Duration of the 2011 Tohoku earthquake ground

Duration of the 2011 Tohoku earthquake ground motions

Hadi Ghofrani & Gail M. Atkinson

Journal of Seismology ISSN 1383-4649 J Seismol DOI 10.1007/s10950-014-9447-y

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Author's personal copy J Seismol DOI 10.1007/s10950-014-9447-y

ORIGINAL ARTICLE

Duration of the 2011 Tohoku earthquake ground motions Hadi Ghofrani & Gail M. Atkinson

Received: 22 May 2013 / Accepted: 8 July 2014 # Springer Science+Business Media Dordrecht 2014

Abstract We compared several different duration measures for the 2011 M9.0 Tohoku earthquake sequence, because empirical duration models are of great interest for purposes of correlation with structural damage, and the Tohoku mainshock was remarkable for its long duration. Among the three considered definitions, RMS duration (McCann and Shah, Bull Seism Soc Am 69: 1253–1265, 1979) is best able to predict the duration within which pulses or groups of pulses of energy arrive; it is particularly suitable for the Tohoku mainshock, for which source complexity caused timeseries with multiple-phase arrivals. Two other considered definitions, both of which tend to underestimate the observed duration, are: (i) duration defined by random vibration theory (RVT); and (ii) the significant duration as defined by the interval between 5 and 75 % or 95 % of the integral of the square of the ground acceleration (known as “Arias intensity” (Arias 1970)) or velocity (known as “energy integral” (Anderson 2004)). In the Tohoku mainshock, significant amplitudes precede the 5 % of the Arias intensity marker; we need to use 0.3 % of the maximum of the accumulated energy as the lower bound marker to appropriately estimate the duration using the significant duration definition. The RVT duration (used in stochastic simulations) can be estimated easily from the 5–75 % of the Arias intensity (significant H. Ghofrani (*) : G. M. Atkinson Department of Earth Sciences, University of Western Ontario, 1151 Richmond Street, London, Ontario, Canada N6A 5B7 e-mail: [email protected] G. M. Atkinson e-mail: [email protected]

duration) definition as the two measures give very similar durations. Overall, the significant duration of ground motions observed during the 2011 M9.0 Tohoku earthquake increases with distance as 0.19Rcd for the horizontal components or 0.33Rcd for the vertical component, where Rcd is the closest distance to the fault plane. By comparison, the duration of four aftershocks (M4.5– 7.7) increases with distance as ~0.10Rhypo where Rhypo is the hypocentral distance. For the mainshock, the distance-dependent slope term is greater, presumably due to the large fault plane size. Keywords Significant duration . Random vibration theory . Root-mean-squared duration . Arias intensity . Tohoku M9.0 mega-thrust earthquake

1 Introduction The 2011 M9.0 Tohoku earthquake and its aftershocks have provided important new quantitative information on ground-motion parameters that will be invaluable in refining seismic hazard analysis and mitigation efforts in subduction-zone regions. The ground-motion characteristics of this earthquake are unique: large peak ground motions at high frequencies, caused by the combined effects of deep earthquake sources having large stress drops, and significant site amplification (Ghofrani et al. 2013); long-duration ground motions which caused widespread and extensive liquefaction (Bhattacharya et al. 2011); and multiple-shock features as the result of a complex source process (Kurahashi and Irikura

Author's personal copy J Seismol

2011). Taking advantage of the rich dataset of strongmotion recordings from the 2011 M9.0 Tohoku earthquake, we previously performed a thorough analysis of site amplification (Ghofrani et al. 2013) and evaluated the capability of the stochastic finite-fault approach for the analysis and simulation of spectral ordinates of this earthquake. Another important parameter that relates to the damage potential of the earthquake is the duration of strong ground shaking. In this study, we examine ground-motion duration, including how it scales with moment magnitude (M) and distance. Peak ground-motion parameters such as peak ground acceleration (PGA), peak ground velocity (PGV), and response spectra are often used as measures of the damage potential of motions. However, it is widely recognized that the duration of shaking influences the degree of damage for a structural system responding in the nonlinear range. The relationship between duration and damage potential is complex, depending on factors such as the structural type, and how damage is measured (Hancock and Bommer 2006). It is generally held that if an earthquake causes large response amplitudes over a short duration impulse, it may not be very damaging because most of the impulse may be absorbed by the inertia of the structure with little resulting deformation, while by contrast a motion with moderate amplitudes but long duration could produce enough load reversals to cause significant damage (Kramer 1996). However, there are also counter-examples to this concept, in which a shorter duration may be more damaging due to the inability of the structure to dissipate energy by damping. One such example is provided by two sets of recordings obtained in San Salvador (El Salvador) from a subduction earthquake in 1982 (Ms7.2) and a local crustal earthquake in 1986 (Ms5.4). The recordings had almost identical Arias intensities (i.e., comparable total energy), but the shorter duration and higher amplitudes of the 1986 event imparted that energy to structures much more rapidly, and led to far greater levels of damage (Bommer and Martinez-Pereira 1999). Duration of shaking also plays an important role in liquefaction, which can lead to significant damage. Liquefaction of soil deposits (i.e., large reduction of shear strength due to pore pressure buildup) may occur if the level and duration of shaking are sufficient to overcome the shear resistance of sandy soils. The shear resistance of soil against liquefaction depends significantly on the number of stress cycles during an earthquake. In the Tohoku earthquake, dissipation of pore pressure may

