The propagation of a seismic rupture on a fault introduces spatial variations
in the seismic wave field surrounding the fault. This directivity effect
results in larger shaking amplitudes in the rupture propagation direction.
Its seismic radiation pattern also causes amplitude variations between the
strike-normal and strike-parallel components of horizontal ground motion. We
investigated the landslide response to these effects during the 2016 Kumamoto
earthquake (Mw 7.1) in central Kyushu (Japan). Although the
distribution of some 1500 earthquake-triggered landslides as a function of
rupture distance is consistent with the observed Arias intensity, the
landslides were more concentrated to the northeast of the
southwest–northeast striking rupture. We examined several landslide
susceptibility factors: hillslope inclination, the median amplification
factor (MAF) of ground shaking, lithology, land cover, and topographic
wetness. None of these factors sufficiently explains the landslide
distribution or orientation (aspect), although the landslide head scarps have
an elevated hillslope inclination and MAF. We propose a new physics-based
ground-motion model (GMM) that
accounts for the seismic rupture effects, and we demonstrate that the
low-frequency seismic radiation pattern is consistent with the overall
landslide distribution. Its spatial pattern is influenced by the rupture
directivity effect, whereas landslide aspect is influenced by amplitude
variations between the fault-normal and
fault-parallel motion at frequencies <2 Hz. This azimuth dependence
implies that comparable landslide concentrations can occur at different
distances from the rupture. This quantitative link between the prevalent
landslide aspect and the low-frequency seismic radiation pattern can improve
coseismic landslide hazard assessment.
Introduction
Landslides are one of the most obvious and hazardous consequences of
earthquakes. Acceleration of seismic waves alters the force balance in
hillslopes and temporarily exceeds shear strength
. Greatly increased landslide rates have been
reported on hillslopes close to earthquake rupture, mostly tied to ground
acceleration and lithology . Substantial
geomorphological and seismological data sets are required to assess the
response of landslides to ground motion, and a growing number of studies have
shed light on the underlying links
e.g.. Several seismic
measures, such as vertical and horizontal peak ground acceleration
PGA;, root-mean-square (RMS) acceleration, or Arias intensity
IA;,
seismic source-moment release, hypocentral depth, and rupture extent
and propagation , correlate with landslide density
.
Landslides concentrate in the area of strongest ground acceleration
, whereas total landslide area decreases from the
earthquake rupture with the attenuation of peak ground acceleration
. This general pattern is modified by
morphometrics (e.g. local hillslope inclination and curvature) and geological
parameters
e.g. lithology, geological structure, and land cover; that influence landslide susceptibility
on top of seismic amplification .
For instance, found that lithology, PGA, and distance from
the rupture plane are important in assessing the distribution of landslides
triggered by the 2008 Wenchuan earthquake (Mw 7.9).
found that hillslope aspect and slope were important
determinants of the landslide distribution resulting from the 2017 Jiuzhaigou
earthquake (Mw 6.5).
On 16 April 2016 at 16:25 UTC central Kyushu was hit by a Mw 7.1
earthquake (Fig. ). The left-lateral dip-slip event ruptured
along the Futagawa and Hinagu faults, striking NW–SE, with a hypocentral depth
of 11 km e.g.. The rupture propagated northeastward and
stopped at Mt Aso. Fault source inversions show a northeast propagation of
the rupture originating under Kumamoto City, with highest slip near the
surface at the western rim of the Aso caldera
e.g..
The earthquake triggered approximately 1500 landslides that
concentrated mainly inside the caldera and the flanks of Mt Aso on the
Pleistocene and Holocene lava flow deposits ,
although most of the terrain near the earthquake rupture is rugged
(Fig. ). Thus, we hypothesize that rupture directivity causes an
asymmetric distribution of landslides around the rupture plane because of
more severe ground motion along the propagating rupture
. Similarly asymmetric landslide
distributions attributed to rupture directivity were reported for the 2002
Denali earthquake Mw 7.9; and the
2015 Gorkha earthquake Mw 7.8;. In the case of
the 1999 Chi-Chi earthquake, speculated that the prevalent
landslide aspects were correlated to the fault movement direction
. These observations indicate that the rupture
process introduces variations on the incoming energy on hillslopes.
Here we link those dominant near-surface seismic characteristics relevant to
the pattern and orientation of coseismic landslides. We investigate the
geological conditions (lithology, aspect, hillslope inclination, topographic
amplification, and soil wetness) in central Kyushu as well as seismic waveform
records from 240 seismic stations within 150 km of the rupture (Fig. ).
The two most prominent seismic effects – well founded in
seismological theory e.g. and documented in empirical
relationships e.g. – are the rupture directivity
and amplitude variations in fault-normal and fault-parallel motion. We
examine whether the geomorphic characteristics around the Aso caldera made
this area more susceptible to landslides than the surrounding topography near
the earthquake rupture or whether rupture effects control the asymmetric
distribution of the landslides. We introduce a ground-motion metric related
to azimuth-dependent seismic energy (i.e. seismic velocity) because these
effects attenuate with increasing frequency and are less captured by
acceleration-based metrics. We conclude by proposing a new ground-motion
model (GMM) that is consistent with the observed coseismic landslide pattern.
The area of Kyushu affected by coseismic
landslides triggered by the 2016 Mw 7.1 Kumamoto earthquake. The
coloured patch is the slip distribution of the rupture model by
, and the dashed box encompasses landslides related to the
triggered event in Yufu (epicentre location after ). The
inset map shows the station network within 150 km of the rupture.
Data
We combine data sets on the response of landslides to the earthquake,
including topography, land cover, geology, seismic waveforms, velocity
structure, near-surface characteristics, and landslide location and planform
(Fig. ).
Topographic and geological features of central Kyushu
with landslides (black dots), landslide-affected area (outer black line),
rupture area (inner black line), hypocentre (black diamond), and mountain
peaks from Fig. (triangles). (a) Hillslope
inclination. (b) Median amplification factor (MAF).
(c) Topographic wetness index (TWI). (d) Geology of central
Kyushu. The most common geological units of the landslides are shown in
(e). For the landslide-affected area the dominant geological units
are in (f). (g) Land cover. Land cover in the landslide
areas is shown in (h) and is shown for the entire landslide-affected area
in (i).
Topographic data
Most topographic data used in this study are provided by the Japan Aerospace
Exploration Agency (JAXA) and its Advanced Land Observing Satellite (ALOS)
project with a horizontal resolution of 1′′ (≈30 m). This digital
surface model (DSM) forms the basis for computing aspect, hillslope
inclination, the median amplification factor MAF;, and
the topographic wetness index . The ALOS project also
provides data on land cover, including anthropogenic influence (sealing and
agriculture) and vegetation, while data on major geological units are from
the Seamless Digital Geological Map of Japan (scale of 1:200000) by the
Geological Survey of Japan.
Topographic amplification of ground motion
Topographic features, such as mountains and valleys, can amplify or attenuate
seismic waves . The largest
ground-motion variations occur on hillslopes and summits, whereas variations
are intermediate on narrow ridges and low on valley floors.
introduced proxies for these topographic site effects, of
which we use the median amplification factor (MAF), based on the topographic
curvature, and the S wave velocity vS travelling at frequency f:
MAF(f)=8×10-4vSfCSvS2f+1,
where CSvS2f is the topographic
curvature convolved with a normalized smoothing kernel based on two 2-D
boxcar functions as a function of vS and f.
