Spatiotemporal distribution of Holocene populations in North America

Significance

We provide the first maps to our knowledge of spatiotemporal paleodemographic growth following human migration into the Americas for the past 13,000 y, using a statistical approach that simultaneously addresses sampling and taphonomic biases. The Canadian Archaeological Radiocarbon Database is sufficiently complete in many areas, demonstrating high correspondence between continental-scale ¹⁴C-inferred population estimates and generally accepted archaeological history. Increases in population density seem robust for eastern and western North America, as well as central Alaska and the region surrounding Cahokia. These results are the first step toward being able to understand continental-scale human impacts on the North American ecosystem during the Holocene as well as demographic growth and migrations in relation to environmental changes.

Abstract

As the Cordilleran and Laurentide Ice Sheets retreated, North America was colonized by human populations; however, the spatial patterns of subsequent population growth are unclear. Temporal frequency distributions of aggregated radiocarbon (¹⁴C) dates are used as a proxy of population size and can be used to track this expansion. The Canadian Archaeological Radiocarbon Database contains more than 35,000 ¹⁴C dates and is used in this study to map the spatiotemporal demographic changes of Holocene populations in North America at a continental scale for the past 13,000 y. We use the kernel method, which converts the spatial distribution of ¹⁴C dates into estimates of population density at 500-y intervals. The resulting maps reveal temporally distinct, dynamic patterns associated with paleodemographic trends that correspond well to genetic, archaeological, and ethnohistoric evidence of human occupation. These results have implications for hypothesizing and testing migration routes into and across North America as well as the relative influence of North American populations on the evolution of the North American ecosystem.

Databases of radiocarbon dates obtained from archaeological sites are used as a source of information about past demographic changes of Holocene populations (1–5). These are typically analyzed using summed probability distributions for a given region, which are interpreted as time series of site occupancy and inferred population tendency. For example, frequency distributions have been used to estimate continental-scale population growth in North America (North America here means the United States and Canada, the extent of our database) over the past 15 ka using the Canadian Archaeological Radiocarbon Database (CARD) (2, 3). Regional studies in North America have highlighted the impacts of environmental change on population size using these methods and documented the impact of human activities on the vegetation (1, 6–10). Although some studies have attempted to investigate between-region demographic changes for particular time periods (11) or map the point distribution for certain time intervals (3), the spatiotemporal distribution of demographic growth through the Holocene in North America after the arrival of humans has not been tracked at a continental scale.

The underlying assumption of the “dates as data” approach is that the frequency of ¹⁴C dates is proportional to population density. This relation improves when aggregated data have a sufficient sample size to apply methods to overcome archaeological sampling bias and taphonomic and calibration effects (12). Summed probability distributions have been used to assess paleodemographic trends in the Americas, Europe, Eurasia, and Australia (e.g., refs. 1, 2, 11, and 13–20) and methodological issues have been discussed at length (see refs. 17, 21, and 22 and Supporting Information for examples). However, to determine and compare demographic patterns in both space and time, a way is needed to estimate the density of archaeological records at a particular time.

We build upon previous work in North America by using an alternative method with data from the CARD and provide the first estimates to our knowledge of the spatiotemporal distribution of the population of North America during the past 13 ka. The kernel method is used to convert the spatial distribution of ¹⁴C dates into estimates of population density at 500-y intervals with a kernel radius (or bandwidth, terms we use interchangeably) of 600 km, a bin size and radius that optimize the macrotemporal and -spatial patterning at continental and geological scales (Supporting Information). The radius of 600 km was chosen based on the optimal bandwidth (595 km) calculated following Scott’s rule for bandwidth selection (23). This bandwidth does not have a simple physical meaning (such as the range of a hunter–gatherer); it is meant to represent where humans could have been (in a probabilistic sense) during the 500-y interval based on the distribution of dates, because each interval represents many generations of humans.

The CARD is unique because it contains 35,905 ¹⁴C dates, 33,756 of which have geographical, chronological, and descriptive information permitting spatial and temporal analysis. The frequency of dates increases exponentially through time, with a maximum frequency around 0.7 ka calibrated B.P. (all future references to dates are in thousands of years calibrated B.P., denoted simply as ka). Previous results (2), as well as our own cross-validation exercises (Supporting Information), indicate that this amount and distribution of dates are sufficiently representative to begin studying past populations at a continental scale (2).

Based on our method, we hypothesize that the resulting intensity distributions (referred to as radiocarbon frequency population estimates, or RFPEs) capture paleodemographic trends because (i) the RFPEs show clear patterns when methods to reduce taphonomic and sampling bias are applied and (ii) the patterns correlate well with preexisting archaeological interpretations of human cultural change across the continent that were derived partly from ¹⁴C dates but also other data sources and inferences, thus representing a partial independent test. The results of this study can be used to model human demographic growth in relation to environmental change and cultural innovation and to determine human impacts on the landscape or fauna at a continental scale, particularly in relation to the early anthropogenic hypothesis (24).

Distribution of ¹⁴C Dates

At the continental scale of this study, the distribution of ¹⁴C dates is the result of human activity that is proportional to population density, although data incompleteness, sampling intensity, and natural taphonomic processes have an effect as well. There are too few dates in the CARD before 13 ka to create reliable population estimates before this time using the methods used in this analysis. Other known issues with the database include a variable temporal and geographic representation, and in particular a lack of data from the southern United States, which limits interpretation of results in this area.

