A method is presented for estimation of seasonally-varying, total loss rate of phytoplankton from time series of satellite-derived phytoplankton biomass data. The loss is calculated as the difference between the (modelled) rate of photosynthesis and the observed, realized rate of change of phytoplankton biomass. A Monte Carlo procedure is used to recover the loss rates. The (biomass-normalized) total loss rate shows a seasonal cycle with values ranging from 0.5 to 3 mg C (mg Chl)−1 h−1 and shows an abrupt shift during the spring bloom. On the other hand, the absolute loss rate increases during blooms, a consequence of the increase in the biomass. The normalized total loss rate can be further expressed as a time-varying fraction of the assimilation number. The fraction lies in the range from 0.2 to 0.8. During the increasing (decreasing) phase of phytoplankton blooming, the ratio of growth to total loss increases (decreases), such that this ratio may have value as an ecological indicator for blooms.
 Although it is simple to measure the rate of phytoplankton photosynthesis in the ocean, it is extremely difficult to measure the corresponding loss rates. The losses include many components, such as respiration, grazing, sinking and natural mortality, each one presenting a major challenge to direct measurement. The sum of the losses, which we may refer to as the total loss rate is an ecologically- and biogeochemically- significant quantity for which we have only limited information [Platt et al., 1991; Siegel et al., 2002].
 One way to estimate photosynthetic rate is through application of remotely-sensed data on ocean colour [Platt et al., 2008]. We may then ask whether the loss rate could be approached in the same way. Serial images, or serial composite images, of chlorophyll concentration allow us to calculate, by difference at successive time steps, the realized rate of chlorophyll increase (difference between growth and total loss). Given time series of photosynthetic rate (estimated from the chlorophyll fields) and realized rates of chlorophyll increase (from the serial chlorophyll fields), we can in principle recover a time series of the difference, namely the total loss rate. This is the method proposed by Platt and Sathyendranath .
 Here we implement the proposed method for estimation of total phytoplankton loss rate using remotely-sensed chlorophyll fields for a 1° × 1° square on the continental shelf of Nova Scotia. We use the Nearest-Neighbour Method (NNM) of Platt et al.  to assign the photosynthesis parameters and estimate primary production, together with a Monte Carlo method to recover the loss rate and its probability density. The results will be relevant to a variety of applications in biogeochemistry and ecology of marine systems [Falkowski and Woodhead, 1992].
 This paper is structured as follows. Section 2 introduces the method and discusses the observations and parameters. Stochastic simulation results are presented in Section 3. Summary and conclusions follow in Section 4.
2. Methods and Sources
 The net change of phytoplankton biomass B (concentration of chlorophyll a) with time t at a given location (x, y, z) is the difference between growth (primary production) and the total loss by respiration, mortality, grazing and sinking, along with a correction for physical processes. It might be formulated simply as
where χ is the ratio of carbon to chlorophyll and Γ corrects for the influence of physical processes. The superscripts indicate normalization to chlorophyll B on the photosynthesis P and the loss L (the variables and parameters are summarized in Table 1), which are both computed as rates of change, in carbon units. Note that this approach is independent of the choice of functional form for LB. For example, we could replace the simple term LBB by a quadratic, μB2 say, and still solve for the loss. The absolute loss rate LBB will not change, but the parameter magnitude (LB or μ) would be different.
