Appendix O: Gridded Seismicity Implementation
Peter M. Powers and Edward H. Field
Introduction
Gridded seismicity sources represent all earthquakes in a forecast that are not associated with
individual faults. Their role is important in that they express future, distributed earthquake
occurrence and account for the fact that many large earthquakes do not occur on known, mapped
faults.
The 2008 NSHM gridded source model (Petersen, et al., 2008), which was adopted in UCERF2,
divides the western US into discrete regions (Fig. 1), treating each as an independent set of
gridded sources with common parameters (e.g. maximum magnitude [Mmax], strike, b-value, focal
mechanism). The individual regions reflect differences in the historic catalog (e.g. quality,
completeness) and tectonic style. Spatially uniform Mmax and b-values and spatially variable avalues, or rates, constrain the magnitude-frequency distribution (MFD) of each source in a region.
Source rates are derived from the declustered catalog in each region that was smoothed using an
isotropic Gaussian kernel with a 50km radius. In California, several sub-regions receive special
treatment: Mendocino, Imperial, Brawley, and the Creeping section of the San Andreas Fault.
These regions either have fixed strike sources, Mmax and b-values that differ from the broader
regional value, or were smoothed using an anisotropic kernel. A primary challenge in building a
gridded seismicity forecast is to avoid double counting at grid nodes that overlap spatially with
faults. The 2008 NSHM addressed this issue by reducing Mmax of sources within 10km of nearvertical faults and over dipping faults.
The focal mechanisms of each source reflect the tectonic style of the parent region, with each
source assigned a weighted combination of strike-slip (SS), reverse (R), and normal (N) motion
where the weights sum to 1. For example, grid sources spanned by the coastal California catalog
have weights of SS=0.5 and R=0.5 whereas sources spanned by the western US extensional
catalog have weights of SS=0.5 and N=0.5. As for implementation, UCERF2 defines sources at
the centers of the 7636 grid nodes in the 0.1˚ by 0.1˚ discretization of California and its buffer
zone (the RELM test region). Because catalog smoothing generates overlap between the NSHMP
source regions, any UCERF2 grid source is a composite of all coincident NSHMP regional
sources with non-zero rate.
UCERF3 is fundamentally different than UCERF2. With the introduction of the Grand Inversion
and fault-system-wide solutions, there are regional constraints on the rate of off-fault earthquakes
that are uniquely balanced with on-fault rates for each solution. We present a model of gridded
seismicity sources that is compatible with UCERF3 Grand Inversion results, avoids double
counting, considers smoothed historic seismicity, and maintains continuity with UCERF2 and the
2008 NSHM.
Gridded source model
As with UCERF2, UCERF3 gridded seismicity sources are spaced at 0.1˚ by 0.1˚ degree nodes
over the RELM testing region and represent all earthquakes other than ruptures large enough to
span the seismogenic depths of modeled faults. The magnitude-frequency distribution (MFD) of
each gridded source is the sum of the MFDs of all sub-seismogenic ruptures (small events) on
faults that intersect the grid node and the MFD of all events unassociated with a particular fault.
UCERF3: Appendix O
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Every Grand Inversion model supplies an implied, total off-fault MFD. The following outlines
the methodology used to partition the inversion-supplied MFD over all grid sources in the model.
Fault-section polygons
The UCERF3 Fault Models (see Appendix A) introduce the notion of fault-zones as defined by
polygons at the earth's surface (Fig. 2). Every fault in a particular Fault Model is meant to be a
proxy for all events that occur within its designated polygon (Appendix A), both seismogenic
(large) and sub-seismogenic (small). The latter are considered here to be part of the gridded
source model. It follows that if the entire area of a grid cell falls within a fault polygon, then it
represents only sub-seismogenic ruptures on the associated fault, implying zero rates of events
above the minimum inversion-based magnitude for that fault. If the entire area of a grid cell falls
completely outside of all fault polygons, then it represents only off-fault seismicity. Grid cells
that fall partly within and partly outside one or more fault polygons get proportional
representation through bookkeeping procedures described below.