As shown in Fig. 1A, some areas are extensively represented (e.g., Wyoming) and others less so (e.g., Texas and Florida). Areas with greater sampling intensity are commonly the result of heightened archaeological interest, mapping objectives, and available funding, whereas others relate to issues involved with the construction of the database, including variable effort in data gathering from available literature (3). This creates a spatial bias that can confound continental-scale maps of population density. To account for this bias and normalize the series of RFPE maps, a spatially varying sampling intensity was computed using ¹⁴C dates from all sites. Using kernel density estimation, a sampling intensity map was then made that treated multiple dates from the same site in each time interval as one and standardized the influence of sites with higher sampling intensities (Fig. 1B; details in Materials and Methods).

Fig. 1.

Locations of the archaeological sites in the CARD and the sampling intensity base map. (A) Distribution of the CARD sites including dated cultural, paleoenvironmental, and unknown material. (B) Sampling intensity base map produced using all geographically distinct cultural and paleoenvironmental dates spanning the full age range in the CARD, such that multiple dates at the same location are considered as one.

There is a tendency in the CARD for a greater number of more recent archaeological sites because geological deposits tend to disappear over time due to normal taphonomic processes (3). We appreciate that taphonomy may vary spatially but apply a spatially constant taphonomic correction to the maps (2, 25) because the undertaking of how, where, and by how much this should be varied requires more study at regional scales. We propose that areas along some coastlines may require an even greater correction due to the loss of accessible archaeological sites as a consequence of sea level rise, or due to reduced sampling of higher elevation stranded shorelines. Similarly, for the Arctic, the kernel function radius of 600 km used to obtain the RFPEs (which is indicative of the spatial extent of the influence of a site on the surrounding intensity) may be too large due to geographic constraints (see Results, Canadian Arctic and Supporting Information).

Data Verification

The only time period where patterns produced using the CARD data may be verified with independent data are just before European contact, when ethnohistoric estimates of the spatial distribution of population have been made (26). Although the absolute numbers have been questioned (27), the relative demographic and spatial patterns should be broadly accurate (26). The RFPE patterns estimated from 0.4 to 0.6 ka (Fig. 2A) generally correspond to those based on precontact estimates of population size (Fig. 2B), with half of the distinct regions in the maps differing by only one level of density. Across a band of the northern United States (and British Columbia), where we have high confidence in the data, the estimates from both sources are similar. In the south where the database is known to be incomplete, and in New England, our estimates of population density are much lower. In the north, an area where the ethnohistoric data were known to be incomplete, our estimates differ by only one or two levels. Despite calibration, taphonomic, or sampling effects, a broad-scale pattern emerges from the data and a reasonable hypothesis is that it is recording a demographic trend. This is the first verification of the accuracy of the CARD data to our knowledge and suggests that the current number of archaeological records in the database, although not exhaustive, is sufficient to produce a general representation of the demographic history of North America (2), if the regional incompleteness of the database is accounted for. Larger datasets and refinements to the method will enhance this kind of modeling, as will updates of the CARD itself.

Fig. 2.

Comparison between population densities at European contact estimated using the RFPE technique and ethnohistorical data. (A) Population density estimates based on the distribution of the CARD sites from 0.4 to 0.6 ka. (B) Population density estimates based on work by Ubelaker (26).

Results

Alaska.

RFPEs fluctuate steadily for the entire study period (Fig. 3 and Fig. S1), which is realistic considering Alaska was the location of repeated migrations into the Americas (28). Estimates are relatively high in northern and central Alaska in comparison with other parts of North America between 7.5 and 13 ka, with a peak at 11.5 ka. This coincides with an abundance of sites associated with fluted points, characteristic of technologies used by Paleoindian groups after 13.5 ka (29). Starting around 6 ka, a second increase is centered on the Aleutian Islands and slowly moves east over the next 1.5 ka. This agrees with archaeological evidence suggesting Aleut peoples colonized the central islands ca. 6 ka (30). The stable but relatively low estimates in Alaska just before the Aleut occupation may be associated with a well-known “hiatus” in human presence (31). Beginning at 4.5 ka, a 1,000-y-long increase is observed, composed of two centers along the northern and southwestern coasts, coinciding with the suspected migration of early Paleoeskimo from Siberia around 4.5 ka (32) and the Arctic Small Tool tradition in the eastern Aleutians (33). Estimates show a decrease for several hundred years and then begin to increase at the northern coast at 2.5 ka and along the west coast (2 ka), until finally the majority of the Alaskan region seems occupied. The development of Thule culture in Alaska (beginning ca. 1 ka) coincides with the increasing RFPE patterns (32, 34).

Fig. 3.

RFPE maps. A subset of RFPEs and the distribution of the associated CARD dates for selected time intervals between 0.5 and 13 ka. Gray areas show the extent of the ice sheets. The entire series of RFPE maps is shown in Fig. S1.

Canadian Arctic.

