Backus averaging and (negative) Q

----- 10 June 2014 -----

Got this from Sven Treitel:

      I thoroughly enjoyed your article in the June issue of TLE. You do offer an intriguing way to compute scattering Q values from velocity logs (what some folks call "extrinsic" attenuation). 
      A question: you obtained your curves by assuming a Backus averaging distance of 46 meters. Would your results and conclusions differ were you to repeat these calculations with a set of additional Backus averaging distances, some larger, some smaller than 46 meters? 
      A remark: what you call "reverse dispersion",. Sheriff's dictionary calls "normal dispersion" (see p. 247 of his dictionary, 4. edition). 
      And another remark: The fact that T(0) can be as large as 2 can be justified on an energy conservation basis, as Larry Lines et al showed in their recent paper---an argument which could be carried over, it seems to me, to the case you make for negative Q. 

My response:

       Glad you enjoyed the article. I chose 46 meter averaging length (L) for this particular well because it calculated out to max P wave frequency of 120 Hz. A bit longer L would be appropriate for a bit lower frequency and the 1/Q values would increase. Using L of the original layer thickness would give no change to vp or vs from sonic readings, thus 1/Q would be zero. Of course this is layer-induced 1/Q, there could also be some intrinsic 1/Q below 100 Hz due to things like patchy saturation or high viscosity pore fluid.
      All of this is, to my mind, a direct parallel with anisotropy. Backus calculates layer-induced anisotropy that combines with intrinsic anisotropy of shale intervals to yield total anisotropy.
      I missed the 'normal dispersion' definition in Sheriff, thank you.
     The philosophical connection between negative Q and T(0)>1 is made in the paper, but perhaps not too clearly.

Sven again:

     You could try to use the 1/Q estimates from your method to correct total 1/Q measurements from the seismic data for "elastic" absorption, thus perhaps leading to better non-elastic, intrinsic 1/Q estimates. As far as i know, the industry has yet to find a good way to separate the two effects, and yours could be an answer.

----- 4 June 2014 -----

Figures from Long-wave elastic attenuation produced by horizontal layering (Liner, 2014, June The Leading Edge).  These are a bit higher resolution than figures that made it into print. 

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

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Here is my original form of Figure 2.

Figure 2 (original form)

Some readers may want to know how to make such a thing. It can be generated from Ashanti's Ajax/Graphviz site by pasting in this code (and hitting the Return/Enter key):

"Sonic [P,S] and Density"->"Backus Theory"
"Avgeraging Length"->"Backus Theory"
"Backus Theory"->Anisotropy
"Backus Theory"->Dispersion
Dispersion->"Constant Q Theory"
"Field Observations"->"Constant Q Assumption"
"Constant Q Assumption"->"Constant Q Theory"
"Constant Q Theory"->"Layer-Induced Attenuation"

Very useful for building  directed graphs using GraphViz originally developed by AT&T Labs.