Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake

Abstract

This is the preliminary report of the energy spectrum of the record during the 2024 Noto Earthquake (at 01/01 16:10
Japan Standard Time). In this report, the ground motion records available from the website of Strong-motion Seismograph
Network (K-NET and KiK-net)1) were used. The main findings are (i) In general, the value of the response spectrum and energy spectrum calculated from the record during the 2024 Noto Earthquake is larger than those calculated from the records during the 1995 Hyogo-ken Nanbu Earthquake and the 2016 Kumamoto Earthquake, in the range where the natural period is larger than 1.5 seconds. (ii) Some of the records during the mainshock of 2024 Noto Earthquake shows the clear directivity. In case of ISK001, ISK002, ISK007, and ISKH02, the direction of the largest cumulative input energy of the major component is closely aligned with the south-east (north-west) directions.

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2024/01/05 Ver.1 (in Japanese), 2024/01/08 English Version (DOI Re-created at 2024/01/11)
Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
Kenji Fujii (Chiba Institute of Technology)
1
Preliminary Analysis of the Energy Spectrum of the Record during the
2024 Noto Earthquake
Kenji Fujii (Chiba Institute of Technology)
1. Introduction
This is the preliminary report of the energy spectrum of the record during the 2024 Noto Earthquake (at 01/01 16:10
Japan Standard Time). In this report, the ground motion records available from the website of Strong-motion Seismograph
Network (K-NET and KiK-net)1) were used.
Note that because the analysis results shown here is a preliminary result (analyzed at 01/03), some further update would
be made.
2. Basic Information
2.1 Epicenter of Earthquake on 1 January 2024
The basic information of the epicenter is shown in Table 2-1 and Table 2-2. Note that the following information was
taken from the Japan Meteorological Agency website2) and F-net website3) (Accessed on 7 January 2024).
Table 2-1 Basic information on the epicenters (according to Japan Meteorological Agency) 2).
Date Latitude (°) Longitude (°) Magnitude (
J
M
) Depth Place name of epicenter
2024/01/01 16:10 37.5 N 137.2E 7.6 Very shallow Noto, Ishikawa Prefecture
Table 2-2 Focal mechanism solution estimated manually (according to F-net website) 3).
Latitude
(°)
Longitude
(°)
Depth
(km)
Strike
(°)
Dip
(°)
Rake
(°)
O
M
(Nm) W
M
37.50N 137.27E 11 52; 204 42; 52 111; 72 2.18×1020 7.5
2.2 Stations
Table 2-3 shows the list of the ground motion records. The numerical values of this table are taken from the K-NET and
KiK-net website1). In this report, the horizontal components (EW and NS components) at the ground level are used. Figure
2-1 shows the location of the epicenter and stations.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
Kenji Fujii (Chiba Institute of Technology)
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Table 2-3 List of the ground motion records.
Station Code Station
Name
Latitude
(°)
Longitude
(°)
JMA
Seismic
Intensity
Distance
(km)
PGA (m/s2)
EW NS
ISK001 Ohya 37.50N 137.18E 6.2 2 14.29 9.04
ISK002 Shoin 37.45N 137.29E 6.2 10 7.07 6.86
ISK003 Wajima 37.39N 136.91E 6.2 28 11.20 14.96
ISK005 Anamizu 37.23N 136.90E 6.5 40 11.46 10.23
ISK006 Togi 37.16N 136.69E 6.6 59 26.78 14.79
ISK007 Nanao 37.04N 136.97E 5.8 55 3.59 3.74
ISK015 Ohmachi 37.23N 136.91E 6.3 40 9.26 9.79
ISKH01 Suzu 37.53N 137.28E 6.2 8 7.48 5.95
ISKH02 Yanagida 37.36N 137.04E 5.8 21 6.17 4.70
ISKH03 Uchiura 37.35N 137.24E 6.3 18 7.72 7.14
ISKH04 Togi 37.19N 136.72E 5.9 55 4.84 6.18
ISKH06 Shika 37.05N 136.82E 5.6 60 7.97 5.73
Figure 2-1 Location of Epicenter and Stations

