Systematic ground motion observations in the
Canterbury earthquakes and region-specific
non-ergodic empirical ground motion modeling
Brendon A. Bradleya)
This paper presents an examination of ground motion observations from 20
near-source strong motion stations during the most significant 10 events in the
2010-2011 Canterbury earthquake to examine region-specific systematic effects
based on relaxing the conventional ergodic assumption. On the basis of similar
site-to-site residuals, surfical geology, and geographical proximity, 15 of the 20
stations are grouped into four sub-regions: the Central Business District; and
Western, Eastern, and Northern suburbs. Mean site-to-site residuals for these subregions then allows for the possibility of non-ergodic ground motion prediction
over these sub-regions of Canterbury, rather than only at strong motion station
locations. The ratio of the total non-ergodic vs. ergodic standard deviation is
found to be, on average, consistent with previous studies, however it is
emphasized that on a site-by-site basis the non-ergodic standard deviation can
easily vary by ±20%.
INTRODUCTION
The 2010-2011 Canterbury earthquake sequence includes the 4 September 2010 𝑀𝑤 7.1
Darfield earthquake (e.g. Gledhill et al. 2011, NZSEE 2010) and three subsequent
earthquakes of 𝑀𝑤 ≥ 5.9, most notably the 22 February 2011 M w6.2 Christchurch earthquake
that resulted in 185 fatalities (NZSEE 2011, Seismological Research Letters 2011). Ground
shaking in the Darfield earthquake resulted in widespread liquefaction in eastern
Christchurch and in isolated areas throughout the region (Cubrinovski et al. 2010), and
substantial damage to unreinforced masonry structures (Dizhur et al. 2010). The 𝑀𝑤 6.2
Christchurch earthquake caused significant damage to commercial and residential buildings
of various eras (Buchanan et al. 2011, Clifton et al. 2011, Kam et al. 2011). The severity and
spatial extent of liquefaction observed in native soils was profound, and was the dominant
cause of damage to residential houses, bridges and underground lifelines (Cubrinovski et al.
a)
University of Canterbury, Private Bag 4800, Ilam, Christchurch, New Zealand
2011). The 13 June 2011 𝑀𝑤 6.0 earthquake caused further damage to previously damaged
structures and severe liquefaction and rockfalls, and similarly for the 𝑀𝑤 5.8 and 𝑀𝑤 5.9
earthquakes on 23 December 2011.
As a result of a high-density of strong motion instruments in the Canterbury region
(Berrill et al. 2011) operated by GeoNet (www.geonet.org.nz) a significant number of high
amplitude near-source ground motion have been recorded during this sequence (e.g. Bradley
2012c, Bradley and Cubrinovski 2011). Of particular note are the large number of strong
motions which have been recorded at the same location over these multiple events. Such a
relatively unique ground motion dataset allows for the opportunity to directly examine
systematic and repeatable ground motion phenomena. Such systematic effects have been
qualitatively noted in the Canterbury ground motions (Bradley 2012b), but significant
additional insight can be gained by quantitative analysis.
One approach to examine systematic ground motion phenomena is based on the use of
empirical ground motion prediction equations (GMPEs) without the so-called ergodic
assumption (Anderson and Brune 1999). Lin et al. (2011), Walling (2009), and RodriguezMarek et al. (2011), among others, represent examples where the non-ergodic ground motion
prediction framework has been utilized to examine the reduction in the standard deviation of
pseudo-acceleration response spectral ordinates due to relaxation of the ergodic assumption.
In this paper, systematic effects in the Canterbury earthquakes are examined within the
non-ergodic empirical ground motion prediction framework. The earthquake events and
strong motion stations considered are first presented, with particular emphasis placed on the
significant ground motion amplitudes considered. The theoretical details of the non-ergodic
ground motion prediction methodology utilized are elaborated upon, and the observed results
from applying this methodology to the near-source strong motions in the Canterbury
earthquake sequence are presented, and compared with previous studies.
Finally, the
application of the developed non-ergodic modification factors are compared with ergodic
predictions and observed ground motions.
EARTHQUAKE EVENTS AND STRONG MOTION STATIONS CONSIDERED
In selecting the events to consider in the Canterbury earthquake sequence, a trade-off
naturally occurs between the number of events and the representative range of the ground
motion amplitudes produced. Considering a larger number of events provides statistically
more robust estimates (i.e. larger sample sizes). However, in order to consider more events
the minimum ground motion intensity obviously has to be reduced. As a result, the overall
dataset becomes increasingly dominated by smaller amplitude ground motions which are
arguably not of primary concern when GMPEs are utilized in probabilistic seismic hazard
analysis (PSHA) for developing design strong motions. Furthermore, there are an abundance
of studies illustrating the lack of correlation between GMPE performance for small and large
magnitude events. As a result of the above considerations, only events above magnitude
𝑀𝑤 4.5, which produced ground motions of engineering significance in the urban
Christchurch area, were considered. This resulted in a set of 10 events, the basic details of
which are given in Table 1. It is to be noted in Table 1 that a maximum source-to-site
distance for recorded ground motions was also utilized to remove those ground motions
which are of low amplitude (approximately PGA>0.01g) as discussed previously.
Table 1: Earthquake events considered
ID
1
2
3
4
5
6
7
8
9
10
Event date
Magnitude,
𝑀𝑤 1
Maximum
source-to-site
𝑚𝑎𝑥
distance, 𝑅𝑟𝑢𝑝
(km)
100
30
30
50
30
50
50
30
50
50
4 September 2010
7.1
19 October 2010
4.8
26 December 2010
4.7
22 February 2011
6.2
16 April 2011
5.0
13 June 2011 (1:01pm)
5.3
13 June 2011 (2:20pm)
6.0
21 June 2011
5.2
23 December 2011 (12:58pm)
5.8
23 December 2011 (2:18pm)
5.9
1
Moment magnitudes obtained from GeoNet (www.geonet.org.nz) regional Centroid Moment Tensor
(CMT) solutions (Ristau 2008).
For the 4 September 2010, 22 February 2011, 13 June 2011, and 23 December 2011 (i.e.
the four largest events) the finite fault inversion studies by Beavan et al. (2011, 2012) were
adopted for defining fault source geometry while moment magnitude, 𝑀𝑤 , was obtained from
regional centroid moment tensor solutions (www.geonet.org.nz; last accessed June 2012),
based on Ristau (2008). For the remaining six smaller events, regional centroid moment
tensor solutions were used to provide the event 𝑀𝑤 , and a finite fault model was developed
assuming the centroid location to be at the center of the finite fault plane along-strike and
down-dip, and using the appropriate magnitude scaling relationship for New Zealand (NZ)
(Stirling et al. 2012) for defining along-strike length and down-dip width. Figure 1 illustrates
the study region and the location of the finite fault models of the 10 different earthquake
events considered.
A total of 20 strong motion stations were considered in the Christchurch region for the
purposes of examining systematic site effects. Ground motion processing was as discussed in
Bradley (2012c) producing useable spectral ordinates from T=0.01-10s. Figure 1 illustrates
the locations of the 20 stations relative to the seismic sources considered, while Table 2
provides a list of the stations and the values of geometric mean PGA which were recorded for
each of the 10 events. Figure 2 provides a histogram of the geometric mean PGA values in
Table 2. It can be clearly seen that the dataset is comprised of a significant number of high
intensity ground motions with values in the range of 0.01-1.41g, and a mean value of 0.183g.
For each strong motion record, PGA and elastic pseudo-acceleration response spectral
ordinates (SA) for vibration periods from 0.01-10s are computed and considered in the
analyses to follow.
Figure 1: Location of the finite fault planes of the 10 considered earthquake events, and the location of
the 20 strong motions at which systematic site effects are examined. Color coding of the finite fault
models is for clarity only.