The curvature is estimated from the DSM ,
and the seismic velocity vS is the average S wave velocity of the
uppermost 500 m from the model by .
Another site effect that influences landslide potential is the local soil or
groundwater content, which can be modelled for uniform conditions to the first
order using the topographic wetness index (TWI) of :
TWI=logActanβ,
where Ac is the upslope catchment area and β is the
hillslope inclination derived from the DSM with filled sinks
.
Ground-motion data
Ground-motion data are from the KiK-Net and K-Net of the National Research Institute for Earth Science and
Disaster Prevention (NIED) of Japan. NIED operates both borehole and surface
stations for KiK-Net, and we use the latter only. The Japan Meteorological
Agency (JMA) also released seismic data from the municipal seismic network
for the largest earthquakes of the Kumamoto sequence. In total, data from 240
stations in Kyushu are available with complete azimuthal coverage within
150 km of the earthquake rupture (Fig. ).
The analysis of seismic waveforms is based on accelerometric data only. Both
the NIED and JMA data are unprocessed, and we follow the strong motion
processing guidelines of . We use both acceleration and
velocity in further processing and integrate the accelerograms to obtain
velocity records. We correct the data with the automated baseline correction
routine by . The JMA accelerometric data further require a
piecewise baseline correction prior to the displacement baseline correction
due to abrupt (possibly instrument-related) jumps
. We use the automated correction for baseline
jumps by von Specht (2019).
An earthquake was triggered approximately 80 km to the northeast in Yufu
32 s after the Kumamoto earthquake Fig. ;. Due to the close succession of the two events,
waveforms of the triggered event interfere with the coda of the Kumamoto
earthquake. We taper the data to reduce signal contributions by the triggered
event. The taper position is based on theoretical travel time differences
between the P wave (vP=5700 m s-1) arrival of the Kumamoto
earthquake and the S wave arrival (vS=3300 m s-1) of the
triggered event. The respective travel paths to the stations are measured
from the hypocentres. Since fewer instruments are located to the northeast
and the triggered event close to the sea, less than 10 % of the data are
strongly contaminated by the triggered event.
NIED hosts the rupture-plane model of , which describes the
slip history on a curved rupture plane (based on the surface traces of the
Futagawa and Hinagu faults) with a total length of 53.5 km and width of
24.0 km (Fig. ). We use the extent and shape of the rupture
plane to estimate the landslide-affected area and to define the rupture-plane
distance rrup, the shortest distance from the rupture plane. We follow the
approach of to identify the asperity from the
rupture-plane model, which is the area with more than 1.5 times the average
slip.
The underground structure in terms of seismic velocities (vP,
vS) and density (ρ) is available for 23
layers down to the mantle in ≈0.1∘ resolution covering all
of Japan; we only consider the layers of the upper 0.5 km to compute a
velocity average for the MAF.
NIED provides data for the subsurface shear wave velocities
(vS30) as well as site amplification factors Samp.
Contrary to vS by , vS30 is
derived for the upper 30 m only and is more suitable for energy estimates,
which require velocities at the surface (recording station). The site
amplification factor Samp describes how much seismic waves are
amplified by, independent of their frequency.
Landslide data
Detailed landslide data are provided by NIED as polygons
(Fig. ), mapped from aerial imagery with sub-metre
resolution at different times after the Kumamoto earthquake. The first data
set contains landslides that were identified between 16 and 20 April, though
the area close to the summit of Mt Aso was not covered. A second data set
was collected on 29 April 2016 and covers those parts of Mt Aso that
remained unmapped. However, the second data set may contain rainfall-induced
landslides, since the rainy season in Kyushu starts in May
, and there was rainfall after the Kumamoto earthquake
and landslides triggered by volcanic activity. We selectively combine the two
data sets for this study, using only those landslides from the second
database, which are also partly present in the first data set. We exclude any
rainfall triggered landslides with this approach, though possibly omitting
seismically induced landslides exclusive to the second database. However, the
area in question is comparatively small to the full extent of the study area,
and the missing landslides are minor in terms of their area.
Several landslides cluster ≈ 80 km to the northeast of the mainshock
in the municipalities Yufu and Beppu (Fig. ), which were hit
by a triggered earthquake . We hypothesize that the distant
northeastern landslides were induced by this triggered event. This also
explains the considerable gap in landslides (≈50 km) between Yufu
and Aso (Fig. ) in otherwise steep topography.
Apart from the special release of landslide data for the 2016 Kumamoto
earthquake, NIED hosts a landslide database for all Japan .
This database covers unspecified landslides of any origin. We extract a
subset from this landslide database to compare it with the landslides
triggered by the Kumamoto earthquake. Contrary to the special Kumamoto
release, only the landslide deposits are mapped as polygons, whereas the
scarps are mapped as lines. We manually define polygons representing the
total landslide area bound by the scarp line and covering the deposit area to
make both data sets comparable and because the landslide source area is
generally not identical to the deposit area.
Total area affected by landsliding
We define the landslide-affected area, in which coseismic landsliding
occurred, as the area spanned by the rupture-plane distance covering
97.5 % of the total landslide area . Thus the
total landslide-affected area is 3968.6 km2 and is within 22.9 km distance
from the rupture plane.
An Mw 7.1 event with a fault length of 53.5 km and an asperity
length of 12.78 km (3 km) results in a landslide-affected area of
3914 km2 (4406 km2) using parameters proposed by . We
derived the event depth of 11.1 km as the moment weighted average of the
rupture model of . Both estimates are consistent with our
area estimate. introduced a topographic constant,
Atopo, relating the total landslide area to the area that
excludes basins and inundated areas. We estimate Atopo from the
ALOS land cover, finding that 97 % of all landslides occurred in areas
without anthropogenic influence, i.e. land with urban and agricultural use,
and water bodies. We exclude water bodies, urban areas – predominantly the
metropolitan area of Kumamoto City, and rice paddies from the topographic
analysis, obtaining an affected area of 3037 km2, i.e.
Atopo=0.68.
Total landslide area
Total landslide area is linked to several earthquake parameters, mostly
magnitude and hypocentre or average rupture-plane depth
. We adopted the relation by to
check for completeness of the total landslide area of 6.38 km2. The
actual total landslide failure plane is likely smaller, as the NIED data set
provides the combined area of depletion and accumulation. The modal hillslope
inclination is estimated at 15∘. Instead of the earthquake magnitude
scaling relation used by , we use the
rupture extent reported by . The area model requires the
average length of the seismic asperities, which globally
assumed to be 3 km. However, derived a relationship of
asperity sizes based on the seismic moment that results in an average asperity
length of 12.78 km for the 2016 Kumamoto earthquake. This length is
consistent with the asperity sizes found by for their
finite rupture model. The estimated landslide area with an asperity length of
3 km results in a predicted total landslide area of 12.90 km2, while
with the magnitude scaled asperity size of , the
landslide area is 3.03 km2. The landslide area estimates with constant
asperity length and moment-dependent asperity length differ by a factor of 2
and 0.5 from the NIED data set, respectively.