have been slow due to the long duration of shaking. The volume of sand boils attributed to this event was massive—it is hypothesized that this could be related to the number of cycles that the soil was subject to during this event (Bhattacharya et al. 2011). In general, we would expect the number of loading cycles to increase with duration, for a fixed frequency content. However, because magnitude and duration are increasing together, the frequency content will be shifting toward lower frequencies as the duration lengthens. This may tend to counteract the expectation of a greater number of cycles for longer durations. Bommer et al. (2006) found that the correlation between duration and number of cycles is actually quite poor. Thus the role of duration in liquefaction may be ambiguous, at least in terms of its relationship to number of loading cycles. In summary, then, duration is an important parameter in both structural and soil response, but must be interpreted with caution, as its influence on damage is not straightforward. In this study, we develop empirical models for several measures of duration, and provide estimates of the uncertainty in the model coefficients; the horizontal and vertical components are considered separately. The selected measures of duration utilize different mathematical definitions and thus may emphasize different attributes of the duration, particularly for a large complex event such as the M9 Tohoku earthquake. The dependence of the duration on soil conditions (e.g., VS30, the average shear-wave velocity in the uppermost 30 m) as well as forearc and backarc travel paths is described. Empirical duration models are also inferred for the aftershocks in an attempt to quantify the influence of source duration (or magnitude). To our knowledge, this is the first study to systematically compare duration measures and their utility for large earthquakes over such a broad magnitude range (M4.5 to M9). Despite the seismological and geotechnical importance of duration, there is no consensus as to how to best describe the duration of ground motion. A review of the proposed definitions is provided by Bommer and Martinez-Pereira (1999). We select the most commonly used duration definitions and compare their applicability to the Tohoku earthquake. In general, the calculated duration of shaking may depend on the frequency band of the assessment, due to a number of factors including frequency-dependent attenuation (Q), possible resonances associated with soil response, and source rupture characteristics. This is why the significant duration of


vertical ground acceleration may be different from that for horizontal acceleration, and why duration may differ for acceleration, velocity, and displacement. Several prediction equations for duration that express frequency-dependent effects have been developed using band-passed accelerograms (e.g., Bolt 1973; Trifunac and Westermo 1977; Mohraz and Peng 1989; Novikova and Trifunac 1994). An advantage of this approach is that the arrival time and duration of each separate strong-motion pulse may be evaluated (Novikova and Trifunac 1994), providing insight into source effects and energy dissipation between strong-motion pulses. Nevertheless, duration parameters that are broad banded in frequency are of great importance for engineering applications; structures respond to the seismic wave train as a whole, and therefore intensity measures for engineering applications need to characterize that broadband excitation (Kempton and Stewart 2006). This is analogous to the use of spectral acceleration, which is likewise defined from the entire accelerogram, to characterize the amplitude of shaking for structural response applications. We therefore focus on broad-banded duration parameters, while acknowledging that these parameters do not express information on frequency-dependent effects. Our motivation for this study is that predictive equations for ground-motion parameters for subduction earthquakes have received far less attention than those from crustal earthquakes, and to the extent of the author’s knowledge there are currently no published equations specifically for the estimation of strong-motion durations from subduction earthquakes. This article examines duration for subduction earthquakes, including in particular the duration for mega-thrust events as large as M9.

2 Brief overview of definitions of strong ground-motion duration The complex source process of the Tohoku earthquake, which produced multiple-phase arrivals in its timeseries, motivated us to examine the commonly used duration definitions to test their ability to characterize the observed S-wave trains in this giant earthquake. It is also interesting to check how these definitions differ from each other. We should emphasize that the duration measured using each specific definition is simply the

value obtained with that measure, and thus there is no single “right” duration. Nevertheless, one may make qualitative judgments as to the extent to which a specific duration measure appears to capture various features of the observations.

2.1 Significant duration The total energy (per mass/per unit weight stored by a set of undamped simple oscillators at the end of an earthquake) of an accelerogram is often represented by the integral of the square of the ground acceleration over the entire length of the signal, which is related to the Arias intensity (Arias 1970). The integral of ground velocity, introduced by Sarma (1970) and referred to as the energy integral by Anderson (2004), can also be used to define duration. The “significant duration” (SD) is defined as the time interval over which some proportion of the total energy or Arias intensity is built-up. The lower limit is typically fixed at 5 % of the cumulative Arias intensity; however, the upper bound varies across different studies (95 % (Trifunac and Brady 1975); 85 % (Herrmann 1985); 75 % (Kennedy et al. 1985; Ou and Herrmann 1990)).

2.2 RMS duration Another definition of strong-motion duration is based on the general trend of the cumulative root-mean-square (RMS) of the ground-motion amplitudes. Cumulative RMS reaches quickly a maximum and then decays slowly to a final RMS value for the entire record (McCann and Shah 1979). The most interesting feature of the cumulative RMS is that it can provide information about the times at which pulses or groups of pulses of energy arrive, particularly for large earthquakes with multiple ruptures. The end of the RMS duration window is defined as the time at which the derivative of the cumulative RMS becomes (and remains) negative. The start of the RMS duration window is estimated following the same procedure as for its end, but using the reverse acceleration time history. There are some cases for which the RMS duration will not peak appropriately at the starting points of significant shaking phases, such as when an earthquake is deep enough to produce a strong P-wave, especially on the vertical component.