The UCERF3 Fault Models define polygons on a per-fault basis, but inversion results and the
UCERF3 forecast consider much smaller subdivisions of each, or ‘fault-sections’. Because faultsection length varies across model branches, we define fault-section polygons programmatically
rather than in the source Fault Models. For each fault-section in an inversion result, we slice the
parent Fault Model polygon at the section endpoints in the average direction of dip (Figs. 3 and 4).
Lines that bisect the angle between adjacent fault-sections were considered, but the resulting
polygons were difficult to manage for faults with complex strikes (e.g. San Cayetano; Fig. 2).
Because the down dip extensions of many faults extend beyond the boundary of their polygon, we
merge the geologically defined polygon with the surface projection of each fault-section at depth
(Figs. 3 and 4). This second step provides polygons for many sections that have no geologic or
surface trace polygon defined in a Fault Model and maintains consistency with the 2008 NSHMP
approach of using surface projections of dipping faults.
Many fault-zone polygons consider additional surface geologic data and have complex outlines
that result in non-singular polygons when sliced. For example, the fault polygon for the Mission
Ridge fault (Figs. 2 and 4) has a splay at its western end. Although the tip of the splay could be
considered to be part of the westernmost section, for the fault-section discretization applied, we
include it with the previous section to maintain polygon singularity, which simplifies
bookkeeping.
MFD distribution
Inputs
The required inputs to the method are a Grand Inversion solution, a spatial PDF of seismicity
rates, and a region of nodes for which sources are to be defined. The Grand Inversion solution
supplies the necessary fault-section data (geometry and length, the parent fault-zone polygon, the
MFD of seismogenic ruptures, and moment rate reduction) and the implied total, regional offfault MFD. The spatial PDF of seismicity rates could be the UCERF3 smoothed seismicity
distribution that uses a variable radius Gaussian smoothing kernel (Appendix M), the UCERF2
smoothed seismicity model, a PDF based on the average off-fault deformation implied by the
deformation models considered in UCERF3 (Appendix C), or any other (Fig. 5). Sources are
generated over the RELM testing region, which includes all of California and a buffer, discretized
at 0.1˚ by 0.1˚ nodes.
UCERF3: Appendix O
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Definitions
For the purposes of this discussion, grid-nodes are indexed over i, and fault-sections are indexed
over k. Other subscript indicators are as follows:
T
Total
L
seismogenic fault ruptures (Large)
S
sub-seismogenic fault ruptures (Small)
U
ruptures Unassociated with a fault
The implied total, regional off-fault MFD supplied by a Grand Inversion solution is:
𝐷𝑇 (𝑚) = ∑ 𝐺T𝑖 (𝑚) = ∑ 𝐺S𝑖 (𝑚) + ∑ 𝐺U𝑖 (𝑚)
where 𝐺T𝑖 (𝑚), 𝐺S𝑖 (𝑚), and 𝐺U𝑖 (𝑚) are the total MFD, the MFD of sub-seismogenic ruptures, and
the MFD of events not associated with any faults, for the ith grid node, respectively. At any grid
node, the latter two components may be zero, but not both. The total MFD for the kth fault
section in a solution is the sum of the rates of seismogenic (large) and sub-seismogenic (small)
ruptures across all magnitudes:
𝐹T𝑘 (𝑚) = 𝐹L𝑘 (𝑚) + 𝐹S𝑘 (𝑚).
The foregoing outlines a method to solve for 𝐹S𝑘 (𝑚), 𝐺S𝑖 (𝑚) and 𝐺U𝑖 (𝑚).
Fault-section sub-seismogenic MFDs
To compute the MFD at each grid node, we first define the sub-seismogenic MFD for each faultsection as:
𝐹S𝑘 (𝑚) = 𝑟𝐷(𝑚 | 𝑚 < 𝑚c𝑘 ).
