Tsunami prediction method for earthquakes withcharacterized source faults(Tsunami Recipe)

English version
Translated in January, 2022.

Earthquake Research Committee, The Headquarters for Earthquake Research Promotion
This document is a translation from Japanese to English of() published in January, 2017.

Tsunami prediction method for earthquakes withcharacterized source faults (Tsunami Recipe)
Table of Contents
Introduction ............................................................................................................................................................................. 1
1. Configuration of characterized earthquake fault model..................................................................................................... 4
1.1 Macroscopic source fault characteristics ................................................................................................................. 4(a) Position and shape of the source fault model ....................................................................................................... 4(b) Size of the source fault model .............................................................................................................................. 5(c) Rake Angle ........................................................................................................................................................... 6(d) Average slip amount ............................................................................................................................................ 6(e) Rupture processes ................................................................................................................................................. 8
1.2 Microscopic source fault characteristics .................................................................................................................. 8(a) Location and number of a large slip zone ............................................................................................................ 9(b) Slip amount and area of a large slip zone........................................................................................................... 10(c) Location and number of a super large slip zone ................................................................................................. 12(d) Slip amount and area of the super large slip zone .............................................................................................. 13
2. Tsunami simulation and uncertainties in the tsunami prediction ..................................................................................... 15
2.1 Tsunami simulations ............................................................................................................................................ 15(a) Computational area and grid spacing ................................................................................................................... 15(b) Discretization of topographic and bathymetric data........................................................................................... 15(c) Crustal movement and distribution of initial water height ................................................................................. 16(d) Baseline tidal level ............................................................................................................................................. 16(e) Simulations for tsunami propagation and run-up ............................................................................................... 16(f) Boundary conditions ........................................................................................................................................... 18(g) Coefficient of roughness .................................................................................................................................... 18(h) Length of computation time and computational time step interval .................................................................... 19
2.2 Uncertainties in tsunami predictions .................................................................................................................... 20
3. Validation of the characterized earthquake fault model .................................................................................................. 21
4. Appendix: Validation of the tsunami prediction method for earthquakes with characterized source faults (Tsunami
Recipe) ............................................................................................................................................................................. 22
5. References ....................................................................................................................................................................... 25

Introduction
In the past, many large earthquakes have occurred in marine areas around Japan, including the 2011 off the Pacific coastof Tohoku Earthquake, and the accompanying tsunamis have caused serious damage.
Figures 1 and 2 show recorded trace heights from past major tsunamis plotted onto the map of Japan. Past impacts in theregion can be observed from these figures. However, there are limits to historical documentation, and some past tsunami
records may be either unclear, or the tsunamis may not be recorded in the first place. Therefore, predictions of future tsunami
possibility, created using records from the past, may be affected. A concept and approach that compensate for this limitation,and that take the associated uncertainty into account, are needed.
Note that trace heights shown in Figures 1 and 2 include the tsunami height, trace height, and run-up height, as definedin Figure 3.

Due to the short period of data recording(100 Years), there is potential for future
tsunamis of a type that has not occurred overthe last 100 years to be under-estimated.

Figure 1. Trace heights and location records for tsunamis over 50 cm that have occurred since 1896

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Note that tsunamis that reached theshore but records are either unclear
of not available are excluded from
Note that tsunamis that reached thethe figure.
shore but records are either unclear or
not available are excluded from thefigure.

Figure 2. Trace height and location records for tsunamis over 50 cm that occurred in or before 1895

Tide station
Tsunami height
Inundation
depth
Trace
heightRun-upheightNormal tide level

Figure 3. Relationship between the tsunami height at a tidal station, the inundation depth, the trace height, and the runup height (from the Japan Meteorological Agency website)

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The Earthquake Research Committee (ERC) has compiled a long-term evaluation of potential future earthquake
occurrences (i.e. long-term evaluation), and the committee has plans to also evaluate associated tsunami risks, based on thislong-term evaluation. In general, for a future earthquake, the slip distribution of a fault cannot be determined with certainty,
and this is why this study "Tsunami prediction method for earthquakes with characterized source faults (tsunami recipe)"
has been prepared. In doing so, the authors have considered methods for the prediction and evaluation of tsunamis whichhave been based on many earthquake fault models using simplified parameters, while also accounting for uncertainty.
Tsunami recipe is a method for prediction and evaluation which can be applied by the ERC to evaluate the risk of thevarious tsunami types which accompany earthquakes. The method includes both the largest class of tsunami, and tsunamisof low height and high frequency, which also cause damage; it includes descriptions of how and when the method is to be
utilized and the concepts under which it was derived. A "prediction" refers to a calculated expectation of tsunami height at
the coast after an assumed earthquake, while an "evaluation" refers to an estimate which has taken various uncertainties, suchas the computational errors included in tsunami predictions, into account. While the method is applied mainly to interplate
earthquakes for now, it will be revised in the future to expand the types of earthquakes covered. Given that the knowledge
available for potential future faults and earthquakes, and associated tsunamis, is insufficient, it would be ideal to configure adefault source of prediction variables, by considering and determining the computational approach and potential results now,
for cases where variations in the phenomena and associated uncertainties need to be considered.
In recent years, various institutions have proposed configurations for fault models and computational methods – for thelargest class of tsunamis in particular (e.g. Cabinet Office, 2012; Ministry of Land, Infrastructure, Transport and Tourism,2012).
Figure 4 shows the process used for tsunami prediction and evaluation in the tsunami recipe, as a flowchart, while thefollowing section explains the steps of the chart.