Table 2: Strong motion stations for which systematic site effects are considered and the observed geometric mean PGA values in the 10 considered events
Num events,
Observed geometric mean peak ground acceleration in each event, 𝑃𝐺𝐴 (g)
𝑁𝐸𝑠
Station
1
2
3
4
5
6
7
8
9
10
CACS
D
10
0.197 0.027 0.020
0.21
0.034 0.081 0.136 0.104 0.073 0.083
CBGS
D
10
0.158 0.069 0.270
0.50
0.070 0.183 0.163 0.077 0.157 0.210
CCCC
D
6
0.224 0.119 0.227
0.43
0.134 0.179
CHHC
D
10
0.173 0.089 0.162
0.37
0.146 0.199 0.215 0.115 0.174 0.222
CMHS
D
9
0.237 0.191 0.132
0.37
0.137 0.159 0.178
0.152 0.174
HPSC
E
10
0.147 0.041 0.049
0.22
0.148 0.180 0.256 0.068 0.199 0.264
HVSC
C
10
0.606 0.091 0.111
1.41
0.676 0.455 0.914 0.264 0.306 0.439
KPOC
D
8
0.339 0.013 0.012
0.20
0.052 0.186 0.099 0.067
LINC
D
10
0.437 0.034 0.020
0.12
0.028 0.026 0.065 0.114 0.062 0.073
LPCC
D
9
0.290 0.025 0.018
0.92
0.294 0.146 0.639 0.068
0.437
NBLC
D
6
0.025
0.129 0.232 0.214 0.040 0.201
NNBS
E
8
0.206 0.042 0.039
0.67
0.156 0.239 0.198 0.070
PPHS
D
10
0.221 0.048 0.091
0.21
0.062 0.118 0.122 0.074 0.116 0.138
PRPC
E
9
0.214 0.054 0.087
0.63
0.223 0.299 0.341 0.089 0.290
REHS
D
10
0.252 0.081 0.245
0.52
0.101 0.188 0.264 0.086 0.204 0.254
RHSC
D
9
0.210 0.282
0.28
0.075 0.083 0.194 0.202 0.159 0.159
ROLC
D
10
0.340 0.013 0.022
0.18
0.013 0.036 0.045 0.111 0.102 0.062
SHLC
D
10
0.175 0.072 0.156
0.33
0.116 0.245 0.184 0.076 0.262 0.275
SMTC
D
10
0.176 0.020 0.034
0.16
0.034 0.132 0.085 0.078 0.066 0.148
TPLC
D
10
0.266 0.058 0.032
0.11
0.024 0.037 0.065 0.250 0.068 0.081
1
As defined by the NZ Loadings Standard, NZS1170.5 (2004), i.e. B=rock; C=shallow soil; D=deep or soft soil; E=very soft soil.
Site class1
25
Frequency
20
15
10
5
0
-2
-1
0
10
10
10
Geometric mean peak ground acceleration, PGA (g)
Figure 2: Histogram of the geometric mean PGA values observed at the 20 stations during the 10
earthquake events.
For each event, all strong motion stations within the event-specific maximum distance,
𝑚𝑎𝑥
𝑅𝑟𝑢𝑝
, (Table 1) were used to compute the between-event residuals with respect to empirical
ground motion prediction equations (GMPEs). In some events, for example the 4 September
2010 event, this included several other strong motion stations in addition to those in Table 2,
however in the majority of the smaller 𝑀𝑤 events, only these 20 strong motion stations were
considered.
COMPARISON OF NZ-SPECIFIC GMPE WITH THE CANTERBURY EARTHQUAKES
The representation of (pseudo) spectral acceleration (SA), from event 𝑒, at a single
location s, for the purposes of ground motion prediction, is generally given by:
𝑙𝑛𝑆𝐴𝑒𝑠 = 𝑓𝑒𝑠 (𝑆𝑖𝑡𝑒, 𝑅𝑢𝑝) + 𝛿𝐵𝑒 + 𝛿𝑊𝑒𝑠
(1)
where 𝑙𝑛𝑆𝐴𝑒𝑠 is the (natural) logarithm of the observed SA; 𝑓𝑒𝑠 (𝑆𝑖𝑡𝑒, 𝑅𝑢𝑝) is the median of
the predicted logarithm of SA as given by an empirical GMPE, which is a function of the site
and earthquake rupture considered; 𝛿𝐵𝑒 is the between-event (or inter-event) residual with
zero mean and variance 𝜏 2 ; and 𝛿𝑊𝑒𝑠 is the within-event (or intra-event) residual with zero
mean and variance 𝜙 2 . Based on equation (1), empirical ground motion prediction equations
can provide the distribution of SA as:
𝑙𝑛𝑆𝐴𝑒𝑠 ~𝑁(𝑓𝑒𝑠 , 𝜏 2 + 𝜙 2 )
(2)
where 𝑋~𝑁(𝜇𝑋 , 𝜎𝑋2 ) is short-hand notation for 𝑋 having a normal distribution with mean 𝜇𝑋
and variance 𝜎𝑋2 .
Figure 3 illustrates the PGA observations for the 10 considered events as compared to the
NZ-specific site class D prediction of Bradley (2010, 2013a) (herein “Bradley (2010)”). It is
noted that none of the ground motions from the Canterbury earthquake sequence were used in
the development of the Bradley (2010) GMPE.
Comparisons between predicted and
observed SA ordinates at vibration periods of 0.2, 0.5, 1.0, 1.5, and 3.0 seconds can be found
in Bradley (2013b, Appendix 1). On the basis of Figure 3 it can be seen that the Bradley
(2010) model: (i) exhibits good scaling of PGA amplitudes with source-to-site distance, 𝑅𝑟𝑢𝑝 ,
even for very near-source distances; and (ii) provides a consistent prediction of ground
motions from events of different magnitudes. The same trends for PGA are apparent for SA
at various vibration periods (Bradley 2013b, Appendix 1). In addition, as shown in Figure 18
of Bradley (2013a), the within-event residuals are independent of distance with a standard
deviation that is consistent with the model standard deviation.
It is also worth noting that several large events occurred in quick succession in the
Canterbury earthquake sequence. For example, on 13 June 2011 and 23 December 2011 two
large events (𝑀𝑤 5.3/6.0 and 𝑀𝑤 5.8/5.9, respectively) occurred approximately 80mins apart
(see times in Table 1). Both of these events occurred near the east of Christchurch, which has
soft surficial soil deposits. As a result, it is speculated that the ground motions recorded in
the latter event of these two sequences were affected by the surficial soils having elevated
pore water pressures from the strong shaking in the earlier event. Such speculation would be
expected to result in a general over prediction of short period SA amplitudes and an under
prediction of long period SA amplitudes because of the reduced stiffness of the soft surficial
soils due to elevated pore pressures, which is somewhat evident in the PGA observations in
Figure 3, as discussed in greater detail by Bradley (2013b).
Despite this postulated
phenomena, no explicit account of such effects is considered in the subsequent empiricallybased analyses.
The above results and discussion implies that the Bradley (2010) model provides,
generally-speaking, an unbiased prediction of the 10 considered events as a function of
various model parameters. However, Figure 3 also alludes to the significant variability in the
observations for a given set of model parameters. A large portion of this variability arises
due to systematic features of the specific sites at which the ground motions were recorded;
features which are not adequately captured in the simple parameterization used in GMPEs.
The subsequent sections therefore examine the between- and within-event residuals obtained
for the Bradley (2010) GMPE in order to ascertain these site specific systematic effects.
B
C
D
E
Peak ground acceleration, PGA (g)
Bradley (2010)
Be/=-1.214
-1
10
B
C
D
E
-2
10
0
Bradley (2010)
B /=1.31
16 April 2011
e
-1
10
B
C
D
E
-2
0
10
0
10
-1
10
B
C
D
E
-2
0
10
1
23 December 2011 (12:58pm)
Bradley (2010)
Be/=-0.299
-1
10
B
C
D
E
-2
10
0
10
1
10
Source-to-site distance, Rrup (km)
0
1
10
Source-to-site distance, Rrup (km)
Bradley (2010)
Be/=-0.217
22 February 2011
0
10
-1
10
B
C
D
E
-2
10
0
1
10
Source-to-site distance, Rrup (km)
0
10
Bradley (2010)
Be/=-0.108
13 June 2011 (1:01pm)
-1
10
B
C
D
E
-2
10
0
10
10
Source-to-site distance, Rrup (km)
0
10
-2
10
1
Bradley (2010)
Be/=-0.579
B
C
D
E
10
10
Source-to-site distance, Rrup (km)
13 June 2011 (2:20pm)
10
-1
10
10
Source-to-site distance, Rrup (km)
0
Bradley (2010)
Be/=-0.434
19 October 2010
1
10
10
0
10
10
Peak ground acceleration, PGA (g)
Peak ground acceleration, PGA (g)
0
26 December 2010
2
10
10
10
Peak ground acceleration, PGA (g)
1
10
Source-to-site distance, Rrup (km)
Peak ground acceleration, PGA (g)
0
Peak ground acceleration, PGA (g)
-2
10
Peak ground acceleration, PGA (g)
-1
10
1
10
Source-to-site distance, Rrup (km)
0
10
21 June 2011
Bradley (2010)
Be/=-0.124
-1
10
B
C
D
E
-2
10
0
10
Peak ground acceleration, PGA (g)
Peak ground acceleration, PGA (g)
10
10
Peak ground acceleration, PGA (g)
Bradley (2010)
Be/=-0.092
4 September 2010
0
1
10
Source-to-site distance, Rrup (km)
0
10
23 December 2011 (2:18pm)
Bradley (2010)
Be/=-0.492
-1
10
B
C
D
E
-2
10
0
10
1
10
Source-to-site distance, Rrup (km)
Figure 3: Comparison of the Bradley (2010) GMPE (for site class D) with PGA observations from the 10 considered events
including values of the normalized between-event residual, 𝜹𝑩𝒆 /𝝉.
