Appendix N: The Grand Inversion
1
Appendix N: Grand Inversion Implementation and Exploration of Logic-Tree
Branches
Morgan Page (USGS), Ned Field (USGS), and Kevin Milner (USC)
Introduction
The purpose of the “grand inversion” is to solve for the long-term rate of all possible (“on-fault”)
ruptures in California. The rates of these ruptures are constrained by fault slip rates,
paleoseismic event rates observed in trenches, magnitude-frequency distributions (MFDs)
observed in seismicity (and adjusted to account for background seismicity), as well as other a
priori and smoothing constraints described below.
The inversion methodology we use is based on a methodology proposed by Andrews and
Schwerer (2000) and further developed by Field and Page (2011). Below we describe 1) the
constraints used in the inversion, 2) the simulated annealing algorithm, 3) the rules used to define
the set of ruptures on the system of faults, and 4) details of a suite of models produced by the
inversion algorithm.
1. Setting up the Inversion: Data and Constraints
The inversion methodology solves for the rates of ruptures that are consistent with the data and
constraints described below. Some of these constraints differ depending on the inversion model
branch; for example the CHAR (characteristic) branch solution is constrained to be as close to
UCERF2 rates as possible, while the G-R (Gutenberg-Richter) model branch is designed to have
a spatially-smooth, approximately Gutenberg-Richter magnitude distribution in space.
Constraints used for all branches:
1) Slip Rates. The average slip in each rupture a section participates in, multiplied by the rate of
each rupture, must sum to the long-term slip rate for that section, as given by the deformation
model. This constraint is applied to each fault subsection. Each slip-rate constraint is
normalized by the target slip-rate (so that misfit is proportional to the fractional difference, rather
than the absolute difference, between the model slip rates and the target slip rates), with the
exception of slip rates below 10-4 m/yr, which are normalized by 10-4 m/yr. This prevents some
extremely low slip rates (some deformation models have slip rates as low as 10-13 m/yr) from
dominating the misfit. The slip rates on fault subsections with zero or NaN deformation-model
slip rates are minimized (with a weight of 10-4).
The slip rates used in this constraint have been reduced from the slip rates specified in the
deformation models to account for subseismogenic-thickness ruptures and aseismicity (more
details on this are given in the main report).
2) Paleoseismic Event Rates. The total rate of all ruptures a section participates in, multiplied
by the probability each rupture is paleoseismically visible (Appendix I), must sum to the mean
paleoseismic event rate for that section. This constraint is applied to a total of 34 fault
subsections that are in the paleoseismic database (see Appendix G).
Appendix N: The Grand Inversion
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Additional CHAR branch constraints:
The CHAR (characteristic / close to UCERF2) branch is designed to have rates that are as close
as possible to UCERF2 ruptures rates, while satisfying the slip-rate and paleoseismic event rates
better and eliminating the magnitude “bulge” (overprediction relative to seismicity MFD) that
existed in the UCERF2 model.
1) A priori Rupture Rates. For this branch, we apply an a priori rupture-rate constraint to force
rupture rates to be close to UCERF2 rates. Of the solutions that come closest to satisfying the
other constraints, we choose the one that is closest (L2 norm) to equivalent UCERF2 rates (the
mapping of UCERF2 ruptures to the equivalent inversion ruptures is far from trivial; details are
available upon request, but we are confident that conclusions here do not depend on any
subjectivity in this mapping). This constraint is only applied to ruptures that have a mapping to a
UCERF2 rupture, so it does not affect new faults that have been added to the fault database since
UCERF2.
2) Magnitude Distribution. The on-fault target magnitude distribution is found by reducing the
total magnitude distribution (for which we use the 1850-present magnitude distribution found
from seismicity, see Appendix L) by the amount of off-fault moment in each deformation model.
For the CHAR branch, this reduction of the total magnitude distribution is accomplished by
reducing the b-value below magnitude 7.6 (which corresponds to the maximum “background”
earthquake allowed in UCERF2 – in C-zones), as shown in Figure 1.
For each branch, the magnitude-distribution constraint is applied as an equality constraint up to
and including magnitude 7.85, and applied as an inequality constraint (with the target on-fault
MFD serving as an upper bound) above M7.85. Thus the inversion is not constrained to exactly
equal the target MFD at magnitudes for which the true magnitude-frequency distribution may
“roll-off”. As it is not well known what the MFD looks like at high magnitudes, we allow the
inversion to choose whatever “roll-off” at high magnitudes is consistent with the other
constraints.
For each branch, this constraint is applied for 2 regions: Northern California and Southern
California.
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Figure 1. Schematic of magnitude-frequency distributions for the CHAR (Characteristic) branch. The total target
magnitude frequency distribution (found from seismicity, see Appendix L) is shown in black. The total on-fault
MFD is shown in purple. It is found by reducing the total target MFD below the bilinear transition magnitude of
7.6 to account for off-fault moment (the moment-rate difference between the black and purple lines is set to be
equal to the seismic off-fault moment for each deformation model). This MFD is further reduced to account for
subseismogenic ruptures (more details on this are given in the main report) to give the cyan line, which is the
MFD target for the inversion, and represents all on-fault ruptures that rupture the full seismogenic width. This
MFD target is applied as an equality constraint below the MFD transition magnitude of 7.85 and an inequality
(upper-bound) constraint above M7.85. Finally, the implied background MFD, which includes both off-fault and
subseismogenic-thickness on-fault ruptures, is defined as the black minus the cyan line, and is shown in grey.
