wave field, he demonstrated that the correlation of the signals recorded at two stations separated by a distance r can be written: (1) is the azimuthal average of the correlation , c(ω) is the phase velocity and Jn is the Bessel function of order n. (2) and are the recorded signals at two stations separated by distance . A typical distance between two stations or aperture can be of the of 100m. But trees, streets, buildings and other obstacles can result in an irregular shape array and it is not always possible to have equal length aperture and good azimuthal coverage at the same time. For such imperfect arrays the method proposed by Bettig et al. (2001) is used, grouping pairs of stations along rings of finite thickness. A modified equation is used, and the interested reader is referred to the original paper (Bettig et al., 2001). This modified version of SPAC is called MSPAC but for reasons of simplicity in this report we’ll call it just SPAC. SPAC inversion The goal of SPAC inversion is to infer the parameters of the soil structure (Vs values) from the measured SPAC ratios. Assuming an ambient vibration wave field made mostly of surface waves, the SPAC curves are linked to the dispersion curve through equation (2). The dispersion curve is defined for the case of vertically heterogeneous 1D models and its inversion is a classical problem (Hermann, 1987). Recently, the Neighbourhood Algorithm (NA) developed by Sambridge (1999a,b) has been used for the inversion. This explores the parameter space more thoroughly than conventional methods and is the method used in the present work. The misfit for the NA is calculated by comparing the theoretical curves obtained from equation (2) and the SPAC data curves: (3) : SPAC ratio of data curves at frequency fj and for ring i : SPAC of calculated curves at frequency fj and for ring i : observed variance for the sample at frequency fj and for ring i nR: number of rings nFi: number of frequency samples for ring i nFk: number of frequency samples for ring k
wave field, he demonstrated that the correlation of the signals recorded at two stations separated by a distance r can be written: (1) is the azimuthal average of the correlation , c(ω) is the phase velocity and Jn is the Bessel function of order n. (2) and are the recorded signals at two stations separated by distance . A typical distance between two stations or aperture can be of the of 100m. But trees, streets, buildings and other obstacles can result in an irregular shape array and it is not always possible to have equal length aperture and good azimuthal coverage at the same time. For such imperfect arrays the method proposed by Bettig et al. (2001) is used, grouping pairs of stations along rings of finite thickness. A modified equation is used, and the interested reader is referred to the original paper (Bettig et al., 2001). This modified version of SPAC is called MSPAC but for reasons of simplicity in this report we’ll call it just SPAC. SPAC inversion The goal of SPAC inversion is to infer the parameters of the soil structure (Vs values) from the measured SPAC ratios. Assuming an ambient vibration wave field made mostly of surface waves, the SPAC curves are linked to the dispersion curve through equation (2). The dispersion curve is defined for the case of vertically heterogeneous 1D models and its inversion is a classical problem (Hermann, 1987). Recently, the Neighbourhood Algorithm (NA) developed by Sambridge (1999a,b) has been used for the inversion. This explores the parameter space more thoroughly than conventional methods and is the method used in the present work. The misfit for the NA is calculated by comparing the theoretical curves obtained from equation (2) and the SPAC data curves: (3) : SPAC ratio of data curves at frequency fj and for ring i : SPAC of calculated curves at frequency fj and for ring i : observed variance for the sample at frequency fj and for ring i nR: number of rings nFi: number of frequency samples for ring i nFk: number of frequency samples for ring k