form:
where K0,z [m−1] – coefficient of attenuation of scalar irradiance PAR0 with depth;
This equation enables the quantum yield at depth z Φfl,z to be determined from measured values of Lint,fl,z, PAR0,z, , K0,z and KL,u,683nm,z. This yield is involved and cannot be expressed by a simple analytical expression. But we can obtain a simpler analytical description by introducing a new magnitude, which, in contrast to the inherent quantum yield of fluorescence Φfl, we shall call the apparent quantum yield of fluorescence . This apparent quantum yield of fluorescence at depth z, , refers not to unit volume but to a column of water of thickness (∞ - z) under unit horizontal area. It is the depth-related mean of the inherent fluorescence quantum yields at depths z' within the layer (∞ - z), Φfl(z'), weighted with exp[-(K0,z + KL,u,683nm,z)⋅(z’-z)], i.e.
If we accept this definition of the apparent quantum yield of fluorescence, Eq. (4) describing the dependence of the radiation Lint,fl,z on the apparent quantum yield of fluorescence at depth z then takes a relatively straightforward form:Eq. (6) we get the dependence of the apparent quantum yield of fluorescence at depth z, , on the values of Lint,fl,z, PAR0,z, , K0,z and KL,u,683,z measured at that same depth:
Finding an expression for the inherent quantum yield is far more complicated. It is not possible to find an appropriate analytical expression that would enable this yield at certain depths to be determined directly from environmental parameters measured at those depths (i.e. Lint,fl,z, PAR0,z, , K0,z and KL,u,683nm,z). It is, however, possible to establish such a dependence if we additionally take into account the values of the apparent quantum yield : these are not measured directly, but are determined indirectly from measurements using Eq. (7). It can be shown that there is a fairly simple relationship between inherent yields and the first depth-related derivative from depth profiles . This relationship, obtained by differentiation and the subsequent transformation of Eq. (5), takes the form:
Substituting this relationship in Eq. (7), we obtain the following operational definition enabling to be determined from measurements of Lint,fl,z, PAR0,z, , K0,z and KL,u,683nm,z as well as values of and its first derivative with respect to depth z, defined additionally for the same depth on the basis of calculations:
where the relative value of the derivative is determined approximately from the apparent quantum yields of fluorescence at two depths calculated according to Eq. (7): zu – above depth z, and zd – below depth z, that is, in accordance with the expression:
The above equations (Eqs. (7), (9)) are operational definitions of the SICF quantum yield. Similar expressions (with slight differences emerging from the fact that they used various approximation methods) were obtained by other authors (see [4,5,24]).
These two differently defined yields of photosynthesis – apparent and inherent – have been calculated in this paper. But the main part of the paper analyses the dependence of the inherent quantum yield on environmental conditions (Eq. (9)). The following two facts lend support to the use of the inherent quantum yield in later analyses. Firstly, this magnitude is formally classified among the inherent optical properties, which describe elementary optical phenomena – in this case the fluorescence of chlorophyll a, which is stimulated by the absorption of solar radiation by the photosynthetic pigments in phytoplankton. It refers to unit volume of sea water, that is, to a single point where we have unequivocally defined values of physical magnitudes (irradiance conditions, temperature, chlorophyll concentration etc.) characterizing the environment and governing the optical processes occurring in it and not, as in the case of the apparent quantum yield, to a column of water of thickness (z - ∞), in which the environmental parameters governing fluorescence do not have unequivocal values and would have to be obtained from appropriate averaging for this water column. Secondly, expression that describes the quantum yield of photosynthesis, used for analysis presented in this paper(see Chapter 4 The principal assumptions and scheme of the model), refers to the unit volume of sea water. It is worth drawing attention to the fact that the apparent quantum yield (Eq. (7)) takes somewhat lower values than the inherent yield. According to our studies the factor differentiating these two yields is on average about .
3. Presentation of the problem
Apart from being dissipated as heat, the solar radiation energy in the sea absorbed by phytoplankton pigments is used either for the primary production of organic matter, the consequence of photosynthesis, or is re-emitted as a result of the fluorescence of chlorophyll. SICF takes place in the band of half-width 5-15 nm in the ca 683 nm wavelength region .
Quantitatively, this alternative consumption of the energy absorbed by phytoplankton pigments during the processes of photosynthesis or SICF is correlated with the conditions in the marine environment. Three sets of factors are involved: (1) the level of PAR irradiance and the light absorption capacity of phytoplankton pigments at different depths in the sea (strictly speaking, the quantity of light absorbed by the photosynthetic pigments of phytoplankton); (2) the trophicity of the basin, the index of which is approximately and conventionally given by the chlorophyll a concentration in its surface waters, and (3) the temperature at different depths. Qualitatively, the nature of the relationship between the quantum yield of fluorescence and the first two sets of factors can be seen on the plots in Fig. 1 and Fig. 2 .
These illustrate the positions of the experimental points of the dependence of at different depths and in different seas on PAR irradiances (Fig. 1(a)) and surface chlorophyll a concentrations (Fig. 2(a)). In Fig. 1(b) and Fig. 2(b) the same relations are shown for data averaged in differently defined subsets.