3) Minimum Rupture Rates (waterlevel). To avoid assigning zero rates to ruptures, a
minimum “waterlevel” is assigned to each rupture rate. The original inverse problem Ax=d with
x ≥ xmin, where xmin are the minimum rupture rates, is mapped to a new inverse problem with
simple nonnegativity constraint, Ax=d, x ≥ 0. The “new” data vector is given by d=d - Axmin
and the rupture-rate mapping is given by x+xmin=x. The nonnegativity constraint for the
mapped inversion problem is strictly enforced in the simulated annealing algorithm, which does
not search any solution space that contains negative rates.
The minimum rupture rates are defined as 1% of the smooth starting solution, which is a
magnitude-dependent waterlevel described below (as it is used as the starting model for the G-R
branch).
Appendix N: The Grand Inversion
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Additional G-R branch constraints:
This branch is designed to have faults that have Gutenberg-Richter-distributed (with b=1)
magnitudes on as small as scale as possible.
1) Magnitude Participation Distribution Smoothness. The purpose of the G-R branch is to
make a model that, while fitting the other constraints, has a magnitude-distribution that, at a
small scale on faults, is as close to a Gutenberg-Richter distribution as possible. A GutenbergRichter distribution of nucleation’s with a b-value of 1.0 implies a uniform (in magnitude)
participation probability for each point on a fault (assuming constant stress drop). Thus, this
constraint is implemented by constraining the magnitude-participation distribution for each fault
subsection to be uniform. This constraint is applied for each fault subsection magnitude-bin: the
rate of ruptures for each magnitude bin, for each fault subsection, is constrained to be close to the
average (over all magnitude bins) participation rate for that subsection.
This constraint is computationally the most demanding (in terms of both memory and
computational time) of all the constraints; for this reason a G-R branch inversion uses 28 GB
while the CHAR branch only requires 1.8 GB of memory. The processor time per simulatedannealing iteration scales approximately the same as the memory usage.
2) Magnitude Distribution. The on-fault magnitude-frequency distribution is found by
uniformly reducing the total target MFD defined from seismicity by the percentage of moment in
the background, for each magnitude bin. Thus the faults are constrained, for Northern and
Southern California, to have a total MFD that is Gutenberg-Richter, allowing for a “roll-off” at
high magnitudes.
3) Minimum Rupture Rates (waterlevel). The minimum rupture rates are defined as 1% of the
smooth starting solution, which is a magnitude-dependent waterlevel described below (as it is
used as the starting model for the G-R branch).
Additional UNCONSTRAINED branch constraints:
This branch is not entirely unconstrained, but rather not over-constrained (as it lacks the
regularizing rupture-rate constraint of the CHAR branch, and magnitude participation
smoothness constraint of the G-R branch). Due to stochastic nature of the simulated annealing
algorithm, this model will give a different solution each time as there are multiple models that fit
the data. This branch allows for the exploration of which features on the long-term rates are
robust outcomes of the data constraints.
1) Magnitude Distribution. The magnitude-distribution constraint is applied the same way as
for the G-R branch (see above).
2) Nonnegativity Constraint. Rupture rates cannot be negative. This is a strict constraint that
is not included in the system of equations but is strictly enforced in the simulated annealing
algorithm, which does not search any solution space that contains negative rates.
Appendix N: The Grand Inversion
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With the exception of the nonnegativity constraint and the MFD inequality constraint, these
constraints are linear and can be combined to form a matrix equation of the form Ax=d. Our task
is to set up this matrix equation and solve for x, the rate of all ruptures, given A and d.
Starting models for branches
In addition to the inversion constraints described above, we also specify a starting model, which
is a set of initial ruptures rates given to the simulated annealing algorithm. For the CHAR
(characteristic) branch, the starting rates are the UCERF2 ruptures rates (or more accurately,
UCERF2 rates mapped onto the inversion rupture set – we find the closest analogues to UCERF2
ruptures given our discretization of the fault system). This means that for most ruptures in the
UCERF3 rupture set the starting rate for the CHAR branch will be zero (since there are no
corresponding UCERF2 ruptures for most ruptures in the UCERF3 rupture set, since UCERF2
had less than 10,000 on-fault ruptures and the UCERF3 rupture set has 205,254 ruptures for
Fault Model 3.1).
For the G-R branch, we use a “smooth starting solution” whose rates sum to a Gutenberg-Richter
distribution (with a b-value of 1.0 and an a-value set from the total deformation model moment).
Rupture rates within each 0.1 magnitude-unit bin are proportional to the minimum-slip-rate
subsection in each rupture.
As a final processing step before the simulated annealing, the starting solutions for both the
CHAR and G-R branches are modified for any magnitude bins where they exceed the MFD
constraint rate (all rates within any magnitude bin that exceeds this constraint are uniformly
reduced to that the total rate in that bin will equal the constraint rate). This step speeds up the
annealing time required to find acceptable solutions.
The starting model for the UNCONSTRAINED branch is a rate of zero for all ruptures.
Table 1. Inversion Parameter Settings.
Setting for
CHAR
branch
Setting for
GR branch
Setting for UNCONSTRAINED
branch
Details
relative Paleo Rate
Weight
1
1
1
Weight of paleoseismic rate constraint
relative to slip-rate constraint.
Normalized by event-rate standard
deviation.
relative MFD Equality
Constraint Weight
10
10
10
Weight of MFD-equality constraint
relative to slip-rate constraint.
