Improved Estimation of F-Ratio in Natural Phytoplankton Assemblages

Deep-Sea Research I 46 (1999) 1793}1808

Improved estimation of f-ratio in natural phytoplankton assemblages Marc Elskens! *, Leo Goeyens!, Frank Dehairs!, Andrew Rees", Ian Joint", Willy Baeyens! !Laboratory of Analytical Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium "NERC Centre for Coastal and Marine Sciences, Plymouth Marine Laboratory, Prospect Place, The Hoe, Plymouth PL1 3DH, UK Received 2 February 1998; received in revised form 19 October 1998; accepted 28 December 1998

Abstract

Statistical properties in non-linear regression models of f-ratio versus nitrate relationships were investigated using a case study conducted at the European continental margin (Project OMEX). Although the OMEX data "t within the family of empirical models introduced by Platt and Harrison (1985) Nature 318, 55}58), it is shown that the discrepancy between experimental and "tted values is larger than expected, assuming that the data are accurate. Since the ambient ammonium concentration plays a leading role in regulating new production, the present analysis was extended to include explicitly the e!ect of ammonium on f-ratio versus nitrate plots. Experimental results based on controlled ammonium additions were used to express the f-ratio as a function of both nitrate concentration and ammonium inhibition, i.e. H f(,- ?).I(,& @). The estimation behaviour of the data set/model combination was analysed by testing the appropriateness of various model functions for f H and I. The best "t was obtained with a sigmoid curve, for which mean values of the random #uctuation are almost commensurate with the estimated uncertainty of the measurements with natural phytoplankton assemblages. Precision of the f-ratio estimates was further assessed from contour diagrams of constant likelihood. The signi"cance, validity limits of the estimated parameters, and the relevance of the proposed model as a predictive tool are discussed. Overall, this empirically determined model results in improved precision of f-ratio estimations. ( 1999 Elsevier Science Ltd. All rights reserved.

* Corresponding author. Tel.: 0032-2-6292716; fax: 0032-2-6293274. E-mail address: [email protected] (M. Elskens)

1. Introduction

Knowledge of the relationship between nutrient supply and phytoplankton production is essential for understanding the major processes in marine ecosystems. It de"nes the sequestration of atmospheric CO and its transport into deep water via the biological pump (Longhurst and Harrison, 1989). Dugdale and Goering (1967) pro- vided an outstanding model, which distinguishes new and regenerated primary production, according to the origin of the nitrogenous nutrient taken up by phytoplankton. This model assumes a steady state in the surface layer, which relates the concentration of the limiting nutrient to the corresponding speci"c uptake rate and the loss rate of phytoplankton (Dugdale, 1967). The major nitrogen stock in the ocean is nitrate, which is assimilated by phytoplankton in the euphotic zone and incorporated into the food chain. The supply rate of nitrate to the euphotic zone regulates primary production (Dugdale and Goering, 1967) and determines the maximal export rate due to settling of particles (Eppley and Peterson, 1979). In order to address the spatiotemporal variability in nitrogen uptake by the autotrophic community and particle export to the deep sea, various models, using f-ratio as a key parameter, have been suggested (Eppley and Peterson, 1979; Platt and Harrison, 1985; Dugdale and Wilkerson, 1989). The f-ratio, usually de"ned as the ratio of nitrate uptake to the uptake of nitrate and ammonium (plus occa- sionally urea), increases asymptotically with increasing nitrate concentration. It is assumed that this relation applies throughout the World Ocean (Harrison et al., 1987; Sathyendranath et al., 1991; Vezina, 1994) but breaks down in oceanic regions where nitrogen does not limit primary production, such as the high-nutrient low-chlorophyll areas (HNLC, Minas et al., 1986). Recently, sea-surface temperatures have been used as a proxy to model surface nitrate distributions, as well as total and new production (Dugdale et al., 1997). The latter approach provides an interesting tool to extend productivity estimates to large ocean basin surfaces, and to integrate productivity over entire growth seasons. It must be emphasised, however, that f-ratio versus surface nitrate concentration always shows considerable scatter, and the relationship is signi"cantly a!ected by additional parameters such as the availability of other nutrients and phytoplankton species composition. The present study aims to reduce the uncertainty associated with the estimation of f-ratio and provides con"dence regions for the f-ratio versus nitrate relation. Since most of the models discussed are non-linear, con"dence regions and intervals were calculated according to the likelihood method. Although this method is computation- ally demanding, it produces con"dence limits with observed coverages that are closer to the nominal probability than those obtained using linear methods (Donaldson and Schnabel, 1987). This study is an attempt to investigate both the nitrate and ammonium uptake regimes, since recent work has shown that nitrate uptake may be substantially reduced in the presence of ammonium, even at nanomolar concentrations (Wheeler and Kokkinakis, 1990; Harrison et al., 1996). The data used to investigate these relationships are nutrient distribution pro"les and nitrogen uptake data obtained during a 3-year study in the northeastern Atlantic Ocean margin (Elskens et al., 1997; Rees et al., 1999). M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1795