It is evident from the figures that at low levels of PAR irradiance, the quantum yield of fluorescence tends to rise with increasing irradiance, but decreases abruptly when certain critical threshold values are exceeded. Moreover, values of are highest in oligotrophic oceanic waters (where chlorophyll concentrations are low) and fall dramatically with increasing trophicity . Hence, in the eutrophic waters of the Baltic Sea they may be several times and even several tens of times lower than in the Atlantic.
The dependence of on the water temperature, temp is rather weak and difficult to define (Fig. 3 ). Typically, there is a slight drop in the value of with increasing temp, which can be seen from the averaged relationship in Fig. 3(b). But this is not a universal feature, particularly in the case of recorded at high levels of irradiances in surface waters. In such a situation the tendency is often reversed, with the quantum yield of fluorescence increasing with increasing temp.
It was not possible to capture these subtleties (not shown in Fig. 3(b)) in a standard statistical analysis of our limited empirical database; this was feasible only with more complex analyses. The world literature boasts a number of papers covering various aspects of the problems involving the effects of environmental factors on phytophysiological processes in phytoplankton, including photosynthesis and fluorescence (e.g [1,3–5,26–30].). These problems are better understood where the relationship between photosynthesis and the above-mentioned environmental factors are concerned, whereas knowledge of the dependence of fluorescence on these factors is still somewhat rudimentary. To give an example: mathematical model formulas of the dependence of the quantum yield of photosynthesis in phytoplankton on the most important factors governing phytoplankton growth in the sea have been derived and presented in several papers, including some with the present author’s participation (e.g [31–41]. and others). In contrast, no such general mathematical description in relation to fluorescence has yet been derived. As marine photosynthesis and fluorescence are alternative means of deactivating the excitation energy in phytoplankton pigments, the analysis was based on the existing model of the quantum yield of photosynthesis, derived previously by our research group [38,40]; efforts were made to modify it in order to construct a mathematical description of the relationship between SICF of phytoplankton and environmental factors.
4. The principal assumptions and scheme of the model
4.1 Assumption 1
This model of the fluorescence yield makes use of the following mathematical expression, derived with the participation of the present author, for the quantum yield of the alternative process to fluorescence, i.e. photosynthesis (after );
It is the product of the maximum possible yield of photosynthesis (equal to 0.125 mol atC Ein−1 or 1 Ein Ein−1) and five dimensionless factors taking values from 0 to 1. Analogous factors for the quantum yield of fluorescence are described below (with the Eq. (13)). Their mathematical relationships with environmental factors were established by means of empirical studies and are given and discussed in detail: the factor fa in  see also , factor in , and the other factors in , see also 
The second of these papers  addressed the dependence, applicable to all oceans, of the changes in the quantum efficiency of photosynthesis caused by the variability in the trophic index , described by the factor . The first paper  analysed the links between the quantum yield of photosynthesis and environmental factors in oceans taking four independent variables into consideration: underwater irradiance, water trophicity, water temperature and nutrient content. The third paper  presented analyses of Baltic waters in which only first three variables were taken into account; the nutrient content was omitted. Given the empirical material available, this paper used all the relationships obtained from analyses performed for Baltic conditions except for the description of factors and fa, which, because of their greater universality and their not being directly dependent on the nutrient content, was taken unchanged from the results obtained for oceanic conditions.
4.2 Assumption 2
It is well known that the intensity of chlorophyll fluorescence in phytoplankton has two components: the fluorescence constant and the variable fluorescence (e.g [42–45].). It was therefore assumed that the quantum yield of fluorescence could also be expressed as a sum:
where is the quantum yield associated with the fluorescence constant , and is the quantum yield associated with the fluorescence variable .
The first of these components is always measurable, regardless of whether the reaction centres in the photosynthetic apparatuses of plants (PS2 RC) are open or closed. On the other hand, the variable is detectable only when PS2 RC are closed and fluorescence is taking place instead of photosynthesis.
4.3 Assumption 3
As in the case of photosynthesis, the mathematical expression for the variable component of the quantum yield of fluorescence can be written as the product of its theoretically possible maximum value (equal to 1 or 1 Ein Ein−1) and five dimensionless factors:
Three of the five dimensionless factors ffl,i appearing in Eq. (13) have the same significance (and similar values) as the corresponding factors in expression Eq. (11) describing the quantum yield of photosynthesis . They are:
- - - the non-photosynthetic pigment absorption effect factor. The excitation energy of photoprotecting pigments is not passed on to the reaction centre, so it can be neither utilized for photosynthesis nor re-emitted in the ca 683 nm spectral band by the chlorophyll in that centre. Hence ;
- - - the inefficiency factor in energy transfer and charge recombination. This describes the disturbances to the function of PS2 RC, preventing the uptake of excitation energy from the pigments, which could later be used for photosynthesis or be re-emitted in the form of chlorophyll fluorescence in the ca 683 nm band. Hence ;
- - - the factor describing the reduction in the portion of functional PS2 RC as a result of photoinhibition (). These centres are damaged, as a result of which, as before, there is no uptake of energy that could later be utilized for photosynthesis or be re-emitted in the form of chlorophyll fluorescence in the ca 683 nm band. Hence .