Normalized by target rate for each
magnitude bin.
relative MFD Inequality
Constraint Weight
1000
1000
1000
Weight of MFD-inequality constraint
relative to slip-rate constraint.
Normalized by target rate for each
magnitude bin.
Bilinear Transition
Magnitude
7.6
N/A
N/A
Below this magnitude, the b-value for the
on-fault MFD is reduced to allow for
background seismicity. This represents
the maximum moment of the background
for the CHAR branch.
Parameter name
Appendix N: The Grand Inversion
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relative Participation
Smoothness Constraint
Weight
0
1000
0
Weight of magnitude-participation
constraint relative to slip-rate constraint.
Units of rate/magnitude-bin/yr.
participation Constraint
Magnitude Bin Size
N/A
0.1
N/A
Width, in magnitude units, of magnitude
bins for magnitude-participation
constraint.
relative Rupture Rate
Constraint Weight
100
0
0
Weight of rupture-rate constraint relative
to slip-rate constraint. Units of rate/yr.
Applied only to nonzero rates.
a Priori Rupture
Constraint
UCERF2
solution
N/A
N/A
Rates used for rupture-rate constraint.
minimum Rupture Rate
Fraction
0.01
0.01
0
Fraction of minimum rupture-rate basis
that is applied as a minimum rupture-rate
(waterlevel).
minimum Rupture Rate
Basis
Smooth
starting
solution
Smooth
starting
solution
N/A
Rates used for minimum rupture rates
(waterlevel).
initial Rupture Model
UCERF2
starting
solution
Smooth
starting
solution
Zero rate for all
ruptures
Starting set of rupture rates for simulated
annealing algorithm.
Magnitude-Frequency
Distribution (MFD)
Constraints
Bilinear
distribution
G-R
G-R
Target MFD for on-fault ruptures: Total
region MFD target less off-fault
seismicity.
MFD Transition
Magnitude
7.85
7.85
7.85
Magnitude to switch from MFD equality
constraint to MFD inequality constraint.
2. The Simulated Annealing Algorithm
The simulated annealing algorithm solves the nonnegative least squares problem Ax=d with the
additional constraint Aineqx ≤ dineq (this last constraint is due to the MFD inequality constraint).
This algorithm simulates the slow cooling of a physical material to form a crystal. It has the
following steps:
1. Set x equal to initial solution x0. For the CHAR branch, the starting solution is mapped
UCERF2 ruptures rates; for the G-R branch, it is the “smooth starting solution” described above,
and for the UNCONSTRAINED branch it is a vector of all zeros.
2. Lower the “temperature”, T, from 1 to 0 over a specified number of iterations. We lower the
temperature linearly, although different approaches to annealing specify different functions for
the temperature. The more iterations (i.e., the more slowly the temperature is lowered), the
better the final solution will be. Over each simulated annealing iteration, the following is done:
a. One element of x (one rupture rate) is chosen at random. This element is then
perturbed randomly. It is here that the nonnegativity constraint is applied – the
perturbation function is a function of the rupture rate and will not perturb the rate to a
Appendix N: The Grand Inversion
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negative value. Unlike some simulated annealing algorithms, our algorithm does not use
smaller perturbations as the temperature is lowered (this was tested but did not result in
faster convergence times).
b. The misfit for the perturbed x, xnew, is calculated, and from this the “energy” of that
solution: Enew = (Axnew- d)2+Eineq, where Eineq is additional energy from the MFD
inequality constraint: Eineq = (min (Aineqxnew- dineq, 0))2. The weight on the inequality
constraint is set quite high so that a solution that violates it faces a significant penalty;
thus in practice this is a strict inequality constraint.
c. The transition probability, P, is calculated based on the change in energy (between the
previous state and the perturbed state) and the current temperature T. If the new model is
better, P=1. Therefore, a new model is always kept if it is better. If the new model is
worse, it is sometimes kept, and this is more likely early in the annealing process when
the temperature is high. If E < Enew, P = e(E-Enew)/T.
3. Once the annealing schedule is completed, the best solution x found during the search (the
solution with the lowest energy) is returned. (Note that this is a departure from “pure” simulated
annealing, which returns the last state found. In some cases the final state will not be the best
solution found, since occasionally solutions are discarded for worse solutions).
Simulated annealing works similarly to other nonlinear algorithms such as the genetic algorithm.
One advantage of simulated annealing is that there is a mathematical basis for it: besides the
analogy to annealing a physical material, the simulated annealing algorithm will find the global
minimum given infinite cooling time (Granville et al. 1994).
There are several advantages of this algorithm in contrast to other approaches such as the
nonnegative least-squares algorithm. First, the simulated annealing algorithm scales well as the
problem size increases. (In fact, it would not be computationally feasible for us to use the
nonnegative least-squares algorithm to solve a problem of this size.) It is designed to efficiently
search a large parameter space without getting stuck in local minima. Next, quite importantly,
for an underdetermined problem the simulated annealing algorithm gives multiple solutions (at
varying levels of misfit depending on the annealing schedule). Thus both the resolution error
(the range of models that satisfy one iteration of the data) and the data error (the impact of
parameter uncertainty on the model) can be sampled. Finally, simulated annealing can allow us
to include other nonlinear constraints in the inversion apart from nonnegativity; in our case we
incorporate the MFD inequality constraint, which is a nonlinear constraint that cannot be easily
incorporated into the perturbation function.