Figure 4. Flowchart for tsunami prediction and evaluation using the tsunami recipe31. Configuration of characterized earthquake fault model
A "characterized earthquake fault model" is a source fault model that represents the characteristics of source faults
through major parameters. In the tsunami recipe, "source fault" refers to the fault of an earthquake that causes a tsunami. Modelparameters are configured by considering macroscopic source fault characteristics that represent the location, shape, and sizeof the source fault overall, as well as microscopic source fault characteristics that mainly represent the unevenness of the slip
distribution of the source fault. In a tsunami recipe, a characterized earthquake fault model is expressed by combining one ormore rectangular element faults, whose sides are parallel to the sea floor.
The approach to configuration of source fault characteristic parameters, as explained here, is shown specifically toconstruct a characterized earthquake fault model for a tsunami that occurs due to an assumed earthquake. For considerationof a characterized earthquake fault model, the concept of "Strong ground motion prediction method for earthquakes withspecified source faults (" Recipe")" (Earthquake Research Committee, 2016), was utilized as a reference.
In particular, as represented by many interplate earthquakes, information on past earthquakes are available, dependingon the marine area. This information can be utilized in configuring the characteristic earthquake fault model. As explainedin Section 3, the model is validated, as needed, by using data related to past tsunamis, and source fault characteristic
parameters are revised if needed. In such a situation where a characteristic earthquake fault model is revised, an investigation
may sometimes take place by going beyond the tsunami recipe, as source fault characteristic parameters that can explain
observational data for a tsunami are required. In this case, some of the relevant data from past tsunamis (such as the size ofthe earthquake, the source area of the tsunami, the tsunami record, trace height, damage, etc.) may not be consistent with otherdata, and data needs to be utilized by assigning priorities, depending on the objective, such as prediction or evaluation.
A similar consideration is also underway in the field of nuclear safety assessment, where a characterized wave source
model, that takes broad tsunami characteristics into account, is being investigated (e.g. Sugino et al., 2014; Japan Society of
Civil Engineers, 2016)
1.1 Macroscopic source fault characteristics
The following parameters are configured as the macroscopic source fault characteristics for the source fault model:•

Position and shape of the source fault model (length, width, strike and dip angles)•

Size of the source fault model (fault area, seismic moment)•

Rake angle•

Average slip amount•Rupture process(a) Position and shape of the source fault model
Position, length, width, strike, and dip angles of a source fault model are configured based on the fault parameters foran assumed earthquake, if the parameters are estimated. For an interplate earthquake, the source fault model is configured
to correspond to the upper surface of the oceanic plate, estimated by seismic surveys and hypocenter distribution.
If the evaluation result for position and shape of the source area is shown by a long-term evaluation, the result should
be noted.

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(b) Size of the source fault model
The size of the source fault model (fault area, seismic moment) is configured by one of the following methods.
Method 1: Clearly define the range of the source fault by position and shape in the source fault model, calculate thearea equivalent to that range, and estimate the seismic moment using the empirical relationship between the seismic
moment and the fault area.
Method 2: For an earthquake in which determining the seismic source fault is difficult, configure the seismic moment,
and configure the area of the tsunami source fault by using an empirical relationship between the seismic moment and thefault area.
To establish the empirical relationship between the seismic moment M0 (N ∙ m), and fault area S (m2), use an equationthat considers the local characteristics, if some level of past earthquake data is available. For example, a local relationshipbetween seismic moment M0 (N ∙ m) and fault area S (m2) may be configured by estimating an average stress drop Δσ(MPa) for an earthquake that occurs in the area coverage based on equation (1) (Eshelby, 1957), which assumes a circular
failure surface, similar to Kanamori and Anderson (1975).
M0 = 16/(7π3/2) ∙ Δσ ∙ S 3/2(1)

On the other hand, if little or no past earthquake data is available, an empirical relationship for a seismic moment M0(N ∙ m) that shows an average characteristic, and fault area S (m2), is used (e.g. Sato, 1989; Yamanaka and Shimazaki,
1990; Murotani et al., 2008, 2013).
However, caution needs to be exercised in selecting the range of values for seismic moment and fault area within thedata set from which the empirical relationship was developed.

For example, working in a manner similar to Yamanaka and Shimazaki (1990), configuring an empirical relationshipsuch that equation (1) holds by using seismic moment M0 (N ∙ m) and fault area S (m2), as organized by Sato (1989) forinterplate earthquakes that occurred on the Pacific side around Japan and adding more data to it, yields a result of M0 =1.77×106 ∙ S 3/2 (Δσ = 4.3 MPa) (Fujiwara et al., 2015). The value is set at Δσ = 3.0 MPa by the Committee for Modeling aNankai Trough Megaquake (second report) (Cabinet Office, 2012), and these values can be referenced.
However, the empirical relationship between seismic moment and fault area represents an average relationshipestablished with data from several past earthquakes, and individual earthquakes may vary around this relationship. For thisreason, caution needs to be exercised over the fact that prediction results contain a similar variance when predicting future
tsunamis using an empirical relationship, and that the empirical relationship itself is still at the research stage.
Empirical relationships in which the seismic moment is proportional to a power of 3/2 of a fault area, as shown in theequation, correspond to the fact that the ratio of length and width of a fault, and the average stress drop at the source fault, isconstant, regardless of the size of the earthquake. However, the width of a fault is restricted by the depth of the upper andlower ends, and the ratio of length and width may not be constant in a giant-size earthquake so that the empirical relationshipbetween seismic moment and fault area may be different. In fact, the empirical relationship between seismic moment andfault area in relatively large inland crustal earthquakes (over magnitude Mw 6.5) is known to differ from the relationship forsmaller size earthquakes (e.g. Irikura and Miyake, 2001; Tajima et al., 2013). Observational data is limited for giant-size

5interplate earthquakes in which the width of a fault reaches an upper limit. Some studies have suggested that seismic moment
can be explained, within the range of variance, by the relationship where a seismic moment is proportional to the power of
3/2 of a fault area, for cases including giant earthquakes that have occurred globally since 1950 (Murotani et al., 2013). Other
studies have proposed a relationship in which a seismic moment is the square of the fault area, for earthquakes of a certain
size (Tajima et al., 2013), ultimately leading to multiple interpretations of the relationship. On this basis, referencing multiple
concepts is a preferred approach for configuring a source fault model for great and giant earthquakes.
Some studies have pointed out that the relationship between observed seismic moment and fault area is different between
an interplate earthquake and an intraplate earthquake (e.g. Iwata and Asano, 2011). They have suggested that caution should
be exercised over what earthquake was used for the base data set, when using an equation that takes locality into account, byutilizing past earthquake data from the location. Estimated values for observed seismic moment and fault area should also
be treated carefully, by recognizing that the values vary depending on the data used for analysis, such as ground motion,
tsunami, and geodetic data.
The relationship between moment magnitude Mw, and seismic moment M0 (N ∙ m), is expressed by the following equation.
log M0 = 1.5 ∙ Mw+9.1(c) Rake Angle
In the case of an interplate earthquake, estimate rake angle from a momentum vector relative to the plate motion, and theshape of the plate boundary. If the values are presented for a long-term evaluation, note them accordingly.

