> I disagree. The longer term LOD variation was removed in in the inset, thus reducing the fluctuation extremes from 2 to 1.

OK on the image you showed it is not indicated that something had been taken away, the diagram header (Excess Length-of-day (LOD)) is exactly the same for the top as for the middle panel.

In the paper you cited it is written:
>One sees that the interannul LOD variation is mainly caused by the anomalous mass transport (mostly in the east-west wind field) of the Southern Oscillation in the tropical Pacific-Indian Ocean.

Here the author writes about interannual LOD but nothing indicates that data been taken away. It is in this paper:
http://www.earth.sinica.edu.tw/~bfchao/publication/eng/2003-Chao-EOS-Geodesy%20is%20not%20just%20for%20static%20measurements%20any%20more.pdf

where it is finally written:
>the blue curve is VLBI measurement, after removal of the
seasonal terms due to mass transports of meteorological origin and tidal forces;

So it is rather the short-term data (the annual) that had been taken away).
Anyways do you know where he has the long-term LOD data from?
The figure 1 caption just says:
>The blue curve is the entire ∆LOD data set that human kind ever acquired
(the post-1960 densification of data resulted from the advent of the atomic clock);

There is currently a discussion on
https://plus.google.com/+TimothyGowers0/posts/GNdRfNqcZw9
about a paper, where nobody seems to know were the long-term LOD is coming from.