 Research article
 Open Access
 Published:
Modelling medieval masonry construction: taxaspecific and habitatcontingent Bayesian techniques for the interpretation of radiocarbon data from MortarEntrapped Relict Limekiln Fuels
Heritage Science volumeÂ 9, ArticleÂ number:Â 113 (2021)
Abstract
Using data from simulated and actual case studies, this paper assesses the accuracy and precision of Bayesian estimates for the constructional date of medieval masonry buildings, generated from the radiocarbon evidence returned by different assemblages of woodcharcoal mortarentrapped relict limekiln fuel (MERLF). The results from two theoretical studies demonstrate how Bayesian model specifications can be varied to generate a chronologically continuous spectrum of distributions from radiocarbon datasets subject Inbuilt Age (IA). Further analysis suggests that the potential for these distributions to contain the date of the constructional event depends largely upon the accuracy of the latest radiocarbon determination within each dataset, while precision is predicated on dataset age range, dataset size and model specification. These theoretical studies inform revised approaches to the radiocarbon evidence emerging from six culturally important Scottish medieval masonry buildings, each of which is associated with a woodcharcoal MERLF assemblage of different botanical character. The Bayesian estimates generated from these radiocarbon datasets are remarkably consistent with the historical and archaeological evidence currently associated with these sites, while age range distributions suggest the IA of each MERLF assemblage has been constrained by the taxaspecific and environmentally contingent lifespans and postmortem durabilities of the limekiln fuel source. These studies provide further evidence that Bayesian techniques can generate consistently accurate chronological estimates for the construction of medieval masonry buildings from MERLF radiocarbon data, whatever the ecological provenance of the limekiln fuel source. Estimate precision is contingent upon source ecology and craft technique but can be increased by a more informed approach to materials analysis and interpretation.
Introduction
Scholars across the world often face significant challenges in ascribing constructional dates to masonry buildings with enough precision to enable meaningful interdisciplinary environmental or historical discourse. The independent evidence returned by radiocarbon analysis of mortarentrapped relict limekiln fuel (MERLF) fragments can usefully inform these chronological ascriptions, and the sedimentary context within which these materials survive presents some valuable characteristics to facilitate interpretation of that data. Firstly, the durability of lime mortars can allow them to survive insitu for hundreds and sometimes thousands of years in upstanding masonry contexts, even if the walls within which they were deposited have been incorporated in later buildings or the structure has become ruinous [1,2,3]. Secondly, mortar compositions are historically and environmentally contingent, and material contrasts noted through field survey and labbased analyses can inform relative phasing interpretations, even where direct stratigraphic relationships are absent (e. g. [4, 5]). And thirdly, the widespread use of wood fuel in preindustrial limeburning practices has often resulted high concentrations of woodcharcoal MERLF fragment inclusions, and this material has wellrecognised radiocarbon dating potential [6, 7]. These characteristics of durability, distinctiveness, relative ubiquity, and radiocarbon dating potential are all underpinned by a rapid postdepositional mortar set, which not only allows an unequivocal archaeological association between the mortarâ€™s constituent components and the surrounding masonry fabric [8], but also precludes infiltration by other materials in later periods. It is crucial for wider interpretation that mortar phasing is correctly understood, and all radiocarbon measurements are accurate [9], but thereafter the determinations returned by all woodcharcoal relict limekiln fuel fragments are expected to calibrate to dates which are no later than the initial deposition of the masonry mortar within which they were entrapped [10,11,12,13].
The extent to which these radiocarbon determinations might calibrate to periods which predate the construction of the identified masonry phase will be defined by various interrelated factors. The size of the laboratory measurement error margin and the section of the atmospheric calibration curve to which the determination relates are important factors in defining chronological precision, but an allowance must also be made for any â€˜inbuilt ageâ€™ (IA) which might separate the average age of the annual tree rings in the MERLF sample (the cells of which rapidly cease exchanging carbon with the surrounding environment after formation) from the constructional event of interest [14,15,16,17,18]. The â€˜bridging periodsâ€™ [14] which might contribute to the IA of a sample must be considered on a contextbycontext basis and might include postlimekiln factors such as lime transport, mortar maturation and building construction times. But in the absence of historic evidence for the use of preprepared charcoal to fuel limekilns [19], the most significant bridging periods are likely to be associated with the loss of outer tree rings from the sample during postmortem rot, wood conversion, combustion, and/or mortar mixing: periods limited by the prekiln growth and storage ages of the selected wood fuels and essentially defined by ecological parameters [7].
The potential IA of every surviving MERLF sample is influenced by its taxaspecific and habitatcontingent lifespan (constraining growth age), habit (influencing trunk and branch wood availability), and postmortem resilience (constraining storage age). These factors are closely interrelated, since higher resource environments and faster growing taxa are almost universally associated with relatively short lifespan plants characterised by low density wood with poor postmortem resistance to wooddestroying fungi, and low resource environments and slower growing taxa are widely associated with longer lifespan plants characterised by higher density wood with better postmortem resistance to wooddestroying fungi [20,21,22,23,24,25]. In general, therefore, an inverse correlation between the growthrate of the wood fuel source and the potential IA of the MERLF assemblage generally pertains, and this is determined by taxon and habitat.
These relationships have implications for the archaeological resource at different ecological scales. At the broadest scale, IA potential may be determined by the class of the parent tree and the character of the surrounding biome, since gymnosperms are often characterised by extremely long lifespans [26] and boreal forests can be associated with (postmortem) coarse woody debris many hundreds of years old [27, 28]. Cold, sheltered and phytogeographically marginal environments are also associated with increased tree longevity in temperate woodlands although, with maximum tree lifespans of 3â€“400Â years pertaining across many oldgrowth northern hemisphere deciduous forests [29], dendroecological metadata suggests this is generally more limited than records of particularly (probably very slowgrowing) ancient individuals might suggest. Even within this reduced range, however, interspecies contrasts in lifespan and postmortem resilience can be considerable. In the Atlanticinfluenced environments of the UK, dense and comparatively slowgrowing angiosperms such as Quercus sp. mature at around 150Â years old, can live to over 500Â years, and demonstrate high postmortem durability; while fastergrowing shorterlived genera such as Betula sp. generally mature at around 60Â years old, rarely reach 100Â years old, and are very rapidly destroyed by fungal attack after death [30,31,32]. At a finer scale, field reports suggest Betula pendula has a much longer lifespan in the colder climates of central Scotland than elsewhere in these islands [31], and yet the lifespan of â€˜selfcoppicingâ€™ Corylus avellana stems (which generally extend to 30â€“50Â years in England) [30], can be limited to 12â€“15Â years on some thin western Scottish soils [33]. Consideration of woodland ecologies in the immediate locality is often useful for evaluation of IA potential, therefore, and particularly in a country such Scotland which is crossed by numerous Northern European phytogeographical boundaries [34, 35].
Scale of analysis is also important for our interpretations of how the carbon might be distributed within a potential IA range. The distribution of carbon within a woodland exploited for limeburning fuel will be determined by a complex mix of historicallycontingent ecological processes, but although the IA potential of an evenaged stand will tend to increase over time as the population matures and productivity will naturally decline in later years as mortality increases, metadata gathered from longterm monitoring projects across the globe suggests that aboveground mass growth rates (and so carbon gain) in individual trees generally increases continuously throughout their lifespans [24, 36,37,38]. Indeed, it has been reported that only 6% of oldgrowth forest in the western USA is comprised of trees with trunks of over 100Â cm diameter, and yet this small population contributes a remarkably high â€˜33% of the annual forest mass growthâ€™ [36]. This allometric evidence is consistent with reports that more than 50% of a tree stemâ€™s volume typically derives from the outer 30% of its annual rings [39], and its average radiocarbon age is likely to be approximately a third of its overall lifespan [10] (Fig.Â 1), while the destruction of old wood during trunk hollow formation [40] and replacement of older branches [41] is likely to sharpen carbon distributions in mature trees still further. Providing the assemblage is representative of the woodland source, therefore, ecological parameters suggest most woodcharcoal samples are likely to calibrate to dates which are equal to or only slightly earlier than their date of deposition [13, 17, 42].
Quantifying the relationship between these radiocarbon determinations and the constructional date of the building from which they were removed is particularly important in historic contexts where wider political, cultural, and environmental processes are understood with reasonably high chronological precision. That the IA of a woodcharcoal fragment is contingent on its botanical character has long been recognised, and half a century ago Tjalling Waterbolk (1971) suggested samples could be usefully divided into three groups whereby: Group A materials deriving from twigs and outermost tree rings would present a radiocarbon â€˜time differenceâ€™ which was negligible (< c. 20Â years); Group B materials deriving from shortlifespan wood species would present time differences measurable in decades (20â€“100Â years); while Group C materials deriving from longer lifespan species might result in time differences of over a century [10. See also 11].^{Footnote 1} Waterbolk (1971) also acknowledged that the archaeological context from which a sample was recovered might suggest an association with a lower IA Group, and proposed that charcoal fragments from hearths or ovens were likely to be narrowly distributed since â€˜firewood for daily consumption would have consisted mostly of very young woodâ€™Â [10],^{Footnote 2} although the historiography of MERLF materials research reveals contrasting approaches to this issue. Rainer Bergerâ€™s (1992; 1995) analysis of materials from various preRomanesque chapel sites in Ireland, for example, began from a premise that shortlived wood was deliberately selected for limeburning fuel and the calibrated radiocarbon determinations returned by single (probably bulk) MERLF samples were thereby reported as direct standalone constructional estimates [43, 44]. In clear contrast, a reinterpretation of large radiocarbon datasets from masonry buildings in Africa and Asia began from a premise that storage age and reuse in individual woodcharcoal fragments were â€˜indeterminableâ€™ chronological factors, whose interpretation required analysis of multiple samples from each phase [17, 12,Â 45].^{Footnote 3}^{,}^{Footnote 4} Notably, this latter assertion followed an article by Patrick Ashmore (1999) which highlighted the dangers of bulking multiple charcoal fragments in a single sample and the challenges of establishing charcoal residence times on archaeological sites [18]. But in both instances highlighted above, the radiocarbon distribution of each sample assemblage is not expected to reflect the ecology of a natural woodland source (both are essentially defined by anthropological or technical processes), and there is a concomitant lack of botanical information on the character of the materials under consideration (see also [46,47,48]).
