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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