Here we use the total off-fault MFD as a reference, truncated below the minimum seismogenic
magnitude for the fault-section:
𝑚c𝑘 = min[𝑚 | 𝐹L𝑘 (𝑚) > 0],
and scaled by a value r (Fig. 6). This scale value is determined one of two ways:
a) Moment balancing – r is set such that 𝐹S𝑘 (𝑚) has exactly the moment rate
previously removed from the fault-section for sub-seismogenic ruptures. This
produces a perfectly moment-balanced model from the perspective of fault slip
rates.
b) Matching observed rate – r is set such that 𝐹S𝑘 (𝑚) has the same cumulative
rate of events as implied by the smoothed seismicity model (see Appendix M)
inside the fault-section polygon. This produces a model that matches the
observed rate of events.
The latter requires an inventory of fault-section participation in each node, which we denote
𝑝𝑘→𝑖 = 𝑝𝑖 [𝐴𝑖∩𝑘 / ∑𝑖∩𝑘 𝐴𝑖∩𝑘 ]
UCERF3: Appendix O
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where the ratio of the area formed by the intersection of the ith node and kth fault-section, 𝐴𝑖∩𝑘 ,
to the sum of the intersecting areas of all fault-sections in the node is scaled to a fraction of the
total participation of the node
𝑝𝑖 = [⋃𝑖∩𝑘 𝐴𝑖∩𝑘 ]𝐴−1
𝑘 .
The total participation of a node is the ratio of the area of a union of all intersecting fault-section
polygons and the total area spanned by the node, 𝐴𝑘 . In the equations above, summation or union
implied by 𝑖 ∩ 𝑘 is over all k fault-sections that intersect the ith grid node.
The participations of a fault-section in various nodes determine the fraction of the smoothed
seismicity model, 𝑆𝑖 (𝑚), at each node that is allocated to a fault-section, that is, the MFD used as
the basis for the scale factor r is ∑𝑖∩𝑘 𝑝𝑘→𝑖 𝑆𝑖 (𝑚).
We represent 𝑝𝑘→𝑖 in this way to accommodate grid nodes that are intersected by multiple,
overlapping fault-sections and avoid over-allocating events from nodes in the smoothed
seismicity model. For example, in the case where half a node is spanned by a single fault-section,
𝑝𝑘→𝑖 is 0.5. If, however, three overlapping sections individually span 60%, 40%, and 20% of a
node and in aggregate the sections cover 60% of a node, then 𝑝𝑘→𝑖 is 0.3, 0.2, and 0.1,
respectively.
Gridded source MFDs
Once the sub-seismogenic MFDs are calculated for each fault-section, the total regional MFD of
events not associated with faults is
𝐷U (𝑚) = 𝐷(𝑚) − ∑ 𝐹S𝑘 (𝑚).
Because we removed events from specific grid nodes, we rescale the smoothed seismicity PDF
based on the non-participating fraction of each node
𝑃𝑖 = 𝑃̂𝑖 (1 − 𝑝𝑖 )/ ∑ 𝑃̂𝑖 (1 − 𝑝𝑖 )
where 𝑃̂𝑖 is the original input spatial PDF of seismicity.
𝐺𝑈𝑖 (𝑚) = 𝑃𝑖 𝐷U (𝑚)
is the MFD of unassociated events at each grid node and the sub-seismogenic MFD at each grid
node is the sum of the participating portions of the sub-seismogenic MFDs of each fault-section
that intersects a grid node:
𝐺S𝑖 (𝑚) = ∑𝑘∩𝑖 𝑝𝑖→𝑘 𝐹S𝑘 (𝑚).
Here, the summation implied by 𝑘 ∩ 𝑖 is over all i nodes that intersect the kth fault-section. The
node participation of each fault section, 𝑝𝑖→𝑘 , is defined as the ratio of the fault-section area that
intersects a node to the total area of the fault-section polygon:
𝑝𝑖→𝑘 = 𝐴𝑖∩𝑘 /𝐴𝑘 .
