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I calculated for the entire series and the first and second half of the series, the mean and standard deviation (SD) of the tree ring index described above. In addition I plotted the difference series for that index measure for each duplicated core. The tabled and graphed results are shown below along with the R code I used for the calculations.

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The Schweingruber duplicates did not have complete overlap of measured tree rings for some reason that is unknown to me. The series also did not have long lived trees with older tree ring ages as the Yamal series did. Nevertheless, the SDs were more uniform than in the Yamal series and did not vary significantly with tree ring age. What was similar to the Yamal series was that tree rings measured for the same tree and same year had hugely varying TRW. The graphs presented below show that not only do the differences vary from year to year but also trend up and down over extended portions of the entire difference series, not unlike what we expect to see in the chronology.

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Based on my Yamal and Schweingruber analyses I have to have answered the following questions before taking seriously any conclusions coming out these TR chronologies:

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Do dendroclimatologists show and discuss the results of replicated tree cores?

Do dendrochronologists pay proper attention, or at least attempt, to establish CIs for their chronologies?

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]]>download.file(“http://data.climateaudit.org/data/tree/russ035w.rwl.tab”,”temp.rwl”,mode=”wb”);

load(“temp.rwl”);dim(tree) # 3872 4

Fact=factor(tree$id)

levels(Fact)

[1] “878011” “878012” “878021” “878022” “878031” “878041” “878042” “878051”

[9] “878052” “878061” “878062” “878071” “878072” “878081” “878091” “878092”

[17] “878101” “878102” “878121” “878122” “878131” “878132” “878151” “878152”

[25] “878161” “878162” “878171” “878172” “878181” “878182” “878191” “878192”

[33] “878201” “878202”S878011= tree[grep(“878011″,tree[,1]),]

Merge=merge(S878011, S878012, by.x=”year”, by.y= “year”)

Mean=(Merge[,4]+Merge[,7])/2

Diff=Merge[,4]-Merge[,7]

Index=Diff/Mean

Len=length(Index)

sd(Index,na.rm=TRUE)

mean(Index, na.rm=TRUE)

sd(Index[1:Len/2], na.rm = TRUE)

sd(Index[(Len/2+1):Len], na.rm = TRUE)

mean(Index[1:Len/2], na.rm = TRUE)

mean(Index[(Len/2+1):Len], na.rm = TRUE)

range(Merge[,1])

range(Merge[,3])

Min=min(Merge[,1])

TS=ts(Index, start=Min)

ts.plot(TS, main=”Series by Differencing Duplicate Cores

+ from the Schweingruber Chronology”, ylab=”(TRW1-TRW2)/((TRW1+TRW2)/2)”, xlab=”Years”)

Curiously though, the average of the annual COV’s that I get from Excel for the entire L00131-4 series is 0.25 (this is the average of [each year’s SD divided by that years RW]) – Sadly I cannot yet understand enough R to work out whether I am doing it differently to you -it could also be one of those Excel ‘dragging down’ slip-ups that I get quite often ðŸ™‚ ]]>

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The first table below shows those results and the expected differences of tree ring widths with tree ring age. The second table shows the standard deviations and means for the difference series using the measured tree ring widths (in place of deltas) between replicated samples from the same tree.

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From the tree ring width differences one can see that the variations are influenced by the younger tree ring ages being bigger and tending to give a larger standard variation, but not entirely as the smaller older tree rings in some of the trees from the second half of the difference series had standard deviations larger than the younger larger rings from the first half of the difference series.

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By the nature of the delta calculation one would expect if the error in measurement were the same regardless of tree ring width (age) that the variation would be higher for the smaller (and older tree ring) widths because one is dividing the same difference by a smaller number. Just by looking at all the tree ring width data the foregoing situation would appear to explain some of the older versus younger tree ring differences in variation but not all of it. In fact some of the effects for young versus old tree rings would appear to be unique to some of the trees with replicated samples.

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To get a better feel for the affect of average tree ring width on the variation between old and young tree ring ages and without including the RCS expected growth curve, I simply divided all the standard deviations (SD) for the entire, first half of and second half of series by the average tree ring widths of the replicated cores used. This would be in line with dividing the tree ring widths by the RCS growth curve in calculating deltas used in the Yamal chronology. When that adjustment is made, as shown in the third table below, the same relationships as found when using delta differences are apparent and can be shown by calculation, i.e. the regression of SD divided by TRW versus tree age for the entire series is significant and the differences between the first and second half of the series where SD is divided by the TRW are also significant.

Finally, when one observes the large standard deviations, in the tree ring widths (as noted previously when looking at deltas) of the difference series of replicated cores, relative to the absolute tree ring widths, one has to ponder how meaningful results can be extracted â€“ and particularly with older tree ring ages.

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]]>I looked at the (raw) annual ring widths (in excel) for L00131-4 and the annual SD of the widths seems to increase when the trees put down larger rings – their coefficient of variation looks pretty trendless with time. (though there are some spikes in the series when a ring or two has a width of zero).

