Geophysical Journal International
Geophys. J. Int. (2012) 188, 559–569

doi: 10.1111/j.1365-246X.2011.05272.x

Instrumental evidence of normal mode rock slope vibration
Jan Burjánek,1 Jeffrey R. Moore,2 Freddy X. Yugsi Molina2,3 and Donat Fäh1
1 Swiss

Seismological Service, ETH Zürich, Sonneggstrasse 5, CH-8092 Zürich, Switzerland. E-mail: burjanek@sed.ethz.ch
Institute, ETH Zürich, Sonneggstrasse 5, CH-8092 Zürich, Switzerland
Norwegian Geological Survey, Postboks 6315 Sluppen, NO-7491 Trondheim, Norway

2 Geological
3

Accepted 2011 October 17. Received 2011 October 3; in original form 2011 June 30

SUMMARY
A unique field experiment was performed to constrain the seismic response of a large, potentially unstable rock slope in the southern Swiss Alps. Small-aperture seismic arrays were
deployed to record ambient vibrations both inside and outside of the mapped instability boundary. The recordings were analysed by means of the high-resolution f–k method, site-to-reference
spectral ratios and time–frequency dependent polarization analysis. All three methods indicated that the wavefield within the potentially unstable rock mass is dominated by normal
mode motion (standing waves) rather than horizontal propagation of seismic waves. Both
fundamental frequency and relative amplification could be recovered from ambient noise data.
The observed amplification is strongly directional, and the maximum amplification is oriented
perpendicular to open tension cracks mapped at the ground surface. Our results highlight site
response characteristics relevant for analysis of earthquake-triggered rock slope failures.

1 I N T RO D U C T I O N
Earthquakes are among the main triggers of landslides, and coseismic slope failures contribute significantly to earthquake-induced
damage in areas of large topographic relief (Marano et al. 2010).
Although a number of studies exist on the subject (reviewed, e.g.
by Keefer 2002), most of these are focused on systematic documentation of past events. Quantitative evaluation of a slope’s susceptibility to ground shaking is still typically based on simplified
models, which ignore interactions between incoming waves and the
topography or internal structure of the landslide mass. On the other
hand, detailed numerical modelling of these phenomena requires
a complex geophysical model, which can only be constrained by
extensive field investigations at potential slope instabilities (e.g.
Danneels et al. 2008; Gischig et al. 2011).
The objective of this study is to analyse the seismic response of
a large, potentially unstable rock slope by means of ambient vibrations. The use of ambient vibration measurements has become
increasingly attractive in a broad range of seismological disciplines
at different scales. For example, noise tomography (i.e. noise crosscorrelation) has improved the resolution of regional crustal velocity
models (Bensen et al. 2009). Small-aperture seismic arrays are commonly employed to record ambient noise and characterize the shear
wave velocity profile in shallow sediments (Tokimatsu 1997). The
so-called Nakamura’s method (i.e. H/V method) is often used to
identify the resonant frequency of a soil column from single-station
noise measurements (Nakamura 1989). Finally, eigenfrequencies of
civil structures can be also extracted from records of ambient vibrations (e.g. Michel et al. 2010). Although the applications listed share
 
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one main advantage, relatively simple and efficient data acquisition,
the underlying assumptions are very different. The key distinction
is the content of ambient noise. For example, in noise tomography
studies an incoherent diffusive wavefield is usually assumed (e.g.
Curtis et al. 2006). On the contrary, coherent surface wave propagation is usually considered to dominate noise in shallow sediments
(Bonnefoy-Claudet et al. 2006). Apparently, noise characteristics
can be fundamentally different for different frequency bands. For
example, Roten & Fäh (2007) showed that ambient noise measured
in a deep sedimentary basin consisted of both standing waves of
the entire basin and of surface waves propagating in the shallowest
layers.
Several studies have also been conducted in the past decade regarding analysis of ambient vibration recordings on unstable hill
slopes. Havenith et al. (2002) estimated resonant frequencies of
a rock slope instability using single-station H/V measurements.
Danneels et al. (2008) used small-aperture seismic arrays to measure dispersion of surface waves and subsequently invert for the
shear wave velocity profile in the area of an active earth flow.
Burjanek et al. (2010) estimated amplification factors and identified a range of resonant frequencies for a large unstable rock slope
by noise polarization analysis. Renalier et al. (2010) applied a noise
tomography method to resolve the shear wave velocity structure of
a landslide composed of clayey soil deposits. Finally, Lévy et al.
(2010) monitored the resonant frequency of an unstable rock column by analysis of the noise power spectrum, reporting measurable
changes prior to column collapse. Such a variety of studies reflects
the different origins of ambient vibrations in unstable slopes, and
consequently the variety of related subsurface structures.

