Linear method: Operation.

The ‘linear method’ (Eqn 1) has been commonly used to estimate specimen abundances in a population, whereby concentrations of thousands or even millions of specimens (per unit mass or volume) can be estimated from only a few hundred. This method involves the identification and counting of individual specimens, and their assignment to two (or more) specimen categories: markers () and targets ( or, for multiple target categories, ).

Following [31], this procedure can be illustrated by an agricultural analogy. A farmer introduces 1000 black sheep into a large flock of white sheep of indeterminate size. After a modest interval, during which the black and white sheep subpopulations mingle (assuming the sheep do not discriminate based on colour), the farmer herds the sheep into a line. The farmer then counts a sample of sheep until the number of white sheep (the targets, or ) reaches 100. Assume that among this random sampling, the farmer also counted ten black sheep (the markers, or ). By extrapolating the white-to-black sheep ratio (10:1) to the whole flock, the farmer estimates the total white sheep population as 10,000.

This can be considered a linear procedure, whereby specimen counts (both and ) are collected simultaneously and sequentially. In the simplest case, this is conducted until a pre-determined minimum value of is reached (although one could also use pre-determined values). Stopping when a predetermined number of target specimens is reached defines a random “window” of a sample area (e.g., microscope slide) within which we now have a random number of markers. In the microfossil case study, the target population is the total number of microfossils in a given sediment or sedimentary rock, and the marker population is the total number of introduced exotic markers (e.g., Lycopodium spores). Since all specimens in a sampled area are examined individually and assigned to a count category, the linear method tends to provide accurate target identifications.

The highest precisions for concentration estimates are achieved when the target specimens and exotic markers are equal in the population (i.e., ; Regal & Cushing [61]). We recover this result as a by-product of mathematical analyses comparing the efficiency of the two methods (see S1 File). However, for combined () specimen counts of several hundred (e.g., ≥ 500), a ‘target-to-marker ratio’ (or , following the terminology of [31]) close to 2:1 provides the most efficiency by striking the optimum balance between precision and data collection effort [31]. For this target-to-marker ratio (), count sizes greater than 500 provide sharply diminishing returns with increased effort (translating to increased data collection time). Similarly, Price et al. [51] recommended that the target-to-marker ratio should be < 5:1 (or ). In cases where the target and marker ratios are near-equivalent (for single target counts), precise concentrations can typically be achieved with target counts of less than 300 [49].

(Note: unless stated otherwise, we assume that the targets are more common than the exotic markers; hence, we utilise the term ‘target-to-marker ratio’ in place of ‘common-to-rare ratio’. If, however, the markers are more common than the targets, see the relevant section in the S1 File.)

Linear method: Precision estimates.

To gauge the precision of concentration estimates using the linear method, concentration total error values () were calculated with the formula (from [34], updated with terms defined by [31]):

(2)

where is the total standard error of the linear method concentration estimate (in %). In this formula, the variables , and are as in Eqn 1, while is the proportional sample standard deviation of the number of exotic markers per dose (see S1 Table).

The standard deviations ( and ) come from the underlying assumption that all specimens (targets and markers) are independently distributed according to a Poisson point process (i.e., they are distributed uniformly at random over the study area and, hence, their distributions are independent of each other). These assumptions provide a valid statistical framework for many data in the biological [62] and physical [63] sciences. The standard deviations are then divided by the respective number of counted specimens to yield proportional standard deviations. Note that since the number of targets () (or, in some cases, the number of markers []) is pre-determined in the linear method, the inclusion of error terms for both may seem counterintuitive. However, the error term coming from the fixed quantity can be understood as the contribution to the error from the variable size of the linear method window determined by the random locations of the counted specimens.

Since the specimens are counted sequentially, neither density nor homogeneity (defined here as the variance of the specimen density across the study area) will impact the precision of this method. However, these factors will certainly affect the ‘data collection effort’, which is reflected by the time spent during counting (e.g., at the microscope, sparsely populated slides will take a longer time to achieve a given count size). We have not provided a comprehensive discussion of the effects of inhomogeneity, since we do not yet have quantitative data to test this; however, this would serve as the basis for future investigation.

Precision of linear method concentration was also estimated by calculating confidence intervals for . For these, the parametric approach by Maher (1981, p. 179) was implemented, the relevant parameters and formulae for which are summarised in S1 Table. A minor change to the confidence interval calculation was made to adapt it to sample mass (rather than volume), following [49]. Maher’s [31] calculations encompass the uncertainty in size (in units of volume or mass) and number of samples. In the microfossil case study below, the masses were considered true values, and standard errors were set to 0.1 g (equivalent to the precision limits of the instrument used for measuring mass).

Field-of-view subsampling (FOVS) method: Operation.

The FOVS method is a count data collection approach that we have designed as an alternative, or complement, to the linear method. This area-based sampling is distinct from the linear method above, for which both target () and marker () specimens are counted in a continuous sequence. Instead of linear, stepwise counts of all individual specimens, the FOVS method obtains large abundance data sets by extrapolating abundances from surface area subsamples.