Parallelization of the Simulated Annealing Algorithm
In order to tackle the additional computational demands of the statewide inversion, a parallel
version of the simulated annealing algorithm has been implemented. This algorithm runs the
serial version of simulated annealing (as described above) over a number of processors for a
given number of subiterations. Then the best solution among these is kept and redistributed over
the processors; this process repeats until convergence criteria (a target misfit, a given number of
total iterations, or an allotted annealing time) is satisfied.
Appendix N: The Grand Inversion
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The algorithm scales well up to 20-50 nodes, but adding nodes beyond this does not improve
performance. Using the parallelized algorithm on a cluster results in average speedups of 6-20
relative to the serial algorithm. Graphs of simulated annealing “energy” vs. time (lower energy
corresponds to a lower misfit) for a range of models and cluster configurations are shown in
Figure 2.
The solutions presented here were run on the HPCC cluster at the University of Southern
California; CHAR-branch models are annealed on 8-core machines for 8 hours,
UNCONSTRAINED-branch models for 1 hour. The G-R-branch models require significantly
more memory and annealing time; they are run on a cluster of 23 24-core machines for 8 hours.
Figure 2. Speedup results for the parallelized simulated annealing algorithm depend on problem size and the
amount of constraints on the solutions. Each line shows the average of 5 runs using the number of nodes shown
in the legend. More negatively-sloped lines indicate faster convergence speed.
3. Defining the set of possible fault-based ruptures
A final a priori step that must be done before running the inversion is defining the set of possible
ruptures. The faults are first discretized into subsections, as described in the main report. These
subsections are for numerical tractability and do not represent geologic segments. Subsections
are linked together to form ruptures, the rates of which will be solved for by the inversion. It is
worth noting that here a “rupture” is defined as an ordered list of fault subsections it includes; it
has no defined hypocenter. All viable ruptures are generated from the digitized fault sections
that satisfy the following rules:
1) All fault sections connect within 5 km or less.
This rule is consistent with the maximum jump distance in the Wesnousky database of
earthquake surface ruptures (Wesnousky, 2008).
2) Ruptures cannot include a given subsection more than once.
3) Ruptures must contain at least 2 subsections on any involved fault section.
Each fault section is discretized into subsections that have a length of at least half the
seismogenic width, so this rule results in ruptures with an along-strike length at least as
large as the seismogenic-thickness.
4) Ruptures can jump between 2 different fault sections only at their closest point in 3D.
This prevents ruptures from “leap-frogging” back-and-forth between two parallel faults.
5) The maximum azimuth change between neighboring subsections is 60 degrees.
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6) The maximum azimuth change between the first and last subsection is 60 degrees.
7) The maximum cumulative rake change (summing over each neighboring subsection pair) is
180 degrees.
Note: The current setting of the rake filter allows for Garlock to Northern San Andreas
ruptures, which is a departure from the filtering rules used in previous UCERF3 reports.
8). The maximum cumulative azimuth change (summing over each neighboring subsection pair)
is 560 degrees.
This filter, known colloquially as the “squirrelly-ness” filter, prevents ruptures that have
highly non-linear traces. This filter is needed in the UCERF3 fault models in particular
because of the high density of faults that could otherwise link in the Ventura and LA basins.
An example of the type of rupture excluded by this filter is shown in Figure 3. Without this
filter, the number of ruptures would reach hundreds of millions and the inversion would be
computationally intractable. Increasing the maximum cumulative azimuth change
significantly increases the number of ruptures; the current value is set to keep the number of
ruptures low while being high enough to allow near “wall-to-wall” San Andreas ruptures.
9) Coulomb filter: At each branching point (defined as place where a rupture could “jump” from
one fault sections to another, whether or not it takes this jump) the Coulomb probability (defined
as the ratio of a given stress change at a branching point relative to the cumulative stress change
of all possible jumps at that branching point) must be greater than 0.1, unless the sum of
Coulomb stresses at each branching point for that rupture is greater than 1.5. (For more details on
the Coulomb methodology, see Appendix J.)
We are applying the Coulomb modeling to branches only because otherwise some ruptures
with no jumps at all between different parent sections (some of which were included in
UCERF2) were excluded. The minimum stress exclusion ceiling (the value of 1.5 mentioned
above) was added to prevent exclusion of kinematically-favored jumps at branching points
where many possible linking-options led to a low Coulomb probability. Examples of
ruptures allowed by the above filters but excluded by the Coulomb filter are shown in Figure
4.
Some ruptures allowed by the above rules will be highly improbable due to slip-rate differences
along strike (as the slowest sections limit how often different fault sections can rupture together).
The inversion will naturally satisfy this consideration since it is constrained to satisfy the
deformation-model slip rates on each fault subsection.
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Figure 3. Example of rupture excluded by maximum cumulative azimuth change filter (subsections in rupture
are shown in red).
Figure 4. Examples of ruptures excluded by the Coulomb filter.
The above rules define a total of 205,254 ruptures for Fault Model 3.1. By comparison, our
UCERF2 mapping has (for a smaller set of faults) 7773 on-fault ruptures.
One important remaining question is to what extent ruptures allowed by the above rules may
require further penalty. One could impose an “improbability” constraint to any ruptures that are
deemed possible but improbable; these constraints a numerically equivalent to an a priori
rupture-rate constraint with a rupture rate of zero. These constraints can be weighted as well, so
some rupture rates could be penalized more harshly (that is, contribute more to the misfit if they
are nonzero) than others. Currently, this improbability constraint is not implemented for the
UCERF3 rupture set as we do not have a viable model (that is, a model ready for implementation
and agreed to be useful by Working Group members) for the constraint. It is also not clear that
Appendix N: The Grand Inversion
11
this constraint is needed.