Landslide concentration is defined as landslide area per area at a given
distance band . For the seismic processing, we consider
the rupture-plane distance rrup based on the rupture model
instead of the hypocentral distance or the Joyner–Boore
distance .
Ground motion and seismically induced landslidingCoseismic landslide displacement
The sliding-block
model of is widely used to estimate coseismic hillslope
performance e.g.. The model
estimates the permanent displacement on a hillslope affected by ground
motion. established a relation for hillslope displacement
in terms of the maximum velocity at the hillslope for a single rectangular
pulse, vmax (m s-1):
ds=vmax221Aay-1A,
where A is the magnitude of the acceleration pulse and ay
(m s-2) is the yield acceleration, which is the minimum pseudostatic
acceleration required to produce instability. For downslope motion along a
sliding plane, ay is related to the angle of internal friction,
ϕf, and the hillslope inclination, δ, by
ay=gtanϕftanδsinδ=g(FS‾-1)sinδ,
with the average factor of safety FS‾.
characterized unstable hillslopes – related to both rainfall and earthquakes
– by a safety factor of FS<1.5.
An upper bound for the displacement, ds, is based on two ground-motion parameters :
dmax=PGAayPGV2ay,
where PGA (m s-2) and PGV (m s-1) are the peak ground
acceleration and velocity, respectively. Thus, the coseismic hillslope
performance can be characterized by velocity and acceleration. In the
following sections, we derive a ground-motion model based on the
acceleration-related Arias intensity and the velocity-related radiated
seismic energy.
Ground-motion metrics
Though PGA is the most widely used ground-motion metric in geotechnical
engineering, the Arias intensity IA is widely
used to characterize strong ground motion for landslides:
IA=π2g∫T1T2a(t)2dt,
where g=9.80665 m s-2 is standard gravity and T1 and T2 are
the times where strong ground motion starts and cedes (the acceleration
a(t) has units of m s-2 and the Arias intensity has units of
m s-1). The Arias intensity captures both the duration and amplitude of
strong motion. Empirical relationships between IA and
ds in terms of earthquake magnitude and epicentre distance have
been developed e.g..
Since PGA and IA are related to each other
e.g. and the hillslope displacement depends on both
velocity and acceleration (Eqs. and
), it is reasonable to characterize velocity similarly to
Arias intensity. The velocity counterpart to IA is IV2, the
integrated squared velocity :
IV2=∫T1T2v(t)2dt.
The squared velocity is also used in radiated seismic energy estimates. The
quantity jE is the radiated energy flux of an earthquake and
estimated by , , and :
jE=ρcSamp2e-krrupIV2,
where ρ and c are the density and seismic wave velocity at the
recording site and Samp is the site specific amplification
factor. The distance from the rupture is given by rrup, and k is
a term for path attenuation and effects of transmission
and reflection . The attenuation constant k is also
influenced by anisotropy and structure heterogeneity
. The full definition of the energy flux
includes two terms for compressional waves (c=vP) and shear waves
(c=vS). The radiated energy of an earthquake, ES,
results from the integral over the wavefront surface:
ES=∬jEdA,
where A is the area of the surface through which the wave passes at the
recording station and represents the geometrical spreading.
The radiated seismic energy ES describes the energy leaving the
rupture area and is related to the seismic moment :
ES=Δσ2μM0,
where Δσ is the stress drop, μ the shear modulus, and M0
the seismic moment. We make use of this relation when considering the
magnitude-related term in the ground-motion model. Since most seismic energy
is released as shear waves, we apply the shear wave velocity at the recording
site (vS) to the entire waveform, i.e. we assume that all waves
arrive with velocity vS at a site. This assumption has the
advantage that it does not require a separation of the record into P and
S waveforms, simplifying the computation. In Appendix we show
from a theoretical perspective that using a uniform vS has only a
small impact on the overall energy estimate. The site-specific correction
term for the energy estimate E^ based on Eqs. () and
() becomes
E^=A^ρvSSamp2e-krrupIV2.
While ES is the radiated seismic energy at the source, E^
is estimated from the velocity records at a station and only approximates
ES. Therefore, E^ may differ from the true and unknown
radiated energy ES. Several assumptions
characterize E^:
All energy is radiated as S waves in an isotropic, homogeneous medium.
Geometrical spreading is corrected for an isotropic, homogeneous
medium.
Since IV2 (Eq. ) depends on the radiation pattern, E^ depends on
the azimuth.
Attenuation is homogeneous.
Surface waves are not considered.
Site amplification is frequency-independent.
Below, we investigate the azimuthal variation in the energy estimates to
characterize the radiation pattern.
The estimated wavefront area A^ is related to the rupture extent and
rrup, and A^ corresponds to a simplified version of the
wavefront area approximation by and :
A^=2WL+πrrup(L+2W)+2πrrup2.
The extent of the rupture is assumed to be rectangular with length L and
width W. Equation () describes a cuboid with rounded corners
and with only half of its surface considered because no energy flux is
assumed to be transmitted above ground.
While the geometrical spreading correction is expressed analytically as the
wavefront area A^, we estimate the attenuation parameter k.
Attenuation changes with distance, as a power law at short distances
<150 km; and longer distances are not considered. An
empirical attenuation relationship is
lnY=C+krrup,
where Y is
Y=A^ρvSSamp2∫T1T2IV2,
i.e. the logarithm of the energy estimate without the attenuation term e-krrup from Eq. (). The dummy variable C is only used
for estimating k and not in the final correction for attenuation. A
distance-independent form of the Arias intensity, i.e. corrected for
geometrical spreading and attenuation, is defined by
IA,A=A^Samp2e-krrupIA,
where k is determined by Eq. () and setting Y=IAA^. The corrected Arias intensity IA,A is the
acceleration-based counterpart to E^.
(a) Far-field spectrum after
. The spectrum can be read as displacement (red), velocity
(black), and acceleration (blue). (b) The squared Brune spectrum
corresponds to the frequency sensitivity of velocity-based IV2 (blue) and the
acceleration-based IA (black).
Low-frequency effects, like directivity, are better captured with a velocity-based metric (e.g. azimuth-dependent energy estimate) than an acceleration-based metric (Arias intensity) alone.
In terms of the Fourier transform, the sensitivity of acceleration at higher
frequencies becomes apparent, as the Fourier transform of the time derivative
of any function is
F(f˙(t))=iωF(f(t)),
and thus scales with frequency in the spectrum. The frequency sensitivity of
IV2 and IA is related to the squared spectrum given the
metrics. For example, in Fig. we show the different spectral
sensitivities of IV2 and IA for a theoretical seismic source
spectrum . IV2, and thus E^, has a higher sensitivity
to lower frequencies than IA. The low-frequency part of the
spectrum can be accounted for by considering IV2 in a ground-motion model.
Landslide-related ground-motion models
The basic form of landslide-related ground-motion models for Arias intensity
is based on earthquake magnitude M and distance from the earthquake rupture
re.g.:
lnIA=p1+p2M+p3lnr.