To illustrate the differences between the linear and FOVS methods, let’s consider a modified version of the agricultural analogy described above (see ‘linear method: operation’). As before, the farmer introduces a known number of black sheep into a white sheep flock of larger but indeterminate size. This time, however, the farmer adopts a novel perspective: examining the flock from satellite. Even at this scale, the black sheep are clearly distinct from the white sheep. Using a consistent field-of-view size (e.g., 100 m²), the farmer first establishes the mean and standard deviation of white sheep (the targets, or ) from a subsample of fields of view across the meadow. This ‘calibration count’ is followed by a second count (the ‘extrapolation count’), whereby another independent subsample of fields of view from the same meadow is collected, but this time counting only the distinctive and less common black sheep (the markers, or ). Given the relative rarity of black sheep, the farmer can quickly scan a large quantity of fields of view, building a large sample size of black sheep markers during this second stage. The farmer can then infer an enormous sample size of white sheep targets by combining the mean number of white sheep per field of view in the first count with the ratio of black-to-white sheep provided by the second count.

Area-based subsampling techniques have been applied widely across disparate sciences to infer population statistics [64], but—to our knowledge—never before to organic microfossil assemblages (cf. [16]). For example, quadrat sampling is a standard method used in field ecology, whereby the specimens within a set of representative areas are counted, from which populations can be extrapolated across a broader study area [65,66]. We propose that the microscope field of view (FOV) can be utilised as the unit of surface area, in place of the quadrat. For valid results, the areas of the subsamples or ‘fields of view’ should be consistent for all counts of a given assemblage.

The essential difference between standard quadrat methods and the FOVS method proposed herein is the addition of exotic markers of known quantity (Eqn 1). This is because quadrats cannot be applied directly to samples of target specimens (e.g., cells, microorganisms) that have been isolated from their original populations (e.g., a patient, a body of water or a sedimentary stratum). This process of isolation results in specimen distributions and concentrations that are very different to those in the natural systems from which they were sampled. In such isolated contexts, the observed proportions of targets to markers provide an accurate assessment of total target concentrations within a population from which a sample has been taken. If the additional uncertainty derived from the field-of-view statistics is offset by the larger sample sizes, this combination of concentration estimation and surface area subsampling holds the potential for precise absolute abundances with minimal sampling effort.

The FOVS method involves the following sequence of steps, which are summarised in Fig 1; see S1 Table for equations and terms. (Note: The following procedure assumes that the FOVS method has been determined as the most appropriate for the given population [Fig 1, steps 1, 2]. For more details on the method determination process, see discussion in ‘choosing the superior count method’).

1. A. Examine the assemblage to determine whether the targets or markers are more common. (For the following example, the target specimens, , are assumed to be more common. If markers, , are more common than , see the S1 File.)
2. B. Conduct a series of ‘calibration counts’ for the target specimen type (Fig 1, step 3). These consist of counts of all target specimens in a subsample of fields of view (where is the total number of calibration-count fields of view). To avoid overrepresenting the number of target specimens per unit area, exclude half of the specimens on the field-of-view margins (e.g., on one side of the field of view; cf. [67]).
3. C. Calculate the mean () and standard deviation () of the target specimens per field of view from the calibration counts (Fig 1, step 4).
4. D. Conduct an ‘extrapolation count’ of new fields of view (Fig 1, step 5), whereby two values are collected simultaneously: 1, the number of extrapolation-count fields of view (); and 2, the number of rare specimens (typically, the exotic markers, ). To ensure that the calibration counts are representative, the regions chosen for both the calibration- and extrapolation-count fields of view should be located close to each other on the microscope slide. In this way, the specimen density and heterogeneity will be approximately equivalent (Fig 2A).
5. E. Organic microfossil concentration can then be calculated (Fig 1, step 6). Assuming the target specimens were the common specimens during the calibration counts, then an approximate value for can be calculated by multiplying the mean number of targets per field of view from the calibration counts () by the counted number of fields of view from the extrapolation counts (). This extrapolated value of for the extrapolation count is denoted , i.e.,

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Fig 2. Field-of-view subsampling (FOVS) method demonstrated with a schematic microscope slide of organic microfossils.

(A) Organic microfossil slide, with specimens dispersed across a coverslip. For valid subsampling: 1, the spatial range of the extrapolation-count fields of view (FOVs) should match those of the calibration counts; 2, calibration counts should be conducted with representative fields of view (e.g., both x- and y-axes, or randomly distributed); and 3, prevent edge effects by avoiding counts near the coverslip margins. (B) A typical field of view of an organic microfossil (palynological) slide with potential targets and markers; red and blue arrows: terrestrial microfossils, fossil wood fragments (red) and a fossil plant spore, Playfordiaspora crenulata (blue); green arrows: exotic markers (modern Lycopodium clavatum spores); distinctive optical features of the markers are necessary for accurate and rapid data collection using the FOVS method; scale = 50 μm (sample S090320, Bonneys Plain-1, 301.56 m).

https://doi.org/10.1371/journal.pone.0320887.g002

(3)

The subscript in the term indicates that the target specimens are the most common specimen type, and subjects of the calibration counts. The total number of exotic markers from the extrapolation counts () can then be inserted directly into a modified version of Eqn 1:

(4)

where we denote the estimated organic microfossil concentration from the FOVS method by to distinguish it from the linear method concentration estimate (; see Eqn 1). (The subscript “” for denotes that the target specimens are the foci of the calibration counts; if this is not the case, see the S1 File).