2. Methods

2.1. Data collection and treatment

The data were obtained between March 1993 and October 1995 during seven cruises on board R.V. Belgica, R.R.S. Discovery, R.R.S. Charles Darwin and R.V. Valdivia conducted as part of the Ocean Margin Exchange (OMEX) project 1993} 1995. Special care was exercised in computing nitrogen uptake rates with respect to the isotope mass balance (Collos, 1987), excess tracer addition (Harrison et al., 1996), and isotope dilution e!ects (Glibert et al., 1982). Full details of the method used are set out in publications by Elskens et al. (1997) and Rees et al. (1999). In the present study, only measurements made in the upper 20 m layer are considered, yielding a set of about 200 data distributed among "ve sites located across the ocean margin of the northeast Atlantic (Joint et al., 1999): an area representative of the Celtic Sea Shelf (493}503N103}10.53W), the OMEX 1 site on Goban Spur (113}123W), the OMEX 2 site (123}12.73W), the OMEX 3 site (133}16.53W) and La Chapelle Bank (473}483N 63}83W).

2.2. Estimation of the probable error awecting measurements with natural assemblages

To gauge whether the value of the residual mean variance is su$ciently small in a given non-linear regression model (see below), it is helpful to determine the e magnitude of the experimental uncertainties (&) that can a!ect the measurements. An estimate of the standard deviation was "rst derived from the variability associated with regression lines for the nutrient standards. This method yields a conservative l ( \ ( l estimate of the standard deviation with values of 0.05 M for 0 [NO ] 10 M l ( > ( l and 0.04 M for 0 [NH ] 1 M. The variability of N abundance measurements was estimated from replicate determinations when available. Samples with N abundance ranging from 0.5 up to 3%, as generally observed in this study, show a coe$cient of variation of about 5%. Propagation of these experimental uncertainties on N-uptake rates and f-ratio was estimated according to Miller and Miller (1988). e ' l It follows that the mean (&) is 0.082 (CV 15%) for nitrate 1 M, but is relatively higher at lower nutrient concentrations, reaching 0.053 (CV 39%) when both nitrate and ammonium are below 0.1 lM.

2.3. Non-linear regression modelling

Computerised non-linear curve "tting was used throughout this study to obtain the parameters of the independent variables that give the best "t between the anticipated equation and the data. The "tter used a Marquardt}Levenberg algorithm, which seeks the observed and predicted values of the dependent variable by an iterative procedure (SigmaPlot software, Jandel Scienti"c). Unless otherwise stated, 1796 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808

e the stochastic term () in the model functions refers to an additive error assumption. The loss function to minimise is L h " + ! h RSS( ) (>G f (XG; )) , G h where RSS( ) is the residual sum of squares, >G the dependent variable, XG the explanatory variable(s) and h a vector of p parameters to be estimated. The goodness of "t was assessed by comparing the magnitude of the residual variance: s"RSS(hK )/(n!p), amongst various model functions (Healy, 1984; Ratkowsky, 1990), where RSS(hK ) corresponds to the least-square estimate of the p parameters and n is the sample size. e e It was considered to be satisfactory when () approached (&). However, additional information is required to determine which of several compet- ing models is the most appropriate. This was achieved (i) by examining the residuals e after "tting the models: the normality assumption for () being ascertained by the Kolmogorov}Smirnov test (Bradley, 1968) and (ii) by computing asymptotic standard errors on the parameter estimates, as well as parameter dependencies (SigmaPlot software, Jandel Scienti"c): variance of the parameter, other parameters constant D"1! . variance of the parameter, other parameters changing Parameters with dependencies near 1 are strongly interdependent. This may indicate that the equations used are overparameterised. It is noteworthy that the greater the number of parameters, the greater is the extent of non-linearity e!ects. As a basic principle, it is better to "nd non-linear regression models that exhibit close-to-linear behaviour, i.e., with almost unbiased, normally distributed and minimum variance estimators, but at the same time low parameter correlations (Ratkowsky, 1990). Finally, it is important to provide precision on the estimated parameters. Statistical techniques for obtaining this information include the use of con"dence regions and intervals. Regions are used to make a joint statement about the set of parameter values, while intervals apply to individual parameters separately. Since RSS(hK )in non-linear regression models are generally biased estimators of the true parameters, the most commonly used methods based on linearisation are inappropriate (Beale, 1960). Comparison of the leading methods to approximate con"dence regions and intervals in non-linear least squares has indicated that the likelihood method is robust and provides exact con"dence regions (the coverage probability equals the nominal one), when models are close-to-linear in their behaviour (Donaldson and Schnabel, h hH 1987). Likelihood con"dence regions for parameters (2N) are those values for which hH ! hK ) RSS( ) RSS( ) s pFN L\N \?, h while the likelihood con"dence interval for the jth parameter H is included between h !hK the points that maximise ( H H) and for which: hH ! hK ) RSS( ) RSS( ) s F L\N \?. M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1797

