Semblance processing of the data

As mentioned above, a semblance method is used to look for sources. First we create a 3-D grid of points in the subsurface. Then for each point in the grid, knowing the velocity model, we compute the moveout trajectory that energy originating from a source at that point would follow. The semblance is computed along that trajectory, and averaged over time.

A simple model of the seismic record is assumed:

$$S_{i}(t)=\sum_{k=1}^N{x_{k}(t-t_{ki})}+n_i(t).$$

Here x_k is the signal generated by source k, t_ki is the time delay, determined by velocity model and the positions of the source k and geophone i, and n_i is spatially uncorrelated noise.

An additional assumption is that signals from different sources are mutually uncorrelated and that signals from any source are well correlated from geophone to geophone.

This model is obviously too simplistic to give an accurate description of the real situation, but the idea is that if there is a powerful enough source beneath the array, it will be possible to detect it using this model.

Assuming that the velocity model of the area is known, a simple coherency measure, the semblance coefficient, can be computed:

$$Sem(x,y,z)={\sum_t{(\sum_i{S_i(t+t_i)})^2}\over {M\sum_t{\sum_i{S_i^2(t+t_i)}}}},$$

Here x, y, and z are the coordinates of a point in the medium, t_i is the travel time from that point to geophone i, and M is the number of channels.

The semblance coefficient is convenient because of the following property: If seismograms are absolutely uncorrelated, then it equals 1/M, and in the case of good correlation the semblance tends toward 1. That means that if there is a source of seismic energy at some particular point in the medium, we can expect to detect a large semblance value at that point.

We will apply this method to microseismic noise data from the Iceland experiment. But first we will describe the experiment and use some software developed at SEP to analyze the ambient wavefield.