Field-of-view subsampling (FOVS) method: Precision estimates.

While both absolute abundance methods (linear and FOVS) estimate the population values from subsamples, the accuracy of common specimen abundances using the FOVS method depends on the representativeness of the fields-of-views during the calibration counts. This entails a different source of error compared to the linear method. As such, the calculation of total error in Eqn 2 needs to be modified as follows:

(5)

where is the total standard error of the concentration mean for the FOVS method (in %). In this function, and are as in Eqn 2, while is the proportional sample standard deviation for the target specimens from the calibration counts and is the number of fields of view during the calibration counts. For further calculations and descriptions of terms, see S1 Table.

Note that we have scaled by the so-called correction, which is needed to ensure an unbiased estimation of the standard error of a sample mean with an underlying normal distribution (without this correction, the standard error is expected to understate the true value). This appears to be a classical result, with the earliest unambiguous reference we can find being [68] (eqn 16 therein). This correction is an artefact of the square root taken to obtain a standard deviation, and so if one uses the variance then it is not needed. However, standard deviation is more commonly cited in the literature, and so we utilise the more widely used standard deviation formulation in Eqn 5, but include the conservative correction in the term (see the S1 File).

Owing to their large sample sizes, standard sampling theory [64] tells us that the abundances of marker specimens per dose and the mean number of common specimens in the calibration-count fields of view () both approximate Gaussian (or ‘normal’) distributions. Hence, their error functions—represented by the relationships between terms and , and and , respectively—take similar forms in Eqn 5. This differs from the uncertainty associated with the target specimens () in Stockmarr’s [34] total error estimate for the linear method (Eqn 2), where a single measurement for is taken and the statistics are assumed to have a Poisson distribution. However, given the typically low absolute abundances of the rare specimen () counts per field of view (often between 0.2 and 5), the uncertainty estimate associated with still likely follows a Poisson distribution. This type of distribution is particularly appropriate for discrete count data that are very rare (e.g., have a relatively low probability of occurrence within a given field of view; [69]). To demonstrate this, we present simulated data sets of rare specimen counts (mean number of rare specimens per field of view: 0.9, iterations: 100,000), and compare the probability distributions of two counts (Fig 3): 1, a single field of view (); and 2, the sum of 15 fields of view (). The red curves are the Poisson approximations, showing a very good visual fit, and the sum of squared residuals of 1x10^-7 for the fit to the single field-of-view histogram and 1x10^-5 for the total of 15 calibration-count fields of view. This allows us to conclude that the Poisson error estimate for the rare grain type is a suitable approximation.

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Fig 3. Histograms of counted marker specimens for an assemblage with rare markers; in this case, target-to-marker ratio [] = 30; expected number of rare grains per field of view () = 0.9; the number of simulations was 100,000.

These histograms are compared to the Poisson probability mass function (red curves) with rate given by the actual mean number of markers per field of view calculated from the simulations. (A) Histogram for a single random field of view. (B) Histogram for the sum of 15 random fields of view.

https://doi.org/10.1371/journal.pone.0320887.g003

As in the case of the linear method, multiple target specimen types may be assigned for the FOVS method (e.g., ). For the FOVS method; however, the means per field of view () and standard deviations () of each target will need to be collected during the calibration counts. Combined with the markers () from the extrapolation counts, and following the sequence of steps outlined above (see ‘field-of-view subsampling (FOVS) method: operation’), their individual concentration estimates could then be separately derived using Eqn 4. The number of appropriate targets is dependent on the relative abundances of these targets in the assemblage (i.e., the evenness of the assemblage). Should these additional targets be particularly rare, then their calibration counts will likely result in large standard deviations; hence, their predicted total error estimates will be correspondingly large.

It is predicted that for routine data collection effort (e.g., total counts of several hundred) the FOVS method will provide superior concentration precision (lower total error values) to the linear method for assemblages with suboptimal target-to-marker ratios (e.g., ). This is because in cases with disproportionately abundant common specimens, there will be diminishing returns in their additional counts [31], at which point, the key limiting factor in concentration precision will be the number of rare specimens. The power of the FOVS method lies in its ability to detect greater absolute abundances of rare specimens with minimal additional sampling effort, thus mitigating this issue. However, if the target relative abundance is exceedingly high—resulting in a very large target-to-marker ratio (e.g., —then it is predicted that not even the FOVS method will salvage satisfactorily precise concentrations without impractical sampling efforts, as the precision still relies on the absolute number of exotic markers counted. We test these predictions below by comparing the efficiencies of the FOVS and linear methods.

Linear method.

For the linear method, we note that the expected amount of effort is directly proportional to the number of specimens that we count in a window. To see this, we substitute into Eqn 6 to measure linear method effort, thus obtaining

(9)

We can then substitute (which reflects the degree of effort for each specimen in the linear method) into Eqn 2 to give us an expression for the precision as a function of the amount of effort:

(10)

where is the error contribution from the marker doses (; e.g., Lycopodium spore tablets). As we increase the data collection effort (), we naturally expect an increase in precision (i.e., a decrease in error, ). Our derivations of Eqn 10 (see the ‘analysis of precision as a function of effort and the ratio of common and rare grains’ section of S1 File) reveal: 1, there are steeply diminishing returns from increased effort with the linear method, as error decreases with the square of the effort (see S1 Eqn); and 2, that the highest precision estimates of the concentration are achieved when (i.e., ; S2 Eqn), thus confirming the result by Regal & Cushing [61] mentioned in the introduction.

