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).