The maps show a rapid increase in RFPEs in coastal northwestern Canada, including the Mackenzie Delta, Banks Island, and western Victoria Island around 6.5 ka, and an increase shortly thereafter on Ellesmere Island (Fig. S1). This predates the Independence I (35) and Early Pre-Dorset (36) occupations by several thousand years and may be due to known dating problems from Arctic sites (37) and temporal smoothing of the maps. However, these dates support recent genetic evidence suggesting there was human activity in the general region around 6,000 y ago (38). By 4 ka, there are signs of occupation across the entire Canadian Arctic, and at 2.5 ka, site density increases to the south. This is due to data from Dorset culture sites that have been found from Victoria Island in the west to Greenland and Newfoundland in the east (39). RFPEs on Victoria Island as well as the area surrounding Foxe Basin are likely underestimated, because Pre-Dorset archaeological evidence for the latter suggests a population was centered on this region around 2.5 ka (40). An increase in RFPEs along the southernmost Thule migration route from Alaska to western Hudson Bay beginning at 0.75 ka is expected in the context of history of the Thule people (41). A site in Nunavut at 9.5 ka, in an area that was glaciated at this time, causes an isolated population center in the central Arctic, which persists until 8.1 ka, at which point it is joined with the population further west (Fig. S1) and by which time the ice had retreated. The spatial extent of this population is a product of the kernel radius chosen (600 km); a larger radius would merge this population with its western neighbor (Fig. S2). Although these dates seem inconsistent with the ice extent (which is, however, based on radiocarbon data), it is reasonable to hypothesize that humans were found along the edge of the ice at this time.

British Columbia and the Ice-Free Corridor.

Before and during the Last Glacial Maximum, human migration into the Americas via an interior route was not possible due to the presence of ice sheets. At ∼13.5 ka an ice-free corridor opened between the Laurentide and Cordilleran Ice Sheets (42, 43). Genetic and cultural evidence suggests a nomadic pre-Clovis population would have been the first to migrate northward into the corridor from further south around 10.5 ka, following a previous migration south via a Pacific Ocean coastal route (29). In the maps, a high-density population is observed in the ice-free corridor two millennia before 10.5 ka (Fig. S1). RFPEs remain high until 12 ka, when the Cordilleran Ice Sheet retreated and westward human expansion into this new terrain likely occurred. Around 11.5 ka, RFPEs increased along the entire west coast of British Columbia. This timing is in line with archaeological findings of maritime communities living on the outer coast of Haida Gwaii (44) and is consistent with the hypothesis that Pleistocene–Early Holocene sites in outer coastal regions would today be underwater and thus poorly sampled archaeologically. A mirror isostatic effect, in which glacial loading depresses areas under ice sheets while raising offshore areas beyond the ice margin, strands shorelines along the mainland at much higher elevations, which leads to undersampling (45). At 7 ka, a second increase at the coast is observed, followed by an increase in the corridor. After 5.5 ka, populations seem to increase in both locations, potentially due to the stabilization of relative sea level on the coast and the development of new trap and tool technologies east of the Rocky Mountains (46). In the Late Holocene, the entire province of British Columbia was densely populated and remained so until 1 ka, reflecting the demographic success of complex hunter–gatherer cultures.

Eastern United States.

Beginning around 11 ka, there was an increase in RFPEs in the southeastern United States, which grew larger and expanded northward then westward after 9.5 ka (Fig. S1). This interpretation is plausible in light of numerous archaeological studies indicating a strong presence of Paleoindians in the southeast before 10 ka (47). RFPEs decreased east of the Appalachians after 9.5 ka and were centered in Missouri at 8.5 ka, where the presence of Paleoindian populations has been confirmed (48). Estimates fluctuated until 4.9 ka when populations grew to the east and west of the Appalachians as well as in the Middle Atlantic and New England regions. After 3.5 ka, RFPEs greatly increased in the central part of the eastern temperate deciduous forest, where studies of numerous sites document intense human occupation in the Late Archaic and Early Woodland periods [e.g., Dunbar Cave, Tennessee (49)]. After 2 ka, the highest estimates were in the Ohio–Kentucky region (Cahokia and environs), and this persisted until the time of contact.

Atlantic Canada.

Between 12 and 13 ka, the maps show a relatively large Paleoindian population in Atlantic Canada. The Debert site (Nova Scotia) is the only dated site in the CARD at this time, but the dates from this site are consistent with one another and are accepted in archaeological literature (50). The RFPEs decreased following this, as temperatures and dryness increased during the Younger Dryas (51). RFPE values fluctuated and began to increase around 10.5 ka, consistent with archaeological evidence from Debert (50). At 8.5 ka, RFPEs tripled in Newfoundland and Labrador (Fig. 3) with the onset of early Maritime Archaic settlement along the coast [5.5–8 ka (52)]. Estimates are relatively low in the Maritime Provinces because many sites were presumably inundated by sea level rise. There is a slight increase in RFPEs during the Middle Archaic in eastern Quebec and central Labrador, and at 4.5 ka there was a migration into these provinces. A larger population seems to have spread across eastern Hudson Bay and the coastline of Labrador between 3.2 and 4 ka during the late Archaic. The influence of Arctic-adapted Paleoeskimo populations (e.g., Groswater and Dorset sites), the Meadowood complex, and Late Woodland populations are observed during the last 3 ka, although these values may be overestimated due to a higher number of sites being divided by a low sampling intensity in more recent times. RFPEs decreased in central Quebec but remained high in Newfoundland, Labrador and the area surrounding New Brunswick until 1.5 ka, when estimates increase over Atlantic Canada.

Central and Western United States.

The oldest maps contain two centers of relatively high estimates in Arizona and on the Colorado–Kansas border, areas where paleoecological records confirm the presence of Paleoindians (53). At 12 ka, these centers shifted to Texas, although this area remains anomalous for the entire record due to the relatively large number of dates in the Early- and Mid-Holocene compared with the more recent past (Fig. S1). By 10 ka, an increase appeared in California and persisted until 7.5 ka, which is likely associated with offshore communities exploiting marine resources (54). Also during this time there was a slight increase in RFPEs in Montana, which became joined with larger populations living in the ice-free corridor. The latter half of the Holocene was characterized by an intensification in Idaho, which shifted toward the coast at 4 ka.