This form is widely used . In engineering
seismology, ground-motion models usually have an additional distance term for
anelastic attenuation:
lnIA=c1+c2M+c3r+(c4+c5M)ln.
This is a modified version of the model template by . While
Eqs. () and () share some parameters (p1,
c1 and p2, c2), the geometric spreading term includes not only
distance dependence (p3, c4) but also has a magnitude-dependent
component (c5). In addition, anelastic attenuation is included as well
(related to c3) in Eq. (). The template of
relates to the majority of GMMs in engineering seismology.
Models of this kind address strong motion in the context of landsliding
. The incorporation of anelastic attenuation
is less common in landsliding GMMs and not mentioned in these studies but is
included in more recent studies .
We exchange the magnitude term from Eq. () with a
site-dependent energy term, assuming that landsliding is more related to the
energy of incoming seismic waves than to the moment at the source. We replace
moment magnitude by the logarithm of energy (Eq. ), since energy
is proportional to the seismic moment M0 (Eq. ). Based on
the site-dependent energy estimate E^, we propose the model
lnIA=c1+c2lnE^+c3r+(c4+c5lnE^)lnr.
The five coefficients are inferred by non-linear least squares
e.g.. We use the rupture-plane distance
(rrup), i.e. the shortest distance between a site and the rupture
plane.
Rupture directivity model
In the NGA-west2 guidelines , the directivity effect is
modelled by isochrone theory or the azimuth
between epicentre and site . We use the latter approach
and model directivity for estimated energy and corrected Arias intensity in a
simplified way:
20lnE^θ=lnE^0+aEcos(θ-θE),21lnIA,A,θ=lnIA,A,0+aIcos(θ-θI),
where E^0 and IA,A,0 are the offset (average), aE and
aI the amplitude of variation with azimuth, and θE and θI
are the azimuths of the maximum. The definition of θ is similar to
that of for the angle measured between the epicentre
and the recording site, with the difference of being measured clockwise from
the north. The azimuths of the maximum, θE and θI, are free
parameters because (1) the rupture is assumed to have occurred on two faults
and has thus variable strike and (2) the event is not pure strike-slip and
has a normal faulting component. We therefore do not expect a match between
the rupture strike and θE and θI. The geometrical spreading
is already incorporated in the energy estimate as a distance term
.
Model for fault-normal-to-fault-parallel ratio
The ratio of the response spectra of the horizontal sensor components is a
function of oscillatory frequency fosc.
Distribution of hillslope inclination and MAF.
The left column shows (a) hillslope inclinations and
(c) MAF within the landslide-affected area (green) and within the
landslide areas (black). The right column presents (b) hillslope
inclinations and (d) MAF in different segments of the landslide
areas which is expressed as relative height. Segments towards to the toe
(relative height 0.0–0.5) are in green, and segments towards the crown are in red (relative
height 0.5–1). The solid line is the mean, and the dashed lines enclose the
95 % uncertainty range. The concept of relative height is illustrated for
the Aso Ohashi landslide in (e). MAF<1 indicates
attenuation and MAF>1 amplification of seismic waves due to
topography. The cyan line in (d) highlights MAF=1, i.e.
no amplification or attenuation.
(a) Landslide concentration with
(a) rupture distance rrup and (b) asperity
distance rasp of the Kumamoto earthquake landslides. The rate
parameter of the exponentially decaying landslide concentration is
estimated by maximum likelihood. The distances to the four peaks shown in
Fig. are given. Densities change little with distance
metric, as highlighted by the similar kernel density estimates and the
near-identical rate parameter estimates λ^. The landslide
concentration for Mt Aso depends more on the distance metric than for the
other three locations. The more distant mountains have very similar
concentrations despite differences in distances (in particular Mt Otake).
However, when compared to Fig. , Mt Shutendoji has a higher
landslide concentration than Mt Kinpo and Mt Otake, despite being the
farthest away.
Kernel density estimates of azimuth and distance
of (a) landslide concentration of the coseismic landslides,
(b) concentration of landslide-susceptible terrain with hillslope inclinations >19∘, and
(c) landslide concentration of unspecified landslides. Azimuth and
distance are with respect to the asperity centroid. The marginal densities
with respect to azimuth are shown in blue as outer ring. The densities are
normalized to their maxima.
Spatial distribution of landslides.
(a) Coseismic landslides. The total landslide area at a location is
shown as a colour-coded smooth function in the background; (b) same
as in (a) but for unspecified landslides within the landslide-affected area of the Kumamoto earthquake.
The north and east components (E, N) of the sensor are rotated to be
fault-normal (FN) and fault-parallel (FP) with fault strike ϕ.
22FN=Ecosϕ-Nsinϕ,23FP=Esinϕ+Ncosϕ,24FN/FP=SAFN(fosc)SAFP(fosc).
The response spectra are calculated from accelerograms after
, with a damping of ζ=0.05.
The amplitudes of waves parallel to rupture propagation differ from waves
normal to propagation on top of the directivity effect. This variation
depends on the azimuth and is larger only at high periods. The fault-normal
response amplitude is larger than the fault-parallel response if directed
parallel or antiparallel to the rupture. We model the ratio similar to
:
ln(FN/FP)=(b1+b2foscb3cos(2(θ-θR))H(b1+b2foscb3),
where parameters bi describe the relationship of the oscillatory frequency
to the ratio, θ is the azimuth (Eq. ), and
θR is the azimuth of the maximal ratio. The ratio azimuth is as subject
to assumptions as is its counterpart θE. The Heaviside function
H(⋅) avoids negative values in the model, which would be equivalent to
an undesired phase shift in the cosine term.
We introduced a functional form for oscillatory frequency dependence with
four parameters in Eq. (). We did not introduce a distance term
and apply the model only to data with rrup≤ 50 km.
ResultsTopographic analysis
Landslides occurred mostly in tephra layers (Fig. a, b) covered
by forests (Fig. d, e) and predominantly along the NE rupture
segment. Nearly all landslides concentrated on hillslopes with a steepness
between 15 and 45∘ and an MAF ≈1 (Fig. a,
c). Hillslope inclination and the MAF were higher towards the landslide crown
(Fig. b, d), indicating a progressive landslide failure
starting from the crown, consistent with numerical simulations by
. The TWI is linked to land cover and is highest in areas
with rice paddies (Fig. i). Terrain with landslides has a
uniformly low TWI, thus we cannot evaluate the hydrological impact on the
earthquake-related landslides e.g..
Most landslides originated at locations with amplified ground accelerations
and steep hillslopes and ran out on flatter areas with less amplified ground
acceleration. Landslides – interpreted as shear failure – start as mode II
(in-plane shear) failure at the scarp and mode III (anti-plane shear) failure
at the flanks . At later stages of
the landslide rupture, mode I (widening) failure can also occur in the
process . Simulations of elliptic landslides by
show that either the most compressive or the most tensile
stresses are parallel to the major axis of the landslide, coinciding with the
average landslide aspect. show for
several Japanese landslides that peak forces were aligned parallel to the
long side of the landslides; shows from waveform
inversion for the Mt Meager landslide that force and acceleration were
parallel to the long side of the landslide source area.
