Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

Abstract

Summary Various parameterizations have been suggested to simplify inversions of first arrivals, or P −waves, in orthorhombic anisotropic media, but the number and type of retrievable parameters have not been decisively determined. We show that only six parameters can be retrieved from the dynamic linearized inversion of P −waves. These parameters are different from the six parameters needed to describe the kinematics of P −waves. Reflection-based radiation patterns from the P − P scattered waves are remapped into the spectral domain to allow for our resolution analysis based on the effective angle of illumination concept. Singular value decomposition of the spectral sensitivities from various azimuths, offset coverage scenarios, and data bandwidths allows us to quantify the resolution of different parameterizations, taking into account the signal-to-noise ratio in a given experiment. According to our singular value analysis, when the primary goal of inversion is determining the velocity of the P −waves, gradually adding anisotropy of lower orders (isotropic, vertically transversally isotropic, orthorhombic) in hierarchical parameterization is the best choice. Hierarchical parametrization reduces the tradeoff between the parameters and makes gradual introduction of lower anisotropy orders straightforward. When all the anisotropic parameters affecting P −wave propagation need to be retrieved simultaneously, the classic parameterization of orthorhombic medium with elastic stiffness matrix coefficients and density is a better choice for inversion. We provide estimates of the number and set of parameters that can be retrieved from surface seismic data in different acquisition scenarios. To set up an inversion process, the singular values determine the number of parameters that can be inverted and the resolution matrices from the parameterizations can be used to ascertain the set of parameters that can be resolved.

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

“…In our previous work (Kazei & Alkhalifah, ), we performed an analysis of the radiation patterns describing the scattering of P waves on orthorhombic parameters in an isotropic, homogeneous‐medium background. Here, we present a similar analysis for different types of scattering.…”

“…The phase accumulated while traveling through a background medium is dropped in our analysis so that we can focus on the local scattering features. Therefore, the scattered wavefield δ U depends only on the incident plane wave slowness vector s , the scattered plane wave slowness vector g , the respective polarizations ς and ξ , the frequency ω , and the perturbation of the parameters δ m = ( δρ ( x ), δ c ( x )) T : The coefficient R , well‐known for its arbitrary point scatterers, is called the scattering function (Calvet et al, ; Eaton & Stewart, ; Kazei & Alkhalifah, ; Shaw & Sen, ), and is defined as While R is independent of the background medium, which can have up to triclinic‐anisotropy complexity, eight different angles determine the directions of polarization and propagation for its incident and scattered wavefield (there are four directions, featuring two angles each). Thus, there are many possibilities for polarization and propagation relations in a general anisotropic background as well as for velocities of incident and scattered waves; here, we consider scattering in an isotropic background.…”

“…We start with scattered P waves as they are the easiest to handle and by far the most popular in FWI applications. By substituting equations and into equation , and projecting the scattered wavefield U onto the direction of g (from the perturbation to the receiver), we obtain the approximate amplitude of the scattered P ‐ P waves (Kazei & Alkhalifah, ) as where …”

Full-waveform inversion in acoustic orthorhombic media and application to a North Sea data set

“…Indeed, Oh and Alkhalifah (2016b) and Kazei and Alkhalifah (2018) analyze the scattering radiation patterns of the nine orthorhombic elastic parameters; six of them are the acoustic parameters that we are interested in here, and the three others are v s1 , γ 1 , and γ d . Their studies reveal strong trade-offs between v s1 and η 1 , as well as between η d and γ d when using PP and PS surface seismic data.…”

“…This extension has resulted in challenges related to the trade-offs between the model parameters in our inversion process. The trade-offs between the model parameters and their resolution limits have motivated several studies investigating suitable parameterizations for our inversion schemes Operto et al, 2013;Alkhalifah and Plessix, 2014;Alkhalifah, 2016;Kamath and Tsvankin, 2016;Kazei and Alkhalifah, 2018). Most of these studies rely on the analysis of the radiation (scattering) patterns based on the Born approximation (Wu and Aki, 1985;Panning et al, 2009).…”