Discussion

If known issues with the database are accounted for, RFPEs correspond with archaeological, paleoenvironmental and genetic evidence for human migration and dispersal into and across North America (Table 1). The maps (Fig. 3, Fig. S1, and Movie S1) show the importance of proximity to the coast for human settlement as well as areas where human activity seems to have been continuous throughout much of the Holocene (northeastern North America, Alaska, and western Plains). The maps also show examples where sampling bias (i.e., the quality of the CARD data) affects the RFPEs. For example, the Kivalliq Region (west of Hudson Bay) and the Ungava Region show an overestimation beginning ca. 4 ka. This is connected to selective archaeological work in the region and an extensive research program into the local precontact history (56–58) combined with comparatively few dates in the CARD before this interval.

Table 1.

Summary of the major patterns seen in the RFPE maps and archaeological history for the different regions mentioned in the text with references

Region	Time period, ka calibrated B.P.	RFPE patterns	Archaeological history	Refs.
Alaska	7.5–13	Increase in north, central	Fluted point technology	29
	6	Increase on Aleutian Islands	Aleut colonization	30
	4.5–5.5	Increase at northern and southwestern coast	Migration from Siberia, Arctic Small Tool tradition	32, 33
	1	Increase	Thule	32, 34, 39
Canadian Arctic	2.5	Increase in southern Arctic	Dorset culture
	Modern–0.75	Increase along southern Thule migration route	Thule migration beginning ca. 0.8	41
British Columbia/ corridor	11.5	Increase along west coast	Maritime community occupation	44
	5.5	Second increase along west coast and ice-free corridor	Shift in technology	46
Eastern United States	8.5	Increase in Missouri	Paleoindian occupation	48
	4.9	Increase in mid-Atlantic and New England	Late Archaic and Early Woodland occupation	49
	Modern–2	Increase in Cahokia	Population rise with maize agriculture	55
Atlantic Canada	10.5	General increase	Debert culture	50
	8.5	Large increase in Newfoundland and Labrador	Early Maritime Archaic	52
Central/western United States	7.5	Increase in California	Marine exploitation	54

We have shown that aggregated continental data exhibit macrodemographic patterns comparable to known regional histories; in addition, use of the CARD permits a continental-scale analysis. At 12.5, 9, 4.5, and 1 ka, the overall pattern is close to generally accepted culture-historical reconstructions (29, 41, 44, 49, 52, 54). There was an overall increase in continental population estimates from 12.5 ka, as expected, followed by a decrease between 0.5 and 1 ka due partly to the taphonomic correction, but also reflecting the decreased use of ¹⁴C dating in the context of historical methods and a lessened archaeological interest in recent periods. The distribution at ca. 0.8–1.2 ka provides a broad example of both current and ancestral population distribution because the majority of currently understood large-scale migrations had occurred by this period, and comparatively little migration has occurred since, with the exception of the Athapaskan migration into the southwest.

These results illustrate the value in applying advanced statistical methods to aggregate ¹⁴C data from archaeological databases. The accumulation of known ¹⁴C data into facilities such as the CARD should be a priority, because this offers an opportunity to summarize previous research and provide a context for regional and local studies. “Dates as data” (7, 12) is a relevant means of investigating and modeling demographic, and thus historical, trends across long periods of time and continental areas. The kernel density method, in combination with statistical approaches to reduce sampling and taphonomic bias, holds potential for accommodating known issues in ¹⁴C datasets that cannot be handled by other statistical analyses developed so far (e.g., averaging the number of observations). These results suggest that the CARD is a highly useful archive of paleodemographic data that can be used to investigate subjects such as migration routes into and across North America as well as a valuable tool for studies linking anthropogenic impacts with post-ice age faunal extinctions, ecosystem decline, and changing environmental and climatic conditions.

Materials and Methods

The CARD (3) contains 35,905 radiocarbon dates derived from cultural and paleoenvironmental material collected from 9,149 geographically distinct archaeological sites. The selection of dates for this study depended on completeness of the database (locations for 77 dates are missing), location (61 dates are from Russia), dating information (2,071 entries are missing normalized ¹⁴C ages, normalized ¹⁴C errors, or both), and classification (cultural, paleoenvironmental, etc.). Descriptions of the cultural association and type of material dated and general comments about the entry were also considered, although dates were not eliminated from the study if this information was missing. This resulted in 29,609 cultural dates that we used to create the RFPE maps and 4,087 dates from paleoenvironmental (or unknown) material (Fig. 1A). The ¹⁴C ages calculated from the CARD (1) have been converted to calendar years (based on CALIB v.5.0.2 and IntCal04) using the median of the 2 sigma ages of the calibration curve (3). Because the maps are based on intervals of 500 y, any error when using the median as an estimate is much less than the sample interval.

