What paper are you reading in particular? Nothing of your message makes sense without pointing out the exact paper. The code he published... who is he?
My opinion on terrain algorithms, condensed to be as simple as possible

1. Small organic patches: brute force
   
2. Generic terrains: hierarchical geomipmapping
   
3. Large/streaming/infinite terrain: geometry clipmaps
   

In GeoMipmapping the deviation is simply a function of the distance between the interpolated value and the missed sample.
Geometry clipmaps don't care about the deviation, they exploit Nyquist Theorem so nobody really cares.

As a start, when you read stuff about chunked LOD, it's probably about stuff that fits in RAM. Most people just assumes terrain data will fit. I have myself tried an algorithm which blew 150+ MiB of RAM with no remorse.
The algorithm you pulled out is quite old but still good at its core. I think you might be missing part of the picture, it appears the deviation is a result of a preprocessing step. I'm not going to look at the code, but I suspect he's doing something like:
[attachment=10178:terrain_error_maybe.png]
That is, the error is probably the length of either the blue lines or the yellow lines. I general Discrete LOD starts from finding an error we want to allow for each mesh level and removes every vertex which contribute to correctness less than we want. That is something like
[source lang="java"]foreach vertex
if(abs(value - LinearNeightbours()) < EPS) discard;[/source][font=arial,helvetica,sans-serif](that would evaluate the length of blue lines, this code is just food for thought). The biggest error is stored in the .CHU file (I'm guessing that's the "terrain files" you're referring to). If you think at it, you might see this is not really as easy as it seems. [/font][font=arial, helvetica, sans-serif]For this reason, I suggest to look at either more modern approaches OR easier approaches. [/font]In particular, I suggest to look at geomipmapping, which works in reverse: instead of selecting vertices to drop based on a error metric, it drops vertices using a simple logic (every odd vertex) and obtains a deviation from there. For densely tessellated height fields that works rather well. The various chunks can still page to disk or assembled in hierarchical fashion (albeit most people using geomipmapping does not seem to be aware of that. [font=arial, helvetica, sans-serif]Geomipmapping is older than your paper but I think it provides better value as it probably takes less to implement and avoids dealing with not-quite-regular grids. Think at it.[/font]

For first I figured out I've written something really wrong in my previous message. Meshes are progressively refined from coarsest to finest and not vice versa (I wrote this hoping it would help you understand the process but I guess it's not valueble).

1. I looked geomipmapping but as you said it can still be used but a little outdated.
   
2. And as far as I know chunked lod allows you to create bigger terrains than geomipmapping
   
3. What I don't understand about his error metric is he says every successive lod level has half of its parent error. Meaning Error(L+1) = Error(L)/2. How does he ensure that or does he just assume that it is like that for simplicity?
   
4. And I really appreciate if you have any suggestions about other modern approaches.
   

1. No. I said it is old. Not outdated. No discrete LOD system producing static geometry is truly outdated. By contrast, most continuous LOD schemes are outdated in my opinion. Let me be clear: Geomipmapping is NOT better, it's just easier to deal with IMHO.
   
2. This is a common misconception. Again, most people will never have to deal with the paging problem. Nothing prevents geomipmapping from being paged to disk. Terrain clipmaps suffers from a common misconception: the algorithm itself is based on "fitting a grid on the terrain signal", while the paging is a basically orthogonal problem but if you ask around everyone will say terrain clipmaps are natively paged. They are not completely wrong either as this is what the term "clipmap" is typically used for.
   
3. (I assume you're talking about Geomipmapping) It's a direct consequence of doubling the sampling frequency. Draw it on paper on 1D.
   
4. I already listed three methods in my 1st message. But I'd add one. The possibility of traversing the heightmap with raycasting.
   

My experience with CDLOD was terrible. The algorithm takes more preprocessing time than anything else I know about as well as more runtime resources. At that cost, the benefit over well-estabilished algorithms is not clear to me. The paper is written in such a way one wonders how much of it is true, how much is snake oil, how much I couldn't understand and how much is written with no clear understanding of the notions involved. Last time I moved those objections I experienced consistent fanboyism (mainly by a single user, who clearly didn't read my messages, let alone explain anything). In my opinion, its usefulness is unproven. You can use the search functionality to find what I have written (keyword VTF) I must admit however that it had quite some sarcasm in it, in retrospect, that was an error by my side.