ANISOTROPIC VELOCITY PARAMETERS

In this section I will propose a simple parameterization of angle-dependent velocity appropriate for relatively weak anisotropy with a vertical axis of symmetry. Anisotropy will be assumed to derive from layered media that may be isotropic on a fine scale but which will appear to be anisotropic on the scale of larger seismic wavelengths. The anisotropic parameters will be described as a sum of smooth basis functions, with no more spatial variations than necessary to explain the data. The form will allow easy linearization of slowness with respect to the model coefficients.

The parameterization of velocities should have enough degrees of freedom to describe all plausible models; the particular numerical formulation of this parameterization is less important. Our methods should be able to request the velocity at any point and in any direction from an essentially continuous model.

The model should also be differentiable, to allow the efficient optimization of raypaths. If discontinuities are included in the model, then it should be possible to smooth these values numerically to calculate gradients on different scales. I will also require that the resolution of the model should be adjustable at any time, even during optimization of the model. Adjustable resolution will allow the optimization to converge first on the smoothest, most reliable components of the velocity model. As the accuracy of estimated raypaths improves, then more detail will be allowed in velocity models.

No convenient explicit equation exists to describe group velocity as a function of angle. Instead, I will use an approximation with enough degrees of freedom to explain the data well and still adequately span the same range of functions allowed by the exact theory. Tomography will have a limited ability to estimate arbitrary changes in velocity with angle. An approximate equation will have a form that is easy to optimize and yet describe the most important variations in velocity with angle. The errors introduced by analytic approximation are intended to be much smaller than errors introduced by inaccurate traveltimes. We might fit data first with more approximate curves and introduce refinements only when sensitivity improves.

The following equation parameterizes group velocities as a function of group angle $\phi$, measured from the vertical axis of symmetry:

$$\begin{eqnarray} V(\phi)^{-1} &= V_x^{-1} \sqrt{ 1+2 \eta \cos^2 (\phi ) \sin^2... ...+ \eta \cos^2 (\phi ) \sin^2 (\phi) + \epsilon \cos^2 (\phi ) ] .\end{eqnarray}$$ (1)

See the appendix for a fuller justification. This equation will have several advantages for tomography. The equation describes reciprocal velocity, or slowness, as an easily linearized function of three variables $V_x^{-1}, \eta , \epsilon$.The parameters of anisotropy $\eta$ and $\epsilon$ have small magnitudes, on the order of 0.05. The parameter $\epsilon$ controls an elliptical stretch, and $\eta$ controls the bulge of anellipticity.

Slowness is integrated as a function of distance to give traveltimes. The horizontal velocity V_x is well defined by a physical experiment and is measured accurately from surface or crosswell experiments; the velocity V_z along the vertical axis of symmetry is much less well determined. The first parameter $\eta$ in the square root is determined second best by surface experiments Alkhalifah and Tsvankin (1994); Tsvankin and Thomsen (1994) because it specifies the difference between a normal moveout (NMO) velocity and V_x according to equation (21). Finally, the parameter $\epsilon$expresses the third and least well determined part of anisotropy, giving the fractional change of vertical velocity according to (19).

The magnitude V_x will be allowed to change most arbitrarily, in many dimensions. Because this anisotropy is assumed to be a layered phenomenon, $\eta$ will only be allowed to change vertically (or perpendicular to layering). Because $\epsilon$ is so poorly determined by surface data, I will assume that it has a single global value which can be chosen to fit well ties when available. Or one may attempt to predict $\epsilon$ from the other two parameters by observing the correlation in values that are produced by equivalent media calculations from well logs Backus (1962).

We can express the continuous velocity functions as a scaled sum of smooth basis functions. For example, if $\vec{ \bf x}$ is an arbitrary Cartesian coordinate, then we can express the slowness $V_x^{-1}( \vec{ \bf x} )$ as a linear function of discrete parameters $\vec{ \bf s} = [ s_i ]$, using smooth basis functions $S_i ( \vec{ \bf x} )$: 

$$V_x^{-1} ( \vec{ \bf x} ) = \sum_i s_i S_i ( \vec{ \bf x} ) = \vec{ \bf s} \cdot \vec{ \bf S} ( \vec{ \bf x} ) .$$ (2)

If $\vec{ \bf \hat z}$ is the unit vector in the direction of the vertical axis of symmetry, then we can express the function $\eta ( \vec{ \bf x} )$ as a linear function of scale factors $\vec{ \bf \eta} = [ {\eta}_i ]$ and smooth one-dimensional basis functions E_i ( z ):  

$$\eta( \vec{ \bf x} ) = \sum_i {\eta}_i E_i ( \vec{ \bf x} \c... ...a} \cdot \vec{ \bf E} ( \vec{ \bf x} \cdot \vec{ \bf \hat z}) .$$ (3)

The third parameter $\epsilon$ is already a single constant.

Let us designate this discrete set of velocity parameters as a single vector $\vec{ \bf v} = [ \vec{ \bf s} , \vec{ \bf \eta} , \epsilon ]$. The different elements of $\vec{ \bf v}$ are understood to have different scales, and will assume different variances during optimization. The approximate magnitude of V_x in (2), and thereby of $\vec{ \bf s}$, is easily anticipated from a quick glance at traveltimes over certain short distances. Theory Backus (1962) can easily anticipate reasonable magnitudes for $\eta$ and $\epsilon$for equivalent layered media.

Now we can write the continuous group velocity explicitly as a function of these discrete parameters  

$$V( \vec{ \bf x} , \phi ) ^{-1}= [ \vec{ \bf s} \cdot \vec{ \... ...z}) \cos^2 (\phi ) \sin^2 (\phi) + \epsilon \cos^2 (\phi ) ] .$$ (4)

A finite perturbation is easily linearized as

$$\begin{eqnarray} \Delta [ V( \vec{ \bf x} , \phi ) ^{-1} ]= &[ \Delta \vec{ \bf ... ... \cos^2 (\phi ) \sin^2 (\phi) + \Delta \epsilon \cos^2 (\phi ) ] .\end{eqnarray}$$ (5)

Unperturbed parameters in this equation are understood to remain at their reference values. Reference velocities will be iteratively updated and relinearized by a Gauss-Newton optimization algorithm described later.