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Key words: Wavelet transform; Site effects; Wave propagation.

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Figure 1. Photograph of the Walkerschmatt study site (view to the West from the opposite valley flank), and its location within the Matter valley (relief map at
right). Seismic stations were deployed in the area surrounded by the dashed rectangle (see Fig. 2 for aerial photograph). Vertical scale is roughly represented by
the double-headed arrow. The exposed eastern flank of the valley is recognizable from the relief map. The total elevation difference between mountain peaks
and the adjacent valley bottom locally exceeds 3000 m.

In this paper, we follow and extend the approach introduced
in our recent work at the Randa rock slope instability (Burjanek
et al. 2010; henceforth BU10). BU10 proposed that the wavefield
within an unstable rock mass may be dominated by normal mode
vibration rather than horizontal propagation of seismic waves. To
test this hypothesis, a new field experiment was performed at a
different rock slope instability in the same region. Ambient vibration
recordings were processed by means of time–frequency dependent
polarization analysis (as in BU10), site-to-reference spectral ratios
(SRSR) and the high-resolution f–k array method (Capon 1969). All
three methods supported the hypothesis, and allowed us to retrieve
both the fundamental frequency and relative amplification.

2 S T U DY A R E A
The area of canton Valais is the most seismically active region in
Switzerland and home to some of the greatest topographic relief in
the Alps. On average, a magnitude 6 earthquake occurs every one
hundred years (Fäh et al. 2011). Numerous unstable rock slopes
in the Matter valley area have been identified (Yugsi 2010), and a
number of coseismic slope failures have been observed in previous

earthquakes (Fritsche et al. 2006; Fritsche & Fäh 2009; Fritsche
et al. 2010). Since the region is relatively accessible, it has been
extensively studied and a solid background of relevant geological information is available (e.g. Yugsi 2010). Therefore, it is a
favourable area for studying seismic triggering of landslides and
site effects in unstable, hard-rock slopes.
The study site for this project, named ’Walkerschmatt’, is a small
topographic bench bounded by steep cliffs located near the village
of St Niklaus in the Matter valley (Fig. 1). It lies 6 km north of
the Randa rockslide, which was the subject of our previous study
(BU10). The potential slope instability consists of an estimated
800 000–1 000 000 m3 of crystalline rock, located within an elevation range between 1855 and 2130 m a.s.l. (Yugsi 2010). The stepped
meadow at the top of the instability is transected by a network of
open fractures (often with aperture >1 m) bearing record of past
or ongoing deformation (see later Fig. 6). Bedrock is composed of
competent orthogneiss (the so-called Randa augengneiss), a metamorphosed Permian porphyritic alkaline to subalkaline granitic intrusion constituting the core of the Penninic Siviez-Mischabel nappe
(Bearth 1964), with persistent foliation (shown in black in Fig. 2)
dipping gently into the slope to the S–SW (mean dip direction/dip:
240/21). The local fracture network is composed of two persistent,
 
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 Evidence of normal mode rock slope vibration

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Figure 2. Configuration of the two seismic arrays (red: Array 1; yellow: Array 2; blue: Reference station). Different symbols distinguish between fully
operational stations (triangles), stations without GPS time synchronization (squares), and failed stations (circles). Red, blue and orange lines represent open
fractures belonging to sets J1, J2 and J3, respectively (dashed lines: inferred, solid lines: mapped). Black symbols show foliation orientation (lines parallel to
strike, triangles point to dip direction). Solid black lines are contour lines with 10 m spacing.

steeply dipping joint sets J1 (132/85) and J2 (024/83) and one less
persistent set J3 (067/72). Trace lengths of discontinuities in sets
J1 and J2 are on the order of several tens of metres. Open fractures
within the instability preferentially follow discontinuities belonging
to sets J1 and J2 (red and blue lines in Fig. 2, respectively). Tension
fracture aperture in the unstable rock mass increases towards the
edges of the ridge, from the west (stable area) to the east and northeast. To the southeast, a few isolated unstable blocks can be found.
A fault zone with orientation 036/41 is thought to act as the basal
limit of the instability (not visible in Fig. 2). Precision differential
GPS (GNSS) measurements conducted at the site over 10 months
between 2008 and 2009 were inconclusive (displacements below
the computed accuracy level), which suggests very slow rates of
active deformation (Yugsi 2010).

stable areas of the slope. The geometry of the second array (WAL2)
was more irregular, aimed at covering the remainder of the study
site. A reference station was deployed on a small plateau within
the stable portion of the slope, recording continuously throughout
the day. GPS time synchronization failed at two stations WAL101
and WAL211, so precise time reference for these measurements is
not available. Station WAL208 of the second array was placed on
an individual rock block detached from the main part of the slope
by an open fracture (aperture ∼1 m at the surface). Furthermore,
recordings from stations WAL104, WAL108 and WAL205 were all
unusable due to a sensor cable malfunction.

