We’ve all had enough experience with the merry adjusters to know that just because someone “adjusts” something doesn’t necessarily mean that the adjustment makes sense. A lot of the time, the adjuster arm waves through the documentation and justification of the adjustment. Today I’m going to work through aspects of one of the most problematic adjustments in this field: the MBH99 adjustment for CO2 fertilization – something that some of us have wondered about for a couple of years. In this particular case, there are a number of crossword puzzles and I’d welcome solutions to these puzzles, as I’ve pondered most of these issues for a long time and remain stumped. I’m not discussing here whether CO2 fertilization exists or not, but merely whether the Mannian approach is a reasonable approach to the issue, trying first merely to understand what the approach is.
I’m working towards assessing the impact of the new Abaneh Sheep Mountain data on the Mannian-style results – and the impact on the Mann and Jones 2003 PC1 (used by Osborn and Briffa 2006; IPCC AR4; etc.)
The MBH99 adjustment is explained as follows (and this discussion immediately precedes the log CO2 argument that I just posted on):
A number of the highest elevation chronologies in the western U.S. do appear, however, to have exhibited long-term growth increases that are more dramatic than can be explained by instrumental temperature trends in these regions. Graybill and Idso  suggest that such high-elevation, CO2-limited trees, in moisture-stressed environments, should exhibit a growth response to increasing CO2 levels. Though ITRDB PC #1 shows signfi cant loadings among many of the 28 constituent series, the largest are indeed found on high-elevation western U.S. sites. The ITRDB PC#1 is shown along with that of the composite NT series, during their 1400-1980 period of overlap (Figure 1a). The low-frequency coherence of the ITRDB PC#1 series and composite NT series during the initial four centuries of overlap (1400-1800) is fairly remarkable, considering that the two series record variations in entirely different environments and regions. In the 19th century, however, the series diverge. As there is no a priori reason to expect the CO2 effect discussed above to apply to the NT series, and, furthermore, that series has been verified through cross-comparison with a variety of proxy series in nearby regions [Overpeck et al, 1997], it is plausible that the divergence of the two series, is related to a CO2 influence on the ITRDB PC #1 series.
There are obviously other possible approaches to adjusting for CO2 fertilization if one is worried about that. One possible method would be to do a correlation of the growth pulse to CO2 and work with the residuals. This would be an approach consistent with age detrending methods in dendrochronology and you’d think that Mann might have tried something like this. (I’ll discuss this on another occasion.) Here is the Figure 1a from MBH99 showing the “target” Jacoby series, the PC1 in need of adjustment and the adjusted PC1.
MBH99 Figure 1a. Comparison of ITRDB PC#1 and NT series. (a) composite NT series vs. ITRDB PC #1 series during AD 1400-1980 overlap. Thick curves indicate smoothed (75 year low-passed) versions of the series. The smoothed “corrected” ITRDB PC #1 series (see below) is shown for comparison.
The above graphic is pretty muddy. Here is a re-drafting of the graphic to show the features a little more clearly. In making this graphic, I’ve used data versions stored at the former Virginia Mann FTP site. In the directory TREE/COMPARE, it contains nearly 100 versions of different series, which I visit from time to time to try to decode. I’ve made a little push recently (and thus the post before I forget.) The light blue series is a plot of a rescaled version of the AD1000 Mannomatic PC1 (together with a smooth); the grey series is a plot of an average of Jacoby chronologies (together with a smooth) and the red dashed line is the adjusted PC1 (in a smooth version). I can sort of mock up the rescaling of the PC1 but the rescaling principles remain a mystery as discussed below.
Continuing a quick overview of the adjustment before looking at the details, Mann deducts the smoothed Jacoby average from the smoothed Mannomatic PC1 to get a “residual”, shown in MBH98 Figure 1b (left panel below). He then carries out another smoothing operation on these already smoothed residuals to get a “secular trend” – which he compares to CO2 increases (already discussed.) This “secular trend” is then backed out of the Mannomatic PC1 to get what Mann calls a “fixed” PC1 – although I think that the term “adjusted” is more apt. In the right panel, as a short exercise, I’ve done a similar comparison between the incorrectly Mannomatic PC1 and the AD1000 correlation PC1. The AD1000 network is already heavily dominated by bristlecones, so the difference between the correlation PC1 (or covariance PC1) and the Mannomatic PC1 isn’t large for this network – an observation that we made in MM2005 (EE). (The difference is large for the AD1400 network, as has been endlessly discussed.) What’s intriguing here is that the “adjustment” required to reduce the Mannomatic PC1 to the correlation PC1 is almost exactly the same amount as the vaunted “CO2 adjustment” although it gets implemented in the 20th century rather than the 19th century.
The Jacoby “Target”
The most obvious question in this exercise is: why is the Jacoby series selected as a “target” against which to adjust for CO2 fertilization? If there were regional differences between California and northern Canada in the timing of emergence from the Little Ice Age, then spurious correlations might easily arise between low-frequency series. Wouldn’t it make more sense to use something like the Briffa et al 1992 temperature reconstruction (a reconstruction referred to in the Ababneh thesis)? What would be the impact of using some other target? I doubt that I’ll get to some of these analyses as the data from the Ababneh thesis is going to make some of these speculations very moot. From a statistical point of view, in assessing “confidence” for these calculations, it’s just that one has to remember that the Jacoby target has been selected from a larger population and was not selected randomly.