If the data on the rake angle are not available, the type of fault in the interplate earthquake is treated as a reverse fault,
and the rake angle is assumed to be 90° (Earthquake Research Committee, 2016).
The rake angle in a source fault of an actual earthquake varies depending on the location on the fault plane.

(d) Average slip amountThe relationship between the average slip amount, D (m), and seismic moment, M0 (N ∙ m), for the entire source fault
can be expressed using the area of the source fault, S (m2), and modulus of rigidity μ (N/m2):
D = M0 / (µ ∙ S )(2)

The modulus of rigidity is assumed to be a standard value for the medium in the area where a source fault is distributed,
and is calculated by the following equation, using the density, ρ (kg/m3), of the earthquake occurring layer, and S wave
velocity β (m/s):
μ=ρ∙β2(3)

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Standard values for the modulus of rigidity of the medium in the marine areas around Japan can be referred to – such as
from a resource by the Japan Society of Civil Engineers (2016). The modulus of rigidity of the medium around the plateboundary surrounding Japan, based on the Japan integrated velocity structure model (provisional version) (Earthquake
Research Committee, 2012), is shown in Table 1.
Table 1. Modulus of rigidity of the medium around plate boundaries surrounding Japan
Depth (referencesclassification by theJapan Society of Civil
Engineers, 2016)
Depth of the entire fault
plane is less than 20 km

Physical values by the Japan
integrated velocity structure model(interim version)

Modulus ofrigidity

Physical value ofthe upper crust

β = 3.4 km/s
ρ = 2.7 g/cm3→μ = 3.12×1010N/m2
Physical value ofthe lower crust

3.5×1010
N/m2

β = 3.8 km/s
ρ = 2.8 g/cm3→μ = 4.04×1010N/m2
Values will be an intermediate of
the above.
Depth of the entire fault
plane is deeper than 20 km

Physical valueof the mantle
β = 4.5 km/s
ρ = 3.2 g/cm3

6.5×1010
N/m2→μ = 6.48×1010 N/m2
Entire fault plane isValues will be an intermediate of
the above.

spread over above and5.0×1010
N/m2

below the 20 km depth

No cases have been found where the average slip of a giant earthquake significantly exceeded 10 m, although average
slips for giant earthquakes are being studied currently.
A study that has estimated the modulus of rigidity near a plate boundary by an analysis of many interplate earthquakes
has shown that it increased with depth (Bilek and Lay, 1999). When the shallow part of the plate boundary is included in thesource fault, caution should be exercised, owing to the fact that a large slip, relative to the size of the earthquake, may occur
as a result of a small modulus of rigidity. In cases where unsolidified sediments are found near the trench axis, deformationof the sediments also contributes to a greater tsunami (for example, Fukao, 1979). An earthquake that causes a large tsunami
relative to the size of the ground motion is generally called a tsunami earthquake, but its generation conditions are currently
under research, as various causes have been proposed.

7

If tsunami magnitude, Mt, is available as an earthquake size information, in addition to the position and shape of thesource fault in a tsunami earthquake, Sections 1.1 (b) and (d) can be omitted, and the average slip can be adjusted such thattsunami magnitude, Mt, calculated from the tsunami height, etc. by tsunami simulation, can match that information.
(e) Rupture processes
While there is diversity in the rupture processes of an earthquake, a characterized earthquake fault model that expresses
the characteristics of a source fault by major parameters typically assumes that a slip across the entire source fault occurs
simultaneously and instantaneously. However, caution should be exercised on the impact of the difference in rupture
velocity on the tsunami simulation.
Rupture velocity is generally faster than the propagation velocity of a tsunami, and therefore is calculated by assumingthat the slip across the entire source fault occurred simultaneously. For example, recent studies have shown that thedifference in rupture velocity had little impact on the waveform of the tsunamis from either the 2003 Tokachi-oki
earthquake, or the 2014 Iquique earthquake (Tsushima et al., 2012; Gusman et al., 2015). On the other hand, the impactof the difference in rupture velocity on the tsunami waveform cannot be ignored in cases where a tsunami that occurs on alarge source fault, or a source fault near the coast, is calculated in detail. When taking a finite rupture velocity into account,arrival time of the tsunami is typically slower, while maximum wave height is large in the direction of propagation and small
on the opposite side, compared to the case in which rupture velocity is assumed to be infinite (e.g. Kawata et al., 2006; Suppasri
et al., 2011). Development of a method for calculating finite rupture velocity is a challenge for the future.
There is a directivity to energy propagation in a tsunami that accompanies an earthquake, and in general, the majority ofthe energy is propagated in the direction of the short axis of the source fault, while a minority of the energy propagates along
the long axis (Kajiura, 1970; Ben-Menahem and Rosenman, 1972). When accounting for the impact of rupture propagation,
directivity of the energy propagation in a tsunami should be noted, and its impact needs to be taken into consideration along
the direction in which the larger amount of energy propagates.
In a case where a slip occurs instantaneously, rise time (the time for which slip continues at the location where the fault
surface exists) becomes 0.

1.2 Microscopic source fault characteristics
Microscopic characteristic parameters for a source fault model include the following:

Location and number of the large slip zone



Slip amount and area of the large slip zone

A large slip zone is configured to account for the impact of heterogeneous slip distribution on the occurrence of a tsunami
and is an area with large slip compared to the background. At its simplest, a two-stage slip distribution is assumed here, where
the slip amount in the large slip zone and the background zone are uniform.
In one case where the fault rupture nearly reached the trench axis, a very large slip occurred near the trench axis, andcaused a very large tsunami (e.g. Satake et al., 2013). In order to account for the impact of such a slip on the occurrence of atsunami, a super large slip zone, where the slip amount is even greater, is assumed within a large slip zone defined along thetrench axis. In this case, the following parameters are configured, in addition to the parameters discussed above:8



Location and number of the super large slip zone



Slip amount and area of the super large slip zone

For the simplest case, a three-stage slip distribution is assumed here, where the slip amount in the super large slip zone,
the large slip zone, and the background zone, are uniform.
Note that the Cabinet Office (2012) allocated conceptual large slip zones and super large slip zones to the largest class oftsunami, when considering government measures to be allocated against disaster prevention.

