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Modeling spatio-temporal earthquake dynamics
using generalized functional additive regression
Alexander Bauer1, Fabian Scheipl1, Helmut K¨uchenhoff1and Alice-Agnes Gabriel2
1Department of Statistics, LMU Munich, Germany
2Department of Geophysics, LMU Munich, Germany
Alexander.Bauer@stat.uni-muenchen.de
Data and Research Question
Setting
We analyze artificial earthquake data derived from large-scale computer simulations based on a 1994 earthquake in
Northridge (USA). In each of 135 simulations, the (isotropic) absolute ground velocity [m/s] was measured at
6146 virtual seismograms with a temporal resolution of 2Hz.
This project marks the first time that physics-based simulations of earthquakes are combined with modern statistical
methods. Apart from gaining new insights in the geophysical processes regression models could in future be used to
predict expected ground movements in earthquake regions.
Main research question
How do the physical conditions at an earthquake fault
affect the surficial ground velocity measured over time?
Challenges
•Very high-dimensional data
•Spatio-temporal functional data
0
1
2
3
4
0 5 10 15
time [s]
ground velocity [m/s]
hypocentral distance
small
medium
large
Typical observations
by hypocentral distance
Figure 1: Left: Categorized mean absolute ground velocity in one simulation over the area under study, darker colours correspond to
increased velocity. Right: Typical observations of absolute ground velocity over time. The initial peak of the ground velocity is delayed and
smaller as hypocentral distance increases.
Simulation setup
The artificial earthquake data were generated using the open-source
software SeisSol (www.seissol.org). In each simulation:
1. Five simulation parameters were pre-set, and
2. absolute ground velocities were simulated solving elastic wave
equations coupled to frictional failure at the earthquake fault.
Influence parameters
The influence parameters are all constant over time.
•soil material ({rock,sediment})
•linear slip weakening distance [m]
•static coefficient of friction
•dynamic coefficient of friction
•direction of tectonic background stress [◦]
•hypocentral distance of seismometer [m]
•elevation of seismometer [m]
•landform at seismometer∗({ridge,plain,valley , . . . })
•moment magnitude [Nm]
∗Categorization into landforms was performed using the Topographic Position
Index (TPI) of Weiss (2001)
1 Modeling process
We use a Generalized Functional Additive Model (GFAM) (see Scheipl et al., 2016)
which is an extension of the GAM model class.
In our case, only the response is functional and we use a Gamma model with log-link.
yi(tl)∼F(µil ,ν) with g(µil ) = β0(t) +
R
X
r=1
fr(Xri ,tl),i= 1,...,n
•yi(tl): Value of functional response observed at time point tl
•F(µil ,ν): Conditional distribution of yi(tl) with conditional expectation µil and
shape parameters ν
•g(·): Link function
•β0(t): Functional intercept
•fr(Xri ,tl): One of Radditive effects with associated covariates Xri and potentially
varying over the functional time domain t
•n: number of functional observations
We use a highly performant estimation algorithm from Wood et al. (2016) to make
estimation of this complex model on such large data feasible. Major advances are:
•a block-wise Cholesky decomposition
•a compressed representation of marginal spline bases
A prediction error based approach was used for tuning basis sizes, resulting smooth effects
were estimated using (tensor product) P-splines.
2 Covariate effects
The hypocentral distance and the dynamic frictional resistance have by far the strongest
effects, with higher values leading to decreased ground velocities for both.
0
5
10
15
20 40 60 80 100
hypocentral distance [km]
time [s]
−2−1 0 1 2 3
estimate
hypocentral distance [km]
20 40 60 80 100
time [s]
0
5
10
15
estimate
−2
−1
0
1
2
0 5 10 15
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Predicted ground velocity
by hypocentral distance [km]
(with 95% confidence intervals)
time [s]
ground velocity [m/s]
20
40
60
80
100
0 5 10 15
0
1
2
3
4
Absolute ground velocity
by dynamic coefficient of friction
(95% confidence intervals)
time [s]
ground velocity [m/s]
0.1
0.3
0.5
Figure 2: Left: Nonlinear, time-varying effect of hypocentral distance as heatmap and 3D surface, and
predictions based on varying hypocentral distances, while other covariates are held constant at realistic
values. Right: Predictions based on varying values of the dynamic coefficient of friction, which has a linear,
time-constant effect of -5.48
3 Model evaluation
●
0
5
0
5
0
5
10
15
0 1 2 3 4 0 5 10 15 0 5 10 15
fitted values time [s] time [s]
residuals
residuals
time [s]
0 75000 150000
Frequency 0 7500 15000
Frequency Residual mean
(−1.00,−0,25] (−0.25,−0.05] (−0.05, 0.05]
( 0.05, 0.25] ( 0.25, 1.00] 0.2 0.4 0.6
Covariance
Residuals vs fitted values
Heatmap of binned points Residuals vs time
Heatmap of binned points Mean residuals over space Autocovariance of residuals
Figure 3: From left to right: Residuals vs fitted values, residuals vs the time domain, residuals vs space,
autocovariance of residuals over the functional domain. The black dot in the third plot marks the epicenter.
10−2
10−1
100
0 5 10 15
time [s]
ground velocity [m/s]
on log10−scale
Small
hypocentral distance
10−2
10−1
100
0 5 10 15
time [s]
ground velocity [m/s]
on log10−scale
Medium
hypocentral distance
10−2
10−1
100
0 5 10 15
time [s]
ground velocity [m/s]
on log10−scale
Large
hypocentral distance
observation prediction
Figure 4: Comparison of model predictions and raw observations for typical observations with different
hypocentral distances.
⇒Spatial residual structure remaining
⇒Predictions in general behave well (70.7% explained null deviance)
4 Conclusion & Outlook
Functional additive regression models are a promising approach for modeling surficial
ground velocity.
Our model
•allows a better understanding of the observed seismological patterns
•adds value to the current seismological discussion of how important precise
determination of specific physical parameters is
•offers predictions which could in future replace computer-intensive earthquake
simulations
Secondary finding
Moment magnitude can be predicted very well using the simulation parameters (98.2%
explained null deviance)
Future research
The model will be refined further, e.g. by explicitly modeling spatial correlation and by
relaxing the strict assumption of the hypocenter as fixed point source for all earthquakes.
Furthermore model performance will be examined for additional earthquakes.
References
Bauer, A. (2016). Auswirkungen der Erdbebenquelldynamik auf den zeitlichen Verlauf der Bodenbewegung. MA thesis. Ludwig-Maximilians-Universit¨at, Munich, Germany. Available: https://epub.ub.uni-muenchen .de/31976/
Scheipl, F., Gertheiss, J., Greven, S. (2016). Generalized functional additive mixed models. Electronic Journal of Statistics,10.1, 1455 – 1492.
Weiss, A. (2001). Topographic position and landforms analysis. Poster presentation, ESRI user conference, San Diego, CA, 200.
Wood, S.N. et al. (2016). Generalized additive models for gigadata: modelling the UK black smoke network daily data. Journal of the American Statistical Association. DOI: 10.1080/01621459.2016.1195744.
IWSM Groningen 2017
