This post is an expansion on some points made to date, relative to the issue of removing non-climatic trends from tree rings, the second of the three issues I’m addressing in the series, as described in part one. If you understand that issue, then you may not need to read this. There are some new graphs here, in order to try to illustrate better the concept of geometric vs biological effects on long term trends. If you haven’t ready any of the previous posts in this series, this one may or may not prove useful.

To summarize the essential point, we have to remove the geometric trend in ring widths that results strictly from geometric considerations, and also any additional biological effects that result from the tree varying the total amount of wood it lays down in each ring. These two effects result from entirely different causes but they both constitute long term trend noise, and so need to be removed.

First, here’s what the purely geometric trend in ring widths looks like, without a biological or climatic effect. It’s just a simple inverse square root function of tree size (y = x^(-0.5)):

The change is steepest near the beginning because relative tree size (i.e. tree size as a function of size at the beginning of any year) is changing the most rapidly there. Clearly, this effect is going to cause problems if not removed, especially when the tree is small. Here’s what this same effect looks like when applied to a group of cores from a site, with steadily increasing, and then decreasing climate trends of about the same magnitude as used in previous posts (3 units), and with some added inter-annual climate variation:

If we remove the geometric effect, by using ring* areas *instead of ring widths those two graphs transform to these two, respectively:

[Note that the previous four graphs are for groups of cores, from trees which start their lives at perfectly spaced intervals throughout the analysis period, and there is a climate trend–the graphing of individual cores would reveal the climatic trend effects more clearly.]

These two climatic trends produce these paleoclimatic estimates for two variants of the ICS and RCS methods respectively, and the Biondi “C” method:

[Here I’ve smoothed the long term trends using loess smoothing, instead of the strictly linear trend lines shown in previous posts. These curves reveal some of the multi-decadal variation in estimates that result from the randomness of the inter-annual (year to year) climate variations. The true, linear climatic trend is also shown by the thin, straight black line however]

All of the previous graphs have lacked a true biological effect. Such effects on ring areas might reasonably look like either of the following (again, for a group of cores, originating at even spacing throughout the time period, and with high inter-annual climate variation):

The first of these biological effects represents a mild unimodality, almost verging on the asymptotic, whereby tree radial growth rates are maximal in mid-life and both rise to, and decline from from that point, gradually. The second shows a stronger unimodal effect, in which growth rate increases rapidly until about age 50, then declines steeply, and then more gradually, toward zero. These two effects will produce the following two paleoclimatic trend estimates, respectively: Mild unimodal effect. Strong unimodal effect.

Results from the ICS methods are included; these haven’t been shown until now. Clearly, the ICS methods have no ability at all to capture long term trends, which is why I haven’t shown them. But that fact, which was first addressed 20 years ago, when the RCS method was re-introduced, hasn’t stopped people from continuing to use ICS methods for trend estimates, right up to the present. The “C” method of Biondi and Qaeden, a relatively new method, is also shown, and it too does very poorly. For the RCS methods, results are generally worse when biological effects are added, especially when these effects are strongly unimodal (third linked pdf graph above), but even without them the trend estimate errors are still large, especially when trends are negative (first pdf graph).