3 FIELD EXPERIMENT

4.1 Method

12 three-component velocity sensors with an eigenperiod of 5 s
were used together with nine digital seismic systems (six channels
per instrument, GPS time synchronization) to record ambient vibrations on the slope. The field experiment was carried out over 1 d in
the beginning of 2009 August. The total time span of the experiment
was less than 5 hr. Compass-oriented sensors were placed on metal
trihedrons in shallow holes, removing superficial unconsolidated
soil. The error of the sensor orientation was less than ±5◦ . The precise positions of the sensors were measured with differential GPS.
We progressively deployed two arrays with apertures up to 200 m
(Fig. 2). The first array (WAL1) had an almost linear configuration, traversing uniformly the anticipated transition from unstable to

Time–frequency polarization analysis is based on the combination
of complex polarization analysis (Vidale 1986) and the continuous wavelet transform (CWT). We first use a CWT for signal
time–frequency decomposition, and then apply polarization analysis on the complex wavelet amplitude for each time–frequency pair.
Accurately, one should use the term ’scale’ (timescale decomposition) when speaking about a CWT. However, scale can be related
to Fourier frequency (e.g. Farge 1992), and here we use the terms
scale and frequency interchangeably. The particle motion is then
characterized for each time and frequency by an ellipse, which is
generally tilted in 3-D Euclidian space. Especially the orientation
of the major axis and the ellipticity are of primary interest in our

 
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4 T I M E – F R E Q U E N C Y P O L A R I Z AT I O N
A N A LY S I S

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Figure 3. Particle motion is characterized for each time step and wavelet
scale as an ellipse; we analyse the three parameters indicated: strike ( ), dip
(δ) and ellipticity (tan ε).

structed over time for each frequency (Fig. 4a). Polar plots were then
adopted for the presentation of final results, which illustrate combined angular and frequency dependence (Fig. 4b). We found that
histograms of ellipticity (for a single frequency band) typically resulted in an asymmetric broad peak, which was represented well by
a beta distribution. Parameters of the beta distribution could be estimated for each frequency by standard statistical methods (e.g. the
maximum likelihood method). In this study, we applied the methodof-moments, where the parameter estimate is based on the sample
mean and variance. The estimated mode values are interconnected
by a line (e.g. Fig. 5c) to illustrate the variation of ellipticity with
frequency. The curve can be considered as a central value, whereas
histograms in the background represent the scatter. Since the mode
estimate is not based on histograms, it is not affected by the choice
of bin size, and smoothly maps the frequency dependence.
4.2 Results

analysis. These parameters can be described by three angles (see
Fig. 3): (1) azimuth of the major axis, or strike ( ), measured in
degrees from north; (2) tilt of the major axis, or dip (δ), measured in
degrees downward from the horizontal and (3) ellipticity, tangent of
the angle ε denoted in Fig. 3, defined as the ratio between the length
of the semi-minor and semi-major axes. All three polarization parameters vary with both time and frequency. The time–frequency
resolution is generally controlled by the choice of mother wavelet
utilized in the CWT (e.g. Torrence & Compo 1998). The Morlet
mother wavelet, a complex exponential modulated by a Gaussian
envelope, was adopted in this study. The time–frequency resolution
in this case is then simply controlled by changing the width of the
Gaussian envelope. The presented method has the advantage of being more general than commonly used azimuthally dependent H/V
spectral ratios (e.g. Del Gaudio & Wasowski 2007; Panzera et al.
2011).
We did not observe any systematic trends of retrieved parameters with time in our data set. Therefore we assume that observed
ambient vibrations are quasi-stationary (i.e. noise properties do not
change systematically on the timescale of the experiment—a few
hours), and analyse the relative occurrence of polarization parameters. In particular, histograms of polarization parameters were con-

All recordings were processed in this analysis, including those from
WAL101 and WAL211 that lacked time synchronization since an
absolute time reference is not necessary for polarization analysis.
The duration of recordings ranged from 60 to 90 min. Results for
three stations (WALref, WAL209 and WAL204) are presented in
Fig. 5. The locations of the three selected stations follow the anticipated transition from stable to unstable parts of the rock slope as
described in Section 2. Here we discuss certain polarization features,
especially with respect to the boundaries of the slope instability, and
within the context of three frequency bands: 0.2–1 Hz, 1–2 Hz and
2–30 Hz. In the lowest frequency band (0.2–1 Hz), noise polarization follows a roughly northeast–southwest trend (strike   ∼ 45◦ ),
remains primarily in the horizontal plane (dip δ ∼ 0◦ ), and the
ellipticity (∼0.3) does not follow any specific trend with respect
to frequency. This same pattern also applies for most other stations, with some exceptions (WAL109, WAL110, WAL203 and
WAL208). These stations do not have much in common, except the
proximity of stations WAL109 and WAL110 (separation distance
15 m), and have greater amplitudes in this frequency range (shown
in Section 5). Nevertheless, histograms are relatively fluctuating
for this lowest frequency band, suggesting that even longer recording times may be required for future experiments. In the highest

