Valerie-Anne Nicolas

Bertrand Gilles

Laurent Lemarchand

Lionel Marcé

Bruno Castel
email: bruno.castel@isismpp.fr

Maintenance Testing of Mixed-Signal Boards: the FCB case study

Keywords: Maintenance testing, mixed-signal board testing, modelling, automatic test pattern generation, diagnosis

We present an ongoing work in the domain of mixed-signal board maintenance testing, supported by an industrial case study. We propose a method providing a semi-automation and an help for the board maintenance testing and diagnosis stages. It is validated by the implementation of a prototype tool.

MOTIVATION

The test is an activity which takes place all along the life cycle of a component under different forms. Testability analysis during the design stage, then, during other stages (development, production, maintenance), conception and generation of test data, application of test data and finally analysis of results obtained. The development of communication and multimedia technologies, the constant use of onboard technologies (transports, mobile telephony, ...) require more and more efficient and safe mixedsignal components. There is therefore a need for more adapted, reliable and safe tests. In this context, it is crucial to master more efficiently the testing stages of mixed-signal boards.

A large amount of work has already been done concerning testing during conception phases (design for testability) and production. The main part of these works deals with digital boards, some others with analogue boards, and quite a few with mixed-signal boards [START_REF] Michael | Essentials of Electronic Testing for Digital, Memory and Mixed-signal VLSI[END_REF][START_REF] Mark | Roberts "An Introduction to Mixed-Signal IC Test and Measurement[END_REF]. As a result, tools exist for digital boards testing, analogue boards testing, or some specific electronic components testing. One may notice that, surprisingly, not much interest has been thrown into testing during the maintenance stage. Indeed, the maintenance process has its own specificities and testing problematic, different from these of conception or production. Our work is related to this aspect and concerns more particularly the maintenance testing of mixed-signal boards.

First, we explain the specificity of maintenance testing. Then, we sketch the main ideas of our method in this context. Next, some more details on the method and its application via a prototype for mixed-signal boards testing are given. Results on an industrial case study are then presented. A discussion on future work ends the paper.

MAINTENANCE CONTEXT

The maintenance stage is one of the step constituting the life cycle of a manufactured product, which begins after the development/production cycle, and thus does not belong to it. This cut is not artificial and explains why methods conceived for the development stage are not well suited to maintenance. For example, when a failure occurs in a product in use for several years, it is often long and difficult to repair it. Although it is always possible to throw away a faulty component/board and just replace it by a new good one in the case of cheap large series, there are still some more complex, particular technology, valuable components/boards that really need to be repaired. Actually, in most cases, specifications and test data for the product are missing.

It is also rare that the development/production team for this product is still there to help. The one in charge of the diagnosis and repair has then to face a complex problem, often in an empirical way, with no tool available to guide or automate at least a part of his work.

In maintenance, one may distinguish preventive maintenance from corrective maintenance. Corrective maintenance is, as in the above example, to be able to deal with failures when they occur, sometimes very quickly. Preventive maintenance consists in checking periodically the correct behaviour of components, and may seem more time-consuming than corrective maintenance. However, by anticipating incorrect behaviours, it allows to guarantee a higher level of confidence in the product and thus a lower need for corrective maintenance.

As far as testing is concerned, corrective maintenance may be viewed as a local approach whereas preventive maintenance is more global. In the first case, tester is interested in one behaviour (to reproduce the target failure), and in the second case, he has to check the set of all behaviours of the board. Independently of their nature, these two kinds of maintenance are in great need of generic testing methods and tools.

The method we introduce next was first designed for corrective maintenance needs, and offers a well-adapted precision level according to this aim. In adds, it may also constitute an help for preventive maintenance.

MAIN IDEAS

Our final goal is to propose a method, supported by a (semi-)automatic tool, providing an help to the board maintenance testing and diagnosis stages. We thus have to formalise the testing and diagnosis data and processes, taking into account the industrial context and practices. Two main aspects are to be considered. First, mixed-signal boards testing practitioners have some know-how related to their experience and skills in the testing process. We think that this knowledge has to be included in the tool. Second, the tool must match the background of testing practitioners in order to be really useful and used. This implies formalisms which are intuitive or well-known to testing engineers. Because of the lack of tools dealing with mixed-signal components, we pay special attention to the modelling and testing of mixedsignal boards. For more precise diagnosis, the method provides some functions that test more specifically the internal behaviour of a functional block (branch, state and path testing and coverage in an FSM context).

METHOD AND PROTOTYPE

We have implemented the ideas illustrated on Figure 1 in a prototype tool. This prototype provides a Graphical User Interface (GUI) allowing high level topological description of a mixed-signal board. In addition, the GUI includes some facilities for the choice of a testing strategy, for the description of the board-ATE connection and for the description of data (signals) flow.

A mixed-signal board is modelled as a set of connected functional blocks (each block identifies a function that can either be digital, analogue or mixed-signal). A functional block expresses a function, and does not need to correspond exactly to one of the physical components of the board. Indeed, most of the time it involves a set of such physical components. The description of the board is thus made at two levels: the board level and the block level as illustrated by Figure 3. The prototype proposes different ways for defining functional blocks. The first one is dedicated to digital functions, the second to analogue or mixedsignal functions, and the third to black box functions.

define a digital function, the user describes the behaviour of the block using a finite state machine (FSM) formalism. In the second case, the user selects a function in a library of analogue and mixed-signal common components. Due to the specificity of maintenance testing, the prototype also allows black box functions. This kind of definition has to be used when absolutely no information on the board behaviour is available. In practice, it is therefore possible to learn from a known good board a test data set characterising the board. This description is used in a black box definition. Even though using this description reduces the power of the method at the board level, it is some time mandatory to use it.

To generate more accurate test data than that obtained just from a board behaviour specification, we use testing strategies in add of the functional description of the board. A testing strategy expresses behaviours or faults that have to be specifically targeted by the generated test data set. The prototype relies on some basic predefined testing strategies (this usefully benefits from crossfertilisation with testing practitioners), but it also allows to describe and use its own specific testing strategy.

From an internal point of view, the different kinds of blocks (digital, analogue or mixed-signal) are treated homogeneously. They are each associated to a functional model and a test model, expressed as communicating finite state machines (cf. Figure 3). Actually, the test model is automatically inferred from both the functional model and the chosen testing strategy (excepted in the case of black box components where the functional and test models are the same, the testing strategy being already part of this model). One can view the test model as a particular instantiation of the functional model. Nevertheless, both models are needed to achieve automatic test data generation.

We define the generated board test data set as the optimized union of all blocks test data sets. A block test data set is obtained by transition coverage of its test model and full upstream and downstream propagation through other blocks functional models to primary inputs and outputs. The aim of the optimization is to reduce the size of the test data set.

The problem of test data generation is faced using constraint logic programming (CLP) and classical algorithms for finite state machines (transition coverage, state coverage, path coverage). In the prototype, test data are represented in a symbolic way, using ranges of values instead of one instantiated data. It is essential to work on a symbolic representation of data for the validity and efficiency of the approach. The instantiation of test data is only obtained at the end, on demand from the user, or automatically to derive a test program for a specific ATE.

Most of the prototype is written in C++, excepted the part concerning testing strategies that is implemented using constraint logic programming, with the ECL i PS e solver [START_REF] Cheadle | Eclipse: An introduction[END_REF].

The prototype has been validated on two first industrial case studies: the "Tachy board" [START_REF] Gilles | Towards a New Modelling of Mixed Signal Boards For Maintenance Testing[END_REF] and the "Filter Command Board", which are mixedsignal boards. We detail the application of the method on the "Filter Command Board" in the next section.

CASE STUDY

We first give a description of the board involved in the case study. Then, applying our method, we present the board modelling and board testing results.

Board description

The "Filter Command Board" (FCB) is a mixedsignal board (depicted in Figure 4) and is part of a telecommunication system. We describe shortly the inputs/outputs and the functionality of this board.

FCB has the following inputs/outputs signals:

• one serial digital input (SDI),

• one clock input (CK),

• one copy command (Copy),

• one transfer command (E/D),

• one serial digital output (SDO)

• eight digital outputs (DO),

• sixteen analogue outputs (AO1 and AO2). FCB mixed-signal part is made of two commutation amplifiers CA1 and CA2. Each amplifier has height digital inputs and height analogue outputs. The commutation table for each input/output is given in Table 1.

CA1 CA2 0 3 V 3 V 1 -200 V -400 V

Table 1 : The commutation table

The main function of the FCB is to command weight filters connected to the commutation amplifiers outputs A01 and A02. Digital outputs DO and SDO command other modules in the system.

Board modelling

We now present the method internal representation of the board, that is the functional models corresponding to the different blocks constituting the board. We begin with the modelling of the board inputs, next the modelling of the blocks involved in the digital part of the board, the modelling of the blocks of the mixed-signal part, and finally the modelling of the board outputs. Test strategies and test models are also discussed.

Data

The modelling of data (board inputs) is particular. Data are not functional blocks of the board. However, to generate test data, we need to model them. This also allows to model characteristics of sources and generators of a specific ATE in order to produce a test program. Thus, each board input has a functional model (but no test model as it does not correspond to a board component).

Figure 5 shows the modelling of the two input clocks (the main one and Copy), using a discrete time representation. "A ? m" stands for reception of message m from FSM A and "A ! m" for emission of message m to FSM A.

