1959) and Willis, Tauber, and Munnich (1960), and subsequently has ... and Ralph, 1972; Clark and Renfrew, 1972; Damon, Long, and Wallick,. 1972 ...... chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 549-569.
{Rr11)foc;ARI oN,
Vor, 24, No. 2, 1982, P 103-150]
Radiocarbon 1982
CALIBRATION OF RADIOCARBON DATES :
Tables based on the consensus data of the Workshop on Calibrating the Radiocarbon Time Scale JEFFREY KLEIN*, J C LERMAN**, p E DAMONX*, and E K RALPH'S A calibration is presented for conventional radiocarbon ages ranging from 10 to 7240 years IMP and thus covering a calendric range of 8000 years from 6050 BC to AD 1950. Distinctive features of this calibration include 1) an improved data set consisting of 1154 radiocarbon measurements on samples of known age, 2) an extended range over which radiocarbon ages may be calibrated (an additional 530 years), 3) separate 95% confidence intervals (in tabular from) for six different radiocarbon uncertainties (20, 50, 100, 150, 200, 300 years), and 4) an estimate of the non-Poisson errors related to radiocarbon determinations, including an estimate of the systematic errors between laboratories. INTRODUCTION
quite generally accepted that "conventional" radiocarbon dates need to be "calibrated" because of temporal variations in the radiocarbon content of atmospheric carbon dioxide. The discovery of this phenomenon was made largely by the pioneering work of de Vries (1958; 1959) and Willis, Tauber, and Munnich (1960), and subsequently has been carried on by more than a dozen radiocarbon laboratories worldwide (for a review see Damon, Lerman, and Long, 1978). The assessment of these variations relies on the measurement of "C activity in samples of known age. Dendrochronologically dated wood has proved to be an ideal material for such measurements, and currently all radiocarbon calibrations are based on measurements of 14C activity in wood. The longest chronology extant is that of the bristlecone pine, resulting from the efforts of Schulman (1956) and Ferguson (1969; 1970; 1972). It reaches continuously to 8681 years ago, and to 8580 years ago with sufficient material to allow radiocarbon dating. This work includes measurements on wood as old as 8000 years. Many calibrations have appeared during the past 13 years (Suess, 1979; 1970a; 1967; Clark, 1980; 1979; 1975; McKerrell, 1975; Damon et al, 1974; Ralph, Michael, and Han, 1973; Switsur, 1973; Michael
It
is now
Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104. ** Laboratory of Isotope Geochemistry, Arizona, Tucson, Arizona 85721
Department of Geosciences, University of
103
104
Jeffrey Klein,
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Lerman, P E Damon, and E K Ralph
and Ralph, 1972; Clark and Renfrew, 1972; Damon, Long, and Wallick, 1972; Wendland and Donley, 1971; Lerman, Mook, and Vogel, 1970; Ralph and Michael, 1970; Stuiver and Suess, 1966). Although all reflect similar long-term changes in atmospheric radiocarbon concentrations, they differ significantly in their treatments of shorter period variations. This diversity of available calibrations and the apparently conflicting results obtained when calibrating dates using one in preference to another has resulted in a suspicion on the part of many archaeologists regarding calibration, in particular, and radiocarbon dating, in general. Consequently, in 1978, it was suggested to the USA National Science Foundation that it was time to attempt a consensus among the divergent efforts of the many laboratories then involved in calibration research. With this as a goal, a workshop was held in Tucson, Arizona in early 1979, entitled, "Workshop on the Calibration of the Radiocarbon Dating Time Scale" (Damon et al, 1980; Michael and Klein, 1979). This work is largely the implementation of the decisions reached at that meeting. The Workshop participants decided to provide a calibration table suitable for the calibration of individual or "single" radiocarbon dates. A "single radiocarbon date" is defined as any radiocarbon date that is not associated with another radiocarbon date by a tight, independently determined relative chronology. Such a chronology is exemplified by tree rings, where the number of intervening rings determines the relative ages of samples, and by stratified samples, where the rate of stratification is known independently of the radiocarbon ages of samples contained therein. Included in the category of "single radiocarbon dates" are series of dates from samples thought to be coeval, or series in which the temporal sequence, or even the relative ages of its members is unknown. A second decision of the participants of the Workshop was to provide the "user" with a realistic assessment of the precision of calibrated dates. A consideration of many factors is necessary in the estimation of this precision. These include the precision with which the sample's activity has been measured, involving not just the "counting" statistics quoted by the measurement laboratory, but also an estimate of the true reproducibility of the measurement, ie, the degree to which a particular result can be repeated by the same laboratory or any other laboratory on subsequent measurements. In addition, there is the precision to which the calibration function is known near a particular calendric date. This depends on the quantity and quality of data used in the construction of the calibration. Finally, there is the "shape" of the calibration "curve" in the region in which it is being employed. This factor is often the most influential in determining the magnitude of the uncertainty of a calibrated date, and although its importance has been recognized for some time (Renfrew and Clark, 1974; Grey and Damon, 1970) it is often ignored in the routine calibration of dates. These objectives were implemented by providing a range of calibrated dates, representing the 95% confidence interval, for each radiocarbon age of specified precision. An advantage of specifying an interval,
105 Calibration of Radiocarbon Dates rather than a midpoint and uncertainty, hinges on the fact that many confidence intervals are asymetrically related to the value obtained from simply calibrating the 14C date without consideration of uncertainties. THE DATA
This calibration is based on the 14 C activity measurements performed by the radiocarbon laboratories at the Universities of Arizona, Groningen, California at La Jolla, Pennsylvania, and Yale, on 1154 samples of dendrochronologically dated wood, principally Pinus longaeva and Sequoia gigantea (bristlecone pine and giant sequoia). The data set consisting, for the most part, of an updated version of previously published data (current data sets in preparation by individual laboratories), was prepared for the "Workshop on the Calibration of the Radiocarbon Dating Time Scale." Only measurements on samples of wood containing 20 or fewer rings were used in this work so as not to attenuate significantly through averaging, variations occurring on the time scale of the order of 100 years. Beyond this consideration, no selection of the data was undertaken.