NON-ERGODIC GROUND MOTION PREDICTION
The Bradley (2010) GMPE, and GMPEs in general, is based on the ergodic assumption,
i.e. the time-averaged behavior of a random process at a given location is the same as the
space-averaged behavior at given instants in time.
Practically speaking, the ergodic
assumption is invoked when ground motion records from different locations around the globe
are combined without discrimination for the purposes of ground motion prediction at a single
location. Several studies have illustrated that the ergodic assumption generally leads to an
over-prediction of ground motion uncertainties because it combines variability in source, path
and site effects from different tectonic regions and sites, and some of this variability may in
fact be systematic in a site-specific context (Anderson and Brune 1999).
The ergodic
assumption can be relaxed by considering that the between- and within-event residuals, given
in Equation (1), are no longer purely random variables with zero-mean, but systematically
depart from this for a given earthquake source (or source region) and given site of interest.
The contemporary non-ergodic methodology for considering such effects employed here
has also been adopted by Lin et al. (2011), Walling (2009), and Rodriguez-Marek et al.
(2011), among others, and notational convention herein generally follows Al Atik et al.
(2010).
Firstly, the between-event residual, 𝛿𝐵𝑒 , is separated into a systematic event
location-to-location (herein simply ‘location-to-location’) residual (for location l), 𝛿𝐿2𝐿𝑙 , and
0
0
a ‘remaining’ between-event residual, 𝛿𝐵𝑒𝑙
(i.e. 𝛿𝐵𝑒 = 𝛿𝐿2𝐿𝑙 + 𝛿𝐵𝑒𝑙
). Secondly, the within-
event residual, 𝛿𝑊𝑒𝑠 , is separated into a systematic ‘site-to-site’ (sometimes also referred to
as ‘station-to-station’ (Rodriguez-Marek et al. 2011)) residual (for site s), 𝛿𝑆2𝑆𝑠 , and a
‘remaining’ within-event residual, 𝛿𝑊𝑒𝑠0 (i.e. 𝛿𝑊𝑒𝑠 = 𝛿𝑆2𝑆𝑠 + 𝛿𝑊𝑒𝑠0 ). As a result, Equation
(1) can be re-written as:
0
𝑙𝑛𝑆𝐴𝑒𝑠 = 𝑓𝑒𝑠 (𝑆𝑖𝑡𝑒, 𝑅𝑢𝑝) + (𝛿𝐿2𝐿𝑙 + 𝛿𝐵𝑒𝑙
) + (𝛿𝑆2𝑆𝑠 + 𝛿𝑊𝑒𝑠0 )
(3)
It should be noted that no consideration here is given to a path-specific effect in the
within-event residual, as done so by Lin et al. (2011), for example. The principal reason for
this omission is that only near-source recordings from moderate-to-large magnitude
earthquakes are considered. As a result, ray paths from different sub-faults in the idealized
rupture plane can be quite different. This makes determination of the specific ray path to be
used in consideration of spatial correlation and path-specific effects non-unique.
Furthermore, since only events in the relatively small geographic region of Canterbury are
considered, any systematic path effects are likely to be simply represented in the systematic
location-to-location residual. Finally, it is noted that all 10 earthquake events considered
here are taken to be from the same ‘Canterbury’ region so that all events are considered to
have the same location-to-location residual, 𝛿𝐿2𝐿𝑙 . Subsequent analyses illustrate that this
residual is not a function of event magnitude.
BETWEEN-EVENT RESIDUAL AND ITS COMPONENTS
As noted in Equation (3), the between-event residual, 𝛿𝐵𝑒 , is considered to be comprised
of two parts. The systematic location-to-location residual can be computed as the mean value
of 𝛿𝐵𝑒 from all the events considered:
𝑁𝐸
(4)
1
𝛿𝐿2𝐿𝑙 =
∑ 𝛿𝐵𝑒
𝑁𝐸
𝑒=1
where 𝑁𝐸 is the number of events (i.e. 𝑁𝐸 = 10 in this study).
For each event the
‘remaining’ portion of the between-event residual, 𝛿𝐵𝑒0, can then be computed from:
0
𝛿𝐵𝑒𝑙
= 𝛿𝐵𝑒 − 𝛿𝐿2𝐿𝑙
(5)
0
By definition, since 𝛿𝐿2𝐿𝑙 is the mean of 𝛿𝐵𝑒 , 𝛿𝐵𝑒𝑙
has zero mean. In addition to their
0
mean values, both 𝛿𝐿2𝐿𝑙 and 𝛿𝐵𝑒𝑙
are uncertain. The uncertainty in 𝛿𝐿2𝐿𝑙 results from the
fact that it is computed from a finite number of events, and hence its variance can be
computed from:
2
𝑉𝑎𝑟[𝛿𝐿2𝐿𝑙 ] = 𝜏𝐿2𝐿
=
𝑙
𝜏̂ 2
𝑁𝐸
(6)
where 𝜏̂ 2 is the sample variance of the between-event residuals, 𝛿𝐵𝑒 . The variance in the
‘remaining’ between-event residual can be computed simply from statistical inference of the
0
values of 𝛿𝐵𝑒𝑙
:
𝑁𝐸
2
𝜏𝑙0
=
0
]
𝑉𝑎𝑟[𝛿𝐵𝑒𝑙
1
0 2
)
=
∑(𝛿𝐵𝑒𝑙
𝑁𝐸 − 1
𝑒=1
(7)
WITHIN-EVENT RESIDUAL AND ITS COMPONENTS
As noted in Equation (3), the within-event residual, 𝛿𝑊𝑒𝑠 , is considered to be comprised
of two parts. The systematic site-to-site residual can be computed for each site (i.e. strong
motion station) as the mean value of 𝛿𝑊𝑒𝑠 from all the events considered:
𝑁𝐸𝑠
1
𝛿𝑆2𝑆𝑠 =
∑ 𝛿𝑊𝑒𝑠
𝑁𝐸𝑠
(8)
𝑒=1
where 𝑁𝐸𝑠 is the number of events at site 𝑠. Since not all events are recorded at the same set
of locations then 𝑁𝐸𝑠 ≤ 𝑁𝐸, and the value of 𝑁𝐸𝑠 for each site is given in Table 2. Once
𝛿𝑆2𝑆𝑠 has been computed, the ‘remaining’ within-event residual can be computed as:
𝛿𝑊𝑒𝑠0 = 𝛿𝑊𝑒𝑠 − 𝛿𝑆2𝑆𝑠
(9)
By definition, since 𝛿𝑆2𝑆𝑠 is the mean of 𝛿𝑊𝑒𝑠 , 𝛿𝑊𝑒𝑠0 has zero mean. In addition to their
mean values, both 𝛿𝑆2𝑆𝑠 and 𝛿𝑊𝑒𝑠0 are uncertain. The uncertainty in 𝛿𝑆2𝑆𝑠 results from the
fact that it is computed from a finite number of events, and hence its variance can be
computed from:
𝑉𝑎𝑟[𝛿𝑆2𝑆𝑠 ] =
2
𝜙𝑆2𝑆𝑠
𝜙̂ 2
=
𝑁𝐸𝑠
(10)
where 𝜙̂ 2 is the sample variance of the within-event residuals, 𝛿𝑊𝑒𝑠 . The variance in the
‘remaining’ within-event residual can be computed simply from statistical inference of the
values of 𝛿𝑊𝑒𝑠0 :
𝑁𝐸𝑠
2
𝜙𝑠0
1
= 𝑉𝑎𝑟[𝛿𝑊𝑒𝑠0 ] =
∑(𝛿𝑊𝑒𝑠0 )2
𝑁𝐸𝑠 − 1
(11)
𝑒=1
NON-ERGODIC PREDICTION
Having characterized the components of the between- and within-event residuals and
their sub-components in the previous section it is now possible to obtain the mean and
variance of the non-ergodic GMPE. The mean value of 𝑙𝑛𝑆𝐴𝑒𝑠 is given by:
𝐸[𝑙𝑛𝑆𝐴𝑒𝑠 ] = 𝑓𝑒𝑠 + 𝛿𝐿2𝐿𝑙 + 𝛿𝑆2𝑆𝑠
(12)
0
] = 𝐸[𝛿𝑊𝑒𝑠0 ] = 0. Note that, in general, the index l in 𝛿𝐿2𝐿𝑙 is obtained based
since 𝐸[𝛿𝐵𝑒𝑙
on the geographic location of site s and therefore 𝑆𝐴𝑒𝑠 and 𝑓𝑒𝑠 omit this index for brevity.