Ultimately, where the radiocarbon determination returned by a MERLF sample might be associated with significant IA, then circumscription of the chronological period within which the constructional event took place requires a comparative approach. Indeed, multidisciplinary approaches to MERLF radiocarbon data can very usefully bracket the period of building construction where historical or archaeological evidence postdating that event is also available, since these are effective terminus post quem (TPQ) and terminus ante quem (TAQ) dates, and where the constructional date is already known with some precision then the IA of the MERLF sample or assemblage can be closely quantified [49]. Where multiple radiocarbon determinations are available, however, then Bayesian and other statistical techniques can be used to generate a comparative â€˜standaloneâ€™ constructional estimate from the radiocarbon and phasing evidence aloneÂ Â complete with upper limits which are independent of historical evidence [7]. Once again, such statistical interpretations are generally predicated on a binary distinction between short or longlived organic materials, although that is most often defined by the distribution of the determinations within the radiocarbon dataset, rather than the botanical character of the samples within the charcoal assemblage. Where samples deposited in a single event have returned determinations so narrowly distributed that a chisquare type test suggests the dataset is statistically consistent at 5% significance, then the samples can be regarded as effectively contemporaneous, and a more precise Combine average can be generated which is assumed to directly represent their date of deposition [50]. Where determinations from a particular depositional context are not statistically consistent at 5% significance, however, then a higher level of IA must be suspected, and a different approach is required. In the OxCal calibration programme used throughout this paper, measurements from datasets subject to IA can be grouped into model phases framed by probability distributions known as Start and End Boundaries; and the position of this latter event at the end of the phase (or between phases in a multiphase scenario) may be accepted as a reasonable estimate for the completion of the constructional event [51]. Generating a Last distribution will also provide a probability estimate for the last determination within the series, however, and these measurements can be further constrained to reflect our prior belief that the dataset should be exponentially distributedâ€”in line with allometric data. OxCal offers two main methods by which this might be achieved: a Tau Start Boundary can be selected to impose an exponential â€˜priorâ€™ distribution on the whole phase; or each individual determination can be tagged with an Outlier Probability linked to a separate exponentially distributed Charcoal Outlier Model [13]. The default Charcoal Outlier Model in OxCal is specified with a 1000Â year timeconstant to encompass the mean lifespan of an extremely longlived assemblage, but the logarithmic scale of the model is defined by the actual distribution of the radiocarbon determinations in each dataset and, providing sufficient independent determinations are available, the lowest IA materials are expected to be steeply distributed very close to the exponential asymptote [13, 17].
The assumptions which underpin these interpretive schemes and their general application for MERLF analysis are open to challenge. Ashmoreâ€™s [18] thesis demonstrated that very shortlived charcoal materials retrieved from various excavated contexts have sometimes returned problematically early radiocarbon determinations, but the storage age potential of MERLF materials is likely to be more limited where wood (rather than charcoal) has routinely been used as a limekiln fuel, and long residence times are largely irrelevant in a mortared masonry context where intrusion is effectively precluded. Recent studies have reported evidence that bark evidence does occasionally survive on woodcharcoal MERLF fragments, and some of the assemblages associated with these fragments have also returned radiocarbon determinations which are statistically consistent at 5% significance [52], but historical, archaeological, and radiocarbon evidence from across northern Europe and elsewhere suggests that limekilns were often charged with mixtures of different wood taxa which can return statistically inconsistent radiocarbon datasets [7]. In an important piece of work evaluating the accuracy and precision of â€˜standaloneâ€™ constructional estimates generated using Bayesian techniques in OxCal, Michael Dee and Christopher Bronk Ramsey (2014) concluded that the Charcoal Outlier Model approach generates the most consistently accurate End Boundary estimates from woodcharcoal datasets, whilst the exponential prior approach generated more precise but occasionally inaccurate distributions [17]. Where the number of determinations from a particular phase is more limited, however, then the Charcoal Outlier approach tends to generate very broad positively skewed End Boundary distributions, which can seem incongruous against the comparatively low mean lifespans of most temperate woodland taxa. The TPQ role performed by woodcharcoal MERLF radiocarbon determinations can be of huge value for multidisciplinary interpretation and the effect of model selection on upper limits may be much less important where an early and convincing TAQ is available to truncate these distributions (Fig.Â 2aâ€“f). Standalone estimate precision, however, is vital for increased interdisciplinary (rather than multidisciplinary) discourse. And while there is also some evidence that reducing the exponential timeconstant of the Charcoal Outlier Model to reflect the more limited source material lifespans can constrain End Boundary distributions [49], persistent contrasts in precision with estimates using the exponential prior approach and binary approaches to statistical consistency do not appear to reflect a continuous spectrum of potential IAs predicated on variation in woodland ecologies. Importantly for this paper, however, the accuracy and precision of constructional estimates generated using different Bayesian approaches can also be evaluated using simulated datasets in theoretical models, without initial reference to architectural or historical evidence.
This paper describes a reevaluation of these Bayesian frameworks, with a concern to further characterise statistical relationships which might pertain between the different limekiln fuel resources exploited during the construction of medieval masonry buildings, and the archaeological potential of any surviving MERLF materials. Following the approach developed by Dee and Bronk Ramsey [17], this work is predicated on Bayesian analysis of simulated and actual radiocarbon datasets subject to varying levels of IA, although a broader range of simulated singlephase datasets, model specifications and generated estimates are considered here. Indeed, two theoretical studies centred on a single date in the medieval period will demonstrate how a chronologically continuous spectrum of distributions can be generated from radiocarbon datasets of varying lifespans and sizes, and different Bayesian model specifications can therefore be employed to maximise constructional estimate precision whilst retaining accuracy. These theoretical results will then inform the modelling approaches applied to the published radiocarbon datasets from six culturally important Scottish medieval buildings (CS16), each of which is associated with a MERLF assemblage of contrasting botanical character. Highlighting remarkable levels of consistency between the radiocarbon, historical and ecological data, the paper will conclude that the age range of an insitu MERLF assemblage does often appear to be constrained by the taxaspecific and environmentally contingent lifespans and postmortem durabilities of the limekiln fuel source.
Method
The methodologies presented below detail the processes by which data were generated in two theoretical studies and six case studies. Following previous authors [17], the theoretical studies are essentially circular and pragmatic. Exponentially distributed sets of calibrated radiocarbon dates associated with different IA mean lifespans (hereafter IAÏ„) have been simulated from a particular â€˜true eventâ€™ calendar date, and these datasets have then been constrained within various singlephase Bayesian models to assess how the accuracy and precision of newly generated distributions are affected by model specification. In contrast to that previous work, however, a calendar date of 1250 AD has been selected for the true event in each theoretical study, since this correlates with a relatively monotonic section of the radiocarbon calibration curve and occupies a central chronological position relative to the case studies presented later in the paper. The error margin ofâ€‰Â±â€‰35Â years on each simulated date was also selected to more closely reflect the data associated with these case studies. An increased range of dataset IAÏ„ and model specifications has also been employed, while End Boundary and Last distributions are evaluated for constructional estimate accuracy and precision. All datasets and models have been generated using OxCal 4.4 [51] and are calibrated with the IntCal20 atmospheric calibration curve [53].
Theoretical studies
In Theoretical Study 1 (TS1), multiple sets of twenty simulated calibrated dates subject to varying levels of IA were generated from a theoretical true event date of 700 BPâ€‰Â±â€‰35Â years (1250 AD). Twenty independent datasets were generated in TS1, with four separate datasets each subject to IAÏ„ specified at 10, 50, 100, 200 and 500Â years. The number of dates in each dataset which included the true event date has been counted, dataset age ranges have been estimated using the OxCal Difference function to compare the earliest and latest simulated dates [51], and the actual mean lifespan of each dataset has been calculated by finding the sum of the mean values from each individual simulated date. Each dataset has then been situated in a singlephase Bayesian model framed by Start and End Boundaries and subject to a range of different specifications. Run at default Oxcal settings to allow a reasonably fast turnaround of results, these include: (i) a Combine model; (ii) an exponential prior/no outlier model; (iii) an exponential prior/modified Charcoal Outlier Model (with a timeconstant modified to the same scale as the IAÏ„ specified for the simulated dataset); (iv) an exponential prior/default Charcoal Outlier Model; (v) a uniform prior/no outlier model; (vi) a uniform prior/modified Charcoal Outlier Model (with a timeconstant modified to the same scale as the specified IAÏ„ of the simulated dataset); and (vii) a uniform prior/default Charcoal Outlier Model. In the third and fourth run of each dataset/modelling approach combination, a Last distribution was also generated.Â All models associated with TS1 are presented in Additional file 1.
In Theoretical Study 2 (TS2), multiple simulated datasets of varying size and subject to varying levels of IA were generated from a theoretical true event date of 700BPâ€‰Â±â€‰35Â years. Fortyfive independent datasets were generated in TS2 with three separate datasets associated with an IAÏ„ specified to 10, 50, 100, 200 and 500Â years and including fifteen, ten and five simulated dates. As in TS1, the number of dates in each TS2 dataset which included the true event date has been counted, the age range has been estimated using the OxCal Difference function to compare the earliest and latest simulated dates, and the actual mean lifespan of each dataset has been calculated by finding the sum of the mean values from each simulated date. These datasets were included in singlephase models framed by Start and End Boundaries and subject to three model specifications including: (i) an exponential prior/no outlier model, (ii) an exponential prior/modified Charcoal Outlier Model (with a timeconstant modified to match the IAÏ„ specified for the simulated dataset), and (iii) a uniform prior/default Charcoal Outlier Model. All these TS2 models have been run at default settings to allow a reasonably fast turnaround of results, and all include a Last distribution.Â All models associated with TS2 are presented in Additional file 2.
In Case Sudies 16 (CS16), the radiocarbon data from six Scottish medieval buildings with woodcharcoal MERLF assemblages comprised of contrasting taxa are reevaluated. This includes Castle Fincharn main block (CS1), Aros Castle northwest block (CS2), Castle Roy enclosure and tower (CS3), Lochindorb Castle primary enclosure (CS4), Achanduin castle enclosure and hall (CS5), and Lismore Cathedral nave (CS6). The MERLF assemblages associated with the first five of these studies are dominated by charcoal fragments which displayed no surviving terminal ring, bark, or sapwood boundary evidence, and each has been published elsewhere in some detail. Full details of CS6 await publication but is included here since the MERLF assemblage included a Corylus sp. fragment with some terminal ring evidence. The distributions of each of these radiocarbon datasets was investigated using the Ward and Wilson (1978) chisquare type test [50], and an age range was calculated using the Difference function to compare the earliest and latest dates available [51]. The data from each site were then included in a series of singlephase Bayesian models. Run at 1Â year resolution and 20,000 Kiterations, these include: a Combine model; an exponential prior/no outlier model; an exponential prior/modified Charcoal Outlier Model; an exponential prior/default Charcoal Outlier Model; and a uniform prior/default Charcoal Outlier Model. The modified Charcoal Outlier Model IA timeconstants specified in CS16 have been estimated from published data regarding tree mean lifespan and wood postmortem resilience data (Table 1). Mean lifespans are rarely reported so working values have been calculated at 33% of reported maximum lifespans, in line with allometric data, added to which an estimate of postmortem resilience has been derived from the resistance to wooddestroying fungi according to the (1â€“5) durability scale applied by British and European Standards [54] and other published reports (Table 1). Where the datasets returned by mixedtaxa assemblages are statistically consistent at 5% significance (CS5) then these are tagged to a single Charcoal Outlier Model with a timeconstant modified to reflect the lowest IAÏ„ samples, and where these datasets are not statistically consistent at 5% significance (CS2 and CS6) then the highest IAÏ„ data is used. A Last distribution has been generated in all case study models, and the Last and End Boundary distributions compared with various potential TPQ and TAQ dates (from other types of historical, archaeological or architectural evidence) using the Order function [51]. All models associated with these case studies are presented in Additional file 3.Â In the interest of brevity, presentation of wider evidence relating to these buildings is kept to a minimum, and readers are encouraged to follow the cited references for more detailed information.