Clk Copy

Data ! top R2 ! top2 The transfer command modelling is shown Figure 7, and sends either 0 or 1.

Valid

Enable ! 1 Enable ! 0

Digital part

The digital part of the board is modelled by four communicating finite state machines. For simplicity, we model each serial three 8-bit buffers by a single 24-bit buffer.

The first one (R1, cf. Figure 8) is the functional model of the shift register. Its input comes from the serial input (Data) and its outputs go to the storage register (R2).

Mixed-signal part

The mixed-signal part of the board is modelled by 16 instances of two communicating finite state machines (CA11..CA18, CA21..CA28).

The first one (cf. Figure 11) is the functional model of the CA1 commutation amplifier. Its input comes from the 3-state buffer (Enable) and its output goes to the analogue measurement points (MP_ANALOGi).

Figure 12 shows the functional model of the CA2 commutation amplifier, which behaves the same manner.

On this example, one may notice that CA1 and CA2 have only two behaviours each. Their functional models are explicitly exhaustive and thus correspond to their test models (coverage of all kinds of behaviours, with no associated specific fault).

Outputs

As we did previously for board inputs, we have to model the board outputs even if they are not real functional blocks. It is needed to generate test data, and it also allows to describe ATE measurement tools characteristics in order to produce a test program.

Figure 13 presents the analogue measurement point functional model. MP_ANALOGi is just waiting for data #i coming from one commutation amplifier (CA1 or CA2).

Concerning the digital part, functional models express behaviours in an implicit way, and they do not include information on specific faults to detect. Thus, we choose to add some testing strategies on the digital measurement points. Figure 14 illustrates the "stuck-at" strategy on a digital measurement point. MP_NUMi is forced to receive at least one 0 and at least one 1 from the 3state output buffer (Enable).

MP_ANALOGi

CAji ? x Finally, after optimized union of all blocks test data sets, the board test data set is:

Board test data set = {DT 24 , DT' 24 , DT''}.

This test data set is made of three sequences, the first two of size 24, the third one of size 1. It covers all the possible values for the 25 outputs, and the particular case of the transfer command set to 0.

CONCLUSION AND FUTURE WORK

We have presented a method for mixed-signal board testing, designed to face the specificities of maintenance testing. Some aspects have been validated such as the functional approach at all levels, the homogeneous modelling formalism (for digital, analogue and mixed-signal functions) and the testing strategies. The prototype is already able to generate expected test data (thus completing the first part of the method, cf. Figure 1) for the boards involved in the case studies.

Further work is needed to validate our complete approach and prototype, and its real impact on diagnosis. In particular, the second part of the method (depicted in Figure 2) is not yet implemented and needs an important effort as it implies the construction of a physical board, dedicated to the application of our method on a specific ATE.

Future work includes more experimentation of the method on other types of boards, but first results are encouraging, reinforcing the idea that at least semi-automatic testing of mixed-signal boards is feasible. This kind of tool would respond to a real need in the industrial community.

Figure 1 and

 1 Figure 1 and Figure 2 sketch our complete method process flow.

Figure

  Figure1shows how to generate a test data set for a board given some testing strategies and a description of the (correct) board. This test data set may then be automatically translated into a test program expressed in an Automatic Test Equipment (ATE) language. The test program is

Figure 2 :Figure 2

 22 Figure 2: The board testing process flowFigure2presents how to obtain a test report by running the test program on the maybe faulty board, using an ATE. The first information in the test report is the verdict: the board passed the tests or not. If not, for each failed test, the test report indicates the real and expected results, and the possibly faulty functional blocks. This ends the testing stage and is the departure point of the diagnosis stage. Depending on the required grain of diagnosis, one can stop there and just replace by good ones the components corresponding to faulty blocks. For more precise diagnosis, the method provides some functions that test more specifically the internal behaviour of a functional block (branch, state and path testing and coverage in an FSM context).

Figure 3 :

 3 Figure 3: The architecture of the prototype

Figure 4 :

 4 Figure 4: The Filter Command board

Figure 5 :Figure 6 :

 56 Figure 5: The main clock signal and the copy clock signal functional models

Figure 7 :

 7 Figure 7: The transfer command functional model

3 Figure 12 :

 312 Figure 11: The CA1 commutation amplifier functional (and test) model

FigureFigure 14 :

 14 Figure 13: The analogue measurement point functional model

The test data set generation process flow

  [START_REF] Michael | Essentials of Electronic Testing for Digital, Memory and Mixed-signal VLSI[END_REF] shows how to generate a test data set for a board given some testing strategies and a description of the (correct) board. This test data set may then be automatically translated into a test program expressed in an Automatic Test Equipment (ATE) language. The test program is the test data set instrumented with ATE primitives and an oracle. The oracle compares the real outputs of the test with the expected ones, and is thus able to decide and list which tests fail.

		Testing
		Strategies	Automatic Test Pattern Generator (ATPG)	Test Data Set / Test Program
	Digital	Board
	Functions	Description
	(FSM)
	Black Box	Analogue and Mixed-Signal
	Defined	Parametric Functions
	Functions
	Test Data Set / Test Program Figure 1: Automatic Test Equipment (ATE)	Test Report
	Tested Board
	(DUT)

Figure 10: The 3-state output buffer functional model

			Data ? bit [bit modifies reg1]
		R1 Wait		R1 OK
				R2 ! reg1
	Figure 8: The shift register functional model
	The storage register (R2, cf. Figure 9), cadenced
	by the Copy command, receives data from the shift
	register (R1) and sends it to the 3-state output
	buffer (Enable).
			C op y ? to p2	R2 Sync	R 1 ? re g 1 [r e g 1 m o d ifi e s re g 2 ]
	R2 Wait				R2 OK
				Enable ! reg2
	Figure 9: The storage register functional model
	The 3-state output buffer (Enable, cf. Figure 10),
	receives data from the storage register (R2). It
	sends it to the commutation amplifiers (CA1,
	CA2) and to the digital measurement points
	(MP_NUMi) only if the transfer command (Valid)
	is 1.
	Enable Init	R 2 ? re g2	[r eg 2 m od ifi es re g3 ] Va lid ? 0	Enable Wait	V a lid ? 1	Enable OK
		CA1i(i:1..8) ! reg3(i), CA2i(i:1..8) ! reg3(i+8),MP_NUMi(i:1..8) ! reg3(i+16)

ACKNOWLEDGMENTS

We wish to thank Michel Le Goff from ISIS-MPP for his availability and his willingness to clarify matters on the industrial practices involving mixed-signal board maintenance testing.


Markus Schartau
email: markus.schartau@gkss.de

Michael R Landry
email: mlandry@ucsd.edu

Robert A Armstrong
email: rarmstrong@notes.cc.sunysb.edu

Armstrong

Density estimation of plankton size spectra: A reanalysis of IronEx II data

Keywords: plankton size, kernel density estimation, parametric size density, plankton community structure, iron fertilisation Plankton size, nonparametric kernel density estimation, parametric size density, plankton community structure, iron fertilization 11 pt, Complex Script Font: 11 pt, German Germany 11 pt, Complex Script Font: 11 pt, Portuguese Portugal 11 pt, Complex Script Font: 11 pt Formatted: Portuguese Portugal

Many critical processes of ecosystem function, including trophic relationships between predators and prey and maximum rates of photosynthesis and growth, are size-dependent. Size spectral data are therefore precious to modellers because they can constrain model predictions of size-dependent processes. Here we illustrate a multi-step statistical approach to create size spectra based on a reanalysis of plankton size data from the IronEx II experiment, where iron was added to a marked patch of water and changes in productivity and community structure were followed. First, bootstrapping was applied to resample original size measurements and cell counts. Kernel density estimation was then used to provide nonparametric descriptions of density versus size. Finally, parametric distributions were used to obtain parameter estimates that can more easily be applied in models. A major advantage of this approach is that it provides confidence envelopes for the density distributions. Our analyses suggest three basic distributional patterns of cell concentration versus logarithm of equivalent spherical diameter (ESD) for individual taxa. Composite size-densities of heterotrophs and photoautotrophs reveal important aspects of the coupling between protist grazing and the phytoplankton community.

INTRODUCTION

Most oceanic primary production and consumption occurs in the microbial base of the food web among bacteria and protists [START_REF] Calbet | Phytoplankton growth, microzooplankton grazing and carbon cycling in marine systems[END_REF]. Trophic interactions within this microbial base are potentially complex (Azam and Malfatti, 2007); but the general structure suggests that size-based analyses and interpretation are needed (Armstrong, 2003a;2003b). The ability to construct, analyse, and predict changes in size distributions may therefore allow a more precise description of interactions within plankton communities, where larger consumers typically eat smaller prey [START_REF] Hansen | The size ratio between planktonic predators and their prey[END_REF].

Interest in the size structure of aquatic systems has been stimulated by the introduction of technologies (Coulter Counter, flow cytometer, image analysis microscopy) for efficient enumeration of particles and organisms by size, and has sparked the development of additional novel approaches and instruments (Optical Plankton Counter, Underwater Video Profiler, FlowCAM, ZooScan) to fill measurement gaps.