FOVS method.

We would like to obtain an estimate of the improvement in FOVS method precision () for increased effort (), as we did for the linear method. However, we cannot do this directly, since the FOVS method involves two independent variables that determine the amount of effort: 1, the number of calibration fields of view (), which increases ; and 2, the number of extrapolation-count fields of view (), which increases . Yet, if we can find the optimal ratio of calibration- and extrapolation-count fields of view, then we will reduce our two-dimensional problem to a single dimension. Ultimately, this was done by employing a standard optimisation technique: calculate the derivative of the FOVS method error function (Eqn 5) and set it equal to zero (see ‘FOVS method optimisation’ below).

To make this derivative tractable, however, we will assume that the error term for the common specimens , which will typically be the target specimens, is well-approximated by the Poisson result . Given that this approximation is quite good for small numbers (as shown in Fig 3), we expect that this Poisson assumption will not introduce any error into our determination of the optimal field-of-view count ratio, since any small deviations will be erased when the numbers of calibration-count and extrapolation-count fields of view are rounded off to the nearest integer. The assumption may introduce a small degree of error into the formula for choosing between the FOVS and linear methods (see ‘choosing the superior count method’); however, our predictions for this choice match the data generated in the simulations (see S1 File). Thus, we have confidence that the assumption is sound for this purpose, especially given the large differences in values expected in real assemblages.

So, with the assumption of a Poisson distribution for the target grain error term, we make the replacement , using the definition of in Eqn 3. (Note: is the number of targets counted during the calibration counts.) Also, using the estimate for the average density of marker grains in each extrapolation-count field of view , we make the following replacement: . Substituting these into Eqn 5 gives us

(11)

where we have included the arguments of the function to explicitly denote that this is a two-dimensional function, and to make the following calculations clearer.

From the results of the derivations (see S1 File for details of this procedure), the increase in precision decreases with the square of the effort (S3 and S4 Eqns). This is analogous to the linear method (S1 Eqn) and, once again, the error depends in a non-trivial way on the target specimen density () and the target-to-marker ratio ().

FOVS method optimisation: The FOVS method error depends in large part on the numbers of both calibration-count fields of view () and extrapolation-count fields of view (). To find the optimal ratio of these counts, first we solve Eqn 7 for

(12)

and then insert this equation for into Eqn 11 to obtain

(13)

For a fixed amount of effort, we can now find the number of calibration fields of view that minimises this error (). This is a two-dimensional optimisation problem, and is calculated by taking the derivative of with respect to (Eqn 13) and setting it to zero:

(14)

Substituting into Eqn 12, we can then find the corresponding optimal number of extrapolation-count fields of view ()

(15)

By dividing Eqn 15 by Eqn 14, we can provide the ratio of the optimal numbers of extrapolation-count () to calibration-count () fields of view. The sampling effort () cancels out, leaving us with the optimal field-of-view count ratio () in the following simple formula:

(16)

This is perhaps one of the most useful quantities for the practitioner of the FOVS method. It tells us that the primary variable in determining the most efficient number of field-of-view counts for a given assemblage is the ratio of target to marker specimens (). When , we have , and so we should choose an equal number of calibration- and extrapolation-count fields of view. However, in the more typical scenario where there are greater numbers of targets than markers (), then the proportion of extrapolation-count fields of view should increase as the square root of .

The optimal field-of-view ratio () depends to a lesser degree on the density of target specimens across the study area () and an observer’s counting habits (reflected by ). Crucially, all three of these variables can be estimated during the calibration counts. To maximise the utility of the FOVS method for routine concentration estimates, we recommend utilising the relatively simple metric in Eqn 16 during the calibration counts for determining the optimal ratio of extrapolation- to calibration-count fields of view. The optimal field-of-view count ratio can be easily calculated by inserting the relevant variables directly into the user-friendly interface we have provided (link here: https://github.com/Palaeomays/FOVS_vs_linear_methods.git; additional information in S1 Text). Note that the optimal ratio of extrapolation- to calibration-count fields of view will not tell a practitioner how many of each should be counted for a given precision; this is discussed below (see ‘achieving a targeted precision’).

Lastly, if we wish to validly compare the efficiency of the two methods, we can now utilize the optimal field-of-view counts to characterise the FOVS method’s relationship between error vs effort. Substituting the optimal values and into Eqn 11 gives the FOVS method precision as a function of the effort

(17)

which is analogous to the linear method error vs effort function in Eqn 10. In the supplementary material (see ‘analysis of precision as a function of effort and the ratio of common and rare grains’), we used S1 and S5 Eqns to show that the FOVS and linear methods both have the same “one on error-squared” decrease in error for increasing effort. So, we need to analyse the methods more closely to determine which is superior; these analyses are provided in the following section.

Linear method.

Starting at a fixed point on the virtual study area, a contiguous region of pre-determined height and variable length was marked out (Fig 5). The length was modified for each virtual study area to encompass a specified number of target specimens (). This provided our random “window” within which the number of markers () were also counted. These data were then used to produce the following simulated parameters of the linear method: target specimen concentration (; Eqn 1), standard error (; Eqn 2) and sampling effort (; Eqn 6).

