To picture the higher Chladni patterns on a sphere, imagine a round balloon full of water in zero-gravity, then tweak it various ways to make it vibrate in different modes. The first order Chladni pattern just corresponds to inflating or deflating the balloon, so that the whole surface rises or falls together.

On a circle with equally spaced reference points, the Chaldni patterns are just Fourier harmonics plus a zero frequency constant term. With unequal spacing, the first several patterns are basically harmonic, but then they get irregular, depending on the spacing.

But on a bounded violin back, Antarctica, or SW US, even the first order Chaldni pattern is constrained by the boundary. See cool pictures on Wikipedia, Chaldni Patterns.

I’ll try to do a post on the spherical patterns when I’m done with the 23 other things I’m in the middle of!

I think this study makes appropriate and cautious use of PCA.

I didn’t provide the data link as a challenge to any ‘point about archiving’. You stated in an earlier comment that it would be difficult to perform your analysis without the data, so I pointed it out. You certainly don’t need a site list to replicate Li et al. and their PCA, which is the central topic of this post and my comment. I read their paper, and I read your post, and I disagree with your assertion that Li et al. are misinterpreting unphysical ‘Chladni patterns’. That’s all.

I don’t plan to pursue this until they archive their data.

Taking the long view, this could be a fitting epitaph. But not for our host, for the people who didn’t listen and shut him, Ross and so many others out. It will all change, I have no doubt – one funeral at a time.

The leading empirical orthogonal function (EOF) of NADA, which accounts for 23.5% of the total variance, shows a distinct moisture pattern heavily loaded over southwest North America (Methods and Fig. 1a), a region strongly affected by ENSO variability15, 16. The explained variance of the first EOF for the southwest North America domain rises to 49.5% (Supplementary Fig. S3), and its principal component (PC) has a correlation of 0.99 with that for the whole of North America over the past 1,100 years. This confirms that NADA PC1 dominates variability in southwest North America.

The authors then list several procedures to establish the leading (North American) PC as ENSO-sensitive:

The fidelity of NADA PC1 in representing ENSO variability is demonstrated in five ways: (1) NADA PC1 is significantly correlated with equatorial Pacific sea surface temperatures17 (SSTs) during the instrumental period (Fig. 1c). For example, its correlation with the January–March (JFM) ENSO Niño3 index for 1870–2002 is 0.51 (P<0.001) (Supplementary Fig. S4). (2) NADA PC1 is significantly correlated with a modern coral record from Palmyra Island in the central tropical Pacific13, at r=−0.58 for the common period 1891–1994 (P<0.001) (Supplementary Fig. S5). After adjusting relict coral U/Th dates within the analytical error windows13, we found possible correlations between North American drought and tropical corals that persisted throughout the past millennium (Methods and Fig. 1d). Considering the independent nature of these proxy records, this agreement is remarkable, strongly indicative of a coherent relationship between ENSO and North American drought over the past millennium. (3) NADA PC1 is highly correlated with, but is significantly longer than, the existing ENSO reconstructions (Supplementary Table S1). Most of these reconstructions are largely, although not entirely, independent, as they only share a few common tree-ring data from southwest North America. This suggests that NADA PC1 represents reasonable estimates of ENSO variability over the past millennium. (4) Separate composite analysis of NADA for periods of large and small PC variability over the past millennium shows nearly identical patterns (Supplementary Fig. S6). This pattern stability suggests that the westerly waveguide is stable in tropical forced stationary waves, and that NADA PC1 represents the modulation of ENSO itself. (5) Separate EOF analysis shows that the first EOF of NADA is heavily loaded over southwest North America both before and after the 1976/1977 climate regime shift18, 19, and the PC is highly correlated with tropical eastern Pacific SSTs for both periods (Supplementary Fig. S7). Although there exist some differences in the EOF pattern and correlation with tropical SSTs between the two periods, these results indicate that moisture variability in southwest North America has been sensitive to ENSO variability both before and after the 1976/1977 climate regime shift. The relationship to ENSO is further confirmed by the SST correlation pattern for 1870–2002 (Fig. 1c). Therefore, teleconnections of canonical eastern Pacific ENSO (hereafter simply ENSO) are probably stable over southwest North America. On the basis of these validations, we consider that NADA PC1 represents a continuous record of interannual ENSO variability during the past millennium.

It doesn’t appear to me that the authors are recklessly interpreting some random or artificial mode as an ENSO indicator. They have been pretty thorough about establishing an ENSO connection.

The tree-ring data is archived here. It’s a gridded data set composed of many separate tree-ring records:
ftp://ftp.ncdc.noaa.gov/pub/data/paleo/drought/NAmericanDroughtAtlas.v2/

Steve: before you uncritically assume that there is any physical meaning to eigenvectors from tree ring networks, I urge you to read prior CA posts on Chladni patterns, starting form the earliest http://climateaudit.org/tag/chladni/. In my opinion, the evidence is quite convincing in previous situations. If sites are not distributed in a spatially uniform or random pattern, the eigenvector resulting from PCA from spatially autocorrelated sites will have a different appearance than Chladni patterns on a simple geometric shape, but that doesn’t mean that the eigenvector has any physical meaning. I’ve discussed such cases in the past. Unfortunately, the issue doesn’t seem to be well understood by practitioners – otherwise, Steig et al wouldn’t have gotten confused on the matter.

Steve: in the case of Steig et al, the authors incorrectly interpreted the Chladni patterns as having physical significance and used this incorrect interpretation to retain only a few PCs. It mattered in that case – not that you or anyone else admitted it.

Right now, I don’t know whether the effect on Li et al is as material as on Steig et al. There is no list of sites that they used or data archive. Without such information, it’s hard to analyse and I don’t plan to pursue this until they archive their data.