Calibrated date ranges in each theoretical and case study radiocarbon dataset are expressed as cal AD or cal BC at 95% and 68% confidence using upright text and have been rounded out to 10Â yearsÂ [12].^{Footnote 5} Modelled age range, End Boundary, and Last distributions are reported as Highest Posterior Density (HPD) interval date ranges at both 68% and 95% probability with median values, and these estimates have been rounded out to 5Â years and are presented in italics. Generated date ranges in both theoretical studies are regarded as accurate when they include the true event date from which each simulated dataset was generated (i.e. 1250Â cal AD). The agreement indices returned by each model are considered [51, 55], but individual measurements which fall below the accepted 60% threshold of compatibility have not been removed from theoretical models since to do so would bias results [56].
Results (theoretical studies)
Theoretical Study 1 (TS1)
The twenty simulated datasets generated in TS1 are all exponentially distributed (e.g. FigureÂ 3). There is relatively little variation between the latest simulated date in each dataset, and all include the 1250 AD true event at 95% confidence (Table 2). Increased dataset IAÏ„ is associated with some decrease in latest date age, however, and with a decrease in the number of dates containing the true event at 95% confidence (from all 20 dates in 10Â years IAÏ„ dataset M1a run 1, to only a single date in 500Â years IAÏ„ dataset M1e run 4). An average of 19 simulated dates include the true event in 10Â year IAÏ„ datasets, 13 in 50Â year IAÏ„ datasets, 11.5 in 100Â year IAÏ„ datasets, 5 in 200Â year IAÏ„ datasets (M1d), and 2.5 in 500Â year IAÏ„ datasets.
Age range variation in the TS1 datasets is mostly predicated on variation in the earliest simulated dates within each dataset (Table 2). Increased IAÏ„ is generally associated with an increase in earliest date age, an increase in date range, and an increase in date range variation. The earliest calibrated dates in all the datasets specified with 10Â years IAÏ„ are situated in the second millennium AD and age range variation in this group is limited to between â€“Â 35â€“225Â years (M1a run 1) and 20â€“250Â years (M1a run 2). This 10Â years IAÏ„ group includes the only dataset in TS1 which falls into minus values, and this is the same dataset in which all simulated dates include the 1250 AD true event. In contrast, all earliest dates in the datasets specified with 500Â years IAÏ„ are situated in the cal BC period, with age range variation between 1315â€“1640 yearsÂ (M1e run 3) and 3205â€“3465Â years (M1e run 1). Dataset age ranges in these two lowest and highest IAÏ„ groups are distinctive, and the datasets throughout the study are generally consistent with this trend, but there is considerable overlap between individual datasets in adjacent 50, 100 and 200Â years IAÏ„ groups.
All dataset mean lifespans are earlier than the 1250 AD true event date in TS1 and these values also generally increase in age and variation with increased IAÏ„ specification (Table 2). The mean lifespans of the datasets specified with 10Â years IAÏ„ are narrowly distributed between 1224 and 1242Â cal AD, whilst the datasets specified with 500Â years IAÏ„ present mean lifespans situated in the first millennium AD between 491 and 780Â cal AD. The average lifespans within each group are all consistent with this trend and close to expected valuesâ€”1229Â cal AD (10Â years IAÏ„ specified), 1183Â cal AD (50Â years IAÏ„ specified), 1156Â cal AD (100Â years IAÏ„ specified), 1039Â cal AD (200Â years IAÏ„ specified), and 670Â cal AD (500Â years IAÏ„ specified)â€”although there is some overlap in the lifespans of individual datasets in adjacent groups specified to 50, 100 and 200Â years IAÏ„.
There is a clear relationship between the IAÏ„ specified, the distribution of simulated dates, and the date range of each dataset generated in TS1. All four datasets in the 10Â years IAÏ„ group (M1a runs 1â€“4) pass the Ward and Wilson (1978) chisquare type test and have generated accurate Combined dates, although three of these models contain four individual dates with low agreement indices (Aiâ€‰â‰¤â€‰60%) and present low overall Combined Agreement Indices (Acomb) (Table 3). The exception is the 10Â years IAÏ„ dataset in which all 20 simulated dates include the true event date (M1a run 1), which has a mean lifespan of 1242Â cal AD (the highest in the study and very close to the 1250 AD true event) and has only returned one low Ai. All sixteen simulated datasets specified with 50, 100, 200 and 500Â years IAÏ„ fail the Ward and Wilson (1978) test. The four datasets specified to 50Â years IAÏ„ (M1b runs 1â€“4) and one specified to 100Â years IAÏ„ (M1c run 1) have generated Combined date ranges, although these are all too early. The three remaining models associated with datasets specified to 100Â years IAÏ„ (M1c runs 2â€“4) and all models specified with 200 and 500Â years IAÏ„ datasets have failed to generate a Combined distribution at all.
Outwith the Combine models, all TS1 models except one present overall agreement indices which are above the 60% threshold (Table 4). The exception is associated with a uniform prior/no outlier modelling approach to a 50year IAÏ„ dataset (M1b, run 4), which presents an Overall Agreement Index of 58.9%. The number of individual dates with low Agreement Indices decreases with increasing IAÏ„, and with models employing an exponential prior specification. All three uniform prior approaches generate higher numbers of individual low Agreement Index distributions than their exponential counterparts from 10 and 50Â years IAÏ„ datasets. The uniform prior/no outlier approach contains the most low Agreement Index distributions in TS1, with a maximum of three presented from a 10Â year IAÏ„ dataset (M1a run 3). Individual low Agreement Indices are limited to a single dates in all exponential prior modelling approaches (Table 4).
98% of all TS1 models (118/120) have generated End Boundary HPD intervals which are accurate at 95% probability, and 85% (102/120) are also accurate at 68% probability (Table 5). Consistency of End Boundary accuracy at 68% probability is inversely proportional to dataset IAÏ„; decreasing from 96% of models associated with datasets specified with 10Â years IAÏ„ to 58% of models associated with datasets specified with 500Â years IAÏ„. Consistency of End Boundary accuracy also varies with model specification and a broad correlation with prior distribution is evident; with 85â€“90% of models with exponential priors generating accurate End Boundary estimates at 68% probability, reducing to 70â€“85% of models with uniform prior distributions. Overall, the uniform prior/default Charcoal Outlier Model approach has generated the least consistently accurate estimates, with 70% (14/20) of all End Boundary HPD intervals associated with this specification including the true event date at 68% probability; ranging from 75% (3/4) of datasets specified to 10Â years IAÏ„, to 50% (2/4) of datasets specified to 100Â years and 500Â years IAÏ„. The two End Boundary estimates in this study which are inaccurate at 95% probability are also associated with the same 500Â years IAÏ„ dataset (M1e run 3) and with uniform prior/Charcoal Outlier Model approaches. The exponential prior/Charcoal Outlier Model approaches have generated the most consistently accurate End Boundary distributions in TS1; with this specification generating accurate End Boundaries at 68% probability from 90% of all datasets, and accurate End Boundaries at 95% probability from all datasets (100%). Notably, all models employing an exponential prior have generated accurate End Boundary HPD intervals at both 95% and 68% probability from all datasets specified with 10, 50 and 100Â years IAÏ„ (Table 5).
Inaccurate End Boundary distributions at 68% probability in TS1 can be either earlier or later than the true event (Table 4). All inaccurate estimates generated by uniform prior models are late, including both of those which are inaccurate at 95% probability, whilst exponential prior models have generated End Boundary HPD intervals which are both too early and too late at 68% probability. The exponential prior/no outlier modelling approach to 50Â years IAÏ„ and 200Â years IAÏ„ datasets are the only two modellingdataset combinations in TS1 which have generated End Boundary average medians that are earlier than the true event date of 1250 AD.
All Last HPD intervals generated in TS1 are accurate at 95% probability (Table 6), although this would not have been the case had Last distributions been generated in runs 1 and 2. Last HPD interval accuracy at 68% probability is inversely proportional to the specified dataset IAÏ„; and falls sharply from 100% accuracy in models associated with 10Â years and 50Â years IAÏ„ datasets, to 58% accuracy in those generated from datasets specified to 500Â years IAÏ„. Most model specifications have generated accurate Last HPD intervals at 68% probability from 90% of all datasets. This is reduced to 80% for the uniform prior/default Charcoal Outlier Model approach, although this difference relates to one extra inaccurate date only.
End Boundary precision in TS1 is inversely proportional to specified dataset IAÏ„ for all modelling approaches (Table 7). Overall, End Boundary median age is also inversely proportional to dataset IAÏ„ across the study, although that trend is much clearer in models with uniform priors (Table 8). End Boundary precision and median age also vary with model specification; such that the imposition of an exponential prior distribution generally increases relative End Boundary precision and median age (whether or not a Charcoal Outlier Model is also used), whilst the introduction of a Charcoal Outlier Model generally decreases relative End Boundary precision and median age (whatever prior distribution is employed) (Tables 7 and 8). Decreasing the Charcoal Outlier Model timeconstant has generally increased End Boundary median age in lower IAÏ„ TS1 datasets (whatever the priors) and slightly increased End Boundary precision in models associated with uniform priors (Tables 7 and 8). These effects are cumulative; such that variation in model specification has a much more significant on End Boundary precision and median age when associated with higher IAÏ„ datasets.
Last HPD interval precision in TS1 is also inversely proportional to dataset IAÏ„ for all modelling approaches (Table 9). Last precision also varies according to model specification: decreasing on association with a Charcoal Outlier Model (whichever prior distribution is used) and increasing on association with exponential priors and lower IAÏ„ datasets. In general, these two factors have a cumulative effect, although the Last distributions generated by the uniform prior/no outlier model approach display a lower decrease in precision with increased dataset IAÏ„ than other modelling approaches and are thereby associated much greater comparative precision at higher datasets IAÏ„s. Modifying outlier model timeconstant has no clear effect on Last precision and there is no clear relationship between Last median ages and dataset IAÏ„. Importantly, all modeldataset combinations in TS1 have generated Last distributions which are more precise and have earlier median values than their corresponding End Boundaries, and this contrast is also more significant with increasing dataset IAÏ„ (Tables 7, 8, 9 and 10). 37% (11/30) of the modeldataset combinations in TS1 have returned Last distributions with average medians of 1250 AD or earlier (Table 10).