Figure 4. (a) Example histogram representing the relative occurrence of strike (azimuth) of ambient vibrations for a single wavelet scale, which is related to
Fourier frequency. The dashed line represents a corresponding uniform distribution, or no preferential orientation. (b) Polar plot representation of histograms
for different frequencies. The colour scale represents the relative frequency of occurrence (the height of the histogram for a given direction), and frequency
changes along the radius from 0.2 to 30 Hz as indicated. The single tick within the colour scale represents the level of a uniform distribution.
 
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Figure 5. Results of the polarization analysis for three stations: WALref (a–c), WAL209 (d–f) and WAL204 (g–i). Colour scale represents the relative frequency
of occurrence (see Fig. 4) of strike (a, d, g), dip (b, e, h) and ellipticity (c, f, i). Estimated maxima of the ellipticity distributions (assuming a beta distribution)
are connected by a solid black line.

frequency range (2–30 Hz), results vary significantly from station
to station, reflecting small-scale geological heterogeneity at the
site. Most interesting is the intermediate frequency range (1–2 Hz).
The reference station (WALref) shows only weak polarization—
relatively flat and broad maxima centred around f = 1.6 Hz,   =
130◦ , δ = 10◦ and a slight increase in ellipticity (∼0.4). Moving downhill to WAL209 (see Fig. 2), the polarization peak becomes more pronounced, shifts slightly (f = 1.6 Hz,   = 120◦
and δ = 5◦ ), and a local minimum appears in ellipticity at 1.6 Hz
(Fig. 5f). Finally, moving farther into the suspected unstable area at
station WAL204, ellipticity drops to almost zero at 1.6 Hz and the
peak becomes even more pronounced (f = 1.6 Hz,   = 120◦ and
δ = 5◦ ).
Histograms of polarization strike ( ) in the intermediate frequency range (1–2 Hz) are shown for all stations in Fig. 6, together
with mapped tension cracks traversing the site. The histogram peak
around 1.6 Hz oriented at ∼120◦ is present for all stations, however with different intensity. The direction 120◦ is approximately
perpendicular to the trace of observed surface cracks. The peak
intensity of strike (f = 1.6 Hz and   = 120◦ ) increases systematically moving towards the east (towards the cliff). Note the relatively
 
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sharp transition in peak intensity between station pairs WAL107 and
WAL207 (separation distance 35 m), and WAL102 and WAL103
(separation distance 15 m). A similarly sharp change in noise polarization across the boundary of a slope instability was also observed
in our previous study (BU10), where the precise location of the
border was known from years of monitoring. Thus, the difference
in noise polarization response between these two station pairs may
reveal the boundary between stable and unstable parts of the rock
slope (shown by red arrows in Fig. 6). Moreover, noise polarization
is weaker to the south (WAL103–111) as compared to the north
(WAL201–206, WAL107). These two areas are separated by a persistent crack mapped at the surface (indicated by blue arrows in
Fig. 6), which divides the instability into two structural compartments.
5 SITE-TO-REFERENCE SPECTRAL
R AT I O S
5.1 Method
Direct comparison of ground motion recordings between sites has
proven to be a useful tool in a number of studies investigating

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Figure 6. Distributions of ambient vibration azimuths in the frequency range of 1–2 Hz (polar plots, see Fig. 4 for explanation). Open cracks mapped at the
ground surface are shown as red lines. The blue and red arrows indicate key surface cracks that delimit areas of similar polarization patterns.