3. Results and discussion

3.1. The f-ratio algorithm

The plot of f-ratio versus ambient nitrate for the available OMEX data (Fig. 1) has the aspect of a saturating exponential, which "ts within the family of empirical models introduced by Platt and Harrison (1985). The equation is

"a ! !a \ #e f-ratio (1 exp( [NO ]) (), (1)

e a a where () is an additive random error. The parameters and of this exponential equation (Table 1), derived from the OMEX data, agree well with the range of estimated values for other marine environments (Harrison et al., 1987; Sathyen- dranath et al., 1991). There is considerable scatter, however, especially in the region of the curve where nitrate is low. For ambient nitrate concentrations up to 0.1 lM, the corresponding f-ratio varies between (0.05 and 0.5. This variability is larger than expected from propagation of random and mutually independent errors on concentration and N abundance measurements. Although estimates of the experimental e uncertainties on f-ratio, (&) (see Methods section) are proportionally higher at low l e levels of nitrate (0}0.1 M), they consistently remain below (), except at the asymp- e totic end of the curve (Fig. 1). Furthermore, the assumption that () values are normally distributed with mean zero and constant variance must be rejected at the 0.05 level of signi"cance (Table 1). The conclusion from this analysis is that the nitrate concentration alone is a poor predictor of the f-ratio, and that a model with one single explanatory variable might be inadequate to provide seasonal estimates of large-scale average new production in this ocean margin ecosystem. Obviously, the exportable fraction of primary production ( f-ratio, Eppley and Peterson, 1979) depends largely on the e$ciency of nutrient regeneration in the surface layer and its variation in space and time (Harrison, 1992). In terms of nitrogen #uxes, the conceptual model of Dugdale and Goering (1967) and Eppley and Peterson (1979) has clearly illustrated that the uptake of regenerated nitrogen, mainly ammonium, plays a key role in the biogeochemical nitrogen cycle. Moreover, it is a basic tenet of nitrogen utilisation studies that ammonium is generally the preferred substra- te for most phytoplankton in the World Ocean (Smith and Harrison, 1991) and that ammonium inhibits the uptake of nitrate (Dortch, 1990). Therefore, the present analysis was extended to include explicitly the e!ect of ammonium on f-ratio versus nitrate relationships.

3.2. Ewect of ammonium

Additional incubation experiments were carried out to assess the e!ect of ammonium additions on the nitrate uptake rate (Elskens et al., 1997). The results indicate rapid initial reduction of nitrate uptake, even at ammonium levels as low as 0.1 lM. However, inhibition of nitrate uptake was rarely complete, even at ammonium 1798 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808

Fig. 1. Relationship between nitrate and f-ratio for (A) the whole range of nitrate concentrations measured in the OMEX Project and (B) for the low nitrate concentrations below 1 lM. The continuous line is given by Eq. (1), and the dashed line represents the 95% likelihood con"dence regions. concentrations above 1 lM. The best agreement between calculated and experimental data was obtained with a model based on the equation for non-competitive inhibition, as suggested by Harrison et al. (1996):

I *NH o "o*! ! -NO NOA1 #* B, (2) K* NH M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1799

Table 1 Statistics of non-linear regression models "tted to f-ratio. Parameter values are means$asymptotic standard errors. D represents values for the parameter dependency, s the residual variance, RSS(hH) the residual sum of squares corresponding to the 95% likelihood con"dence regions, and K the maximum absolute deviation of the Kolmogorov}Smirnov test which must be compared to the critical value of 0.062 at the 0.05 level of signi"cance

a a b b hH e Eq. s RSS( ) K (P)