It is not even clear what constitutes a “multi-fault” rupture in the case of a complex, possibly
fractal, connected fault network. One way to quantify the amount of “multi-fault-ness” that may
not be ideal but at least lacks ambiguity is to simply define “multi-fault” in the context of the
names assigned to faults in the database. For this definition of multi-fault, we count all sections
of a fault such as the San Andreas fault as one “named fault” even though the individual sections
have different names (e.g., Carrizo, Mojave North, etc.). Using the inverted rupture rates for the
UCERF3 reference model that we describe below in Section 4, 13% are on multiply-named
faults. This increases to 17% for ruptures with M≥6.5 and to 45% for ruptures with M≥7. We
can compare this number to the ruptures in the Wesnousky database of surface ruptures
(Wesnousky, 2008); of these, 50%, or 14 out of 28, are on multiply-named faults. So by this
albeit simple metric, the solutions given by the inversion algorithm are not producing more
multi-fault ruptures than are seen in nature.
It is important to note that many of the truly multi-fault ruptures seen in nature may in fact not be
part of our model at all because they could link up known, mapped faults with unknown faults
(or faults that are known but not sufficiently studied as to be included in our fault model). Our
model completely separates “on-fault” ruptures from background earthquakes; there are no
ruptures that are partly “on-fault” and partly “off-fault”.
Without the improbability constraint, the inversion is not likely to give a lower rate to a rupture
that is only moderately kinematically compatible (but allowed by our filters) relatively to a more
kinematically-favored rupture, unless kinematically less-favored jumps are also disfavored by
slip-rate changes. This can be seen in the Figure 5, which compares Coulomb probabilities to
normalized subsection pair rates (the rate at which a pair of subsections ruptures together divided
by the total rate at which they rupture) for neighboring subsection pairs. There is a positive
correlation between these variables (which would imply that the inversion is naturally doing
something similar to Coulomb penalties), but it is not particularly strong. Then again, the most
egregious subsection pairs that have very low Coulomb probabilities at branching points have
already been excluded.
Appendix N: The Grand Inversion
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Figure 5. The probability that 2 subsections will rupture together given by Coulomb modeling vs. the probability
given by the UCERF3 reference model. There is some positive correlation, but the inversion is currently not
directly constrained to minimize the rates of ruptures with low Coulomb-probability subsections, beyond
branching points (those ruptures are excluded from the rupture set entirely).
4. Modeling Results
In order to make sense of the inversion results and understand how and why they are different
from UCERF2, we present a series of models between UCERF2 and our new UCERF3 models.
First, we show the UCERF2 model mapped into our rupture-set framework, and show how well
this model matched its own constraints. Next, we show a model that uses UCERF2 ingredients
but is generated with UCERF3 inversion methodology – we will see how this model fits the
UCERF2 constraints and what changes to the model are a result of methodological changes
(rather than changes to data). Finally, we will show models generated with UCERF3 ingredients
and UCERF3 methodology. These models use the newer fault models (which have more faults
than UCERF2), new deformation models, different magnitude-area and slip-length relations, and
updated paleoseismic constraints and seismicity constraints.
UCERF2 Reference Model
The first model we present uses mean UCERF2 on-fault rupture rates, mapped onto our rupture
set. We set the magnitudes of ruptures to their mean UCERF2 values and assume a tapered
along-strike slip distribution for each rupture. In the plots below we compare the UCERF2
model to data that was used to constrain the model: slip rates from Deformation Model 2.1,
paleoseismic event rates from UCERF2, and the total statewide magnitude-frequency
distribution found from 1850-2007 seismicity rates. UCERF2 matches slip rates well on a faultaveraged basis; however, on a smaller scale the model overpredicts slip rate in the center of
sections and underpredicts slip rate at fault ends. The UCERF2 deformation model had uniform
slip rates for each fault section, yet the model does not fit these uniform rates. This is caused by
two factors: 1) the “floating” ruptures on each fault section, which overlap more in the center of
the sections, and 2) the tapered slip distribution we assume for the ruptures. In addition, the
Appendix N: The Grand Inversion
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UCERF2 on-fault magnitude distribution, when combined with the background MFD, led to an
overprediction of events around M6.5.
UCERF2 ingredients, annealed with the UCERF3 inversion algorithm
We next apply UCERF3 inversion methodology to the data that was used in the UCERF2 model.
This model is able to fit the slip-rate data, paleoseismic event-rate data, and magnitudefrequency distribution constraints better than the UCERF2 model. In particular, we are able to
eliminate the “bulge” problem in UCERF2 – and overprediction of earthquakes around M6.5.
Through testing changes to the rupture set, we determined that fitting the MFD constraint and
thus eliminating this “bulge” required multi-fault ruptures not included in UCERF2. By
allowing faults to “link up”, moment that was leading to the M6.5 bulge could be put in largermagnitude ruptures.
Figure 6. Slip-rate misfit for UCERF2 (left) and UCERF2 ingredients fit with the inversion algorithm (right). The
UCERF3 methodology is able to better-fit the slip rates on a subsection-by-section basis. The UCERF2 solution
tends to underpredict at fault section ends and overpredict the section centers, although it is moment-balanced
on a section-averaged basis. Note that Mendocino and the San Andreas creeping section were not part of the
UCERF2 on-fault solution, so those misfits should be disregarded for the UCERF2 reference model.
