Power law variation

The velocity (either $V_p$ or $V_s$ ) at depth $z_i$ is given by

$$V_i=V_0 ((z_i+1)^\alpha-(z_0+1)^\alpha+1)$$ (4.2)

where $z_0$ is the top of the layer considered, $V_0$ is the velocity at $z_0$ , $\alpha$ is the power-law exponent, generally varying between 0 and $1$ . The substraction in equation 4.2 is necessary if the power law variation is used for deep layers ($z_0 > 0$ ). Like the linear profiles, the function $V_i(z)$ is divided into a fixed number of homogeneous sub-layers. Setting the exponent $\alpha$ as a parameter is not a good choice, because it generates models with an uncontrolled maximum velocity. The situation is even worse if several heterogeneous layers are used in the same structure. A better solution is to set the top ($V_0$ ) and the bottom ($V_n$ ) velocity as two distinct parameters. For the conditional neighbourhood algorithm, the simple condition $V_0<V_n$ is introduced. For the standard neighbourhood algorithm, $V_0$ and $dV$ are the parameters, $V_n$ being equal to $V_0+dV$ . $\alpha$ is calculated by solving the following equation:

$$f(\alpha)=(z_n+1)^\alpha-(z_0+1)^\alpha-\frac{V_n-V_0}{V_0}=0$$ (4.3)

A few iterations with the bissection method are generally necessary. There is always only one solution between 0 and 1 because $f(\alpha)$ is monotonously increasing. Other iterative methods are not appropriate.

If the thicknesses of the sub-layers are constant, the power law variation is badly sampled. Very high velocity jumps are observed for the first sub-layers. Thus, it is better to impose a constant velocity jump from one sub-layer to the next one, equal to $\frac{dV}{n}$ . The depth of the top and of the bottom of each sub-layer is then easily calculated from

$$z_0, \ldots, \left( i\frac{dV}{V_0}+(z_0+1)^\alpha\right)^{1/\alpha}-1,\ldots,z_n$$ (4.4)

Inside each sub-layer, for the sake of simplicity, we set the velocity of the sub-layer to the value of the analytical power law function at the middle of the sub-layer. Hence,

$$V_i=V_0\left[\left(\frac{z_{i-1}+z_i}{2}+1\right)^\alpha-(z_0+1)^\alpha+1\right], i=1,\ldots,n$$ (4.5)

To summarize, from the thickness of the layer and $dV$ (or $V_n$ ), it is possible to define in a unique way the individual thicknesses of each sub-layer and their velocities. An intermediate computation is necessary to obtain the value of the exponent. The exponent $\alpha$ can be recalculated from the thicknesses and the velocities of the two first sub-layers by solving the equation

$$g(\alpha)=(z_2+1)^\alpha-\frac{V_2}{V_1}(z_1+1)^\alpha+\left(\frac{V_2}{V_1}-1\right)(z_0+1)^\alpha=\frac{V_2}{V_1}-1$$ (4.6)

$g(\alpha)-\left(\frac{V_2}{V_1}-1\right)$ is also monotonously increasing and have only one root between 0 and 1. It is solved by bissection.

Figure 4.17: Inversion with a three-layer model with heterogeneous layers, with prior information about the depth of basement. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Dispersion curves corresponding to models of figures (a) and (b). The black dots are the theoretical dispersion curve used as the target curve during inversion.

Figure 4.18: Inversion with a three-layer model with heterogeneous layers. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Dispersion curves corresponding to models of figures (a) and (b). The black dots are the theoretical dispersion curve used as the target curve during inversion.

The parameters for a layer with power law gradient are $V$ (either $V_p$ or $V_s$ ), $dV$ ( or $V_n$ ), and the thickness $H$ (or $z_0$ and $z_n$ , the depth of the top and of the bottom of the layer). The number of sub-layers is only a tuning parameter.

An example of the use of layers with a power law variation in the inversion is shown in figure 4.17. This is the same case as in figure 4.10 where the two first homogeneous layers are replaced by layers with power law variations. The number of fixed layers is five in each case. Two parameters are added to the parameter space (making a total of 10 parameters) of the inversion plotted in figure 4.10. The $V_p$ variation across the layers can vary between 0 and 2000 m/s. The minimum achieved misfit is similar to the homogeneous case, but the posterior uncertainty on the second layer is larger in these later inversions.

This kind of layer is also tested with a large band dispersion curve (from 0.2 to 20 Hz) in figure 4.18. Compared to figure 4.6, the uncertainty are slightly increased.

Figure 4.19: Comparison of three type of parameterizations ($V_s$ profiles): (a) inversion with a three-layer model with homogeneous layers, (b) inversion with a three-layer model with gradient layers, and (c) ten layers of fixed thicknesses, accepting low velocity zones.