local amplification due to site effects. To ensure that the observed
variability in ground motion is caused by only changes in site conditions, one has to remove the variability resulting from source directivity and path effects (e.g. Andrews 1986). If the distance to the
source is sufficiently large (compared to the measurement separation), radiation patterns and path effects can be neglected (Borcherdt
1970). Such comparison is often conducted from earthquake recordings, where the distance to the source is well constrained. In
particular, amplitude spectra of recordings obtained at a site with
known conditions (the reference site) are used to normalize spectra calculated at other sites. These so-called SRSR then represent
frequency-dependent amplification functions with respect to the
reference area. In analyses of ambient vibrations, precise sources
are generally not known and the distant-source assumption cannot be validated. However, SRSR calculated using noise recordings
can still be representative of local site conditions (e.g. Irikura &
Kawanaka 1980; Roten et al. 2006; Roten & Fäh 2007).
Here we briefly describe the process of SRSR estimation used in
this study. Time-series of ambient vibrations were split into 50 per
cent overlapping windows of 100 s length and tapered with a cosine
window. For each time window, the Fourier amplitude spectrum was
calculated and smoothed using the window proposed by Konno &
Ohmachi (1998) with a bandwidth b = 40. Finally the geometrical
mean of ratios for all windows was determined.
5.2 Results
Horizontal components of all recordings were rotated by 120◦ , as
this was the direction of maximum polarization identified from
the previous noise polarization analysis. Recordings from stations
WAL101 and WAL211 were not processed, since these were missing

an absolute time reference. The resulting SRSR for three components (120◦ , 210◦ and vertical) are shown in Fig. 7. Most of the
presented SRSR have reasonable, expected variation, tending towards unity at low frequencies. This is not the case, however, for
station WAL110, which has more energetic horizontal components
at lower frequencies (0.2–0.5 Hz). Recordings from the nearby station WAL109 also show greater amplitudes in this frequency range,
suggesting that this anomaly is not caused by instrument malfunction. Station WAL110 was located precisely at the top of the cliff
(Fig. 2), and it was more exposed to the wind than other stations.
Moreover, a strong long-period signal on horizontal components
might be related to local tilts (Forbriger 2006). Nevertheless, longer
recording times would be necessary for detailed analysis, so at the
moment we cannot determine the exact origin of this anomaly.
A local maximum of the SRSR (120◦ component) is observed
at 1.6 Hz for all stations, however, the amplitude of the spectral
peak is different from station to station. Interpolated maps of SRSR
at 1.6 Hz are presented in Fig. 8; station WAL110 was excluded
from this interpolation. A clear spatial trend is present in the 120◦ component of noise. Spectral ratios increase towards the east, reaching values of 7 in the northeast corner of the site where the strongest
polarization was observed (see Fig. 6). On the other hand, 210◦ component spectral ratios show values <3 in the eastern part of
the slope. No amplification is observed on the vertical component.
Three distinct regions of the rock slope were identified in the preceding polarization analysis in the intermediate frequency range
(1–2 Hz): (i) the western portion of the slope—presumably stable,
with weak polarization; (ii) the northeastern area—presumably unstable, with strong polarization and (iii) the southeastern portion—
also presumably unstable, with intermediate polarization. The
schematic borders between these three compartments are shown
 
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 SRSR

SRSR

SRSR

SRSR

SRSR

SRSR

Evidence of normal mode rock slope vibration
16
8 WAL102
4
2
1
0.5
0.25
16
8 WAL106
4
2
1
0.5
0.25
16
8
4
2
1
0.5 WAL110
0.25
16
8 WAL202
4
2
1
0.5
0.25
16
8 WAL206
4
2
1
0.5
0.25
16
8 WAL209
4
2
1
0.5
0.25
0.2
0.5 1 f0 3 5
Frequency (Hz)

WAL103

WAL105

WAL107

WAL109

WAL111

WAL201

WAL203

WAL204

WAL207

WAL208

0.2

WAL210

0.5

1 f0 3 5
Frequency (Hz)

10

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20

120 component
210 component
Z component
10

20 0.2

0.5

1 f0 3 5
Frequency (Hz)

10

20

Figure 7. Site-to-reference spectral ratios for two horizontal components (120◦ and 210◦ azimuth) and the vertical component (Z) of ground motion. Dashed
black line denotes the identified fundamental frequency of 1.6 Hz.

Figure 8. Interpolated map of site-to-reference spectral ratios at the fundamental frequency of 1.6 Hz for three components of ground motion [(a) 120◦ azimuth,
(b) 210◦ azimuth and (c) vertical]. Grey bold lines delimit the three structural compartments (denoted by Roman numerals) identified from polarization analysis.
Map coordinates are given in metres of the Swiss coordinate system.

in Fig. 8. The relative amplification factors on the 120◦ component
can be linked to the individual compartments (I: SRSR ∼ 1–3; II:
SRSR ∼ 3–7 and III: SRSR ∼ 3–5). On the other hand, there is no
evident change in relative amplification between compartments II
and III for the 210◦ component of motion (SRSR ∼ 2–3), however
amplification here is still greater than in compartment I (SRSR ∼
1–2). Spectral amplification on the vertical component shows no
difference between the three compartments (SRSR < 2).
We can also identify other local maxima in the calculated spectral ratios (i.e. at 2.3, 3.7 and 5.3 Hz), which are present at almost
 