Theoretical Study 2 (TS2)
Most TS2 datasets present convincingly exponential distributions, and 91% (41/45) contain at least one simulated date which includes the true event (Table 11). Increased IAÏ„ specification in this study has generally resulted in datasets with earlier latest dates, a decreased number of dates which include the true event, increased age ranges, increased age range variation, earlier earliest dates, and earlier mean lifespans. No latest dates in TS2 are later than the true event at 95% probability. Decreased dataset size has also resulted in earlier latest dates, however, and at very high IAÏ„ this can result in no true dates at all. There is no clear correlation between dataset size and percentage of true dates or mean lifespans for a given IAÏ„ across the study, but earliest dates tend to decrease in age with decreased dataset size and thereby age range and age variation also decrease (since this is mostly driven by the early dates) (Fig.Â 4aâ€“c).
The inverse correlation between dataset IAÏ„ and the age of the latest date is exaggerated by dataset size. There is a strong correlation in TS2 between the specified IAÏ„ and the number of accurate dates in each dataset, with 94â€“98% of all determinations including the true event date at 95% confidence in datasets specified with 10Â years IAÏ„, and 67% (6/9) of these 10Â years IAÏ„ datasets are completely dominated by such accurate simulated dates in all three dataset sizes. At the other end of the IAÏ„ spectrum, only 6â€“13% of datasets specified with 500Â years IAÏ„ are dates which include the true event at 95% confidence and four datasets in the study do not contain any accurate simulated dates at all (MRRR1d run 3; MRR1e run 1; MRRR1e runs 2 and 3). Although there is no clear relationship between number of true dates and dataset size for each IAÏ„ specified, the earliest latest dates in all five IAÏ„ groups are presented by five date datasets, and the latest latest dates have been generated within a 15 date dataset in four of the five IAÏ„ groups. Three of the four datasets which do not contain an accurate date are limited to 5 dates (the other is a 10 dater) and three of the four are 500Â years IAÏ„.
Age range and age range variation across TS2 generally increases with increased IAÏ„, and with dataset size within each IAÏ„ group (Table 11). Narrowly distributed averages of âˆ’Â 5â€“225Â years (MRRR1a) to 10â€“240Â years (MR1a) are presented in datasets specified to 10Â years IAÏ„, and much more widely distributed averages of 1040â€“1350Â years (MRRR1e) to 2225â€“2490Â years (MR1e) are presented in datasets specified to 500Â years IAÏ„. The average age ranges presented by the 50, 100 and 200Â years IAÏ„ datasets are consistent with this trend, although there is some overlap between individual datasets from all these adjacent groups. The 5 date 100Â years IAÏ„ group is particularly notable since this contains two datasets with extraordinarily narrow age ranges of âˆ’Â 10â€“225Â years (MRRR1c run 2) and 10â€“225Â years (MRRR1c run 1), but also extends up to 450â€“625Â years (MR1c run 2). The narrowest age ranges are presented by 5 date datasets in four of the five IAÏ„ groups, whilst the widest age ranges are presented by 15 date datasets in three of the five IAÏ„ groups. Age range variation is mostly predicated on the earliest simulated date in each dataset, which generally increase in age with increased IAÏ„ specification and dataset size; varying from 1053â€“1273Â cal AD in the 10Â years IAÏ„ dataset (MRR1a, run 2), to 1416â€“1224Â cal BC in the 500Â years IAÏ„ dataset (MR1e, run 2). Where datasets include very low numbers of dates, and few which include the true event date, then the exponential distribution becomes less visible and more sigmoidal distributions are apparent (Fig.Â 4aâ€“c).
Mean lifespans are all earlier than the true event date and generally increase in age with the increased IAÏ„ specification and age range in TS2 (Table 11). Without rounding out these are all close to expected values. Depending on dataset size, the study presents average mean lifespans of: 1211â€“1231Â cal AD in the group specified with 10Â years IAÏ„ (average 1223 rather than 1240 AD); 1185â€“1199Â cal AD in the group specified with 50Â years IAÏ„ (1191 rather than 1200 AD); 1136â€“1150Â cal AD in the group specified with 100Â years IAÏ„ (1141 rather than 1150 AD); 1000â€“1076Â cal AD in the group specified with 200Â years IAÏ„ (1041 rather than 1050 AD); and 699â€“790Â cal AD in the group specified with 500Â years IAÏ„ (717 rather than 750 AD). Average mean lifespans for each level of specified IAÏ„ are all distinct from adjacent groups although, apart from the 500Â years IAÏ„, there is some overlap between the mean lifespans of individual datasets with those of all other adjacent groups.
All TS2 models have returned Overall Agreement Indices that are above the 60% threshold (Table 12). 12% (16/135) of these models contain at least one simulated date with a low Agreement Index (Ai) and 4% (6/135) contain more than one such date. Individual dates with low Agreement Indices are more strongly associated with lower IAÏ„ and larger datasets, with no instances in the 500Â years IAÏ„ or 5 date dataset groups. All six models associated with more than a single low Ai date are associated with two 15 date datasets specified with 10Â years IAÏ„ (MR1a, runs 2 and 3, all three model specifications).
99% of all models in TS2 (133/135) have generated accurate End Boundary HPD intervals at 95% probability, and 84% (113/135) are also accurate at 68% probability (Table 13). Consistency of End Boundary accuracy varies across the study with dataset IAÏ„, model specification, and dataset size although these relationships are not straightforward. The lowest IAÏ„ datasets have produced the most consistently accurate estimates at both 95% and 68%, but there is no clear overall trend in the relationship between these variables in the rest of the study: all End Boundary HPD intervals generated from datasets specified with 10Â years IAÏ„ are accurate at both 68% and 95% probability; all models associated with the 50Â year IAÏ„ datasets are also accurate at 95% probability, but only 7O% of these estimates are accurate at 68% probability; 78% of models associated with the 100Â years and 200Â years IAÏ„ groups have returned accurate estimates at 68% probability, but accuracy at 95% probability is down to 96% in both cases; while the 500Â years IAÏ„ group has returned 100% accuracy at 95% probability and 93% accuracy at 68% probability. The exponential prior/modified Charcoal Outlier Model approach has presented the most consistently accurate End Boundary HPD intervals across the study, with all models (45/45) generating accurate estimates at 95% probability and 87% (39/45) generating accurate estimates at 68% probability; the uniform prior/default Charcoal Outlier Model approach is also 100% accurate at 95% probability, and 80% (36/45) of these models are accurate at 68% probability; the exponential prior/no outlier approach has presented accurate estimates in 84% of models at 68% probability but is the only approach to present inaccurate End Boundary HPD intervals at 95% probability. Overall consistency of End Boundary accuracy in TS2 is inversely proportional to dataset size: 100% (45/45) of all modelled End Boundaries associated with 5 date datasets are accurate at 95%, and 89% (40/45) are also accurate at 68% probability; 98% (44/45) of all modelled End Boundaries associated with 10 date datasets are also accurate at 95%, and 84% (38/45) are also accurate at 68% probability; and 98% (44/45) of all modelled End Boundaries associated with 15 date datasets are accurate at 95%, and 78% (35/45) are also accurate at 68% probability.
95% of all models in TS2 (128/135) have generated accurate Last distributions at 95% probability, and 81% (109/135) are also accurate at 68% probability (Table 13). Consistency of Last distribution accuracy varies across the study with dataset IAÏ„, model specification, and dataset size although these relationships are not straightforward. The lowest IAÏ„ datasets have produced the most consistently accurate estimates at both 95% and 68%, but there is no clear overall trend in the relationship between these variables in the rest of the study. All Last distributions generated from datasets specified with 10Â years.
IAÏ„ are accurate at both 68% and 95% probability; all models associated with the 50year and 100year IAÏ„ datasets are also accurate at 95% probability, but accuracy at 68% probability decreases to 81% and 89% respectively. 81% of the 200Â years IAÏ„ groups have returned accurate estimates at 95% probability, and 75% of these distributions are also accurate at 68%. 93% of the Last distributions generated from the 500year IAÏ„ datasets are accurate at 95% probability, but only 59% of this group remains accurate at 68%. The uniform prior/default Charcoal Outlier Model approach has generated the most consistently accurate Last distributions across TS2 and is the only model specification to generate accurate Last distributions for all (45/45) TS2 datasets at 95% probability. 87% (39/45) of these estimates are also accurate at 68% probability. Consistent accuracy is slightly lower for distributions generated using the exponential prior/modified Charcoal Outlier Model approach, with 96% of Last distributions accurate at 95% probability and 82% (37/45) at 68% probability; whilst the exponential prior/no Outlier approach has generated the least consistently accurate Last distributions overall, with 82% (37/45) of models at 95% probability and 73% (33/45) at 68% probability. In direct contrast with the End Boundary data, overall consistency of Last Distribution accuracy in TS2 is proportional to dataset size: with 98% (44/45) of all generated Last distributions generated from 15 date datasets are accurate at 95%, and 78% (35/45) also accurate at 68% probability; and 96% (43/45) of all modelled End Boundaries associated with 10 date datasets are accurate at 95%, and 78% (35/45) are also accurate at 68% probability; and 91% (41/45) of all modelled End Boundaries associated with 5 date datasets are accurate at 95%, and 80% (36/45) are also accurate at 68% probability.
End Boundary precision in TS2 relates strongly to dataset IAÏ„, dataset size and model specification (Table 14). End Boundary precision is inversely proportional to dataset IAÏ„, and the continuous spectra presented by all three dataset sizes for this parameter is notable. In the 15 date models there is a continuous spectrum in End Boundary distributions at 68% probability from 33Â years (10Â years IAÏ„) to 277Â years (500Â years IAÏ„), and at 95% probability from 53Â years (10Â years IAÏ„) to 627Â years (500Â years IAÏ„). In the 5 date models there is a continuous spectrum at 68% probability from 52Â years (10Â years IAÏ„) to 537Â years (500Â years IAÏ„), and at 95% probability from 113Â years (10Â years IAÏ„) to 1593Â years (500Â years IAÏ„). End Boundary precision in TS2 is proportional dataset size; and reducing datasets from 15 to 5 dates more than doubles the age range of End Boundary HPD intervals in all models at 95% probability (whatever the specified dataset IAÏ„), although the contrast between 10 and 5 date datasets accounts for much of this increase. End Boundary precision also varies according to model specification; with the exponential prior/no outlier approach almost invariably presenting the most precise End Boundary HPD intervals at both 68% and 95% probabilities (the exception here being the models generated from 10year IAÏ„ datasets) and the uniform prior/default Charcoal Outlier Model approach has always presented the least precise End Boundary date ranges from each dataset. The exponential prior/modified Charcoal Outlier Model approach is generally situated between these two ranges, but closer to the other more precise exponential modelling approach. Each of these effects is cumulative, such that the effects of decreased dataset size and broader model specification on End Boundary precision increases with dataset IAÏ„.