(a) Location and number of a large slip zone
In order to express the diversity of slip distribution for future earthquakes, several characterized earthquake fault
models have been configured with different large slip zone positions.
A large slip zone takes a shape that aligns with the area of the characterized earthquake fault model, and a single area isdefined in the model. However, for a long-term evaluation, multiple large slip zones may be configured for giant earthquakesthat rupture multiple segments at once.
For example, a large slip zone is configured such that it overlaps along the strike direction of the source fault for
approximately half the length of the large slip zone, while deep, middle, and shallow parts are defined in the dip direction (asshown in Figure 5; however, a total of n × 3 patterns are configured, if n patterns are configured in the strike direction). An
evaluation method that takes differences in probability of occurrence of each model into account should be addressed in the
future.
Intermediate part

Shallow part

Strike direction

Deep partDip
direction

Figure 5. Examples of configured large slip zones: Orange areas are large slip zones, and yellow areas represent thebackground zone
In a case where the result of tsunami simulation does not change significantly by altering the location of a large slip zone
for relatively small earthquakes, multiple characterized earthquake fault models, where different locations of a large slip zone
are configured, can be represented by a single model, once the differences in the results of predictive calculations due todifferences in the location of a large slip zone have been verified.
If the source fault is shallow, and land is located immediately above the fault, the difference in the location of the large
slip zone may have a significant impact on the result of a tsunami simulation. When conducting detailed evaluation of atsunami in such a location, conditions for configuration of the location of a large slip zone need to be investigated individually,
to account for the impact of having a large slip zone in such a diverse location on the evaluation point.

9Given that the coupling rate of the fault plane is higher than that of the surroundings, in an area that becomes a large slip
zone when an earthquake occurs, the location in which slip deficit is large can be assumed to be the large slip zone if the
distribution of slip deficit can be estimated based on crustal movement data. For example, recent studies based on theanalysis of observed crustal movement data have reported that an area with a large slip deficit prior to a great earthquake
became the source area for a great earthquake and slipped significantly (Hashimoto et al., 2009; Nishimura, 2012). In thecase of an interplate earthquake, a gravity anomaly data may also be referenced, since a corresponding relationship has been
pointed out between an area with a negative gravity anomaly and the location of a large slip zone in the source area of theinterplate earthquake (e.g. Wells et al., 2003; Sugiyama, 2004; Song and Simons, 2003).
When estimating the distribution of a slip deficit off the coast from a location with terrestrial crustal movement data, aconstraint condition exists whereby the slip deficit near the trench axis is set to 0 in some cases to avoid an unstable solution– and therefore, caution is advised when using such data.

(b) Slip amount and area of a large slip zone
Slip amount for a large slip zone, DL (m), is represented as an average slip, D (m), of the entire source fault, multipliedby ξL, and the value of ξL is defined based on the area ratio per slip amount, using analysis based on observational datafrom past, great earthquakes.
DL =ξL ∙ D(4)Next, a large slip zone, SL (m2), is represented as the product of the area, S (m2), for the entire source fault, multipliedby rL, and the value of rL is defined, based on the area ratio per slip amount, by the result of an analysis based on
observational data from past great earthquakes.

SL = rL ∙ S(5)Area of the background zone, SB (m2), can be calculated by the following equation:
SB = S − SL

(6)

Slip amount DB (m) for the background is calculated by the following equation, which uses the area, SB (m2), and seismic
moment, M0B (N ∙ m), of the background, where the latter is obtained by subtracting the seismic moment M0L (N ∙ m) of thelarge slip zone from the seismic moment M0 (N ∙ m) for the entire source fault. The value of the modulus of rigidity, μ (N/m2),
is obtained from equation (3) in Section 1.1 (c).
M0L = µ ∙ DL ∙ SL

(7)

M0B = M0 − M0L(8)

DB = M0B / (µ ∙ SB)

(9)

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By substituting equations (2) and (4)–(8) into equation (9), the following equation is obtained as the relationship between
the slip amount, DB (m), in the background zone, and the average slip, D (m), for the entire source fault.
DB = (1 − ξL ∙ rL) D / (1 − rL)(10)

Based on tsunami inversion analysis of past great earthquakes, Figure 6 shows the area ratio (cumulative area ratio) of
the element fault with slip amount over a certain ratio, against the entire fault, presented in descending order of the normalized
slip amount of element faults. The slip amount in a characterized earthquake fault model is expressed as a step function, andeach parameter should be configured such that they approximate the result of analysis.
For example, the function in Figure 6 is generally approximated by configuring the parameters such that ξL = 2, and rL =0.3. As a result, in this example, a large slip zone would be configured such that the slip amount was double the average slip
for the entire source fault and the area was 30% of the entire source fault.
In "Strong ground motion prediction method for earthquakes with specified source faults" ("Recipe") (Earthquake
Research Committee, 2016), the average slip for the entire asperity is also defined as double the average slip for the entire
seismic source fault.

Large slip zone

Background zone

Figure 6. Distribution of normalized slip ratio and cumulative integrated ratio for great earthquakes. The bold black linein the figure represents the parameters for the characterized earthquake fault model
The size of earthquakes in the legend is shown as M, by the Japan Meteorological Agency, and Mw by analysis.

11(c) Location and number of a super large slip zone
Super large slip zones are defined to consider extremely large slips near a trench axis, and a single area can be definedwithin a single large slip zone in a manner such that it is adjacent to the upper end of the large slip zone in a dip direction.
A super large slip zone typically features a shape that aligns with the large slip zone.
In order to account for diverse slip distributions, large slip zones and super large slip zones are configured in a mannersuch that a large slip zone overlaps for approximately half the length of the super large slip zone in the strike direction (asDip direction

shown in Figure 7).

Strike direction
Trench axis

Figure 7. An example of configuration of a super large slip zone: Red represents the super large slip zone, orange
represents the large slip zone, and yellow represents the background zone.