2.3 RVT duration Using random vibration theory (RVT), Vanmarcke and Lai (1977) proposed another definition for strong-motion duration. They used the theory of stationary Gaussian random functions to predict the most probable value of the ratio of peak to RMS motion (called peak ratio), during the steady strong-motion interval. The peak ratio can be estimated using the following equation (Cartwright and LonguetHiggins 1956): Z N e o ymax 1 ∞n ¼ pffiffiffi 1− 1−ξexp −z2 dz ð1Þ yrms 2 0 where ymax is the peak ground-motion parameter (e.g., peak ground acceleration), and yrms is the root-mean-square ground-motion parameter; the ratio of ymax/yrms is known as the peak factor. In Eq. 1, ξ=Nz/Ne and Nz, Ne are the number of zero crossings and extrema, respectively (Boore 2003). The number of zero crossings and extrema are related to the frequencies of zero crossings (fz) and extrema (fe) and to duration (T) by the equation: N z;e ¼ 2ef z;e T

ð2Þ

where frequencies are related to the zeroth, second, and fourth moments of the squared spectral amplitude. In Eq. (1), yrms is also dependent on duration (T) through the following relation: . 1=2 yrms ¼ m0 T ð3Þ where m0 is the 0th spectral moment. The RVT duration can be computed as the value of T which satisfies Eq. (1).

3 Strong ground-motion data and processing The strong-motion data used in this study were collected from the National Research Institute for Earth Science and Disaster Prevention (NIED) networks of Japan. We used data from both K-net (Kyoshin network) and KiKnet (KIBAN kyoshin network). The K-net consists of 1,043 strong-motion seismographs sited on the ground surface, covering the whole of Japan, while the KiK-net consists of 692 strong-motion observation stations

installed both on the ground surface and at the bottom of boreholes. The locations of K-net and KiK-net stations are shown in Fig. 1. For all records of K-net and KiK-net, we follow the same processing procedure, which includes windowing, correction for baseline trends and band-pass filtering. We have applied noncausal, band-pass Butterworth filters with an order of 4. The selected frequency range of analysis is 0.1 to 13.50 Hz. The lower frequency limit was selected by inspecting many records; this value is appropriate to produce well-shaped displacement time series, with a flat displacement spectrum at low frequencies. The upper limit is chosen by considering the cut-off frequency of the seismograph response spectrum (15 Hz). The focal mechanism information for the aftershocks (Table 3) is gathered from the Fundamental Research on Earthquakes and Earth's Interior Anomaly (F-NET: http://www.fnet. bosai.go.jp). For each record, we have calculated the duration based on several definitions. The significant duration is calculated for two cases, corresponding to 5–75 % and 5–95 % of the Arias intensity, or the energy integral. After calculating the cumulative RMS function, the RMS duration of each time-series is also calculated from the time interval between the points where the first derivative of the cumulative RMS function becomes and remains negative in its sign. Knowing the RMS values of each record (yrms) and its maximum value (ymax), Eq. 1 for the RVT duration can be solved numerically. Figure 2 shows a set of example time series at several distances, with varying characteristics, for which different duration measures are compared (these stations are shown as black squares in Fig. 1). In this example, we use the same starting point for the RVT and significant duration (5 % of the Arias intensity), because RVT indicates the length of strong motion, not the starting point. Due to the insignificant difference of duration values for the three components (as shown in more detail later), we use and show the results for the vertical ground motions. It is clear from this figure that the RVT and 5–75 % duration definitions result in similar duration values, which reflect the duration of a single dominant phase. The RMS and 5–95 % durations are better able to estimate the total duration including multiple phases (i.e., full wave train), whereas the 5–75 % interval captures the more dominant body-wave part of the signal.

Author's personal copy J Seismol Fig. 1 The spatial distribution of all KiK-net and K-net stations (black dots) that recorded the Tohoku event. The star is the epicenter of the earthquake. The major tectonic boundaries—the trench and the volcanic front—are represented by dashed black lines. A hatched rectangle shows the fault plane obtained from the GPS Earth Observation Network System (GEONET) data analysis

45o

K-net & KiK-net stations Volcanic front Trench GSI’s Fault Plane L = 400 km W = 150 km Strike = 202 deg FKSH17 Dip = 18 deg Depth = 10 km Epicenter of 2011 M9.0 Tohoku Earthquake

40o

IWTH03 MYGH04

35o

IBRH10

130o

135o

140

5%-95%

5%-95%

Acceleration (cm/s2)

5%-75%

5%-75% RVT

RVT

IWTH03 PGA = 135.61 (cm/s2) Rcd = 108.5 km

IBRH10 PGA = 221.88 (cm/s2) Rcd = 93.0 km

0

0

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-140 RMS

560

Acceleration (cm/s2)

145o

RMS

RMS

225

140o

RMS

125

5%-95%

5%-95%

5%-75%

5%-75%

RVT

RVT

MYGH04 PGA = 401.34 (cm/s2) Rcd = 91.2 km

0

FKSH17 PGA = 121.52 (cm/s2) Rcd = 101.9 km

0

-560

-125

0

100

200

Time (s)

Fig. 2 Comparing different duration definitions (RMS, Significant, and RVT) at selected stations (rectangles in Fig. 1) that recorded the Tohoku earthquake. The stations show different characteristics including: a single pulse (first phase is not visible),

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0

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200

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Time (s)

dominant first phase, two distinct phases, and dominant second phase. The light gray curve overlaid on each time-series is the accumulated squared acceleration normalized by the PGA; dashed lines show RMS window


4 Results 4.1 Comparison of different duration definitions It can be seen from Fig. 2 that different duration definitions result in different duration estimates for the same record. Among them, the RMS duration is best able to pick the window that contains the significant shaking for all phases, for all of the sample stations; we note that RMS values are in agreement with those that would be obtained by a subjective pick of where the strong shaking starts and stops, by visual inspection. To gain insight into the robustness of this definition in determining the duration of strong ground motions during the Tohoku earthquake, we compare the duration models with RMS duration in Fig. 3. For all range of durations (or distances correspondingly), RVT and significant durations produce a lower estimated value than does the RMS duration. Looking at Fig. 2 in more detail, we see that significant amplitudes are observed at stations before the time that is indicated by 5 % of the cumulative integral of acceleration-squared. To investigate further, we picked the S-arrivals at 249 stations from K-net and 244 stations from KiK-net and calculated the ratio of accumulated acceleration-squared before the picked start of the S-wave to the total acceleration-squared of the record. The ratios show a mean value of ~0.3 % for both K-net and KiK-net stations. We performed a similar 300