To reduce sampling bias in the CARD, which is a result of inconsistencies in regional- (e.g., Canada vs. United States) and local-scale (e.g., Wyoming) sampling strategies and intensities, a sampling intensity map (Fig. 1B) was created using kernel density estimation. Only geographically distinct sites (n = 7,754) were used to create Fig. 1B; the amount of sampling (intensity) is estimated by considering only the geographic location of each site so that multiple dates at the same site have no influence. All archaeological sites were used in this step regardless of the age of the dated material or its classification (i.e., cultural, paleoenvironmental, or unknown). This map is interpreted as the density of sites, in other words, the distribution of sampling sites in space, and not the number of ¹⁴C dates per site. If this remained unaccounted for, regions with higher sampling intensity would always be associated with elevated population estimates.

The underlying assumption of this approach is that a greater number of dates is indicative of a higher population density (3, 59). To produce the RFPE maps, a grid of points every 0.1° between 23–85°N and 173–51°W was created to define the area of interest (North America) in ESRI ArcMap 10.1. This grid, in combination with ArcGIS shapefiles of the extent of glacial ice at 1,000-y intervals (60), was used to differentiate between glaciated and inundated areas and land. The period from 0.5 to 13 ka was chosen due to the lack of CARD dates before and after this date range. To visualize patterns in older time periods we would need to use a larger smoothing radius, but this tends to overgeneralize the resulting patterns (Fig. S2). The data were examined in 500-y intervals as a compromise to avoid aggregating too many dates and having too few to produce a reliable estimate, with consecutive intervals overlapping every 100 y.

To temporally smooth the data within each interval and account for ¹⁴C dating errors, dates that do not occur directly within the interval are considered if they occur within 400 y before or following the interval. The dates within an interval are given a weight (w) of 1, whereas the dates within ±400 y of an interval are given a weight of <1. The weighting is achieved by taking a date that occurs outside of an interval and computing the temporal distance to the interval boundary and dividing the result by 400 (i.e., dates that occur at a greater distance from an interval boundary will have a smaller weight). Dates within 400 y before or following an interval are considered because they can still have an influence on population intensity (i.e., the estimated density of people at a given time and location is influenced by how many people were there before them) and to account for possible dating errors. When multiple dates exist for a single site within an interval, each date is considered separately.

Kernel density estimation, a method previously applied to archaeological data from Europe (61, 62), is used to produce the RFPE maps. An Epanechnikov kernel with a bandwidth fixed at 600 km (Fig. S3) is chosen, although the Gaussian kernel method and adaptive bandwidth options were also examined (Fig. S4). First, raw population estimates are produced. Generally, these estimates are highly correlated to the sampling intensity because a greater sampling effort produces a larger number of dates, which results in higher population estimate (2). Therefore, producing RFPE maps based on raw estimates does not truly reflect human activity due to this sampling bias. A more reliable estimate is obtained when the raw population estimates are divided by the sampling intensity (Fig. 3 and Fig. S1). In doing so, RFPEs can be interpreted as the relative frequency of dates per site, which should better reflect population intensity than only considering the number of dates. A similar approach has previously been used to account for a temporal bias (2). The values on the maps have been rescaled to 1 and are plotted in a logarithmic scale, with each color being approximately three times that of the previous color. This is done to maintain a consistent scale across all time intervals, and to make temporal changes more apparent, especially during earlier intervals. The resulting maps were corrected using a taphonomic formula that had previously been applied to the CARD data (1, 25).

To assess the quality and completeness of the CARD, we ran two additional analyses (Figs. S5–S8). In the first, individual dates within the database flagged as “anomalous” (i.e., too young or old) were removed (n = 1,419, or 5% or the original data points) and a new series of maps were made. This is representative of a worst-case scenario in which the dates with the highest probability of negatively influencing the RFPEs are removed, thus addressing the potential effect of “bad dates” on the quality of the maps. In the second analysis, a new series of maps is made after 50% of the dates (n = 17,953) had been randomly selected and removed, to determine whether or not the amount of data in the CARD is spatially representative. Figs. S6 and S8 show a very high degree of similarity between the original RFPE maps and the two new series despite the missing data, putting to rest our concerns regarding the state and usability of the database and its applicability in spatiotemporal analyses. A supplementary test was performed using two subsets of calibrated ages based on the IntCal04 and IntCal13 calibration curves (Fig. S9). This test was done to determine if changes made between IntCal04 and IntCal13 would have an effect on the calibration results. The resulting calibrated ages were nearly identical allowing us to use a previously calibrated version of the CARD based on CALIB v.5.0.2 and IntCal04 (1–3) (see Supporting Information for additional discussion).

The methodology is composed of five major steps that include the calculation of the (i) RFPEs, (ii) taphonomic correction, (iii) sampling intensity, (iv) edge correction, and (v) a comparison with ethnohistorical data. RFPEs are computed as a ratio between preliminary RFPEs and sampling intensity. Similar techniques based on kernel density estimation are used in steps i and iii but they differ in two important ways. In step i (RFPE calculation), multiple dates at a single site within a given time interval are considered individually, and only cultural dates are used. In step iii (sampling intensity calculation), only geographically distinct sites are considered such that multiple dates at the same location are considered as one, and all cultural and paleoenvironmental dates of all ages in the CARD are used.