Next, there are some differences between the Jacoby target used in MBH99 and the reconstruction published by Jacoby and D’Arrigo (1989, 1992). The MBH Jacoby version goes 200 years earlier than the Jacoby-d’Arrigo reconstruction, which began only in 1601, not in 1400.) The figure below compares the Jacoby and d’Arrigo reconstruction (scaled to match the MBH99 version) and the MBH99 target (light grey).
Next, the Jacoby chronologies used in MBH99 were obsolete even at the time – a point made in MM03. For example, Mann used a TTHH chronology with values back to 1459 while the archived chronology goes back only to 1530. In the final archive, no chronology is presented for a period that does not contain at least 3 cores: Mann used an early version which included a period prior to 1513 where there was only one core. In MBH, Mann stated that they carried out quality control on their chronologies to ensure that there was a match between the measurement data and the chronology. In the case of one of the Jacoby chronologies (Churchill), there was negligible correlation according to this test. A couple of years ago, I wrote to Rosanne D’Arrigo observing that the Churchill chronology could not obtained from the archived measurement data. She said that the archived chronology was wrong and said that she would replace it at ITRDB; however, this has not taken place two years later.
In the 15th century portion of the Mann version, a very old Coppermine chronology is used (values from 1428-1977). I inspected the measurement data and determined that this data set only had one tree (two cores) with values prior to 1500. This portion of the chronology does not meet normal QC standards for the minimum number of trees to form a chronology (including standards said to have been applied in MBH – a minimum of 5). The only Jacoby chronology contributing to the earliest part of the 15th century is Gaspé – about which much has been said in the past. In the earliest portion of the Gaspé chronology, there is only one tree.
A more fundamental problem with the 11 Jacoby chronologies is the risk that the selection has been contaminated by cherrypicking. At the NAS Panel, D’Arrigo told an astonished panel: you have to pick cherries if you want to make cherry pie. In Jacoby and D’Arrigo 1989, they reported that they used (and archived) only the 10 most “temperature sensitive” treeline series (to which they added Gaspé known to have a pronounced HS shape.) If you take a network of 36 red noise series with relatively high autocorrelations (as Jacoby chronologies) and pick and average the 10 with the biggest 20th century trends, you’ll get something that looks like the Jacoby average. A proper evaluation of the statistical significance of “adjusting” the Mannomatic PC1 to such a target is needless to say not discussed in the underlying literature.
The Divergence Problem is also an issue. One of the series in the target Jacoby average is Twisted Tree Heartrot Hill (TTHH), which has been recently updated (but not archived). Davi et al (including D’Arrigo) observed a Divergence Problem at this site (see Jacoby category -scroll back).
In replication terms, it is possible to arithmetically calculate the MBH version of the Jacoby average and, in this case, the chronologies as used and the average were all digitally available at one point. The issues here pertain primarily to the significance and selection of the target for the coercion.
Smoothing the Jacoby Average
Now, a little Mannian puzzle. How is the smooth of the Jacoby average done? In this case, we have the input data and the output data, so it’s possible to see if we can match the result. In the caption to Figure 1b, Mann says that you used a “75 year low-passed” filter. In tree ring jargon, a “75 year” filter is often used to describe a filter with gaussian weights that is long enough to pass 50% of the variance through 75 years (see Meko’s notes on this). In the figure below, I tried a 75-year gaussian filter and get something that looks similar to the Mannian result, but not the same. I’ve experiment with other filters and other end period approaches – reflection padding and Mannomatic padding yield worse results. Lowess filters don’t improve things. So I don’t know. I’ve archived materials for people to test: see columns 1 and 2 in the data set. A smoothing point to bear in mind: why is the delta calculated after smoothing: does it matter if the deltas are calculated first and smoothing done later?
I also checked circular padding in the smooth – it was a little improved but didn’t resolve the problem.
Smoothing the Rescaled Mannomatic PC1
There’s a tricky re-scaling step in the PC1 illustration, which is the next step in the logic, but before discussing it, I’m going to discuss the smoothing step for the Mannomatic PC1, just in case this helps decode the smoothing methods. Here the calculation starts with the Mannomatic PC1 (known to be “incorrectly” calculated – not that I believe that “correctly” calculated PC1 necessarily means anything.) Once again, I’ve tried a 75-year gaussian smooth and find that it is related to the archived version, but isn’t quite the same. I’ve provided materials (cols 3-4) for puzzle solvers.