12(d) Slip amount and area of the super large slip zone
Slip amount, DS (m), in the super large slip zone is represented as an average slip, D (m), of the entire source fault,
multiplied by ξS. The slip amount, DL’ (m), in the large slip zone is represented as an average slip, D (m), of the entire
source fault, multiplied by ξL’, ξS and ξL’ values are defined based on the area ratio by slip amount, from analysis resultson observed cases of past giant earthquakes.
DS = ξS ∙ D(11)DL’ = ξL’ ∙ D(12)Next, the super large slip zone, SS (m2), is represented as the product of the area of the source fault, S (m2), and rS, andthe large slip zone, SL’ (m2), is represented as the product of the area of the source fault, S (m2), and r L’. Then rS and r L’
values are defined, based on the area ratio by slip amount from analysis based on observed cases of past giant earthquakes.

SS = rS ∙ S(13)SL’ = rL’ ∙ S(14)The area of the background zone, SB’ (m2), can be calculated by the following equation:
SB’ = S − SS− SL’(15)

Slip in the background zone, DB’ (m), is calculated by the following equation, using the area of the background zone, SB’
2(m ), and the seismic moment of the background zone, M0B’ (N ∙ m). The latter parameter is obtained by subtracting theseismic moment of the super large slip zone, M0S (N ∙ m), and the seismic moment of the large slip zone, M0L’ (N ∙ m), fromthe seismic moment for the entire source fault, M0 (N ∙ m). The value of the modulus of rigidity μ (N/m2) is the value obtained
from equation (3) in Section 1.1 (c).
M0S = µ ∙ DS ∙ SS

(16)

M0L’ = µ ∙ DL’ ∙ SL’(17)

M0B’ = M 0 – M0S − M0L’

(18)

DB’ = M0B’/ (µ ∙ SB’)(19)

Equation (20) is then obtained as the relationship between the slip amounts in the background zone, DB’ (m), and theaverage slip, D (m), for the entire source fault, by substituting equations (2) and (11)–(18) into equation (19):DB’ = (1 − ξS ∙ rS− ξL’ ∙ r L’) D / (1 − rS − r L’)

13

(20)

Based on tsunami inversion analysis of past giant earthquakes, Figure 8 shows the area ratio (cumulative area ratio) of
the element fault with a slip amount over a certain ratio, against the entire fault, presented in descending order of thenormalized slip amount of element faults. The slip amount in a characterized earthquake fault model is expressed as a step
function, and each parameter should be configured such that they approximate the results of analysis.
For example, the function in Figure 8 is generally approximated by configuring parameters as ξS= 4, rS= 0.1, ξL’ = 2, andr L’ = 0.2. This means that, in this example, the super large slip zone could be configured such that the slip amount was 4times the average slip for the entire source fault, the area was 10% of the entire source fault, while the large slip zone could
be configured such that the slip amount was double the average slip for the entire source fault, and the area was 20% of theentire source fault (i.e. the super large slip zone and large slip zone together account for 30% of the entire source fault).

Super large slip zone

Large slip zone

Background zone

Figure 8. Distribution of normalized slip ratio and cumulative integrated ratio for giant earthquakes. The bold black line
represents the parameters for the characterized earthquake fault model
The size of earthquakes in the legend is shown as M by the Japan Meteorological Agency, and Mw by the result of theanalysis adopted.

14

2. Tsunami simulation and uncertainties in the tsunami prediction
This chapter explains the method of tsunami simulations and its uncertainties. In this method, a topography model isdeveloped for marine and terrestrial areas first, and then crustal movements and the distribution of initial water height arecalculated, based on a source fault model configured in "1. Configuration of characterized earthquake fault model". Also,
the propagation and run-up of a tsunami are calculated to predict the coastal tsunami height. Results from tsunami simulations
based on numerical computation include uncertainties such as computational errors, and the section explains this factor as
well.
2.1 Tsunami simulations(a) Computational area and grid spacing
Computational area and computational grid spacing in tsunami simulations are configured so that tsunami behaviors
such as refraction, reflection, and run-up, etc. are calculated with good accuracy, while taking features such as the size of
the wave source area and topography of the sea floor and shores into account.
The ideal computational grid spacing for tsunamis is known to be less than 1/20 of a single wavelength of a tsunami(Hasegawa et al., 1987). Tsunami wavelengths become shorter as water becomes shallower, and nesting would be the ideal
approach where areas with different computational grid spacing (that vary depending on the wavelength and topography of
the tsunami) are connected and computed simultaneously. When nesting, a method in which computational grid spacing is
reduced by a ratio of 1/3 or 1/2 is used, since short wavelength components that occurred in a small area may be trapped
within the small area without propagating to a larger area.
Computational grid spacing needs to be configured appropriately for the required computational accuracy. It has been
configured as 1350 m, 450 m, 150 m, and 50 m from the open ocean to coastal areas in the appendix case, since its objective
was to provide an overview of tsunami height across Japan.
When calculating the run-up, computational grid spacing in a terrestrial area needs to be configured smaller byapproximately 10 m (e.g. Ministry of Land, Infrastructure, Transport and Tourism, 2012). However, no significant difference
has been observed, in terms of either the coastal level or the maximum water level in particular, due to differences in terrestrial
computational grid spacing (Murashima et al., 2006).
(b) Discretization of topographic and bathymetric data
The topographic and bathymetric data have a significant impact on the results of calculation for the propagation andrun-up of tsunamis, and are therefore discretized appropriately, based on the latest surveys and studies. The topographic
data are typically produced from high-precision elevation data, obtained by airborne LiDAR survey, etc.
Generally, elevation of the terrestrial area is determined using the Tokyo Peil (T.P.) as the reference plane, and the heightof the sea surface in the marine area (tidal level) is determined using the datum line (D.L.) established for each tidal station
as the reference plane. Tsunami predictions need to be calculated by treating marine areas and terrestrial areas as a whole,and the topographic and bathymetric data used generally uses the T.P. as the reference plane.
Coastal structures are usually included as needed.