Because 𝑆𝐴𝑒𝑠 has a lognormal distribution then it follows that the median value of 𝑆𝐴𝑒𝑠 can
be obtained as the exponential of the mean of 𝑙𝑛𝑆𝐴𝑒𝑠 :
𝑀𝑒𝑑𝑖𝑎𝑛[𝑆𝐴𝑒𝑠 ] = 𝑒𝑥𝑝{𝐸[𝑙𝑛𝑆𝐴𝑒𝑠 ]}
(13)
In examining the systematic effects, 𝛿𝐿2𝐿𝑙 and 𝛿𝑆2𝑆𝑠 , it is useful to consider a
‘systematic amplification factor’ which is the ratio of the non-ergodic to ergodic median
GMPE predictions. Taking the ratio of Equation (13) and Equation (2) gives:
𝑀𝑒𝑑𝑖𝑎𝑛[𝑆𝐴𝑒𝑠 ]𝑛𝑜𝑛𝑒𝑟𝑔𝑜𝑑𝑖𝑐
= 𝑒𝑥𝑝⁡(𝛿𝐿2𝐿𝑙 + 𝛿𝑆2𝑆𝑠 )
𝑀𝑒𝑑𝑖𝑎𝑛[𝑆𝐴𝑒𝑠 ]𝑒𝑟𝑔𝑜𝑑𝑖𝑐
(14)
In terms of the prediction variance, making the conventional assumption that the different
residuals are uncorrelated, the variance of the non-ergodic prediction can be obtained as:
2
2
2
2
𝑉𝑎𝑟[𝑙𝑛𝑆𝐴𝑒𝑠 ] = (𝜏𝐿2𝐿
+ 𝜏𝑙0
) + (𝜙𝑆2𝑆𝑠
+ 𝜙s0
)
𝑙
(15)
For each of the between-event, within-event, and total residuals, a standard deviation
reduction factor can be computed for the ratio of the non-ergodic and ergodic standard
deviations:
2
2
𝜏𝐿2𝐿
+ 𝜏𝑙0
𝑙
√
𝑅𝐹𝜏 =
𝜏2
𝑅𝐹𝜙 = √
2
2
𝜙𝑆2𝑆𝑠
+ 𝜙𝑠0
𝜙2
2
2
2
2 )
(𝜏𝐿2𝐿
+ 𝜏𝑙0
) + (𝜙𝑆2𝑆𝑠
+ 𝜙𝑠0
𝑙
√
𝑅𝐹𝜎𝑇 =
𝜏 2 + 𝜙2
(16)
(17)
(18)
OBSERVED SYSTEMATIC EFFECTS FROM THE CANTERBURY
EARTHQUAKES
BETWEEN-EVENT RESIDUAL
Figure 4 illustrates the computed values for 𝛿𝐵𝑒 as a function of SA vibration period for
the 10 events considered (specific results for PGA are not shown herein as they are
numerically equivalent to those for SA(0.01)). Also shown in Figure 4 is the systematic
location-to-location residual, 𝛿𝐿2𝐿. It can be seen that for short vibration periods (𝑇~ <
0.3𝑠) the value of 𝛿𝐿2𝐿 is approximately zero, illustrating that the Bradley (2010) GMPE is,
on average, unbiased for these short vibration periods, across the events and strong motion
stations considered. However, as the vibration period increases the value of 𝛿𝐿2𝐿 increases.
Bradley (2012c) and Bradley and Cubrinovski (2011) have suggested that greater than
predicted SA amplitudes at long periods in the 4 September 2010 and 22 February 2011
events, respectively, could be the result of: (i) near-source forward directivity; (ii) nonlinear
response of soft surficial soils; (iii) basin-induced surface waves; and (iv) inherent model bias
as a result of a limited amount of reliable ground motion records at long vibration periods.
While all these points are plausible on a single ground motion observation by observation
basis, the observations in Figure 4 are based on sites in the Canterbury region located at
various azimuths from 10 different earthquake events. Firstly, forward directivity rupture
effects would not systematically affect sites at the range of azimuths considered, and such
effects would not be significant for smaller magnitude events. Secondly, as the majority of
the stations considered are located on the Canterbury alluvial deposits, nonlinear response of
surficial soils may be of importance, since only ground motions from moderate-to-large
magnitude earthquakes at close distances were considered (e.g. the average PGA of the
considered motions 0.183g (i.e. Figure 2)), and also basin-induced surface waves are likely of
importance. Finally, while inherent model bias is a possibility for very long periods (i.e. 𝑇 >
5𝑠), it is unlikely at shorter periods (i.e. T=1s), and therefore this is not considered as a
significant factor in the observed departure from zero in Figure 4.
Between-event residual,  Be
1.5
1
Single event
Mean for all events, L2L
0.5
0
-0.5 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
Figure 4: Computed between-event residuals, 𝜹𝑩𝒆 , and the location-to-location residual, 𝜹𝑳𝟐𝑳.
The dependence of the between-event residuals as a function of event magnitude for five
different vibration periods is illustrated in Figure 5. It can be seen that there is no apparent
(i.e. statistically significant) trends in 𝛿𝐵𝑒 as a function of event magnitude. The only
noteworthy observation is the relatively large 𝛿𝐵𝑒 values from the 𝑀𝑤 5.0 event on 16 April
2011. Without further investigation, the reason for this is not immediately apparent. Hence,
for a given vibration period, the assumption that the location-to-location residual is a constant
(i.e. is not dependent on event magnitude) is reasonable.
Between-event residual,  Be
1.0
0.5
0.0
PGA
SA(0.2)
SA(0.5)
SA(2.0)
SA(5.0)
-0.5
-1.0
4.5
5.0
5.5
6.0
6.5
Magnitude, Mw
7.0
7.5
Figure 5: Variation in between-event residuals with magnitude for five different vibration periods.
WITHIN-EVENT RESIDUALS
By considering the within-event residuals at each strong motion station site during the 10
events, systematic site effects can be ascertained via Equation (8).
Bradley (2013b,
Appendix 1) provides the within-event residuals as a function of vibration period for all 20
strong motion station sites considered, and because of space limitations only a subset are
presented here.
Figure 6a-c illustrate, for example, the within-event residuals and the
systematic site effect at CBGS, LINC, and HVSC, respectively. It can be seen in Figure 6a
that the ground motions observed at the CBGS (Canterbury Botanic Gardens) site is on
average similar to that predicted by the median of the Bradley (2010) model, with 𝛿𝑆2𝑆𝑠
being close to zero for all vibration periods. Figure 6b illustrates that LINC (Lincoln)
generally has ground motion observations which are lower than that predicted by the Bradley
(2010) model, with 𝛿𝑆2𝑆𝑠 generally slightly less than zero. Finally, Figure 6c illustrates that
HVSC (Heathcote Valley) generally has ground motion observations which are significantly
greater than predicted at short periods and less than predicted at long periods, consistent with
the identified basin-edge effects at this location (Bradley 2012a).
Figure 6a-c illustrate that different strong motion stations have systematic site specific
residuals, 𝛿𝑆2𝑆𝑠 , which depart from zero. Figure 6d illustrates these 𝛿𝑆2𝑆𝑠 residuals for all
20 strong motion stations considered, as well as the median, 16th and 84th percentiles of these
𝛿𝑆2𝑆𝑠 values. It can be seen that, on average, the values of 𝛿𝑆2𝑆𝑠 for the 20 strong motion
stations considered are very close to zero. This is largely expected, given that the betweenevent residual is used to give a within-event residual which is random with approximately
zero mean (Lindstrom and Bates 1990). However, Figure 6d serves to clearly illustrate that
while the Bradley (2010) model employing the ergodic assumption is generally unbiased, a
significant amount of the observed variability is the result of systematic site effects.
2
Individual record
1.5
1
0.5
0
-0.5
-1
Mean for all records
S2Ss
-1.5
-2 -2
10
-1
0
10
10
Vibration period, T (s)
10
2
Station: HVSC
1.5
1
0.5
0
-0.5
-1
-1.5
-2 -2
10
2
Station: LINC
1.5
1
0.5
0
-0.5
-1
-1.5
-2 -2
10
1
(d)
Site-specific residual,  S2Ss
Within-event residual,  W es
(c)
(b)
Station: CBGS
Within-event residual,  W es
Within-event residual,  W es
(a)
-1
0
10
10
Vibration period, T (s)
10
1
1
All stations
0.5
0
-0.5
-1 -2
-1
0
1
-1
0
10
10
10
10
10
10
Vibration period, T (s)
Vibration period, T (s)
Figure 6: Within-event residuals, and the site specific effect, 𝜹𝑺𝟐𝑺, at:CBGS (Christchurch Botanical
Gardens); LINC (Lincoln); HVSC (Heathcote Valley); and (d) site specific effect, 𝜹𝑺𝟐𝑺, for all stations
considered.
