Computation

The ellipticity is defined by the ratio $\frac{r_1(z_0)}{r_2(z_0)}$ where $r_1(z_0)$ and $r_2(z_0)$ are the factors appearing in equation (3.24). This ratio can be calculated from the terms of matrix $R(z_0)$ as shown by equation (3.35). As detailed in section 3.1.4, the matrix $R(z)$ is never completely calculated during the dispersion curve computation and values of $r_{12}(z_0)$ and $r_{11}(z_0)$ are not available. However, it is possible to calculate the ratio $\frac{r_{12}(z_0)}{r_{11}(z_0)}$ from sub-determinants $r(z_0) \left\vert \begin{array}{cc}1 & 2 \ a & b\end{array}\right\vert$ as shown here below.

From the computation of the dispersion curve we know that $r(z_0) \left\vert \begin{array}{cc}1 & 2 \ 1 & 2\end{array}\right\vert\approx 0$ . The approximation comes from the fact that the dispersion curve is solved numerically with a finite precision. Here, the problem is assumed to be perfectly solved, and the approximation is dropped in the following equations. For simplicity, the $z_0$ dependency is also dropped

$$r_{11}r_{22} = r_{12}r_{21}$$

$$r \left\vert \begin{array}{cc}1 & 2 \ 1 & 3\end{array}\right\vert = r_{11}r_{23}-r_{13}r_{21}$$ (3.41)

$$ir \left\vert \begin{array}{cc}1 & 2 \ 1 & 4\end{array}\right\vert = ir \left\vert \begin{array}{cc}1 & 2 \ 2 & 3\end{array}\right\vert = r_{12}r_{23}-r_{13}r_{22}$$

It is useful to mention that $r \left\vert \begin{array}{cc}1 & 2 \ 1 & 4\end{array}\right\vert$ is imaginary as demonstrated by equation (3.35). We multiplied by $i$ as a real value is internally computed. The solution of the system of equations (3.41) is

$$\frac{r_1(z_0)}{r_2(z_0)}=\frac{r_{12}(z_0)}{r_{11}(z_0)}=i \frac... ...right\vert}{r \left\vert \begin{array}{cc}1 & 2 \ 1 & 3\end{array}\right\vert}$$ (3.42)

Thus, for elastic waves in a layered model, this ratio is an imaginary number either positive (prograde) or negative (retrograde). These terms come from the analogy between a rolling ball and the particle motion

For a half space, using equations (3.42) and (3.31),

$$\frac{r_1(z_0)}{r_2(z_0)}=i \frac{k(l_n-2 \hat k_n \hat h_n)}{\hat h_n(l_n-2k^2)}$$ (3.43)

From $r_{11}r_{22} = r_{12}r_{21}$ and equation (3.31), it comes that

$$4 k^2 \hat k_n \hat h_n = l^2_n$$ (3.44)

And equation (3.43) reduces to

$$\frac{r_1(z_0)}{r_2(z_0)} = -i \frac{2 \hat k_n k}{l_n} = -2 i \frac{\sqrt{1-(V_r/V_s)^2}}{2-(V_r/V_s)^2}$$ (3.45)

which is the classical formula given by Tokimatsu (1997) for a half space.

It is always a negative imaginary number and $r_1(z_0)$ and $r_2(z_0)$ are out of phase by 90$^\circ$ with each other. The particle motion at the surface is then always retrograde elliptical for a half space. In general, only the real absolute amplitude of the ellipticity is shown on a log-log plot.

Equation (3.42) proves that the ellipticity can be calculated at a very low cost once the dispersion has been correctly computed. However, the results are stable and reliable only if the dispersion problem (equation (3.34)) is sufficiently solved. Taking into account the error, equation (3.42) transforms into

$$\frac{r_1(z_0)}{r_2(z_0)}=i \frac{r \left\vert \begin{array}{cc}1... ...{r_{11}(z_0)r \left\vert \begin{array}{cc}1 & 2 \ 1 & 3\end{array}\right\vert}$$ (3.46)

where $e$ is the remainder of equation (3.34). Its magnitude is not constant because the iterations of the root search algorithm are stopped when the Rayleigh slowness (or the velocity) is estimated with a relative precision of $10^{-7}$ . No test are performed on the absolute value of the remainder. Calculating the error on the ellipticity value is never done because the total error depends upon particular terms of matrix $R(z_0)$ , not fully computed here. In most cases, experience has proved that the problem is sufficiently solved with a $10^{-7}$ relative precision on the dispersion curve. An exception to this rule is shown in the next section for a three-layer model where a $10^{-50}$ relative precision is necessary. Such computations are possible with numbers having more than 50 significant digits handled by the ARPREC library (Bailey (2004)).

In contrast to section 3.1.7, the misfit computation is presented after the sensitivity study because a better understanding of the particular shape of the ellipticity curve is necessary to define the misfit.