Last distribution precision in TS2 relates strongly to dataset IAÏ„, dataset size and model specification (Table 14). Last distribution precision is inversely proportional to dataset IAÏ„ (generally at least doubling between 10year and 200year IAÏ„ datasets at both 68% and 95% probability), and directly proportional to dataset size (Last distributions generated from 5 date datasets are generally 1.5 times broader than 15 date datasets). Again, the difference in Last distribution precision between 10 and 5 dates is considerable and accounts for much of this overall contrast. In general, exponential prior approaches are associated with increased relative precision and Charcoal Outlier Model approaches with decreased precision. All three of these parameters have a cumulative effect on Last distribution precision, such that the 15 date 10Â years IAÏ„ datasets modelled using the exponential prior/no outlier approach have generated Last distributions with an average precision of 50Â years at 95% probability and 30Â years at 68% probability; whilst the 5 date 500Â years IAÏ„ datasets modelled with using the uniform prior/default Charcoal Outlier Model approach have generated Last distributions with an average precision of 1593Â years at 95% probability and 537Â years at 68% probability.
Average End Boundary median values in TS2 vary from 1213Â cal AD (MRRR1d; exponential prior/no outlier) to 1450Â cal AD (MRRR1e; uniform prior/default Charcoal Outlier). End Boundary median values relate directly to model specification; whereby exponential prior distributions are associated with increased median age and the Charcoal Outlier Model approaches are associated with decreased median age (Table 15). End Boundary median values generally decrease in age, and variability increases, with reduced dataset size and increasing IAÏ„.
Average Last distribution median values in TS2 vary from 1158Â cal AD (MRRR1e; exponential prior/no outlier) to 1305Â cal AD (MR1e; uniform prior/default Charcoal Outlier) (Table 15). These median values relate directly to model specification; whereby exponential prior distributions are generally associated with higher median ages and the Charcoal Outlier Model approaches with lower median ages. Last median values generally increase in age with reduced dataset size, however, and (although more complex) with increasing dataset IAÏ„. This is clearest in the smaller datasets.
Case Study 1 (CS1)â€”Castle Fincharn main block.
Documentary evidence suggests some kind of secular building was constructed in the MidArgyll settlement of Fincharn between 1240 and 1296 AD, or more certainly 1240â€“1308 AD [7, 66, 67]. Castle Fincharn has never been excavated, however, and architectural interpretations of the upstanding but fragmentary 2â€“3 storey structure surviving on the site have varied from the 13^{th} to the sixteenth century. An assemblage of MERLF fragments removed from this building included Quercus sp., Betula sp. and Corylus sp., consistent with regional vegetational histories, and radiocarbon analysis of five widely spaced single entity Corylus sp. samples returned determinations which calibrate to dates ranging from 1050â€“1270Â cal AD (95% confidence; SUERC54793) to 1220â€“1380Â cal AD (95% confidence; SUERC54796) (Table 16; Fig.Â 5).
This 5 date MERLF radiocarbon dataset is statistically consistent at 5% significance level (Tâ€²â€‰=â€‰3.5, Tâ€²(5%)â€‰=â€‰9.5, Î½â€‰=â€‰4); generating a Combine distribution of 1220â€“1275Â cal AD (95% probability; Fincharn Castle; Additional file 3: Sect.Â 3.2) and an age range Difference of âˆ’Â 40 to 220Â years (95% probability; Fincharn Range; Additional file 3: Sect.Â 3.1) (Table 16). The Last and End Boundary distributions generated from the dataset range vary between 1230â€“1285Â cal AD (95% probability) probably 1245â€“1280Â cal AD (68% probability; Fincharn Lowest IA MERLF 1; Additional file 3: Sect.Â 3.3), and 1230â€“1335Â cal AD (95% probability) probably 1245â€“1295Â cal AD (68% probability; Castle Fincharn Construction Completed 4; Additional file 3: Sect.Â 3.6) (Table 17). This includes an exponential prior/modified Charcoal Outlier Model approach specified with a 20Â year timeconstant consistent with the Corylusdominated character of the analysed MERLF assemblage (Table 1) and character of the local woodland (Additional file 3: Sect.Â 3.4).
The upper end of the Combine distribution generated from this Castle Fincharn dataset is not inconconsistent with the historical evidence relating to the site but is relatively early. All Last and End Boundary distributions are also consistent with historical evidenceâ€”ranging from: 90% after TPQ and 100% before TAQ (Fincharn Lowest IA MERLF 1; Additional file 3: Sect.Â 3.3) to 96% after TPQ and 89% before TAQ (Castle Fincharn Construction Completed 4; Additional file 3: Sect.Â 3.6). There is little variation in lower limits of all these End Boundary and Last ranges but decreased upper limit ages in End Boundary HPD intervals generated by models which include Charcoal Outlier Models at 95% probability reduces precision and consistency with historical evidence. All Last ranges are more precise than the latest calibrated date (SUERC54796) and, with an estimateTPQ/TAQ probability sum of 195%, the Last distribution generated by the exponential prior/modified Charcoal Outlier Model approach is the most consistent with currently available historical evidence (Table 17; Fig.Â 6).
Case Study 2 (CS2)â€”Aros Castle northwest block
Documentary evidence suggests some kind of castle building was constructed on the Dun Aros site before 1385, and current arthistorical typologies suggest the bar traceried arcuate windows in the upstanding 2â€“3 storey northwest block first emerged in Scotland at Elgin Cathedral after 1270 AD [68]. The site has never been excavated and the relationship between the northwestblock and the adjacent enclosure is currently unknown, but architectural comparanda also suggests a late 13thâ€“14th date for this former building is likely. An assemblage of MERLF samples removed from this structure during a wider programme of buildings and materials analysis included fragments of Betula sp., Corylus sp., Fraxinus sp. and Quercus sp. These taxa are consistent with regional vegetational histories, and radiocarbon analysis of five widely spaced single entity Betula and Corylus samples returned determinations which calibrate to dates ranging between 1170 and 1280Â cal AD (95% confidence; SUERC82567) and 1290â€“1410Â cal AD (95% confidence; SUERC62566) (Table 18; Fig.Â 7).
The current 5 date dataset associated with this building is not statistically consistent at 5% significance level (Tâ€²â€‰=â€‰22.3, Tâ€²(5%)â€‰=â€‰9.5, Î½â€‰=â€‰4), but generated a Combine distribution of 1270â€“1295Â cal AD (95% probability; Aros Castle; Additional file 3: Sect.Â 3.8) and an age range of 35 to 190Â years (95% probability; Aros Range; Additional file 3: Sect.Â 3.7) (Table 18). The End Boundary and Last distributions generated from the dataset range between 1290â€“1395Â cal AD (95% probability) probably 1295â€“1380Â cal AD (68% probability; Aros NW Lowest IA MERLF 1; Additional file 3: Sect.Â 3.9), and 1290â€“1595Â cal AD (95% probability) probably 1310â€“1425Â cal AD (68% probability; Aros NW Block Construction Completed 4; Additional file 3: Sect.Â 3.12) (Table 19). This includes an exponential prior/modified Charcoal Outlier Model approach specified with a timeconstant of 50Â years (Additional file 3: Sect.Â 3.10) consistent with the longestlived fragments of Betula sp. (here probably B. pubescens) (Table 19; Fig.Â 8).
All estimates generated from this dataset are consistent with the arthistorical, architectural, and documentary evidence relating to the building and site, although the Combine distribution is very early. Variation in the lower limit of all other generated distributions is limited to five years at 95% probability (1295â€“1300Â cal AD), and hence all are 100% after TPQ. Decreases in the upper limit age (and median) of these Last and End Boundary ranges decreases precision and consistency with historical evidence. With an estimateTPQ/TAQ probability sum of 193%, the Last distribution generated using the exponential prior/no Outlier approach is most consistent with the available arthistorical and historical evidence, presenting a date range very similar to latest dataset date (SUERC62566; 1290â€“1410Â cal AD at 95% probability) (Table 19; Fig.Â 8).
Case Study 3 (CS3)â€”Castle Roy enclosure and tower
The Speyside lordship of Abernethy emerges into the surviving documentary record in 1226 AD and the castle enclosure currently surviving on the site of the Castle of Abernethy (now known as Castle Roy) displays an arcuate entrance which is unlikely to have been constructed before 1150 AD [69]. Excavation suggests this substantially upstanding masonry structure is the earliest building on the site, and an extensive assemblage of insitu MERLF fragments removed from the upstanding castle enclosure included Quercus sp., Betula sp. and Pinus sp. This is broadly consistent with the vegetational history of the region and five widely distributed single entity fragments of Betula and Pinus returned radiocarbon determinations which calibrate to dates ranging between 990 and1160Â cal AD (95% confidence; SUERC75745) and 1040â€“1260Â cal AD (95% confidence; SUERC75746) (Table 20; Fig.Â 9).
This 5 date dataset is statistically consistent at 5% significance (Tâ€²â€‰=â€‰7.2, Tâ€²(5%)â€‰=â€‰9.5, Î½â€‰=â€‰4), but generates a Combine distribution with poor agreement of 1040â€“1170Â cal AD (95% probability; Castle Roy; Additional file 3: Sect.Â 3.14) and an age range of âˆ’Â 75 to 210Â years (95% probability; Roy Range; Additional file 3: Sect.Â 3.13) (Table 20). The Last and End Boundary distributions generated from the dataset during this study range between: 1050â€“1225Â cal AD (95% probability) probably 1155â€“1220Â cal AD (68% probability; Roy Lowest IA MERLF 1; Additional file 3: Sect.Â 3.15); and 1055â€“1415Â cal AD (95% probability) probably 1080â€“1290Â cal AD (68% probability; Castle Roy Construction Completed 4; Additional file 3: Sect.Â 3.18) (Table 21), and this includes an exponential prior/modified Charcoal Outlier Model specified with a timeconstant of 100Â years (Additional file 3: Sect.Â 3.16), consistent with the shortestlived fragments of Betula sp. (here probably B. pendula; Table 1).
The extreme upper end of the Combine distribution is consistent with the architectural evidence but is very early. Variation in the lower limits of the generated End Boundary and Last distributions is limited to five years between 1050 and 1055Â cal AD at 95% probability, and all End Boundary and Last distributions are consistent with the available architectural and historical evidence relating to the building and site. With an estimateTPQ/TAQ probability sum of 173%, the Last distribution generated using the exponential prior/no outlier approach is most consistent with this evidence, presenting a date range very similar to latest date (SUERC75746) at 68% probability, but considerably more precise at 95% probability (Table 21; Fig.Â 10).