As one of the principal objectives of this analysis has been to understand more fully the nature and causes of the variability of radiocarbon dates, the data were examined carefully for signs of non-random errors. Much to our surprise and despite previous findings to the contrary (Damon, Lerman, and Long, 1978; Clark, 1975; Damon, 1970), there is significant evidence of systematic differences between the laboratories represented. Of the five laboratories, one shows an average systematic difference of approximately six per mil, roughly 50 radiocarbon years, significant at less than the 1% level. The other four laboratories agree within experimental uncertainties. Independent comparisons with a sixth laboratory have resulted in similar conclusions (Stuiver, pers commun, 1981). Systematic differences were determined by calculating residuals of each data set with respect to the calibration function calculated on the combined data set. If no systematic differences had existed, then the sum of residuals would have been consistent with zero for all laboratories; it was not. A table of these differences was reported earlier (Klein et al, 1980), and is included here with slight modifications (see Table 1). Since it is unlikely that the systematic errors between other radiocarbon laboratories are, in general, less than those encountered here (International Study Group, submitted for publication), we decided to leave the data as they were and to include the uncertainty related to interlaboratory standardization within the calibration uncertainty. CONSTRUCTION OF TABLES
Though the method used to construct this calibration has been outlined elsewhere (Ralph and Klein, 1979; Klein et al, 1980) and will be described in more detail in a forthcoming article, it is briefly described here. The procedure may be divided into three steps: a "global" regression which describes the long period (of the order of a few thousand
Jeffrey Klein,
Lerman, P E Damon, and E K Ralph years) secular changes in the atmospheric 14C concentration; a series of short term intervals called "shingles" which describe variations of a few hundred years; and finally, the construction of the table itself from the combination of these functions. First, paired dendrochronologic ages and radiocarbon ages are scaled logarithmically so that each ranges over the interval [-1,1]. This is done to avoid the pathology common with polynomial regressions, namely the dominance of measurements at large values of the independent variable in the determination of the coefficients of the function. Next, each measurement is weighted by an estimate of the inverse of its variance. But, as it is widely accepted that the uncertainties quoted by radiocarbon laboratories, based only on counting statistics, are underestimates of the "true" variability, the laboratory uncertainties were increased under the following assumptions: 1) the additional sources of variance are independent of the Poisson error of the activity measurement; 2) this added variance is of approximately the same magnitude for samples of similar age; 3) these "extra" components increase with the age of the sample, as demonstrated by the poorer reproducibility of radiocarbon dates for older samples (Currie and Polach, 1980; Pearson et al, 1977; Clark, 1975; Curre, 1972). Consequently, the "counting" variance was increased by an additive term which was allowed to be a slowly increasing function of the age of the sample, hence: 106
Wi
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C
1
= +
40+
xl 150
This has the effect of increasing the smallest error to approximately 60 years for samples less than 1000 years old, and to approximately 115 years for samples with ages greater than 6000 years. These figures compare favorably with the error estimates of Otlet et al (1980), viz: 50 years for samples less than 5000 years and 100 years for samples less than 10,000 years old, and the estimates of Clark (1975), viz: 50 years for samples less than 3000 years and 95 years for samples with ages greater than 3000 years. Finally, the weighted, scaled radiocarbon ages are least squares regressed against their calendric (dendrochronologic) ages using a polynominal basis to obtain the long period trend curve. Polynominals were chosen since 1) a sample's radiocarbon age is, to first order, linearly related to its chronologic age, and 2) though the difference between a sample's uncalibrated age and its true age is bounded, and described reasonably well by a sine function (Damon, Long, and Wallick, 1972; Houtermans, 1971), a polynomial fit is better. With Fisher's F-test as a criterion, the "best fit" was determined to be a polynominal of order six. Because of its low order, this function is insensitive to short-period variations in the 14C inventory and, for the most part, reflects variations resulting from changes in the earth's magnet-
107 Calibration of Radiocarbon Dates is field. (See, eg, Sternberg and Damon, 1979; Lingenfelter and Ramaty, 1970; Damon, 1970; Bucha, 1970; Lal and Venkatavaradan, 1970; Suess, 1970b.) This function and the data are plotted in Figure 1. The second step involves a piecewise Fourier analysis of the residuals around the polynominal regression. A piecewise regression, ie, one that divides the data into a number of similar intervals instead of considering the data set as a whole, was adopted because of several distinctive features observed in the variations of atmospheric 14C. Such characteristic changes are represented by the variations in 14C concentration occurring during the Sporer, Maunder, and Wolf minima (Stuiver and Quay, 1980a; 1980b; Damon, Long and Grey, 1966); by those occurring in the sixth millennium BP (de Jong, Mook, and Becker, 1979; de Jong and Mook, 1980), and by the peaks at 200 years, 150 years, etc, observed in the power spectra of Fourier analyses performed by various investigators (Nef tel, Oeschger, and Suess, 1981; Suess, 1980; Lazear, Damon, and Sternberg, 1980; Siegenthaler, Heimann, and Oeschger, 1980; Houtermans, 1971). Damon (1977) has noted that although characteristic periods appear in the spectral analyses of atmospheric 14C, their phase relationships