One can see that the annealed model fits UCERF2 slip rates significantly better than the
UCERF2 reference model (more particularly, the inversion can fit these slip rates on a smaller
scale). The inversion simultaneously is constrained to fit the paleoseismic event rates as well,
and improves the squared residuals by a factor of 2 relative to UCERF2 (Although it should be
noted, the model does have more outliers than UCERF2. This could be improved by increasing
the event-rate constraint weight, but it does not seem to be a problem in models with UCERF3
ingredients.)
Appendix N: The Grand Inversion
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Appendix N: The Grand Inversion
15
Figure 7. Paleoseismic Event Rates for the UCERF2 Reference Model (top) and UCERF2 Annealed Model
(bottom). This plot shows paleoseismically visible participation rates along all fault sections that have a
paleoseismic trench in red; the mean event rate and 95% confidence bounds is shown in black. The inversion is
able to improve upon the paleoseismic fits of UCERF2; the sum of squared residuals for the annealed model is
half that given by UCERF2.
Figure 8. Comparison of statewide magnitude-frequency distributions for the UCERF2 reference model (left) and
UCERF2 ingredients inverted with UCERF3 methodology (right). Note that the UCERF2 reference model has
aftershocks added back in, and neglects aleatory magnitude variability, which was a part of the UCERF2 model.
The UCERF2 model overpredicted rates in some magnitude bins (as the on-fault MFD and background together
will overpredict the total seismicity MFD shown in black); by contrast, the annealed model on the right is
explicitly constrained to match the cyan target MFD which has been constructed so that, when background
seismicity is added, the total MFD of the model will match a G-R distribution with a-value and b-value
constrained by historical seismicity (black line).
The on-fault MFD distribution for the inverted model shown above shows considerable
difference from UCERF2. This is by construction; however, this difference leads to a significant
difference in the on-fault event rates in the M6.5-7.5 range. This can be seen in the participation
plots below. These plots show the rate at which each fault subsection participates (rather than
nucleates) in a rupture of a given magnitude; we compare this to UCERF2 participation rates and
see considerable differences stemming from the different magnitude-frequency distributions fit
by the models.
Appendix N: The Grand Inversion
16
Figure 9. Participation rates for the model applying UCERF3 methodology to UCERF2 ingredients divided by
participation rates for the UCERF2 reference model for M6-7(left), M7-8 (center), and M≥8 (right). UCERF2 has
higher event rates, particularly for M6.5-7.5, which is reflective of the magnitude “bulge” problem in this model.
UCERF3 CHAR-branch Reference Model
Finally we present our UCERF3 reference branch. This branch is not intended to represent the
preferred branch, but rather a reference with which to compare models using other branches on
the logic tree. The logic-tree choices for the reference model are shown in the table below.
Importantly, the reference branch assumes that 50% of off-fault deformation is aseismic. As
discussed in the main report, this is one of several moment adjustments we test in order to bring
the deformation model moments into alignment with the historical seismicity rates.
Table 2. Logic Tree Settings for UCERF3 Reference Model.
Fault Model
3.1
Deformation Model
Average of Geologic and Average Block
Model (ABM)
Inversion Branch
CHAR (Characteristic / close to UCERF2)
Magnitude-Area Relation
Ellsworth-B
Slip-Length Relation
Ellsworth-B
Shape of Slip Distribution Along Strike
Tapered
Moment Rate Adjustment
50% of off-fault deformation modeled as
aseismic
Appendix N: The Grand Inversion
17
As shown below, this model fits its constraints quite well. Slip-rate fits are on the whole good,
although the model does underpredict the total on-fault target moment (this is the moment given
by the deformation model, less creep and subseismogenic-rupture moment) by 6.5%. This
underprediction is primarily concentrated on the Imperial and Cerro Prieto faults near the
California-Mexico border. The slip rates on these faults are typically underfit by the inversion
models due to the MFD constraint (we have tested that removing this constraint leads to good fits
on these faults). Both faults have high slip rates and would require many moderately-sized
ruptures to satisfy these slip rates due to their isolation (the Cerro Prieto is not close enough to
other faults to link with them, and the Imperial fault, while it can link to the north with the
Brawley seismic zone, has a much higher slip rate than this fault). The rates of moderately-sized
earthquakes are limited strongly by the MFD constraint.
Figure 10. Slip-rate misfit for the UCERF3 reference model.
The paleoseismic fits for the UCERF3 reference model are shown in Figure 11. All 32
constraints are fitting within their 95% confidence bounds (in fact, we are perhaps overfitting
these data due to the generous error bars).
Appendix N: The Grand Inversion
18
Figure 11. Paleoseismic event-rate misfit for the UCERF3 reference model. This plot shows paleoseismically
visible participation rates along all fault sections that have a paleoseismic trench in red; the mean event rate
and 95% confidence bounds is shown in black.
On-fault and implied off-fault magnitude distributions for this model are shown in Figure 12.
The model fits its target MFD quite well; at high magnitudes it tapers off (which does not
contribute to the MFD misfit since the MFD constraint becomes an inequality constraint above
M7.85). The background MFD is similar, but smoother, than the background MFD in UCERF2.
Appendix N: The Grand Inversion
19
Figure 12. California-wide magnitude-frequency distributions for the UCERF3 reference model. The MFDs shown
are as follows: Black – the total target MFD defined from seismicity, Blue – the on-fault MFD found by the
inversion, Cyan – the target on-fault MFD used in the inversion MFD constraints, Grey – the implied background
MFD (black total target minus blue on-fault MFD), Pink – the UCERF2 background MFD (for comparison). As this
is a CHAR (Characteristic / Close to UCERF2) branch model, the on-fault MFD is constrained to deviate from the
total MFD below M7.6 in order to leave enough moment for the background seismicity.