The MFD at any grid node is thus the sum of the sub-seismogenic and unassociated MFDs for the
node:
UCERF3: Appendix O
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𝐺T𝑖 (𝑚) = 𝐺S𝑖 (𝑚) + 𝐺U𝑖 (𝑚).
Verification
As a check, the sub-seismogenic MFDs of all fault-sections and the independent MFDs of each
grid node sum to the original inversion-supplied total off fault MFD
𝐷𝑇 (𝑚) = ∑ 𝐹S𝑘 (𝑚) + ∑ 𝐺U𝑖 (𝑚),
as do the independent MFDs and the sub-seismogenic MFDs of all fault-sections partitioned over
the relevant grid nodes:
𝐷𝑇 (𝑚) = ∑ 𝐺S𝑖 (𝑚) + ∑ 𝐺U𝑖 (𝑚).
Focal mechanisms
The UCERF2 method of assigning focal mechanisms to sources on a per region basis with
customization for several small sub-regions provides an adequate baseline for the application of
focal mechanisms to UCERF3 sources. However, there may be considerable value added by
including data from known focal mechanisms in the UCERF3 forecast. The focal mechanisms of
historic earthquakes express geologic structure at a point and could be used to reduce reliance on
customized sub-regions. Of course, focal mechanism data available in catalogs is often highly
uncertain.
Kagan and Jackson (1994) describe a procedure for forecasting earthquakes and their associated
focal mechanisms (and likely preferred nodal planes) at grid nodes in a region with some
uncertainty. When applied to a focal mechanism catalog for California spanning the years 1800 to
2007 (Wang et al., 2011), the method yields a model of likely fault rupture nodal planes at 0.1˚ by
0.1˚ spaced grid nodes and associated uncertainties (Heimer, et al., in review). The uncertainties
are not entirely straightforward in that the values are the standard deviation of the weighted 3-D
rotation angles required to bring pairs of focal mechanisms into alignment. For any pair of focal
mechanisms there exists a pole about which a single rotation will bring them into alignment; the
maximum angle of this rotation is 120˚ (Kagan, 2007).
This focal mechanism uncertainty data presents several problems. Although the uncertainties tend
to be low in the vicinity of major faults, especially those that that have been involved in large
earthquakes, they are also low where the catalog data are sparse. It is therefore difficult to use the
uncertainties as the basis for weighting catalog focal mechanisms and the NSHMP approach.
We recommend exploring two branches in the context of hazard: (1) the UCERF2 focal
mechanism model and (2) a model based on the preferred nodal planes of the Kagan and Jackson
(1994) and Weimer et al. (in review) analyses that ignores published uncertainties. To implement
the UCERF2 focal mechanism model, we compute the focal mechanism weights at each grid
source from the weights in any coincident UCERF2 sources, considering the rate of each
contributing source. When used as inputs to a hazard model, gridded sources are typically treated
as having unknown strike. If the UCERF2 model is given full weight, some accommodation for
the small areas of fixed strike sources must be made. Alternatively, the focal mechanism data
branch, if more heavily weighted, may achieve the same result without requiring special zones.
UCERF3: Appendix O
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Discussion
In UCERF3, gridded sources are tightly coupled with inversion results and the UCERF2 and the
2008 NSHMP approach of using a fixed set of gridded sources is not wholly compatible.
However, this historic source model is a necessarily baseline to which UCERF3 results must be
compared and so we recast the UCERF2 model as an input to the UCERF3 gridded source model
generator. To do this, we create a spatial PDF of seismicity rate based on the a-values of the
various regional UCERF2 grid sources, and ignore any spatial variations in M max and b-value
(Fig. 5). Comparisons of participation of rates of different magnitude events between UCERF2
(translated to the UCERF3 framework) and two UCERF3 branches highlight the differences
between models that require further consideration and analysis (Fig. 7).
References
Hiemer, S., Jackson, D. D., Wang, Q., Kagan, Y. Y., Woessner, J., Zechar, J. D., and Wiemer, S.