are different depending upon the section of the 8000-year record analyzed. With this in mind, it seemed prudent to divide the entire time period into short segments and consider the fluctuations individually in each. Consequently, the calendric time scale was divided into 28 shingles, each 500 years long, and each overlapping the previous and next shingle by 250 years (50% overlap each end, 100% overlap for the entire shingle). Two Fourier analyses were carried out to a minimum period of 65 or 110 years, depending on the number of measurements in the shingle. The minimum periods were chosen with consideration of the attenuation factors predicted by various models for changes in the atmospheric 14( activity resulting from changes of various durations in the production-forcing function (Oeschger et al, 1975; Houtermans, 1966). Such models predict attenuation factors on the order of 25 times for variations in production lasting less than 100 years. The result of these procedures is shown in Figure 2. Two analyses were performed in order to assess the effects of outlying points on the calibration function. The first analysis used the unmodified data base as described in the section on data, whereas the second analysis used a "winsorized" data set in which the residuals used for winsorization were taken with respect to the function calculated on the unmodified data. "Winsorization" is a process which reduces the effect of a few aberrant measurements by limiting the effect on the mean of a single outlying point to less than '2.56s/n, where s is the standard error estimated from the fourth quintile of the variance of the data, and n is the number of points in the interval. Winsorization, as employed here, is described elsewhere (Dixon, 1960). Winsorization was used instead of a simple rejection of "outlying" points for the following reasons: 1) the maximum rate of change of the 14C concentration is not certain, and although it appears that changes of the order of a few per mil per year seem to be
Jeffrey Klein,
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109 Calibration of Radiocarbon Dates the rule (Stuiver and Quay, 1980b; Burchuladze et al, 1980; Lerman, 1970a; 1970b; Lerman et al, 1969; Lerman, Mook, and Vogel, 1967), it seemed preferable not to establish an arbitrary criterion for the rejection of suspect measurements, and 2) in the assessment of the "true" errors associated with radiocarbon dates, the rejection of measurements with large residuals furthers the practice of underestimating the scatter in the data. Another problem is caused by unequal residuals at the ends of the regression intervals (endpoint effects) and this was eliminated by using a cosine weighted average of the overlapping functions. This weight is equal to one in the center of the interval and zero at the ends, producing a final calibration function that is both continuous and differentiable. The combined uncertainty of the calibration and the "true" uncertainty of the data are estimated by averaging the residuals of the data around the final calibration function, using the following formula: 11
{(yi
- Y1)2
0-21}
I (n
-
a)
shingle
where the y(i) are winsorized, but the a-(i) are the unmodified laboratory estimates of the measurement uncertainty, and n is the number of measurements in the 500-year interval. The assumption is that Var(y
- y) = Var(y) - Var(y),
which is the natural decomposition, assuming the independence of
y
Fig 1. The composite "workshop data set" is plotted against the 6th order polynominal regressed on the logarithmically scaled data. Calendric age minus conventional radiocarbon age is the ordinate; the calendric age is the abscissa. Positive values represent radiocarbon ages that are too young (too recent) and, consequently, atmospheric concentrations were greater than that of the standard atmosphere of 1890. Laboratories are identified by the following symbols: p = Arizona; Q = Pennsylvania; = Yale; + = Uppsala. Error bars are laboratory = La Jolla; X = Groningen; estimates of uncertainties calculated from counting statistics. The equation of the trend line in logarithmically compressed cordinates is:
)
an x
1
11=0
where
log10(X;) + $, the dendrochronologic age in years before cients are defined by: x1 = x1 is
a
a = a, a,
_ =
a3
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-2.024200 -0.023469
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= -1.249500 = 0.641460 = 0.591000 _ -0.344350
a,; = 1.205700 0.143050 ;13 = The predicted radiocarbon age (in years before y1i is obtained from y;, using the formula:
y' = exp
AD
AD
-a a
1975
and with T,1, = 5730 years);
Jeffrey Klein, J C Lerman, P E Damon, and E K Ralph and y. In fact, this is not the case for linear regression which always leaves residuals correlated with the original data, but this correlation has little effect on the value of this procedure in determining the magnitude of the combined uncertainty of the calibration and the true measure110
ment variability. Finally, the calibration tables were derived from the composite calibration function and the combined error of the calibration and the quoted error of the radiocarbon date being calibrated. This was done by adding together the variance of the calibration (which includes not only the error of the calibration proper, but also an estimate of the nonPoisson error associated with a typical radiocarbon date) and the variance
CALIBRATED DATE (RD-BC) Fig 2. Graphic representation of the period covered by the calibration tables. The ordinate is the conventional radiocarbon age in years BP (1950 used as origin, ages calculated using the 5568-year half-life); the abscissa is the calendric date in years AD-BC. The same data set as in Figure 1 is plotted, but the data here have been winsorized as described in the text. The function includes both the trend analysis and the Fourier analysis of the residuals around the trend. If conventional radiocarbon years were equivalent to calendric years, all the data would fall on the diagonal line; that they do not is readily apparent. The maximum deviations between uncalibrated conventional radiocarbon dates and calendric dates occur ca 5200 Be.
