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CALIB makes the conversion from radiocarbon age to calibrated calendar years by calculating the probability distribution of the sample's true age. Graphics and a variety of options are available through the program's menus. The program can be cited by the published description of a previous revision (Stuiver and Reimer, 1993) or the on-line version (Stuiver et al., 2005). References to the calibration dataset used should be cited as found in the program output.

## Introduction

The calculation of the radiocarbon age of the sample assumes that the specific activity of the

^{14}C in the atmospheric CO_{2}has been constant. However, early in the history of radiocarbon dating it was recognized atmospheric^{14}C was not constant (de Vries, 1958). A calibration dataset is necessary to convert conventional radiocarbon ages into calibrated years (cal yr). By measuring the radiocarbon age of tree rings of known or other independently dated samples it is possible to construct calibration datasets. The need for a consensus radiocarbon calibration dataset led to the first internaltionally agreed upon calibration in 1982 (Klein et al., 1982). Since that time the calibration datasets have continued to be extended and improved. The IntCal working group established a set of criteria for calibration datasets (Reimer et al. 2002). The current calibration datasets including IntCal04, Marine04, and SHCal04 have been presented for ratification at the 18th International Radiocarbon Conference and are recommended for general use until further notice (Reimer et al., 2004). These datasets are described in Reimer et al. 2004, Hughen et al. 2004, and McCormac et al. 2004 and summarized in Section 2. The single year dataset UWSY98 (Stuiver et al 1998) is also still in use for short-lived samples 350 14C BP.## Preliminary considerations

### Choosing a calibration dataset

For most non-marine radiocarbon samples from the Northern Hemisphere, the IntCal04 curve is the preferred choice. For short-lived (< 10 year), high precision samples (σ< 30 yr) younger than 350

^{14}C yr BP, the single year calibration curve may be used for higher time resolution comparisons although a moving average of 2 to 3 years is recommended to reduce the noise and hence the number of cal ranges. For Southern Hemisphere samples SHCal04 is available back to 11,000 cal BP (McCormac et al., 2004).For marine samples such as shells, corals, fish etc. the Marine04 curve should be used. Since this dataset represents the "global" ocean, no marine reservoir correction should be made to the sample radiocarbon age prior to calibration, however a regional difference ΔR should be input as discussed in Section 2D. Age calibration of samples composed of both marine and terrestrial carbon is discussed in Section 2E. Post-AD 1950 samples cannot be calibrated with CALIB. A calibration program for post-nuclear testing samples is available at http://www.calib.org

### Adjustments for systematic offset

Ideally, before calibrating a radiocarbon age, the age should be adjusted for the laboratory systematic offset if any. Although international calibration efforts point towards the possibility of laboratory offsets (for summary, see Scott et al., 1998), specific laboratories have not been identified. The adjustment for any systematic offset is not incorporated in this program.

### Lab error multipliers or additional lab variance

The standard deviation reported with a radiocarbon age may be based on count rate statistics only. The user must decide whether to use the quoted laboratory error, or increase the quoted error to account for other sources of variance by either applying a lab error multiplier k or adding variance f² (yr²). Thus the sample standard deviation σ becomes either k·σ or (σ² + f²)

^{½}. The calibration curve sigma σ_{c}is automatically added in both cases to give the total sigma of the radiocarbon age prior to its cal age transformation.A lab error multiplier is based on the overall reproducibility and should be supplied by the individual laboratory. The added variance option can be used by calculating the f value from:

σ

_{t}= (σ_{s}² + f²)^{½}where σ

_{s}is the quoted standard error and σ_{t}is the total error. Of course, &sigma_{t}/σ_{s}equals the error multiplier, which can be entered directly.### Reservoir correction for marine samples (e.g. shells, corals, fish etc.)

Organisms from marine (and lacustrine environments) have been exposed to different levels of

^{14}C than their counterparts in the atmosphere. The marine calibration incorporates a time-dependent global ocean reservoir correction of about 400 years. To accommodate local effects, the difference ΔR in reservoir age of the local region of interest and the model ocean should be determined. Guidelines for the determination of ΔR values are given in Stuiver and Braziunas, 1993. A global database of values is kept at http://www.calib.qub.ac.uk/marine. The marine calibration dataset must be selected for calibrating these samples.Reservoir deficiences R for lacustrine samples are typically determined from the age of the topmost sediments (e.g. Stuiver, 1970), or comparisons between lacustrine and terrestrial samples (e.g. Hutchinson et al., 2004). Sample ages are calibrated with the atmospheric dataset due to comparably rapid exchange rates. The user must subtract the reservoir deficiency R from the sample radiocarbon age in this case.

### Percent of marine carbon

For samples derived from a mixture of marine and terrestrial carbon, such as bones of humans or animals with a mixed marine (fish, mollusk, plankton etc.) and terrestrial (grain, grass, other terrestrial animals etc.) diet, the percent of marine carbon should first be determined from other means, such as ethnohistorical accounts, dental pathology, archaeological evidence, or stable isotope composition (e.g. DeNiro and Epstein, 1978; Ambrose and Norr, 1993; Molto et al., 1997). The ΔR value must be determined as for marine samples above. The "mixed" marine and Northern or Southern atmospheric calibration curve should be selected for calibrating these samples.