RVT duration (Vanmarcke and Lai, 1980) Significant duration (5-75%) Significant duration (5-95%) Significant duration (0.3-95%)

Strong motion duration [s]

250

200

4.2 RVT versus significant duration One of the interesting results from Fig. 3 is that the RVT and 5-75 % duration estimates provide similar duration values. This is important because calculating the integral in Eq. 1 is cumbersome. As RVT is the basic definition used in stochastic simulations (Boore 1983, 2003), its possible substitution with the significant duration (5– 75 %) is helpful; it makes the calculations of appropriate durations for stochastic applications easier. To check this correlation, we plot the RVT versus the 5–75 % significant durations of the KiK-net stations (both surface and borehole data) in Fig. 4. It is clear that the significant and RVT durations show similar trends at distances ≤200 km, with the former being less scattered. This finding is similar to that of a previous study by Raoof et al. (1999) for earthquakes in Southern California. We conclude that the RVT duration is approximately equivalent to the 5–75 % significant duration measure. 4.3 Duration model for the Tohoku earthquake as a function of distance and local site conditions

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100

50

0

analysis for the upper bound of the S-windows. The ratio of the accumulated acceleration-squared at the end of the S-window is about 96 % of the total, which is comparable to the common value of 95 % used in the significant duration. This suggests that appropriate bounds for significant duration are 0.3–95 % of the cumulative acceleration-squared, if we wish to include the entire signal from the start of the strong portion of shaking. The significant durations calculated based on the new sets of bounds are plotted versus the RMS durations in Fig. 3. It is clear from this figure that the significant duration calculated as 0.3–95 % of the cumulative acceleration-squared agrees well with the RMS durations.

0

50

100

150

200

250

300

RMS duration (McCann and Shah, 1979) [s]

Fig. 3 Comparison of RMS and estimated duration parameter based on several definitions (RVT and significant durations) using KiK-net surface vertical ground motions

The total duration (Ttotal) of the observed/recorded ground-motion on the surface is usually described as the sum of: (i) source duration; (ii) the lengthening of the wave train with increasing distance due to dispersion effects; and (iii) site effects on duration. Thus, the total duration is a function of earthquake magnitude, distance, and site variables: T total ¼ T source þ T path þ T site þ σ

ð4Þ

Author's personal copy J Seismol EW [surface]

NS [surface]

UD [surface]

200 RVT dur. Significant (5-75%) Mean RVT+ _1

T = T0 + b.R

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100

50

0 0

100

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400

500 0

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500 0

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Rcd (km)

Rcd (km)

Rcd (km)

EW [downhole]

NS [downhole]

UD [downhole]

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500

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200 RVT dur. Significant (5-75%) Mean RVT+ _1

T = T0 + b.R

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500 0

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Rcd (km)

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Rcd (km)

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500 0

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Fig. 4 The total duration of ground motions from the Tohoku earthquake recorded on KiK-net stations (surface and borehole) as a function of closest distance to fault (Rcd). Light gray dots in the background are duration values estimated using the RVT method;

their corresponding mean values are shown by white squares with ±1 standard deviation error bars. Dark circles show significant duration (5–75 %)

where Tsource is source duration (e.g., Hanks and McGuire 1981; Boore 1983), Tpath represents the distance dependence, Tsite represents the site dependence factors, and σ is the standard deviation of the residuals. Using the rich database of strong ground motions recorded during the Tohoku earthquake, we can examine the contributions of source, path, and site to the total duration.

The least-squares coefficients of Eq. (5) along with their corresponding standard errors are provided in Table 1. Note that any effects of site on duration are not considered at this stage, but would map into either the c1 or c2 terms. We examine the possible site contributions later. The values indicated by Eq. (5) are greater than those obtained using empirical relations for significant duration as developed in the previous studies of Bommer et al. (2009), Kempton and Stewart (2006), and Abrahamson and Silva (1996). However, the relations developed by Esteva and Rosenblueth (1964) are in reasonable accord with the significant duration (5– 75 %) of vertical acceleration (SDa5–75%[ver]) of the Tohoku earthquake (Fig. 5). Looking specifically at the distance-dependent duration of motion term (coefficient c2), it is slightly larger than typical values for this parameter (e.g., 0.10 for earthquakes with M3.1–6.7 in Southern California from

4.3.1 Source and path duration In Fig. 5, we plot the RMS and significant durations as a function of Rcd (closest distance to fault rupture) considering all KiK-net and K-net surface ground motions at Rcd ≤200 km. Both duration measures appear to follow a linear relationship: T total ¼ c1 þ c2 ⋅Rcd

ð5Þ


Total Duration [s] T = T0 + b.R

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91.55(_ +4.94) + 0.318(_ +0.036)Rcd 39.84(_ +4.60) + 0.333(_ +0.033)Rcd SDa5-95[ver] RMS Kempton & Stewart (2006) [M9.0 - Rock] Abrahamson & Silva (1996) Esteva and Rosenblueth (1964)

example, based on Lee (2012), among others, we infer a source duration of about 80–107 s for this event. This source duration is equivalent to the time over which 75– 90 % of the total slip was released. Another estimate of source duration is that based on corner frequency (fc) of the Fourier amplitude spectrum. Simple seismic source models (e.g., Hanks and McGuire 1981; Boore 1983) suggest that source duration (Tsource) is inversely related to fc:

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T source ¼ f c−1

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0

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ð6Þ

We computed the corner frequency (fc) of the source model according to the objective equation introduced by Andrews (1986): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uZ ∞ u V 2 ð f Þdf u 1 u 0 uZ fc ¼ ð7Þ 2π t ∞ 2 D ð f Þdf

400

Closest distance (Rcd) [km]

Fig. 5 Duration model as a function of closest distance to fault for the vertical component of ground motions at K-net and KiK-net (surface) stations. The best fit lines are shown in black (solid for RMS duration, dashed for significant duration)

0

where V( f ) and D( f ) are the Fourier velocity and displacement spectra, respectively. Averaged over all components, we obtain fc ≈0.014±0.008 Hz; there is no significant difference between surface and borehole stations. The corresponding estimate of Tsource ≈70 s is in good agreement with other methods of estimating source duration.

Raoof et al. 1999; 0.09 for the M8.1 Tokachi-Oki mainshock from Macias et al. 2008; and 0.05 for moderate events in eastern North America from Atkinson 1993). The relatively large increase of the duration with distance may be due to the large magnitude of the Tohoku event and its multiple-event nature. An estimate of the source duration (Ttotal at R=0) can be obtained from the intercept of the best fit lines in Fig. 5, as ≈100 s. This agrees with estimates based on source time functions reported for the Tohoku earthquake. For

4.3.2 Variability due to forearc and backarc travel paths The backarc region of Japan is affected by a low-Q, low-velocity mantle wedge, which filters out the high-

Table 1 The least-squares coefficients of source and path durations (Eq. 5) along with their corresponding standard errors for all KiK-net and K-net surface ground motions (at Rcd ≤200 km) Duration

Horizontal

Vertical

σha

σva

c1

c2

c1

c2

SDa5–95%

56.49±4.57

0.19±0.03

39.84±4.60

0.33±0.03

0.14

0.15

SDa5–75%

35.61±2.82

0.13±0.02

25.93±2.77

0.21±0.02

0.13

0.14

SDv5–95%

84.54±5.00

0.29±0.04

113.44±4.17

0.05±0.03

0.10

0.08

SDv5–75%

43.66±2.75

0.17±0.02

57.48±2.23

0.04±0.02

0.10

0.08

RVTa

17.62±3.72

0.20±0.03

5.024±4.707

0.35±0.03

0.21

0.26

RVTv

12.46±5.68

0.46±0.04

47.08±6.55

0.34±0.05

0.20

0.18

RMSa

103.74±4.56

0.19±0.03

91.55±4.94

0.32±0.04

0.08

0.09

RMSv

126.15±6.16

0.26±0.05

147.29±6.40

0.01±0.05

0.09

0.10

For the cells with italic font, the regression coefficient is not significantly different from zero (P>0.05); two-tailed t test for 95 % significance level a

Standard deviation of residuals for horizontal (σh) and vertical components (σv), respectively


frequency content of ground motions (Ghofrani and Atkinson 2011). For this reason, stations in the backarc region might be expected to have longer significant durations compared to forearc stations. We explore the trends of residuals (observed−calculated duration based on coefficients in Table 1) for forearc and backarc stations in Fig. 6. We use residuals for the significant duration measure, as based on the integral of the squared ground acceleration (SDa5–95%), for horizontal components. Based on the Student’s t test, the means of these two populations of residuals are not significantly different from zero. We therefore conclude that the durations are not significantly different for forearc and backarc travel paths.

the statistical significance of site effects on duration by defining: T site ¼ log10 ðT obs Þ−log10 ðc1 þ c2 ⋅Rcd Þ

ð8Þ

in which we have assumed that Tsite are the residuals from the parameterization of Eq. 5. Tsite represents the change in ground-motion duration due to site effects, after removing the source and path components of duration. The logarithm of duration is used because duration is assumed to be log-normally distributed (Kempton and Stewart 2006). We then seek a relationship for Tsite as function of site condition: T site ¼ c4 þ c3 ⋅log10 ðV S30 Þ

ð9Þ

4.3.3 Variability due to site effects It may be expected that soft soil or basin sites would have longer durations than rock sites, due to effects such as resonance within sedimentary layers or trapping of waves within basins. A comprehensive review as to what extent this feature has been captured in existing predictive equations for duration can be found in Bommer et al. (2009). We checked for site effects on the duration by plotting the residuals of Eq. (5) for the significant duration as a function of VS30, as shown in Fig. 7. An initial observation is that, for the horizontal components, the residuals for SDa5–95% appear to decrease with increasing VS30. However, this trend is not noticeable for the vertical components, nor was it observed for the other duration definitions. We examined

100

The results are summarized in Table 2. Considering the P value of the t test for intercept (c4) and slope (c3) of the regression models, it can be inferred that site stiffness has negligible effect on the vertical component duration, nor are the effects significant for the horizontal components, for either the RVT and RMS duration definitions. The site stiffness does have an effect on both the acceleration-based (i.e., the Arias intensity) and the velocity-based (i.e., the energy integral) significant durations for the horizontal component. The contradictory results from different measures concerning the significance of site stiffness on duration are puzzling. Previous studies would suggest that duration increases on soft thick deposits with low VS30 values (e.g., Bommer et al. 2009; Dobry et al. 1978).

(a) Horizontal

100

(b) Vertical -0.03(_ +0.04)Rcd + 2.9(_ +5.3) (forearc) -0.01(_ +0.03)Rcd + 2.9(_ +5.3) (backarc)

50

Residuals (obs. - cal.)

Residuals (obs. - cal.)