111 Calibration of Radiocarbon Dates of the particular date. The square root of this "total" variance was added to and subtracted from the composite calibration function, producing an uncertainty band in '4C activity representative of the 95% confidence interval for a single determination of the 1`1C activity in a sample of given age. This was converted to an uncertainty interval in calibrated '4C age age by determining the range of calendric dates for which the post-industrial the was consistent (see Figure 3). With the exception of period, multiple calibration intervals were found to be statistically unjustifiable. Consequently, after combining the variances associated with the calibration and those associated with an individual date, the bound-
Fig 3A.
CALIBRATED DATE (RD-BC)
of 20, 100, Fig 3A-G. Calibration limits (monotonic) for radiocarbon uncertainties include The data are the same as in Figure 2. The error bands systematic differences both the error of the calibration and an estimate of the possible between laboratories. primarily for The 90% confidence intervals plotted in these graphs are intended shorter than those obusers with multiple dates and will provide calibration intervals date, first locate the radiocarbon age tained from the tables. To calibrate a radiocarbon draw a horizontal line (parallel to the (BP 1950) on the ordinate (vertical axis), then the x-axis of the interabscissa) through the calibration curves. The projection onto gives the calibrated sections of this line with the "curves" of appropriate uncertainty years. range of the date. Note that each graph spans 1400 radiocarbon 200, and. 300 years.
Jeffrey Klein, J C Lerman, P E Damon, and E K Ralph
112
ing functions were made monotonic in calendric age before the calibration interval was determined. In the final table, separate intervals are provided for radiocarbon uncertainties of 20, 50, 100, 150, 200, and 300 years. The table represents the 95% confidence interval for the calibrated date and covers the range from 7240 to 10 BP (radiocarbon years). If we assume that the source of the non-counting error is independent of the counting error and similar for samples of similar age, then the procedure described above properly accounts for this error as well. For samples less than 1000 years BP (radiocarbon) supplementary tables are provided following the main tables. Asterisks in the main table indicate dates for which multiple intervals exist (see Figure 4). The intervals in the main table represent the extremes in range of the multiple intervals in the supplementary tables.
1200
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RD 500
RD 700
CALIBRATED DATE (AD-BC) Fig 3B.
I
RD 900
Calibration of Radiocarbon Dates
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INSTRUCTIONS FOR USING CALIBRATION TABLES
The tables on the following pages are to be employed in the calibration of single radiocarbon dates. One enters the tables with a radiocarbon age (years BP, 5568-year, "Libby," half-life) and uncertainty, and leaves with a 95% confidence interval containing the "true", calendric date. The radiocarbon age, rounded to the nearest 10 years and calcu-
lated using the Libby half-life, determines the row in which the calibrated age is to be found; the uncertainty determines the columns. All dates within the table have been rounded to the nearest five years. Each radiocarbon age is calibrated to a single calendric range for ages greater than 1000 years, though multiple dates are possible for younger samples. Radiocarbon samples with uncertainties between the tabulated values should have their uncertainties rounded to the nearest tabulated value (see table footnote). Hence, a sample with a date of 1960 BP ± 30 would have a calibrated interval of 145 BC to AD 210, whereas 1960 BP ± 40 would range from 155 BC to AD 215. It will normally not be necessary to 2400
2600
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LRLIBFRTED DATE (AD-BC] Fig 3C.
I
500BC
I
I
3008C
Lerman, P E Damon, and E K Ralph interpolate between tabulated ages, as rounding dates to the nearest five years does not significantly affect the calendric interval obtained. Negative values in the body of the table represent BC dates; positive, All dates; and -1/1 represents the transition year between 1 BC and AD 1 (omitted in the widely-adopted chronology of Dionysius Exiguus (ca 525)). Occasionally, there are large "jumps" in the length of the calibration intervals as read from the table, eg, between 1920 and 1930 13p ± 20 or between 1770 and 1780 BP ± 150 years. These are caused by "flat" regions in the calibration, ie, periods when the 14C in the atmosphere has decreased at a rate greater than 1.2 per mil per 10 years, allowing multiple calendric ages for a single 1}C activity. In other calibrations, these periods have often been handled by assigning several calendric dates to a single radiocarbon age. However, as described previously, the ability to distinguish these as separate periods vanishes when the uncertainties of the calibration and radiocarbon activity measurement are considered. Reference to the calibration graphs should clarify this. 114
Jeffrey Klein,
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CRLIBRRTED DATE [AD-BC) Fig 3D.