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 2-2 presents the primary and shear wave profiles for the different stations. The soil properties for each station are
available from the K-NET and KiK-net website1). As shown in this figure, the shear wave velocity (
s
V
) at ISK002 (K-NET
Shoin), ISK005(K-NET Anamizu), ISK007(K-NET Nanao) is smaller than other stations.
Figure 2-2 Primary and shear wave profiles.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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2.3 Recorded Acceleration
Figures 2-3 through 2-14 show the ground motion time histories. In these figures, the EW and NS components at the
ground motion surface are shown.
Figure 2-3 Ground motion time histories (ISK001: K-NET Ohya)
Figure 2-4 Ground motion time histories (ISK002: K-NET Shoin)
Figure 2-5 Ground motion time histories (ISK003: K-NET Wajima)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 2-6 Ground motion time histories (ISK005: K-NET Anamizu)
Figure 2-7 Ground motion time histories (ISK006: K-NET Togi)
Figure 2-8 Ground motion time histories (ISK007: K-NET Nanao)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 2-9 Ground motion time histories (ISK015: K-NET Ohmachi)
Figure 2-10 Ground motion time histories (ISKH01: KiK-NET Suzu)
Figure 2-11 Ground motion time histories (ISKH02: KiK-NET Yanagida)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 2-12 Ground motion time histories (ISKH03: KiK-NET Uchiura)
Figure 2-13 Ground motion time histories (ISKH04: KiK-NET Togi)
Figure 2-14 Ground motion time histories (ISKH06: KiK-NET Shiga)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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3. Response Spectrum
3.1 Pseudo-Acceleration, Pseudo-Velocity and Displacement Spectrum
The pseudo-acceleration spectrum (
()
pA
ST
), pseudo-velocity spectrum (
()
pV
ST
) and displacement spectrum (
()
D
ST
)
are calculated and compared herein. Figure 3-1 shows an isotropic linear one-mass two-degree-of-freedom model4) used
in this study. The EW and NS components are applied in the X- and Y-direction, respectively. The damping ratio is set to
0.05.
Figure 3-1 An isotropic linear one-mass two-degree-of-freedom model4)
For the comparisons, the ground motion records of the Kobe JMA Observatory observed in the 1995 Hyogo-ken Nanbu
Earthquake (KMO) 5), and KiK-NET Mashiki in the 2016 Kumamoto Earthquake (MSK)1) are also analyzed.
Figure 3-2 and Figure 3-3 compare the calculated response spectrum. From these figures, the following conclusions
can be made.
 In general, the period of at the peak of
()
pV
ST
(
p
eak
T) is relatively larger: in Figure 3-2, the peak period
p
eak
T of
ISK001 (K-NET Ohya), ISK002(K-NET Shoin), ISK005(K-NET Anamizu) is 2.60 seconds, 3.00 seconds, and 1.46
seconds, respectively, while in Figure 3-3, the peak period
p
eak
T of ISKH01 (KiK-NET Suzu) is 1.94 seconds.
 The response spectrum of ISK001, ISK002 and ISK005 are notably larger than those of KMO and MSK, in the range
the natural period (T) is larger than 1.5 seconds, as shown in Figure 3-2. Similarly, the spectrum of ISKH01 is also
notably larger than those of KMO and MSK, in the range T is larger than 1.5 seconds, as shown in Figure 3-3.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 3-2 Response spectrum (damping ratio: 0.05) (1)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 3-3 Response spectrum (damping ratio: 0.05) (2)

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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3.2 Energy Spectrum
Next, the cumulative input energy spectrum (
()
I
VT) and the maximum momentary input energy spectrum (
()
E
VT
Δ)
are calculated herein. As described below, an isotropic linear one-mass two-degree-of-freedom model is used for the
analysis. The damping ratio is set to 0.10. The cumulative input energy spectrum (
()
I
VT) is calculated as
2
IIBI
VEm=, (3.1)
()
0
d
t
IBI gX gy
Em axaydt=− +

. (3.2)
Here, IBI
Em
is the cumulative input energy per unit mass, d
t is the duration of the ground motion record.
While the maximum momentary input energy spectrum (
()
E
VT
Δ) is calculated as
,max
2
EBI
VEm
Δ=Δ . (3.3)
Here, ,maxBI
E
mΔ is the maximum momentary input energy per unit mass, which is defined as the maximum value of
BI
EmΔ calculated from Equation (3.4) over the course of the seismic event.
()
tt
BI gX gy
t
Em axaydt
+Δ
Δ=− +