Case Study 4 (CS4)â€”Lochindorb Castle enclosure
A very narrow 1258â€“1279 AD constructional date has been widely accepted for initial construction of the upland Moray castle of Lochindorb on the basis that the building was constructed by John Comyn before a 1279 reference to â€˜Robert of Lochindorbâ€™ [7, 70]. A limited assemblage of insitu MERLF samples removed from the earliest upstanding phase of the enclosure was completely dominated by fragments of Quercus sp., although this genus is not consistent with relict seminatural woodland populations of PinusBetula surviving locally. Five widely spaced single entity Quercus sp. samples with no terminal ring evidence were selected from this phase for radiocarbon analysis, and these returned a wide distribution of calibrated dates ranging between 550 and 380Â cal BC (95% confidence; SUERC75752) and 1160â€“1270Â cal AD (95% confidence; SUERC75747) (Table 22; Fig.Â 11).
This 5 date dataset is not statistically consistent at 5% significance (Tâ€²â€‰=â€‰2014, Tâ€²(5%)â€‰=â€‰9.5, Î½â€‰=â€‰4), generating an age range of 1555 to 1925Â years (95% probability; Lochindorb Range; Additional file 3: Sect.Â 3.19) and failing to generate a Combine distribution (Additional file 3: Sect.Â 3.20) (Table 22). The Last and End Boundary distributions generated from this dataset range between 1175â€“1270Â cal AD (95% probability) probably 1195â€“1260Â cal AD (68% probability; Lochindorb Lowest IA MERLF 1; Additional file 3: Sect.Â 3.21), and 1200â€“3010Â cal AD (95% probability) probably 1235â€“1890Â cal AD (68% probability; Construction Lochindorb Castle 4; Additional file 3: Sect.Â 3.24) (Table 23). This includes an exponential prior/modified Charcoal Outlier Model specified with a timeconstant of 300Â years (Additional file 3: Sect.Â 3.22) consistent with the Quercus sp. dominated character of the MERLF assemblage (Table 1).
All generated Last and End Boundary estimates are consistent with the available historical evidence. With an estimateTPQ/TAQ probability sum of 114%, the Last distribution generated using the exponential prior/modified Charcoal Outlier Model approach is the most consistent with this other evidence, and this distribution is later and somewhat broader than the latest dataset date (SUERC75747) (Table 23; Fig.Â 12).
Case Study 5 (CS5)â€”Achanduin Castle enclosure and hall
Surviving charter evidence suggests the upstanding castle at Achanduin on Lismore was constructed between 1240 and 1310 AD, whilst a Balliol coin recovered during excavation beneath the castle courtyard has been highlighted to suggest this constructional period may be constrained to a very narrow 1292â€“1310 AD period [49, 71]. A very limited assemblage of insitu MERLF fragments removed from the upstanding essentially singlephase building was comprised of Quercus sp. and Betula sp., consistent with regional vegetational histories, and radiocarbon analysis of one Quercus and two Betula fragments returned determinations which calibrate to between 1180â€“1290Â cal AD (SUERC62547) and 1260â€“1390Â cal AD (SUERC62546) at 95% confidence (Table 24; Fig.Â 13).
This 3 date dataset is statistically consistent at the 5% significance level (Tâ€²â€‰=â€‰4.0, Tâ€²(5%)â€‰=â€‰6.0, Î½â€‰=â€‰2), generating a Combine distribution of 1265â€“1295Â cal AD (95% probability; Achanduin Castle; Additional file 3: Sect.Â 3.26) and an age range of 0 to 155Â years (95% probability; Achanduin Range; Additional file 3: Sect.Â 3.25) (Table 24). The Last and End Boundary distributions generated from the dataset range between 1270â€“1385Â cal AD (95% probability) probably 1275â€“1305Â cal AD (68% probability; Achanduin Lowest IA MERLF 1; Additional file 3: Sect.Â 3.27), and 1275â€“1820Â cal AD (95% probability) probably 1280â€“1450Â cal AD (68% probability; Achanduin Castle Construction Completed 4; Additional file 3: Sect.Â 3.30) (Table 25), and this includes an exponential prior/modified Charcoal Outlier Model specified with a timeconstant of 50Â years (Additional file 3: Sect.Â 3.28), consistent with the shortestlived Betula fraction of the MERLF assemblage (Table 1).
The generated Combine distribution is not consistent with the archaeological evidence at 68% probability. All Last and End Boundary distributions generated are consistent with the available archaeological and historical evidence, with lower limits varying from 1265 to 1275Â cal AD and precision and median age decreasing with uniform prior and Charcoal Outlier Model specifications (Table 25). With an estimateTPQ/TAQ probability sum of 141%, the Last distribution generated using the exponential prior/modified Charcoal Outlier Model approach is the most consistent with this other evidence, and this distribution is similar to the latest dataset date (SUERC62546) at 95% probability but much more precise at 68% probability (Table 25; Fig.Â 14).
Case Study 6 (CS6)â€”Lismore Cathedral nave
The earliest surviving contemporary reference to a church building which can be reasonably related to the site of Lismore cathedral dates to 1314 AD, although other historical evidence suggests the diocese was formally erected between 1192 and 1214 AD [72,73,74]. An upstanding medieval church chancel on the site has been ascribed to a range of 13^{th} to fourteenth century dates and highlighted to illustrate the challenges faced by architectural historians in ascribing more precise dates to western Scottish masonry buildings of this period [75]. An assemblage of MERLF samples removed during excavation of the more fragmentary nave and western tower included fragments of Alnus sp., Betula sp., Corylus sp. and Quercus sp. consistent with local vegetational histories, and 3 samples of Corylus and Quercus from the earlier nave returned a range of radiocarbon determinations calibrating to between 1030â€“1219Â cal AD (95% confidence; SUERC75732) and 1290â€“1400 (95% confidence; SUERC75727) (Table 26; Fig.Â 15). The Corylus MERLF sample (SUERC75727) within this small assemblage retained some probable terminal ring evidence.
This 3 date dataset is not statistically consistent at 5% significance (Tâ€²â€‰=â€‰48.9, Tâ€²(5%)â€‰=â€‰6.0, Î½â€‰=â€‰2) but has generated a Combine distribution with poor agreement of 1250â€“1275Â cal AD (95% probability; Lismore Cathedral Nave; Additional file 3: Sect.Â 3.32), and an age range of 125 to 345Â years (95% probability; Lismore Range; 4.31) (Table 26). The Last and End Boundary distributions generated from the data range between 1295â€“1400Â cal AD (95% probability) probably 1300â€“1395Â cal AD (68% probability; Lismore Nave Lowest IA MERLF 1; Additional file 3: Sect.Â 3.33), and 1320â€“1905Â cal AD (95% probability) probably 1325â€“1905Â cal AD (68% probability; Lismore Cathedral Nave Complete 4; Additional file 3: Sect.Â 3.36). This includes an exponential prior/modified Charcoal Outlier Model specified with a timeconstant of 300Â years (Additional file 3: Sect.Â 3.34), consistent with the longestlived Quercus fraction of the assemblage (Table 1).
The late 13^{th}century Combine date is consistent with available historical evidence. All Last and End Boundary distributions are later than the historical TPQ (100% probability), but the extent to which these distributions predate the documentary TAQ varies between 24% (Lismore Nave Lowest IA MERLF 1; Additional file 3: Sect.Â 3.33) and 0% (Lismore Cathedral Nave Complete 4; Additional file 3: Sect.Â 3.36). With an estimateTPQ/TAQ probability sum of 124%, the Last distribution generated using the exponential prior/no outlier approach is the most consistent with this other evidence, and is very similar to the latest dataset date (SUERC75727) (Table 27; Fig.Â 16).
Discussion
The theoretical studies
Variation in the datasets generated from the same model parameters during these theoretical studies highlights that radiocarbon date simulation is a random probabilistic process, and multiple datasets are therefore required to examine how this variability affects the estimates generated using different modelling approaches. Sixtyfive exponentially distributed datasets of between five and twenty simulated dates were randomly generated from a true event of 1250 AD for the two main theoretical studies considered in this paperâ€”TS1 and TS2.
None of these TS1 or TS2 datasets contain dates which are later than the true event at 95% probability, and the number of dates in a single dataset which contain the true event at 95% probability varies from twenty to zero (Tables 2 and 11). Almost all datasets contain at least one date which includes the true event, and this includes all 15 date and 20 date datasets and all datasets with a specified IAÏ„ of 100Â years or less. Increasing dataset IAÏ„ has increased the age of the latest date in both studies and thereby resulted in datasets with a lower fraction of dates which include the true event (Table 28). There is no convincing relationship between fraction of accurate dates and dataset size in these studies, although a drop off is apparent between 10 and 5 date datasets (Table 28) and some small very high IAÏ„ datasets are completely dominated by inaccurately early dates (Table 11).
Dataset age ranges in these studies are proportional to IAÏ„ and size (Table 29). The only 20 date dataset to present an age range with minus values was a 10Â years IAÏ„ dataset in which all simulated dates contained the true event (TS1 M1a run 1), but three smaller datasets also present minus age range values, including a 5 date 100Â years IAÏ„ dataset (TS2 MRRR1c run 2) within which four dates include the true event (Tables 2 and 11). With some dataset age ranges ranging across thousands of years, it is clear that single determinations from theoretical assemblages subject to IAÏ„ do not always directly represent the true event, while the inaccuracy of all Combine distributions generated from 20 date datasets of 50Â years IAÏ„ or more illustrates that the unweighted averaging of datasets subject to considerable IA does not directly represent this event either. It is salient, however, that the Combine approach can generate accurate distributions from datasets specified with 10Â years IAÏ„, at least (Table 3).
251 (98%) of the 255 models in TS1 and TS2 have generated accurate End Boundary HPD intervals at 95% probability, and 215 (84%) of these are also accurate at 68% probability (Table 30). Accurate End Boundary estimates have been generated with almost identical consistency in TS1 and TS2, and by the three main models included in both studies (Table 31). End Boundary and Last distributions generated from datasets with very low IA lifespans (10Â years IAÏ„) are more consistently accurate in both studies, but no general relationship between accuracy and dataset IA or dataset size was noted elsewhere (Tables 13, 31 and 32). The most consistently accurate End Boundary estimates in both theoretical studies were generated by models specified with both exponential priors and Charcoal Outlier Models (Table 32). This was the only model specification to generate accurate End Boundary HPD intervals at 95% probability from all datasets in both studies, and in both studies 87â€“90% of these estimates were also accurate at 68% probability. No change in End Boundary accuracy resulted from modifying the Charcoal Outlier Model timeconstant (to match that of the specified dataset lifespan) in 20 date models with exponential priors, but an increase in accuracy is evident in the less precise uniform prior approaches (Table 5).
All Last distributions are slightly earlier than the End Boundaries generated from the same datasets in both theoretical studies. These contrasts are more marked in 95% probability distributions and increase with increasing dataset IAÏ„, decreasing dataset size, and uniform prior and Charcoal Outlier Model specifications. The Last distributions generated in the 5 date to 15 date datasets of TS2 are slightly less consistently accurate than the corresponding End Boundaries (Table 30) and, in further contrast, there is some minor evidence that Last accuracy is proportional to dataset size. Overall, the most consistently accurate Last distributions in TS2 were generated by the uniform prior/default Charcoal Outlier Model approach; and this was the only model specification to generate accurate Last distributions at 95% probability from all TS2 datasets, with 87% of these also accurate at 68% probability (Table 13). That these accuracy percentages are identical to those reported for exponential prior/Charcoal Outlier Model End Boundary distributions is salient and will be returned to below.