Plankton size distributions, often pieced together from measurements optimised for different size ranges or types of organisms, are typically represented as size spectra, where the main attribute is the slope on a biomass-normalised logarithmic scale [START_REF] Sheldon | Size distribution of particles in the ocean[END_REF]. Numerous studies have established the usefulness of plankton size spectra for summarizing the inherent structural properties of a system's biota, and for comparing and contrasting system types and temporal/spatial variability within systems (e.g., Rodriquez and Mullin, 1986;[START_REF] Gaedke | Ecosystem analysis based on biomass size distributions: A case study of a plankton community in a large lake[END_REF][START_REF] Cavender-Bares | Microbial size spectra from natural and nutrient enriched ecosystems[END_REF][START_REF] Rinaldo | Cross-scale ecological dynamics and microbial size spectra in marine ecosystems[END_REF]San Martin et al., 2006a;[START_REF] Reul | Control of microplankton size structure in contrasting water columns of the Celtic Sea[END_REF]. Understanding the size structure of lower trophic levels has similarly provided mechanistic insights into the bioenergetic constraints and habitat partitioning of foraging predators (e.g., [START_REF] Kerr | Prey availability, metaphoetesis, and the size structures of lake trout stocks[END_REF][START_REF] Rykaczewski | Influence of ocean winds on the pelagic ecosystem in, upwelling regions[END_REF]. Parallel developments in spectral analyses of plankton communities and allometric theory have further established a basis for inferring biomass and energy/mass flux, trophic efficiencies, and vital rates (growth, mortality) from biomass distributions in aquatic food webs (e.g., [START_REF] Kerr | Theory of size distribution in ecological communities[END_REF][START_REF] Platt | Organisation in the pelagic ecosystem[END_REF][START_REF] Moloney | General allometric equations for rates of nutrient uptake, ingestion, and respiration in plankton organisms[END_REF][START_REF] Zhou | Population dynamics theory of plankton based on biomass spectra[END_REF] 3 density functions, for example, using a normal probability density function (pdf) with two parameters (mean and variance) for each species or group. Such parametric pdfs could be used in describing plankton interaction by convolving parametric predator-and prey-density functions (Armstrong, 2003a). To derive parametric size-density functions from observation is not straightforward. Historically, microscopical data have been plotted as discrete histograms to combine species information with size measurements. However, this approach produces results of uneven quality, particularly when cell abundance is low and when sampled water volumes are small.

In this study, we describe a sequential procedure in which we first derive nonparametric kernel density estimates and then fit parametric functions to the nonparametric distributions. We illustrate this analysis with size-based data for plankton taxa observed during the IronEx II mesoscale iron-enrichment experiment in the eastern equatorial Pacific (Landry et al., 2000a). Our data reconstruction reveals taxon-specific details of community size responses to iron fertilisation and grazing impacts. More generally, the approach supports size-structure inter-comparison of disparate data sets and the development of size-based plankton models.

METHOD

Our primary data are plankton counts, cell lengths and widths, distinguished by taxonomic group.

We analysed the data using kernel density estimators [START_REF] Parzen | On estimation of a probability function and mode[END_REF][START_REF] Wegman | Nonparametric probability density estimation density estimation methods[END_REF][START_REF] Silverman | Density estimation for statistics and data analysis[END_REF], which provide smooth nonparametric plankton size-densities, even from sparse discrete measurements. In addition, we assessed confidence limits of these nonparametric density estimates.

The confidence limits were obtained by applying a bootstrapping approach [START_REF] Efron | An introduction to the bootstrap[END_REF], where original plankton counts and size measurements were re-sampled with the objective of separating size structure information from observational noise. Plankton counts and corresponding size measurements were assumed to follow specified error distributions: Poisson distribution for counts, and Uniform, and Gaussian distributions for size measurements. For each set of original and re-sampled data, we computed a nonparametric representation of density versus size. Computations are extensive, but the resample series (ensemble) of nonparametric estimates allowed us to calculate mean size densities for each plankton group and to specify their confidence limits. We then fit unimodal and bimodal normal distributions functions, on a logarithmic scale, to mean density estimates of dominant plankton groups. In the sections below we describe the assumptions inherent in our bootstrapping approach, provide details for the optimum bandwidth selection needed for the kernel density estimation, and define a misfit function for fitting parametric functions to plankton size-density estimates. Figure 1 depicts the steps in these analyses. During the IronEx II project in the eastern equatorial Pacific [START_REF] Coale | A massive phytoplankton bloom induced by an ecosystem-scale iron fertilization in the equatorial Pacific[END_REF], data on species identity, abundance, and size were collected at a series of stations, both inside and outside a mesoscale iron-fertilised patch. At all stations, samples were collected from the mixed layer (15 m) and analysed as described in Landry et al. (2000a). Briefly, small volume (3 ml) samples were preserved with 0.9% paraformaldehyde and analysed by dual-beam flow cytometry for populations of Prochlorococcus, Synechococcus, and eukarytoic phytoplankton. Samples of 20-50 ml were preserved with 1% gluteraldehyde, stained with DAPI, and prepared as slides (0.2-µm filters) for analysis by epifluorescence microscopy. At least 2,000 cells per slide were enumerated and sized in specific recognizable categories (species, genera, or classes) according to the method of [START_REF] Chavez | Growth rates, grazing, sinking, and iron limitation of equatorial Pacific phytoplankton[END_REF], with the vast majority of cells viewed at 1000x magnification, but with large (>40 µm) and rare types counted on a larger portion of the slide at 400x. For the present analysis, picophytoplankton cell categories were grouped for data processing, with direct size measurements of Synechococcus and eukaryotes (abundance estimates from microscopy and flow cytometry were in good agreement). For Prochlorococcus, we assumed a mean size of 0.735 µm, similar to that of [START_REF] Blanchot | Picophytoplankton in the equatorial Pacific: vertical distributions in the warm pool and in the high nutrient low chlorophyll conditions[END_REF]. Figure 1a shows an example of original cell counts and the corresponding length measurements of centric diatoms observed at stations within the iron-fertilised patch. To compare plankton responses, we combined measurements from stations inside the fertilised patch into one data set, and those from stations outside the fertilised patch into a second ("control") set.

Accordingly, we did not resolve variations due to temporal and spatial variability inside and outside the patch.

Resampling of lengths, widths, and counts

Obtaining simultaneous data on cell abundances, size, and taxonomic identity requires laborintensive microscopy; and it is therefore seldom possible, within time and effort constraints, to replicate microscopic analyses. On the other hand, we desire confidence limits for nonparametric size-density estimates, while considering uncertainties in counts and size measurements. We calculated these confidence limits from an ensemble of kernel density estimates. The ensemble of densities was determined using resample data sets, which were obtained by randomly resampling the original data set with replacement (bootstrapping). In addition to the original data set, the bootstrapping produces sets of pseudo-data data having the same statistical properties as the data. In total, we produced one hundred resample data sets from each primary data set.

The absolute accuracies of microscope measurements of cell length and width depend on the magnification chosen. By examining the data from IronEx II, we found that the precision of length 1 shows the size dependency in precision, rounded to the nearest 1, 2, 5, 10, 20, 40 and 100 µm. The precisions in length and width measurements increase with increasing size. We imposed uniform error distributions whose interval was equal to the resolution of the size range. For sizes below 2 µm, we assumed a normal error distribution with a standard deviation of 0.2 µm. While length and width were resampled, we approximated cell thickness based on different shape factors, which varied among plankton groups. For approximation of cell thickness we adopted the conversion factors used in Landry et al. (2000a), who related width to thickness according to [START_REF] Chavez | Growth rates, grazing, sinking, and iron limitation of equatorial Pacific phytoplankton[END_REF]. Cell volumes were calculated using the formula for a scalene ellipsoid, combining the two semi-axes (measured or resampled) length and width with the calculated thickness. Cell volumes were then rescaled to equivalent spherical diameter (ESD) of a sphere.

A single plankton count can be interpreted as a discrete event during a certain time period of measurement, where each cell is found within a microscopic sub-sample that has been taken from a larger water volume. If no event is found, the sub-sample may have been too small to capture cells that were highly dispersed at this station. On the other hand, if a cell is counted, it may not be representative of the larger water volume and an extrapolation to cell concentration (cells per litre) may be biased. Thus, a single plankton count within a sub-sample should be regarded as a rare event, and a repetition of this event should remain rare during resampling. The Poisson error distribution is an appropriate description of uncertainty when accounting for such rare events. As individual counts were resolved in the data set, we therefore imposed the Poisson expected value of "one". We also obtained zero counts while drawing from the Poisson distribution (Fig. 1b). The resampled data were then extrapolated to cell concentrations in units of cells per litre (Fig. 1b,c) as in Landry et al. (2000a), accounting for different water volumes filtered and varying portions of the slide areas analysed. Individual cell counts correspond to concentrations on the order of 10 2 cells per litre (Fig. 1c).

Nonparametric density estimation

Histograms are often used to depict plankton size densities. Defining the bins of histograms requires defining points of origin and interval widths, subjective choices that can yield different distributional patterns. Such subjectivity can be justified; but it does hinder inter-comparison among studies. In the present study, we chose a different statistical approach, based on a density estimation method first proposed by [START_REF] Parzen | On estimation of a probability function and mode[END_REF] and now commonly termed "kernel density estimation". In comparison to other density estimation techniques, the kernel density estimator generally provides greater detail of distribution structure [START_REF] Wegman | Nonparametric probability density estimation density estimation methods[END_REF][START_REF] Silverman | Density estimation for statistics and data analysis[END_REF] A kernel density estimate is derived from the sum of symmetric probability density functions (kernels) defined around each data point. For each data point, the kernel can be regarded as a weighting function for the observation. For example, Figure 1d shows all kernels of the resampled set depicted in Figure 1c. The summation over all kernels leads to a smoothed distribution Fig. 1e, which can then be normalised to fulfil criteria of a probability density function (pdf). In plankton ecology, the utility of kernel density estimation has not been widely appreciated.