Chas, you happened to select a tree that appears to have little trend in SD with tree ring age as noted in the table above and thus I would expect the tree rings widths (before division) to show little trend also. L00311,2,3,4 appears to be atypical of the trees with replicate samples.

I used the differences in deltas because the Yamal chronology uses that measure in its time series plots. You are correct, however, that the RCS growth rate curve remains in play when I take differences. When I difference the same tree ring for the same tree for the same year but different core sample I am looking at:

Delta1,x â€“ Delta2,x = (TRW1,x â€“ TRW2,x)/RCSx with RCSx varying with tree ring age, but at a much slower rate at the older tree ring ages. Instead of speculating about this, however, I need to do the same analysis using TRW for the replicated tree cores.

Thanks, Chas, for pointing out the possible sources for variations.

I also wanted to look at the Schweingruber data where 2 cores per tree is the norm.

]]>I looked at the (raw) annual ring widths (in excel) for L00131-4 and the annual SD of the widths seems to increase when the trees put down larger rings – their coefficient of variation looks pretty trendless with time. (though there are some spikes in the series when a ring or two has a width of zero).

If the delta is arrived at by division then a higher theorised (cf. actual) RW in the RCS curve might tend to reduce the delta’s SD ? ]]>

Based on my findings for same tree and same year differences, I should have reiterated in my previous post that the most varied deltas (and unexplained at that) would likely appear to be occurring in the modern era of the Yamal chronology. That era has older trees with the older tree rings. I think that much of that variation is captured in the CIs I posted in Post #500 above – its just that further analysis puts us a step closer to understanding why.

]]>.

I divided the difference time series for all 13 trees that had replicate core samples into 1st and 2nd halves and calculated the standard deviation (SD) and mean for each half of the series. I then did a calculation to determine whether the 1st and 2nd half differences for SD and mean were statistically significant using the 95% CIs. The results are listed and tabled below.

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1st versus 2nd half for SD: Ave Difference = -0.214; SD of Difference = 0.211; n = 13; df = 12; 95% CI = -0.341 to -0.086

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1st versus 2nd half for mean: Ave Difference = -0.075; SD of Difference = 0.115; n = 13; df = 12; 95% CI = -0.145 to -0.006

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Since the 1st and 2nd half differences of the difference series are statistically significant for SD and mean (the CIs do not include 0), the tree ring age does appear to be the critical factor when comparing tree ring deltas for the same tree and same year. Why this should be so and at the level of variation shown by this analysis is a question that I would think that the Yamal authors would have discovered and at least attempted to answer before going on with any further interpretation of the Yamal chronology.

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It is important to remember that these difference comparisons take the issue and uncertainty of the RCS expected growth rate (with tree ring age) entirely out of the picture – as that effect is cancelled in the differences.

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A special thanks to you, Romanm for your clear commenting – it is 95% your code ðŸ™‚

Its not much real use, but maybe it gives a feel of how an inverse exponential curve fits each tree’s growth. (Note that there is a parameter ‘s’ that allows the curve’s origin the freedom not pass through 0,0 ,if it chooses). It is a unusual version of an inverse exponential, I know, but it has the slight advantage that ‘b’ is the time taken to reach 50% of the asymptotic size.

-Some trees seem to be positively exponential and the fitting fails.

download.file(“http://www.climateaudit.org/wp-content/yams.tab”, “yams.tab”, method=”auto”,

quiet = FALSE, mode = “wb”,cacheOK = TRUE)

load(“yams.tab”)

ids = unique(yamal$id)

for (i in 1:length(ids)) {

onetree.rw=yamal[(yamal$id==ids[i]),”rw”]

onetree.age=yamal[(yamal$id==ids[i]),”age”]

onetree.growth= cumsum(onetree.rw)/1000

par(ask=TRUE) ### ASK B4 PLOTTING NEXT CHART

y=onetree.growth

x=onetree.age

##WRAP THE NLS IN ‘TRY’ TO STOP IT JUMPING OUT OF THE LOOP IF THERE IS A FITTING ERROR

nlmod <-try(nls(y~ (A*(1-(.5^((x+s)/b)))),start=list(A=150,s=5,b=50),trace=FALSE),TRUE)

plot(x,y,main=ids[i])

###ONLY PLOT A CURVE IF THERE ISNT A FIT ERROR:

if(class(nlmod)!=”try-error”)lines(x,predict(nlmod),col=2)

}

##

To see a plot (or to see the next plot) one has to press ‘Enter’

]]>.

Adj R^2 = 0.44; Trend Slope = 0.13 standard deviation per 100 years of age; SE = 0.04 standard deviation per 100 years of age, t stat = 3.25.

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The residuals, line and normal probability plots are shown below.

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From these calculations the age of the tree then would appear to affect the measurement error of the tree ring widths (TRW) or alternatively have varying TRW with the coring radius. Therefore I ask the question: Are we confounding a physiological tree age issue with a measurement one? Notice also that if one can extrapolate the variation problem into the ages of the trees that mostly make up those in the modern era of the Yamal chronology, we are looking at very large errors. I do not see that the variation is affected by tree ring age which implies that the variation is caused by a measurement error that is more pronounced in older trees.

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