Calibration of Radiocarbon Dates
115
CALIBRATION INTERVAL FOR SAMPLES WITH UNCERTAINTIES GREATER THAN 300 YEARS
The following procedure should be employed in calibrating ages of samples with radiocarbon uncertainties greater than 300 years. First, 60 years should be subtracted from the uncertainty of the date to be calibrated. This is to remove the uncertainty of the calibration, which is automatically added into the range in the tables. Then, the resultant uncertainty should be added to and subtracted from the radiocarbon age of the sample, producing two ages which are looked up in the calibration table, under the columns headed by sigma=20 years. The calibration interval is formed from the extremes of the intervals obtained from the table. That is, the lower limit of the interval [older limit] is equal to the lower limit of the calibration interval for the radiocarbon age plus the modified uncertainty. Similarly, the upper limit [younger limit] is the upper limit of the calibration range for the radiocarbon age minus the modified uncertainty. As an example, consider the calibration of 3200 ± 400 years. First, subtract 60 years from 400 to obtain 340 years, which,
CALIBRATED DATE (AD-BC) Fig 3E.
116
Jeffrey Klein, J C Lerman, P E Damon, and E K Ralph
alternately added to and subtracted from the sample's radiocarbon age produces 3540 and 2860, respectively. Looking up the appropriate limits for these two ages, the interval 2110 to 875 BC is obtained. CALIBRATION OF DATES BEYOND TABULATED VALUES
At this time, the only data set of sufficient quality to provide retrospective assessment of atmospheric 1C to a precision suitable for calibration consists of measurements on wood. This is largely because of the stringent requirements for a sample suitable for this purpose. The sample must 1) be independently datable, 2) contain carbon that is reliably associated with atmospheric 1C at the date of the sample formation, and 3) contain sufficient quantities of carbon for an accurate activity measurement. Beyond the existing range of dendrochronologically dated wood, we must rely either on samples of inferior quality (shorter or less certain chronology, or of smaller size, frequently containing too little carbon to
CALIBRATED DATE (AD-BC) Fig 3F.
Calibration of Radiocarbon Dates
117
obtain an accurate date), or on "secondary" sources that estimate the constancy of cosmic rays from the measurements of other radionuclides, or from the inferred strength of the earth's magnetic field from archaeomagnetism. The consensus of these sources suggests that the cosmic ray flux reaching the earth and producing 14C has probably remained constant to within ±10% over the past 50,000 years or more (Vogel, 1980; Barbetti, 1980; Forman and Shaeffer, 1980; Stuiver, 1971). A 1O0/ uncertainty in a radiocarbon concentration represents an 800-year uncertainty in age, regardless of the age of the sample. Consequently, the current "best estimate" of the date of a sample older than 8000 years BP is obtained by assuming a constant atmospheric concentration of the 14C, and using the 5730 half-life to calculate the date. An uncertainty of 1000 years, or the measurement uncertainty quoted by the laboratory, whichever is larger, would constitute a reasonable estimate of the uncertainty for the calendric age of the sample.
LRLIBPRTED DATE (RD-BC) Fig 3G.
118
Jeffrey Klein,
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CONCLUDING REMARKS
It is the intent of the participants of the Workshop that this should be the first in a series of "consensus" calibrations, updated as warranted by improvements in the data base. At present, 1132 measurements of 14C activity have been made on samples of bristlecone pine, the maximum age of which is 8000 years BP. There are 60 samples of wood currently being dated by the radiocarbon laboratories at the Universities of Arizona, California at La Jolla, Pennsylvania, and Washington which will extend the calibration another 550 years. An additional piece of wood, containing 500 rings, is still undatable dendrochronologically but from preliminary radiocarbon measurements appears to be approximately 9000 to 10,000 years old (Ferguson and Graybill, 1981). Another piece of wood, containing only 200 rings, also antedates the present master chronology.
CALIBRATED DATE (AD-BC) Fig 4. The first 1400 radiocarbon years. Similar to the graph in Figure 3A, here, however, the calibration function is not monotonic, and corresponds to the supplementary tables for the most recent 1000 years. Note that for several ages, multiple calendric intervals are possible for a single radiocarbon age.