10
1
The dependence of the within-event residuals on ground motion intensity is illustrated in
Figure 7 based on the PGA of each ground motion. The dependence based on spectral
acceleration ordinates at each vibration period provided similar insight, and therefore is not
explicitly presented here. 18 of the 20 stations considered illustrated trends similar to Figure
7a and Figure 7b, where no apparent trend of 𝛿𝑊𝑒𝑠 with PGA is evident. In contrast, Figure
7c and Figure 7d illustrate that at LPCC and KPOC there is a clear trend of high-intensity
ground motions resulting in larger 𝛿𝑊𝑒𝑠 residuals. For LPCC, this trend is seemingly
apparent across all vibration periods, while for KPOC this trend is only apparent at short-tomoderate vibration periods (i.e. 𝑇 ≤ 1𝑠). These results for 18 out of 20 stations (such as
shown in Figure 7a Figure 7b) demonstrate that, in general, the consideration of nonlinear
soil effects in the Bradley (2010) GMPE (which is based on Chiou and Youngs (2008)) is
1.5
(b)
Within-event residual,  W es
0.15
0.2
Station: CACS
0.5
0
-0.5
-1
S2S
-1.5 -2
10
(c)
0.1
PGA (g)
Within-event residual,  W es
0.05
1
-1
0
10
10
Vibration period, T (s)
10
(b)
2
0.2
0.4
PGA (g)
1.5
0.6
Station: LPCC
0.5
0
-0.5
-1
-1.5
S2S
-1
0.1
0
10
10
Vibration period, T (s)
0.15
0.2
1
0.25
PGA (g)
0.3
0.35
Station: CHHC
0.5
0
S2S
-0.5
-1
0
10
10
Vibration period, T (s)
10
1
2
0.8
1
-2 -2
10
1.5
-1 -2
10
1
Within-event residual,  W es
Within-event residual,  W es
(a)
1.5
0.05
0.1
0.15
0.2
PGA (g)
0.25
0.3
Station: KPOC
1
0.5
0
-0.5
S2S
10
1
-1 -2
10
-1
0
10
10
Vibration period, T (s)
Figure 7: Ground motion intensity dependence of within-event residuals at: (a) CACS (Canterbury Aero
club); (b) CHHC (Christchurch hospital); (c) LPCC (Lyttelton Port); and (d) KPOC (Kaiapoi North
school).
10
1
consistent with the observed strong ground motions. Hence, for 18 of the 20 stations
considered, the assumption that the within-event residual, 𝛿𝑊𝑒𝑠 , is constant for a given
vibration period is adequate. For the LPCC and KPOC stations 𝛿𝑊𝑒𝑠 should be considered
explicitly as a function of the ground motion intensity for the purpose of non-ergodic ground
motion prediction. It is noted that the intensity-dependence of 𝛿𝑊𝑒𝑠 at LPCC and KPOC
may be indicative that the site classification of these sites (i.e site classes B and E, as given in
Table 2) provides a poor prediction of their true site response for the range of ground motion
intensities examined.
SYSTEMATIC
MEDIAN
AMPLIFICATION
FOR
SPECIFIC
SUB-REGIONS
IN
CHRISTCHURCH
The location-to-location and site-to-site residuals (𝛿𝐿2𝐿𝑙 and 𝛿𝑆2𝑆𝑠 , respectively)
presented in the previous sections, in combination with Equation (12), allows for non-ergodic
site-specific prediction of ground motions at the 20 strong motion stations from earthquakes
in the Canterbury region. For the purposes of developing design ground motions for the
Christchurch region it also desirable to develop predictions which can be utilized over broad
sub-regions of Christchurch, rather than only at a specific site location.
This section
examines the 𝛿𝑆2𝑆𝑠 residuals in order to develop such sub-region predictions.
Systematic site-to-site residuals for various sub-regions of Christchurch
Based on the examination of the 𝛿𝑆2𝑆𝑠 residuals as a function of vibration period,
surfical geology, geographical proximity, and the authors judgment, it was considered
reasonable to group 15 of the 20 considered stations into four different subgroups: (i) Central
Business District (stations CBGS, CCCC, CHHC, REHS; (ii) ‘Western suburbs’ (stations
CACS, TPLC, ROLC, LINC); (iii) ‘Eastern suburbs’ (stations SHLC, PRPC, HPSC, NNBS,
KPOC); and (iv) ‘Northern suburbs’ (stations PPHS and SMTC). The remaining 5 stations
did not exhibit characteristics which, in the author’s opinion, allowed them to be grouped
easily, and are discussed subsequently.
Figure 8 illustrates the site-specific residuals for various stations by sub-region, as well as
the mean residual for each sub-region. Figure 8a illustrates that the four sites in the central
business district (CBD) have similar 𝛿𝑆2𝑆𝑠 residuals with vibration period, with residuals of
approximately zero for vibration periods less than 0.3s, then increasing over the range of
T=0.3-4.0 s. The value of the residuals over this range are similar for three sites (CBGS,
CCCC, CHHC), but the residuals for the REHS station between vibration periods of T=0.42.0s are notably larger, inferred to be the result of several meters of peat deposits at this
location (Canterbury Geotechnical Database 2011).
Figure 8b illustrates the site-specific residuals at four locations on the western extent of
the Christchurch urban area, in which the surficial soils are comprised primarily of gravelly
deposits (Brown and Weeber 1992, Cubrinovski and McCahon 2011). In contrast to the
𝛿𝑆2𝑆𝑠 residuals in the CBD region (Figure 8a) it can be seen that the site-to-site residuals at
these locations lie typically in the range of -0.5 to 0.0, except for T>4s, where the effects of
the deep sedimentary basin generally lead to 𝛿𝑆2𝑆𝑠 >0.
1
REHS
0.5
Mean
0
-0.5
-1 -2
10
(c)
(b)
Central Business District
(CBGS, CCCC, CHHC, REHS)
Site-specific residual,  S2Ss
Site-specific residual,  S2Ss
(a)
-1
0
10
10
Vibration period, T (s)
10
(d)
Site-specific residual,  S2Ss
Site-specific residual,  S2Ss
Eastern suburbs
(SHLC, PRPC, HPSC, NNBS, KPOC)
KPOC
0.5
0
Mean
-0.5
-1 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
Western suburbs
(CACS, TPLC, ROLC, LINC)
0.5
Mean
0
-0.5
-1 -2
10
1
1
1
1
-1
0
10
10
Vibration period, T (s)
10
1
Northern suburbs
(PPHS, SMTC)
0.5
Mean
0
-0.5
-1 -2
10
-1
0
10
10
Vibration period, T (s)
Figure 8: Site-specific residuals for various sub-regions: (a) Central Business District(CBD); (b) ‘Western
Suburbs’; (c) ‘Eastern Suburbs’; and (d) ‘Northern Suburbs’.
Figure 8c illustrates the site-to-site residuals in the eastern suburbs of urban Christchurch,
as well as Kaiapoi to the north of urban Christchurch (Figure 1). While there is a slightly
greater variability among the 𝛿𝑆2𝑆𝑠 residuals across these five stations than for the
10
1
aforementioned results in Figure 8a and Figure 8b, it can be seen that residuals, on average,
have a mean value close to zero across the range of vibration periods. The large 𝛿𝑆2𝑆𝑠
residual for the KPOC station for T>4s is the result of the deep sediments in the Pegasus
basin, which results in strong wave-guide effects, as clearly observed in the 4 September
2010 Darfield earthquake (Bradley 2012c) which contributed to large liquefaction in various
areas of Kaiapoi (Cubrinovski et al. 2010) despite a relatively large distance from the rupture
source.
Figure 8d illustrates the site-to-site residuals of the PPHS and SMTC stations located in
the northern suburbs of urban Christchurch. For T<0.2s it can be seen that these sites exhibit
average 𝛿𝑆2𝑆𝑠 residuals in the range of -0.5 to -0.3. Between T=0.2-0.7s the residuals, on
average, increase from approximately -0.5 to 0.2; and for T>0.7s the residuals vary, on
average, between 0.1 and 0.3.
Locations which don’t conform to sub-region categories
The RHSC (Riccarton High School) station is a relatively standard site class D site, but
with relatively strong SA amplitudes in the range 𝑇 = 0.2 − 0.4𝑠, which is approximately the
natural period of the site’s shallow surficial soils. The CMHS (Cashmere High School),
HVSC (Heathcote Valley), and LPCC (Lyttelton Port) stations are all affected by the shallow
depth of the strong impedance contrast with the Banks Peninsula volcanics, which underlie
the alluvial deposits at the surface, and respond in ways that cannot be generalized among
them. Finally, the NBLC site, while being located in the eastern suburbs, is actually located
on a sand dune region, and therefore exhibits a short period site response more a-kin to a
stiffer soil site. The systematic 𝛿𝑆2𝑆𝑠 residuals for these sites are presented subsequently.