Kernel Density Estimation

The kernel density technique is one of the most common nonparametric methods used for quantifying the spatial behavior of a point pattern of sites (63). Surrounding each site is an individual kernel function that can be interpreted as a “hat” with a specific radius r (or bandwidth) that sits on top of the site. The considered site has an influence on the intensities at all locations within a circular area with radius r centered on the site. In our study, the radius was defined as 600 km (Fig. S3), a value close to the optimal radius (595 km) calculated using Scott’s rule (23). Choosing a higher or lower radius would lead to an over- or undersmoothing of the maps (Fig. S2). Each site is considered individually using the following Epanechnikov kernel function (23):

$$K\left(u\right)=\frac{2}{\pi }\left(1-u^{T}u\right)I_{\left\{u^{T}u\le 1\right\}},$$ [S1]

where u = (u₁, u₂) is a point on the map. This kernel is transformed to have the radius r and additionally scaled by two values: (i) the weight w (mentioned in Materials and Methods) for computing the RFPE (not for sampling intensity) and (ii) the edge correction e (see below). For each possible location on the map, the values of the scaled kernels that cover this location are summed to produce the intensity function λ (·), that is,

$$\lambda \left(u\right)=\frac{1}{r^2}{\displaystyle \sum }_{i=1}^n{w}_i{e}_{i}K\left(\frac{x_i-u}{r}\right),$$ [S2]

where u = (u₁, u₂) is again a point on the map. For the computation of the sampling intensity, x₁,…, x_n denote the locations of all geographically distinct CARD sites and w_i is always 1. When computing the RFPEs, x₁,…, x_n are the sites with dates in the corresponding interval (±400 y), where multiple dates from one site are summed individually. Possible modifications of kernel density estimation are discussed below.

Taphonomic Bias

The taphonomic bias inherent in the CARD, due to long-term loss of samples through time in the fossil record, is addressed by computing the mean age of each interval and using

$$n_t=5.726442\times {10}^6{\left(t+2176.4\right)}^{-1.3925309},$$ [S3]

where $n_t$ is an indicator for the number of ¹⁴C dates in the CARD surviving from time t, or the mean age of the interval in question. This taphonomic loss rate was developed based on ice core and geologic volcanic activity records and operates on a removal-through-time basis (25). We assume that the archaeological remains used to derive the CARD data share degradation characteristics similar to these geological contexts (2, 17, 64, 65), and thus that ¹⁴C dates are subject to the same rate of taphonomic loss. Based on this, the resulting value will be higher for more recent intervals and lower for older ones. The whole map is globally divided by the value computed from the taphonomic equation (25); values of RFPEs from older intervals are too low due to taphonomic loss and are divided by a lower value than the earlier intervals (2). We are aware that the application of this correction may be suboptimal in some areas where taphonomic loss is probably lower (e.g., the Arctic, where permafrost effectively preserves archaeological materials) or in later time intervals (e.g., between the present and 0.5 ka when archaeological interest declines and historical information is used instead of ¹⁴C dates for dating) and may lead to decreased RFPEs. This could potentially be addressed via regional analyses but this was beyond the scope of this study.

Edge Correction

Edge corrections are necessary because RFPEs would always decrease toward the coast (because dates near the coast have less of an influence). The circular area around a site with radius r can potentially intersect with an area where no population is possible, for example water or ice. If this is the case, then this location would have a lower capacity for human population and thus a lower influence on the RFPE function. In other words, a kernel has a volume within which the population can occur; if half of the circular area around the site occurs over water then the volume available for population is reduced by half. Ideally, every individual kernel will have an equal influence on the intensity. Therefore, the numerical integral of the kernel function over land is calculated, and the value of the kernel at each influenced location is divided by this numerical integral (denoted as e above) so that each corrected kernel has an equal influence on intensity (i.e., kernels that are at least partially over water or ice are divided such that they have the same influence as those completely over land). If half of the kernel is over water, the volume over land is only half of the other kernels over land, so this is scaled by 2 to allow the part of the kernel over land to have the same influence as any other kernel over land. When computing RFPEs, water and ice are taken into account for the edge correction, whereas for the sampling intensity only water is considered.

Estimating Population Density at Contact: Comparison with Results from Ubelaker (1992)

The same techniques used to produce the sampling intensity and RFPE maps are used to produce an estimate of population size following European contact (Fig. 2A). This is compared with ethnohistorical documentation of North American tribal area population estimates (Fig. 2B) (26, 66), which is largely based on a compilation of population estimates made by multiple authors and early explorers (67). The maps (Fig. 2) have been rescaled and generalized to correspond to early representations of tribal areas (26). In this case, data for Canada are a very generalized, and the estimates for “Greenland” are used in this study for northern Canada.

Shape and Bandwidth of the Kernel Density Estimator

A variety of kernel functions exist for use in spatiotemporal analyses. The Epanechnikov kernel, which is used in this analysis, and the Gaussian kernel are two methods that are frequently used. The latter, however, has an unbounded support (i.e., dates would have an influence on the whole continent). The Epanechnikov kernel complements our approach of assuming each date influences an area within a certain radius of its location. It is generally accepted that the choice of the kernel function has only a minor effect on the resulting estimate and its statistical properties. Another reason for choosing the Epanechnikov kernel is that this kernel minimizes the approximate mean integrated square error of the kernel density estimator, which is a desirable statistical property (63).