Calibration of Radiocarbon Dates
119
Perhaps within the next few years, these pieces will be linked with the present 8681-year chronology extending it to beyond 11,000 years ago. Still other chronologies are being developed both in this country and in Europe. The University of Washington has made activity measurements on nearly 2000 years of Douglas fir (Stuiver and Quay, 1980a; Stuiver and Quay, 1981). A second bristlecone chronology, 3200 years long, has been established on wood found in Nevada (Graybill, pers commun, 1982). Several floating chronologies are being developed in Europe (Becker, 1979; 1980; Beer et al, 1979; Lambert and Orcel, 1979; Pilcher et al, 1977) and it is likely that within the next few years it will be possible to connect them with existing recent chronologies. When this is done, they will be valuable in checking and reinforcing the USA chronologies. Even now, they are of some value after their age has been fixed using "wiggle matching" (see eg, Clark and Sowray, 1973) because these data sets are of high quality and their combined use (although not done in this work) with the calibration data set strengthens and reduces the errors of the current calibration. ACKNOWLEDGMENTS
Special thanks are due j W Tukey and R M Clark for their many suggestions that have resulted in significant improvements in the algorithms originally presented to the Workshop. Thanks are due as well to C W Ferguson and the directors and staffs of the radiocarbon laboratories responsible for the activity measurements on these known-age samples, without which this work would not have been possible. We would also like to acknowledge the assistance of the operations staff at the University of Arizona's Computer Center, especially Jackie Dombrowski and Barry Shaede for their unfailing help, particularly during the preparations of the graphs presented here. The patience of those who have waited for the final publication of this calibration, even delaying their own work in some cases, also should not be forgotten. And finally, we would like to thank the National Science Foundation for their support of this publication through their Grant BNS-8022250, and for their support, since 1956, of the bristlecone-pine project, under the direction of C W Ferguson, through various grants, most recently EAR 78-04436 and EAR-8018687 and for their support of the USA laboratories involved in calibration-related research. The US Department of Energy should also be acknowledged for their recent support of the bristlecone-pine project with Contracts EE-78-A-28-3274 and DE-AC02-8 1EV10680. REFERENCES
Barbetti, Mike, 1980, Geomagnetic strength over the last 50,060 years and changes in atmospheric 14C concentration: emerging trends, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 192-199. Becker, Bernd, 1979, Holocene tree ring series from southern central Europe for archaeological dating, radiocarbon calibration, and stable isotope analysis, in Berger, Rainer and Suess, H E, eds, Radiocarbon dating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, p 554-565. 1980, Tree-ring dating and radiocarbon calibration in south-central Europe, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 219-226.
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C
Lerman, P E Damon, and E K Ralph
Beer, J, Giertz, V, Moll, M, Oeschger, Hans, Riesen, T, and Strahm, C, 1979, The contribution of the Swiss lake-dwellings to the calibration of radiocarbon dates, in Berger, Rainer and Suess, H E, eds, Radiocarbon dating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, p 566-584. Bucha, V, 1970, Influence of the earth's magnetic field on radiocarbon dating, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 501-511. Burchuladze, A A, Pagava, S V, Povinec, P, 'rogonidze, G I, and Usacev, S, 1980, Radiocarbon variations with the ,1-year solar cycle during the last century: Nature, v 287, p 320-322. Clark, R M, 1975, A calibration curve for radiocarbon dates: Antiquity, v 49, p 251-266. 1979, Calibration, cross-validation and carbon-14, I: Royal Statistical Soc Jour, ser A, v 142, no. 1, p 47-62. 1980, Calibration, cross-validation and carbon-14, II: Royal Statistical Soc Jour, ser A, v 143, no. 2, p 177-194. Clark, R M and Renfrew, Colin, 1972, A statistical approach to the calibration of floating tree-ring chronologies using radiocarbon dates: Archaeometry, v 14, p 5-19. Clark, R M and Sowray, A. 1973, Further statistical methods for the calibration of floating tree-ring chronologies: Archaeometry, v 15, no. 2, p 255-266. Currie, L A, 1972, The evaluation of radiocarbon measurements and inherent statistical limitations in age resolution, in Rafter, T A and Grant-Taylor, T, eds, Internatl radiocarbon dating conf, 8th, Proc: Wellington, Royal Soc New Zealand, p 598-611. Currie, L A and Polach, H A, 1980, Exploratory analysis of the international radiocarbon cross-calibration data: consensus values and interlaboratory error, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 3, p 933-935. Damon, P E, 1970, Climatic versus magnetic perturbation of the atmospheric C-14 reservoir, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 571-593. 1977, Solar induced variations of energetic particles at one AU, in White, 0 R, ed, The solar output and its variation: Boulder, Colorado, Colorado Assoc Univ Press, p 429-448. Damon, P E, Ferguson, C W, Long, Austin, and Wallick, E I, 1974, Dendrochronologic calibration of the radiocarbon time scale: Am Antiquity, v 39, p 350-366. Damon, P E, Lerman, J C, and Long, Austin, 1978, Temporal fluctuations of atmospheric 14C- causal factors and implications: Ann Rev Earth Planetary Sci, 457-494. Damon, P E, Lerman, J C, Long, Austin, Bannister, B. Klein, Jeffrey, and pLinick, T W, 1980, Report on the workshop on the calibration of the radiocarbon dating time scale, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 3, p 947-949. Damon, P E, Long, Austin, and Grey, D C, 1966, Fluctuation of atmospheric C'4 during the last six millennia: Jour Geophys Research, v 71, no. 4, p 1055-1063. Damon, P E, Long, Austin, and Wallick, E 1, 1972, Dendrochronologic calibration of the carbon-14 time scale, in Rafter, T A and Grant-Taylor, T, eds, Internatl radiocarbon dating conf, 8th, Proc: Wellington, Royal Soc New Zealand, p 44-59. Dionysius Exiguus, ca AD 525, see Bickerman, E J, 1968, Chronology of the ancient world: Ithaca, New York, Cornell Univ Press. Dixon, W J, 1960, Simplified estimation from censored normal samples: Ann Math Statistics, v 31, p 385-391. Ferguson, C W, 1969, A 7104-year annual tree-ring chronology for bristlecone pine, Pinus aristata, from the White Mountains, California: Tree Ring Bull, v 29, no. 3-4, p 1-29. 1970, Dendrochronology of bristlecone pine, Pious aristata: establishment of a 7484-year chronology in the White Mountains of eastern-central California, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 237-245. 1972, Dendrochronology of bristlecone pine prior to 4000 Be, in Rafter, T A and Grant-Taylor, T, eds, Internatl radiocarbon dating conf, 8th, Proc: Wellington, Royal Soc New Zealand, p Al-A10. Ferguson, C W and Graybill, D A, 1981, Dendrochronology of bristlecone pine: Terminal rept NSF Grant EAR-78-04436 & DOE no. EE-78-A-28-3274. Forman, M A and Schaeffer, 0 A, 1980, Cosmic ray intensity over long time scales:
Preprint.
