Thickness, $V_p$ , and $V_s$

For each layer of the ground model, the considered parameters are: the thickness ($h$ ), and the velocities $V_p$ and $V_s$ . The density is generally not inverted here as its influence on the dispersion curve is usually small compared to the other parameters' one (section 3.1.8).

In this section, we make use of the standard neighbourhood algorithm developed in Fortran by Sambridge (1999a). For each generated parameter set, a misfit value must be calculated by the forward algorithm, even if the parameters do not fulfil with physical and prior conditions. With the original inversion code, it is not possible to reject a particular model. The wrong model might be discarded by returning an arbitrary high misfit to the neighbourhood algorithm. However, we prove hereafter that it is an inefficient method, especially when the number of parameters is increasing. Assuming a parameter set $P_1 \ldots P_n$ , when there is only one physical condition of the type $P_i<P_j$ , there is one chance over two to get a good model. From the combinatorial probabilities, if the number of conditions increases up to $m$ , the chance of getting one good model reduces to $\frac{1}{2^m}$ . Typically, for a three-layer model, the number of parameters is 8 and the number of physical conditions of the type $P_i<P_j$ is also 8. Hence, the probability of generating one good model is 1/256. Usual values for the tuning parameters of the neighbourhood algorithm are $it_{max}$ =100, $n_s$ =100, and $n_r$ =100 to generate 10000 models. In most cases, three iterations are thus necessary to get at least one good model. At the next iteration, 100 new models are generated in the 100 best cells. Hence, one new model is added close to the good model and 99 other models are still selected in the wrong regions of the parameter space. Finally, very few good models are obtained and the good regions of the parameter space are poorly investigated. All the wrong models are stored by the neighbourhood algorithm and all of them are included in the computation of the Voronoi geometry. As the number of models is increasing, the rate of the model generation is always decreasing, slowed down by useless wrong models. At the end of our work, we developed a modified neighbourhood algorithm that takes into account the model rejection in an efficient way (sections 2.4). However, this study is based on the standard algorithm which requires an appropriate parameter transformation in order to avoid generating wrong ground models. This part is covered in this section.

The thicknesses of the layers may take whatever positive value. Thus, the transformation is just a linear scaling from $[0,1]$ to $[h_{min}, h_{max}]$ . The layer thicknesses may also be set by specifying the absolute depth of the bottom of each layer ($z_i$ ). In this case, the user must avoid overlapping of the depth ranges which may induce negative thickness. As for thickness, it also reduces to a linear scaling from $[0,1]$ to $[z_{min}, z_{max}]$ . Mixtures of both types of position parameters are not possible in the developed software.

$V_p$ and $V_s$ are linked by Poisson's ratio. For geological materials, Poisson's ratio is always between 0 and 0.5. Hence, $V_p$ and $V_s$ must satisfy the following inequalities $0<V_s<\frac{\sqrt{2}}{2}V_p=0.707 V_p$ . There are two alternatives to parameterize $V_p$ and $V_s$ satisfying the conditions, which both make use of ratio $V_{sp}(\nu)=\frac{V_s}{V_p}$ :

1. Calculating $V_s$ from the first parameter with a scaling from $[0,1]$ to $[V_{s,min}, V_{s,max}]$ . The second parameter is the ratio $V_{sp}$ scaled to $[V_{sp,min}, V_{sp,max}]$ where $0<V_{sp,min}$ and $V_{sp,max}<\frac{\sqrt{2}}{2}$ .
2. Calculating $V_p$ from the first parameter with a scaling from $[0,1]$ to $[V_{p,min}, V_{p,max}]$ . The second parameter is the ratio $V_{sp}$ with the same limits as in the last case.

The first option is more intuitive because $V_s$ has the greatest influence on the dispersion curve. However, the generated $V_p$ values range from $\sqrt{2}V_s$ to $\infty$ or to any value above common real observations. Secondly, the prior probabilities of $V_s$ and $V_{sp}$ are uniform on the user specified range. Considering parameters independently ($V_s$ or $V_{sp}$ ), it means that the whole parameter space is equally investigated. From the parameterization point of view, every model has an equal chance to be taken at random. However, considering $V_p$ , it is the ratio of two uniform random variables $V_s$ and $V_{sp}$ , and its density of probability is far from being constant over the user specified range. Thus, some $V_p$ values have more chances to be generated by the neighbourhood algorithm than others. Because $V_p$ is not always well constrained by the dispersion curve, the parameterization may artificially orientate the inversion towards particular models rather than exploring the whole parameter space. From the user point of view, the $V_p$ profile may appear better constrained than it is really.

On the other hand, taking the second option, $V_p$ profiles are uniformly investigated. Because $V_s$ is relatively well constrained by the dispersion curve, the influence of the parameterization is only sensitive at the beginning of the process. Once the area of solution is delineated, the bias introduced by the non-uniform probability becomes negligible. Also, the range for $V_p$ is fixed by the user and no abnormal $V_p$ value is generated. $V_s$ values are always less than $\frac{\sqrt{2}}{2}V_p$ . For models with a reasonable number of layers (up to three or four), this option is probably the best one and it has been chosen in the software implementation tested in the next sections.

For a stack of layers, a common condition is the absence of low velocity zones or a monotonous increasing profile. This aspect is studied in section 4.3 for a stack of $N$ layers. The increasing of velocity with depth may be parameterized by setting the velocity increment at each interface as parameters ($P$ ), and $(V_p)_i=(V_p)_{i-1}+P$ . $V_s$ is calculated as above with the values of $V_{sp}$ . Low velocity zones may still appear on $V_s$ profiles. When necessary, they may be avoided by multiplying the final misfit by a penalty factor, function of the magnitude of the low velocity zone. This technique works only for a reduced number of layers (up to three or four), for reasons probably similar to the ones detailed in the introduction of this section. We use it in the simple parameterizations hereafter.