. (3.4)
Here, t and tt+Δ are the beginning and end time of a half cycle of structural response, respectively. In this study, t
and tt+Δ are defined as the time when the absolute (vector) value of the displacement
()
dt is at a local maximum6).
The absolute value of
()
dt is expressed as
() () ()
{}
()
{}
22
dt t xt yt== +d. (3.5)
The conditions for
()
dt at a local maximum are expressed as:
() () () () ()
() () () () () ()
{}
()
{}
22
0: 0
0: 0
d t xt xt yt yt
d t xtxt yt y t xt yt
=+=

<+++<



    
. (3.6)
Figure 3-4 compares the calculated cumulative input energy spectrum (
()
I
VT). From this figure, the following
conclusions can be made.
 In most of the records observed in this earthquake, the calculated
()
I
VT is larger than that of KMO and MSK in
the range where the natural period (T) is larger than 1.0 seconds. Exceptionally, the calculated
()
I
VT of ISK006
is smaller than that of KMO and MSK in the range where the natural period (T) is larger than 0.5 seconds.
 The calculated
()
I
VT of ISK002 and ISK005 exceed 6.0 m/s in the interval T is between 1.0 to 4.0 seconds.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 3-4 Cumulative input energy spectrum.
Figure 3-5 shows the calculated maximum momentary input energy spectrum (
()
E
VT
Δ
). From this figure, the following
conclusions can be made.
 In most of the records observed in this earthquake, the calculated
()
E
VT
Δ
is larger than that of KMO and MSK in
the range where the natural period (
T
) is larger than around 1.2 to 1.8 seconds. This trend is similar to the
()
I
VT
spectrum shown in Figure 3-4. Meanwhile, the calculated
()
E
VT
Δ
of ISK006 is smaller than that of KMO and MSK
in the range where
T
is larger than 0.5 seconds.
 The calculated
()
E
VT
Δ
of ISK002 and ISK005 is close to 4.0 m/s in the interval
T
is larger than 1.5 seconds.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 3-5 Maximum momentary input energy spectrum.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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4. Characteristics of Energy Spectrum
4.1 Principal Axis of the Horizontal Ground Motion Based on the Cumulative Energy Input
In this section, the principal axis of the horizontal ground motion based on the cumulative energy input of each record
is calculated following the author’s previous study7). Figure 4-1 shows the definition of the principal axis of the horizontal
ground motion. In this figure, the ξ-axis is the major axis which maximizes the cumulative input energy per unit mass,
while the ζ-axis is the minor axis which is orthogonal to ξ-axis. The angle between ξ- and X-axis is referred to as
()
ET
ψ
.
Figure 4-1 Definition of the principal axes of the horizontal ground motion.
The cumulative input energies per unit mass in the ξ- and ζ-direction ( I
E
m
ξ
and I
E
m
ζ
, respectively) and the angle
between ξ- and X-axis (
()
ET
ψ
) are calculated as
0cos sin cos sin
0sin cos sin cos
IEEIXXIXY EE
IEEIXYIYY EE
Em EmEm
Em EmEm
ξ
ς
ψψ ψψ
ψψ ψψ
−


=


−

 , (3.7)
() ()
() () () ()
{}
() ()
0
0
0
1
2
d
d
d
t
IXX gX
t
IXY gX gY
t
IYY gY
Em atxtdt
Em atytatxtdt
Em atytdt
=−



=− +



=−








, (3.8)
12
1Tan
2
IXY
E
IXX IYY
Em
EmEm
ψ
−
=− 
−

. (3.9)
The equivalent velocities of
()
I
VT
ξ
,
()
I
VT
ζ
are calculated as
() ()
2, 2
IIII
VT EmVT Em
ξξςς
==
. (3.10)
The ratio of the equivalent velocities in the horizontal major and minor axis (
()
EI
RT
) is defined as
() () ()
EI I I
RT VTVT
ςξ
=. (3.11)

