Discussion and Conclusions

Three processing methods have been tested to retrieve the dispersion properties (dispersion curves or auto-correlation curves) on a two-layer model from simulated noise array measurements: the f-k method, the high-resolution f-k method and the spectral auto-correlation technique. Only the vertical components are processed and the dispersion (or auto-correlation) curves are inverted to obtain the $V_s$ profile. The first conclusion is that several array apertures have to be used to construct the dispersion (auto-correlation) curves in the appropriate frequency range. From the inverted velocity profile point of view, all three methods have almost the same efficiency for this synthetic case. The $V_s$ profile is correctly retrieved down to about 25 m which is the depth of the interface.

A special attention is paid to the limited reliability specific to each array. Based on the knowledge of the true dispersion curve, we conclude that the wave number limits deduced from the theoretical array response are consistent with the capabilities of the f-k method. Outside those limits, the calculated curves may exhibit strong bias. The high-resolution f-k method is sometimes more efficient than the f-k approach in defining the dispersion curve but no definitive and systematic improvement may be found. Like the auto-correlation method, the high-resolution method can be seen as complementary technique confirming the results of the f-k method.

No method is able to retrieve the velocity below the interface at 25 m. This limited penetration depth is a direct consequence of the high-pass filtering effect of the ground structure on the vertical component. This characteristic is a strong limitation of the method for assessing the local amplification factor in earthquake engineering, which depends upon the value of the velocity contrast.

Figure 6.19: Inversion of Love and Rayleigh fundamental modes for perfect dispersion curves. (a) $V_s$ models. (b) Calculated Love dispersion curves. (c) Calculated Rayleigh dispersion curves. The black dots are the target Rayleigh dispersion curves. The grey dots are the target Love dispersion curve.

Further improvement of the technique should consider horizontal components which are richer in low frequency waves than the vertical ones (figure 6.2). This alternative is tested in section 5.1.2 where the Love dispersion curve measured at low frequency combined with the Rayleigh dispersion curve allows an extention of the reliability of the inversion towards deeper layers. We compute the theoretical Love and Rayleigh dispersion curves for the ground model used in simulations. The Rayleigh dispersion curve is cut between 2.5 and 8.5 Hz as observed in figure 6.2 while the Love dispersion curve is supposed to be known only between 1.5 and 2.5 Hz, in the vicinity of the fundamental resonance frequency. The theoretical Love samples used for the inversion are represented with grey dots, and the Rayleigh samples with black dots in figures 6.19(b) and 6.19(c). These two curves are jointly inverted with five independent runs and the results are shown in figure 6.19. Only the $V_s$ profiles of the generated models are shown in figure 6.19(a). The corresponding calculated dispersion curve for Love and Rayleigh are shown in figure 6.19(b) and 6.19(c), respectively. Compared to the Rayleigh-wave inversions, the combined Love- and Rayleigh-wave inversion correctly retrieves the $V_s$ value below the main velocity contrast at 25 m. This result stresses out the interest of developing techniques of Love wave extraction from noise array measurements.