-0.07(_ +0.04)Rcd + 5.9(_ +5.2) (forearc) -0.01(_ +0.03)Rcd + 5.8(_ +5.2) (backarc)

0

-50

forearc -100

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backarc 100

forearc 150

Rcd (km)

200

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backarc 100

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Rcd (km)

Fig. 6 Duration residuals (SDa5–95%[obs]−SDa5–95%[cal]) separated for forearc and backarc stations. The values in parentheses are standard errors of the mean


Residuals (log10obs. - log10cal.)

Horizontal Components

0.5 SD5-95 (Acc.) Mean + _

0.25

0

-0.25

-0.5 100

200

300

1000

Vertical Components

0.5 SD5-95 (Acc.) Mean + _

0.25

0

-0.25

-0.5 100

200

300

1000

0.5 SD5-95 (Vel.) Mean + _

0.25

0

-0.25

-0.5 100

2000




Residuals as a function of VS30 (Rcd < _ 200 km)

200

300

1000

2000

0.5 SD5-95 (Vel.) Mean + _

0.25

0

-0.25

2000

-0.5 100

200

300

VS30 (m/s)

1000

2000

VS30 (m/s)

Fig. 7 Residuals (log(significant duration)−log(predicted significant duration based on coefficients in Table 1)) based on Arias intensity (acceleration-based measure, left) and energy integral

(velocity-based measure, right) as a function of VS30. Top horizontal components and bottom vertical component

To investigate further, we look at the effect of depth-tobedrock as an additional site variable. The depth-tobedrock, defined by the depth to a layer with VS ≥ 760 m/s, or to a significant impedance contrast between surface soil deposits and material with VS ∼760 m/s, is obtained for each site from its velocity profile (Ghofrani et al. 2013). On Fig. 8, we plot the duration residuals from Eq. 8 as a function of depth-to-bedrock. For depth-

to-bedrock values from 10 to 100 m, there is a trend to increasing residuals (longer durations) with increasing depth on the horizontal component, more clearly seen in acceleration than velocity. We thus conclude that there is some evidence for increasing duration due to deeper soil deposits, more related to soil depth rather than stiffness. Finally, we examine whether spatial mapping can shed further light on the residual trends noted for the

Table 2 Variability due to site effects as a function of VS30 (Eq. 9) for both horizontal and vertical components (at Rcd ≤200 km)

For the cells with italics font, P>0.05, the coefficient is not significantly different from 0 Numbers are standard errors

Duration

Horizontal

Vertical

c3

c4

c3

c4

SDa5–95%

−0.18±0.04

0.43±0.10

−0.11±0.04

0.26±0.11

σha

σva

0.13

0.14

SDa5–75%

−0.10±0.04

0.24±0.10

−0.04±0.04

0.08±0.10

0.14

0.14

SDv5–95%

−0.06±0.03

0.15±0.07

−0.08±0.02

0.20±0.06

0.10

0.08

SDv5–75%

−0.09±0.03

0.21±0.07

−0.08±0.02

0.20±0.06

0.10

0.08

RVTa

−0.14±0.06

0.32±0.15

−0.12±0.07

0.24±0.19

0.21

0.26

RVTv

−0.06±0.06

0.11±0.15

0.04±0.05

−0.14±0.13

0.20

0.18

RMSa

−0.07±0.02

0.16±0.06

−0.06±0.03

0.13±0.06

0.08

0.09

RMSv

−0.11±0.02

0.27±0.06

−0.18±0.03

0.45±0.07

0.08

0.09


SD5-95 (Acc.) Mean + _

0.25

0

-0.25

-0.5 1



0.5

10

100

Vertical Components

0.5 SD5-95 (Acc.) Mean + _

0.25

0

-0.25

-0.5 1

10

0.5 SD5-95 (Vel.) Mean + _

0.25

0

-0.25

-0.5

1000

100

1



Residuals as a function of Depth-to-bedrock (Rcd < _ 200 km) Horizontal Components

1000

Depth-to-bedrock (m)

10

100

1000

0.5 SD5-95 (Vel.) Mean + _

0.25

0

-0.25

-0.5 1

10

100

1000

Depth-to-bedrock (m)

Fig. 8 Residuals for significant duration in log units (log(significant duration)−log(predicted significant duration based on coefficients in Table 1)) based on: Arias intensity (acceleration-based

measure, left); and energy integral (velocity-based measure, right) as a function of depth-to-bedrock. Top horizontal components and bottom vertical component

acceleration-based significant duration. Figure 9 contours these duration residuals in space. We note that the residuals are mainly clustered in three regions. Large negative duration residuals are mostly clustered in and around the Ibaraki prefecture at the largest plain in Japan, Kanto Plain, which consists of broad alluvial lowlands; this region reflects largely southward propagation paths. In this region, however, there is also a cluster of positive residuals in the Chiba, Tokyo, and Kanagawa prefectures (i.e. Tokyo bay region). The long-duration ground motions observed in these areas caused severe liquefaction effects in the Tokyo Bay area, although relatively low levels of PGA (~150–230 gal) were recorded in this area. This phenomenon could be possibly explained by basin effects; waves diffracted from the edge of the basin and direct S-waves which are propagating vertically from the basin bottom constructively interfere and elongate the time-series (e.g., Irikura et al. 1996; Field 1996; Kawase 1996; Kawase et al. 1998; Pitarka et al. 1998). The third group of residuals is largely positive and clustered in the Tohoku and Honshu regions. The largest positive residuals are in

