In this paper, we employed polynomial chaos (PC) expansions to understand earthquake rupture model responses to random fault plane properties. A sensitivity analysis based on our PC surrogate model suggests that the hypocenter location plays a dominant role in peak ground velocity (PGV) responses, while elliptical patch properties only show secondary impact. In addition, the PC surrogate model is utilized for Bayesian inference of the most likely underlying fault plane configuration in light of a set of PGV observations from a ground-motion prediction equation (GMPE). A restricted sampling approach is also developed to incorporate additional physical constraints on the fault plane configuration and to increase the sampling efficiency.

One of the most important challenges seismologists and earthquake engineers
face in designing large civil structures (e.g., buildings, dams, bridges, power
plants) and response plans, especially in highly populated cities prone to
large damaging earthquakes, is the reliable estimation of ground-motion
characteristics at a given location. Ground-motion prediction equations
(GMPEs), which are one of the most important elements for probabilistic
seismic hazard analysis (PSHA), are designed for this purpose. These are
obtained from regression analysis by fitting a dataset (empirical and
simulated) and are mainly expressed in terms of site conditions,
source–site distance (e.g., rupture distance or Joyner–Boore distance, denoted
as

The Joyner–Boore distance is defined as the shortest distance from a site to the surface projection of the rupture plane.

), magnitude, and mechanism, although other terms such as directivity and hanging-wall effect are also consideredMany efforts have been made to characterize the seismic ground-motion
considering both real and simulated data. For example, using real data, five
research groups under the Pacific Earthquake Engineering Research Center Next
Generation Attenuation (PEER NGA) project derived GMPEs for shallow crustal
earthquakes considering an extensive database of recorded ground motion

In this study, we investigate the level of complexity needed in kinematic
rupture models of magnitude 6.5 strike-slip events to produce
ground motion similar to a reference GMPE. To this end, we utilize the polynomial chaos (PC)
approach

This paper is organized as follows. In Sect.

A magnitude

Example of fault plane configuration. The red star denotes hypocenter location, and the ellipse is the asperity with Gaussian slip distribution inside. The slip distribution is tapered in the area between the dashed and solid rectangles.

A virtual network of

Velocity model used in this study, modified from

PGVs at a virtual network of

Parameters governing fault plane configurations.

PC expansions

An open-source toolkit
for the PC framework is available at

Each random parameter vector

Let

To determine the expansion coefficients (

The
corresponding source code is available at

Following

We first validate our PC surrogate models for PGVs at all stations. To this
end, we introduce a second independent source model simulation ensemble
(again an 8000-member LHS set

Relative

Figure

PC-predicted PGV distributions at two selected stations (as
indicated in Fig.

Apart from the above error estimates, the convergence of PC surrogate models
with respect to truncation order is also investigated from a statistical
point of view. Figure

Comparison of PGV distributions predicted by the source model (blue
solid curve) and PC surrogate model (red dashed curve), respectively, at
selected stations (as indicated in Fig.

We finally compare distributions of PC and source model predictions; see
Fig.

The PC surrogate models obtained in the previous section provide immediate
access to prediction statistics, as given by Eqs. (

The interested reader is referred to

Comparison of PC statistics (based on uniform distribution assumption of the canonical PC random parameters) with GMPE results. Solid black curve denotes the median GMPE prediction, while the dashed lines are GMPE standard deviation bounds. Note that log scales are used in the plot.

The conditional mapping from canonical PC random variables (

First-

Figure

First-order sensitivity indices with respect to grouped parameters.

To better illustrate the above sensitivity observation, we divided the
parameters into the following two groups (

In this section, we utilize a Bayesian approach

To formulate the Bayesian problem, we start with Bayes' formula:

Recall that the PC prediction variability seems to decrease with

The prior distribution of

We rely on the adaptive metropolis MCMC approach

Posterior probability distributions of prediction uncertainty parameters (each PDF curve is scaled to have unit peak height for better comparison).

As mentioned above, we exploit the PC surrogate models in Bayesian inference
analysis and update the posterior distribution of random parameters
(

Similarly, we examine the sampling chains of PC random parameters

Prior (dashed black, derived from uniform

Figure

One needs to be cautious about the Bayesian inference results discussed
above. From the physical point of view, the spatial distribution of those
stations (see Fig.

Inferred fault plane configuration with MCMC chain starting from the “symmetric” counterpart configuration.