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all stations. Interpretation of these anomalies is not straightforward,
since the directions of maximum amplification are not consistent
across the array and the spatial pattern is complex. The decrease in
spectral ratios above 15 Hz may be explained by local amplification
at ∼20 Hz at the reference site (see Figs 5a–c). Finally, the resolved
spectral ratios have remarkably similar shapes for several different
station combinations (e.g. WAL103–105, WAL207–209-210 and
WAL201–202). Such clustering may reflect additional separation
of the slope into even smaller structural compartments or even individual blocks. However, objective interpretation of this complexity

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Figure 9. Apparent velocities estimated by picking maxima in the wavenumber plane for (a) subarray in the suspected unstable area of the slope (stations
WAL201, 202, 203, 204, 206), (b) subarray in the suspected stable area of the slope (stations WAL207, 208, 209, 210, ref). The semblance value (level of
correlation) is indicated by the colour scale. The resolution limit estimated from the array response function is shown by a black solid line. Fundamental
frequency f0 identified by polarization and SRSR analysis is shown as a dashed red line.

would require an even denser measurement grid and more detailed
geological model.
6 A R R AY P R O C E S S I N G
Small-aperture seismic arrays have been widely used for characterization of the shear wave velocity structure in the shallow
crust (<1 km; Tokimatsu 1997; Okada 2003; Bard et al. 2010).
The method is particularly useful in a layered soil medium,
where dispersive surface waves dominate the noise wavefield. The
frequency–wavenumber (f–k) method is commonly used to retrieve
phase velocity dispersion curves of surface waves, which are input
for inversion of velocity structure. The aim of the f–k processing
applied in this work was to identify waves propagating across the
array.

6.1 Method
Generally the f–k method is based on a frequency–wavenumber
power spectrum estimate. Assuming that the wavefield is composed
of plane waves, it is possible to estimate apparent phase velocities and wave propagation directions by picking local maxima in
the power spectrum (Lacoss et al. 1969). The power spectrum estimate is reliable only for a specific interval of wavenumbers. This
interval is defined by the resolution limits of the array, which are
controlled by array geometry. In principle, the lower resolution limit
(minimum resolvable wavenumber) is related to the largest interstation distance (array aperture), whereas the upper resolution limit
(maximum wavenumber) is related to the minimum interstation distance (station spacing). The larger and denser the array, the broader
the interval of wavenumbers that can be analysed. Estimation of
resolution limits has been analysed in number of previous studies
(e.g. Woods & Lintz 1973; Kind et al. 2005; Wathelet et al. 2008).
Moreover, the maximum likelihood estimate of power spectrum
introduced by Capon (1969) can considerably improve the array
resolution limits (Asten & Henstridge 1984; (Zywicki 1999), as
will also be demonstrated in this paper. However, this improvement
depends strongly on the characteristics of uncorrelated noise in the
data set, which can usually not be anticipated. An extension to the
three components of motion was proposed by Fäh et al. (2008) and
Poggi & Fäh (2010).
In this study, we applied both classical (Lacoss et al. 1969) and
high-resolution (Capon 1969) f–k power spectral estimate methods (so-called beamforming methods), as implemented by Wathelet

et al. (2008). Recorded time-series were divided into overlapping
windows of 100 s duration. A consecutive set of narrow-band filters
(characterized by central frequencies f cent ) were applied to trace the
frequency dependence. Wavenumber power spectra were then estimated for each time window and frequency band, and the location
(kopt ) of the strongest peak in the spectrum was identified. Finally,
an apparent velocity cap was calculated as cap = 2πf cent / kopt  .

6.2 Results
We performed f–k analysis on recordings obtained by Array 2, since
the geometry of Array 1 was almost linear (see Fig. 2) with a strongly
asymmetric response. We focused on a subset of stations consisting
of WAL201, WAL202, WAL203, WAL204 and WAL206, all located
in the area of the slope near the cliff face and delimited by a distinct
surface crack (Fig. 2). Horizontal components of recordings were
rotated by 120◦ as in the previous analysis of SRSR. High-resolution
f–k estimates of apparent velocities for the 120◦ component are
plotted in Fig. 9(a). The level of signal correlation across the array
(semblance) is indicated by the colour scale. Note the sharp increase
in apparent velocity centred at ∼1.6 Hz (red dashed line), which
was identified as a potential fundamental frequency of the slope
instability from SRSR analysis. The apparent velocities of these
highly correlated signals (semblance close to one) do not drop below
4 km s–1 in the narrow band between 1.4 and 1.7 Hz for any time
window, and most of the points lie in the interval between 6 and
30 km s–1 . The lower resolution limit of the array is depicted by
a solid black line, as estimated from the array response function
following Wathelet et al. (2008). Note that all points picked below
3 Hz are beyond this resolution limit, so the observed increase in
apparent velocities could be artificial. In the following, however, we
argue that this is not the case.
An example of resulting wavenumber power spectra for a single
time window is presented in Fig. 10. Power spectra estimated by
the high-resolution f–k (HRBF) for three different frequency bands
(centred on 1.4, 1.6 and 1.8 Hz) are depicted in Figs 10(a)–(c), respectively. Power spectra estimated by the classical f–k (BF) method
are depicted in Figs 10(d) and (e) for two frequency bands, 1.4 and
1.6 Hz, respectively. The array response function, defined as the
response of the array to a vertically propagating plane wave (e.g.
Woods & Lintz 1973), is depicted in Fig. 10(f) for comparison.
Resolution limits as defined by Wathelet et al. (2008) are indicated
by the black dashed line. Open circles in the centre of the plots correspond to an apparent velocity of 4 km s–1 . Estimates obtained by
 