Relative precision in End Boundary and Last distributions across both theoretical studies consistently decreases with increasing dataset IAÏ„, decreasing dataset size, and the imposition of a uniform prior distribution or Charcoal Outlier Model, and each of these factors has a cumulative effect. This is illustrated by the average precision of Last and End Boundary distributions generated using the exponential Prior/modified Charcoal Outlier Model approach in TS1 and TS2 where increasing dataset IAÏ„ above 100Â years or reducing dataset size below 10 dates has considerable impact, even though these datasets were generated independently (Tables 33 and 34). Last distributions are generally more precise than End Boundary estimates at both 68% and 95% probability, and thereby variations in dataset IAÏ„, dataset size, and model specification have a reduced impact. Conversely, given that reducing the Charcoal Outlier Model timeconstant appears to have increased precision in broader 20 date TS1 End Boundary distributions subject to uniform priors and high dataset IAÏ„, it is reasonable to expect that timeconstant variation would also have a greater effect where dataset size is very reduced and generated estimates broad (see below).
The older latest dates associated with smaller and higher IAÏ„ datasets in TS1 and TS2 (Tables 2 and 11) generally result in End Boundaries with older lower limits, and the implications of this relationship for estimate accuracy are clearly illustrated where that latest dataset date does not contain the true event. In TS2 dataset MRRR1d run 3, for example, the latest date (MRRR5d) is too early at 68% confidence (1020â€“1160Â cal AD) and 95% confidence (990â€“1160Â cal AD), and thereby the End Boundary and Last distributions generated by all models at 68% probability and the Last distributions generated by both exponential prior approaches at 95% probability are also too early (Table 12). No latest dataset dates are inaccurately late at 95% probability in these theoretical studies, but in TS1 dataset M1e run 3 the latest date (M5e) is too late at 68% confidence (1260â€“1300Â cal AD), and the End Boundary distributions generated by all models from this dataset are also too late at 68% probability (Table 4). Indeed, the End Boundaries generated by both uniform prior/Charcoal Outlier Models from this dataset are also too late at 95% probability, and these are the only two inaccurate End Boundary HPD intervals at 95% probability in TS1.
The data associated with these examples also illustrate how model specification can mitigate against latest date variation, since the earlier distributions generated by models with exponential priors are generally more accurate where the latest dataset date is relatively late, whilst the older estimates generated by models incorporating the Charcoal Outlier Model are generally more accurate where the latest dataset date is relatively early. In TS2 dataset MRRR1c run 3, for example, the latest simulated date contains the true event at 95% confidence (1220â€“1380; MRRR1c) but is too late at 68% confidence (1260â€“1300Â cal AD; MRRR1c; Additional file 2: Sect.Â 2.9.10); and in this instance both Charcoal Outlier modelling approaches have generated inaccurately late End Boundaries at 68% probability, and only the exponential/no outlier approach has generated an accurate End Boundary at 68% probability. In contrast, the only two inaccurate End Boundaries at 95% probability in TS2 are too early (MR1c run 3 and MRR1d run 1) and, predictably therefore, these distributions are associated with comparatively early dataset latest dates and exponential prior/no outlier modelling approaches.
These processes also have implications for distribution selection; since the earlier Last distributions are often more accurate than End Boundaries where a comparatively late latest date pertains (e. g. TS2, MR1b run 2, exponential prior/no outlier) while End Boundaries are more accurate where an early latest date pertains. Indeed, given the preponderance of relatively early dates, this might explain the comparatively greater consistency of End Boundary distributions overall. The estimates generated from the TS2 dataset MRR1d run 1 illustrate how these processes affect both distribution and model selection, since the latest date (MRR3d) contains the true date at 95% confidence (1040â€“1270Â cal AD) but is too early at 68% confidence (1050â€“1230Â cal AD) (Table 11), and thereby: the End Boundary generated by the exponential prior/no outlier modelling approach is too early at 95% probability; the End Boundaries generated by both exponential prior models are too early at 68% probability; the Last distributions generated by both exponential prior models are also too early at 95% probability; and the Last distributions generated by all three modelling approaches are too early at 68% probability (Table 12). The uniform prior/Charcoal Outlier Model approach, however, has generated accurate End Boundary estimates at both 95% and 68% probability from this high IAÏ„ dataset, as well as an accurate Last distribution at 95% probability.
Ultimately, the Last and End Boundary distributions generated by different model specifications form a continuous chronological spectrum: from the very early and precise Last distributions generated from large low IAÏ„ datasets by the exponential prior/no outlier modelling approach; to the later and more imprecise End Boundary distributions generated from small high IAÏ„ datasets by the uniform prior/Charcoal Outlier Model approach. The evidence presented in TS2 also suggests this spectrum correlates with the accuracy of estimates generated from datasets subject to different IAÏ„/size characteristics: wherein the exponential prior/no outlier approach has generated the most consistently accurate Last distributions from datasets with an IAÏ„ which is lower than 100Â years (where latest dates are likely to be relatively late), and the uniform prior/Charcoal Outlier approach has generated the most consistently accurate Last distributions from datasets of 100Â years IAÏ„ and above (where latest dates are likely to be relatively early) (Table 13). Unsurprisingly, given their precision, the proximity of the Last distribution median values to the true event date follows this same pattern (Table 15), while the accuracy threshold between these contrasting approaches is slightly broader in the End Boundary distribution evidence. Both exponential prior approaches present the most consistently accurate End Boundary distributions from datasets with an IAÏ„ lower than 100Â years in TS1 and TS2, and approaches which include a Charcoal Outlier Model are more consistently accurate from 200Â years IAÏ„ and above (although this breaks down at 500Â years IAÏ„ in TS1) (Table 31). That a considerable overlap between these theoretical Last and End Boundary spectra also pertains is clearly illustrated in the TS2 results, wherein the most consistently accurate End Boundaries have been generated by the exponential prior/Charcoal Outlier Model approach, whilst the most consistently accurate Last distributions have been generated by the uniform prior/Charcoal Outlier Model approach. Indeed, both approaches have generated accurate estimates from all datasets at 95% probability and 87% of all models at 68% probability, while closely comparable average precision and median values between these different distributions confirms they overlap considerably (Tables 13, 15 and 17).
These results are consistent with those presented by previous authors. If the 500yrs IAÏ„ datasets are disregarded, then the uniform prior/Charcoal Outlier Model approach does indeed generate more consistently accurate End Boundary distributions than the exponential prior/no outlier approach [17], with this latter approach once again returning some precise but inaccurate End estimates at 95% probability (Table 16). These inaccuracies are limited to datasets above 50Â years IAÏ„, however, and the End Boundary estimates generated by the exponential prior/Charcoal Outlier Model approach are more consistently accurate overall. In the above studies this is even evident where dataset IAÏ„ range is much reduced, and most particularly so with lower IAÏ„ datasets where the less accurate and less precise uniform prior approach has been specified. These data, therefore, also support previous MERLF analysis protocols which had promoted a binary shortlived (exponential prior) and longlived (Charcoal Outlier Model) approach [7]; but allows an increased role for exponential prior model specifications and further understanding of how these relate to Combine and Charcoal Outlier Model approaches. The accuracy of the Combine distributions generated from 10Â years IAÏ„ datasets is also resonant of Waterbolkâ€™s (1971) Group A samples which, he suggested, would extend up to 20Â years [10]. Ultimately, these theoretical results provide a less binary Bayesian framework which can inform our interpretations of the datasets returned by MERLF materials from the six Scottish medieval case study buildings.
The case studies
The compositions presented by these case study MERLF assemblages confirm that a range of different locally available tree taxa were exploited for limekiln fuel in Scotland during this period, and the maximum age ranges presented by the resulting radiocarbon datasets are generally consistent with the taxaspecific and habitatcontingent IAÏ„ of those woodland sources. This includes: the radiocarbon dataset associated with the Corylus sp. dominated assemblage from Castle Fincharn, which generated an age range of âˆ’Â 30 to 220Â years consistent with a 5 date mean lifespan of 10Â years IAÏ„ or less; the dataset associated with the Betula sp. dominated assemblage from Castle Aros which generated an age range of 35 to 190Â years consistent with a 5 date mean lifespan of 50Â years IAÏ„ or less; and the smaller dataset associated with the Quercus sp. dominated assemblage from Lismore Cathedral nave, which nevertheless generated an age range of 125 to 345Â years consistent with a 5 date mean lifespan of 200Â years IAÏ„ or less (Table 35).
The association of some taxa or environments with comparatively high mean lifespans does not of course preclude the exploitation of shorterlived or immature wood, where growth habits and woodland population dynamics allow. Indeed, although the case study results presented here are biased by an analysis strategy which privileged the selection of shorterlifespan taxa, it is evident from the statistical consistency and relatively narrow age ranges presented by the mixed assemblages from Castle Roy (CS3) and Achanduin Castle (CS5) that shortlived fragments of long lifespan taxa were also included in limekiln charges (Table 35). The only case study dataset which generated an age range broader than expected is associated with the Quercus sp. dominated assemblage from Lochindorb Castle, which returned a dataset age range (1555 to 1925Â years) consistent with a 5 date mean lifespan of over 500Â years IAÏ„. This is improbable, and when the single very early radiocarbon determination (SUERC75752) is manually excluded from the model, the age range of the remaining dataset (10â€“220Â years; Additional file 3: Sect.Â 3.37) is consistent with a 5 date dataset of 50Â years IAÏ„ or less.
Preexisting historical, architectural, and archaeological evidence has situated initial building construction at the six case study sites in the same long 13thcentury period, and within chronological periods ranging between 18 and 122Â years (Table 36). The relationships between this evidence and the masonry buildings under consideration are indirect and open to challenge, and these periods are much broader than the true event date from which the simulated datasets were generated in TS1 and TS2. The lack of terminal ring evidence in five of the case study assemblages, moreover, introduces a bridging period between these datasets and the constructional date which does not apply to those theoretical studies. However, all case study datasets contain at least one latest date which is consistent with the preexisting evidence from other disciplines, five of the six present latest dataset dates which extend past their potential historical TAQs, and none are inconsistently late (Table 36). This evidence suggests the volume of material missing material from these MERLF fragments (and so the extent of that bridging period) is quite limited and, although small dataset sizes limit the interpretive potential of their unmodelled distributions (Figs.Â 5, 7, 9, 11, 13, 15), these case study datasets generally contain high fractions of consistent dates.