(1) 0.68$0.02 1.5$0.14 **0.0222 4.6 0.121 D"0.19 0.19 ** (1)}(4) 1.2$0.30 2.0$0.20 0.08$0.07 0.77$0.07 0.0176 3.6 0.132 D"0.99 0.26 0.99 0.91 (1)}(4)! 0.95$0.07 2.1$0.21 0.24$0.06 1 0.0180 3.7 0.120 D"0.91 0.24 0.92 * (4)}(5) 0.48$0.04 * 0.26$0.8 0.90$0.16 0.0151 3.2 0.048 D"0.29 * 0.94 0.94 (4)}(5)! 0.46$0.03 * 0.31$0.02 1 0.0150 3.1 0.055 D"0.07 * 0.07 * (4)}(6) 1.5$0.27 0.47$0.07 0.19$0.04 0.92$0.04 0.0144 3.0 0.053 D"0.20 0.38 0.90 0.91 (4)}(6)! 1.5$0.26 0.45$0.07 0.22$0.02 1 0.0144 3.0 0.060 D"0.13 0.33 0.29 *

! b " In these model functions, the "tting procedure was carried out with 1.

* where NH is the amount of ammonium added, K* the half saturation constant for o inhibition, I the maximum inhibition achieved, -NO the nitrate uptake rate in the oH presence of ammonium, and -NO the maximum uptake rate at zero addition of ammonium. Mean values for K* and I , "tted from the additional inhibition experiments, were 0.55 (SE"$0.3) lM and 1 (SE"$ 0.3), respectively (Elskens et al., 1997). They are in agreement with several results published on inhibition of nitrate uptake under l controlled ammonium conditions. Price et al. (1994) found 0.40 M for K* and 1 for I in the nitrate-rich equatorial Paci"c. Recently, Harrison et al. (1996) observed ( l lower K* values ( 0.05}0.2 M) in oceanic waters of the North Atlantic and values as high as 0.6 lM in samples taken at some neritic stations. The authors also observed ( that nitrate uptake was rarely completely inhibited (i.e. I 1). Eq. (2) allows the f-ratio to be expressed as a function of both the nitrate and ammonium concentrations:

f-ratio"f H I #e , (3) (,-_?) (,&_@) () where f H is assumed to be an empirical function of the nitrate concentration and a a a vector of unknown parameters [ , 2, G] to be estimated, and [NH>] I" ! 1 b #b , (4) [NH] 1800 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 where I is the fractional inhibition term, [NH] the ambient ammonium concentra- b " b " tion, K*/I and 1/I . I was reparameterised for inference since the estimation behaviour of a Michaelis}Menten equation is improved when both parameters appear in the denominator (Ratkowsky, 1986). Fitting the experimental data with the combination of Eqs. (1) and (4) allows for a signi"cant reduction of the mean square residuals. Yet the results show high a b b e parameter dependencies for , and , while the normality assumption for () must be rejected at the 0.05 level of signi"cance (Table 1). Since Eq. (1) has rather poor estimation properties, other model functions for f H that are close to linear in their behaviour were investigated.

3.3. Model validation

The simplest model satisfying most of these requirements is a sigmoid curve with a a single parameter : H" ! ! \ ? f 1 exp( [NO ] ). (5) Only a slight improvement over Eq. (5) was achieved by using a more complicated model, such as for example a two-parameters logistic function (Table 1): H" # !a \ !a \ f [1 exp( ([NO ] ))] . (6) In comparison with Eqs. (1) and (4), the parameter estimates are improved (the asymptotic standard errors are smaller compared with the estimates themselves), and e there is no evidence for a lack of normality of () in both models (Table 1). a b Eqs. (4) and (5) yield rather narrow likelihood con"dence intervals for , and b b values but with an interval for that is not signi"cantly di!erent from 1 (95% CI: b b 0.77}1.53). Therefore, when is set equal to 1 in Eq. (4), the intervals for (95% CI: a 0.27}0.36) and (95% CI: 0.40}0.54) become also smaller without a!ecting the goodness of "t (Table 1). Similar conclusions can be drawn for the other models. Under these conditions, it should be noted that all three models provide comparable b l estimates for with values ranging from 0.22 to 0.31 M. In view of the ecological signi"cance of the derived kinetic parameters, it should be stressed that a model with two adjustable parameters for I (Eq. (4)) yields K* and I values that are comparable to those of Harrison et al. (1996). However, such a model exhibits overparameterisation: variations of K* can be counterbalanced by #uctuations of I at contours of equal residual sum of squares. This results in a deterioration of the estimation behaviour of the model function with con"dence regions for the parameters that deviate from the ellipsoidal shape, and asymmetric con"dence limits for the individual parameters. In contrast, a model with only one b adjustable parameter, , provides a K* value that is close to the one obtained in the same area from incubation experiments with controlled ammonium additions (Elskens et al., 1997; Table 1). On the other hand, it can be demonstrated that replacing the Michaelis}Menten formulation in Eq. (4) by the linear term " !b > I (1 [NH ]), as has been suggested by Dortch (1990), is less adequate whatever model is used for f H. Similar conclusions can be drawn for models relating nitrate M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1801