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567

Figure 10. Normalized estimates of wavenumber power spectra for different frequencies close to the fundamental frequency (1.6 Hz) determined by both
high-resolution beamforming (a, b, c) and classical beamforming (d, e) methods. The black dashed line marks the resolution limit obtained from the array
response function. Black and red open circles mark the apparent velocity of 4 km s–1 . Power focuses towards zero wavenumber at 1.6 Hz for HRBF (b),
overcoming the resolution limits shown in (f).

the classical f–k method are very similar for both frequencies (1.4
and 1.6 Hz), and almost undistinguishable from the array response
function. On the other hand, estimates made by the high-resolution
f–k method show clear improvement in resolution, which allows us
to distinguish differences between the three frequency bands (1.4,
1.6 and 1.8 Hz). Note the considerable focusing of power towards
zero wavenumber at 1.6 Hz. A similar pattern is also found for other
time windows (not presented here). Thus, the increase in apparent
velocities at ∼1.6 Hz presented in Fig. 9(a) is not artificial.
To summarize, observed apparent velocities of 6–30 km s–1 indicate that ground motion is generally in-phase at 1.6 Hz within
the array. Although such high apparent velocities may indicate vertically propagating plane waves, this is not an entirely plausible
interpretation in our case. The apparent velocities drop dramatically in the frequency band between 1.6 and 2 Hz (Fig. 9a), and
such strong dispersion is not realistic for body waves. The f–k analysis was also performed using other subarray station combinations
(WAL207, WAL208, WAL209, WAL210 and WALref) in the suspected stable area of the slope. Results are presented in Fig. 9(b).
We also performed f–k analysis on other velocity components (210◦
and vertical). No similar sharp increase in apparent velocities was
identified in either case. At frequencies above 3 Hz, no clear dispersion curve could be determined with f–k analysis. This indicates
mechanical separation between different blocks in the unstable part
of the slope, preventing the propagation of higher frequency surface
waves.

7 DISCUSSION
An increasing number of studies have reported directional site effects in areas of pronounced topography (e.g. Pischiutta et al. 2010;
Panzera et al. 2011), and in some cases in association with potential landslides (Del Gaudio & Wasowski 2011; Moore et al. 2011).
However, the origin of such directionality is still not well understood in most cases. Here we take advantage of the relatively high
 
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density of observational points in our study, and explore combined
interpretation of our results.
Briefly summarizing our main observations:
(1) Recorded ambient noise is polarized in a narrow frequency
band centred at f0 = 1.6 Hz. The preferred orientations are  0 =
120◦ (azimuth) and δ 0 = 5◦ (dip), that is, polarization is mostly
limited to the horizontal plane. Polarization at f0 is strongest within
the limits of the instability nearest the cliff (Fig. 6), where the
particle motion is almost linear (Fig. 5i).
(2) The amplitude of the noise power spectrum increases monotonically towards the cliff at frequency f0 and in the horizontal
direction  0 (Fig. 8).
(3) Ground motion is in-phase within the suspected unstable area
at frequency f0 and in the direction  0 (Fig. 9).
These three attributes are typical for normal mode vibration of
sedimentary basins (Bard & Bouchon 1985; Steimen et al. 2003,
Roten et al. 2006), which is analogous to free oscillation of the
Earth. Therefore, our observations are in agreement with the guiding hypothesis that we observe normal mode rock slope motion
at the Walkerschmatt instability, as also postulated in our previous
study at Randa (BU10). More specifically, we suggest that the unstable rock mass may be viewed as a volume of effectively reduced
elastic moduli (fractured and damaged rock with air-filled voids).
The boundary (both vertical and horizontal) between heavily fractured and more intact rock mass then presents a contrast in seismic
impedance, which traps incoming waves. This results in the development of standing waves (normal mode vibration). Since the cliff
presents a free surface, the greatest amplitudes of motion are found
near the face (analogous to a string fixed at just one end).
However, there is a fundamental difference in comparison to sedimentary basins. Polarization of the basin normal modes has its
origin in basin geometry. For example in 2-D analyses, ground motion can be separated into SH and P-SV contributions with respect
to the axis of the basin (Bard & Bouchon 1985), where SH normal