Table 1. Glossary of Mathematical Notation
Quantity and Description
Biomass, as concentration of chlorophyll a
mg Chl m−3
Irradiance in the Photosynthetically Active Range (PAR)
Ik = PmB/αB
Adaptation parameter of PB − I curve
Total daily PAR
Horizontal eddy diffusivity
Vertical eddy diffusivity
Biomass-normalized total loss rate
mg C (mg Chl)−1 h−1
Primary production rate normalized to biomass
mg C (mg Chl)−1 h−1
mg C (mg Chl)−1 h−1
Horizontal velocity scale
Vertical velocity scale
Scale of horizontal excursion
Scale of mixed-layer depth
Initial slope of the PB − I curve
mg C (mg Chl)−1 h−1 (W m−2)−1
Carbon-to-chlorophyll ratio of phytoplankton
mg C (mg Chl)−1
Correction for physical processes
 Using remotely-sensed data on ocean colour, we can retrieve B. We can also estimate PB and we have general knowledge of Γ and χ. The total loss rate LB is the only unknown quantity in equation (1) and we may solve for it. Equation (1) is solved numerically as a difference equation written as
where subscript n is an index for the time step and Δt = tn+1 − tn is the time interval between two composite images for time tn+1 and tn. We may express LnB explicitly as
The normalized loss rate LnB in equation (3) is computed through the repeated random sampling of B, PB, χ and Γ, a procedure known as the Monte Carlo method. The principle of the Monte Carlo method is to select randomly a large number of points from the ensemble of observations, with equal probability of their being in any part of the ensemble [Forsythe et al., 1977]. The advantage of the Monte Carlo Method is that it gives a quantitative assessment of the uncertainty in LB. The statistical modes, confidence intervals and probability density are thus estimated from a large ensemble of LB. Such information may be difficult to acquire by other methods. Here, some 1000 points are chosen randomly from each ensemble of model variables and parameters for the Monte Carlo calculation of LB using the random number generator UNIFRND in MATLAB statistical toolbox (Mathworks, Inc.). We have tested other sample sizes, and are satisfied that the choice of 1000 points ensures the convergence of the model solution. The carbon-to-chlorophyll ratio χ lies in the range of 30–80 mg C (mg Chl)−1 [Platt et al., 1991; S. Sathyendranath et al., Carbon-to-chlorophyll ratio and growth rate of phytoplankton in the sea, submitted to Marine Ecology Progress Series, 2008] and is assumed to have a uniform probability distribution.
2.2. The Phytoplankton Biomass and SST
 For the biomass B, we use the SeaWiFS data series derived from weekly composite images of chlorophyll concentration. The chlorophyll data were processed using SeaDAS software and the OC4 (v4.3) algorithm (NASA), and then mapped at a 1.5 km × 1.5 km resolution onto a mercator grid. Composite images were then created using SeaDAS/IDL, by taking the average of all valid chlorophyll values for a given latitude/longitude grid point from all remapped files corresponding to the given weekly period.
 The time series of SeaWiFS biomass and AVHRR sea-surface temperature (SST) from 1998 to 2007 are used to construct climatological ensembles for the 1° × 1° square on the western continental shelf of Nova Scotia. The square includes a large number of pixels (≤ 3 × 103) for a given composite image. The square is centered at 43°N, 64.5°W and is located within the Northwest Continental Shelf (NWCS) ecological province [Longhurst, 1995].
 The probability distributions for the climatological ensembles of the SeaWiFS biomass (Figure 2a) are estimated by the kernel density, a smoothed version of the histogram [Silverman, 1986] (Mathworks, Inc.). The greater (lesser) width of the envelope of the normalized kernel density indicates a larger (smaller) inter-annual and spatial variability of biomass in the 1° box. The biomass cycle in the 1° square (Figure 1a) shows the general features of the canonical seasonal cycle of spring and fall blooms. The general concept of the annual cycle of phytoplankton in this area was codified, for example, by Longhurst . A spring bloom develops from the shallowing of mixed-layer depth and the accumulated winter nitrate. A fall bloom is fueled by the sub-pycnocline nutrients when the winter overturn begins.
 We have applied the algorithm of Sathyendranath et al.  to determine whether the phytoplankton community structure at pixels within the 1° square is dominated by diatoms. Figure 1b shows the normalized kernel density estimates of the occurrence of diatoms. There are two peaks in the time series of diatom occurrence associated with the two blooms.
 The normalized kernel density estimates of the SST in the 1° square are calculated from the climatological ensembles of AVHRR SST and are shown in Figure 1c. The seasonal cycle of SST (Figure 1c) has a range of about 18°C and is affected mainly by net sea-surface heat flux with some contributions from advection and mixing [Umoh and Thompson, 1994].
2.3. Assignment of Photosynthetic Parameters
 To compute normalized primary production PB, we first assign the photosynthetic parameters PmB and αB on a pixel-by-pixel basis using the NNM [Platt et al., 2008]. The total daily PB can be written as the functional response of phytoplankton photosynthesis to available light, excluding photoinhibition [Platt et al., 1980, 1991]:
in which I(t) is the photosynthetically-available radiation at time t, D is daylength and the parameters αB and PmB are assigned by searching the archive of parameter data for stations corresponding to the remotely-sensed chlorophyll concentration and sea-surface temperature. The archived parameters were measured at sea by photosynthesis-light experiments using the 14C method [Platt and Jassby, 1976]. The NNM supplies the first-order estimates of the parameters for each pixel. We also tried refinements of the NNM by restricting the search to stations falling in the Northwest Continental Shelf (NWCS) ecological province. This refinement decreased the variances of PmB and αB. However, because there are no observations in the archived data base for January, February and March in the NWCS province, the assignment of the parameters for those months has to be based on data from the previous or the following months.
