Earthquake extraction and correlation energy at Long Beach, California seismic survey

Interpolation

To create these snapshots, we implemented a normalized inverse distance weighting interpolation scheme. First, we normalized the amplitudes of the recordings because we did not want anomalously high amplitudes to dominate the interpolation results. Second, we created a $40$ by $80$ grid of rectangular cells that overlapped with the $8.5$  km north-south by $4$  km east-west region containing the array. To determine which stations would be used for interpolating the response at a given cell, we implemented nearest-neighbor binning.

Rather than average the recordings from each station for a given cell, we apply a weight to the relevant recordings for a given cell and then sum the recordings. The weight is the normalized inverse of the distance between the location of the recording and the center of the cell, having the form

$$\mathbf{u}(\mathbf{x}) = \sum\limits_{i=1}^N \frac{w_i(\mathbf{x})\mathbf{u}_i}{\sum\limits_{i=1}^N w_i(\mathbf{x})} ,$$ (1)

where

$$w_i(\mathbf{x}) = \frac{1}{d_i(\mathbf{x},\mathbf{x}_i)} .$$ (2)

$\mathbf{u}(\bf {x})$ is the interpolated recording at cell $\mathbf{x}$ , $w_i(\mathbf{x})$ is the weight applied to station $i$ when interpolating for cell $\mathbf{x}$ , $\mathbf{u}_i$ is the recording at station $i$ , and $d_i(\mathbf{x},\mathbf{x}_i)$ is the distance between the center of the cell $\mathbf{x}$ and the relevant station at location $\mathbf{x}_i$ . This interpolation scheme weights recordings that are closer to the center of the cell more so than those recordings that are closer to the edges of the cell. The normalization of the weights by the sum of the weights ensures that the amplitudes from cell to cell are relatively similar, particularly because the amplitudes at each station were first normalized.

Earthquake extraction and correlation energy at Long Beach, California seismic survey