Reference of zWeight

[tonapawilliam]

Do you have any literature on zWeight, especially the use of formulas like the one you described for lambda parameter regulation? I have searched several references but I have not found what is the basic formula used to analyze zWeight. Thank you.

[halbmy]

The only reference is Coscia et al. (2011) in equation (2):

Coscia, I., Greenhalgh, S., Linde, N., Doetsch, J. A., Marescot, L., Günther, T., Vogt, T. & Green, A. (2011): 3D crosshole ERT for aquifer characterization and monitoring of infiltrating river water. Geophysics 76(2), G49-G59, doi:10.1190/1.3553003.

[tonapawilliam]

Thank You Sir

[makeabhishek]

May I ask, which reference you followed for lambda regularization?

[halbmy]

Regularization is explained in more detail in the tutorial https://www.pygimli.org/_tutorials_auto/3_inversion/plot_5_Regularization.html

[makeabhishek]

Thanks, yes I followed that. Is there any research article specifically explaining the regularization mentioned in the tutorial. Especially the smoothness constraint matrices considered for unstructured mesh.

[halbmy]

For regularization of , we refer to Günther et al. (2006), the foundation is in more detail explained in Günther (2004) and some other details you can find in Coscia et al. (2011).

Günther, T., Rücker, C. & Spitzer, K. (2006): Three-dimensional modeling and inversion of dc resistivity data incorporating topography – II: Inversion. Geophys. J. Int. 166, 506-517, doi:10.1111/j.1365-246X.2006.03011.x. Günther, T. (2004): Inversion Methods and Resolution Analysis for the 2D/3D Reconstruction of Resistivity Structures from DC Measurements. PhD thesis, University of Mining and Technology, Freiberg, available on http://nbn-resolving.de/urn:nbn:de:swb:105-4152277. Coscia, I., Greenhalgh, S., Linde, N., Doetsch, J. A., Marescot, L., Günther, T., Vogt, T. & Green, A. (2011): 3D crosshole ERT for aquifer characterization and monitoring of infiltrating river water. Geophysics 76(2), G49-G59, doi:10.1190/1.3553003.

[makeabhishek]

What is the recommended approach to choose zWeightin inversion?

[halbmy]

There is no formal approach. The value expresses the expectation to the subsurface, on how much the resistivity varies in the horizontal and vertical directions. This can be between 1 (or even above in rare cases of vertical structures) and low values like 0.1 or 0.01 for strongly layered structures.

[makeabhishek]

What is zWeight in the context of seismic tomography? Just like resistivity range, what could be the possible range in traveltime tomography?

[halbmy]

The parameter controls the regularization, no matter which parameter is inverted. Wasn't this obvious? Please read the tutorials on regularization.

[makeabhishek]

Actually, I was trying to find what values can be given. However, document didn't mention or I didn't find it that it should be in between 0 and 1, not like regularization parameter \lambda. Even while running inversion, giving any value higher than 1 is not throwing any warning or error. Am I right that zWeightneed to be in the range of 0 to 1?

[halbmy]

There is no need to be between 0 and 1. As I explained in my above comment, it can be very reasonably above 1 and of course it can formally also below 0 regarding formula (2) in Coscia et al. (2011). However, if the weight falls below 0, the contribution to the smoothness rises again and there will be a direction with minimum smoothness. So, in fact values below (or even equal to) zero do not make sense although they are possible.