Basically, the first point is that from a direct inversion perspective a linear inverse solution needs to begin with a careful examination/determination of what data is required, in principle to directly invert for that quantity, without any linear or other approximation. Once that data requirement is understood then the meaning of linear inversion is defined as the linear approximate for that quantity, where linear approximate/inverse is the first and linear step in a direct non-linear inverse solution. Only a direct inverse solution ( without linear or other approximations) can provide that clarity and understanding for a meaningful linear inverse. A linear relationship between any data set and sought after subsurface properties, doesn’t represent a starting point for linear inversion- unless that linear relationship is the linear approximate to the nonlinear direct inverse relationship between sought after subsurface properties and measured data.
The second point, is to recognize that if in the inverse problem under consideration the reference medium needs to be elastic, then that calls on the elastic wave equation, it’s integral equation form, that is, the elastic Lippmann -Schwinger equation (LS), and the subsequent/concomitant elastic forward and inverse series solutions. The forward and inverse elastic scattering series involve matrices in both directions. In the inverse elastic series the nonlinear multiplication of data matrices first appear in the first nonlinear( i.e., the quadratic ) step and that data matrix ( with multicomponent sources and receivers) is required for the direct inversion for the mechanical properties in the elastic equation. That required data matrix (for the full nonlinear series) is then the data that appears and defines a direct linear inverse, as the first and linear step in a direct nonlinear solution. The Haiyan Zhang thesis points that out for the elastic parameter estimation problem, and Ken Matson’s ( earlier) thesis similarly requires multi-component data for ocean bottom or on-shore multiple removal. These three examples require an elastic reference medium, the elastic inverse series, with in each case different isolated task subseries of the elastic inverse scattering series. All elastic ISS subseries require multi-component source and receiver data.
These two points are behind several ongoing discussions concerning direct inverse methods, the meaning of linear inversion, and conceptual and practical implications for all seismic amplitude analysis ( for example, velocity analysis and updating, AVO, FWI, reservoir monitoring, time lapse seismic etc) and are a central purpose and goal of the ‘Antidote’ and ‘meaning of linear inversion’ papers. We will be covering this topic in our Seismic Physics Course this Spring 2014 semester at UH, and it will be included in the second volume of "Seismic Imaging and Inversion: Application of Direct Nonlinear Theory" that Bob Stolt and I are writing for Cambridge University Press.