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 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: 1MAF(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 : 2TWI=log⁡Actan⁡β, where Ac is the upslope catchment area and β is the hillslope inclination derived from the DSM with filled sinks .

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.

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.

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): 3ds=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 4ay=gtan⁡ϕftan⁡δsin⁡δ=g(FS‾-1)sin⁡δ, 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 : 5dmax=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.

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: 6IA=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 : 7IV2=∫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 : 8jE=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: 9ES=∬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 : 10ES=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 11E^=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.

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 : 12A^=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 13ln⁡Y=C+krrup, where Y is 14Y=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 15IA,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^.

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 16F(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.

The basic form of landslide-related ground-motion models for Arias intensity is based on earthquake magnitude M and distance from the earthquake rupture r e.g.: 17ln⁡IA=p1+p2M+p3ln⁡r. This form is widely used . In engineering seismology, ground-motion models usually have an additional distance term for anelastic attenuation: 18ln⁡IA=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 19ln⁡IA=c1+c2ln⁡E^+c3r+(c4+c5ln⁡E^)ln⁡r. 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.

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: 20ln⁡E^θ=ln⁡E^0+aEcos⁡(θ-θE),21ln⁡IA,A,θ=ln⁡IA,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 .

The ratio of the response spectra of the horizontal sensor components is a function of oscillatory frequency fosc.

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=Ecos⁡ϕ-Nsin⁡ϕ,23FP=Esin⁡ϕ+Ncos⁡ϕ,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 : 25ln⁡(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.

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.

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.

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,

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).

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. , ).

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.