A fuzzy inference system applied to estimate the error in the recovery of the Green's function by means of seismic noise correlations

A novel fuzzy logic mathematical formulation to obtain accurate synthetic seismograms by means of correlations with noise is presented in this work. Recently, the use of seismic noise, surface waves and seismic ambient information for describing the structure of the Earth has captured the interest of seismologists, geophysicists and in general, the geoscientist community. By means of seismic noise correlations, it is achievable to retrieve some important characteristics of the propagation medium (e.g. the Green's function and wave velocities). However, the precise retrieval values of such characteristics greatly depend on the amount of seismic noise (number of random sources) and even on the difficulty in describing the subsoil that constitutes the propagation media. The impossibility of having exact values of these parameters prohibits the estimation of the error in the recovery of the Green's function by conventional methods, which makes it necessary to choose alternative computational methods, such as fuzzy logic. Firstly, the equations applicable to the seismic noise correlation and its relationship with the Green's function are established, validating with previously published results, and later applying the fuzzy logic to estimate the error in the recovery of the Green's function. In the last section, we propose the design, training and implementation of a fuzzy inference system. To this end, our method is inspired from the Sugeno model, but with modifications that allow the entry of membership functions (in contrast to discrete values) corresponding to the number of environmental sources and to the soil type in question, as merely qualitative values. Based on these, the error in the recovery of the Green's function is calculated.