Analysis of the Spatial Distribution Between Successive Earthquakes

Spatial distances between subsequent earthquakes in southern California exhibit scale-free statistics, with a critical exponent $\delta \approx 0.6$, as well as finite size scaling. The statistics are independent of the threshold magnitude as long as the catalog is complete, but depend strongly on the temporal ordering of events, rather than the geometry of the spatial epicenter distribution. Nevertheless, the spatial distance and waiting time between subsequent earthquakes are uncorrelated with each other. These observations contradict the theory of aftershock zone scaling with main shock magnitude.

Figure 1

Rescaled distribution of jumps, $P_{m,L}(\Delta r)$, for different values of $m$ and $L$. $L=448.5\mathrm{k}\mathrm{m}$ corresponds to the full area where there is just one cell. Values of $\Delta r<2\mathrm{k}\mathrm{m}$ have been discarded due to uncertainties in estimating the epicenters' locations. The solid line with exponent $\delta =0.6$ is shown as a guide to the eye.

Figure 2

The jump distribution, $P_{m,L}(\Delta r)$, for $m=2.4$ and $L=448.5\mathrm{k}\mathrm{m}$. (Orange) circles correspond to the distance between successive earthquakes $\Delta r_i=|{\mathbf{r}}_{i+1}-{\mathbf{r}}_i|$ as in Fig. 1. (Red) squares correspond to $\Delta r_i=|{\mathbf{r}}_{\sigma (i+1)}-{\mathbf{r}}_{\sigma (i)}|$ where $\sigma$ is a randomly chosen permutation of the integers $0<i<N$, giving a shuffling of the catalog as described in the text. The (black) triangles correspond to the distance between any two earthquakes in the same cell. The solid line is a fit to the latter distribution from 2 to 200 km with a critical exponent of ${\delta }_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{f}}=-0.14\pm 0.03$.

Figure 3

The velocity distribution, $P_{m,L}(v)$, for $L=100\mathrm{k}\mathrm{m}$ and different values of $m$: (Blue) crosses and squares correspond to the original data with $v_i=\Delta r_i/\Delta t_i$. (Red) triangles correspond to a randomization $v_i=\Delta r_{\sigma (i)}/\Delta t_i$, where $\sigma$ is a randomly chosen permutation of the integers $0<i<N$. Note that $\Delta r_i>2\mathrm{k}\mathrm{m}$ and $\Delta t_i>60\mathrm{s}$ for all $i$ to avoid any bias due to the uncertainties in the estimates. Inset: Rescaled distributions, using the GR exponent $b=0.95$ and $\eta =1.0$.