As with the UCERF2-ingredients model, this model differs in terms of participation rates from
UCERF2. This is primarily because the on-fault MFD has been constrained to be quite different,
in order to allow for background seismicity while remaining below the total target MFD defined
by historical seismicity rates.
Appendix N: The Grand Inversion
Figure 13. Participation rates for the CHAR-branch UCERF3 reference model (top panels); ratios of these
participation rates for faults in UCERF2 (bottom panels).
20
Appendix N: The Grand Inversion
21
Figure 14. Rates at which section pairs rupture together; each subsection pair that can link in a rupture is
connected by a line, the color of this line gives the rate that they rupture together normalized by the total rate
of ruptures involving those subsections.
We also compare the rates at which subsections rupture together in Figure 14. UCERF3 allows
for many more fault connections than UCERF2; however, due to slip-rate incompatibilities many
ruptures that include these connections have low rates relative to ruptures on single, contiguous
faults.
G-R branch
We now present a G-R-branch model; this model is constrained to have a spatially smooth
magnitude distribution on the faults that is as close as possible to a Gutenberg-Richter
distribution. All other parameter settings for this branch are set as they are in the reference
model presented above. Below we show fits to the data constraints for this model.
Appendix N: The Grand Inversion
22
Figure 15. Slip-Rate misfit of UCERF G-R branch model.
This branch fits the slip rates somewhat worse than the CHAR branch due to the MFD
participation smoothness constraint (which constrains the participation MFD for each
subsection to be approximately uniform in magnitude). It underpredicts the on-fault target
moment by 13.4%. This underprediction is concentrated on the Cerro Prieto and Imperial
faults (as happens with the CHAR branch) as well as the Mendocino section of the Northern
San Andreas Fault. The slip rate on the Mendocino fault is difficult for the G-R branch to fit
because it has a very high slip rate (twice the slip rate as San Andreas - Offshore) with no
tapering at the end or connections to other faults to the west; therefore the MFD
constraints limit the rates of smaller earthquakes that could be used at the western end of
Mendocino to fit the slip rate there.
Appendix N: The Grand Inversion
Figure 16. Paleoseismic event-rate fit for G-R branch model.
23
Appendix N: The Grand Inversion
24
Figure 17. California-wide magnitude-frequency distributions for the UCERF3 G-R model. The MFDs shown are
as follows: Black – the total target MFD defined from seismicity, Blue – the on-fault MFD found by the inversion,
Cyan – the target on-fault MFD used in the inversion MFD constraints, Grey – the implied background MFD
(black minus blue), Pink – the UCERF2 background MFD (for comparison). As this is a G-R (Close to GutenbergRichter) branch model, the on-fault MFD is constrained to deviate from the total MFD uniformly for each
magnitude bin. Note that the implied off-fault MFD (grey line) plotted here is simply the total target (black)
minus the on-fault MFD (blue) and will be truncated at high magnitudes when the background model is
constructed (Appendix O).
The paleoseismic constraints and magnitude-frequency distribution constraints are fit quite well
by the G-R branch.
The participation rates for this model, with UCERF2 comparisons, are shown in Figure 17.
Many of the differences between this model and UCERF2 are controlled by the MFD. This
model assumes that faults are Gutenberg-Richter (and for the fault system as a whole this
constraint is matched quite well as shown in the MFD plots above), so this leads to higher rates
for many faults in the M6-7 range (particularly fast, through-going faults that were given more
moment in larger characteristic ruptures in UCERF2).
Appendix N: The Grand Inversion
25
Figure 18. Participation Rates for the G-R model (top) and compared to UCERF2 (bottom) for different
magnitude ranges.
How “Gutenberg-Richter” is the model? If nucleation rates for a fault section are GutenbergRichter with a b-value of 1 and stress drops are constant with magnitude, incremental
participation rates for that section will be uniform. We thus show participation rates for selected
fault sections; deviations from uniformity represent deviations from G-R behavior. We also
contrast these same sections with our more characteristic reference branch model that is not
constrained to have smooth participation MFDs; the reader will notice that these plots show stark
differences between these two branches.
Appendix N: The Grand Inversion
26
Appendix N: The Grand Inversion
Figure 19. Participation rates for selected fault subsections for the Char-branch reference model (left) and G-R
branch model (right). Incremental distributions are shown in blue and cumulative distributions are shown in
black. Smooth incremental distributions represent behavior that is closer to Gutenberg-Richter behavior.
Unconstrained branch
The unconstrained branch lacks some of the regularization present in the CHAR and G-R
branches as it has no rupture-rate constraint or magnitude participation smoothness constraint.
Due to the stochastic nature of the simulated annealing algorithm and the underdetermined
nature of the inversion problem for this branch, each unconstrained model will have a different
set of rupture rates. We can use this branch to probe which elements of the inversion are robust
to this resolution uncertainty.
We generate hundreds of models for this branch; the data inputs are the same and the logic-tree
parameter choices are the same as the reference model. Thus differences between these models
is not due to the data inputs or parameter choices. Each model fits the constraints well; however
individual models find different combinations of rupture rates to satisfy these data.
Figure 20. Standard deviation of participation rates divided by mean participation rates, for a suite of 1000
Unconstrained-branch models.