FOVS method.

Firstly, an initial FOVS method sampling effort () was chosen that was as close as possible to the linear method sampling effort () of the same virtual study area. This level of effort was used to estimate the optimal numbers of calibration fields of view () and extrapolation-count fields of view () for the FOVS method (see ‘FOVS method optimisation’ above for details). Then the total number of common specimens was counted from this optimal number of calibration-count fields of view, from which we calculated the sample mean () and proportional sample standard deviation () of the common specimens. (Note: in all simulated cases, the number of targets was equal to or greater than markers []; so, the targets [] were the foci of the calibration counts, and .) Next, the number of rare specimens (= markers [ in all simulated cases) were counted from the corresponding optimal number of extrapolation-count fields of view (). With these data, we then calculated the following parameters for the FOVS method: target specimen concentration (; Eqn 4) and standard error (; Eqn 5) and the sampling effort (; Eqn 7).

Note that we experimented with the parameters for the linear method, so that the level of effort required for the FOVS method kept the total number of fields of view to a suitably small number, specifically . (Our simulated virtual study areas can have a maximum of 1089 non-overlapping fields of view.) By keeping the maximum sampled area much smaller than the total study area, we can avoid significant “finite population effects”, whereby too many samples from a finite area will produce distorted statistical estimates. In the extreme case, if the total sample count contains the entire population, then the sample standard deviation will be zero. To ensure we make precise comparisons between the estimates from the two methods (linear and FOVS) with the computer simulations, we included a “finite population correction” (FPC; see [64], p. 83; details in S1 File).

Simulated data sets.

For each set of parameters, we ran the simulation 1,000,000 times, producing data sets of the concentration, total error and effort for each iteration of both methods (see S4–S14 Tables). We then computed the mean values of: 1, the estimated concentrations (; Eqn 1); 2, the total errors ( and ; Eqns 2 and 5), recalling that we have no error contribution from the marker doses; 3, the standard deviation from the known exact concentrations (; S17 Eqn); and 4, the required data collection effort ( and ; Eqns 6 and 7). For maximum precision of the simulations, additional statistical corrections and effort standardisation across the different methods were included (see S1 File).

The simulations were generated in Matlab v.R2020a. The code to generate these simulations (S2 File), and instructions on how to implement and modify the code, can be downloaded here: https://github.com/Palaeomays/FOVS_vs_linear_methods.git. For each instance of the simulation, the user chooses:

• the number of targets and markers on the slide ( and , respectively);
• the error contribution from the marker doses ();
• the number of effort units ();
• the field of view transition factor (); and
• the number of iterations of the simulation.

Processing, imaging and sample details.

All empirical data analysed herein derive from one drill core within the Tasmania Basin, southeastern Australia: Bonneys Plain-1 (41° 46’ 27.69“S, 147° 36’ 13.35”E; Fig 7). The target strata were part of the upper Parmeener Supergroup and, although these strata lack precise age control, they correlate to the upper Permian (Lopingian) to Lower Triassic Series [71]. For organic microfossil processing and comparisons of linear and FOVS count methods, 18 samples were chosen at random stratigraphic heights throughout the drill core (see S3 Table for details). The assemblages consisted of a range of shallow marine to coastal plain palaeoenvironments ([71]; C.R. Fielding, pers. comm.). Inorganic mineral content was removed by digestion with hydrochloric acid followed by hydrofluoric acid. Prior to acidification (following [51]), a number of tablets of Lycopodium clavatum spores were added; these spores served as the exotic markers in this case study. Tablets were produced by the Department of Geology, University of Lund, Sweden. Details of the tablets (including means and uncertainty estimates of Lycopodium spores per tablet and batch numbers), the specific quantities of Lycopodium tablets () added to each sample and spore quantity estimates of each tablet (, , ) are provided in S3 Table. No sieving, heavy liquid separation or oxidation was performed for these ‘kerogen’ residues. The resultant residues were mounted on glass slides, and glass coverslips were sealed with epoxy. A summary of these various aspects of palynological processing techniques was provided by Riding [72]. These residues of ‘sedimentary organic matter’ (sensu [73]) are the targets of palynofacies analysis (sensu [74]) and are reflective of the undissolved particulate organic carbon (sensu [75,76]) of a sediment sample’s ‘total organic carbon’ (or TOC).

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Fig 7. Geographic and geologic contexts for the Permian–Triassic organic microfossil assemblages.

(A) Map of Australia. (B) Geological map of the Tasmania Basin with location of target well core succession (Bonneys Plain-1) and approximate distributions of Permian-Triassic sedimentary strata ([70]; 1:500,000 Digital Geological Data reproduced from Mineral Resources Tasmania (http://www.mrt.tas.gov.au/) under a Creative Commons Attribution 3.0 Australia license, © State of Tasmania).

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By targeting mudrock facies of continental environments (e.g., lacustrine, fluvial, coastal), these microfossil assemblages were likely the result of low-energy, suspension deposition proximal to their sources (e.g., [77–81]). This is reflected by the high proportions of plant-derived phytoclasts in most assemblages. Moreover, estimating microfossil concentrations from sedimentary organic matter assemblages should avoid sieving, oxidation and heavy liquid separation steps, each of which has strong potential for introducing unintentional biasing effects ([49]; see reviews [72,82]).