Fig. 2. Distribution of nutrient concentrations (A) and f-ratio (B) at Goban Spur in spring, summer and fall. The continuous line in B is given by the combination of Eqs. (4) and (5).

" uptake and f-ratio to the corresponding ratio of nutrients, e.g. RNO \ \ # > [NO ]/([NO ] [NH ]), which implies competitive inhibition (Harrison et al., 1987; Collos, 1989). From these analyses, it may be concluded that the introduction of ambient ammonium concentration as a second explanatory variable reduces signi"cantly the mean square residuals when compared to Eq. (1). Fig. 2 illustrates how Eqs. (4) and (5) "ts the e experimental data: mean values of the random #uctuation () around the "tted line are e almost commensurate with the estimated uncertainty (&) on the measurements on board. Considering the experimental uncertainties, the goodness of "t is satisfactory, and only slight improvement will result from the use of a more complicated model. The model proposed here (Eqs. (4) and (5)) can, therefore, be considered as an acceptable compromise between data acquisition, predictive power and statistical properties.

3.4. Signixcance and validity limits of f-ratio estimates

A major objective of the present study was to evaluate the precision of estimated f-ratios. Therefore, likelihood con"dence regions were calculated for the set of 1802 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808

Fig. 3. 95% upper and lower likelihood con"dence limits for f-ratio estimates as function of both nitrate and ammonium concentrations. For the sake of clarity, only lines corresponding to 0.05, 0.1 and 1 lM ammonium were drawn. parameter values introduced in Eqs. (4) and (5). A family of f-ratio versus nitrate curves was generated, each of them corresponding to a given ammonium concentration and being characterised by its con"dence region. Fig. 3 represents the upper and lower limits of the joint con"dence regions corresponding to the individual curves. When ammonium concentrations are not available, the model provides an upper limit for the f-ratio estimate, which corresponds to a situation of ammonium depletion. It should be noted that if all regenerated N-#uxes equal zero, then the f-ratio should be 1, whatever the ambient level of nitrate. However, this seems to be unrealistic in "eld studies with natural phytoplankton assemblages. Therefore, more realistic estimates of f-ratio require reliable ammonium values for the regional or the seasonal data set. Fig. 4 shows the results obtained by applying the model to data obtained during two zigzag transects across the continental margin. The area covered was from Goban Spur to the Southern part of the Gulf of Biscay in September 1993 and April 1994, respectively. During both cruises, surface nitrate was measured continuously (samp- ling frequency of 3 min), while surface ammonium was determined at "xed stations. f-ratio was computed using Eqs. (4) and (5), and the precision of the estimates was assessed by considering the standard deviation associated with the nutrient determinations (Table 2). Obviously, larger experimental uncertainties result in broader con"dence limits for the predicted f-values. The comparison between Fig. 4A and B shows that f-ratio is not a mere image of the nitrate distribution. For instance, M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1803

Fig. 4. Mapping of nitrate concentrations and f-ratio along the continental margin during (A) September 1993 and (B) April 1994. f-ratio was computed with Eqs. (4) and (5) using horizontal pro"les of nitrate and ammonium concentrations as described in the text. 1804 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808

Fig. 4b. M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1805

Table 2 95% con"dence limits for f-ratio estimates using Eqs. (4) and (5) and contours of constant likelihood. Nutrient values are means, and standard deviations were calculated as described in the Method Section 2

