Sequence-slotting is a technique for combining, in an optimal fashion, two sequences of data (such as pollen cores) into a single consolidated sequence. Often the aim of such an exercise is not just to see how the cores match, but also to relate the depth scale of one core to the depth scale of the other.
Version 1.6 of my PC-based sequence-slotting program PCSLOT automatically does such matching of depth scales, using a technique known as "Partial Path Length Interpolation". How this works may be illustrated by the test data included with the program (in self-extracting file PCSLT16Z.EXE, available via anonymous ftp from the INQUA file boutique).
In this example, the two sequences are the first 40 levels from well-logs at Bartasovsky No. 1 well (Sequence A) and Kisling No. 5 well (Sequence B) in Kansas. In this article, we ask: how can we relate the depth scale of Sequence B to that of Sequence A?
In this case, the optimal slotting starts off as
A1 A2 A3 A4 A5 B1 A6 A7 .....A14 B3 B4 B5 B6 B7 A15...
Clearly, element B1, the first element in Sequence B, is slotted between A5 and A6, and must therefore be located somewhere between levels 5 and 6 of Sequence A. But where exactly? It is tempting to put B1 at position 5.5, "halfway" between A5 and A6, but we can do better than this. It turns out that the "distance" from A5 to B1 is 2.491, while the total "distance" from A5 to B1 and on to A6 is 4.635. Since the ratio 2.491/4.635 equals 0.537, B1 ought to be located 0.537 of the way between A5 and A6, that is, at position 5.537 on the "A scale". This is the essence of the interpolation technique.
The same argument can be applied for any B, or a sequence of B's, lying between consecutive A's in the combined slotting. For example, there are 5 B's between A14 and A15; applying the same sort of interpolation as for B1, we see that these 5 B's are NOT equally spaced between A14 and A15. The first four are closer to A14 than A15.
PCSLOT Version 1.6 does this interpolation for all B's located in the combined sequence between the first and last element of Sequence A. The output in this case starts off as follows.
ESTIMATED POSITION OF SEQUENCE B
RELATIVE TO SEQUENCE A
(based on Interpolation of Partial Path Length (PPL))
Seq./Level A B as A PPL
A1 1.0 0.000
A2 2.0 4.180
A3 3.0 6.368
A4 4.0 9.263
A5 5.0 15.134
B1 5.537 17.625
A6 6.0 19.769
A7 7.0 21.536
B2 7.758 27.053
A8 8.0 28.814
A9 9.0 29.678
A10 10.0 31.779
A11 11.0 33.962
A12 12.0 36.125
A13 13.0 41.661
A14 14.0 45.890
B3 14.118 48.629
B4 14.189 50.269
B5 14.290 52.622
B6 14.415 55.528
B7 14.638 60.710
A15 15.0 69.114
...... Lines omitted here ......
A39 39.0 351.510
B39 39.212 352.306
B40 39.313 352.682
A40 40.0 ** 355.261
** CPL for optimal slotting
The "B as A" entry of 5.537 for B1 means that B1 should be located at
position 5.537 on the A scale. Similarly, B3 is located at position
14.118 on the A scale, that is, 0.118 of the distance between A14 and
A15.
Hence, if the depths down Sequence A are known, the corresponding depths down Sequence B can be readily computed. Alternatively, if the time scale down Sequence A is known, then the corresponding time scale for Sequence B can also be reconstructed, as well as the resulting sedimentation rates.
The "Partial Path Length" (PPL) is defined as the total "distance" along the optimal combined sequence from the start of that sequence up to and including the current element of the combined sequence. The PPL has the properties that:
Where there is a single B or a block of B's between successive A's, the location of those B's relative to the numbering of elements of sequence A is obtained by linear interpolation of the between-element distances as given by the PPL values. In this way we can distinguish between those B's which happen to be very close to an A and those which are more distant, as judged by the distance measure being used.
Even if the depth scale for neither sequence is known, the PPL can be used to provide a useful visual representation of the relationship between the two sequences. This is done by plotting the Partial Path Length (PPL) versus the "Combined Path Position" (CPP), using different symbols for A's and B's. The CPP gives the position, from 1 to (M+N), of each element in the combined sequence.
In regions where this graph is fairly steep, the slotting is well determined or tightly controlled. This is because the PPL is increasing rapidly, which means the successive points in the combined sequence are relatively distant from one another. Conversely, flat spots in this graph indicate regions where the points are close to one another, and where any changes to the slotting would have a minimal effect.
Figure 1 shows the corresponding "CPP Plot" for these sequences, with elements of Sequence A shown as open circles, and Sequence B as filled circles. This plot clearly shows the blocking or bunching of the A's and B's, as well as the different slopes in different parts of the combined sequence. These slopes indicate that slotting is "loose" up to about position 35 in the combined sequence, then gets much tighter (as shown by the steeper slope) up to position 65, and then is loose again towards the end.
Clark, R. M. 1993. Assessment of sequence-slotting. INQUA - Commission for the Study of the Holocene, Working Group on Data-Handling Methods Newsletter 9:5-10.
Clark, R. M. 1993. A new version of PCSLOT. INQUA - Commission for the Study of the Holocene, Working Group on Data-Handling Methods Newsletter 10:21-22.
Clark, R. M. 1994. Corrections and extensions to PCSLOT. INQUA - Commission for the Study of the Holocene, Working Group on Data-Handling Methods Newsletter 12:21.