Our analysis joins recent studies by [START_REF] Sanvicente-Añorve | The use of kernel density estimators to analyze length-frequency distributions of fish larvae[END_REF] and [START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF] in applying kernel density estimation to plankton size spectra. We complement the method with a bootstrapping approach that relates uncertainties in plankton measurements to confidence intervals of the density estimates (Fig. 1f,g).

Kernel density estimation of plankton size

ESD data (esd ij ; µm, with index i for every observation of plankton group j) were calculated from length and width data that were originally rounded to discrete sizes (Table 1), depending on the magnifications used for microscopy. After resampling of cell counts and sizes (Fig. 1b) we calculated cell concentrations (c ij ; cells L -1 ), multiplying counts ( n ij ) by an extrapolation factor ( E ), Figure (1c). The total cell concentration ( C j tot ) is the sum of M individual cell concentrations:

C j tot = c ij i =1 M j ∑ (esd ij ) = E⋅ n ij i =1 M j ∑ (esd ij ) (1)
The size distribution ( ) f j ) of a single plankton group can be described as the product of total cell concentration and the pdf of ESD:

) f j = C j tot ⋅ pdf j (ESD). (2)
By definition, a pdf is required to integrate to one. For our size range of interest, 0 to 327 µm, the following equation fulfils that constraint:

pdf 0 µm 327 µm ∫ ⋅ dESD = 1.
(3) Kernel density estimation theory allows one to construct a nonparametric plankton size distribution that is "smooth" (i.e., continuous with continuous derivatives of all orders). A smooth density estimate ( ˆ f j ) can be normalised simply by dividing by total cell abundance (cells L -1 ); the result can then be interpreted as a data-based, nonparametric pdf. Since phytoplankton size measurements extend over a range of three orders of magnitude, we apply the kernel density estimator on a logarithmic scale. As used traditionally in aquatic ecology, log-scale analysis implicitly accounts for the doubling of cell volume during the cell cycle, allowing each species to have the same width on the log scale, rather than giving larger species larger apparent variances.

For kernel density estimation, we first transform ESD to a dimensionless size variable s:

s = ln ESD s 0       = ln ESD 1 µm       . (4)
A kernel density estimate for a pdf is constructed by adding together all individual kernel functions (here Gaussian distribution functions on a logarithmic scale, Fig. 1d). Each Gaussian kernel is centred around ln(esd ij / s 0 ), the normalised and log-transformed observation of esd ij , with index i for each observation within plankton group j. The Gaussian kernel itself is a smooth distribution that integrates to one. Each kernel is then multiplied by the corresponding cell concentration (c ij ; cells L -1 ). Therefore the sum of Gaussian kernels has to be normalised (dividing by the total cell concentration, C j tot ) in order to yield again a probability density estimate for plankton group j:

pdf est (s) j = 1 C j tot c ij h j ⋅ 2π ⋅ exp i =1 M j ∑ s -ln(esd ij /s 0 ) ( ) 2 2⋅ h j 2         (5)
The width of each kernel distribution is expressed by the smoothing parameter h j , called bandwidth. When a Gaussian kernel is chosen for the transformed variable s [equation ( 5)], the bandwidth h j , which remains constant for a single plankton species or group, resembles a standard deviation on a logarithmic scale. Overall, it is the summation over a series of kernels that yields a final distribution where each data point is represented by itself, which is the major difference from the application of a histogram. The main task in kernel estimation theory is the selection of an optimum bandwidth, one that minimises bias while maximising variance information (bias-variance trade-off). Briefly, a best bandwidth is identified if the estimate of the pdf at any data point can also be represented by a density estimate from neighbouring points. For example, if we remove one data point and if the bandwidth were too broad, the entire pdf would spread over too large a range, and the pdf estimate at the location of the missing point would be too small (biased estimate). On the other hand, if the bandwidth were too narrow, the pdf would consist of narrow spikes. The pdf values would have large variations within the size range of interest (too much variance). In the worst case, the pdf would become zero at the missing point. Figure 2 depicts an example of bandwidth selection where a reference density, which was described by five data points, has to be recovered by a density estimate from the four neighbouring data points. Optimum bandwidth selection provides reasonable density predictions for any data point removed.

A common approach to kernel density estimation is to specify a single bandwidth that remains constant throughout the analysis of a data set. Criteria exist for choosing an optimal constant bandwidth, mostly derived from practical considerations and based on normality assumptions. For example, assuming a priori that a final density estimate ˆ f can be expressed in terms of a normal density distribution facilitates the approximation of an optimal constant bandwidth (finding an asymptotic solution that describes the estimator's trade-off between variance and bias). Making such assumptions about unknown density distributions for bandwidth selection are often referred to as plug-in methods.

Pragmatic plug-in methods for choosing constant bandwidths have been shown to produce good results for size-structure analyses of fish larvae [START_REF] Sanvicente-Añorve | The use of kernel density estimators to analyze length-frequency distributions of fish larvae[END_REF] and for the estimation of size diversity [START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF]. Our approach is similar, but we adopt an ansatz, discussed by [START_REF] Härdle | Nonparametric and Semiparametric Models[END_REF], who refined a rule-of-thumb bandwidth method originally suggested by [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF]. Our bandwidth estimate h j is inferred from the variance information of the normalised and log-transformed ESD observations, s ij = ln(esd ij / s 0 ). To reduce the sensitivity of such an estimate to outliers, an inter-quartile range ( R j ) was calculated by subtracting the upper 75% quartile from the lower 25% quartile:

R j = s ij 0.75M [ ] -s ij 0.25M [ ] (6)
Given the inter-quartile range R j together with the standard deviation ( ˆ σ j ) of s ij , the rule-of-thumb bandwidth can then be calculated as:

h j = 1.06⋅ min ˆ σ j , R j 1.34       M j -1/ 5 (7)
with M j being the total number of data points for each plankton group. The factors 1.06 and 1.34 are derived from statistical considerations, assuming that the nonparametric density estimates resemble normal distributions [START_REF] Härdle | Nonparametric and Semiparametric Models[END_REF]. Numbers of data points for each plankton group, together with their observed size ranges, of this study are provided as Supplementary Material.

Fitting parametric distribution functions to plankton size-density.

For parametric density functions, we chose normal density functions on logarithmic scales to characterise the patterns suggested by the nonparametric density estimates. We hypothesize that each plankton group identified during measurement may actually consist of more than one species, or different clones if each resolved "type" represents a species. Plankton size distributions for each "type" are assumed to arise from a mixture of cells whose component populations are subject to different intrinsic growth parameters on the species level and are exposed to external variability due to micro-scale physics, nutrient availabilities, and plankton interactions. As we are concerned with the dominant peaks of the densities, we restrict our analysis to resolving no more than two peaks.

Therefore, the general form of the probability density function is bimodal ( pdf 2 ), defined as an affine combination of two Gaussian functions on a logarithmic scale:

pdf (s,q,a 1 ,b 1 ,a 2 ,b 2 ) = (1 -q)⋅ N(s,a 1 ,b 1 ) + q⋅ N(s,a 2 ,b 2 ) = 1 2π (1 -q) b 1 exp - s -a 1 ( ) 2 2b 1 2         + q b 2 exp - s -a 2 ( ) 2 2b 2 2                 (8)
The unimodal function ( pdf 1 ) was derived simply by setting q=0 in equation ( 8) and optimising a 1 and b 1 only. The parameters ( a 1 ,a 2 and b 1 ,b 2 ) are the means and variances of the normalised and log-transformed variable (s). The first mode of the pdf is characterised by its amplitude (1q) ≥ 0 at location s = a 1 and its width b 1 ; similarly, the second mode is specified by its amplitude (q) at s = a 2 and width b 2 .

Optimum parameter values were obtained by maximum likelihood estimation. Although the notion of comparing two continuous distributions (as it is the case for our nonparametric and the possibly be achieved by incorporating the optimum bandwidth, defined before, as correlation length scale. Nevertheless, the proof of using the optimum bandwidth for computations of covariances requires an in-depth analysis that goes beyond the scope of this study. Furthermore, the optimum bandwidths differ, although slightly, between resample data sets of each plankton group. This means that an ideal data-model comparison would also require data-model fits for each resample data set. The average of the ensemble of parameters estimates for equation ( 8) would then describe a perfect fit. To keep computations moderate we assumed a conditional probability that imposes independence between size measurements and describes deviations between the parametric function [equation ( 8)] and the nonparametric kernel density estimate [equation ( 5)] at points corresponding to ESDs in the original data sets. Our conditional probability can be defined as a likelihood ( L ):

prob( pdf est (s i ) | q,a 1 ,b 1 ,a 2 ,b 2 , pdf ) = L = 1 ε i ⋅ 2π ⋅ exp i =1 M ∏ - pdf est (s i ) -pdf (s i ) ( ) 2 2⋅ ε i 2         (9)
This likelihood measures the fit to the nonparametric density estimate (pdf est ) at the points of measurement (s i ), given the parameter values ( q,a 1 ,b 1 ,a 2 ,b 2 ) for the parametric function (pdf)

[equation ( 8)]. The likelihood's variance information (ε i 2 , which is the uncertainty in our nonparametric density estimate) was calculated from the ensemble of density estimates, according to our bootstrapping approach. In practice, maximum likelihood estimates are obtained when the global minimum of the negative logarithm of the likelihood (-ln(L)) is found. Rather than seeking for the minimum of the negative logarithm of the likelihood we calculated the Akaike information criterion (AIC), which also accounts for the additional degree of freedom of the parametric function used to produce the fit. In our case the AIC was determined as follows:

AIC = 2⋅ (-ln(L) + P) = 2 - ln( 1 ε i ⋅ 2π ) - pdf est (s i ) -pdf (s i ) ( ) 2 2⋅ ε i 2         i =1 M ∑ + P         (10)
with M being the number of data points (Supplementary Material). P is the number of parameters needed for the parametric functions. We distinguished between AIC 1 with P = 2 for the unimodaland AIC 2 with P = 5 for the bimodal function. Because of the last term in equation ( 10) the AIC Formatted: Font color: Auto, English U.S.