 The 10-year time series of αB and PmB within the 1° square are acquired using the NNM and are then grouped to create climatological ensembles of αB and PmB. The normalized kernel density estimates for the assimilation number PmB (Figure 1d) show that PmB has a roughly 5-fold variation throughout the year, with highest values in the summer and early autumn. The seasonal variations of PmB for the Scotian Shelf closely follow changes in temperature [Bouman et al., 2005; Platt et al., 2008]. However there is a steep decrease of PmB from 4 to 2 mg C (mg Chl)−1 h−1 during the spring bloom period. The normalized kernel density estimates for the initial slope αB (Figure 1e) show that αB varies between 0.05 and 0.2 mg C (mg Chl)−1 h−1 (W m−2)−1 throughout the year with an abrupt change from 0.1 to 0.05 mg C (mg Chl)−1 h−1 (W m−2)−1 during the spring bloom. Both PmB and α values tend to be low at moderate and high chlorophyll concentrations, consistent with a dominance of diatoms [Bouman et al., 2005]. The strong seasonal variation and the amplitude of the assigned photosynthetic irradiance (PB− I) parameters are consistent with the results of field experiments [Platt and Jassby, 1976; Bouman et al., 2005], which further supports the NNM.
 From PmB and αB we can derive another important quantity, namely the photoadaptation parameter Ik = PmB/αB. The normalized kernel density estimates for Ik (Figure 1f) show that Ik has two cycles throughout the year, which may be related to the absolute value of local change of total daily photosynthetically available radiation (PAR) with time. Large (small) values of Ik are representative of phytoplankton communities photoadapted to high (low) light. Changes in community composition may also be involved.
 The irradiance in the photosynthetically active range I(t) is calculated from the total daily PAR Ipar and daylength D, and is given by Platt et al. 
in which the total daily PAR Ipar is derived from an empirical relationship between the SeaWiFS total daily PAR climatology, daylength and latitude [Platt et al., 2008]. The total daily PAR in the 1° square is highest in the middle of the year and lowest in January and December, whereas the absolute rate of change of total daily PAR with time is maximal in March and September.
2.4. Correction for Physical Processes
 The correction term that accounts for the effect of physical processes, Γ, includes advection and mixing and is expressed as
where u, v and w are velocity components, and Kh and Kv are the horizontal and vertical eddy diffusivity respectively. Here we use a simple scale analysis to determine the approximate magnitude of individual terms. The labels under the braces in equation (6) are the typical scales for each term. Roman capital letters U, W, X and Zm in the labels refer to the scaled variables: horizontal and vertical velocities, and horizontal and vertical excursions respectively.
 The horizontal velocity scale U is about 0.1 m s−1 on the Scotian Shelf, which agrees with consistent estimates from drifter and altimetric data [Han et al., 2002; C. Tang, unpublished results, 2008]. The synoptic time scale for weekly images is about 106 s, thus the horizontal excursion during this time is about 105 m. The vertical velocity scale of 10−6 m s−1 is taken from previous estimates for the open-ocean Ekman upwelling [Umoh and Thompson, 1994]. The vertical excursion scale is taken as the mixed-layer depth (10–100 m). A typical value of vertical eddy diffusivity Kv is about 10−4 m2 s−1, and the horizontal eddy diffusivity Kh is on the order of 5 × 102 m2 s−1 [Umoh and Thompson, 1994]. We substitute these values into equation (5) and estimate the upper bound of Γ, which is on the order of 10−6 s−1. Since advection and mixing could be either sources or sinks for the local change of phytoplankton biomass, we give a range of Γ between −10−6 s−1 (−3 × 10−3 h−1) and +10−6 s−1 (+3 × 10−3 h−1), and we assume that the probability distribution for Γ is uniform. It should be noted that the maximum correction for physical processes is still one order of magnitude smaller than the primary production, implying that the estimated total loss may have a typical error no greater than 10% due to the effect of physical processes.
