Upscaling in orthorhombic media: Behavior of elastic parameters in heterogeneous fractured earth

2019

JGR Solid Earth

Full‐waveform inversion (FWI) optimizes the subsurface properties of geophysical Earth models in such a way that the modeled data, based on these subsurface properties, match the observed data. The anisotropic properties, whether monoclinic, orthorhombic, triclinic, or vertical transversally isotropic (VTI), of the subsurface, be it a fractured reservoir or the core‐mantle boundary, are necessary to describe the observed wave phenomena. There are no principal limitations on the complexity of the anisotropy that can be inverted using FWI. However, the question remains—what kind of anisotropic descriptions of the elastic properties of the Earth can or cannot be inverted reliably from seismic waveforms? We reveal the resolution that can be achieved through reconstructions of each elastic parameter by building vertical resolution patterns from the scattering radiation patterns of body waves. A visual analysis of these patterns indicates “trade‐offs,” that is, perturbations of parameters that have the same reflection‐based scattering patterns as other perturbations. Each trade‐off leads to an apparent ambiguity in the inversion, which must be addressed by additional assumptions, constraints, or regularizations. For orthorhombic media, we find the exact trade‐offs that exist between parameters. Our parameterization isolates the VTI parameters, and therefore, we also obtain trade‐offs for every scattering mode for VTI media. We discover that only monoclinic parameters are recoverable from the first‐order scattering of monotypic waves. We summarize the trade‐offs in tables for easy reference. This paper is intended to be useful for researchers setting up anisotropic FWI problems, and interpreting or controlling the quality of such inversions.

Section: Introductionmentioning

confidence: 99%

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

Kazei

2019

JGR Solid Earth

“…The elastic properties of most hydrocarbon reservoirs can be wellapproximated using orthorhombic anisotropy (Schoenberg & Helbig 1997;Tsvankin 1997;Ivanov & Stovas 2016) on the scale of common exploration seismic wavelengths. Systems of orthogonal fractures or vertical layering within a reservoir can be described by macroscale orthorhombic media; see Fig.…”

Section: Introductionmentioning

confidence: 99%

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

Kazei

Geophysical Journal International

2018

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.

“…The elastic wave equation for orthorhombic (ORT) media (Tsvankin ; Albertin et al . ; Ivanov and Stovas ) can be expressed by

ρ \frac{\partial^{2} u}{\partial t^{2}} = \nabla \cdot σ + {boldf}_{s}

with

\begin{matrix} true \\ right σ = ((), \begin{matrix} σ_{x x} \\ σ_{y y} \\ σ_{z z} \\ σ_{y z} \\ σ_{x z} \\ σ_{x y} \end{matrix}) = ((), \begin{matrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{matrix}) ((), \begin{matrix} ε_{x x} \\ ε_{y y} \\ ε_{z z} \\ 2 ε_{y z} \\ 2 ε_{x z} \\ 2 ε_{x y} \end{matrix}) = C ε, \end{matrix}

where u and f s denote the particle displacement and seismic source vectors, respectively, and C ij indicates nine elastic stiffness coefficients for the 3D elastic ORT media. The partial derivative wavefields in equation are obtained by taking partial derivative of elastic wave equations with respect to a desired model parameter, p m,n (except density), as follows:

ρ \frac{\partial^{2}}{\partial t^{2}} (\frac{\partial boldu}{\partial})

…”

Section: The Multi‐parameter Inverse Problemmentioning

confidence: 99%

Study on the full‐waveform inversion strategy for 3D elastic orthorhombic anisotropic media: application to ocean bottom cable data

Geophysical Prospecting

2019

A B S T R A C TThe multi-parameter full waveform inversion is an essential tool to estimate subsurface anisotropic properties in a reservoir that may have complex geological behaviour requiring an elastic orthorhombic medium description. For such elastic orthorhombic media, finding a proper inversion strategy to mitigate parameter trade-off and reduce the Null space is crucial considering the large number of parameters describing the medium. We apply our recently developed strategy for orthorhombic medium inversion on synthetic and real ocean bottom cable data, and find the most efficient inversion strategy. At first, we analyse the trade-off patterns in three elastic orthorhombic parameterizations for a hockey-puck-shaped model. We, then, compare the performance of these orthorhombic parameterizations on a 3D synthetic ocean bottom cable data, which are obtained from a channel-shaped model. By interpreting the inverted models based on analytic radiation patterns of the nine elastic orthorhombic parameters in each parameterization, we observe that parameterizations, which have one P-wave velocity, one S-wave velocity and seven dimensionless parameters, can be optimal to recover subsurface isotropic properties in the early stages of inversion. Then, we show that the choice of three anisotropic parameters and their deviations along the horizontal plane helps us mitigate the complexity of the multi-parameter inversion by decoupling anisotropic features, which have vertical and horizontal symmetric axes, respectively. This decoupling of isotropic and anisotropic properties enables us to perform the multi-parameter anisotropic inversion in a multi-stage manner. In addition, for the marine acquisition, we show that the number of parameters can be reduced from 9 to 4, which makes the multi-parameter inversion more practical. Finally, we apply the elastic orthorhombic inversion to real ocean bottom cable data.