On the other hand, the factor describing the effect of irradiance and temperature on photosynthesis differs from the factor , which expresses the effect of these parameters on fluorescence. It is well known that , the classic dependence of photosynthesis on light and temperature (e.g [31,34,47].), also known as the light curve of photosynthetic efficiency at a given temperature, defines the relative number of open PS2 RC, and is consequently proportional to the quantum yield of photosynthesis. But the yield of the alternative process, i.e. fluorescence, is proportional to the relative number of closed PS2 RC, that is, to the factor , which when summed with gives unity. Thus .
Likewise, the trophicity of a water body affects the photosynthetic yield in a different way to the fluorescence yield. So, as in the case of the factors and , factor , describing the relation between the number of functioning PS2 RC and the trophicity of the water body, when summed with gives unity. This means that .
Based on the above assumptions, the mathematical description of the dependence of the variable component of the quantum yield of natural phytoplankton fluorescence on the trophicity of a water body, the light conditions there and the temperature at different depths in the water column is expressed by Eq. (13) and by the partial expressions for the individual compnent and factors given below in set of equations:37,38]
4.4 Assumption 4
Values of the quantum yield of fluorescence associated with the fluorescence component were determined for all the measurement points on the basis of the following equation:
– empirical values of the total quantum yield of fluorescence determined from the measurements mentioned in the Introduction according to the methodology described in 2.3;
In the next step of the analysis the correlation between the set of 1224 measurements of and selected environmental conditions was investigated. The fluorescence yield is best correlated with the trophicity index . Therefore, in the current version of the model, the value of the component is assumed to be dependent solely on the trophicity (see Fig. 4 ), in accordance with the equation obtained by non-linear regression:
5. Validation of the model description; final remarks
The mathematical description of the relationship between the quantum yield of fluorescence and environmental factors, derived and presented in this paper (see Eqs. (12)-(18), (20)), enables its variability under different conditions in the water column down to a depth of ca 60 m to be tracked. Figure 5 gives the modelled dependences of this yield on the light conditions in different trophic types of water, where surface chlorophyll varies from 0.035 to 7 mg m−3 (a), the surface irradiance PAR varies from 300 to 1500 (b), and temp varies from 5 to 30(c). Detailed analysis of these plots reveals certain limitations of the model, however. For example, the modelled values of this yield for superoligotrophic oceanic waters appear to exceed the empirical values (compare Fig. 1 and Fig. 3); conversely, the fluorescence yield calculated for supereutrophic waters is somewhat underestimated. This is also evident from Fig. 6(a) , which compares the fluorescence yield determined using the present model with the corresponding empirical values of based on measurement data. Figure 6(b) shows a histogram of the ratio . The errors of this approximation are given in Table 2 : they are not much greater than those for the earlier model for photosynthesis (standard error factor for photosynthesis x = 1.7 – see , for fluorescence x = 1.74). In the interests of strict accuracy, we should add that the discontinuities and sharp local minima visible on some of the plots in Fig. 5, which are particularly distinct with regard to oligotrophic waters, are not a reflection of reality but represent artifacts due to the imperfections of the statistically approximated functional relationships. The statistical description of these relationships obtained here could, if subjected to statistical analyses of greater penetration, describe the empirical data with greater precision and accuracy. However, such a more detailed analysis would only be meaningful with respect to a much more extensive data set.
To recapitulate, the aim of this work has been achieved. A mathematical model has been derived to describe the dependence of the Sun-induced Chlorophyll Fluorescence on a variety of environmental conditions across a wide range of their recorded natural variability. As mentioned earlier, this model refers to the quantum yield in the surface waters of basins down to a depth of about 60 m. In the case of oligotrophic basins this corresponds to a surface layer of water of approximately half the thickness of the euphotic layer; in supereutrophic waters, this surface layer may be coincident with the whole euphotic layer and possibly witha layer of water even twice as thick. With the formulation of such a model, it will be possible, among other things, to account for the dependence of the quantum yield of fluorescence on environmental factors in fluorescence methods of determining different characteristics of marine plant communities. Given the rather sparse experimental material available, these analyses are of a preliminary nature; they will be repeated in the future on a richer set of data and expanded to include a larger number of environmental factors affecting the fluorescence and photosynthesis of phytoplankton, in particular, the dependence of the quantum yield of fluorescence on the nutrient content in the sea, based on the mathematical description given in .
(,- measured values, - estimated values) - mean systematic error;
- standard deviation (statistical error) of ε; (- mean of ) - mean logarithmic error);
(- standard deviation of the ) - standard error factor; - statistical logarithmic errors.
The study was partially financed by MNiSW (Ministry of Science and Higher Education) within the framework of IO PAN's statutory research and also as part of MNiSW research project N306 1391 33 in 2008-2010. Partial support for this study was also provided by the project “Satellite Monitoring of the Baltic Sea Environment – SatBałtyk” funded by the European Union through European Regional Development Fund contract No. POIG 01.01.02-22-011/09
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