Grey, D C and Damon, P E, 1970, Sunspots and radiocarbon dating in the Middle Ages, in Berger, Rainer, ed, Scientific methods in medieval archaeology: Berkeley, Univ California Press, p 167-182.
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Houtermanns, J C, 1966, On the quantitative relationship between geophysical parameters and the natural radiocarbon inventory: Zeitschr Physik, v 193, p 1-12. 1971, Geophysical interpretations of the bristlecone radiocarbon measurements using the method of Fourier analysis of unequally spaced data Doctoral dissert, Univ Bern. Jong, A F M de and Mook, W G, 1980, Medium-term atmospheric 14C variations, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 267-272. Jong, A F M de, Mook, W G, and Becker, Bernd, 1979, Confirmation of the Suess wiggles: 3200-3700 BC: Nature, v 280, p 48-49. Kleitt, Jeffrey, Lerman, J C, Damon, P E, and Linick, I' W, 1980, Radiocarbon concentration in the atmosphere: 8000-year record of variations in tree rings: first results of a USA workshop, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 3, p 950-961. Lal, Devendra and Venkatavaradan, V S, 1970, Analysis of the causes of C-14 variations in the atmosphere, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 549-569. Lambert, G and Orcel, C, 1979, Dendrochronology of Neolithic settlements in western Switzerland: new possibility for prehistoric calibration, in Berger, Rainer and Suess, H E, eds, Radiocarbon dating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, p 585-590. Lazear, G, Damon, P E, and Sternberg, R S. 1980, The concept of do gain in modeling secular variations in atmospheric 14C, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 318-327. Lerman, J C, 1970a, General discussion of magnitudes of 14C/12C variations (discussion), in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & ,Sons, p 121-126. 1970b, General discussion of the causes of 14Cf 4-C variations (discussion), in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 327-333. Lerman, J C, Mook, W G, and Vogel, J C, 1967, Effect of the Tunguska meteor and sunspots on radiocarbon in tree rings: Nature, v 216, p 990-991. 1970, C-14 in tree rings from different localities, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 275-301. Lerman, J C, Mook, W G, Vogel, J C, and Waard, H de, 1969, Carbon-14 in Patagonian tree rings: Science, v 165, p 1123-1125. Lingenfelter, R E and Ramaty, R, 1970, Astrophysical and geophysical variations in 14C production, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 513-537. McKerrell, H, 1975, Correction procedures for C-14 dates, in Watkins, T, ed, Radiocarbon: calibration and prehistory: Edinburgh, Edinburgh Univ Press, p 47-100. Michael, H N and Klein, Jeffrey, 1979, An international calibration for radiocarbon dates: MASCA Jour, v 1, no. 2, p 56-57. Michael, H N and Ralph, E K, 1972, Discussion of radiocarbon dates obtained from precisely dated sequoia and bristlecone pine samples, in Rafter, T A and GrantTaylor, T, eds, Internatl radiocarbon dating conf, 8th, Proc: Wellington, Royal Soc New Zealand, p 28-43. Neftel, Albrecht, Oeschger, Hans, and Suess, H E, 1981, Secular non-random variations of cosmogenic carbon-14 in the terrestrial atmosphere: Earth and Planetary Sci Letters, v 56, p 127-147. Oeschger, Hans, Siegenthaler, Ulrich, Schotterer, Ulrich and Gugelmann, A, 1975, A box diffusion model to study the carbon dioxide exchange in nature: Tellus, v 27,
-
p 168-192. Otlet, R L, Walker, A J, Hewson, A D, and Burleigh, Richard, 1980, '4C interlabora tory comparisons in the UK: experiment design, preparation and preliminary results, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 3, p 936-946. Pearson, G W, Pilcher, J R, Baillie, M G L, and Hillam, J, 1977, Absolute radiocarbon dating using a low altitude European tree-ring calibration: Nature, v 270, p 25-28. Pilcher, J R, Hillam, J, Baillie, M G L, and Pearson, G W, 1977, A long subfossil oak tree-ring chronology from the north of Ireland: New Phytol, v 79, p 713-729. Ralph, E K and Klein, Jeffrey, 1979, Composite computer plots of 74C dates for treering dated bristlecone pines and sequoias, in Berger, Rainer and Suess, H E, eds, Radiocarbon dating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, 545-553.
