4.2.3 Effect of Spatial Bins, constant β or Variableβ (Strategies 5 and 6)
To determine a reasonable along-strike spatial bin size to use, we first perform a simple test to compare results of increasing sized spatial bins. Smaller bins allow for greater resolution of spatial variability, and are closer to the assumptions of the underlying EGF approach (e.g., Abercrombie, 2015), but in the stacking approach, it is the large number of earthquakes included (e.g., at least 200 as used in AS2007) that provides stability. If the spatial bin size is too small, the number of events will be inadequate to obtain a stable ECS; if the bin size is too big, the spatial variation in ECS will be insufficient to account for heterogeneity.
We select 3 different bin sizes (1x1 km2, 5x1 km2 and 10x1 km2, along-strike between 20 and 30 km, and at 4-5 km depth) from a part of the fault with sufficient earthquakes for stable inversion (Figure 4). We use a narrow depth range to minimize the effects of depth dependence in this test. The three bins include 350, 1100, and 1400 earthquakes respectively, and the smaller bins are subsets of the larger bins. We solve for ECS and source parameters within each bin separately, and then compare the results for common events (that is, the 350 events in the 1km bin).
We find that the ECS from the three spatial bins are almost identical with only minor deviations from the 1 km bin (Figure 4), and the corner frequencies and stress drops are nearly identical for the common earthquakes from bins with different sizes. Only minor deviations occur when the corner frequency approaches or exceeds the frequency limit of the data. This comparison suggests that ECS can be assumed to be constant within a 10 km along-strike bin size to resolve earthquake source parameters. This 10 km bin size is large enough that there are enough earthquakes in most bins to apply the stacking analysis with both along strike and depth varying ECS.
Based on this test, we adopt the following preferred spatial binning strategy:
(1) split the 80km long along-strike fault section into 8 along-strike bins of 10 km;
(2) split the along-strike bins into 1-4, 4-5, 5-8 and 8-15km depth bins (i.e., the same as depth ranges used in Strategies 3 and 4, Section 2.2.2);
(3) only invert spatial bins containing at least 200 events to guarantee enough earthquakes above Mw>0.9 (Figure S4).
We also compare the performance of the original SNSS in Chen & Abercrombie (2020) with the new hybrid SNSS in Figure S5. We repeat the analysis both assuming a constant shear wave (and rupture) velocity (Strategy 5) and allowing shear wave velocity (β ) to vary with depth (Strategy 6). The results are compared to those in the previous inversions. We note that the spatial bins further reduce the magnitude dependence between Mw0.75 and Mw1.75 with slopes reduced to 0.384 and 0.354 with constant and variable β , respectively (Table 1). The depth dependence does not change significantly compared to simple 1-dimensional depth binning (Strategies 3 and 4).