27
Appendix N: The Grand Inversion
28
As shown in Figure 20, on average participation rates on fast-moving, long and contiguous
faults such as the San Andreas and Garlock have less spread among the Unconstrained-branch
inversions than participation rates elsewhere in the model.
Moment-Rate Adjustments
All of the UCERF3 deformation models have significantly higher moment rates than both the
UCERF2 model and historical seismicity (more details on the moment-rate inconsistencies are
given in the main report). Thus the inversion is not able to simultaneously satisfy the fault slip
rates, the off-fault moment, and the historical seismicity a-value. We have tested several
“moment-rate adjustments” to deal with these inconsistencies in the inversion inputs. The
models we presented above have a reference moment-rate adjustment, which is a 50% reduction
in off-fault aseismicity. Even this 50% reduction is not sufficient to match UCERF2 or historical
seismicity moment rates; thus the models presented above still underpredict fault slip rates. For
the Geologic plus Average Block Model deformation model, an 80% reduction in off-fault
moment rate (that is, assuming that 80% of the off-fault deformation is aseismic (!)) is required
to bring moment rates into alignment with UCERF2 moment rates. This more drastic reduction,
which will better satisfy the fault slip rates of this deformation model, is another branch in our
logic tree. We also have three other options in our logic tree: one is to “do nothing”, that is,
invert our inconsistent inputs and obtain a model that does not satisfy these inputs very
accurately. As our default MFD constraint is quite strict, in practice this branch results in
underpredicting the fault slip rates while leaving the background seismicity rates untouched.
Therefore our reference model, with a 50% reduction in off-fault moment and some misfit to slip
rates, is something of an average of our “do nothing” and 80% off-fault moment reduction
branches.
We also have two logic tree branches that assume that the deformation models are correct and
reconcile the data by adjusting the total target magnitude-frequency distribution. One of these
branches assumes that the historically observed seismicity rate (since 1850) is below the true,
long-term seismicity rate, and thus increases the total target MFD by 30%. The other branch
relaxes the MFD constraint weights and allows the inversion model to have a “bulge” – this
model also violates the observed seismicity rates seen since 1850. In addition, the “relax MFD”
bulge will not be Gutenberg-Richter for California as a whole; for this reason we do not prefer
this model, although it is interesting to see what the inversion “prefers” in the absence of a strong
MFD constraint.
Appendix N: The Grand Inversion
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Appendix N: The Grand Inversion
30
Figure 21. Magnitude frequency distributions (left, in the format of Figure 12) and slip-rate misfits (right) for
different moment-rate adjustments.
It is important to note that our “increase MFD” and “relax MFD” branches, unlike the other
moment-rate adjustments discussed, may significantly increase hazard relative to UCERF2 since
these two branches result in overpredicting historically observed moment rates.
Other Logic-Tree Branches
We have scanned solution misfits for other logic tree branches and found that some branch
choices are “preferred” by the inversion, meaning that they consistently have better misfits. For
example, uniform slip along the rupture gives better slip-rate fits than the tapered-slip model.
This appear to be true regardless of the choices on the other branches of the logic tree. Similarly,
the Hanks-Bakun and Shaw-09 magnitude-area relationships perform better than the Ellsworth-B
magnitude-area relationship. This is because they allow for larger slips in high-magnitude
ruptures, which makes it easier to fit the high (relative to seismicity) moment rates in the
deformation models. Some branches do not have a large impact on misfit; the average slip per
rupture model, for example, does not have a big impact on this metric.
Links to more results
We have generated hundreds of models corresponding to different choices in the UCERF3 logic
tree; not all models can be presented here. Plots from these models, as well as solution sets that
can loaded into SCEC-VDO and viewed can be found at the following links:
Characteristic and G-R branch models:
http://opensha.usc.edu/ftp/kmilner/ucerf3/2012_03_28-lots-of-combos/
http://opensha.usc.edu/ftp/kmilner/ucerf3/2012_03_28-production-for-realz/
Mean Unconstrained Model & Statistics of Unconstrained Models:
http://opensha.usc.edu/ftp/kmilner/ucerf3/2012_03_28-unconstrained-run-like-crazy/
UCERF2 reference solutions (for comparison):
http://opensha.usc.edu/ftp/kmilner/ucerf3/2012_03_26-ucerf2_comparisons/
Concluding Remarks
The inversion relies on data inputs and models from many other project tasks to determine the
rates of on-fault ruptures. Many of these inputs will be uncertain and subject to debate.
However, it is important to note that all these uncertainties existed in the old methodology for
determining rupture rates as well. The inversion methodology described here eliminates the need
for voting directly on rupture rates. It provides a means to easily update the model as new data
becomes available. Importantly, the inversion methodology provides a mechanism to
simultaneously fit all the constraints concurrently. This was lacking in UCERF2 – expert
opinion did not simultaneously satisfy slip rates and event rates, and the final model also
presented magnitude distribution problems. The inversion allows all these constraints to be
satisfied to the extent that they are compatible.
Appendix N: The Grand Inversion
31
The inversion can also be used as a tool to determine when a set of constraints is not compatible,
as with the UCERF3 deformation models and historical seismicity rates. This remains one of the
most important uncertainties in our inversion in terms of its impact on hazard. To what extent
more aseismicity is needed in the deformation models or the historical seismicity rates may
underpredict the true long-term rates will largely determine how similar UCERF3 models are to
past models such as UCERF2.
Acknowledgement
Computation for the work described in this manuscript was supported by the University of
Southern California Center for High-Performance Computing and Communications
(www.usc.edu/hpcc).
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