These samples were processed by Global Geolab Limited in Medicine Hat, Canada. Transmitted light microscopy and photomicrography of organic microfossils was conducted with a Zeiss Axioskop 2 transmitted light microscope equipped with a Canon EOS 700D camera. Fluorescence photomicrographs were performed with a Leica K3C camera on a Leica DM2500 LED microscope. Residues and slides have been given the prefix ‘S’ and are housed at the Department of Palaeobiology, Naturhistoriska riksmuseet (NRM), Stockholm, Sweden. The rock core samples from which the microfossil assemblages were derived were collected from the Mornington Core Library, Mineral Resources Tasmania (MRT), Australia. All necessary permits were obtained for the described study, which complied with all relevant regulations; specifically, permission for rock core sampling was provided by MRT. Additional specific methodological details for the different concentration estimates discussed are provided below where relevant.

Count methods.

In these assemblages, the principal goal was to estimate the concentration of terrestrial organic microfossils (). To this end, terrestrial organic microfossil grains were the target specimen type () and the subjects of the calibration counts; spores of Lycopodium clavatum were the exotic marker grains (). The terrestrial organic microfossil populations consisted of (in order of approximate decreasing abundance): wood (=’phytoclasts’, including charcoalified wood), leaves, plant spores, pollen and fungal remains. Fossil resins and animal-derived clasts were negligible. For both count methods, only grains ≥5 μm in diameter were counted; specimens smaller than this could not be consistently identified.

The assemblages in this case study had the following criteria that made them particularly suitable to the FOVS method of data collection: 1, the Lycopodium spores were optically distinct from the other microfossils (e.g., colour, texture, transparency, fluorescence response), even from fossil spores or pollen (Figs 2B, 8); and 2, generally, one of these grain populations was relatively rare (mean target-to-marker ratio for Bonneys Plain-1 was high: ). All fields of view were examined with a 63 × magnification objective. The field-of-view transition effort factor () was measured as approximately twice that of counting a single specimen on one field-of-view (i.e., ). All assemblages included a sampling effort (see below) of > 500. To minimise observer expectation biases, all counts followed the “blind protocol” outlined by Mays & McLoughlin ([30], p. 297); specifically: 1, all slide labels were masked; 2, slide order was randomized; and 3, sample counts were then conducted.

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Fig 8. A typical rectangular field-of-view photomicrograph of an organic microfossil (‘palynological’) assemblage spiked with Lycopodium spore markers.

Specimen S090248, well core MPT-3, Tasmania Basin; scale bars = 100 μm. (A) Greyscale optical light microscopy (with differential interference contrast). (B) Fluorescence microscopy (excitation wavelength: 555 nm); note the strong autofluorescence of the Lycopodium markers (green arrows).

https://doi.org/10.1371/journal.pone.0320887.g008

Linear method: Limitations.

The key limitation of the linear method lies in its high sensitivity to the ratio of target specimens to introduced markers; i.e., the ‘target-to-marker ratio’ (). Regardless of the count type, the highest precisions are achieved when the target-to-marker ratio is close to 1:1 [31,61]. We have demonstrated that, in these optimal conditions (and with reasonable values for other relevant parameters), the linear method shows a marginally superior efficiency (Figs 4 and 6). However, the linear method has a low tolerance for ratio values far beyond this [49,51,85]. This has important practical limitations:

1. A. Reprocessing of samples: For the microfossil assemblages discussed herein, the exotic markers are introduced during sample preparation; hence, before observation at the microscope [72]. However, it is impossible to accurately predict a priori how many markers to add without examining the prepared microfossil assemblages for approximate concentrations of the target specimens. Attempts to circumvent this apparent paradox typically involve undertaking two (or more) preparation phases. For instance, the preparator might: firstly, predict the optimal number of markers to introduce for a given sample (often informed by sedimentary indicators of microfossil concentrations, e.g., sediment grain-size or colour, inferred depositional conditions, total organic carbon analyses); secondly, conduct a preliminary batch of processing; thirdly, examine a subsample of the assemblage to gauge the approximate target-to-marker ratio and, if the ratio is suboptimal, conduct a subsequent phase of processing with a modified number of introduced markers. This may entail several processing rounds. Repeated processing can be costly in time, money and sediment (or sedimentary rock) sample, the latter of which may be irreplaceable; hence, this may be impractical for many potential applications. In principle, processing costs might be mitigated by adding the markers towards the end of the preparation procedure, as this would reduce the number of steps during reprocessing (e.g., acidification, oxidation; [49,50]). However, the late-stage introduction of exotic markers can lead to their overrepresentation, because the early stages of sample preparation demonstrably bias the assemblages [51]. Hence, marker grains should be introduced at the start of the preparation process, resulting in a consistent degree of bias for both markers and targets [51], but also leading to a longer and more costly preparation should the sample need to be processed multiple times.
2. B. High data collection effort: In cases with suboptimal target-to-marker ratios, the key limiting factor in concentration precision will be the low abundances of the rare specimens (typically, these will be the exotic markers). Thus, increasing the number of the rare specimens counted, even by a modest amount, can provide an enormous increase in precision. The ‘brute force’ way to bolster the count of these is simply to undertake a longer data collection phase. However, since precision is the result of multiple factors (Eqns 2 and 5), the relationship between target-to-marker ratios and effort is nonlinear. We have shown (S1 Eqn) that the increase in precision decreases with the square of the effort. Thus, assemblages with higher ratios require inordinate increases in specimen counts—therefore, data collection effort—to achieve the same low degree of uncertainty than assemblages with low ratios ([31,34]) (Fig 6). For example, if seeking a total error of 10% with a target-to-marker ratio of 3:1 (), a density () of 10 targets per field of view, and assuming similar exotic marker errors () and transition-to-count ratios () as those employed in this study, one would need a total effort of c. 627 effort units (calculated from Eqn 21). However, with a target-to-marker ratio of 20:1 (), but all other parameters being equal, an effort of c. 2682 is required to achieve the same degree of precision. Under such conditions, it is entirely reasonable to reprocess the samples. As stated by Maher ([31], p. 188): “it requires very little effort to add a few marker tablets to some sediment [samples]. It does require effort to increase the pollen counts.”
3. C. Unsuitable for multiple concentration targets: A limitation related to the linear method’s sensitivity to target-to-marker ratios is that an ‘optimal’ ratio assumes only one target population. However, there may be more than one target population (e.g., ) for which absolute estimates are being sought, each of which will have its unique concentration value (e.g., ). An assemblage with an ideal target-to-marker ratio for one target type (e.g., pollen, algae, terrestrial organic microfossils) is unlikely to be suitable for other potential target types in the same assemblage, given the narrow window of target-to-marker ratio suitability for the linear method (which, ideally, should be close to ). Hence, the chances of achieving satisfactory concentration precisions for multiple targets using the linear method becomes vanishingly small (without reprocessing for each target population [see point 1 above] or extensive data collection effort [see point 2 above]).

For the reasons above, there will be many instances when either the targets or exotic markers are disproportionately rare. This will result in low precision, reflected by large total error values and confidence intervals. It is worth noting that even in extreme cases where the target-to-marker ratios are exceptionally high or low, the linear method can still inform a qualitative assessment even with large error values. A very high (or very low) target-to-marker ratio is indicative of a very high (or very low) absolute abundance of targets. In this case, the linear method may provide an approximate minimum (or maximum) value for these abundances (e.g., [86]). However, most studies would require more precise estimates. With the linear method, collecting sufficient count data to improve these precisions may require reprocessing, or an inordinate data collection time, both of which may be prohibitive for routine study. The FOVS method aims to improve precision and/or sampling effort for samples with suboptimal target-to-marker ratios.

Field-of-view subsampling (FOVS) method: Limitations.

The resilience of the FOVS method to the target-to-marker ratio circumvents a key limitation of the linear method. As a result, very large, extrapolated data sets can be obtained using the FOVS method, while bolstering the number of rare specimens. Despite the compounded error that this extrapolation entails, our simulated and empirical case studies (see the ‘case study 1—computer simulations: results’ and ‘case study 2—organic microfossils: results’ and sections, respectively) demonstrate generally improved precision for absolute abundance estimates and/or decreased data collection effort. Furthermore, the simulations show that the FOVS method yields a more accurate approximation of precision. However, some limitations should be considered before applying this technique:

1. 1. Unsuitable for relative abundances: The FOVS method is designed for absolute abundance (e.g., concentration) estimates. By counting entire fields of view in a rapid sequence (during the extrapolation-count phase), it extrapolates large data sets by bypassing the collection effort of individual identifications. In contrast, a primary strength of the linear method is the identification of all target specimen types. A byproduct of the latter approach is a robust quantitative data set of the indigenous specimen populations. Hence, the linear method has the potential to provide concentrations for only a small number of targets (those with near-optimal target-to-marker ratios) while providing accurate, concurrent collection of compositional data (or relative abundance data) for the organic constituents of a sediment or sedimentary rock. Such relative abundance data form a cornerstone of pollen [87] and palynofacies [88] analyses. However, depending on the research question, the time-consuming determination of relative population abundances may not be necessary if absolute abundances are the primary goal.
2. 2. Requires visual contrast: In visual observation applications, the FOVS method would work best when there is sufficient visual contrast between exotic markers and the indigenous specimens in the assemblage (Fig 2B). This contrast enables rapid and accurate identification of markers during the extrapolation counts. Without this, thorough observation of each field of view is required to produce accurate results, thus increasing data collection time (specifically, modifying ). The most common exotic markers utilized in organic microfossil studies today, modern Lycopodium spores, have typically undergone acetolysis during preparation which darkens and discolours them [34] (R. Muscheler & Å. Wallin, pers. comm., 2023). This provides them with sufficient contrast in modern or Quaternary assemblages, but can render them more difficult to distinguish from indigenous grains of some deep-time assemblages that have undergone darkening via thermal maturation (e.g., [89–92]). If this is problematic, an alternative method of increasing contrast is to utilize fluorescence microscopy. Owing to the distinctive autofluorescence response of some exotic grains (e.g., Lycopodium spores; Fig 8), the contrast between markers and other grains (particularly in thermally matured assemblages) can be greatly enhanced, thus expediting accurate data collection.
3. 3. Sensitive to heterogeneous spatial distribution: The FOVS method is susceptible to heterogeneity in specimen distribution across the study surface area. This stems from the primary difference between the linear and FOVS methods: the spatial variance of common specimen abundance during the calibration count phase. If the number of specimens per field of view is highly variable during these counts (i.e., high ), the sample’s total error () will increase correspondingly, resulting in decreased precision. To circumvent this issue, it is essential that the spatial ranges of both the calibration and extrapolation counts are near equivalent. In other words, the regions chosen for both the calibration- and extrapolation-count fields of view should overlap or be located close to each other, thus ensuring that the specimen density statistics (means and standard deviations) for both counts will be roughly equivalent (Fig 2A). In the case of organic microfossil slides, we recommend avoiding the slide margins where densities tend to be lower than the medial areas, since these may result in significant edge effects (Fig 2A). Our preliminary Monte Carlo computer experiments have shown that the dispersed nature of the FOVS method counting (as opposed to the contiguous counting in the linear method) may help to compensate for heterogeneity in specimen distribution by sampling a broader region.
   Furthermore, we have acknowledged that the simulated and microfossil datasets have different assumptions about particle distributions. The particles on microscope slides do not necessarily follow random (Poisson) distributions, and may be prone to specimen-specimen interactions, e.g., particle clumping or artificial dispersal. While a range of preparation techniques have been advanced to promote homogenous distributions for microorganisms (settling chambers or surfactants, e.g., [93,94]), this cannot be assured in all cases. Moreover, while such techniques may prevent clumping, they may result in non-Poisson homogeneity. Other particle distributions could be proposed, however we have not yet systematically assessed the effects of distribution heterogeneity and this should be the focus of future efforts.