Characteristic waveforms observed in the
vicinity of the rupture exhibiting rupture directivity effects. Station 93002
is in the forward-directivity region with a large amplitude pulse on the
fault-normal component. Station 93039 is also in the forward-directivity
region but with an offset to the rupture. In this region the fault-parallel
component has higher amplitudes. The station KMM012 is in the
backward-directivity region, and waveforms have longer duration without large
amplitudes. The waveforms are
low-pass filtered at 1.2 Hz.
Energy estimates (E^) over azimuth;
(b) same as in (a) but for Arias intensity with correction
for geometrical spreading (IA,A).
Kernel density estimate of the amplitude ratio of
response spectra of fault-normal and fault-parallel components
(FN/FP) with respect to oscillatory frequency. Beyond
2–3 Hz FN/FP variations cease, as highlighted by our
model and the model by .
Kernel density estimate of
FN/FP with azimuth obtained from response spectra for
three different oscillatory frequency ranges: (a) 0.1–1 Hz,
(b) 1.0–2.5 Hz, and (c)>2.5 Hz. For each plot, our
FN/FP model and the model by are
shown for (a) 0.55 Hz, (b) 1.75 Hz, and (c) 4 Hz.
As in Fig. , amplitudes decrease with increasing oscillatory
frequency.
(a) Aspect and hillslope inclination
distribution within areas of the earthquake-triggered landslides. This
distribution is normalized by the distribution of the aspect of all
hillslopes in the landslide-affected area. The black line denotes the strike
of the Kumamoto earthquake (225∘). (b) Distribution of aspect
and hillslope inclination in the landslide-affected area; (c) same
as in (a) but for unspecified landslides.
Mt Aso and its caldera and Mt Shutendoji had a high density of landslides
(Fig. ), whereas Mt Kinpo and Mt Otake had none, despite
being closer to the epicentre and being comparably close to the rupture
(Fig. ). All these locations have comparable rock type, land
cover, hillslope inclination, and MAFs. Hence, lithology, land
cover, and topographic characteristics are insufficient in explaining the
landslide distribution and concentration with respect to the hypocentre or
the asperity.
The azimuthal density – with respect to the asperity centroid – of the
unspecified landslides follows to some extent the distribution of hillslope
inclinations >19∘ in the landslide-affected area
(Fig. b, c). This similarity shows that the abundance of
unspecified landslides mimics the steepness of topography in the region.
Densities are higher towards Mt Kinpo (W), Mt Otake (WSW), Mt Shutendoji (N),
Mt Aso (E), and the Kyushu mountains (SE). The coseismic landslide
distribution differs completely from the distributions of unspecified
landslides and their surrounding topography (Fig. ),
respectively, as nearly all landslides happened to the northeast of the
epicentre, close to the rupture plane (Fig. ).
identified only 29 landslide
reactivations during the Kumamoto
earthquake. The contrast between the distributions of unspecified landslides
and earthquake-related landslides indicates a contribution by the rupture
process.
Impact of finite source on ground motion and landslides
The results of the seismic analysis are given for waveforms, the basis for
E^ and IA, and response spectra, used for
FN/FP. To the northeast, signals with forward directivity
are shorter in duration, with one or a few strong pulses
(Fig. , top right). Waveforms with backward directivity to
the southwest of the rupture are longer, with no dominant pulse
(Fig. , bottom left). Waveforms parallel to the rupture
have an intermediate duration. Waveforms in either a forward or
backward direction have stronger amplitudes in the fault-normal direction,
whereas waveforms outside the directivity-affected regions have stronger
amplitudes in the fault-parallel direction (Fig. , top
left).
We estimated energies E^ from the three-component waveforms. For the
Arias intensity, both horizontal components are used. The geometrical
spreading A is calculated according to Eq. (), with a rupture
length of L=53.5 km and width of W=24.0 km. Any remaining distance
dependence has been corrected for by estimating and applying the attenuation
parameter k (Eq. )
After the determination of k, E^ and IA,A are considered
distance-independent and can be investigated for azimuthal variations. With a
reference point for the azimuth at the epicentre, E^ shows oscillating
variations in amplitude with azimuth (Fig. a), while
IA,A exhibits a similar amplitude variations over the entire
azimuthal range (Fig. b). The running average based on a von Mises
kernel (κvM=50) of E^ and IA,A shows
increased E^ between 45 and 135∘, i.e. approximately parallel
to the strike. Minimal values of E^ occur in the opposite direction
(200–300∘). The running average of IA,A has several
fluctuations, but these are not as wide and large as those of E^. The azimuthal
variation in E^ indicates the rupture directivity, and the absence of
large variations in IA,A indicates that the directivity effect is
only evident at lower frequencies (compare with Fig. ).
The azimuthal variation in E^ and IA,A is modelled
according to Eq. (). We estimate parameters for two
scenarios:
directivity is assumed, resulting in azimuthal variations where aE
and aI are free parameters,
directivity is not assumed, resulting in no azimuthal variations with aE=aI=0.
The two models are compared with the Bayesian information criterion
BIC; for a least-squares fit:
BIC=nlnN+Nlnσ^2,
where n is the number of estimated parameters (n=4 for the first case and n=2
for the second case), N is the number of data, and σ^2 is the
variance of the model residuals. The model with the smaller BIC is
preferable. The starting values of the parameters are the mean of E^
and IA,A, no azimuthal variation (ad=0), and the azimuths of
the maximum of E^θ and IA,A,θ are set to the
strike of the fault (θE=θI=225∘).
The directivity model for E^ follows the trend of the data and the
running average closer than the model without directivity
(Fig. a). According to BIC, the model with directivity is
preferable (BICdirectivity=-110, BICnodirectivity=-11). In the case of the Arias intensity, the difference in BIC between the two
models is less compared to the azimuth-dependent energy (Fig. b).
Here, the model without directivity is the preferred one
(BICdirectivity=30, BICnodirectivity=22). In
consequence, azimuthal variations in wave amplitudes and energy related to
the directivity effect occur at lower frequencies.
The forward-directivity waves contain a very strong low-frequency pulse
(Fig. ). The pulse amplitude depends on the ratio of
rupture and shear wave velocity and the length of the rupture
. The forward-directivity pulse is superimposed by high-frequency signals in acceleration traces but becomes more prominent in
velocity traces due to its low-frequency nature, i.e.
below 1.6 Hz .
The low-frequency azimuthal variations are also reflected in the spectral
response of the waveforms. Spectral accelerations of stations with
rrup≤50 km were computed from 0.1 to 5 Hz at intervals of
0.01 Hz for the fault-normal and fault-parallel component. The distribution
of FN/FP shows decreasing azimuthal variability with
increasing oscillatory frequency (Fig. ). FN/FP is most variable with
azimuth at low oscillatory frequencies (0.1–1 Hz; Fig. a);
variations are much smaller between 1 and 2.5 Hz (Fig. b)
and nearly absent above 2.5 Hz (Fig. c). This decrease with
frequency is captured by the FN/FP model
(Eq. ; Fig. ). Since our model is an average over
the covered distance, with an average rupture distance of 25.06 km, we
compare it to the FN/FP model of
at 25 km (Figs. , ). Both models show a
similar decay with frequency, with our model predicting a slightly higher
FN/FP. Therefore, the wave polarity ratio related to
rupture directivity is pronounced at lower frequencies and dissipates with
increasing frequency, similar to the azimuthal variations observable in
energy estimates (lower frequencies) but not in Arias intensity (higher
frequencies).
