Elastic parameter estimation

Once we have obtained the elastic $\grave{P}\!\acute{P}$ reflectivity, $R_{pp}(\theta)$, we can consider how to invert for three elastic parameters at each subsurface location.

We begin by making the linear Bortfeld approximation (Bortfeld, 1961, p. 489) to the nonlinear Zoeppritz R_pp coefficient (Aki and Richards, 1980, p. 153). This means that we assume that changes in elastic parameters are linearly related to reflectivity. In practice, this is a good assumption for elastic parameter changes less than about 20% and specular angles less than about 30$^{\circ}$ ($\theta=0^{\circ}$ is a normal incidence reflection). By elastic parameters, we mean the change in elastic property divided by the average elastic property at any given location. In other words, the parameters we estimate are relative changes and are not absolute magnitudes of properties. Hence we will quote dimensionless values of the parameters, like +10%, instead of +0.2 g/cc (density) or +300 m/s (velocity).

The linearized Bortfeld approximation to $R_{pp}(\theta)$ can be written as

 

$$R_{pp}(\theta({\bf x_o})) \approx c_1(\theta)P({\bf x_o}) + c_2(\theta)S({\bf x_o}) + c_3(\theta)D({\bf x_o})$$ (4)

where the $c_i(\theta)$ are known coefficients of specular angle (mathematically determined by choice of elastic parameterization), and $\{P, S, D\}$is the set of three elastic parameters to be found. Generally, P is related to compressional wave properties of the earth, S to shear wave properties, and D to density. At a single location ${\bf x_o}$ we have R_pp defined for several $\theta$ values: as many $\theta$ as there are offsets in the recorded data (recall the constant offset summations). The Bortfeld approximation above can then be written as a linear matrix equation

 

$${\bf Ax} = {\bf y}$$ (5)

where ${\bf A}$ is the matrix of the coefficients $c_i(\theta_j)$, ${\bf x}$ is the parameter vector ${\bf x} = (P,S,D)^{T}$, and ${\bf y}$ is the corresponding vector of R_pp values,

 

$${\bf y} = (R_{pp}(\theta_1),R_{pp}(\theta_2), \ldots, R_{pp}(\theta_j))^{T},\,\,\, j=1,N_{\theta}.$$ (6)

If our original data contain 60 offsets (as in a 120 trace per shot standard marine survey), then we will have 60 $\theta$ values, ${\bf A}$ will be a 60x3 matrix, ${\bf x}$ will be the 3x1 parameter vector, and ${\bf y}$ will be the 60x1 R_pp (data) vector. This is clearly an overdetermined problem since we are using 60 data equations to solve for only 3 parameters. Hence we solve the matrix solution by least squares with a technique known as Singular Value Decomposition (SVD), which decomposes the system into an orthogonal reference frame and reconstructs an optimal solution by eigenvector methods (Lines and Treitel, 1984). We use the spectrum of the 3 eigenvalues to determine the stability of the solution and to estimate the confidence in the inversion result, as discussed later.