To summarise, the FOVS method is designed for absolute abundance estimates of one or more targets with higher precision and/or reduced data collection effort than the linear method. However, it is not optimised for collecting relative abundances of diverse specimen categories. Moreover, to maximise the accuracy and precision of the FOVS method, we recommend: 1, high visible contrast between markers and other specimens; and 2, study area ranges for both calibration and extrapolation counts that are equivalent and homogeneous.

Which technique to use?

Regardless of the count type, the most precise estimates of concentration are achieved when the target specimens (e.g., terrestrial organic microfossils, ) and exotic markers (e.g., introduced Lycopodium spores) are close to equivalent. However, the FOVS method is particularly resilient to disproportionate ratios.

We have provided a test for determining the superior (highest precision and lowest effort) method for a given assemblage (Eqn 20). Highly precise estimates for the necessary parameters of this test—specifically, , and —will only be available after a substantial amount of data collection (e.g., after the calibration counts of the FOVS method; see ‘field-of-view subsampling (FOVS) method: operation’). However, sufficient estimates of two of the parameters (the target-to-marker ratio, , and the target density per field of view, ) will generally be obvious upon cursory inspection of an assemblage. The third parameter (transition-to-count ratio, ) is the time variable, and is dependent on the target types, observation methods, and observer habits, which are largely consistent across assemblages in a study; this factor can be estimated even before inspecting a given assemblage. Hence, ‘ballpark’ values of these parameters will typically be easy to obtain; with these values, we recommend utilising Eqn 20 (Fig 1, steps 1, 2) upon first examination of each assemblage for determining the most efficient count method.

In most cases, the FOVS method is more efficient, except for those with extremely low specimen densities (very low ) and/or where markers and targets are near-equivalent (). We recommend that in cases where target-to-marker ratios approach 1:1, the ‘linear method’ of counting should be preferred. This is because this approach also provides additional data that may be important to the researcher (e.g., relative population abundances), and does not require additional conditions such as high marker contrast, or consistent areas for both calibration and extrapolation counts. In all other cases, the FOVS method should provide superior concentration and precision estimates.

One specific application of the FOVS method we envision: the re-analysis of previously collected samples with suboptimal target-to-marker ratios. Our comparisons of the count methods (summarised in Table 2) show that suboptimal target-to-marker ratios will very likely result in unsatisfactory linear method precisions for most applications. However, samples with these suboptimal ratios may warrant reanalysis using the FOVS method (note: a prerequisite for these reanalyses is the inclusion of exotic markers during original sample preparation). Hence, the FOVS method enables a robust method for replicating past findings and preventing future ‘file-drawer effects’ (whereby null results are dismissed [95,96]). Lastly, we have observed from preliminary experiments with the Monte Carlo simulations that the relative efficiency of the methods is largely dependent on the homogeneity of the specimen distributions. We have not yet quantified these observations, but given that lab-based samples may be quite heterogeneous, we recommend that this important question should be investigated in future work.

The primary considerations in the choice of methods are summarised in Table 2. To aid in the determining the most appropriate count method, we have: 1, provided a step-by-step flowchart (Fig 1); and 2, integrated the relevant calculations into the user-friendly ‘absolute abundance calculator’ application we have provided (see instructions in the ‘software interface for calculating parameters of the linear and FOVS methods’ section). Moreover, we have also supplied the simulation Matlab code with which the reader may experiment to inform their own methodology (S2 File). See S2 Table for a full list of the relevant parameters.