To describe the procedure of how each age-model is constructed, the discussion is limited on the age-model of stalagmite BU-4 for simplicity. Fig. 10.8 illustrates the measured ages with their 1-sigma uncertainty. The straight line links the mean ages of each age that has been determined, whereas "mean" refers to the most likely age according to the statistics of the measurement, though other age-depth relations are also possible solution providing that the ages are within the age uncertainties.
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dft (mm)
age (ka BP)
Fig. 10.8: Illustrated is the age-depth relation of stalagmite BU-4. The ages are pic-tured with the corresponding 1-sigma standard uncertainty and a linear interpolation method is applied.
To construct an age-depth relation, the stratigraphic order of the ages must be fulfilled, i.e., starting with the youngest age at the tip of a spleothem, the subsequent age must be older than the youngest age and the next age must be older than the forgoing. This requirement must be fulfilled by all ages along the growth axis. However, if the domain of values for ages is defined by the 1-sigma range of each individual age along the growth axis and the 1-sigma range of neighbouring ages are overlapping, there are certain age-depth relations possible at which the stratigraphic order of the age-depth relation is not fulfilled; i.e., the age-depth relation has age inversions. Accordingly, a routine must be developed to handle the age inversion of this certain age-depth relation.
10.3.1.1 Routines for age inversions
The first two ages:
To test each age-depth relation for age inversions, the youngest age is used as a tie-point. The subsequent 2nd age of the age-depth relation is than compared to the 1st age.
10. Principal component analysis in speleothem science
(Scenario I ): In the case that the 2nd age of the age-depth relation is younger or equal than the 1st age it is not used for the constructed age-depth relations - "constructed"
age-depth relations shall be defined as the original age-depth relation, that has been tested and corrected for age inversions, respectively. The same procedure is is applied to the 3rd age. (Scenario II ): If the 2nd age is older versus the 1st age it is compared with the 2nd’s following age, the 3rd age. This is due to the reason, that the 2nd age can be older than the 3rd age in case that the 1-sigma ranges of the 2nd and 3rd age are overlapping. Thus, an age-inversion is possible. For this routine the two subsequent ages of the 2nd age, i.e., the 3rd and the 4th age are used, to distinguish between the cases, whether the 2nd and 3rd age are in stratigraphic order or not, and, in case not, if the 2nd age is too old or the 3rd age too young - for the last two cases the 4th age is used. (Scenario IIa): In the case that the 3rd age is older than the 2nd age, i.e., the stratigraphic order is fulfilled, the 2nd age is used for the constructed age-depth relation - note that for this case the 1st and 2nd age are in stratigraphic order. If the stratigraphic of the 2nd and 3rd ages is not fulfilled the 1st and 3rd age are compared. (Scenario IIb): If the 3rd age is older than the 1st age, the 2nd age is not used for the constructed age-depth relation. (Scenario IIc): Is even the 3rd age younger than the 1st age the 4th age is used as a further age reference - this case means technically that the 1-σ range of three succeeding ages are overlapping and the oldest age of the 1-σ range of the 1st age is older than the youngest age of the 1-σ range of the 3rd age.
If the 4th age is older than the 2nd age, the 2nd age is used for constructing the age-depth relation. For a "chaotic" age-depth relation, where the 1-σ ranges of the first four ages are overlapping, it is likely that even the 4th age is younger than the 1st age. The routine is not designed for such complicated age-depth relations, and the original age-depth relation must be checked manually for such complicated age-depth sections.
The third and subsequent ages of the constructed age-depth relation:
When the first two ages of the constructed age-depth relation are determined all fol-lowing ages are determined, by the similar procedure. For the 3rd age of the age-depth relation this means that instead of the 1st age (t1) the 2nd age (t2) of the constructed age-depth relation is the new tie-point. Hence, t1 ⇒t2, t2 ⇒t3, t3 ⇒t4 and t4 ⇒t5. Fig. 10.9 shows the ensemble of age-depth relations based on 1000 MC simulations for stalagmite BU-4. A linear interpolation method has been applied on the constructed age-depth relation.
The universe of the ensemble covers the 1-sigma range of the age-depth relation. At the younger part of the speleothem, where the 1-sigma ranges of the 2nd and 3rd age and the 3rd and 4th age are overlapping, some constructed age-depth relations do not contain the 2nd and 3rd age, respectively. This is due to the reason that an age-inversion has occurred.
10.3.1.2 Interpolation between ages for age-depth modelling
After the constructed age-depth relation is finished, an interpolation method must be applied on it in order to assure that the proxy-depth relation can be translated into a
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