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Formatted: Font color: Auto AIC 1 allows us to assess whether a bimodal function is an improvement compared to the unimodal pdf for a particular plankton group.

RESULTS

The IronEx II data contain 25 plankton groups, among which we distinguished 18 photoautotrophs and 7 heterotrophs. For simplicity, we show details only for those groups that were most abundant or experienced significant impacts from iron fertilisation (Figs. 3 and4). All groups are, however, represented in the composite density estimates (Fig. 5). Density estimates for all 25 plankton groups are provided (with their standard deviations) as Supplemental Material. Tables 2 and3 include information on reconstructing the density estimates with parametric functions [equation ( 8)], with the number of total cell concentration and parameter values, respectively. The last columns depict differences between AIC 2 and AIC 1 of the bimodal and the unimodal fits, respectively.

Nonparametric and parametric density estimates of plankton size

Phytoplankton ---By comparing spectra from the fertilised patch to those from the control sites, we can assess the changes induced by iron addition. By examining the nonparametric density estimates on a logarithmic scale of ESD, we can identify three characteristic distributional patterns: symmetric unimodal, right-skewed, and bimodal (two distinct peaks). Figure 3 shows the responses in size spectra for phytoplankton groups that were abundant within the fertilised patch and that were also observed at the control sites. Some large diatoms, like Rhizosolenia and Thalasissiotrix spp.,

were not found outside the patch (Supplemental Material).

Prymnesiophytes (Fig. 3A), the group that includes coccolithophorids, have a symmetric distribution on a logarithmic scale, and a unimodal pdf suffices to describe the density of the control (Table 2). The dominant mode (the size with highest frequency of cells) is located between ESD = 2.8 µm ( a 1 = 1.03) and ESD = 2.9 ( a 1 = 1.07), with a small shift towards larger cell sizes when the fertilized patch is compared with the control. The parametric fit of the bimodal density function to the control density estimate yields q= 0.04 [equation ( 8)], so that a unimodal distribution can indeed be preferred over a bimodal fit. In contrast, a bimodal fit produces better results for the fertilised patch. Nonetheless, the parametric fit comprises a small contribution of the second mode ( q = 0.1) to the overall density distribution, similar to the control. The apparent improvement is largely due to a better representation of the leptokurtic kurtosis that extends the density estimate up to sizes close to Autotrophic dinoflagellates show a clear bimodal distribution in both the control and the fertilised patch (Fig. 3C). However, the response to fertilisation is different from the structural changes apparent for Pseudo-nitzschia. For dinoflagellates, cell concentration density increases significantly within the first mode, between 2.8 and 3.5 µm ESD (with a 1 = 1.17 and 1.19 for control and patch, respectively). The response in the size range between 5 and 10 (m (second mode) is less expressed. With EMBED Equation.3 a 2 between 1.19 and 1.89, the parametric fits do show two distinct modes, but the amplitudes of the first mode are better resolved both inside and outside the patch. As with Pseudo-nitzschia, some limitation exists in applying the bimodal function, to match the second mode's amplitude of both nonparametric densities. This may possibly be attributed to an overestimate of the width of the second mode ( b 2 = 0.29). The second mode seems wider than the first mode and may include a third mode at the upper end of the distributions for Pseudo-nitzschia and photoautotrophic dinoflagellates. Only if b 2 were smaller could the amplitudes of the function's second mode fit the density estimates while resolving the amplitudes of the first mode equally well.

The fertilisation response of Chaetoceros spp. combines most of the characteristics described before, with a shift of the first mode together with an accentuation of the second (Fig. 3D). These structural changes can be inferred from the optimum values of the parametric fits.

According to small differences in AIC and the better performance of the unimodal fit in the control, a function with a single mode may suffice to describe the shift toward larger cells. The dominant Heterotrophs ---Heterotrophs also showed three different response patterns in size densities that are similar to those of the phytoplankton. By contrast, there are major differences between heterotrophic density spectra and photoautotrophic density spectra in the response of cell concentration to iron fertilisation. Heterotrophic cells are decreased within the fertilised patch for sizes smaller than 5 µm. Cells with ESDs larger than 5 µm were either equally abundant inside and outside the patch or increased after iron fertilisation. Flagellates, the smallest heterotrophic nanoplankton observed, exhibited symmetric unimodal densities both inside and outside the fertilised patch (Fig. 4A). The parametric unimodal density function fits both nonparametric estimates well, as indicated by a positive difference between the AIC (Table 3). The size peak of the flagellates is close to that of the prymnesiophytes, at approximately ESD = 2.9 µm. Although both fill a similar narrow niche in ESD, the abundance of heterotrophic flagellates decreased markedly at the same times that prymnesiophytes increased, suggesting that grazing on heterotrophic flagellates could be preferred to grazing on prymnesiophytes. Alternatively, both groups could have experienced similar grazing pressure, but the specific growth rate of the prymnesiophytes may have exceeded their loss to grazing.

Dinoflagellates are the second most abundant group among the heterotrophs. After iron fertilisation, their total cell concentration decreased slightly, but larger cells (ESD ≥ 8 µm) became more abundant at the same time (Fig. 4B). The size spectrum does not reveal distinct peaks but is Composite density spectra ---In Figure 5, we depict the combined nonparametric density estimates for all plankton groups, distinguishing photoautotrophs from heterotrophs. These composite density spectra cover a wide size range and represent the response of the microplankton community of the eastern equatorial Pacific to the IronEx II fertilisation. Phytoplankton cell densities (Fig. 5A) show that the patch and control distributions diverge about ESD = 5 µm, with patch abundance rising to a distinct peak around 6.5 µm. Cells in the picophytoplankton size range (< 2 µm) hardly change in abundance, but the density slightly declines (Supplemental Material), most likely due to stimulated grazing pressure of microzooplankton in the fertilised patch (Landry et al., 2000b). Cells in the size range of 3 to 5 µm show no noticeable density difference between the patch and control, but sizes larger than ESD = 5 µm in the patch clearly exceed the control concentrations. Heterotrophic dinoflagellates and ciliates seem to benefit from the increased abundance of prey larger than 5 µm, and a significant increase in microzooplankton cell density is evident at ESD = 7 µm. We interpret this cautiously as indicating an approximate predator-prey size ratio of 1.4 (i.e., 7:5 µm), which is roughly consistent with the ~ 1:1 ratio expected for dinoflagellate predators [START_REF] Hansen | The size ratio between planktonic predators and their prey[END_REF]. It is interesting to see how cell densities of Thalasissiotrix spp., appear in significant numbers only within the fertilised patch. These species, with ESDs larger than 30 µm, likely escape microzooplankton grazing. Although large diatoms appear only in small numbers, their contribution to biovolume is substantial (Fig. 5B).

DISCUSSION

Application of kernel density estimator for the analysis of plankton size

Nonparametric kernel density estimation is a widely accepted statistical method in many research fields, including astronomy (e.g. Chen, 1996;[START_REF] Carpenter | 2mass observations of the perseus, orion a, orion b, and monoceros r2 molecular clouds[END_REF], economics (e.g. Aït-Sahalia and Lo, 1998; [START_REF] Herzfeld | Corruption clubs: empirical evidence from kernel density estimates[END_REF], and biomedical research (e.g. Aitchison and Aitken, 1976;[START_REF] Gasser | Nonparametric regression analysis of growth curves[END_REF]. In marine and aquatic sciences, however, kernel density estimation has seldom been 16 standard deviation of size measurements of each plankton group. From our analysis, we learned that such a priori assumptions are not justified because asymmetries are apparent within a taxonomic plankton group. In contrast to [START_REF] Havlicek | Pelagic species size distributions in lakes: Are they discontinuous?[END_REF], we avoided prior assumptions of plankton size distributions by using nonparametric size densities of individual plankton groups, which were then added together to yield composite densities of heterotrophs and photoautotrophs.

Such an approach was outlined by [START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF] to improve the investigation of size diversity. Subsequent analyses of our composite densities may therefore help to address the issue of size-diversity measures.