While we’re thinking about Mannian smoothing, I’ll jump out of sequence and discuss smoothing of the “residuals” to yield a secular trend as discussed above. The legend to MBH99 Figure 1b says that the residuals have been smoothed with a 50-year filter (even though the two predecessor series had already been constructed with 75 year smooths.) The caption to MBH99 Figure 1b (perhaps inconsistently) says that the “secular trend” was calculated “retaining timescales longer than 150 years”. How was this smooth actually calculated? I don’t know. I’ve been able to replicate this result more closely than the other smooths by using a lowess smooth with f=0.3. I doubt that this is what Mann did. As noted previously, there is a big difference in interpretation between Mann’s conclusion that the supposed effect has “leveled” off in the 20th century based on one smooth and the continuing upward trend in the first smooth. Something else that some readers might wish to consider: is it possible that we’re a seeing some Slutsky effect here. (For non-econometricians, the Slutsky effect is a classic and elegant proof from the 1930s that repeated smoothing/averaging operations on time series will push them towards a sine curve.)
I’ve plotted the actual differences between the two series below (for whatever that’s worth) together with a 50-year smooth. I wouldn’t characterize this plot as showing a discrepancy arising in the 19th century and then leveling off in the 20th century. On the right, I’ve limited the plot to the period of the actual Jacoby-D’Arrigo chronology 1601-1980, to minimize the rhetorical effect of the 15th century where there are only 1-2 trees involved. In this case, one would surely characterize the plots not as showing remarkable similarity prior to 1800, but as the bristlecone PC1 gaining relative to the Jacoby reconstruction, with the relative gain reversed only during the HS-pulse of the Jacoby series in the early 20th century.
Re-scaling the AD1000 PC1
Now for a tricky part, that’s annoyed me greatly. In the log CO2 diagram, I’ve showed the mysterious appearance of arbitrary re-scaling with no explanation for what seems to be no more than rhetorical purposes.
In the top panel, I’ve shown the Mannian AD1000 PC1 as originally calculated. I can replicate this calculation from original chronologies (using the incorrect Mannian PC method.) The plot here is from the PC1 at the former Virginia Mann FTP site. It is inverted. PC series have no native orientation and the inversion here doesn’t trouble me. If you want to interpret the PC1 as a weighted average of the underlying chronologies (mostly bristlecones), then you can choose the orientation from positive weights as opposed to negative weights. A second point about the native series is that the PC series is calculated here from an svd algorithm and as a result the sum of squares adds to 1 i.e. the amplitude is very small. Occasionally one sees internet commentary purporting to find a flaw in the red noise simulations in MM (2005 GRL) on the basis that the amplitude of the PC series is “too small”; the amplitude of our simulations is exactly the same as the amplitude of the Mannian PC1 and the people who raise this argument don’t know what they’re talking about (not that that stops them.)
Aside from the inversion of the series, one can see both a rescaling and a recentering, which is not described and I’ve not been able to figure out at all. Another crossword puzzle.
It looks as though the rescaled PC1 has been recentered on some portion of 1400-1900 so that it more or less matches the mean of the “target” Jacoby series in the 1400-1780? period. We’ve seen from the log CO2 diagram that these selections can be opportunistic. If the Jacoby series has been used as a target reference, then it should be possible to find some period over which the Rescaled PC1 mean is equal to the Jacoby mean. As a way of exploring for such a period, I plotted cusums for the rescaled PC1 and the Jacoby target as shown below: they cross for a period of about 1400-1865. Or maybe the reference period doesn’t start in 1400 but some other period. It’s hard to say. But it looks like something like this is going on. Of course, this sort of calculation presumes mathematical precision. Ideas are welcome.
A tougher problem occurs with the rescaling of the PC1. The sd of the PC1 (0.21) is higher over (say) the 1400-1865 period .than the sd of the Jacoby average (0.18). You can see this in the diagrams. So it’s not obvious how the PC1 has been rescaled. Is there some period over which the sd of the target matches the sd of the rescaled PC1? To test this, I did a “cumsd” calculation modeled on the cusum calculation for means. In this case, the two standard deviations match for a reference period of 1400-1600, which is round, but do not match for other periods. Did Mann re-center and re-scale on different periods? If so why? Or is there some other rescaling principle entirely? I have no idea how this rescaling was done and it bugs me.
Black: Cumsd Jacoby NH; red – cumsd – pc1-rescaled. Looks like a round match at 1400-1600.
I’ve collated the following series into a data file in case anyone can solve any of the smoothing or rescaling problems:
2 – Jacoby average, unsmoothed
3 – Jacoby average – Mannian smooth
4 – Mannomatic PC1 – native unsmoothed
5 – Mannomatic PC1 – rescaled unsmoothed
6 – Mannomatic PC1 – smoothed 1400-1980
7 – Mannomatic PC1 – adjusted
8 – Smooth residuals
9 – Secular trend
7 – Mann CO2
8 – Mann log CO2 ratio
Aside from solving the various crossword puzzles, the more fundamental issue is: is this a valid way of adjusting the Mannian PC1? Does it make any sense to try to adjust the PC1 (as opposed to the individual chronologies)? What if the growth pulse in the bristlecones was not related to CO2 fertilization – an effect denied by other writers in the field?
In a subsequent post, I’m going to show the impact of the Ababneh Sheep Mountain chronology on the Mann and Jones 2003 PC1: it’s very dramatic raising some serious questions about exactly what, if anything, is accomplished in these Mannian adjustments.