the northern part of the studied area at Aomori and Iwate prefectures. There are also large positive residuals at Miyagi and Fukushima prefectures. The time-series and acceleration-based significant duration of four selected stations (IWT003, MYH008, IBR003, and CHB024) from different prefectures are compared in Fig. 10. Station IWT003 at the Iwate prefecture and station MYH008 at the Sendai plain recorded very complex and multiple-phase acceleration timehistories. The significant durations for these stations are ~127 and 112 s, respectively. At station IBR003, unlike the stations in Miyagi prefecture (i.e., Tohoku region), only the second phases of arrivals are predominant and the SDa5–95% decreases to ~25 s. Further toward the south, at the Chiba prefecture near Tokyo, at station CHB024, only one wave train is predominant, but the coda portion of time-series show high acceleration peaks riding over a low frequency carrier. This behavior has already been shown as an indicator of soil nonlinearity known as cyclic mobility (Iai et al. 1995; Archuleta 1998; Bonilla et al. 2005). For stations around the Tokyo bay, such as CHB024, the energy did not


Spatial Mapping of the Duration Residuals (Significant duration (SDa5-95%)) (a) Horizontal

(b) Vertical

Volcanic front

40

GSI’s Fault Plane L = 400 km W = 150 km Strike = 202 deg Dip = 18 deg Depth = 10 km

AOMH13 Aomori & Iwate prefectures

40 IWT003

Tohoku region MYG006

Iwate & Miyagi prefectures

38

38 MYGH08 MYG015

Honshu region Fukushima prefecture

GNM002

Ibaraki prefecture

IBR003

36

36

CHB024

0

Negative residuals Positive residuals

100 km

Kanto region Chiba, Tokyo & Kanagawa prefectures

140

142

140

142

Fig. 9 Spatial distribution of significant duration residuals for the observed ground motions ((a) horizontal and (b) vertical components) of KiK-net stations. Sizes of symbols are corresponding to the relative magnitude of the residuals of SDa5–95%. Negative

residuals are shown using yellow circles while positive residuals are shown using red + symbols. The white dashed lines are boundaries of major tectonic regions (i.e., Kanto, Honshu, and Tohoku) in the studied area

dissipate quickly, but lasts practically during the total duration of the record (SDa5–95% ~142 s). According to a field survey by Chiba University (Sekiguchi and Nakai 2011), liquefaction was observed very close to the location of the CHB024 station (Bonilla et al. 2011). The clustering of duration residuals could have a variety of explanations. Possible reasons for smaller-

than-average durations (negative residuals) include the rupture propagation direction; we expect durations to be shorter in the direction of rupture propagation due to forward directivity effects. Positive residuals, by contrast, could be due either to backwards directivity, or due to basin effects. Due to the complexity of the source, path and site effects that contribute to duration, it is difficult to separate these factors in any definitive way.

Author's personal copy J Seismol Fig. 10 Comparison of timeseries and acceleration-based significant duration for selected stations shown in Fig. 9. The light gray curve overlaid on each timeseries is the accumulated squared acceleration normalized by the PGA; orange dashed lines show the 5 and 95 % markers of significant duration window. The numbers in brackets are the PGA values. Second line of information at the end of the timeseries is Rcd and VS30, respectively

Acceleration (cm/s/s)

100 IWT003 (114.8 cm/s/s) 109.9 km - 228.6 m/s

0 SDa5-95%127.59 s

-100


300 MYGH08 (282.9 cm/s/s) 101.1 km - 203.4 m/s

0 SDa5-95%112.38 s


-300 1500 IBR003 (1598.0 cm/s/s) 59.6 km - 292.3 m/s

0 SDa5-95%24.83 s

-1500


200 CHB024 (232.8 cm/s/s) 93.4 km - 231.6 m/s

0 SDa5-95%148.21 s

-200 0

100

200

300

Time (s)

5 Durations of aftershocks It is known that the source component of duration will be magnitude dependent, as larger events have longer source processes. However, it is commonly assumed that the path-component of duration is independent of magnitude. To check for possible path-dependence of duration, we select four aftershocks of the Tohoku earthquake in the range of M4.5–7.7 (Table 3). The length and width of the faults for each aftershock are estimated using the Strasser et al. (2010) empirical relations. It is assumed that the reported hypocenter is located at the center of the fault plane. We obtained the other fault

parameters including focal mechanism parameters, depth, and type of faulting from the Broadband Seismic Network Laboratory (F-NET). Comparison of the calculated significant durations (5–75 % and 5–95 %), and RMS duration measures for the vertical components of the aftershocks timeseries as a function of distance is shown in Fig. 11. From visual inspection of individual time-series, we inferred that the RMS definition appears to be a good representation of the strong shaking window for the aftershocks. For the sake of clarity and brevity, RVT is not shown in Fig. 11, but is similar to SDa5–75% up to magnitude M6.5. For the largest aftershock in the dataset (M7.7),

Table 3 Aftershock parameters Date-Time

lat (°)

lon (°)

M

Depth (km)a

Strikea

Dipa

Rakea

Type of faulting

Length (km)

Width (km)

2011 March 11—15:15

36.11

141.26

7.7

35

26

59

89

Reverse

107

66

2011 March 28—07:24

38.39

142.31

6.5

20

281

67

−101

Normal

22

25

2011 March 12—15:19

39.20

142.50

5.4

32

78

31

−64

Normal

5

10

2011 March 12—10:14

37.30

141.40

4.5

29

77

28

−67

Normal

2

5

a

Focal mechanism parameters (strike, dip, and rake) and depth are from F-NET

Author's personal copy J Seismol 2011/03/12 - 10:14 (h 20km - M4.5)

2011/03/12 - 15:19 (h 10km - M5.4)

80

80 RMS SD5-75% SD5-95%

60 Duration (s)

Duration (s)