The previous inference results are all based on almost complete ignorance of
dependency between hypocenter location and the slip area (asperity). However,
previous studies

In this section, we consider the following restrictions in our inference
analysis:

The elliptical patch is inside the dashed rectangle

The area ratio of the elliptical patch (AR) is between
15 and 29 % of the fault plane area, i.e.,

The elliptical patch is not too elongated, i.e., the axis
ratio

The hypocenter is located outside but near the elliptical
patch, i.e.,

One of the advantages of having previous PC surrogate models (which were
built based on uninformative prior that spans a wide range of feasible
scenarios, i.e., minimal restrictions as in Table

To begin with, we first incorporate the above restrictions into the
Bayesian framework, namely by modifying the previous prior distribution
(Eq.

However, due to the strong restrictions listed above, the support of the
above prior probability distribution (Eq.

Flow chart demonstrating the random sampling process and the
calculation of posterior probability in MCMC. The orange path corresponds to
unrestricted sampling process, whereas the blue path incorporates additional
restrictions on fault plane configurations. Note that

Figure

Prior (dashed black, derived from uniform

Following the same analysis as discussed before, we show the inference
results under restrictions in Fig.

Restricted Bayesian MCMC sample chains of the
hypocenter

Though it is not obvious to see from Fig.

Comparison of PC-predicted PGV responses with aforementioned three inferred fault plane configurations with the reference GMPE curve. Dashed lines are standard deviation bounds of GMPE predictions.

Comparison of PC-predicted PGVs of different inferred configurations
with the reference GMPE curve. Unrestricted-1 and -2 correspond to inferences
in Figs.

We summarize the Bayesian analysis by comparing PC-predicted PGV responses to
the three inferred fault plane configurations discussed above with the
reference GMPE curve (see Fig.

An earthquake rupture model was adopted to explore the stochastic dependence
of ground motion (in terms of PGVs) on random fault plane configurations.
Thanks to the ability to generate two independent source model simulation
ensembles with 8000 members each, we were able to build successful PC
surrogate models to assess PGV responses over the virtual network of

A global sensitivity analysis of PC surrogate models was conducted. The
analysis revealed that the source model PGV response is primarily sensitive
to the hypocenter location, and much less sensitive to properties of the
asperity patch, especially at stations far away from the fault plane (in
terms of the

Our analysis of PGV variabilities indicated that one needs to be cautious
when interpreting PGVs at near-fault-plane stations, as they are more prone
to higher model noise. This is supported by the Bayesian inference analysis,
in which four independent model noise parameters (

We conducted both unrestricted and restricted Bayesian inference analyses to
identify the chosen GMPE reference curve. The key findings are as follows.
(1) Given the station distribution (Fig.

The analyses and findings in this study provide useful insights on how near-source ground shaking (and its variability) depends on random fault rupture configurations. Interestingly, even very simple source models (with elliptical slip patches) are able to generate shaking distributions that well reproduce empirical predictions. To better reproduce the chosen GMPE reference curve, it might be beneficial to consider two or more asperity patches, instead of one in this study, in order to reduce the hypocenter location influence and in return increase the impact of asperity properties. Another potential improvement can be made by refining the station network. As mentioned earlier, the Bayesian inference is primarily limited by the number of available stations at which PGVs are reported. By increasing the number of PGV reporting stations, one may improve the Bayesian inference results (e.g., removing the ambiguity in inferring the elliptical patch location).

The COMPSYN code

Unrestricted mapping – PC random parameter

Let

If the elliptical patch is rotated by

To ensure the resulting elliptical patch is completely confined within the
fault plane, we first find the maximum extent of the ellipse in both

Next, by substituting the above

When the ellipse is not centered at the origin (

The mapping from

We introduce the auxiliary parameter vector

The elliptical patch is inside the dashed rectangle (

The area of the elliptical patch (AR) is between 15 and 29 % of the fault plane area, i.e.,

The elliptical patch is not too elongated, i.e.,

The hypocenter is located outside but near the elliptical patch,
i.e.,

The mapping process is similar to the one in Algorithm

Restricted mapping – auxiliary parameter vector

In this study, HCJ and PMM created the earthquake rupture model and generated both the training and validation ensembles of model simulations for building PC surrogates. The PC-based statistical analysis and Bayesian inference were conducted by GL and OMK. IH provided invaluable insights and advice throughout this work.

The authors declare that they have no conflict of interest.

Research reported in this publication was supported in part by research funding from King Abdullah University of Science and Technology (KAUST). The first author thanks KAUST for all the support during his postdoctoral fellowship. Earthquake rupture and ground-motion simulations have been carried out using the KAUST Supercomputing Laboratory (KSL) and we acknowledge the support of the KSL staff. Edited by: Thomas Poulet Reviewed by: two anonymous referees