Comparison of all sub-regions
Figure 9a illustrates the variation in site-to-site residuals, 𝛿𝑆2𝑆𝑠 , for the four main subregions of Christchurch, as well as the HVSC, LPCC, CMHS, RHSC, and NBLC stations
which don’t conform to these main four sub-regions. As previously noted with respect to
Figure 6d, it can be seen that there is a large variability between the systematic 𝛿𝑆2𝑆𝑠 for the
different sub-regions and other outlying locations. For short vibration periods (𝑇 < 0.2𝑠),
the HVSC station has the largest positive 𝛿𝑆2𝑆𝑠 residual, while the NBLC site has the largest
negative 𝛿𝑆2𝑆𝑠 residual. The LPCC station also has a notable positive residual, while the
North and Western suburbs have notable negative residuals, and all others (i.e. CBD, East,
CMHS, and RHSC) have near zero residuals. For moderate-to-long vibration periods there is
significant fluctuation of the residuals for each sub-region or location as a function of
1
CBD
West
East
North
HVSC
LPCC
CMHS
RHSC
NBLC
0.5
0
-0.5
Systematic amp. factor, exp(  L2L+  S2S)
Systematic site-to-site residuals,  S2S
vibration period.
-1 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
2.5
CBD
West
East
North
HVSC
LPCC
CMHS
RHSC
NBLC
2
1.5
1
0.5 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
Figure 9: Comparison of various sub-regions and other locations in Christchurch: (a) site-to-site
residuals; and (b) systematic median amplification factors from systematic location-to-location and siteto-site effects.
Figure 9b illustrates the systematic amplification factor which would be applied to the
median ground motion prediction (i.e. Equation (14)). Relative to Figure 9a, the most
notable feature is the contribution of the 𝛿𝐿2𝐿𝑙 residual for long vibration periods (i.e. Figure
2), which causes all sub-regions and other locations to have amplification factors greater than
1.0 for long vibration periods. For the Christchurch CBD, in particular, it can be seen that
the amplification factor is approximately 1.0 for T<0.2, then increases to a value of
approximately 1.8 from 𝑇 = 0.5 − 2s, contains a minor localized increase for 𝑇 = 2 − 4𝑠,
and then gradually increases up to values 2.2 for 𝑇 = 10𝑠. Parametric forms of these
systematic effects for the four sub-regions (i.e. CBD, and Western, Eastern, Northern
suburbs) are developed in Bradley (2013b) for use in Canterbury-specific seismic hazard
analysis for the ongoing Christchurch rebuild.
The implications of these amplification
factors for ergodic and non-ergodic prediction are examined in a subsequent section as
compared to ground motion observations.
NON-ERGODIC STANDARD DEVIATIONS
Between-event standard deviations
In order to fully incorporate non-ergodic aspects into a GMPE it is necessary to modify
both its median and standard deviation prediction. Figure 10a presents the ergodic betweenevent standard deviation of the Bradley (2010) model, 𝜏, in comparison with the standard
deviation in the location-to-location residual, 𝜏𝐿2𝐿 𝑙 ; that of the ‘remaining’ between-event
residual, 𝜏0 ; and their SRSS combination. It can be seen that, even with only 10 events, the
value of 𝜏𝐿2𝐿𝑙 is notably smaller than 𝜏0 (i.e. √10⁡times smaller, as per Equation (6)). This is
also true for the within-event residuals, and therefore herein only the combined standard
deviation for the between- and within-event residuals is examined. For 𝑇 ≤ 0.3𝑠 the nonergodic between-event standard deviation is on the order of 85% of the ergodic standard
deviation (i.e. 𝑅𝐹𝜏 = 0.85 in Equation (16)), while for 𝑇 > 1s it becomes an increasing
smaller proportion of the ergodic value, taking a value as small as 60% for 𝑇 = 10s. Hence,
it can be seen that the non-ergodic between-event standard deviation is relatively constant for
the range of vibration periods considered, in contrast to the ergodic standard deviation which
increases notably for long vibration periods. Since long-period ground motion is dominated
by large scale features of the seismic source, and propagation through the earth’s crust and
local sedimentary basins, it is not surprising that gross differences in tectonics and
crustal/basin structure from worldwide events (used in the ergodic model) lead to a notably
larger long period between-event standard deviation than those specifically for the
Canterbury region and Canterbury earthquake sequence (used in the non-ergodic model).
Within-event standard deviations
The within-event standard deviations of the ergodic Bradley (2010) model are a function
of event magnitude and the level of nonlinearity in the site response portion of the model.
Because the variation in the within-event standard deviation due to these factors is notably
Between-event standard deviation
0.7
0.6
(ergodic)
0
0.5
L2L
0.4
l
(20+2L2L )1/2
l
0.3
0.2
0.1
0 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
Figure 10: Ergodic and non-ergodic between-event standard deviation components. The ergodic withinevent standard deviation of Bradley (2010) is magnitude dependent and therefore the average value over
all 10 events is depicted.
larger than the variation in the between-event standard deviation, it is more instructive to
examine the non-ergodic within-event standard deviation on the basis of its ratio to the
ergodic within-event standard deviation (i.e. 𝑅𝐹𝜙 given in Equation (17)).
Figure 11 presents the reduction in the within-event standard deviation as a function of
vibration period for the 20 different strong motion stations which were considered. On
average across the 20 stations, it can be seen that the within-event standard deviation ranges
from 𝑅𝐹𝜙 = 0.65 − 0.85 over the spectrum of vibration periods, with smallest values at long
vibration periods, and largest values at moderate periods (𝑇 = 0.2 − 0.6𝑠). Despite the fact
that, on average, the non-ergodic standard deviation is less than the ergodic standard
deviation, it can be seen that in several instances (both several sites and several vibration
periods) 𝑅𝐹𝜙 > 1, indicating that the non-ergodic standard deviation is greater than the
ergodic standard deviation.
Total standard deviation
Similar to Figure 11, Figure 12 presents the reduction in the non-ergodic vs. ergodic total
standard deviation as a function of vibration period for the 20 different strong motion stations
which were considered. It can be seen that the reduction factor values for the total standard
deviation are very similar to the within-event standard deviation – a consequence of the

Reduction in within-event stdev., RF
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2 -2
10
-1
0
10
10
Vibration period, T (s)
10
1
Figure 11: Reduction in the within-event standard deviation of the non-ergodic model as compared to the
ergodic model for all stations, the subset of CBD stations, and the proposed design parameterization.
fact that the within-event standard deviation is notable larger than the between-event
standard deviation, and hence it is the dominant contributor to the total standard deviation. It
should be noted that the mean, 16th and 84th percentile values of the reduction factors are
presented for the purposes of illustration, but the reduction factors for specific sites or subregions (available from the author on request) should be used if non-ergodic ground motion
prediction is considered for site- or sub-region-specific seismic hazard analysis.
Individual station
Reduction in total stdev., RF

T
1.6
1.4
1.2
1
0.8
0.6
0.4
Mean of all stations
0.2 -2
10
-1
16th and 84th percentiles
0
10
10
Vibration period, T (s)
10
1
Figure 12: Reduction in the ‘total’ standard deviation of the non-ergodic model as compared to the
ergodic model for all stations, the subset of CBD stations, and the proposed design parameterization.
COMPARISON OF NON-ERGODIC STANDARD DEVIATION REDUCTION WITH
PREVIOUS STUDIES
Figure 13 provides a comparison of the non-ergodic total standard deviation reduction
obtained in this study with those of previous studies, which are classified into two types. The
first are so-called ‘single station’ non-ergodic studies in that only the site-to-site systematic
effect (i.e. 𝛿𝑆2𝑆𝑠 ) is considered, such as Lin et al. (2011) (L11), Chen and Tsai (2002)
(CT02), Atkinson
(2006) (A06),
Rodriguez-Marek
et
al.
(2011) (RM11), and
Ornthanammarath et al. (2011) (O11). The second are so-called ‘single-path single-station’
non-ergodic studies that consider systematic site, path and source effects, such as L11, A06
and Morikawa et al. (2008) (M08). Firstly, it can be seen that the ‘single station’ studies
generally result in a smaller mean reduction in total standard deviation than the mean
obtained in the present study.
This is consistent with the notion that this study also
considered the location-to-location systematic effect, 𝛿𝐿2𝐿𝑙 . Secondly, it can be seen that
‘single-path single-station’ studies result in a greater mean reduction in total standard
deviation than the mean obtained in this study. This can be attributed to the additional patheffect considered in such studies, which was ignored here. It may also be a result of the fact
that the aforementioned studies have generally used small amplitude ground motions, so that
additional uncertainty resulting from nonlinear site effects is not scrutinized.
Reduction in total stdev., RF

T
1.6
1.4
Single-station studies
[L11,CT02,A06,RM11,O11]
min-max
range
1.2
16th-84th
percentile
range
Mean
1
0.8
0.6
0.4
0.2
0.01
Single-path single-station studies
[L11, A06, M08]
0.1
1
Vibration period, T (s)
10
Figure 13: Comparison of the reduction in the total standard deviation from non-ergodic consideration in
this study compared with previous studies. PGA results from other studies plotted at T=0.01s.
