Inversion of ray velocity and polarization for elasticity tensor
Introduction
In this paper, we discuss a method of using ray-velocity and polarization measurements to determine the complete set of the density-scaled elasticity parameters that describe a Hookean solid. The problem of determining the twenty-one components of the elasticity tensor from wave propagation has been investigated by many researchers; among them, van Buskirk et al. (1986), Norris (1989), and Dewangan and Grechka (2003). In all cases, the proposed methods relied on using polarizations and wavefront slownesses. We propose a method for finding these components using polarizations and ray velocities. To find the latter quantities, we consider traveltimes measured between a single point source and point receiver, which are directly related to the ray velocities. This method circumvents the need to measure the wavefront slownesses, which – in a seismological context – requires closely spaced sources or receivers.
We also demonstrate that standard seismic measurements of polarizations and traveltimes allow us to obtain uniquely the density-scaled elasticity parameters. In view of the forward problem described in the next section and the inversion formulated in Section 3, we can infer that the relation of the elasticity parameters to the polarization and traveltime measurements is one-to-one in the context of the theory of elastodynamics.
In Section 4, we discuss the error analysis for the proposed method and exemplify it with a numerical example.
Section snippets
Waves and rays in Hookean solids
A Hookean solid is fully described by its mass density and elasticity parameters, which are components of the elasticity tensor appearing in a constitutive equation. The constitutive equation of a Hookean solid is1where σ, c and ε are stress, elasticity and strain tensors, respectively. Strain tensor is a symmetric second-rank tensor given bywhere u and x are the displacement
Inversion for density-scaled elasticity parameters
In this section, we derive expressions that allow us to obtain the density-scaled elasticity parameters using the measured traveltimes and corresponding polarizations. The relationship between the ray velocity, x˙, and wavefront slowness, p, is given by Eq. (2). Using this equation, it is, in general, impossible to express p as an explicit function of x˙ and a. In our formulation, we circumvent this problem by including the measured polarization, A, and expressing p in terms of x˙, a and A.
To
Stability analysis: numerical example
In this section, we discuss the stability of the proposed method; in particular, we discuss the sensitivity of the elasticity parameters to the ray-velocity and polarization measurement errors. To do so, we consider the following density-scaled elasticity tensor.2
The entries of this matrix are related to the components of the elasticity
Conclusions and discussions
We have proposed a method for obtaining the density-scaled elasticity parameters from seismic measurements. These parameters are related explicitly to the wavefront slowness and the polarization via the Christoffel equations. However, considering a point source and a point receiver, we deal directly with the ray velocity rather than the wavefront slowness. These two quantities are related to one another by Eq. (2). In general, this equation cannot be solved explicitly for slowness. Moreover, at
Acknowledgements
We wish to acknowledge insightful comments of Vladimir Grechka and another, anonymous, reviewer, as well as the editorial work of Cathy Beveridge. This research was conducted within The Geomechanics Project. The research of A.B. and M.A.S. was supported also by NSERC.
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