The choice of the bandwidth is a more complex consideration. One option is to use an adaptive bandwidth, that is, a bandwidth that is not globally fixed but varies across space. There are two ways of achieving this. In the nearest-neighbor approach, the bandwidth is chosen based on the location u at which the intensity is computed, that is, we would use r(u) instead of r in Eq. S2. In the variable kernel approach, the bandwidth depends on the current data site, which implies that r(x_i) is written instead of r and the fraction is shifted inside the sum in Eq. S2. Both approaches are quite similar with one conceptual difference. In the variable kernel approach all sites can have a different radius, but these radii remain fixed for all locations u at which the intensity is estimated. In the nearest-neighbor approach, the radius assigned to a certain data site can vary, if different locations u₁ and u₂ are considered. These approaches to bandwidth choice contradict our fundamental logic that each date should have the same influence on a surrounding area with a spatially and temporally fixed radius. The bandwidths for all data sites could be determined adaptively based on all sampling sites, and these bandwidths could be used for all estimations. Then, however, these bandwidths are not adapted to the total number of dates in different time intervals. For these reasons we do not use an adaptive bandwidth approach. To illustrate, RFPE maps created based on the nearest-neighbor and variable kernel approaches are provided (Fig. S4).

A second option is to algorithmically determine a globally fixed bandwidth, in which case likelihood cross-validation is most frequently used. However, this method is sensitive to outliers and to multiple data points and is inefficient when large datasets are considered, making it a poor choice in this study also.

Instead, we opt to set the bandwidth to a fixed value of 600 km following Scott’s rule (23). Our goal was to use the same bandwidth for the sampling intensity function and the RFPEs for all time intervals. This is necessary to maintain comparability through time, to avoid odd “jumps” between different time intervals, and to avoid numerical anomalies when computing RFPEs as ratios. Thus, a bandwidth is computed for each time interval separately using Scott’s rule (see Fig. S3 for a histogram of obtained values). A bandwidth close to the mean value (595 km) seems to be an adequate choice for all time intervals and the sampling intensity.

Sensitivity Analysis

A sensitivity test was performed to determine the effects of varying the kernel function radius. At first, RFPEs computed with a globally fixed radius greater or less than 600 km (Fig. S2C) are considered. A radius of 500 km or smaller is not large enough to capture the overall pattern because often too few sites occur within such a small area (Fig. S2 A and B). When a radius of 800 km or more is used, the overall signal remains but the maps are too smoothed to discern regional patterns (Fig. S2 D–F). It is possible that a radius of 1,000–1,250 km would be ideal for a global scale study, but we recommend ca. 600 km for a continental scale analysis. Likewise, a radius smaller than 500 km would be more suitable for a regional study.

Furthermore, we study the effect of choosing an adaptive bandwidth according to the generalized nearest-neighbor and variable kernel approaches. The concept is similar in both cases; the radius r(u) for each location u in the generalized nearest-neighbor approach as well as the radius r(x_i) for each site x_i in the variable kernel approach are chosen to be inversely proportional to the local density of all sampling sites. These radii are used for computing the sampling intensity and the RFPEs for all time intervals. Note, however, that these radii are not inversely proportional to the site density of the different time intervals anymore, which contradicts the purpose of choosing an adaptive bandwidth. Results show similar population estimates for both approaches (Fig. S4). The maps reveal largely smoothed patterns in regions with a low site density (central Canada) but undersmoothing occurs in more frequently sampled regions (e.g., the southeast or Wyoming). In general, however, both approaches lead to the same population events, although the spatial scale and size of single events can vary slightly. Furthermore, both sequences of maps show the same patterns as in the fixed bandwidth approach, which reinforces the results obtained in this paper.

Quality of the CARD: Removal of “Anomalous” Data

The CARD contains over 35,000 archaeological dates collected by thousands of submitting authors over many years. Although carefully checked, radiocarbon dates are subject to well-known sources of error; for example, a date could be contaminated with younger organic matter or be older due to the incorporation of older carbon. The database could potentially include incorrect dates, and 1,419 dates (of the 29,609 that we used to create the RFPE maps) are currently flagged as “anomalous” in the CARD. A rigorous, nonsubjective method of removing potentially erroneous data from the CARD based on descriptive, geographical, cultural, and other information needs to be developed. Until this can be done, however, we suggest research based on the CARD include all available data, anomalous or otherwise (68). This avoids selectively removing dates based on preconceived notions of paleodemographic trends and eliminates the risk of overcleaning the database. In this study, all available CARD data were included for the purpose of (i) mapping the raw data at a continental scale, a task that has not yet been done, and (ii) identifying aspects of the data that may lead to new insight into the human history of North America. Anomalous data in a local context may not be anomalous in a regional or continental context.

To determine to what extent the dates described as anomalous in the CARD influence the resulting RFPE patterns, we recalculated RFPEs between 0.5 and 13 ka excluding the flagged dates. These dates (n = 1,419, or 5% of the original data points) originated from sites spanning the entire continent (Fig. S5) resembling a random sample of the data in terms of spatial distribution but being a “worst case scenario,” because they have a greater potential of influencing the resulting patterns than a true random sample. It is of interest to note that the date appearing below the glacial ice in the central Arctic was not described as anomalous and is therefore retained in this analysis. The majority of the sites in Fig. S5 were associated with multiple anomalous dates most often contributed by a single author per site. We would emphasize that these data may seem anomalous in the cultural context of the original study but that (i) they may still provide information about population numbers and (ii) they may not be anomalous when studied in the context of other data at regional to continental scales.