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Note that when the ratio
()
EI
RT
is smaller than 0.7, the cumulative input energy in the minor direction ( I
E
m
ζ
) is less
than the half of that in the major direction ( I
E
m
ξ
).
Figures 4-2 through 4-8 shows the comparisons of the cumulative energy spectrum (
()
I
VT,
()
I
VT
ξ
and
()
I
VT
ζ
) and
the maximum momentary input energy spectrum (
()
E
VT
Δ ), the variation of
()
ET
ψ
,
()
EI
RT
, and the ratio
() ()
IE
VT V T
Δ with respect to the natural period (T). From these figures, the following conclusions can be made.
 In most of the records in this earthquake,
()
E
VT
Δ is smaller than
()
I
VT
ζ
. However, in case of KMO and MAS
shown in Figure 4-8,
()
E
VT
Δ is larger than
()
I
VT
ζ
around the natural period T where
()
E
VT
Δ is its peak.
 In the all records, the variation of
()
ET
ψ
with respect to T is notable in the range where T is less than 1.0
seconds. Meanwhile, in the range where T is larger than 1.0 seconds, the records in this earthquake can be divided
into two groups. The angle
()
ET
ψ
of the first group (ISK001，ISK002，ISKH01，ISKH03，ISKH06) is relatively
stable in the range where T is larger than 1.0 seconds. While the angle
()
ET
ψ
of the second group (ISK003,
ISK004, ISK005, ISK006, ISKH02, ISKH04) varies notably in the range where T is larger than 1.0 seconds. In the
past earthquake records (KMO, MAS), the variation of
()
ET
ψ
is relatively small in the range where T is larger
than 1.0 seconds.
 The trend of the ratio
()
EI
RT
depends on the ground acceleration records. In ISK002,
()
EI
RT
is smaller than 0.7
in the range where T is between 0.5 to 1.4 seconds and larger than 2.0 seconds. Similarly,
()
EI
RT
is smaller than
0.7 in the range where T is between 0.7 to 1.1 seconds and between 1.3 to 4.4 seconds in ISKH03. However, the
range where
()
EI
RT
is below 0.7 is limited in the other records in this earthquake. Meanwhile, in KMO,
()
EI
RT
is smaller than 0.7 when T is larger than 0.6 seconds.
 In all records shown herein, the ratio
() ()
IE
VT V T
Δ decreases as T increases. According to the records in this
earthquake, the lower bound of
() ()
IE
VT V T
Δ is around 2. However, in KMO and MAS, the ratio
() ()
IE
VT V T
Δ
is smaller than 2 when T is large. As discussed in the author’s previous study6), the ratio
() ()
IE
VT V T
Δ indicates
the number of cyclic loads. Therefore, the influence of cyclic loading in this earthquake is more pronounced than
KMO and MAS.

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 4-2 Characteristics of energy spectrum (1).

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Figure 4-3 Characteristics of energy spectrum (2).

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Figure 4-4 Characteristics of energy spectrum (3).

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Figure 4-5 Characteristics of energy spectrum (4).

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Figure 4-6 Characteristics of energy spectrum (5).

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Figure 4-7 Characteristics of energy spectrum (6).

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Figure 4-8 Characteristics of energy spectrum (7).

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4.2 Direction of the Major Axis of the Cumulative Energy Input in each Station
Next, the discussion is focused on the direction of the major axis of the cumulative energy input (ξ-axis). Here, the
equivalent velocity
()
I
VT
ξ
is normalized by its maximum value (
()
{
}
max I
VT
ξ
). Figure 4-9 and Figure 4-10 show the
distribution of the normalized equivalent velocity
() ()
{
}
max
II
VT VT
ξξ
with respect to the angle
()
ET
ψ
. In these
figures, plots are added in the range from -180° to -90°, and from 90° to 180°, considering the symmetry.
Figure 4-9 Distribution of the ratio VIξ(T) / max{VIξ(T)} with respect to the angle ψE (1).

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Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
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Figure 4-10 Distribution of the ratio VIξ(T) / max{VIξ(T)} with respect to the angle ψE (2).
From these figures, the following conclusions can be made.
 In case of ISK001, ISK002 and ISKH02 shown in Figure 4-9 and ISK007 shown in Figure 4-10, the direction of the
largest
() ()
{
}
max
II
VT VT
ξξ
is closely aligned with the south-east (north-west) directions.
 However, in case of ISKH01, the direction of the largest
() ()
{
}
max
II
VT VT
ξξ
is closely aligned with the north-
east (south) directions.