### Half-life correction

The accepted half-life of

^{14}C (Libby half-life) for calculating a conventional radiocarbon age is 5568 years (Stuiver and Polach, 1977). If the sample's age was calculated using the half-life of 5730 years, it must be corrected by dividing the 5730 half-life radiocarbon age by 5730/5568 or 1.029. The user must make this correction to the age, if necessary, before using CALIB.### Post-atomic age

Samples with "negative" radiocarbon ages (i.e. samples formed since the mid- 1950's with high initial

^{14}C levels due to nuclear testing^{14}C) cannot be calibrated with this program. This also applies to marine samples when Radiocarbon age minus ΔR is less than 460^{14}C yr BP. A discussion of calbration of post-bomb atmospheric samples is given in Reimer et al. 2004b and the program CALIBomb is available at http://calib.org### Southern Hemisphere correction

There is a separate calibration curve for the Southern Hemisphere atmosphere back to 11,000 cal BP.

### Sample calendar year span

Samples composed of material spanning more than 20 or 30 calendar years are best calibrated with a moving average of the calibration curve. This reduces the detail in the calibration curve, irrelevant in this case, and minimizes the number of ranges and intercepts. Enter a sample age span (i.e. the number of calendar years estimated for sample growth or formation) to use a moving average of the selected calibration curve.

### δ13C correction for isotope fractionation

Conventional radiocarbon ages have been corrected for isotope fractionation by normalizing to = -25‰ PDB or VPDB. CALIB no longer supports the correction for isotope fractionation within the program for the following reason: The δ

^{13}C correction depends on whether the original measurement was a^{14}C/^{12}C ratio (all radiometric and some AMS) or a^{14}C/^{13}C ratio (some AMS systems). An Excel spreadsheet with the formulas for both types of systems is given here. If you don't know which type of measurement was done it is better to contact the laboratory which supplied the radiocarbon date.

## Methods

### Calibrated Probability Distribution Calculation

The probability distribution P(R) of the radiocarbon ages R around the radiocarbon age U is assumed normal with a standard deviation equal to the square root of the total sigma (defined below). Replacing R with the calibration curve g(T), P(R) is defined as

To obtain P(T), the probability distribution along the calendar year axis, the P(R) function is transformed to calendar year dependency by determining g(T) for each calendar year and transferring the corresponding probability portion of the distribution to the T axis.

Probabilities are ranked and summed to find the 68.3% (1 sigma) and 95.4% (2 sigma) confidence intervals and the relative areas under the probability curve for the two intervals calculated.

The total area under the probability curve is normalized to one. For plotting purposes, the probabilities may be "renormalized" so that the maximum probability equals one.

Intercepts with the calibration curve are no longer used because they do not provide a robust indicator of sample calendar age, whereas either the weighted average or the median probability of the probability distribution are more stable estimates (Telford 2003). No single value completely describes the probability distribution function and either the weighted average or median probability may fall where the probability is low. CALIB provides the median probability in the export file, but it is not recommended as a replacement for the cal age ranges or the complete probability distribution.

### Total sigma

Cal age sigma (one sigma range) represents the combined standard deviation in the

^{14}C age, where, depending on whether one has chosen to use a lab error multiplier k or additional lab variance f², the total sigma is given as:one sigma = [(k·σ

_{s})² + σ_{c}²]^{½}or

one sigma = [σ

_{s}² + σ_{c}² + f²]^{½}with σ

_{s}= sample standard deviation and σ_{c}= the curve standard deviation which is interpolated between data points for each intercept with the radiocarbon age R. Marine samples are treated similarly except the user must realize that ΔR, and the uncertainty in ΔR, for each sample are based on its collection location (Stuiver and Braziunas, 1993). The marine total sigma is taken as either:one sigma = [(k·σ

_{s})² + σ_{c}² + (ΔR uncertainty)²]^{½}

or

one sigma = [σ_{s}² + σ_{c}² + (ΔR uncertainty)² + f²]^{½}For "mixed" marine and terrestrial samples of p% marine carbon and (100-p)% terrestrial carbon the total sigma is:

one sigma = [(k·σ

_{s})² + ((1-p/100) · atmospheric σ_{c})²

+ (p/100 · marine σ_{c})²

+ (ΔR uncertainty · p/100)²]^{½}or

one sigma = [(σ

_{s})² + ((1-p/100) · atmospheric σ_{s})² + f²

+ (p/100 · marine σ_{c})²

+ (ΔR uncertainty · p/100)²]^{½}where the covariance between the atmospheric and marine curves has been neglected.

This treatment of ΔR uncertainty assumes that ΔR is independent for each sample. This may result in a slight overestimate of the uncertainty (Jones and Nicholls, 2001). However, given our limited knowledge of the changes in ΔR over time, is not unreasonable.

### Rounding

Cal age results are rounded to the nearest year, which may be too precise in many instances. Users are advised to round results to the nearest 10 years for samples with standard deviations greater than 50 years.

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