 Primary production has several possible fates, including respiration, excretion, grazing and sinking, which may be pooled under the total loss. The Monte Carlo method is used to solve equation (3) and to acquire the climatological ensembles of the total loss, comprising an ensemble of one thousand numerical samples at a given time. The normalized kernel density estimates for the ensembles of the total loss rates (Figure 2a) show that the total loss rates range from 0 to 3 mg C m−3 h−1 and has two peaks during the spring and fall blooms. Resolution of such features is facilitated by the averaging time used in our study. The estimated total loss rates of phytoplankton LBB show more detailed features using weekly time intervals than using monthly averages, in that the weekly composite images can resolve the entire bloom cycle, whereas the monthly images degrade this ability (T. Platt et al., Diagnostic properties of phytoplankton time series from remote sensing, submitted to Estuaries and Coasts, 2008). The annual mean of the total loss rate is about 30 mg C m−3 day−1 and the standard deviation is about 4 mg C m−3 day−1. If we assume the total loss is distributed uniformly in the mixed layer (estimated from the monthly temperature and salinity data set [Tang, 2007]), the annual mean of the total loss rate for the mixed-layer is 445 ± 91 mg C m−2 day−1.
 The normalized kernel density estimates for the normalized total loss rates (Figure 2b) show that the normalized total loss rates vary about 3-fold throughout the year and show a seasonal cycle similar to that of the normalized primary production rate. During the increasing phase of the spring bloom, the normalized loss rates decrease by about 50%, perhaps because of a lower respiration rate in the spring-bloom community (typically diatoms). During the decreasing phase of the spring bloom, the increase of the normalized total loss rates is often associated with an enhanced sinking rate, nutrient depletion, bacterial activity and viral infection [Platt et al., 1991; Pommier et al., 2008; Llewellyn et al., 2008].
 The ratio between the normalized primary production and normalized loss rate PB: LB (Figure 2c) may be useful as an ecological indicator for the pelagic zone. The normalized kernel density for the PB: LB ratio shows that the PB: LB ratio is close to one throughout most of the year, indicating that to first order, phytoplankton photosynthesis is balanced by the total loss. However, during the increasing (decreasing) phase of spring and fall blooms, the PB: LB ratio increases (decreases), suggesting that the balance is temporally disturbed.
 The ratio between the normalized loss rate and the assimilation number LB: PmB (Figure 2d) can also be estimated and may provide a parametrization for the normalized loss rate. The normalized kernel density estimates for LB: PmB ratio show that LB: PmB ratio has a strong seasonal cycle and lies between 0.2 and 0.8 (dimensionless) with lower (higher) values related to the low (high) irradiance throughout most of the year.
 A method for estimation of the total loss rate from remotely-sensed ocean color data was developed using the Monte Carlo method and was illustrated using the data on the Scotian Shelf. The principle of the method is that the loss of phytoplankton can be estimated as the difference between the observed and predicted biomass fields with possible correction for physical processes [Platt et al., 2008]. The absolute loss rate LBB increases during the spring bloom (following the biomass) as is observed, for example, in many sediment-trap studies [Pommier et al., 2008]. The normalized loss rate LB shows a seasonal cycle throughout the year and a pronounced dip at the onset of the spring bloom, followed by a marked increase during the declining phase of the bloom.
 The results of our study are significant from three points of view. First, our calculation provides a robust and rigorous estimation of the total loss rate in an objective manner, since the uncertainties of biomass, PB − I parameters, carbon-to-chlorophyll ratio of phytoplankton, and advection have been taken into account by the Monte Carlo method. Second, the PB: LB ratio provides an index to describe the balance or unbalance between the growth and loss rates. During blooms, the balance will be disturbed. Third, our results may be useful for validation of ecosystem models. The loss rate estimated from observations at sea or laboratory experiments varies significantly, and current knowledge of the processes involved is not sufficient to allow a reliable parametrization [Platt et al., 1991; Siegel et al., 2002]. Our results may provide a way to test whether the cumulative effect of individual loss terms is well represented. The method proposed here may also be used to calculate the new (export) production from remotely-sensed ocean color data. The estimated loss term presented in this study is a near-surface quantity. The estimation of the loss rate for the whole water column, or for the mixed-layer, is also important and remains a subject for future study.
 We are grateful to C. Fuentes-Yaco for providing chlorophyll and SST data, G. N. White III for providing the total daily PAR data, Y. S. Wu, G. Q. Han, H. Maass, H. Bouman and two anonymous reviewers for their useful suggestions and comments. This study was funded by the Canadian Space Agency (GRIP program). This work is a contribution to the NCEO and Oceans2025 projects of NERC (UK).