Orientation of horizontal peak ground
acceleration for the simulated waveforms. The arrow length scales with
magnitude of acceleration. The simulated rupture plane is oriented as the
rupture plane of the Kumamoto earthquake (strike: 225∘, dip:
70∘) and of elliptic shape (grey). The upper side is denoted by the
green line, and the lower half is denoted by black. The rupture process originated at the
hypocentre (red dot) with circular propagation outwards (white arrow).
Distribution of landslides with aspect and rupture distance.
The rupture distance is measured from the model by .
This model does not completely reach the surface, truncating distances below 1 km.
The distribution has been normalized by the distribution of aspect of the affected area.
Ground-motion model for IA. The solid
lines are the model with energy estimates for three different energy levels
as in Fig. a. The inset figure shows, for comparison, the ground-motion model of (green) and landslide concentration density
(red).
The pattern of low-frequency ground motion is well reflected in that of the
landslides. The azimuthal variation in E^ coincides with that of
landslide concentration (Fig. ). Both azimuth-dependent energy and
landslide concentration have a similar trend, with the maximum being parallel to
rupture direction and the minimum strike being antiparallel. The orientation of
maximum FN/FP is also reflected in the landslide aspect.
The northwest and east directions show higher landslide density
(Fig. a). The highest density of landslides has a northwestern
aspect in agreement with maximum FN/FP, both
perpendicular to the strike. The eastward increased density is mostly due to
landslides very close to the rupture. A look at different distances reveals
that the increased density of landslides facing east by southeast is at very
short distances (rrup≤2.5 km; Fig. ), while the
northwest-facing landslides are further away
(2.5km<rrup≤6 km). Only minor landslides are
farther away, with no specific pattern.
The distribution of aspect and hillslope inclination in the landslide-affected area varies little with aspect (Fig. b). The distinct
northwest and east orientation of landslides is not an artefact of the
orientation of the topography in the landslide-affected area
(Fig. a, b). The unspecified landslides in the affected area
have a near-northward aspect and deviate by ≈30∘ from the
earthquake-triggered landslides (Fig. c). This highlights that
the earthquake affects landslide locations (Fig. ) and will
force failure on specific slopes facing in the direction of ground motion
(Figs. , ).
Ground-motion model for Kumamoto
Parameters for ground-motion models.
Model using MwModel using E^c14.0831.453×10-1c21.991×10-1-2.682×10-1c3-2.899×10-5-3.059×10-5c4-4.343×10-1-3.287c58.962×10-38.114×10-2
We derived two ground-motion models for Arias intensity from data with
rrup≤150 km (Table ; Fig. ). One
model incorporates the azimuth-dependent seismic energy (Eq. ).
The other is a conventional isotropic moment magnitude-dependent model
(Eq. ). The decay of Arias intensity with distance for both
models fits the running average well and is proportional to the decrease in
landslide density with distance. Variation in estimated energy is well
covered by the model and spans more than 2 orders of magnitude, resulting in
a variation in Arias intensity of nearly 1 order of magnitude.
The magnitude-based model is nearly equivalent to the energy-based model with
E^=1.2×1015 J. This value is close to the average
energy estimate found from energy estimates of the directivity model from
Eq. () (E^=1.3×1015 J). The
closeness of the two values implies that the magnitude-based model can be
seen as an average over the azimuth of the energy-based model.
Discussion
We provide a framework for characterizing coseismic landslides with an
integrated approach of geomorphology and seismology, emphasizing here the role
of low-frequency seismic directivity and a finite source. Given the
observations of ground motion of the Kumamoto earthquake, two questions
arise:
How specific is the observed ground motion, i.e. is the Kumamoto rupture
particularly distinct?
As a rupture very close to the surface, how much
does seismic near-field motion contribute? The second question arises
because many landslides occurred very close to the rupture plane.
However, it is not possible to separate the observed waveforms into near-,
intermediate-, and far-field terms. To investigate both questions, we
computed theoretical waveforms after , , and
from a circular rupture on an elliptic finite source with constant rupture
velocity in a homogeneous, isotropic, and unbound medium (see Appendix).
Despite the simplified assumptions behind this waveform model, low-frequency
ground motion captures the most prominent features of the observed waveforms.
Simulated waveforms close to the rupture plane change in polarity orientation
towards east–west, while a strong fault-normal polarity appears at larger
distances. A decomposition into a near-field term and combined intermediate-
and far-field term reveals that the former highly contributes to the
ground motion at close distances. The impact of the near-field term may
explain the dominance of east-facing landslides close to the rupture
(Fig. ).
The simulations also demonstrate the effect of directivity on estimates of
radiated energy and Arias intensity. The azimuthal variations in simulated
E^ are similar to the observed variations. The Arias intensity of the
simulations also has azimuthal variations with the same characteristics as
the energy estimate. These variations in Arias intensity are absent in the
observed data, indicating that Arias intensity is more influenced by local
heterogeneities and scattering than the energy estimates, as these are ignored
in the simulations.
The results show that the Arias intensity is not as susceptible to the
directivity effect and variations in fault-normal to fault-parallel
amplitudes as the radiated energy; because of its higher sensitivity
towards higher frequencies, these effects are masked by high-frequency
effects such as wave scattering and a heterogeneous medium. We found that the
radiation pattern related to the directivity effect is recoverable from
energy estimates but not from Arias intensity. This low-frequency dependence
is also seen in the response spectra ratios for FN/FP
where directivity-related amplitude variations with azimuth have been
identified only for frequencies <2 Hz, in agreement with previous work
. We introduced a modified model for Arias
intensity using site-dependent seismic energy estimates instead of the
source-dependent seismic magnitude to better capture the effects of
low-frequency ground motion.
The conventional magnitude-based isotropic model and the azimuth-dependent
seismic energy model correlate with the landslide concentration over distance
(Fig. ). As in it is therefore feasible to
use the ground-motion model to model the landslide concentration,
Pls(IA), by a linear relationship:
lnPls(IA)=aI+bIlnIA.
Azimuthal variations in landslide density correspond to azimuthal variations
in seismic energy and can be described by a similar relationship:
lnPls(E)=aE+bEcos(θ-θE).
For the Kumamoto earthquake data, we estimate aI=2.1, bI=2.6, and
aE=-31.5, bE=2.3. The azimuth-dependent landslide concentration implies
similar landslide concentrations at different distances from the rupture,
thus partly explaining some of the deviation in Figs. and
.
Compared to the model of (Fig. ) our model
uses rupture-plane distance, as opposed to the Joyner–Boore distance
(rJB). When using the hypocentral depth as pseudo-depth, the model
of overpredicts IA both at shorter and longer
distances – irrespective of the pseudo-depth at larger distances. This
misestimate is most likely due to the lack of an additional distant-dependent
attenuation term in their model (Eq. ).