In our study, we imposed constant bandwidths on each plankton group analysed. We also tested an approach where bandwidths of variable size were determined (results not shown) by applying the algorithm for adaptive bandwidth selection of [START_REF] Brewer | A Bayesian model for local smoothing in kernel density estimation[END_REF]. This algorithm relies on Bayesian statistical considerations in that it explicitly considers conditional dependencies between neighbouring kernels. A data point-specific bandwidth was obtained for every kernel (of each s ij )

after multiplying the constant bandwidth of equation ( 7) by a correction factor. Hence, the local (or data-point-specific) correction factor induced bandwidths smaller than the constant one [equation ( 7)] in size ranges where data points are close together, thus allowing better density resolution. In contrast, the correction factor leads to larger bandwidths for data points that are widespread. We found that adaptive bandwidth selection has the benefit of achieving higher resolution within certain size ranges but can be disproportionately affected by outliers at the upper and lower bounds of size-density spectra. Large bandwidths were obtained for minimum and maximum ESDs of individual plankton groups, which artificially enhanced the appearance of leptokurtic kurtosis in our densities. In addition, our resampling strategy tended to mask most of the benefit of higher resolution, while the outlier effects remained. Algorithms for adaptive bandwidth selection are intuitively appealing, but we conclude that a more detailed methodological analysis would be needed to specify conditions where the benefits of adaptive bandwidth selection prevail. Given the large numbers of data points in our study, we restricted ourselves to constant bandwidths. Optimal constant bandwidths were much easier to calculate, which was particularly desirable when used in conjunction with the bootstrapping approach.

Choosing and fitting parametric size-density functions

Through visual inspection of the nonparametric density estimates on logarithmic scales of ESD, we identified three distinct distributional patterns: symmetric unimodal, right skewed, and bimodal.

Using a unimodal normal distribution function to fit the symmetrical patterns seemed an obvious choice. Additional degrees of freedom were introduced when imposing a bimodal function, which An alternative top-down interpretation of deviations from normality can be based on predator impacts: densities might be unimodal in the absence of grazing, with deviations from normality imposed by extensive grazing within a particular size range. This perspective is inspired by the existence of a persistent depression in densities within the iron-fertilised patch between 4 and 5 µm ESD (Figs. 3 and4). Furthermore, disproportionate grazing losses of cells, in the 4-5 µm size range, could explain the bimodal structures depicted for Pseudo-nitzschia (also autotrophic dinoflagellates and heterotophic green dinoflagellates), as well as the densities of Chaetoceros and heterotrophic dinoflagellates. Rather than a "grazing induced" separation into two modes, the asymmetry of the unimodal, but right-skewed, size densities could be due to disproportionate grazing on smaller cells on the left-hand side of the distributions. estimates together with parametric fits that consider a size preference for grazing (ϕ) and a corresponding width for the grazing impact (κ), which remain constant. The distribution of sizeselective grazing acts on all plankton groups that fall within the size range of interest, with only the magnitude of the grazing impact varying among the different plankton groups. The parametric function used can be interpreted as a unimodal density function ( pdf 1 ) that is multiplied by a grazing filter (φ): ˆ

f = pdf 1 ⋅ φ , with φ = A /(κ⋅ 2π )⋅ [1 -B⋅ exp{(s -ϕ) 2 /κ 2 }]. The parameter A
relates the amplitude of the unimodal pdf 1 to the maximum grazing impact (B). Such parametric functions cannot be fit to represent the individual densities within the confidence limits. However, the general characteristics of the parametric fits provide support for a top-down explanation of bimodality in the size range between 2 and 20 µm ESD. The qualitatively good, but statistically non-significant fits of this parametric approach could result from morphological differences of plankton prey that remain unresolved when normalising to ESD. To resolve this issue, we may need a dynamical modelling approach, where convolutions of a grazing kernel function with individual (unimodal) size-densities of plankton groups or species are explicitly considered, similar to the procedure discussed by Armstrong (2003).

Resolving details in plankton size structure and implications for ecological modelling

With the recognized importance of organism size as an organizing trait for plankton community structure, trait-based modelling approaches [START_REF] Litchman | Trait-based community ecology of phytoplankton[END_REF] seem likely to improve descriptions of plankton interactions. Specification of trade-offs among different traits (like size) is important and can link ecosystem function to biogeochemical pathways of organic matter flux (e.g. [START_REF] Verdy | Optimal phytoplankton cell size in an allometric model[END_REF]. For example, trade-offs between nutrient uptake affinity and predation defence [START_REF] Merico | A trait-based approach for downscaling complexity in plankton ecosystem models[END_REF] could be expressed in terms of size-dependencies, including aspects of cellular or metabolic size-scaling (e.g. [START_REF] Irwin | Scaling-up from nutrient physiology to the size-structure of phytoplankton communities[END_REF][START_REF] Litchman | The role of functional traits and trade-offs in structuring phytoplankton communities: scaling from cellular to ecosystem level[END_REF] and adaptation of mean or optimum cell-size [START_REF] Jiang | Adaptive evolution of phytoplankton cell size[END_REF][START_REF] Yoshiyama | Optimal cell size for resource uptake in fluids: a new facet of resource competition[END_REF].

Given these theoretical prospects together with new modeling approaches, the availability of highresolution data of plankton community structure (in terms of size and taxonomic differentiations; Armstrong, 2003b) is of crucial importance. The degree to which details can be resolved depends primarily on the quantity and quality of available data. To produce these data, rigorous procedures for statistical treatment of size measurements are needed. Plankton size data are usually available as histograms that vary in bin width and point of origin. Analysing size data from different investigators or experiments is difficult, as one needs to combine a series of differently binned data.

If continuous densities of plankton size are derived using kernel estimation methods, combining With the bootstrapping approach, as applied in our study, it should be possible to reanalyse historical binned size data and provide continuous density estimates together with confidence limits. This procedure would facilitate inter-comparisons among different data sets and support investigations of species interactions while also providing constraints for size-based plankton models.

The response of plankton community structure to iron addition during IronEx II can be explained by a combination of bottom-up and top-down effects [START_REF] Armstrong | Grazing limitation and nutrient limitation in marine ecosystems: Steady state solutions of an ecosystem model with multiple food chains[END_REF][START_REF] Price | The equatorial Pacific Ocean: Grazer-controlled phytoplankton populations in an iron-limited system[END_REF]. The natural plankton community of the eastern equatorial Pacific is subject to a tight control inherent to the microbial loop, where growth, grazing and remineralisation are strongly coupled [START_REF] Pomeroy | The ocean's food web, A changing paradigm[END_REF]Azam et al., 1983). The bloom develops as a disturbance of the balance between utilisation (growth uptake) of the limiting trace element, iron, and its regeneration via microzooplankton predation. The composite community size distributions reveal a narrow depression between ESD = 4 and 5 µm, which is close to the peaks in densities of Chaetoceros and heterotrophic dinoflagellates outside of the fertilised patch. We speculate that the natural zooplankton community outside the fertilised patch maintains a grazing preference for cells in this size range. Cell concentrations in this range declined after iron addition, whereas abundance of slightly larger cells increased, producing a shift in the peak of Chaetoceros towards larger cells and enhancing the skewness in the distribution of heterotrophic dinoflagellates. At the same time, cell concentrations of Pseudo-nitzschia and autotrophic dinoflagellates increased; but since their size variance includes larger cells that could escape extensive grazing, their size densities expressed a clear bimodality.

From our analysis, it appears that the 4-5 µm ESD size range may represent a sensitive link of predator-prey interaction. At approximately 5-µm ESD, there is a step-like increase in densities of larger cells in the fertilised patch relative to the control (Fig. 5). The slight increase in abundance of photoautotrophic cells smaller than 4 µm ESD has little impact on biovolume, and thus on biomass. The sensitive size ranges in the proximity of ≥5 µm and >40 µm ESD can also be interpreted in terms of the "loophole" concept [START_REF] Bakun | Environmental 'loopholes' and fish population dynamics: comparative pattern recognition with focus on El Niño effects in the Pacific[END_REF][START_REF] Irigoien | Phytoplankton blooms: a 'loophole' in microzooplankton grazing impact[END_REF], in which phytoplankton growth exceeds predation, allowing escape from the tightly linked predatorprey interactions of the microbial food web. However, the loophole concept does not adequately account for bottom-up, allometric size-scaling effects discussed above. Assuming that mineralisation of organic matter and the regeneration of organic-iron compounds regulate the stability and efficiency of the microbial loop, a characteristic size-density pattern of the plankton community is established, as observed at the control stations. The recent model of [START_REF] Yoshiyama | Optimal cell size for resource uptake in fluids: a new facet of resource competition[END_REF] makes the point, for example, that size dependencies extend to the level of the the balance between bottom-up and top-down size-dependent processes needs to be explored using systems models that relate directly to the patterns of size variability observed in natural systems, both in their base states and in response to perturbations such as iron fertilisation. For such modelling efforts, rigorous methods to develop, analyse, and synthesise size-based assessments of plankton community structure are necessary.

SUMMARY AND CONCLUSIONS

Using cell counts and size data from IronEx II, we derived nonparametric characterizations of sizefrequency distributions for several photoautotrophic and heterotrophic plankton groups. Symmetric nonparametric density estimates were described well by unimodal parametric density functions.

Asymmetries, such as right-skewed and bimodal densities, as seen for example in the density spectra of Pseudo-nitzschia and Chaetoceros, could be resolved using bimodal parametric functions. Size-based analysis of the nonparametric densities revealed a significant depression among photoautotrophic and heterotrophic plankton groups in the range of 4-5 µm ESD. Since this characteristic is expressed for both autotrophs and heterotrophs, it seems more likely to be associated with size-selective grazing than with bottom-up effects on primary producers. By imposing parametric functions that represent a size-specific grazing filter, we found qualitatively good agreement to the bimodal nonparametric densities.