In addition to the above comments, Figure 13 also serves to re-iterate the significance of
the variation in the reduction of the total standard which can occur on a site-by-site basis.
Figure 13 illustrates, for example, that while the mean reduction factor is on the order of 0.80
for short periods, the 16th-84th percentile confidence interval is on the range of nearly 0.601.0. Therefore, it is clear that caution should be exercised when considering the use of mean
reduction factors for application to cases which do not have sufficient site-specific data to
directly constrain the standard deviation.
SUB-REGION-SPECIFIC GROUND MOTION PREDICTION
GENERAL CONSIDERATIONS
When considering the sub-region-specific residuals (i.e. the mean of the systematic site
residuals given in Figure 8 and Figure 9a) for use in non-ergodic seismic hazard analysis for
the Canterbury region several considerations are necessary. It should be noted from the
outset that there is no guarantee that at a specific site located within the sub region will
display the same response as that of the instrumented sites upon which the sub-region is
characterized. Despite this, it is the fact that the sub-region factors are based on strong
ground motions that make the use of sub-region factors appealing.
There are several
possibilities by which the consistency of a specific site under consideration could be assessed
with respect to a generic sub region. Firstly, instruments can be deployed at the site to record
ground motions. Since the recorded ground motions will most likely be small amplitude (due
to the infrequent nature of strong motion shaking) then such residuals would ideally be
compared for the same event with those obtained at the pre-existing instrumented sites which
were used to develop the sub-region residuals (since these sites have recorded strong ground
motions). Secondly, geotechnical and geological information (i.e. site characterization) at the
site can be utilized to demonstrate that the site considered is consistent with the properties of
the instrumented sites in the sub region. In either of the above two cases, the additional
uncertainty due to the determination of the consistency of a specific site with respect to a
given sub-region should also be included in the total prediction uncertainty.
Failing either of the two forms of quantitative information sources noted above, it is
possible that sub-region-specific ground motion prediction could be performed by adopting
the Canterbury-specific non-ergodic median prediction for the sub region (i.e. Equation (14))
with a partially non-ergodic variance comprising the non-ergodic between-event variance
2
2
(i.e. 𝜏𝐿2𝐿
+ 𝜏𝑙0
) with the ergodic within-event variance (i.e. 𝜙 2 ). This is the approach which
𝑙
is demonstrated in the following section.
COMPARISON OF ERGODIC AND NON-ERGODIC PREDICTIONS
Having developed non-ergodic modification factors for the Bradley (2010) GMPE, it is
useful to directly examine the relative comparison between these two predictions and
observations from notable scenario ground motions from the Canterbury earthquake
sequence. Figure 14 compares the difference in predictions from the ergodic and non-ergodic
Bradley (2010) model with two ground motions for each of the 4 September 2010 and 13
June 2011 (2:20pm) events. Non-ergodic predictions at the CCCC and REHS stations in
Figure 14a and Figure 14b are based on the ‘CBD’ specific modification; at TPLC based on
the ‘Western suburbs’ modification; and at HVSC based on the HVSC-specific modification
(i.e. Figure 9b). Examination of Figure 14 illustrates that, on average, the non-ergodic
prediction provides a closer comparison to the observations than the ergodic prediction. This
is expected given that the non-ergodic prediction is basically ‘calibrated’ using these same
observations (true validation would require a mutually distinct set of observed ground
motions). However, it is also important to note that the median non-ergodic prediction is not
always closer to the observation on a single event-by-event basis. This is illustrated in Figure
14a for 𝑇 = 0.3 − 1.5𝑠, for example, where the geometric mean of the observation is closer
to the median of the ergodic prediction than the non-ergodic median prediction.
DISCUSSION: COMPARISON WITH CHEN AND FACCIOLI (2013)
Following submission of this paper, Chen and Faccioli (2013) published an analysis of
single-station sigma from a subset of stations in the Canterbury region. There are several
notable differences between the current work and that in Chen and Faccioli which warrant
discussion.
The dataset adopted in this work focuses on large amplitude ground motions of
engineering significance. All 10 considered events in this study utilize finite fault models
(and thus the 𝑅𝑟𝑢𝑝 source-to-site distance). In contrast, Chen and Faccioli utilize a larger
dataset of 65 events (concentrated in the range of M4.2-5.0 and of several tens of kilometers
(Chen and Faccioli 2013)), only one event of which is considered as a finite fault. In terms of
strong motion stations considered, 8 of the 14 stations considered by Chen and Faccioli, are
the same as those from the 20 stations considered in this work.
The Bradley (2010) GMPE, adopted in this work, was specifically developed for NZ
conditions based on the examination of existing foreign GMPEs, and is presently utilized in
NZ PSHA. In contrast, the Faccioli et al. (2010) GMPE has not been previously considered
for PSHA in New Zealand (the evidence presented in Chen and Faccioli does not allow for a
sufficient assessment of its applicability), and the Boore and Atkinson GMPE was illustrated
by Bradley (2013b) to contain several biases with respect to NZ ground motion prediction
(particularly for Mw<5 which comprises the majority of the Chen and Faccioli dataset). In
addition, for both the Faccioli et al. and Boore and Atkinson GMPEs, Chen and Faccioli use
assumed Eurocode site classes and values for the NZ-based site classes without validation. In
the author’s opinion it is necessary that non-ergodic ground motion prediction studies utilize
a GMPE which is unbiased with respect to the considered dataset in order to obtain robust
results.
The site-to-site residuals at stations CHHC, LPCC, and HVSC presented in Chen and
Faccioli are broadly consistent with those obtained here, but differ in details due to the
differences in dataset and adopted GMPE. While Figure 9 of Chen and Faccioli attempts to
consider the dependence of site residuals on three bins of ground motion intensity it should
be noted that the largest bin (PGA>0.5m/s^2, which corresponds to 0.05g) is still for very
small levels of ground motion intensity. Furthermore, in viewing Figure 6 of Chen and
Faccioli it is not clear how many observations are within each of the three PGA bins, and
therefore if the differences observed are statistically significant. The dataset in the present
study has an average PGA of 0.18g and the majority of ground motions are well above 0.05g,
and hence Figure 7 of the present study provides a robust assessment of the effects of ground
motion intensity on site-to-site residuals.
CONCLUSIONS
This study has examined ground motion observations from the most significant 10 events
in the 2010-2011 Canterbury earthquake sequence at near-source sites to scrutinize the New
Zealand (NZ)-specific Bradley (2010) GMPE and develop region-specific modification
factors based on relaxing the conventional ergodic assumption. It was observed that the
location-to-location residual (i.e. systematic feature of the between-event residuals) had
values close to zero for short-to-moderate vibration periods, but became increasingly positive
for 𝑇 >1s, likely indicative of important Canterbury-specific sedimentary basin and nearsurface soil effects which are not adequately accounted for in the Bradley (2010) GMPE. In
general, the site-to-site residuals were found to be independent of ground motion intensity,
however for the LPCC (Lyttelton Port) and KPOC (North Kaiapoi) stations there was an
apparent dependence on ground motion intensity; likely indicating that the nonlinear site
response at these sites is not well characterized by their designated site classes (B:rock and
E:very soft soil, respectively). On the basis of the similar site-to-site residuals, 15 of the 20
stations were adequately grouped into four sub-regions, while the remaining 5 stations did
not fit any of these general sub-regions. The grouping of sites into sub-regions allows the
10
10
-1
-2
Station:CCCC
Rrup=16.3 km
4 September 2010
10
Spectral Acc, Sa (g)
Spectral Acc, Sa (g)
10
0
Geometric mean
observation
Bradley (2010)
non-ergodic
Observation
Ergodic
Non-ergodic
10
10
Bradley (2010)
ergodic
-1
10
0
10
10
1
0
rup
-1
-2
Observation
Ergodic
Non-ergodic
10
-1
13 June 2011 (2:20pm)
10
10
10
-1
-2
-3
Observation
Ergodic
Non-ergodic
10
-1
0
13 June 2011 (2:20pm)
Station:HVSC
Rrup=4.7 km
0
Spectral Acc, Sa (g)
Spectral Acc, Sa (g)
10
10
10
1
Period, T (s)
Period, T (s)
10
Station:REHS
R =15.9 km
4 September 2010
10
0
10
1
10
10
Station:TPLC
Rrup=21.6 km
-1
-2
-3
Observation
Ergodic
Non-ergodic
10
-1
10
0
Period, T (s)
Period, T (s)
Figure 14: Comparison of the ergodic and non-ergodic forms of the Bradley (2010) GMPE prediction
compared to observations: (a) CCCC (Christchurch Cathedral College); (b) REHS (Resthaven); (c)
HVSC (Heathcote Valley); and (d) TPLC (Templeton). Note that both horizontal components and the
geometric mean of the observed response spectra are illustrated.
possibility of non-ergodic ground motion prediction over sub-regions of Canterbury,
rather than site-specific predictions only at strong motion stations. Examination of the
standard deviations in the residuals illustrates that, on average, the non-ergodic standard
deviation is 65-85% of the ergodic standard deviation of the Bradley (2010) model.