 568

J. Burjánek et al.

modes are polarized parallel to the valley axis and are perpendicular to P-SV modes. The strong directional site response observed
in this work, on the other hand, is likely caused by the fractured
nature of the medium. The relationship between polarization and
mapped fractures is apparent in Fig. 6. Predominant polarization of
ambient noise at f0 is in general perpendicular to observed surface
cracks. We suggest that material within the unstable part of the rock
slope is weakened by the cumulative effect of numerous parallel
fractures, many of which can be observed at the surface. Since these
cracks have a preferred orientation subperpendicular to the direction of slope displacement, effective anisotropy in rock mass moduli
arises. Specifically, we suggest that the bulk stiffness of the fractured rock mass drops significantly in the direction perpendicular to
the trace of tension cracks. We do not relate such an effect to inherent seismic anisotropy of the rock (e.g. due to microfracturing), but
rather to the systematic orientation of open macrofractures, which
are common in deep-seated rock slope instabilities. This explanation
is supported by recent numerical modelling of earthquake-induced
ground motion at the Randa rock slope performed by Moore et al.
(2011). Directional amplification and polarization of ambient noise
was recently observed on the flank of Mt Etna (Di Giulio et al.
2009; Rigano et al. 2008), which was also related to the presence
of a fault system.
Excitation of normal mode vibration by ambient noise sources
results in significant local site amplification, with spectral ratios up
to 7 in the unstable area of the slope compared to the reference
station on adjacent stable ground (Fig. 8). Such strong relative amplification cannot be explained by topographic site effects alone,
that is, only by the shape of the topography. Previous numerical
studies suggested that amplification factors rarely exceed 2 for topographic irregularities underlain by a homogeneous half-space (e.g.
Geli et al. 1988; Assimaki et al. 2005). Moreover, the numerical simulation performed by Moore et al. (2011) required introduction of
subsurface heterogeneity (compliant fractures) to explain observed
amplification for an analogous case study at the Randa rock slope
instability. Comparison of SRSR obtained from ambient noise and
regional earthquake data (weak motion) showed good agreement in
analysis of data recorded at the Randa rockslide (Moore et al. 2011).
This is a promising outcome since acquisition of ambient vibration
data is substantially more time- and cost-effective. Nevertheless,
we emphasize that the slope may behave differently under strong
shaking (high-strain motion). For example, high strains may result
in systematic damage to the rock mass (further fracturing), which
would dissipate elastic energy. On the other hand, induced fractures
might generate even stronger amplification and site effects.
At frequencies greater than 2 Hz, we could identify additional
peaks in our analysis of SRSR. Although the resolved amplification
was also directional, polarization orientations were not consistent
across the array and amplification maps present a complex spatial pattern. Moreover, no coherently propagating seismic waves
were identified with f–k analysis at these frequencies. Ground motion above 2 Hz therefore appears to be controlled by vibration of
smaller, localized rock mass compartments within the instability.

8 C O N C LU S I O N S
Measurements presented in this paper support our hypothesis that
ambient vibration of a large, potentially unstable rock slope is dominated by normal mode motion. Since the same conclusion was
previously postulated at another slope instability in the same area
(BU10), we suggest that this phenomenon is not rare. Although the

results presented in this study were obtained using seismic arrays,
polarization analysis could be performed with single-station noise
measurements (point by point). The density of measurement points
is critical for correct interpretation of the slope’s seismic response.
Strong directional amplification was linked to the internal structure
of the rock mass, specifically to systematic fracture opening caused
by slope movement, rather than topography. Detailed interpretation of the low-frequency response (<1 Hz) would require longer
measurement intervals. The resolved fundamental frequency of the
instability could be monitored in the future (as in Lévy et al. 2010),
to asses any potential temporal changes related to internal damage
that may precede failure.

AC K N OW L E D G M E N T S
The authors would like to thank Antonio Rovelli, and an anonymous
reviewer for constructive comments and helpful suggestions. This
research is part of the project COGEAR, funded by the Competence Center for Environment and Sustainability of the ETH Domain (CCES). Wavelet software was provided by C. Torrence and
G. Compo, available at http://paos.colorado.edu/research/wavelets/.
Some figures in this paper were made using Generic Mapping Tools
(GMT) 4 by Wessel & Smith (1998).

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