\ l > l Site/period NO ( M) NH ( M) f-ratio

La Chapelle Bank Sep. 1993 1.6 0.10 0.46}0.62 Apr. 1994 7.8 0.25 0.44}0.59 Celtic Sea Shelf Sep. 1993 0.06 0.23 0.04}0.22 Apr. 1994 4.0 0.05 0.63}0.86 OMEX 1 Sep. 1993 0.24 0.07 0.24}0.42 Apr. 1994 7.0 0.05 0.67}0.92 OMEX 2 Sep. 1993 0.17 0.06 0.20}0.40 Apr. 1994 7.6 0.09 0.61}0.82 OMEX 3 Sep. 1993 0.06 0.05 0.06}0.34

during September 1993 in the vicinity of La Chapelle Bank (47.53}48.53 N and 73}83 W), nitrate and ammonium concentrations were 1.5 and 0.10 lM, respectively, yielding f-ratio up to 0.62; in contrast, in April 1994, although the nitrate concentration was "ve times higher, maximal f-ratio estimates rarely exceeded 0.60, because the ambient level of ammonium was close to 0.25 lM (Fig. 4 and Table 2). Another observation reported by Elskens et al. (1997) and Rees et al. (1999) is that in April 1994 a number of o!-shore stations were sampled before strati"cation occurred. The apparent time-scale di!erence in the growth season maturity is clearly suggested in Fig. 4B with a spot of nitrate-based production (48.53 N}10.53W) expanding along the continental margin towards Goban Spur. In support of this view, Morin et al. (1991) have shown that the nutrient assimilation and phytoplankton development take place sequentially on the Armorican shelf and that a time lag of nearly three months may exist between the initiation of the bloom in South Brittany and in the central part of the continental shelf.

4. Conclusions

The model proposed in the present paper yields a substantial improvement in the f-ratio versus nitrate plots when compared to previous models, since it narrows the e stochastic term (), which is especially important at lower nitrate concentrations. It is known that, as a result of regeneration processes, considerable enhancement of the ammonium availability generally occurs during late spring following the decline of the phytoplankton bloom. This study allows for the introduction of seasonal trends and hence distinguishes between seasonal f-ratio versus nitrate relations. Precision of the 1806 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 f-ratio estimates was assessed with the help of contour diagrams and determinations of the likelihood con"dence regions. Since the model exhibits close-to-linear behaviour, the likelihood method is expected to produce con"dence regions with data coverage approaching the nominal (Donaldson and Schnabel, 1987). Nevertheless, both nitrate and ammonium concentrations are subject to error leading to simultaneous variation of the dependent variable ( f-ratio) and the explanatory variables, which can be important when the errors on nitrate and ammonium are large. Therefore, it should be possible to improve the goodness of "t using low-level nutrient determination methods (Garside, 1982; Brzezinski, 1987). Another source of variability is the fact that the f-ratio is also related to a number of other biogeochemical variables. Herein, this variability is included both in the parameter estimates (e.g., values for the 95% likelihood con"dence intervals) and e in () that is assumed to be randomly distributed. Namely, the rationale for the latter hypothesis is based on the central limit theorem, which states that, for a sample size su$ciently large (n'100), the distribution of a statistic from the sample approaches normal shape even if the distribution of the variable in question is not normal (Bradley, 1968). Our results show, however, that special care should be exercised in selecting suitable parameterisation, as di!erent parameter- isations of the same basic model may di!er greatly in their estimation behaviour (Table 1). Eqs. (4) and (5) are easily implemented and provide a dynamic description of the nitrate and ammonium uptake regimes from routinely measured variables in surface waters. When it is not possible to make "eld measurements, the coupling of satellite data with nitrate and sea surface temperature (SST) relationships might be helpful for ecosystem description and f-ratio estimation (Sathyendranath et al., 1991). Such results will not be as conclusive, however, since omitting the ammonium e!ects reduces the quality and signi"cance of the data. Furthermore, according to Bronk et al. (1994), the current lack of quantitative information on DON #ux and nitri"cation rates within most data sets may be bias (i.e., when these rates are not negligible compared to those of DIN) the use of f-ratio in estimating sources and sinks of biogenic elements.

Acknowledgements

The authors would like to thank the Captains and crews of the R.V. Belgica, R.R.S. Discovery, R.R.S. Charles Darwin and R.V. Valdivia; Roland Wollast and Lei Chou of the UniversiteH Libre de Bruxelles for the scienti"c co-ordination of the OMEX project; and Roy Lowry and Zelko Loncar of the NERC British Oceanographic Data Centre. This work was supported by the European Union in the framework of the Mast 2 Program, contract number MAS2-CT93-0069 (OMEX project), by the Belgian State-Science Policy O$ce, Impulse Program Global Change, contract number GC/03/010, and by the Strategic Research Project 1 of the Plymouth Marine Laborat- ory, a component of the Natural Environment Research Council. M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1807

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