On an earlier occasion, I posted up our updated measurement data from Almagre. I’ve been working on this material in preparation for AGU (Dec 14). Today I’m going to show some initial chronology calculations.
In doing these calculations, I’m trying to reconcile exactly to standard methodology while, at the same time, trying to be a little bit reflective about the statistical meaning of their methods (and porting the software to R as much as possible.) I have done runs in Arstan on our data and on Graybill’s co524. I also corresponded with Rob Wilson on this matter. I had trouble reconciling the archived Graybill chronology with Arstan options and Rob kindly identified the precise Arstan option used by Graybll.
In addition to our collection in 2007, there have been two previous measurement data archives for Almagre: Lamarche in 1968 (co071) and Graybill in 1983 (co524). The Graybill archive includes some (but strangely not all) Lamarche cores. According to my best efforts at concordance, Graybill trees with id over 30 were actually collected by Lamarche. Three of our trees (30,33, 47) matched Graybill trees.
For the calculations here, I’ve collated a data set using the “fresh” Graybill measurements, the original (and more complete) Lamarche measurements and, from our update, I’ve excluded the first two sites (Elk Park, Almagre Base) which are not in the same area as the Graybill samples. I don’t believe that the results are particularly sensitive to the exact collation. (I’ve examined some sensitivities but am still doing analysis.)
In a first pass analysis, I’m using all past and present data (strip bark and whole bark) in order to reconcile results to past results as much as possible and will then look at strip bark-whole bark disaggregation.
The resulting data set has 37 trees and 77 cores.
Arstan detrending is done by first trying to fit a “generalized” negative exponential to each core. A “generalized” exponential has the shape:
There are some interesting numerical analysis issues pertaining to this sort of non-linear fit. I can substantially replicate Arstan results by doing these fits using the R function nls and even more conveniently using nlsList (in the nlme package). I can often obtain convergence when Arstan converence failed and I suspect that they’ve set their iteration a little low relative to their tolerance. A second Arstan option is a negative sloping line if a neg exponential fit fails and then a line through the mean.
An odd feature in Arstan detrending – and I don’t think that it matters much in this example, but doesn’t seem to make any sense – is that the “age” of each core is determined individually even though another core may have established an earlier date for the tree, This is shown in the example below (which has the most cores of any tree on the site). I’ve plotted the ring width information from each core together with the Arstan fit. For example, the green core is fitted as though it’s a new tree even though the blue core has shown that the tree was already at least 150 years old and no longer juvenile at the commencement of the green core. In solid black, I’ve shown the negative exponential fit using all available data for the tree. This seems far more rational than trying to treat each core separately. At the end of the day, I don’t suppose that the decision makes much difference, but detrending on a tree basis (rather than a core basis) seems a safer approach, especially when one is worried about strip bark potential problems. Dendros have been increasingly moving towards standardization on a more regional basis, and standardization at a tree (rather than a core basis) is at least consistent with that trend.
After fitting a curve, the standard dendro procedure is to divide the measured ring width by the fitted width to produce a dimensionless ratio. There are occasional discussions about whether to use residuals, as would be far more conventional in mainstream statistics, but ratios are well-established. I’ve used ratio approaches here in order not to vary too many things at the same time.
A second decision in chronology-making is the decision of whether to transform the data to “normalize” it. Ring width data, even after detrending is typically very non-normal and this is the case here. Willis showed a pretty violin plot in the earlier thread and I’ve applied this below in combination with a QQnorm plot to illustrate the distributions. First here is the distribution of “standardized” detrended ring width ratios calculated according to the above procedures. As you can see, this is highly non-normal with a positive skew.
Cook has initiated the use of power transformations to normalize ring width distributions (a technique nearly always used by Rob Wilson in his work) and an excellent idea. I haven’t explored the criteria that they use to select the power transformation index. I experimented with several different transformations starting with k=0.5 and after a couple of attempts used k=0.375. This resulted in the following distribution. Because of the severe non-normality of the Almagre data, I have the impression that some of the variation in the older chronologies is significantly reduced as a result of power transformation reduction of non-normality artefacts.
First here is a plot comparing a chronology using only updated measurements to a chronology using only Graybill 1983 measurements (excluding Lamarche for now.) This shows a couple of things: that our sampling actually did replicate Graybill’s results (r=0.89 for this period) (I’m not sure whether this is biased upward in the crossdating exclusions, biases from which seem possible to me, but are a large study in themselves). For now, we can say that – whatever interpretation one may put on the final chronology itself, there does seem to be data that can be independently recovered.
First here is a plot showing the original Lamarche and Graybill chronologies, together with our extension to 2007 (without a power transformation). Values were at high levels in the late 19th century and again in the 1950s and have declined since the 1950s reaching more or less average levels in the 1990s-2000s.
The most notable feature on the recent portion of this graphic (and perhaps the entire graphic) is the severely reduced ring widths in the 1840s. Steve Mosher has linked to some references reporting severe drought in Colorado in the 1840s http://www.ncdc.noaa.gov/paleo/pubs/woodhouse2002/woodhouse2002.html
and it’s hard to avoid the iview that the reduced growth in Almagre bristlecones in the 1840s isn’t associated with contemporaneous drought throughout the state – a view which is certainly consistent with the impression of our most knowledgeable botanical observers of moisture limitation at the site.
Here is a blow-up of the same chronology for the 1830-2007 period, covering the low-growth 1840s.
Power Transformation Chronologies
Here is a the power-transformation chronology (k=0.375) for the 1830-2007 period. Much of the variation has been damped down and one is left with an impression of rather limited variation other than for extreme events like the 1840s (the 1920s were also low-growth here).
The correlation of the ring width chronology to the HadCRU3 grdicell (annual) is 0. The first graph shows the correlation of the chronology to monthly temperatures at the nearest USHCN station (Chessman adjusted). In addition to the usual barplot showing the correlations to the current and preceding year, I’ve shown correlations of the ring width to the temperatures in the following year. The most “significant” correlation is between ring width and April temperature of the following year – a “teleconnection” that is appealingly Mannian.
“Reconstruction” of Cheesman Reservoir Temperature
The graphic below shows a “reconstruction” of Cheesman Reservoir July-August temperature done in one of the common dendro ways – by variance matching. The r2 of this “reconstruction” is under 0.01 – not that this precludes a Mannian model. After all, this may teleconnect with temperatures in Bali or Beijing or Rio de Janeiro or Antarctica. 2002 and 2003 were warm summers at Chessman Reservoir, but did not result in exceptional growth.