15(c) Crustal movement and distribution of initial water height
Estimation of the distribution of initial water height generally considers the contribution in vertical direction from thehorizontal crustal displacement of the seafloor slope to the vertical crustal displacement (uplifting and settling) at sea floor
calculated by the source fault model.
Methods for calculating the crustal movement that appears on the surface of an elastic half-space body by the source fault
model include Sato (1971), Sato and Matsu'ura (1973), Yamashita and Sato (1974), and Mansinha and Smylie (1971), whilea program by Okada (1985, 1992) is used widely. Slip amount is uniform in all of these methods, where the theoreticalformula addresses a rectangular fault where one side is parallel to the ground surface. For calculations of the crustal
movement that corresponds to a more realistic source fault model, in which the complex shape of the plate boundary surface
and the slip amount vary by location, the source fault is divided into small rectangular and triangular element faults (Thomas,
1993; Meade, 2007). These small faults are aligned to match the configuration of the upper surface of the oceanic plate asmuch as possible, and the results from individual calculations are combined by configuring the slip amount per element fault.
However, in places where the shape of the upper surface of the oceanic plate changes rapidly, and for boundaries where
the configured slip amount changes significantly, crustal movement of the sea floor obtained by overlapping rectangularfaults depends greatly on the placement of element faults. In such cases, caution needs to be exercised in that actual fault
movement cannot be modeled efficiently – and so an unrealistic crustal movement may be obtained in some cases.
If the contribution of the horizontal crustal movement to the vertical displacement of the sea floor is 𝑢𝑢

, then the initial water height is expressed as 𝑢𝑢𝑢 (Tanioka and Satake, 1996). Here, 𝑢 and 𝑢 are displacementsof the sea floor in a horizontal direction (x-axis direction and y-axis direction, respectively), 𝑢 is the displacement of the sea
floor in vertical direction, and H is the water depth.
If the spatial movement wavelength of the crustal movement has become significantly shorter, compared to thewavelength of tsunami propagation, the hydraulic filter developed by Kajiura (1963) should be applied to the amount ofcrustal movement, and the result should be treated as an initial water height.
(d) Baseline tidal level
Baseline tidal level typically uses the Tokyo Peil (T.P.).
Baseline tidal level can also be set as the mean sea level (M.S.L.), at that point. Tidal conditions should be configured
appropriately, depending on the use of the results from tsunami simulations.
In a tsunami simulation, the tidal level used to configure an assumed tsunami inundation should, in general, be the highwater level (H.W.L.) (Ministry of Land, Infrastructure, Transport and Tourism, 2012).
(e) Simulations for tsunami propagation and run-up
Simulations for the propagation and run-up of tsunamis are based on nonlinear long-wave theory (shallow water
theory), which accounts for advection and friction on the sea floor. However, a linear long-wave theory can also be appliedin a marine area with a deeper water.

16

Most tsunamis occur when a crustal movement involves displacement of the sea floor area from several to a few dozen
meters in the vertical direction, in a location where the sea floor area is less than several kilometers in depth, with a spatial
spread of several dozens to several hundred kilometers in the horizontal direction. These tsunamis can be considered aswaves with a sufficiently large scale in the horizontal direction rather than the vertical direction. On this basis, many aspectsof the tsunami phenomenon can be described by theoretical formulae, applying long-wave theory.
Long wave theory is derived by treating the law of conservation of mass (continuity equation) and law of conservationof momentum (Euler’s momentum equation) as base equations. For nonlinear long-wave theory (shallow water theory),
which accounts for friction at the sea floor and a nonlinear effect by advection that cannot be ignored in a shallow marine
area, the theoretical formula of an integral model, calculated by integrating from the sea floor to the water surface in thevertical direction, is expressed as follows:
𝜕𝜂
𝜕𝑡
𝜕𝑀
𝜕𝑡

𝜕 𝑀
𝜕𝑥 𝐷

𝜕𝑀
𝜕𝑥

𝜕𝑁
𝜕𝑦

𝜕 𝑀𝑁
𝜕𝑦 𝐷

0

𝑔𝐷

𝜕𝜂
𝜕𝑥

(Continuity equation)𝜏𝜌

0(Euler’s momentum equation)

𝜕𝑁
𝜕𝑡

𝜕 𝑀𝑁
𝜕𝑥 𝐷

𝜕 𝑁
𝜕𝑦 𝐷

𝑔𝐷

𝜕𝜂
𝜕𝑦

𝜏
𝜌

0

In the momentum equation, the first term is the local acceleration term, the second and third terms are advection terms,
fourth term is the pressure term, and the fifth term is the friction term. Friction term is generally expressed as the below, witha resistance term for a steady flow based on Manning formula.

/

𝑀√𝑀

𝑁 ,

/

𝑁√𝑀

𝑁 ,

where, η (m) is the wave height, D (m) is the total water depth (=h (m) + η (m)), g (m/s2) is the gravitational acceleration, ρ(kg/m3) is density, τx (N/m2) and τy (N/m2) are frictional forces at the sea floor in the x-axis and y-axis directions, respectively.
M (m2/s) and N (m2/s) are linear flow rates in the x-axis and y-axis directions, respectively. 𝑀

𝐷𝑢 , and 𝑁

𝐷𝑣,

where 𝑢 and 𝑣 are average cross-sectional flow velocities in the x-axis and y-axis directions in the unit of m/s, respectively.
n (m－1/3 ∙ s) is Manning’s roughness coefficient.
The relationship between coordinates and each parameter is as shown in Figure 9.

(Direction of the tsunami)

Figure 9. Coordinate system and parameters of the theoretical formula17

In marine areas deeper than 50 m, the impact in the momentum equation of the advection and friction terms becomes
smaller, and a linear long wave theory that omits these terms can be applied. Figure 10 covers the results of a numericalexperiment in which each term in the momentum equation for shallow water theory was supplied by the one-dimensional
propagation calculation (Imamura et al., 1986). In this case, it can be concluded that, if the only factors involved in
momentum are local terms, and the pressure term is for water depth greater than 50 m, the size of the advection term at depths
shallower than 50 m would be over 5% of the maximum value of the local term, and as much as 50% during a run-up.
Contribution of each term