The recalculation of RFPEs excluding anomalous dates resulted in a series of maps nearly identical to the original maps shown in Fig. S1. We provide a comparison of the recalculated RFPE maps with the original maps in Fig. S6. The overall patterns remain the same throughout the entire study period, at continental to regional scales. The timing of settlement of the High Arctic, Banks and Victoria Islands, the Aleutian Islands, the western continental coastline, and all other major centers of population visible in the maps remain the same. The single exception to this is Newfoundland, where the RFPEs first begin to increase from 0 to 0.17 around 11.8 ka in the original maps, and 10.5 ka in the recalculated maps. Differences are never greater than one order of magnitude, are greater in earlier time intervals, and mostly occur along the coasts. Very small differences are observed (Fig. S6) in northern Alaska (12.5–13 ka), Newfoundland (11–11.5 ka), the Kivalliq region (7–7.5 ka), and Louisiana (7–7.5 and 4.5–5 ka). In each of these cases, the smoothing indicates a slightly earlier occupation in the series of maps based on all available data (i.e., when anomalies are included). After 2 ka, visual differences no longer exist.

The anomalous dates in question were further investigated by choosing three sites and comparing the anomalous dates from those sites with all nonanomalous dates within a 25-km radius. Fig. S7A suggests removing the anomalous dates could potentially show no human presence during the mid-Holocene, which may be the case, or may be simply due to lack of samples. The time of the first arrival in the region would be the same. Removing these dates could also indicate people arrived later in a particular region (Fig. S7B), although the impact of their removal on the maps is small (Fig. S6). In many cases, the removal of anomalous dates would produce no difference in our interpretation of population numbers through time (Fig. S7C). Overall, the number of questionable dates is low compared with accepted dates, their removal has little effect on the general patterns shown in the RFPE maps, and this favors their inclusion in spatiotemporal analyses such as this one. The high degree of similarity between RFPE maps based on all available data and nonanomalous data implies the CARD in its current form is both spatially and temporally representative of paleodemographic trends across North America.

Completeness of the CARD: Randomly Removing 50% of the Dates

Whereas the section above identifies a worst-case scenario in which the RFPE maps are produced following the removal of dates that have been flagged as anomalous, here the database size is reduced to test whether or not the CARD is spatially representative and has a sufficiently large size for all time intervals. To do this we performed a Bernoulli experiment with a success probability of one half for each date and removed all dates showing success, which resulted in 16,894 (around 50%) of the original dates. The RFPEs are then recomputed based on the reduced dataset. Fig. S8 indicates that, despite the smaller number of dates, the spatial patterns remain largely the same, even during the earlier intervals when there are comparatively few dates. Some regional differences are visible; for example, a population center is missing in the ice-free corridor between 11 and 11.5 ka, and Florida shows almost no human presence between 9.5 and 10 ka.

Based on our multiple sensitivity analyses, we propose that the CARD is sufficiently complete for continental-scale spatiotemporal paleodemographic analyses, that individual dates (and even large numbers of dates) do not have a major influence on RFPEs, and nonsystematic errors (e.g., “anomalous” dates) have only a minor influence as well.

Pooled vs. Summing Methods

Recent studies have investigated the use of pooled mean dates for individual site phases (when multiple dates are found at a single site) as opposed to the basic summing method (6, 7, 69). The question is: Are multiple dates due to (i) the original researcher’s simply attempting to obtain replicates, in which case it would bias the population estimates, or (ii) the relative size of the site, and the duration and amount of human activity there, in which case all dates should be considered individually? Although we judge this issue in need of further attention, it was not the purpose of this study to test this, and we refer readers to a prior investigation of this (70). Shennan and Edinborough (70) show small-scale differences in population fluctuations and a shuffling of the peak population period when summed probabilities and pooled means are compared but conclude that the use of a pooling method in lieu of a summing method does not lead to a more straightforward interpretation of ¹⁴C dates.

Calibration Curves: IntCal04 vs. IntCal13

In this study, we use a previously calibrated version of the CARD based on CALIB v.5.0.2 and IntCal04 (1–3) because recalibrating the dates using IntCal13 would make no difference to our results. To demonstrate how little of an effect the changes made between IntCal04 and IntCal13 have on the calibration results (71, 72) we chose a random sample of 200 dates that span the Holocene and calibrated these using IntCal13 and the northern hemisphere calibration curve (removing marine samples that require a marine correction; n = 8) and using 2 SDs. We compared these to the IntCal04 calibrated data we use here (Fig. S9). The two calibrated series are identical except for two points that are from freshwater molluscs, and even these two points differ by much less than our sampling interval.

Robustness of Results

When analyzing RFPEs the question about robustness of inferred results arises. In the previous paragraphs we showed that modifications of the method have only a minor influence on the results. For the broad population patterns, the effect of varying the kernel function is just as small as that of applying an adaptive smoothing parameter. The main regions of population activity stay the same; only the sizes of different population events may vary. A larger effect is obvious if the global bandwidth is varied. An effect on the scale of population events is then visible, whereas affected regions still stay the same. Additionally, we showed that the method is robust in dealing with outliers and anomalous data. If 5% of the dates are considered to be anomalous and dropped, results are almost identical. Even if the CARD is reduced randomly by 50%, our results remain valid—a conclusion that is remarkable in particular for earlier time intervals. This shows the true significance and robustness of the RFPEs. Naturally, the next step would be to further validate our results by using a statistical test for significance of estimated RFPEs (4). However, performing such a test would require a stochastic spatiotemporal model for RFPEs; this is a nontrivial issue and beyond the scope of this paper. The development of an asymptotic test would require extensive theoretical consideration, which is not possible in this context. Bootstrap methods would be appropriate for performing statistical tests but cannot be applied here due to computational (time) restrictions. The development and implementation of a model-based statistical test will be subject to future research.