(in review). A stochastic earthquake source model combining fault geometry, slip rates, and
smoothed seismicity: California. Bulletin of the Seismological Society of America
(submitted).
Kagan, Y. Y. and Jackson, D. D. (1994). Long-term probabilistic forecasting of earth- quakes.
Journal of Geophysical Research, 99(B7): 13685–13700.
Kagan, Y. (2007). Simplified algorithms for calculating double-couple rotation. Geophysical
Journal International. 171: 411-418.
Petersen, M. D., Frankel, A. D., Harmsen, S. C., Mueller, C. S., Haller, K. M., Wheeler, R. L.,
Wesson, R. L., Zeng, Y., Boyd, O. S., Perkins, D. M., Luco, N., Field, E. H., Wills, C. J.,
Rukstales, K. S. (2008). Documentation for the 2008 Update of the United States National
Seismic Hazard Maps. U.S. Geological Survey Open File Report 2008-1128, 2008.
Wang, Q., Jackson, D. D., and Kagan, Y. Y. (2009). California Earthquakes, 1800-2007: A
Unified Catalog with Moment Magnitudes, Uncertainties, and Focal Mechanisms.
Seismological Research Letters, 80(3): 446–457.
UCERF3: Appendix O
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Figure 1. Map of the 2008 NSHMP polygons used to define the boundaries of grid source
regions (in blue, green, red, and black). These regions were used to subdivide the declustered
historical seismicity catalog prior to smoothing. The yellow line marks the boundary of the
RELM testing region, the area spanned by the UCERF2 and UCERF3 earthquake rupture
forecasts.
Figure 2. 3D view (down and to the north) of the Mission Ridge (blue) and San Cayetano
(purple) faults illustrating fault-zone polygon and fault-trace complexity respectively. The dots
are a discretized representation of the down-dip surface of each fault and the solid line is the
border of the polygon representing the area at the earth's surface associated with each fault.
UCERF3: Appendix O
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Figure 3. Schematic diagrams of fault-section polygons. The figure on the left shows a
perspective view of a hypothetical dipping fault composed of three fault-sections with the fault
trace in red, the subdivided fault-zone polygon in blue, and the down dip representation of the
sections in orange. The figure on the right shows a map view of the same with the outlines of the
combined polygons for each section. Areas used for various participation metrics are as follows:
(a) is the participation area of a section in a node and (b) is the total participation area of a node.
Figure 4. Example fault-section polygons. Map view of the Mission Ridge fault showing the fault
trace (red), the subdivided fault-zone polygon (blue), and the surface projections (orange) of the
down-dip representations of each fault-section. For each section, the total fault-zone polygon is
the union of the orange and blue polygons.
UCERF3: Appendix O
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Figure 5. Various spatial-probability-distribution models for off-fault seismicity (values in the
map sum to 1.0). (a) The smoothed-seismicity model developed in UCERF2 (Appendix J in
UCERF2 report). (b) The new smoothed-sesimicity model developed in Appendix M of this
report, which has higher resolution by virtue of applying an adaptive smoothing kernel. (c) The
spatial probability distribution implied by an average of the off-fault moment rates from four
deformation models (Appendix C).
Figure 6. Schematic diagram of the MFDs used in calculating sub-seismogenic MFDs for faultsections. The dashed line is the total off-fault MFD as supplied by an inversion result, 𝐷𝑇 (𝑚).
Using this as a starting point, the total MFD is clipped at the minimum magnitude, mc, for the
seismogenic rupture MFD on a fault section, 𝐹L𝑘 (𝑚). The resulting MFD for 5≤M<mc is then
scaled according to the original moment rate reduction imposed on the section or by the
cumulative rate implied by the area occupied by the fault-section polygon in a smoothed
seismicity model and set as the sub-seismogenic MFD for the section, 𝐹S𝑘 (𝑚).
UCERF3: Appendix O
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Figure 7. Participation rate maps for UCERF2, UCERF3 Reference, and UCERF3 Reference
with the GR a-priori constraint for various magnitude thresholds.
UCERF3: Appendix O
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