Gravity term(pressure term)
Local acceleration term

Friction term

Advection terms

Grid number

Water depth

Figure 10. Comparison of the contribution of each term in the momentum equation using shallow water theory (Imamura
et al., 1986). The maximum value of each term is normalized by the maximum value of a local term, while grid 20 on thehorizontal axis is equivalent to the shoreline.
In a case where a detailed prediction is calculated for a tsunami that propagates for a long distance over a deep sea area,or for a tsunami that propagates over a broad shallow sea area in a near-tsunami, a dispersive wave theory needs to be utilized
to account for dispersion of the tsunami (Iwase et al., 2002; Shigihara and Fujima, 2007).
(f) Boundary conditions
The sea (offshore) boundary is permeated by radiation with a perfect non-reflection, while the land boundary is
typically considered as a run-up.
For a run-up calculation, the presence of water is determined for each computational grid per time step over the courseof calculation, and flow needs to be configured by accounting for the relationship with the water level in adjacentcomputational grids. The methods by Iwasaki and Mano (1979) and Kotani et al. (1998) are frequently used for thiscalculation.
(g) Coefficient of roughness
Bottom friction in marine and terrestrial areas is taken into account by utilizing a coefficient of roughness.
The value of the Manning’s roughness coefficient in a marine area is typically n = 0.025 m- ⅓ ꞏ s. Various coefficients ofroughness have been proposed for terrestrial areas, such as that estimated directly from the results of hydraulic modelexperiments (Fukuoka et al., 1994; Goto and Shuto, 1983), that which uses parameters such as the area occupancy ratio of a18

house and house width (Aburaya and Imamura, 2002), that which provides an equivalent roughness in a uniform manner
per area (Aida, 1977), and that which provides a roughness estimate for various types of land conditions (Kotani et al., 1998).
All of these methods can be considered when configuring the coefficient of roughness for terrestrial areas in a detailed
tsunami run-up calculation.
However, there are issues in applying the Manning formula on a run-up – particularly in the case of a rapidly varying,unsteady flow – and the resistance formula applicable to the tip of a tsunami running up is still a matter for research.
(h) Length of computation time and computational time step interval
The length of computational time for tsunami simulation needs to be configured so that the maximum tsunami height
can be obtained while considering the topography of the tsunami, as well as that of the affected marine and terrestrial
areas. The computational time step interval for tsunami simulations is configured appropriately by considering the stabilityof calculations.
The first wave is not necessarily the largest wave in a tsunami; the second wave often is. Therefore, a sufficient length ofcomputational time should be configured to obtain the maximum height correctly.
Computational time interval should be configured by considering the stability, etc. of calculations against computational
grid spacing, and should satisfy the following conditions (the Courant-Friedrichs-Lewy (CFL) Condition).
∆𝑡

∆𝑥
2𝑔ℎ

In the CFL condition, Δt (s) is the computational time step interval, Δx (m) is the computational grid spacing, hmax (m) is
the maximum water depth, and g (m/s2) is the gravitational acceleration. Actual calculations will involve numerical errors
and the nonlinearity of the phenomena, however, and the computational time interval should be configured sufficiently small
compared to the CFL conditions.

192.2 Uncertainties in tsunami predictions
When configuring the characteristic source fault model, as well as uncertainties regarding the information that can beobtained in advance on the source fault, other uncertainties in tsunami predictions include an inability to obtain sufficient
information on the following:

Location and size of the assumed earthquakeSpatial distribution of the slip amount in the source fault

Results for tsunami simulations obtained when configuring characterized earthquake fault models individually,
involve the following uncertainties:

Errors due to simplification of the source fault model, and expression using the finite number of parametersErrors from numerical calculations, including an error introduced by describing natural phenomena with agoverning equation derived from certain approximations, an error introduced by discretizing the governing
equations, and an error introduced by using an approximation of boundary conditions



Errors contained in the topographic and bathymetric data

When sufficient information on the location and size of the assumed earthquake cannot be obtained, several characterized
earthquake fault models should be configured using different locations and sizes, a tsunami prediction should be calculated
for each model, and the tsunami should then be evaluated while accounting for uncertainties. When sufficient informationon the spatial distribution of the slip in the source fault cannot be obtained, several characterized earthquake fault models
should be configured, using different locations of the large and super large slip zones, as defined in 1.2 (a) and (c), then atsunami simulation should be performed for each model, and the tsunami should be evaluated, while recognizing theuncertainties.

203. Validation of the characterized earthquake fault model
Damage records and observations for tsunamis are often available for the latest seismic activities, since interplate
earthquakes considered in a tsunami evaluation have a short interval of occurrence, from several decades to over a hundredyears. Therefore, the validity of the characterized earthquake fault model may be confirmed, and source fault characteristics
may be revised by comparing such information with tsunami prediction results based on the methods discussed in thisdocument. Items for comparison include estimated values based on tsunami trace height.
The validity of a characterized earthquake fault model can be confirmed by comparing the trace heights of tsunamis
from previous earthquakes with the results from tsunami simulations obtained by applying a characterized earthquake
fault model that corresponds to the earthquake. The model is considered to be valid if spatial features of the trace height
can be explained to an acceptable level by simulations.
If spatial features of the trace height in some areas cannot be explained by tsunami simulations made using thecharacterized earthquake fault model, the model can be nonetheless considered valid in some circumstances. Such a situation
could be if the model provided better explanation for spatial features of the trace height than tsunami prediction results basedon several characterized earthquake fault models that used different configurations of the large and super large slip zone
locations.
If sufficient information on the location and size of previous earthquakes cannot be obtained, the model can still beconsidered valid if a certain level of explanation can be provided for spatial features of the trace height. This may be achieved
using a set of tsunami predictions based on several characterized earthquake fault models with differently configured
locations and sizes.
When comparing trace height records with the results of tsunami simulations, caution should be observed, as a traceheight may be disproportionately high, due to tsunami run-up being affected by the local topography. In addition, it should
be recalled that some data may be affected by the tidal conditions applicable at the time of the trace height observations.

21

4. Appendix: Validation of the tsunami prediction method for earthquakes with characterized source
faults (Tsunami Recipe)
This section compares the results of tsunami predictions for the 2011 off the Pacific coast of Tohoku earthquake (M 9.0) for
which many observation records are available, using calculations based on available records, such as the tsunami trace height, andthe "Tsunami prediction method for earthquakes with characterized source faults" (i.e., this "tsunami recipe"). The tsunami recipe
will be validated by confirming the validity of the characterized earthquake fault model.
Parameters were configured based mainly on "The Long-term evaluation of subduction zone earthquakes along the areafrom off Sanriku to off Boso (ver. 2)" (Earthquake Research Committee, 2011) (i.e. "long-term evaluation").
The source fault model was configured to represent the region from the central Sanriku coast to the Ibaraki coast (including
areas closer to the trench), as this region was identified as the source type of the off the Pacific coast of Tohoku earthquake in thelong-term evaluation. Large slip zones and the super large slip zone were taken into account. The validation was performed witha model that configured the northern edge of the source fault as both the large slip zone and a super large slip zone, that being theconfiguration considered a model most suited for explaining the trace height left by the tsunami from the 2011 off the Pacific coastof Tohoku Earthquake.
Figure A1 and Tables A1 and A2 show an image of the characterized earthquake fault model and its parameters, respectively.