The issue of deriving parametric functions for predator-prey relationships needs to be addressed further. As seen for the size densities of individual plankton groups, the critical size range 4-5 µm ESD appears relatively invariant inside and outside the fertilised patch, while the biovolume of smaller cells is hardly altered little by iron addition. Those plankton groups or species with cells larger than 5-µm ESD yield the major fraction of net biovolume increase after fertilisation, with heterotrophs showing a size-shifted response and an implicit mean predator:prey size ratio suggestive of grazing by dinoflagellates.

In this study, we have shown that kernel density estimation is well-suited to representing important aspects of plankton community structure. The smoothed nonparametric densities are ideal for subsequent analyses of spectra to obtain size functions for growth rates and grazing mortalities, as suggested for example by [START_REF] Zhou | What determines the slope of a plankton biomass spectrum[END_REF]. Since most density estimates yield symmetric distributions outside the fertilised patch, we conclude that the undisturbed microbial plankton community can be well described by unimodal Gaussian functions. Asymmetries are clearly expressed after fertilisation. Only by means of dynamical modelling can we fully resolve the issues of bottom-up and top-down control of the plankton community [START_REF] Armstrong | Grazing limitation and nutrient limitation in marine ecosystems: Steady state solutions of an ecosystem model with multiple food chains[END_REF][START_REF] Price | The equatorial Pacific Ocean: Grazer-controlled phytoplankton populations in an iron-limited system[END_REF], In addition to their use in modelling, kernel density estimates of plankton size can contribute to inter-comparisons among independent studies of plankton community structure and dynamics on different spatial and temporal scales. Such activities are currently hindered by the multiplicity of schemes and scales for size binning in histograms. Use of the bootstrapping approach, in conjunction with the kernel density estimation, provides a statistical analysis procedure that can be utilised to derive continuous size-densities of plankton from historical, binned data. Parametric fit with a unimodel density function ( pdf 1 ) multiplied by a grazing distribution (φ):

FUNDING

ˆ f = pdf 1 ⋅ φ , with φ = A /(κ⋅ 2π )⋅ [1 -B⋅ exp{(s -ϕ) 2 /κ 2 }].
The grazing distribution consists of a preference size for grazing (ϕ=1.52) and a corresponding width for the grazing impact (κ=0.88), which remain constant for all plankton groups: 1) Photoautotrophic dinoflagellates (with a 1 =1.

Formatted

Formatted

Formatted

  differs from a data-model comparison, an appropriate metric should reflect the number of data points. A perfect data-model comparison also demands the consideration of covariances between data points. The calculation of covariances between size measurements could

  negative. The lower the AIC value, the better the fit. The difference between AIC 2 and

  . Apart from these details, the distribution shifts slightly towards larger sizes, which can be adequately described with the unimodal parametric function.The size density changes for Pseudo-nitzschia (previously incorrectly reported as Nitzschia spp. inLandry et al., 2000a) differ from those for prymnesiophytes (Fig.3B). Instead of a slight shift towards larger cells, the distribution switches from a fairly symmetric unimodal density in the control samples to a bimodal distribution within the patch, with two distinct modes at ESD = 2.86 and 7.25 (m (with EMBED Equation.3 = 1.05 and EMBED Equation.3 = 1.98). The bimodality of the nonparametric estimates is well represented by the parametric fit, resolving the separation of the two peaks. The first mode (between ESD = 2 and 3 µm) is similar between control and patch. The second mode around 7 µm is detectable in the control samples, but extremely small. Subsequent analyses (not shown) indicate that two modes are also well expressed in densities based on cell thickness and length rather than on ESD. Cells in the second mode thus seem morphologically similar to those in the first mode but are proportionally larger. The abundance increase of Pseudo-nitzschia spp. is most pronounced compared to other phytoplankton, and resolving the second mode in the parametric function is indeed crucial because it contains a significant biovolume response.

  from ESD = 5 µm to 6 µm. This shift is exaggerated in the parametric density estimate of the unimodal function. The nonparametric density of the iron-patch, suggests the existence of a third and perhaps even a fourth mode for larger ESD. Again, as in the parametric fits of photoautotrophic dinoflagellates and Pseudo-nitzschia, the second mode of Chaetoceros is wide enough ( b 2 = 0.34) to fit the kurtosis at the upper tail and thus resolves the right-side skewness of the nonparametric density estimate. Such density estimates are particularly challenging, since their skewed distributions and their kurtosis are not ideally represented by either unimodal or bimodal functions on log-scale.

  photoautotrophic dinoflagellates and Pseudo-nitzschia.The green dinoflagellates, although containing pigments from prokaryotic endosymbionts, are treated here as heterotrophs. Both density estimates (patch and control) clearly show two modes,

Figure

  Figure4Ddepicts heterotrophic ciliates in the larger size range, where they appear in small numbers. Density estimates could not be made outside the fertilized patch because there were too few data points (three) with independent length and width measurements. Consequently, we could only infer a density estimate from observations made within the fertilised patch. The increase in abundance represents a large fraction of the biovolume increase of the microbial community. An approximately 3-fold increase in ciliate biovolume is reported inLandry et al. (2000a), who suggested a substantial increase in grazing on smaller plankton following fertilisation. A unimodal density function with a 1 = 2.57 and b 1 = 0.38 (peak at ESD = 13 µm) suffices to describe the nonparametric density estimate. No improvements are achieved when optimising the additional three parameters of the bimodal function.

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  ciliates converge around ESD = 15 µm, with ciliates reaching the approximate 2⋅10 3 cells L -1 abundance of dinoflagellates. Large diatoms, mainly Rhizosolenia and

  applied.Sanvincente-Anorve et al. (2003) were first to promote the method for investigations of length-frequency distributions of fish larvae. Their motivation was to allow better characterization of skewness and multimodality in their distributions, which can be difficult to see in scattered, discontinuous histograms.[START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF] provided in-depth comparison of parametric and nonparametric density estimation methods and assumptions for deriving a robust size-diversity model based on the Shannon index. They found nonparametric methods, kernel density estimators in particular, to be superior to fitting parametric functions to size and weight data. Similar to our composite density estimates (Fig.5), they found distributional patterns that can hardly be represented by a single generic parametric function, like a Pareto-based biomass-size density(Vidondo et al., 1997).[START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF] also stressed that most parametric functions that represent the entire plankton community size-structure have no true ecological meaning, apart from exponential-shaped pdfs.A study by[START_REF] Havlicek | Pelagic species size distributions in lakes: Are they discontinuous?[END_REF] compared plankton size-structure patterns among lakes, constructing size distributions of phytoplankton, zooplankton, and fish by summing over a series of Gaussian density functions. Although they referred to their procedure as kernel density estimation, it differed from the conventional approach by imposing parametric pdfs. In our study, we followedSanvincente-Anorve et al. (2003) and[START_REF] Quintana | A nonparametric method for the measurement of size diversity with emphasis on data standardization[END_REF] in using kernel density estimation as a tool to derive nonparametric density estimates. Our analyses are therefore fundamentally different from those of[START_REF] Havlicek | Pelagic species size distributions in lakes: Are they discontinuous?[END_REF] with respect to prior assumptions.[START_REF] Havlicek | Pelagic species size distributions in lakes: Are they discontinuous?[END_REF] assumed that plankton size densities were normally distributed on a linear scale and that each bandwidth of their density function was equivalent to the

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  allow better representation of right-skewed and bimodal size densities. In addition we learned from our analysis that estimating the three additional parameters (q, a 2 , b 2 ) sometimes also resolves kurtosis better (e.g. in heterotrophic dinoflagellates), which is indicated by small estimates of q and larger values of b 2 than of b 1 . This better representation of kurtosis is thus a side effect of fitting the bimodal parametric function to some density estimates. The existence of leptokurtic kurtosis, for example in the distributions of heterotrophic-and green dinoflagellates, appears to be robust. Right-skewed and leptokurtic distributions of natural nano-and microplankton communities were similarly expressed in the data analysed bySabetta et al. (2005), who proposed using Gamma functions to represent these distributional patterns.Campbell and Yentsch (1989) related synchronous and asynchronous cell growth to specific size-distribution patterns. Interestingly, their computations revealed skewed and bimodal densities for an asynchronous cell population, where cell growth is restricted to a short period in the middle of the cellular life cycle. When deriving parametric size density functions for plankton, it is worthwhile considering causal links between population growth, life cycles, and variations of size density. Similarly, distribution functions that combine cell cycle information with "random" Gaussian noise were discussed inArmstrong (2003). In the modelling study of Litchmann et al.(2009), evolutionary stable strategy sizes of plankton were investigated in response to varying pulse periods of nitrogen supply. For intermediate pulse periods (between thirty and forty days) of nitrogen supply, their model yields coexistence of small and larger cells, which generates bimodal size distribution patterns. According to these model predictions, bimodal patterns can be sustained simply by balancing the allometric dependencies of maximum growth rate and nutrient-storage ability. These model findings allow for a bottom-up interpretation of the bimodalities seen in plankton size densities.

Figure 6

 6 Figure 6 depicts bimodal densities of photoautotrophic and heterotrophic plankton density

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  organic matter that serve as resources for microbial consumers. Ultimately,

Figure 1 :

 1 Figure 1: a) -h) The sequential analysis procedure. The example includes resampling of counts and size measurements of centric diatoms observed at stations within the iron-fertilised patch. Original data and resample data are treated throughout the analysis. Details are explained in the Method section.

