Note: For this type of samples you will probably use a scanner resolution between 1600 and 2400 dpi, though in this case the 600 dpi resolution was good enough.

With these normalized curves we then have curves which look much more similar than the plain ring width curves!

Dendrochronologists use various algorithms for the normalization. Two algorithms are named after their originators, i.e. Baillie/Pilcher and Hollstein. You can turn on these algorithms within the CDendro program, though the default algorithm of CDendro is "home brewed" and named "Proportion of last two years growth" (Prop2Yrs). It is indeed related to the other algorithms but in many cases found to be more efficient, i.e. it is somewhat better for finding the correct match (ref 1).

Proportion of last two years growth normalization: For each ring width value this algorithm calculates a value describing this year's growth as a proportion of the last two years growth.

Think of the total growth during the two last years! How much of that (in percent) grew the very last year?
If your last summer was colder and with less rain than the previous summer, then say 40%.
If your last summer was warm and wet, but the previous cold and dry, say 60%.

An example: We have the ring widths w1, w2, w3, w4, w5 ... where w1 is the most recent (youngest) ring.
Calculate normalized values as: w1/(w1+w2), w2/(w2+w3), w3/(w3+w4) ...

Here, the red and green curves are shifted stepwise from left to right along the black and blue curves. For each step we try to quantify how well the red and black curves fit together at that point.

Let the curves lay over each other. After looking at the mismatch you can  set up a score telling how bad it all looks. Then shift the upper curve (drawn on transparent paper) one step (one year) to the right and set up a second score for this mismatch. While stepping and scoring you suddenly see - if you are lucky - a really good match which will give a high score.

In a book on statistics you will find a description of the correlation coefficient. It can be used as a measure of the covariance between two curves, i.e. how well the curves match each other. We could name it a value showing the quality of the matching.

The correlation coefficient is cumbersome to calculate (its algorithm is described below) so you really need a computer for this type of scoring. The plot shown above is actually showing the correlation coefficients calculated from matching our two normalized sample curves shown above.

A coefficient value of 1 means that both curves follow each other exactly.
A value of -1 means that the curves behaves exactly contrary to each other, e.g. when the one curve goes up, the other goes down. Correlation coefficient values are always within the limits -1 to +1!

Note: When comparing ring width curves, we do the correlation coefficient mathematics on the normalized curves! When you document a best value from such a correlation calculus, you should also document the normalization method used, as the requirements on the level of the coefficient to acertain a dating, differs somewhat with the normalization method used (ref 1).

Of course the curves should overlap when we calculate the correlation coefficient. If we shift the curve of one sample so it hangs out a bit to the right of the other curve, it means that we are testing the probability that this sample is younger than the other sample (if later times are to the right in the diagram). It also means that only a part of one curve overlaps the other curve.

How much do these curves have to overlap to make a meaningful calculation?

My experience is that it should be an overlap of at least 50 years!

Anything less than that will give you just a nice coefficient not the basis for a dating. To make me convinced of a dating, I ask for an overlap of about 80-100 years, a high correlation coefficient value and a big step to the next best value.

TTest values are calculated according to the formula below, where n is the number of overlapping years and c is the correlation coefficient value.

The numbers in black color along the bottom of the diagram show the number of trees involved when calculating the mean ring widths, this is sometimes called "sample depth" or "replication".

The numbers in red color represent correlation coefficients calculated for 20 years long segments (blocks) lagged by 10 years.

The gray bar above these represents the so called "trend" ("Gleichläufigkeit"), which is either true (gray) or false (no color). The trend is true when growth increases or decreases for both of the reference curve and the sample being compared.

Standard deviation curves plotted around the reference curve are of great value when inspecting curve matching visually!

The innermost ring widths of a tree are usually the widest rings of that tree, as a young tree grows faster than an old tree. Also - a tree on good soil grows faster than a tree on poor soil! Just adding together ring widths of a certain year and then dividing by the number of trees, is not a good strategy if we want a mean value that mirrors the growth conditions of that year. Then tree rings from then young trees would dominate the calculations.

Detrending of a ring width curve is the process of modifying that curve to remove the effects of tree aging from the ring width data. It also includes a scaling of the ring width curve to compensate for the effects of rich or poor soil.

In principle, the detrending of a ring width curve is done by dividing each ring width value with the mean ring width value of the whole tree. This calculation makes growth curves from different trees comparable though their mean ring width values differ greatly because of different growth conditions.