As in both main theoretical studies, the Last and End Boundary distributions generated from these case study datasets present continuous chronological spectra: from the earliest and most precise Last distributions generated using the exponential prior/no Outlier approach; to the later and broader distributions generated using the uniform prior/Charcoal Outlier Model (Tables 14, 16. 18, 20, 22, 24). It is notable that modifying Charcoal Outlier Model timeconstants to reflect the mean lifespans of the dominant taxa has increased Last and End Boundary precision in all six case studies and, in line with its increased effect on less precise distributions in TS1, this probably reflects the smaller size of these case study datasets. An overlap between the latest Last distributions and earliest End Boundary is again evident in these studies, and this is very clearly illustrated in overlapping estimateTPQ/TAQ percentages.
The sum of these estimateTPQTAQ percentages reflects contrasts in the precision of these different types of evidence, and these vary from Castle Fincharn (whose high sum percentages reflect a low IAÏ„ Corylus sp. dominated MERLF radiocarbon dataset and the moderate precision of the preexisting documentary evidence relating to the wider site) and Lochindorb Castle and Lismore Cathedral nave (which are both associated with relatively high age range Quercus sp. dominated MERLF radiocarbon datasets). Except for the estimate generated by the uniform prior/Charcoal Outlier Model approach to the Lismore Cathedral nave dataset, all Last and End Boundary distributions generated by these different model specifications at 95% probability are consistent with the other evidence relating to these sites, and the proximity of some of these estimateTPQTAQ probability sums to 200% indicates that these Bayesian estimates are closely consistent with that evidence. The consistency of this evidence usefully suggests that the masonry buildings from which these MERLF samples were removed can be reasonably associated with the wider evidence relating to these sites, and where late latest dates have defined comparatively late lower estimate limits (e. g. Aros Castle NW block and Lismore Cathedral nave) then considerable gains in multidisciplinary interpretation precision have been made. It is salient that the datasets associated with both of these sites are high IA and statistically inconsistent at 5% probability, however, and therefore the End Boundary distributions generated from these data are relatively broad.
Recognising how we might retain accuracy while maximising precision through sample selection and model specification requires further comparison with the theoretical and ecological data. Variation in dataset age ranges and levels of statistical consistency across these studies suggests that the assemblages did include materials subject to some IA, and only two of these case study datasets have presented good Combine agreement indices (even including the reduced Lochindorb Castle CS4* dataset) (Table 35). That Combine distributions could be generated at all suggests the IA associated with five of these assemblages is reasonably limited, however, and comparison with the theoretical data from TS1 and TS2 suggests four of the six case studies (or five if the reduced Lochindorb Castle is included) present dataset age ranges consistent with mean lifespans associated with less than 50Â years IAÏ„. Importantly, this situates these fourfive studies below the considerable reduction in estimate precision associated with theoretical datasets subject to IAÏ„s of over 100Â years in TS1 and TS2 (Tables 32 and 34), and all (100%) End Boundary and Last distributions generated from such narrowly distributed 5 date datasets were consistently accurate in TS2 at both 95% and 68% probability (Table 13). The estimate/TPQTAQ sum percentages also illustrate that interdisciplinary consistency generally increases with precision, and the most consistent estimates across all six case studies were Last distributions generated by exponential prior modelling specifications. Indeed, in most cases these Last distributions are very similar to each available latest dataset date, and constructional dates close to the lower limits of these distributions are often most convincing.
The association of higher IAÏ„ and small datasets with earlier latest dates and decreased fractions of accurate dates in TS1 and TS2 (Table 28) suggests that it would be prudent to include a Charcoal Outlier Model within model specifications where radiocarbon datasets are less narrowly distributed, and yet the botany suggests different approaches to the case study evidence are probably required. Both Quercus sp. dominated case study assemblages (Lochindorb Castle enclosure and Lismore Cathedral nave) are statistically inconsistent at 5% significance, but the Lochindorb dataset contains an extremely early determination, high age range, and low fraction of consistent dates. This assemblage is completely dominated by Quercus sp. samples, and although manual exclusion of the early determination has decreased dataset IAÏ„ considerably, this still contains only two dates which are consistent with historical evidence at 95% probability and the latest date does not extend beyond the documentary TAQ at either 68% or 95% probability (Appendix 1: Table 37). Notably, this reduced dataset is also the only example to generate End Boundary estimates which are more consistent with the historical evidence than the earlier and more precise Last distributions, and the End Boundary generated by the exponential prior/modified Charcoal Outlier Model approach to the dataset is the most consistent overall (Appendix 1: Table 38). Archaeobotanical and statistical evidence suggests this approach is less relevant to the Lismore Cathedral nave study since, although the high age range associated with this small dataset is largely predicated on residuality in two early Quercus sp. fragments, the latest date is associated with a fragment of Corylus. Since the uniform prior/Charcoal Outlier Model approach to this dataset has generated the only inconsistent estimate in the data from all six studies (Table 27), it is entirely possible that the calibrated radiocarbon date associated with this latter fragment is relatively late. This latest determination allows the 200Â years or less IAÏ„ of the wider Quercusdominated radiocarbon dataset and its relationship to the constructional event to be interpreted with greater confidence. Notably, this latest radiocarbon determination also calibrates to a date almost identical to the Last distribution generated from the wider dataset using the (most precise) exponential prior/no Outlier approach, and this currently represents the most convincing estimate for the construction of this fabric. The correlation between dataset IAÏ„ and latest date does not hold for the radiocarbon evidence returned by this mixedtaxa assemblage, and the application of a Charcoal Outlier Model approach to this very small dataset is inappropriate.
Conclusion
This paper has presented further evidence that a range of different tree taxa were exploited for limekiln fuel in Scotland during the medieval period, and the range of MERLF taxa surviving from this process are generally consistent with regional phytogeographic distributions. The samples selected from these assemblages for radiocarbon analysis have returned datasets characterised by an array of different age range distributions and most of these are associated with some level of IA. At this stage in the research cycle, however, the IAÏ„ of these materials does appear to be constrained by the taxaspecific and habitatcontingent lifespans and postmortem durabilities of the parent wood fuels. Given that shorter lifespan taxa were generally selected for radiocarbon analysis where possible, it seems probable that the carbon distributions in these assemblages are generally equivalent to the available woodland source. MERLF assemblages can therefore be considered an excellent source of palaeoenvironmental information, with a research potential again underscored by the stratigraphically secure mortar material within which these materials have been entrapped.
The MERLF assemblages considered in this paper are dominated by woodcharcoal fragments without surviving terminal ring evidence, and the range of radiocarbon determinations returned by selected samples suggests that calibrated dates from single determinations, unweighted mean averages of multiple determinations and/or bulk samples, cannot be accepted as direct evidence for the construction of masonry buildings without other forms of evidence. The TPQ role performed by such determinations can be of considerable value for multidisciplinary interpretation, particularly where the radiocarbon evidence is relatively late and documentary evidence is convincing and early. Increasing the potential for these buildings and materials to inform interdisciplinary (rather than multidisciplinary) discourse, however, requires accurate standalone constructional estimates of greater precision.
In the absence of nonresidual intrusive materials, the generation of accurate Last and End Boundary distributions from MERLF radiocarbon datasets subject to significant IA relies more on the accuracy of the latest available determination, than on dataset IAÏ„, dataset size, or model specification. Last and End Boundary precision, however, is very closely related to all three of these parameters; decreasing with increasing dataset IAÏ„, decreasing dataset size, and model specifications which include uniform priors or Charcoal Outlier Models. These factors are interrelated and can be cumulative, and the Last and End Boundary distributions generated by different model specifications thereby present continuous and overlapping spectra. These range from the early and precise Last distributions generated from large lowIAÏ„ datasets by models specified with an exponential prior/no outlier approach; to the later and less precise End Boundary distributions generated from small highIAÏ„ datasets by models specified with uniform priors and the default Charcoal Outlier Model.
Different Bayesian model specifications can therefore be imposed on MERLF radiocarbon data to maximise precision whilst retaining accuracy. The data relating to the 13thcentury events presented in this paper suggests that: (i) Where the IAÏ„ of a dataset is limited in 10Â years or less, then determinations are likely to be statistical consistent at 5% significance and a Combine average distribution is likely to represent an accurate and very precise constructional estimate; (ii) Where the IAÏ„ of a dataset is limited to 50Â years or less, then determinations are unlikely to be statistical consistent at 5% significance and Combine agreement indices will be poor, but the Last distribution generated by a model specified with exponential priors is likely to represent an accurate and reasonably precise constructional estimate; and (iii) Where the IAÏ„ of a dataset is greater than 100Â years, then a Last distribution generated by a model with a Charcoal Outlier Model is likely to generate an accurate but imprecise constructional estimate, while modification of the outlier timeconstant is likely to increase precision where dataset size is limited.
The studies considered in this paper provide further evidence that Bayesian techniques can generate consistently accurate constructional estimates for medieval masonry buildings from MERLF radiocarbon data, whatever the ecological provenance of the limekiln fuel source. Estimate precision is contingent upon source ecology but can be increased by a more informed approach to materials analysis and interpretation. The radiocarbon evidence considered here and elsewhere [7, 76] is biased by the selection of single entity MERLF fragments from shorter lifespan tree taxa, where possible, and most of these have returned determinations consistent with (i) and (ii) above. It seems likely this has enabled the generation of more precise constructional estimates, although in many cases precision might be further increased by expanding these radiocarbon datasets to include higher precision (reduced error margin) analysis of short lifespan MERLF fragments.
Availability of data and materials
All the data relating to this paper are included in the main text and additional file.
Abbreviations
 CS:

Case Study
 ESM:

Electronic Supplementary Material
 HPD:

Highest Posterior Density
 IA:

Inbuilt Age
 IAÏ„:

Mean Dataset Inbuilt Age
 MERLF:

MortarEntrapped Relict Limekiln Fuel
 TAQ:

terminus ante quem (Limit before which)
 TPQ:

terminus post quem (Limit after which)
 TS:

Theoretical Study
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Acknowledgements
The data reconsidered in this paper was generated during PhD research undertaken at the University of Edinburgh and postdoctoral research at the University of Stirling. Within these projects, charcoal identification was undertaken with Dr Mike Cressey (then at CFA Archaeology) and radiocarbon analysis was funded by Historic Environment Scotland. The author is grateful for the comments provided by two anonymous peers and the journal editor, which improved this paper significantly, and for their patience when the original manuscript was withdrawn in 2020 following the emergence of the new calibration curve.
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Additional file 1.
Theoretical Study 1 (TS1).
Additional file 2.
Theoretical Study 2 (TS2).
Additional file 3.
Case Studies 16 (CS16 + CS4*).
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Thacker, M. Modelling medieval masonry construction: taxaspecific and habitatcontingent Bayesian techniques for the interpretation of radiocarbon data from MortarEntrapped Relict Limekiln Fuels. Herit Sci 9, 113 (2021). https://doi.org/10.1186/s40494021005683
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DOI: https://doi.org/10.1186/s40494021005683
Keywords
 Bayesian
 Buildings
 Charcoal
 Ecology
 Masonry
 Medieval
 Outlier
 Radiocarbon