- Interpolation
- Calendar and BP dates
- Radiocarbon calibration
- Reservoir corrections
- Mixed calibration curves
- Calendar dates and Asymmetric dates
- Proportional errors and Factors
- Range calculation
- Combinations and Wiggle Matching
- First and Last Dated Events in a Group
- Offset dates and Age Differences

When integrations or differentiations are carried out they are at the
resolution *r _{c}*.
The details of the interpolation methods (such as methods of rounding
used) have been carefully chosen to give the expected results and
variation from the analytical values are rarely more than a single
year with the standard options.

The files for the calibration curve usually have a different resolution
to the internal storage resolution and so some form of interpolation is
needed.
This can either be linear or a cubic function depending on the setting in
the system options.
The cubic interpolation does not fit a spline function as this is very time
consuming to calculate and can have some undesirable features such as large
excursions between points.
The cubic function used here gives a smooth curve with a continuous first
differential but gives very little overall difference from the linear
interpolation.
The form of the function between two points is simply defined by the four
surrounding points. If *f _{j}* defines the function at

The calibration curve is stored in two arrays one *r _{i}* defining the
radiocarbon age of the tree rings and another

*y _{CAL}* = 1950 -

Thus:

10BP = 1940AD, 11950BP = 10000BC

It should be noted that this does imply a year 0 in the AD/BC sequence which is strictly speaking incorrect. With radiocarbon dates the problem is clearly semantic, with historical evidence it should be borne in mind that age differences across the BC/AD boundary are actually one year larger that they should be. Alternatively negative numbers (BC) should always be taken as the start of the year and positive numbers (AD) as the end. Thus -1 is the start of the first year BC whereas +1 is the end of the year 1AD. The reason for this problem is that in order to keep the internal representation of the numbers consistent it is very difficult to have to deal with a number set which goes from -1 to 1.

In this program the distribution is left normalised to a maximum of 1 rather than the actual probability of any individual year.

**NOTE** This is different to old versions of OxCal (pre 3.2) where
*p _{i}* was simply set to exp(-

See also [Archaeological Considerations]

Solution of this differential equation requires a knowledge of the curve
*R(t)* for all times before t. A linear extrapolation is assumed before
the start of the curve using a gradient estimated from the first half of
the curve *R(t)*. The uncertainties in this are assumed to be ten times larger than
those quoted for the first point in the curve; in practice these assumptions are
unlikely to be significant unless the time constant is very long or you are
considering points close to the start of the calibration curve.

Treatment of the uncertainties is more complicated. If the uncertainties associated with each
point on the calibration curve are assumed to be independent the uncertainties in
the reservoir curve should be smaller. In practice the errors almost certainly
to some extent systematic. They have therefore been treated in exactly the same
way as the concentrations themselves: if *sigma(t)* and *Sigma(t)* are the respective
uncertainties we assume:

If there are also uncertainties in tau the solution of the equations would involve
a double integration which would in practice be very slow. Another algorithm has therefore
been adopted which is to increase the sigma in proportion to the difference between
*R(t)* and *r(t)*. Thus if the uncertainty in tau is *delta _{tau}*:

For the oceans a properly modeled ocean curve should be used
(see Stuiver et al
1998 - marine data). Local
corrections can then be made using a
`Delta_R` correction
term:

See also [Calibration Data]

R

and the proportion of the second curve is *P ± D* then the resultant distribution
is given by:

E

See also [Calibration Data]

*t _{c}* =

*dt _{c}* = -(

Rounding to the nearest *r _{c}* will take place at this stage so you may
notice a slight change in the entered values especially if

Calculation of symmetric probability distributions is simple:

The function used for asymmetric dates is rather more complex:

*p'(t)* = *p(t/f)*

And a proportional error df by using the mapping:

The distribution is then renormalised.

In the program these error factors are normally calculated before each
distribution is reported except in the case of functions such as
`Combine`
which give a resultant distribution when the factor is only applied to the
final result to prevent the systematic errors being reduced in the
combination process.

The probability method (selected for all types of distribution in the
system options) calculates the ranges in a different way (similar to the
method used by van der Plicht 1993).
The elements of the probability distribution array *p _{i}* are sorted by size
and the integral normalised to 1.0.
Starting from the top the array is then integrated until a certain
proportion of the total is achieved
(68.2%, 95.4% or 99.7%) and the level at this point in the distribution
found

If whole ranges are selected from the system options with the probability method a slightly different method is employed in order to generate floruits (see Aitchison et al 1991): the probability distribution is normalised to an integral of 1.0 and then the distribution is integrated from each end until a certain proportion of the curve has been excluded (15.9%, 2.3% or 0.15% from each end); the range defined is then the part of the distribution between these two points.

Integrated distributions (generated by the functions Before and After) define ranges directly from the height of the distribution using the values 0.682, 0.954 and 0.997.

Combinations of probability distributions
(`Combine`)
are simply done by using the Bayesian rules for combinations of
probabilities (see Bayes 1763 and
Doran and Hodgson 1975):
if we have two probability distributions *p _{1}*(t) and

*r(t)* = *p _{1}*(t)

or more generally:

For the purposes of display the maximum of the resultant distributions is always normalised to 1.

If within a group defined for the function
`Combine`
the distributions are given a
`Gap`
*g _{i}* then the combination is performed as:

We can then define a new set of original distributions *p' _{i}* using

*p' _{i}* = r(t+

A very similar method to this is used for wiggle matching using the command
`D_Sequence`,
only difference being the way in which the gap is defined (between each
successive distribution).
A probability distribution *r(t)* is always calculated for the start of the
sequence. This is given by:

and the resultant distributions then calculated using:

In the case of Bayesian wiggle matches and combinations, the program also calculates the chi-squared value for the best fit (ie the highest point on the probability distribution). This is reported in the text log file. For wiggle matching tree ring sequences, where the overall precision can be very high, you use a resolution of one year.

See also [Archaeological Considerations]

and so if a group of events are independent the probability of being after all of them is given by:

This is the distribution (normalised to a maximum of 1) returned by
`After`.
From this a distribution *r'(t)* can be calculated which gives a
probability distribution for the last of the group of events:

*r'(t)* = *d r(t)*/

This is the distribution returned by the request
`Last`
within a phase if MCMC sampling is not needed.

The probabilities of being before a group of events and a distribution for the first event of a phase can be similarly defined.

**WARNING**: These methods assume that the events are entirely independant;
in most cases a much better estimate will be arrived at using MCMC sampling
from a phase which is enclosed within
`Boundary` events.

See also [Archaeological Considerations]

*r(t)* = *p(t-dt)*

If the offset has an error associated with it then *dt+-sigma* the
distribution is given by:

A similar method is used to calculate a probability distribution for
the age difference between two independent distributions (only employed
when MCMC sampling is not necessary) using
`Difference`.

And to shift one distribution by another using
`Shift`:

See also [Archaeological Considerations]