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A link to the mentioned Atlas of the Underworld:

http://www.atlas-of-the-underworld.org/depth-top/

In particular, the image in the article is from http://www.atlas-of-the-underworld.org/Cocos/

The site is designed for specialist, and current version is too technical. I looked at it for a few minutes and read the explanations, but I'm not sure about the interpretation.

It would be nice to have an informal blog post with a layman explanation of each image and what is supposed to show each one. At least, a image edited with paintbrush to add a few ellipses to mark the important parts and arrows with a keyword. Do we have a geologist here?

As a nontechnical person I wonder what they need to say that they didn't say.

We have places around the world where one plate is diving under another plate. Subduction zones. The extrapolation is that the diving plate persists in some form as it dives into the mantle, and can be identified with the position it once occupied on the surface. These pieces of crust are cooler than the mantle they are immersed in. One reason we can track them is they propagate seismic waves at a faster velocity than the surrounding mantle. I'm not a geologist and I welcome feedback on my interpretation of the abstract and other materials.

I understand the concept of what the data shows and how the all this comes about. However, I find it almost impossible to interpret the map. I understood the parent to have the same issue.
Do we know how much of slab movement in the mantle just reflects mantle convection, and how much also reflects slabs' structural integrity? Near the surface, I understand that slabs are load bearing. But below a few hundred km, aren't they melting? Or does the fact that this technique detects them imply that they haven't melted fully yet?
As a layman I don't know what consensus geologists have about your questions. But I think the answer to the last of your questions would be yes. The colder/solid slab is showing up in data gathered about seismic wave propagation speed.

I have the model in mind of an ice cube in water, except with higher viscosity and of course more heat. There would be a penumbra of cold water surrounding the melting cube until a point where there would just be cool water, vaguely cube-shaped. Plus the motion would smear the penumbra but not the remaining solid. Also my guess is there aren't a lot of precise measurements, so the usual blur from that. Like an ice cube in water, if you test the solidity of the cube you find that the cube hasn't softened, just shrunk.

I looked at a few of the plots but only the Cocos plot clearly suggested a 'slab' to my eyes, and probably for the wrong reason: The plot looks like it is divided in half by the blue area. Looking more carefully and reading, I think what I'm seeing is two depths of the same object. The plate subducting under the West coast of the Americas is apparently appearing as an anomaly or slab at two depths.

Thanks. I gather that at least some volatiles bake out of subducting crust during the first 100-200 km. That's what drives subduction zone volcanism. But maybe that increases the melting range for what's left. I have no clue.
It's not clear from the article, but the site makes clear that the 'slabs' are the parts of the map colored in blue.[1]

[1] http://www.atlas-of-the-underworld.org/Atlantis/