Figure 2 :

 2 Figure 2: Example of bandwidth selection. A reference density (dashed line) is described by five data points (asterisk and square symbols); our aim is to recover a density estimate (solid line)after removing one data point (square symbol). "Bias" refers to the deviation between the dashed line and the solid line at the location of the missing point; "variance" refers to the variability of the density estimates that might be gotten at that point, e.g., from different bootstrap estimates. See text for further explanation.

Figure 3 :

 3 Figure 3: Cell concentration densities of the four dominant taxonomic phytoplankton groups during the IronEx II experiment. A) Prymnesiophytes, B) Pseudo-nitzschia, C) Photoautotrophic dinoflagellates, and D) Chaetoceros spp. . Shaded areas include nonparametric density estimates of the mean and the 95% confidence limits (± 2 standard deviations). The horizontal axis of equivalent spherical diameter (ESD) is on a logarithmic scale (base 10). ESD can be backtransformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to a linear scale according to ESD=exp(s).

Figure 4 :

 4 Figure 4: Cell concentration densities of microzooplankton during the IronEx II experiment. A) Heterotrophic nano-flagellates, B) Heterotrophic dinoflagellates, C) Heterotrophic, green dinoflagellates with endosymbionts, and D) Ciliates. Shaded areas include nonparametric density estimates of the mean and the 95% confidence limits (± 2 standard deviations). The horizontal axis of equivalent spherical diameter (ESD) is on a base 10 logarithmic scale. ESD can be backtransformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to a linear scale according to ESD=exp(s).

Figure 5 :

 5 Figure5: Composite size-spectra: Plankton densities of the photo-autotrophic and heterotrophic plankton communities were obtained by summing corresponding mean nonparametric density estimates respectively. A) Cell concentration densities inside (solid lines) and outside (dashed lines) the iron fertilised patch. B) Corresponding densities of biovolume (as indicator for biomass) inside (solid lines) and outside (dashed lines) the fertilised patch.

Figure 6 :

 6 Figure 6: Nonparametric density estimates and their 95% confidence limits (shaded).

  h) The sequential analysis procedure. The example includes resampling of counts and size measurements of centric diatoms observed at stations within the iron-fertilised patch. Original data and resample data are treated throughout the analysis. Details are explained in the Method section. 215x279mm (150 x 150 DPI) Example of bandwidth selection. A reference density (dashed line) is described by five data points (asterisk and square symbols); our aim is to recover a density estimate (solid line) after removing one data point (square symbol). "Bias" refers to the deviation between the dashed line and the solid line at the location of the missing point; "variance" refers to the variability of the density estimates that might be gotten at that point, e.g., from different bootstrap estimates. See text for further explanation. 421x310mm (72 x 72 DPI) Cell concentration densities of the four dominant taxonomic phytoplankton groups during the IronEx II experiment. A) Prymnesiophytes, B) Pseudo-nitzschia, C) Photo-autotrophic dinoflagellates, and D) Chaetoceros spp. . Shaded areas include nonparametric density estimates of the mean and the 95% confidence limits (± 2 standard deviations). The horizontal axis of equivalent spherical diameter (ESD) is on a logarithmic scale (base 10). ESD can be back-transformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to a linear scale according to ESD=exp(s). 246x188mm (150 x 150 DPI) Cell concentration densities of microzooplankton during the IronEx II experiment. A) Heterotrophic nano-flagellates, B) Heterotrophic dinoflagellates, C) Heterotrophic, green dinoflagellates with endosymbionts, and D) Ciliates. Shaded areas include nonparametric density estimates of the mean and the 95% confidence limits (± 2 standard deviations). The horizontal axis of equivalent spherical diameter (ESD) is on a base 10 logarithmic scale. ESD can be back-transformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to a linear scale according to ESD=exp(s). 246x188mm (150 x 150 DPI) Composite size-spectra: Plankton densities of the photo-autotrophic and heterotrophic plankton communities were obtained by summing corresponding mean nonparametric density estimates respectively. A) Cell concentration densities inside (solid lines) and outside (dashed lines) the iron fertilised patch. B) Corresponding densities of biovolume (as indicator for biomass) inside (solid lines) and outside (dashed lines) the fertilised patch. 222x103mm (150 x 150 DPI) Nonparametric density estimates and their 95% confidence limits (shaded). Parametric fit with a unimodel density function ( ) multiplied by a grazing distribution (φ): , with . The grazing distribution consists of a preference size for grazing (ϕ=1.52) and a corresponding width for the grazing impact (κ=0.88), which remain constant for all plankton groups: 1) Photoautotrophic dinoflagellates (with a1=1.44, b1=0.2, A=130, B=0.998), 2) Pseudo-nitzscha (a1=1.44, b1=0.27, A=90, B=0.998), 3) Chaetoceros spp. (a1=1.75, b1=0.27, A=30, B=0.96), and 4) green dinoflagellates (a1=1.59, b1=0.27, A=65, B=0.99). 247x188mm (150 x 150 DPI) Cell concentration densities of four phytoplankton groups as observed during the IronEx II experiment, outside of the fertilise patch (control, gray shaded) and inside of the iron-fertilised patch (patch, red shaded): A) Pico-plankton is a combination of Synechococcus, eukaryotes, and Prochlorococcus, B) Phaeocystis, C) miscellaneous pinnate diatoms, and D) Cryptomonads. Shaded areas depict the 95% confidence limits (mean ± 2 • standard deviations) of the nonparametric density estimates. Horizontal axis of equivalent spherical diameter (ESD) is on logarithmic scale with base 10. ESD can be back-transformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to linear scale according to ESD=exp(s). The corresponding density estimates are included in the supplementary phytoplankton data files. 251x193mm (150 x 150 DPI) Cell concentration densities of four different diatoms observed during the IronEx II experiment, outside of the fertilise patch (control, gray shaded) and inside of the iron-fertilised patch (patch, red shaded): A) Nitzschia bicapitata, B) Thalassiothrix, C) Thallassionema, and D) Tripodoneis. Shaded areas depict the 95% confidence limits (mean ± 2 • standard deviations) of the nonparametric density estimates. Horizontal axis of equivalent spherical diameter (ESD) is on logarithmic scale with base 10. ESD can be back-transformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to linear scale according to ESD=exp(s). The corresponding density estimates are included in the supplementary phytoplankton data files. 252x193mm (150 x 150 DPI) Cell concentration densities of three different diatoms and of auto/mixo-trophic ciliates, as bserved during the IronEx II experiment, outside of the fertilise patch (control, gray shaded) and inside of the iron-fertilised patch (patch, red shaded): A) miscellaneous centric diatoms, B) Thalassiosira, C) Rhizosolenia, and D) autotrophic ciliates. Shaded areas depict the 95% confidence limits (mean ± 2 • standard deviations) of the nonparametric density estimates. Horizontal axis of equivalent spherical diameter (ESD) is on logarithmic scale with base 10. ESD can be back-transformed from the normalised, logarithmic variable (s=ln(ESD/1.0 µm)) to linear scale according to ESD=exp(s). The corresponding density estimates are included in the supplementary phytoplankton data files. 252x189mm (150 x 150 DPI)

  kernel density estimator is less susceptible to the intuitive specifications needed for histograms. Also, if the density estimate is smooth rather than discontinuous, the intermediate step of kernel density estimation provides an ideal basis for devising parametric functions.

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  This research was conducted as part of the NSF grants OCE 03-24666, OCE 99-11765, and OCE 04-17616. The reanalysis was partially funded by the Helmholtz Association (Germany) PACES programme "Polar regions and Coasts in a changing Earth System".

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Table Captions Table 1 :

 Captions1 Resolutions of microscopic size measurements during the IronEx II experiment.

Resampled data sets were drawn from uniform error distributions based on the microscopic resolution within each length and width size range.

Table 2 :

 2 Best parameter estimates for dominant phytoplankton groups, given unimodal and bimodal parametric probability density functions (pdf 1 and pdf 2 ). Total cell concentrations are equal to the integral of the nonparametric densities of the original data. Differences in Akaike information criterion (unimodal, AIC 1 ; bimodal, AIC 2 ) compare the goodness of the parametric fits for each plankton group.

Table 3 :

 3 Best parameter estimates for dominant zooplankton groups, given unimodal and bimodal parametric probability density functions (pdf 1 and pdf 2 ). Total cell concentrations are equal to the integral of the nonparametric densities of the original data. Differences in Akaike information criterion (unimodal, AIC 1 ; bimodal, AIC 2 ) compare the goodness of the parametric fits for each plankton group. Plankton groups observed and analysed. The different groups are separately listed as autotrophic and heterotrophic plankton. The number of actual counts is equal to the number of data points considered during our reanalysis. The numbers of enumerated cells were converted to cell concentrations, depending on sample volume and on the count area on the microscope slide, see text for details. Length and width were measured during microscopy. Cell thickness was calculated, as described in the text.

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http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

ACKNOWLEDGEMENTS

We thank Sara Tanner and Susan Brown for their efforts and care in producing the plankton community data set for IronEx II. We also thank Carsten Lemmen for assistance during the implementation of C-code into MATLAB Mex-files. We are thankful for comments by Darcy Taniguchi. We greatly appreciate supportive and constructive comments made by four anonymous reviewers.

Legends of Supplementary Material

Suppl. Figure 1