122
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J
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Ralph, E K and Michael, H N, 1970, MASCA radiocarbon dates for sequoia and bristlecone pine samples, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 619-624. Ralph, E K, Michael, H N, and Han, M C, 1973, Radiocarbon dates and reality: MASCA Newsletter, v 9, p 1-20. Renfrew, Colin and Clark, R M, 1974, Problems of the radiocarbon calendar and its calibration: Archaeometry, v 16, p 5-18. Schulman, E, 1956, Dendroclimatic changes in semiarid America: Tucson, Univ Arizona Press. Siegenthaler, Ulrich, Heimann, Martin, and Oeschger, Hans, 1980, ''C variations caused by changes in the global carbon cycle, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th. Proc: Radiocarbon, v 22, no. 2, p 177-191. Sternberg, R S and Damon, P E, 1979, Sensitivity of radiocarbon fluctuations and inventory to geomagnetic and reservoir parameters, in Berger, Rainer and Suess, H E, eds, Radiocarbon (lating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, p 691-717. Stuiver, Minze, 1971, Evidence for the variation of atmospheric "C content in the Late Quaternary, in Turekian, K K, ed, The late Cenozoic glacial ages: New Haven, Yale Univ Press, p 57-70. Stuiver, Minze and Quay, P 1), 1980a, Changes in atmospheric "C attributed to a variable sun: Science, v 207, p 11-19. 1980b, Patterns of atmospheric 14C changes, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 166-176. 1981, Atmospheric "C changes resulting from fossil fuel COQ, release and cosmic ray flux variability: Earth Planetary Sci Letters, v 53, p 349-362. Stuiver, Minze and Suess, H E, 1966, On the relationship between radiocarbon dates and true ages: Radiocarbon, v 8, p 534-540. Suess, H E, 1967, Bristlecone pine calibration of the radiocarbon time scale from 4100 Be to 1500 Be, in Radioactive dating and methods of low-level counting: Vienna, IAEA, p 143-151. 1970a, Bristlecone pine calibration of the radiocarbon time-scale 5200 BC to the present, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 303-311. 1970b, The three causes of the secular C-14 fluctuations, their amplitudes and time constants, in Olsson, I U, ed, Radiocarbon variations and absolute chronology, Nobel symposium, 12th, Proc: New York, John Wiley & Sons, p 595-604. 1979, A calibration table for conventional radiocarbon dates, in Berger, Rainer and Suess, H E, eds, Radiocarbon dating, Internatl radiocarbon conf, 9th, Proc: Berkeley/Los Angeles, Univ California Press, p 777-785. 1980, The radiocarbon record in tree rings of the last 8000 years, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 200-209. Switsur, V R, 1973, The radiocarbon calendar recalibrated: Antiquity, v 47, p 131-137. Vries, Hessel de, 1958, Variation in the concentration of radiocarbon with time and location on earth: Koninkl Nederlandse Akad Wetensch Proc, ser B, v 61, p 94-102. 1959, Measurement and use of natural radiocarbon, in Abelson, P H, ed, Researches in geochemistry: New York, John Wiley & Sons, p 169-189. Vogel, J C, 1980, Accuracy of the radiocarbon time scale beyond 15,000 BP, in Stuiver, Minze and Kra, Renee, eds, Internatl radiocarbon conf, 10th, Proc: Radiocarbon, v 22, no. 2, p 210-218. Wendland, W M and Donley, D L, 1971, Radiocarbon-calendar age relationship: Earth Planetary Sci Letters, v 2, p 135-139. Willis, E H, Tauber, Henrik and Miinnich, K 0, 1960, Variations in the atmospheric radiocarbon concentration over the past 1300 years: Radiocarbon, v 2, p 1-4.
Calibration of Radiocarbon Dates
TABLE
123
1
SYSTEMATIC DIFFERENCES OBSERVED BETWEEN LABORATORIES
Average deviation from
mean
Laboratory
(%)
+3.0 ± +2.7 ±
Arizona (A) Groningen (GrN) La Jolla (Lj) Pennsylvania (P) Yale (Y)
1.7 1.5
-3.2 ± 1.1 +3.4 ± 2.5 +3.2 ± 2.0
TABLE 2
MAIN CALIBRATION TABLES (P 124)
(See instructions in text and in footnote below) Look up under nearest tabulated value radiocarbon elates with uncertainties between tabulated values, hence:
for sigma =
0-
35 75
3676 - 125 126 - 175 176 - 250 251 - 350 350
look up under:
=
20
= 50 = Q= Q= a= use the procedure described in the text r
* in body of table indicates multiple calibrated ranges exist for these dates. See supplementary tables.
f of rey Klein, J
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