This post is kind of long. It is designed to show one (relatively simple but representative) example of the kinds of variation in results that can/will arise from the variation in tree ring analysis procedures, using a tree core sample of very similar age structure to that of the Briffa et al. (2013) Yamal data.
I’ll apply a tree growth model (as in previous posts), with two different tree age/size effects and two different ring response variables, to this data set. The data set is a truncation of the full Briffa et al. Yamal data, one in which I select 1/3 of the years, and 1/3 of the cores in that data set, creating thereby one that is (1/3)^2, i.e. 1/9, the size of it. The cores and years used are a random sample, and the only reason I do this truncation is because my growth model (coded in R) appears to run ~ exponentially slower as the data size increases linearly, and the full Yamal data matrix covers 2770 years and 615 tree cores, really slowing it down. So, I pare it down so I can run a number of different simulations, but on data that has the same critical characteristics as the full Yamal data set, i.e. in the number of trees of different ages, and their timing of occurrence (years of first and last ring).
I could set the program up to systematically explore a whole number of possible variations, including variation in the climatic trend magnitude, shape and direction, various different tree age/size effects, different tweaks of the RCS method, different levels of stochastic variation in growth and climate, and on and on, but for now this more simple exercise will suffice. And bear with me please as I display separate graphs for separate model runs/situations instead of presenting something a little more concise–just don’t have the time for that re-coding right now.
I noted previously that, if one has a perfectly structured tree age sample, that the RCS detrending method works very well in removing the tree age/size effect. The problem is that no data sets ever meet that perfect criterion, and in fact most of those archived at the ITRDB are not even close and will therefore almost completely fail to recognize any long term environmental trend signal present in the tree rings. That “perfect” age structure is one in which every calendar year is very well represented by all possible tree ages/sizes (and conversely, every particular ring age occurs across the full range of calendar years under investigation). As I noted in the two previous posts, the sub-fossil trees used in the Yamal and Polar Urals have the potential for getting the sample closer to that perfect age structure than do the live trees sampled by typical sampling practice at typpical tree ring sites, but that it is nevertheless not clear how far from perfection the sample is, especially for Yamal, where the mean cambial age per year is favorable but the mean year of occurrence of each ring (“cambial”) appears to be much less so.
The bottom line is it’s not clear, apriori, how well these two data sets will return a known environmental signal, and so I want to test that.
So, the truncated data set is over 900 years long, comprised of over 200 cores. Most of the cores are < 200 years long, the mode being around 100-120 years. I impose a monotonic climatic trend that is exponentially increasing with time. I test a situation in which there is no tree/age size effect at all (perhaps mimicking wood density measures, which show a much smaller age/size effect than does ring size (some time later I'll get into the evidence I have for that–it’s not based on anything in the literature, though the literature does confirm the finding, generally)). Then I test a situation in which there is a rather mild unimodal effect (i.e. trees of intermediate age/size grow the fastest). No extreme or unrealistic age/size effects are tested. Moreover, I completely remove variation in other factors that could cause complications. This equates to (1) no inherent variation in inter-tree growth rates (due to genetics, site factors etc.), (2) no random variation in growth whatsoever (thereby completely removing any effects due to small sample size, because there is no stochasticity), and (3) no uncertainty in tree age/size due to missed early rings and/or inaccurate estimates thereof. Lastly, I vary the ring measure used as the response variable: ring width versus ring area, and I vary the stiffness of the smoothing spline that is used to create the RCS “Regional Curve” (very stiff versus very flexible, based on the two (and only two) such conditions that have been used in previously published studies).
The following eight graphs are identical in structure and show estimated chronologies (thinner black lines) graphed along with the true, known environmental values (thicker black lines) resulting from the following full factorial design:
1. No age/size effect, ring widths as response variable, flexible spline
2. No age/size effect, ring widths as response variable, stiff spline
3. No age/size effect, ring area as response variable, flexible spline
4. No age/size effect, ring area as response variable, stiff spline
5. Mild age/size effect, ring widths as response variable, flexible spline
6. Mild age/size effect, ring widths as response variable, stiff spline
7. Mild age/size effect, ring area as response variable, flexible spline
8. Mild age/size effect, ring area as response variable, stiff spline
Note: that the darker line is not a fit to the chronology estimate but rather is the known environmental state variable. Note also that when I say “mild age/size effect”, that is after the purely geometric trend due to increasing tree size has been removed, because the model works on ring areas, not ring widths,and only ring widths are subject to the purely geometrically driven decrease as a function of tree size. The full age/size effect, including the geometric effect, is stronger, showing the typical rapid decline in ring width in early years followed by a leveling off of ring size in later years; this effect is present in all ring width analyses here.
So, a few things are obvious right away. The first is that every single reconstruction is different. The second is that one method is clearly superior to all others (Fig. 3), although all of them have some–often decent to good–ability to capture the long term climatic trend. The biggest problem in most cases shown here differs from what I showed in previous posts on general analytical problems in the field, wherein there was an essentially universal inability for sites with age/size structures typical of actual tree ring sites archived at the ITRDB to return even a semblance of a known environmental trend. Here the greater problem is often the false creation of multi-decadal scale variation that does not in fact exist (remember, there is NO stochasticity in these model runs; everything is deterministic). In one case (Fig. 8), this false variation is really extreme. Note also that the response variable used does make a difference to the results, a fact which is not necessarily intuitive by any means. Note also that which response variable is most accurate depends on other analytical factors; ring widths give consistent results but always overestimate the magnitude of the short time-scale variation, whereas ring areas are sometimes extremely accurate at all scales, but other times highly inaccurate.
Obviously, when tree ring studies are conducted, one desires to get good estimates at all temporal scales, because all are important, even if the emphasis may vary. We want to neither under-estimate the long term trend, nor over-estimate the shorter term variations. Again, this is a limited set of tests here, but they they should illustrate the important points well enough.