Resonant multiple wave scattering in the seismic response of a city

When a seismic wavefield impinges on the foundation of a building, the building vibrates and generates waves in the subsoil. In a city, different buildings interact with each other through the scattered waves. The detailed description of the wave propagation in this coupled city - soil system is a complex problem. Instead of solving this problem for a particular city configuration, a statistical description of the city is applied and the limit of a city of infinite size is considered. This leads to a model of the coupled city - soil system, where the buildings are modelled as resonant scatterers that are uniformly distributed at the surface of a deterministic, horizontally layered elastic half-space that represents the soil. The equations that govern the interaction between the city and the soil now become a set of stochastic equations. Based on these equations, the Dyson and Bethe - Salpeter for the configurationally averaged field and field correlation are formulated. The solution of the single scatterer problem is used to obtain an approximate solution of these equations that allows us to quantify the change of the mean site response through the presence of the city and the ratio of the coherent and incoherent response. Furthermore, the influence of the city on the duration of the seismic records is estimated by the approximate solution of the non-stationary Bethe - Salpeter equation. The results obtained for the configurationally averaged field quantities are validated by means of results for the seismic response of a deterministic model of a city quarter of Mexico City.