The use of the MAF instead of curvature alone provides a proxy by how much a
seismic wave is amplified (or attenuated) for a given wavelength and
location. We show that both hillslope inclination and the MAF tend to be lower
towards the landslide toe (Fig. ). This effect is linked to
the convention that landslide polygons cover both the zone of depletion and
accumulation. consider the tephra layers rich in halloysite
to be the main sliding surfaces indicating shallow landslides
. When relating coseismic landsliding to the seismic rupture,
only the failure plane of the landslide matters because this is the
hillslope portion that failed under seismic acceleration.
noted, for example, that landslide susceptibility and safety factor
calculation depend on whether the entire landslide or only parts – scarp
area or area of dislocated mass – are considered. The reconstruction of the
landslide failure planes is limited to statistical assessments of landslide
inventories . However, failure may have likely
originated close to the crown and then progressively propagated downward the
hillslope because MAF>1 indicates an amplification of ground
motion towards the crown of the landslides.
Coseismic landslide locations have a uniformly low topographic wetness index,
indicating that hydrology may have added little variability to the pattern of
the earthquake-triggered landslides; at least we could not trace any clear
impact of soil moisture on the coseismic landslide pattern .
Conclusions
We investigated seismic waveforms and resulting landslide distribution of the
2016 Kumamoto earthquake, Japan. We demonstrate that ground motion at higher
frequencies controls the isotropic (azimuth-independent) distance dependence
of Arias intensity with landslide concentration. In addition, ground motion
at lower frequencies influences landslide location and hillslope failure
orientation, due to directivity and increased amplitudes normal to the fault,
respectively. Topographic controls (hillslope inclination and the MAF) are
limited predictors of coseismic landslide occurrence because areas with
similar topographic and geological properties at similar distances from the
rupture had widely differing landslide activity
. Nonetheless landslides concentrated only to
the northeast of the earthquake rupture, while unspecified landslides have
been identified throughout the affected region.
We introduced a modified model for Arias intensity using site-dependent
radiated seismic energy estimates instead of the source-dependent seismic
magnitude to better model low-frequency ground motion in addition to the
ground motion at higher frequencies covered by the Arias intensity.
Compared to previous models widely used in landslide-related ground-motion
characterization our model is based on state-of-the-art ground-motion models
used in engineering seismology, which have two different distance terms, one
for geometrical spreading and one for along-path attenuation. The latter is
rare in landslide studies e.g.. Our results
emphasize that the attenuation term should be considered in ground-motion
models, as the landslide concentration with distance mirrors such
ground-motion models.
The effect of the earthquake rupture on the rupture process of the landslides
results in landslide movements parallel to the strongest ground motion. Due to
the surface proximity of the earthquake rupture plane, near-field ground
motion influences the aspect of close landslides to be east–southeast. The
intermediate- and far-field motion of the earthquake promoted more landslides
on northwestern exposed hillslopes, an effect that overrides those of steepness
and orientation of hillslopes in the region.
We highlight that coseismic landslide hazard estimation requires an
integrated approach of both detailed ground-motion and topographic
characterization. While the latter is well established for landslide hazard,
ground-motion characterization has been only incorporated by simple means,
i.e. without any azimuth-dependent finite rupture effects. Our results for
the Kumamoto earthquake demonstrate that seismic waveforms can be reproduced
by established methods from seismology. We suggest that these methods can
improve landslide hazard assessment by including models for finite rupture
effects.
Data availability
KiK-Net and K-Net data are accessible at
http://www.kyoshin.bosai.go.jp/. The JMA special release seismic
waveform data are accessible at
http://www.data.jma.go.jp/svd/eqev/data/kyoshin/jishin/index.html.
Coseismic landslide data are available at
http://www.bosai.go.jp/mizu/dosha.html. Unspecified landslides are
available at
http://dil-opac.bosai.go.jp/publication/nied_tech_note/landslidemap/gis.html.
The VS30 site amplification data are available at
http://www.j-shis.bosai.go.jp/map/JSHIS2/download.html?lang=en. Seismic
velocities and density after Koketsu et al. (2012) are available as part of
the JIVSM data set at
http://www.eri.u-tokyo.ac.jp/people/hiroe/link.html. The ALOS 30 m DSM
is available at https://www.eorc.jaxa.jp/ALOS/en/aw3d30/index.htm. The
ALOS land use data are available at
https://www.eorc.jaxa.jp/ALOS/en/lulc/lulc_index.htm. The seamless
geological map of Japan is available at
https://gbank.gsj.jp/seamless/download/downloadIndex_e.html. All data
are free of charge, and data sources were last accessed on 19 March 2019.
Synthetic waveforms from displacement of a finite rupture
We illustrate the generation of ground displacement as a discontinuity across
a rupture fault e.g..
The displacement for any point x at time t is given by
ui(x,t)=∬Σcjkpq∂Gip(Dj(ξ,t))∂xqnkdΣ,
where c is the fourth-order elasticity tensor from Hooke's law, G is
Green's function describing the response of the medium,
D(ξ,t) is the displacement on the fault with area Σ and
coordinates ξ, and n is the fault-normal vector.
Summation over i, j, p, and q is implied.
While the surface integral is carried out numerically, the derivatives of
Green's function for an isotropic, homogeneous, and unbound medium can be
solved analytically:
A2∂∂xqGipDjξ,t=15γiγpγq-3δipγq+δiqγp+δpqγi4πρr4A2a⋅∫rαrβDjξt-ττdτA2b+6γiγpγq-δipγq+δiqγp+δpqγi4πρα2r2Djξ,t-rαA2c-6γiγpγq-2δipγq+δiqγp+δpqγi4πρβ2r2Djξ,t-rβA2d+γiγpγq4πρα3rD˙jξ,t-rαA2e-γiγpγq-δipγq4πρβ3rD˙jξ,t-rβ,
where
r=x-ξandγi=xi-ξir,
and δij is Kronecker's delta. The terms in Eq. ()
are commonly separated into groups with respect to their distance r. In
Eq. () is the near-field (NF) term; as its amplitude decays
with r-4, it affects the immediate vicinity of a rupture only. Terms
with a distance attenuation proportional to r-2 are called
intermediate-field (IF) terms for P waves (Eq. ) and S waves
(Eq. ). The remaining two terms are the far-field (FF) terms
for P waves (Eq. ) and S waves (Eq. ) with a
decay proportional to r-1. A major difference between the NF and IF
terms and the FF terms is that the former depend on the slip on the rupture,
and they are the cause for static and dynamic displacement, whereas the
latter are functions of the time derivative of slip and result in dynamic
displacement only.
Set-up of the rupture model. Grey ellipse
represents the rupture: light grey area is unruptured, medium grey area is
slipping, and the dark grey area is after slip arrest.
The slip function in time is related to the Yoffe function
of and , with rise time T. We use the slip distribution
of to describe the amplitude distribution of the slip on
the rupture as well as the elliptical fault shape and rupture propagation
from . The slip amplitude is given by
Dξ=D01-ξ1-pϵL2L22-ξ2W22,
where D0 is the maximum displacement at the centre of the fault, L and
W are the length and width of the fault with eccentricity
ϵ=1-WL2, and p determines whether
the rupture starts at the focus at the front of the rupture plane
(strike-parallel, p=1) or at the focus at the end (strike-antiparallel,
p=-1). The rupture originates in one of the two foci and propagates
radially away from the source with constant velocity ζ and terminates
when it reaches the rupture boundary. The slip vector s^
describes the orientation of the displacement D(ξ) on the
fault plane. We follow the definition of n^ and s^ in terms of fault strike ϕs, dip δ, and rake λ from
:
n^=-sinδsinϕssinδcosϕs-cosδ,s^=cosλcosϕs+cosδsinλsinϕscosλsinϕs-cosδsinλcosϕs-sinλsinδ.
