$P-S_V$ refracted waves

Classical refraction (Mota (1954)) is achieved with sources placed at the two ends and at the middle of the line. Sources are located at sensor positions in order to control the time reference. The first P-wave arrival times are picked on the signals. If the ground structure is made of inclined homogeneous layers with increasing velocity with depth, the traveltime-distance plot allows the geometry and the seismic velocity of the layers to be retrieved (Mota (1954)). With a limited set of data (24 values maximum) and considering the experimental uncertainties which can be high in noisy conditions, the solution is rarely unique and several $V_p$ profiles may fit the data in a similar way.

Figure 1.3: Reference model for refraction synthetic traveltime-distance plot. (a) Traveltime-distance plot for three sources. (b) Interfaces of the reference model and the ray paths with the minimum traveltimes. (c) Velocity profile at horizontal distance 0.

Figure 1.4: Inversion of synthetic traveltime-distance plot. (a) Traveltime-distance plot for three sources calculated for the generated models. (b) Interfaces of the generated models. (c) Generated velocity profiles at horizontal distance 0.

With the aim of extracting the different solutions explaining the experimental traveltimes in an objective way we developed a simple method based on the neighbourhood algorithm (Sambridge (1999a), chapter 2). The method, the principles of which are identical to the inversion of dispersion curves (Wathelet et al. (2004)), generates two random one-dimensional $V_p$ profiles with a fixed number of layers, which define a model with inclined layers. The $V_p$ value within each layer is randomly chosen inside an interval defined from a prior knowledge of the geological structure. For each generated model, the ray paths are analytically calculated using the Snell-Descarte refraction law for inclined interfaces and the traveltimes are computed for all source-receiver distances. The experimental time-distance values are compared to the calculated ones using the following misfit function:

$$misfit=\sqrt {\frac{1}{n} \sum_{i=0}^n \left( \frac{t_{exp}-t_{calc}}{t_{err}} \right)^2}$$ (1.13)

where, $t_{exp}$ is the experimental arrival time corrected by the initial time delay, $t_{calc}$ is the calculated arrival time for the current model, $t_{err}$ is the phase picking error or equal to $t_{exp}$ if no error estimation is available, and $n$ is the number of receivers. The experimental error, which depends upon the sharpness of the P-wave arrival and the signal to noise ratio, is manually estimated. This method was tested with success on synthetic models with constant velocity layers and dipping interfaces, using two shots made in opposite directions. This technique is used for our real test case in section 6.2.1 on page 6.2.1.

The method is illustrated for a reference model with three layers (figure 1.3(b)). The constant velocity inside each layer is represented in figure 1.3(c) by the $V_p$ profile measured on the left side of the model (distance=0). The thin black lines in figure 1.3(b) are the ray paths with the minimum traveltimes corresponding to the plot of figure 1.3(a). Figure 1.3(a) is the traveltime-distance plot obtained with three sources placed at the two extremities and in the middle of the section. Two inversion processes are launched generating 30,000 models among which 18,000 have a misfit lower than 0.02. The lowest misfit found is 0.00063. The generated models are shown in figure 1.4(b) and 1.4(c). The corresponding traveltime-distance plots are visible in figure 1.4(a). Considering 0.02 as an acceptable misfit, the depth of the first interface is correctly retrieved but the depth of the second one is poorly constrained by the refraction experiment, unless a very high precision can be achieved while picking the arrival phase of the distant receivers. From figure 1.4(c), the velocity is correctly inverted down to 16 m. Below 16 m, if all models with a misfit lower than 0.02 are equally acceptable, any velocity between 1000 and 4000 m/s is equally valid to explain the experiment results.