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4.3 Direction of the Maximum Momentary Input Energy
Next, the discussions are focused on the direction of the displacement increment during a half cycle of structural
response corresponds to the maximum momentary energy input. In this study, this direction is referred to as the direction
of the maximum momentary input energy. Figure 4-11 shows examples of the displacement orbit.
Figure 4-11 Examples of the displacement orbit.
From this figure, the angle of incidence the direction of the maximum momentary input energy with respect to X-axis
(
ψ
) is calculated as the incidence of the dotted line AB with respect to X-axis, as
()()
()()
11
Tan Tan
BA
BA
yt t yt
yy
xx xt txt
ψ
−−

+Δ −

−
=− =− 

−+Δ−

 
. (3.12)
Figure 4-12 and Figure 4-13 show the distribution of the ratio
() ()
{}
max
EE
VT VT
ΔΔ
with respect to the angle
()
T
ψ
.
Here, the equivalent velocity
()
E
VT
Δ
is normalized by its maximum value (
()
{}
max
E
VT
Δ
). From these figures, the
following conclusions can be made.
 In case of ISK001, ISK002 and ISKH02 shown in Figure 4-12 and ISK007 shown in Figure 4-13, the direction of the
largest
() ()
{}
max
EE
VT VT
ΔΔ
is closely aligned with the south-east (north-west) directions.
 The distribution of
() ()
{}
max
EE
VT VT
ΔΔ
with respect to the angle
()
T
ψ
shown in Figure 4-12 and Figure 4-13
is similar to the distribution of
() ()
{}
max
II
VT VT
ξξ
with respect to the angle
()
E
T
ψ
shown in Figure 4-9 and
Figure 4-10.

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Figure 4-12 Distribution of the ratio VΔE(T) / max{VΔE(T)} with respect to the angle ψ (1).

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Figure 4-13 Distribution of the ratio VΔE(T) / max{VΔE(T)} with respect to the angle ψ (2).

2024/01/05 Ver.1 (in Japanese), 2024/01/08 English Version (DOI Re-created at 2024/01/11)
Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake
Kenji Fujii (Chiba Institute of Technology)
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5. Conclusions
In this report, the response spectrum and energy spectrum of the record during the 2024 Noto Earthquake (at 01/01
16:10 Japan Standard Time) are calculated. The following conclusions can be made.
 In general, the value of the response spectrum and energy spectrum calculated from the record during the 2024 Noto
Earthquake is larger than those calculated from the records during the 1995 Hyogo-ken Nanbu Earthquake and the
2016 Kumamoto Earthquake, in the range where the natural period is larger than 1.5 seconds.
 Some of the records during the mainshock of 2024 Noto Earthquake shows the clear directivity. In case of ISK001,
ISK002, ISK007, and ISKH02, the direction of the largest cumulative input energy of the major component is closely
aligned with the south-east (north-west) directions.
Note that these conclusions are the author’s conclusions at the time of writing this report (7 January 2024).
Acknowledgements
First, the authors wish to express respectful regret for those who suffered losses in Ishikawa and Niigata prefectures
because of these earthquakes, and hope that they can overcome this disaster as soon as possible. Ground motions used in
this study were taken from the websites of the Japan Meteorological Agency, and National Research Institute for Earth
Science and Disaster Resilience (NIED).
Reference
1) Strong-motion Seismograph Network (K-NET and KiK-net) website: https://www.kyoshin.bosai.go.jp/kyoshin/，
Accessed on 3 January 2024.
2) Japan Meteorological Agency website: https://www.data.jma.go.jp/multi/quake/index.html?lang=en, Accessed on 7
January 2024.
3) NIED F-net website: https://www.fnet.bosai.go.jp/top.php?LANG=en, Accessed on 7 January 2024.
4) Fujii K., Murakami, Y. (2021). “Bidirectional Momentary Energy Input to a One-mass Two-DOF system,” in the
Proceedings of the 17th World Conference on Earthquake Engineering. Sendai, Japan
5) Japan Meteorological Agency website: https://www.data.jma.go.jp/svd/eqev/data/kyoshin/jishin/index.html ，
Accessed on 3 January 2024.
6) Fujii, K. (2021). Bidirectional Seismic Energy Input to an Isotropic Nonlinear One-Mass Two-Degree-of-Freedom
System. Buildings. 11, 143.
7) Fujii K (2022), Influence of the angle of seismic incidence of long-period pulse like ground motion on an irregular
base isolated building. Frontiers in Built Environment. 8, 1034166.
Preliminary Analysis of the Energy Spectrum of the Record during the 2024 Noto Earthquake © 2024 by Kenji Fujii is
licensed under Attribution 4.0 International