60

RMS SD5-75% SD5-95%

40

20

40

20

0

0 60

120

180 Rcd (km)

240

300

60

2011/03/28 - 07:24 (h 31km - M6.5)

180 Rcd (km)

240

300

2011/03/12 - 15:19 (h 10km - M5.4)

200

250 RMS SD5-75% SD5-95%

RMS SD5-75% SD5-95%

200 Duration (s)

150 Duration (s)

120

100

50

150

100

50

0

0 60

120

180 Rcd (km)

240

300

60

120

180 Rcd (km)

240

300

Fig. 11 Comparison of significant durations (SD5–75% and SD5–95%) with the RMS duration for vertical components as a function of distance

cumulative energy was needed to match the RMS and significant duration for the mainshock, for calculating the significant duration of aftershocks, we kept the lower marker at the more-common value of 5 %.

the RVT duration under-predicts the SDa5–75% duration significantly (by a factor of ~2). In all the selected aftershocks, the RMS duration and SDa5– 95% are in reasonable agreement. Although we observed that a wide window of 0.3 to 95 % of the

The distance-dependent duration slope (c1) and the source term (c2): T = c2 + c1.R 0.5 hypocentral distance closest distance Macias et al (2008) 2003, M8.1 Tokachi-Oki Kempton & Stewart (2006) [Rock]

0.3

100

Intercept (c2)

Slope (c1)

0.4

hypocentral distance closest distance Macias et al (2008) 2003, M8.1 Tokachi-Oki Kempton & Stewart (2006) [Rock at R = 1 km]

0.2

10

0.1

0

1 4

5

6

7

8

Moment magnitude (M)

9

4

5

6

7

8

9

Moment magnitude (M)

Fig. 12 Slope (left) and intercept (right) for RMS duration, expressed as T=c2 +c1 ·Rcd, showing the effects of magnitude on these parameters


For each aftershock, we fit the best line to the SDa5–75% durations with the general functional form of c1 ·R+c2 where R is considered to be either hypocentral distance or closest distance to fault. The slope and intercept of the fitted lines (c1 and c2), which represent the path-dependent duration term and the source term, respectively, are plotted as a function of moment magnitude (M) in Fig. 12, for both the mainshock and the aftershocks. We also added one more point (diamond symbol), taken from Macias et al. (2008), to enrich the observations for large magnitudes. This point is the duration of the 2003 M8.1 Tokachi-oki earthquake as obtained by Macias et al. (2008) from plots of shear-wave duration (up to 90 % of the signal energy) versus distance. The dependence of source duration on M is well known and expected. From Fig. 12, it can be seen that logarithm of the source duration increases linearly with M. For the distance-dependent term, the largest event has a steep slope, but only when parameterized in terms of Rcd; this could be because of the large fault plane which ruptured during the Tohoku earthquake (~400 km×~150 km). Another possibility for the large distance-dependent duration term for this event could be its multiple pulse nature. Despite the lack of large earthquakes (M≥8) in the dataset used by Kempton and Stewart (2006) to develop prediction equations for significant duration, their models are in remarkably good agreement with the results obtained in this study, as indicated in Fig. 12.

6 Conclusions In this study, we examined ground-motion duration, including how it scales with moment magnitude (M) and distance, for the Tohoku earthquake and its aftershocks. The overall conclusions can be summarized as follows: &

& &

The significant duration (5-95 %) of the M9.0 Tohoku earthquake can be expressed as 56.5(±4.6) +0.19(±0.03)Rcd, or 39.8(±4.6)+0.33(±0.03)Rcd for the horizontal and vertical components, respectively. The 5–75 % Arias intensity duration (significant duration) is approximately equivalent to RVT duration, at least for distances up to 350 km. The RMS duration is larger than significant or RVT durations for large earthquakes with multiple ruptures where the time-series consists of pulses or

&

&

&

groups of pulses of energy; in such cases the RMS duration is superior in expressed the full observed length of the strong shaking window. From the statistical analysis of residuals, it is concluded that the durations are not significantly different for forearc and backarc travel paths; the means of these two populations of residuals are not significantly different from zero. Based on the regression analysis of the significant duration as a function of site condition, it can be inferred that site stiffness has negligible effect on the vertical component duration, nor are the effects significant for the horizontal components, for either the RVT and RMS duration definitions. The site stiffness does have some effect on the significant duration for the horizontal component. There is some evidence, albeit weak, that the depth of the site influences duration. Spatial mapping of significant duration residuals clearly shows that the residuals are clustered in space, with large negative residuals mostly in and around the Kanto plain. Positive residuals are predominant at the northern stations of the studied area and also around the Tokyo Bay. At the southern part of the Kanto region (i.e., Chiba, Tokyo, and Kanagawa prefectures), time-series may be elongated due to basin effects.

Acknowledgments We thank Julian Bommer for his insightful comments that helped improve the manuscript. Ground-motion data and site information for this study were obtained from the KiK-net and K-net networks. This study was funded by the National Science and Engineering Research Council of Canada.

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Data and Resources The strong-motion records obtained by the Kyoshin network (“Knet”) and KIBAN kyoshin network (“KiK-net”) are now both reported at: http://www.kik.bosai.go.jp/ (last accessed March 2014). The focal mechanism information for the aftershocks is gathered from the Fundamental Research on Earthquakes and Earth's Interior Anomaly (F-NET: http://www.fnet.bosai.go.jp/, last accessed March 2014). The fault plane parameters of the Tohoku earthquake obtained from the GPS Earth Observation Network System (GEONET) data analysis (http://www.gsi.go.jp/, last accessed March 2014). All figures were prepared using the graphics software package CoPlot (www.cohort.com, last accessed March 2014).