The displacement vector D in Eq. () is given by
D=D(ξ)s^.
We consider an isotropic medium, and the elasticity tensor
c from Eq. () is
cjkpq=δjkδpqλM+(δjpδkq+δjqδkp)μM,
where λM and μM are the Lamé constants of the isotropic
medium:
λM=ρ(vP2+2μM),μM=ρvS2.
We set λM=μM, resulting in the widely observed relation
vP=vS3.
With the assumptions outlined above it is possible to calculate the
displacement of an earthquake at location x with 12 parameters
(Fig. ):
fault size and orientation, including length L, width W, strike ϕs, and dip
δ;
material, including first and second Lamé constants λM and
μM and
density ρ (alternatively: compressional and shear wave velocities
vP and vS and density ρ);
rupture and slip, including rupture velocity ζ, slip D0, rise time
T, rake λ and rupture orientation with respect to strike ϕs
and rupture orientation parameter p.
The fault size and displacement of earthquakes are correlated with each other
and are scaled to the magnitude. The number of parameters reduces to 10
(9 if the Lamé constants are equal) when scaling relations
e.g. are used in combination with the
seismic moment M0. The moment can be expressed as
M0=μMAD‾,
with shear modulus (second Lamé constant) μM, the rupture area – here
an ellipse – A=π4LW, and average displacement D‾, which
follows from Eq. () as D‾=23D0.
The results are not strictly comparable to observed data due to the model
simplicity. The computed amplitudes will be smaller than observed values
because no free surface is assumed. Assuming a free surface would nearly
double the amplitudes from wave reflection as well as the amplifications
from wave transmissions (from high- to low-velocity zones). Only direct waves
are computed, and effects of reflections of different layers are not covered
due to the isotropy and homogeneity. Corresponding waveforms – in particular
surface waves – are not exhibited. However, the purpose of this model is to
show (1) the general behaviour of waveforms in the vicinity of a rupture,
which is dominated by direct waves, and (2) how amplitudes distribute
relatively in space.
Radiated seismic energy estimation
Ratio between the approximate and exact energy
estimates for different P wave velocities in the medium. The exact estimate
assumes that P and S waves arrive at different velocities at the recording
site, while the approximate estimate assumes that all waves arrive with shear
wave velocity at the site. This approximation introduces only a minor
underestimation, since most radiated energy is released as S waves. The
distance variation arises from the different distance and velocity
dependencies of the intermediate-field terms and the far-field terms.
The exact calculation of radiated seismic energy is challenging. One
simplifying assumption is that all waves arrive at the site with shear wave
speed, which is a very good approximation for the far-field term. The
reasoning can be justified from a theoretical perspective: for most earth
media the ratio between the P wave velocity α and S wave velocity
β is
αβ=3.
From this and Eqs. () and (), it follows that the
amplitude of compressional waves is ≈133 of the
shear wave amplitude. If we say that the P wave train has a similar duration
as the S wave train, then the energy contribution of the P waves with respect
to the S waves becomes (133)2=127. The total
energy of a signal is Etotal=EP+ES,
and can be estimated by
E^total=αSaIV2α+βSaIV2β,
with the integrated squared velocity (IV2) for P and S waves from Eq. (),
the P and S wave velocities αP and αS at the
recording site, and a constant a covering the remaining factors which are
identical for both terms (compare with Eq. ). If we express the
energy contribution of P waves in terms of S waves, we can summarize the
above relation to
B4E^total=aIV2βαP27+βSB5=aIV2ββS327+βSB6≈aIV2ββS27+βS.
The last expression is differs only by 2.6 % from the exact term. While
slightly underestimating the energy, this approximate definition of using
αS instead of βS does not require the identification of P and
S waves. This is useful, since at short distances the S wave train is
usually inseparable from the P wave train.
At shorter distances, the intermediate-field term needs also to be taken into
consideration. The amplitude of the intermediate term decays with r2
(Eqs. , ), while the far-field amplitude decays with
r (Eqs. , ). That is, the amplitude scales by
distance and velocities and thus the IV2 are
B7IV2α=α-4r-2r-1+α-12,B8aIV2β=β-4r-2r-1+β-12.
Again by replacing all P wave terms by S wave terms, the total energy becomes
B9E^total=αSaIV2α+βSaIV2β=ar-2αSα-4r-1+α-12B10+βSβ-4r-1+β-12,=ar-23-3βSβ-4r-1+3-1β-12B11+βSβ-4r-1+β-12.
With the assumption that αS=βS, Eq. () becomes
E^totalappr≈ar-2βSα-4r-1+α-12B12+βSβ-4r-1+β-12,=ar-23-4βSβ-4r-1+3-1β-12B13+βSβ-4r-1+β-12.
The ratio between the approximation and the exact solution is
E^totalapprE^total=3-4r-1+3-1β-12+r-1+β-123-3(r-1+3-1β-1)2+r-1+β-12.
The two limits with respect to distance are
B15limr→0E^totalapprE^total=3-4+13-3+1B16≈0.932,B17limr→∞E^totalapprE^total=3-5+13-6+1B18≈0.974.
The second limit is identical to the far-field case derived above. The two
limits show that even in the range of the intermediate-field term, the energy
estimate deviates little when assuming that all waves arrive with βS
at the recording site. A comparison of the approximate energy estimate and
the exact estimate as a function of distance and velocity is shown in
Fig. .
Author contributions
SvS and UO devised the main conceptual ideas and performed the numerical calculations.
All authors contributed to the design and implementation of the research and
to the writing of the paper.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
We highly appreciate the help of Tomotaka Iwata and Kimiyuki Asano for
providing links to additional seismic data from the municipal and NIED
networks and for several helpful discussions on the specifics of the data. We are
sincerely grateful to Takashi Oguchi, Yuichi Hayakawa, Hitoshi Saito, and
Yasutaka Haneda for the field trip to the Aso region and fruitful
discussions. We also thank Hendy Setiawan, Tao Wang, and Qiang Xu for
reviewing and helping to improve the paper. Thanks to Arno Zang, John
Anderson, and Odin Marc for various discussions and comments.
Sebastian von Specht, Ugur Ozturk, and Georg Veh acknowledge support from the
DFG research training group “Natural Hazards and Risks in a Changing World”
(grant no. GRK 2043/1). Ugur Ozturk is also supported by the Federal Ministry
of Education and Research of Germany (BMBF) within the project CLIENT II –
CaTeNA (FKZ 03G0878A). The article processing
charges for this open-access publication were covered by a
Research Centre of the Helmholtz Association.
Review statement
This paper was edited by Ulrike Werban and reviewed by Hendy
Setiawan, Tao Wang, and Qiang Xu.
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