However, on a site-by-site basis this average ratio can easily vary by ±0.2 units.
10
1
ACKNOWLEDGEMENTS
Financial support provided by the New Zealand Earthquake Commission is greatly
appreciated.
Constructive comments from both Adrian Rodriguez-Marek and Fabrice
Cotton, who alerted the author to the recent work of Chen and Faccioli (2013), are greatly
acknowledged.
REFERENCES
Al Atik, L., Abrahamson, N., Bommer, J. J., Scherbaum, F., Cotton, F., Kuehn, N., 2010. The
Variability of Ground-Motion Prediction Models and Its Components, Seismological
Research Letters, 81, 794-801.
Anderson, J. G., Brune, J. N., 1999. Probabilistic seismic hazard analysis without the ergodic
assumption, Seismological Research Letters, 70, 19-28.
Atkinson, G. M., 2006. Single-Station Sigma, Bulletin of the Seismological Society of
America, 96, 446-455.
Beavan, J., Fielding, E., Motagh, M., Samsonov, S., Donnelly, N., 2011. Fault Location and
Slip Distribution of the 22 February 2011 Mw 6.2 Christchurch, New Zealand, Earthquake
from Geodetic Data, Seismological Research Letters, 82, 789-799.
Beavan, J., Motagh, M., Fielding, E. J., Donnelly, N., Collett, D., 2012. Fault slip models of
the 2010–2011 Canterbury, New Zealand, earthquakes from geodetic data and observations
of postseismic ground deformation, New Zealand Journal of Geology and Geophysics, 55,
207-221.
Berrill, J. B., et al., 2011. The Canterbury Accelerograph Network (CanNet) and some results
from the September 2010, M7.1 Darfield Earthquake, in 9th Pacific Conference on
Earthquake Engineering: Auckland, New Zealand p. 8.
Bradley, B. A., 2010. NZ-specific pseudo-spectral acceleration ground motion prediction
equations based on foreign models, Department of Civil and Natural Resources Engineering,
University of Canterbury, UC Research Report 2010-03, Christchurch, New Zealand. 324pp.
http://ir.canterbury.ac.nz/handle/10092/5126
Bradley, B. A., 2012a. Ground motion and seismicity aspects of the 4 September 2010 and 22
February 2011 Christchurch earthquakes, Technical Report Prepared for the Canterbury
Earthquakes
Royal
Commission,
62pp.
http://canterbury.royalcommission.govt.nz/documents-by-key/20120116.2087
Bradley, B. A., 2012b. Ground motions observed in the Darfield and Christchurch
earthquakes and the importance of local site response effects, New Zealand Journal of
Geology and Geophysics, 55, 279-286.
Bradley, B. A., 2012c. Strong ground motion characteristics observed in the 4 September
2010 Darfield, New Zealand earthquake, Soil Dynamics and Earthquake Engineering, 42,
32-46.
Bradley, B. A., 2013a. A New Zealand-specific pseudo-spectral acceleration ground-motion
prediction equation for active shallow crustal earthquakes based on foreign models, Bulletin
of the Seismological Society of America, 103.
Bradley, B. A., 2013b. Systematic ground motion observations in the Canterbury earthquakes
and region-specific non-ergodic empirical ground motion modelling, Department of Civil and
Natural Resources Engineering, University of Canterbury, UC Research Report 2013-03,
Christchurch, New Zealand. 74pp.
Bradley, B. A., Cubrinovski, M., 2011. Near-source Strong Ground Motions Observed in the
22 February 2011 Christchurch Earthquake, Seismological Research Letters, 82, 853-865.
Brown, L. J., Weeber, J. H., 1992. Geology of the Christchurch urban area, Geological and
Nuclear Sciences 110pp.
Buchanan, A., Carradine, D., Beattie, G. J., Morris, H., 2011. Performance of houses during
the Christchurch earthquake of 22 February 2011, Bulletin of the New Zealand Society for
Earthquake Engineering, 44, 342-357.
Canterbury
Geotechnical
Database,
2011.
https://canterburyrecovery.projectorbit.com/SitePages/Home.aspx, (last accessed: July
2011).
Chen, Y.-H., Tsai, C.-C. P., 2002. A New Method for Estimation of the Attenuation
Relationship with Variance Components, Bulletin of the Seismological Society of America,
92, 1984-1991.
Chiou, B. S. J., Youngs, R. R., 2008. An NGA Model for the average horizontal component
of peak ground motion and response spectra, Earthquake Spectra, 24, 173-215.
Clifton, C., Bruneau, M., MacRae, G. A., Leon, R. T., Fussell, A., 2011. Steel Structures
Damage from the Christchurch Earthquake Series of 2010 and 2011, Bulletin of the New
Zealand Society for Earthquake Engineering, 44, 297-318.
Cubrinovski, M., et al., 2011. Geotechnical Aspects of the 22 February 2011 Christchurch
Earthquake, Bulletin of the New Zealand Society for Earthquake Engineering, 44, 205-226.
Cubrinovski, M., et al., 2010. Geotechnical reconnaissance of the 2010 Darfield (Canterbury)
earthquake, Bulletin of the New Zealand Society for Earthquake Engineering, 43, 243-320.
Cubrinovski, M., McCahon, I., 2011. Foundations on deep alluvial soils, University of
Canterbury, Christchurch, Technical Report Prepared for the Canterbury Earthquakes Royal
Commission, 40pp.
Dizhur, D., Ismail, N., Knox, C., Lumantarna, R., Ingham, J., 2010. Performance of
unreinforced and retrofitted masonry buildings during the 2010 Darfield earthquake, Bulletin
of the New Zealand Society for Earthquake Engineering, 43, 321-339.
Gledhill, K., Ristau, J., Reyners, M., Fry, B., Holden, C., 2011. The Darfield (Canterbury,
New Zealand) Mw 7.1 Earthquake of September 2010: A Preliminary Seismological Report,
Seismological Research Letters, 82, 378-386.
Kam, W. Y., Pampanin, S., Elwood, K. J., 2011. Seismic performance of reinforced concrete
buidlings in the 22 Febraury Christchurch (Lyttelton) Earthquake, Bulletin of the New
Zealand Society for Earthquake Engineering, 44, 239-278.
Lin, P.-S., Chiou, B., Abrahamson, N., Walling, M., Lee, C.-T., Cheng, C.-T., 2011.
Repeatable Source, Site, and Path Effects on the Standard Deviation for Empirical GroundMotion Prediction Models, Bulletin of the Seismological Society of America, 101, 22812295.
Lindstrom, M. J., Bates, D. M., 1990. Nonlinear mixed effects models for repeated measures
data, Biometrics, 46, 673-687.
Morikawa, N., Kanno, T., Narita, A., Fujiwara, H., Okumura, T., Fukushima, Y., Guerpinar,
A., 2008. Strong motion uncertainty determined from observed records by dense network in
Japan, Journal of Seismology, 12, 529-546.
NZS 1170.5, 2004. Structural design actions, Part 5: Earthquake actions - New Zealand.
Standards New Zealand: Wellington, New Zealand. p. 82.
NZSEE, 2010. Special Issue: Preliminary observations of the 2010 Darfield (Canterbury)
Earthquakes, Bulletin of the New Zealand Society for Earthquake Engineering, 43, 215-439.
NZSEE, 2011. Special Issue: Preliminary observations of the 2011 Christchurch Earthquake,
Bulletin of the New Zealand Society for Earthquake Engineering, 44, 181-430.
Ornthammarath, T., Douglas, J., Sigbjörnsson, R., Lai, C., 2011. Assessment of ground
motion variability and its effects on seismic hazard analysis: a case study for iceland, Bulletin
of Earthquake Engineering, 9, 931-953.
Ristau, J., 2008. Implementation of routine regional moment tensor analysis in New Zealand,
Seismological Research Letters, 79, 400-415.
Rodriguez-Marek, A., Montalva, G. A., Cotton, F., Bonilla, F., 2011. Analysis of SingleStation Standard Deviation Using the KiK-net Data, Bulletin of the Seismological Society of
America, 101, 1242-1258.
Seismological Research Letters, 2011. Focused Issue on the 22 February 2011 Magnitude 6.2
Christchurch Earthquake, Seismological Research Letters, 82.
Stirling, M., et al., 2012. National Seismic Hazard Model for New Zealand: 2010 Update,
Bulletin of the Seismological Society of America, 102, 1514-1542.
Walling, M., 2009. Non-ergodic probabilistic seismic hazard analysis and spatial simulation
of variation in ground motion. University of California, Berkeley. p. 263.