Figure A1. Characterized earthquake fault model: Red represents the super large slip zone, orange represents the large slip
zone, and yellow represents the background zone

22

Table A1

Table A2

Parameters for macroscopic source fault characteristics

Parameters for microscopic source fault characteristics

configured based on the tsunami recipe

configured based on the tsunami recipe

Item

Value

Item

Value

Fault area S (km2)

1.05×105

46.0

Seismic moment M0 (Nm)

6.1×1022

Average slip for the super large slip
zone DS (m)1.05×104

Moment magnitude Mw

9.1

Area of the super large slip zone SS(km2)

Average stress drop Δσ (MPa)

4.3

Average slip for the large slip zone
DL (m)

23.0

Modulus of rigidity μ (N/m2)

5.0×1010

Area of the large slip zone SL (km2)

2.11×104

Average slip D (m)11.5

Average slip for the backgroundzone DB (m)3.3

Area of the background zone SB(km2)

7.38×104

At this point, tsunami trace height predictions based on the tsunami recipe were relatively high compared to the actual traceheights left by the tsunamis. The reason for this is that the variable used was Mw 9.1, which was obtained by using an empirical
relationship derived from the seismic moments and fault areas for earthquakes that have occurred on the Pacific side of Japan(Fujiwara et al., 2015), as shown in Table A1. Therefore, the average stress drop was adjusted to be Mw 9.0, to match the M 9.0
actually observed for the 2011 off the Pacific coast of Tohoku Earthquake, and the tsunami predictions were re-calculated, after
re-configuring the parameters, as shown in Tables A3 and A4.
Table A3

Table A4
Macroscopic and microscopic source fault characteristics

Source fault characteristics parameter adjusted to be Mw 9.0

parameters adjusted to be Mw 9.0

Item

Value

Item

Value

Fault area S (km2)

1.05×105

Average slip for the super large slip
zone DS (m)32.0
1.05×104

22

Seismic moment M0 (Nm)

4.2×10Moment magnitude Mw

9.0

Area of the super large slip zone SS(km2)

Average stress drop Δσ (MPa)

3.0

Average slip for the large slip zone
DL (m)

16.0

Modulus of rigidity μ (N/m2)

5.0×1010

Area of the large slip zone SL (km2)

2.11×104

Average slip D (m)8.0

Average slip for the backgroundzone DB (m)2.3

Area of the background zone SB(km2)

7.38×104

Trace heights left by tsunamis were based on the data of The 2011 Tohoku Earthquake Tsunami Joint Survey (TTJS) Group(http://www.coastal.jp/ttjt/, Oct. 3, 2012) and those that matched the following conditions were extracted (Korenaga et al., 2013).
Trace reliability grade A, as defined by The 2011 Tohoku Earthquake Tsunami Joint Survey (TTJS) Group.

Trace height at or over 50 cm, and coastal upstream distance known within 100 m

23



If there are structures nearby that cannot be expressed on a 50 m computational grid, the computational grid of thecoast and trace point found were found to be within 50 m of each other

Comparisons of trace heights extracted from Aomori Prefecture to Chiba Prefecture (●), and calculated heights for thecomputational grid for the coastal areas closest to the trace height locations (◆) are shown in Figure A2. The height calculatedby the Mw 9.0 model was able to match the trace heights, in general, and the validity of the characterized earthquake fault
model can be confirmed by this result.
However, the trace height was not matched very well by the calculated height in some areas, particularly in Fukushima
Prefecture, for example, which highlights the limit of the characterized earthquake fault model. This point needs to be kept inmind when using the characterized earthquake fault model

Oma Zaki

Shiriya Zaki

Oma ZakiTrace heightCalculated height

Shiriya Zaki

Hachinohe

Hachinohe

Kuji

Kuji

MiyakoMiyakoKamaishi

KamaishiKesennuma

Kesennuma

Onagawa

Onagawa

Ishinomaki

Ishinomaki

Matsushima

Matsushima

Soma

Soma

OnahamaOnahama

Hitachi

HitachiKashima
Katsuura

Trace heightCalculated height

KashimaInubo Saki

Katsuura

Nojima Zaki

Inubo Saki

Nojima Zaki

Figure A2. Comparison of actual trace heights and calculated heights obtained by using the characterized earthquake fault
model; the calculated heights in the left figure were obtained using the Mw 9.1 model, while those for the right figure were obtainedusing the Mw 9.0 model.

24

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Rake angle

Verification witha past records

Seismic moment of the large
slip areas M0L

Eq. (7)

Average slip of the large
slip areas DL

Area of the large slip areas SL

When a past tsunami record is available

Eq. (4)Eq. (5)Characteristics of the large slip area

Microscopic source fault
characteristics

Characterized source fault modelNo

Yes
Goal of tsunami evaluation

Revise a characterized source fault model

Eq. (8)

Seismic moment of the
backgrournd areas M0B

Eq.(9)

Average slip of the
background areas DB

Area of the background areas SB

Eq. (6)

Characteristics of the background area

Figure F1. Flowchart used for configuration of source fault characteristic parameters for interplate earthquakes (large slip zones configured)

Tsunami simulation with a characterized
source fault model

Rigidity μ

Average slip D

Eq. (2)

Seismic moment M0

Eq. (1)

Area of the source fault S

Shape of a source fault

Macroscopic source fault
characteristics

Long-term evaluation, Source faults of past events

Rake angle

Seismic moment of thelarge-slip area M0L’

Eq. (17)

Average slip of the large
slip areas DL’

Area of the large slip areas SL’

Characteristics of the large slip area

Microscopic source fault
characteristics

Yes

Revise a characterizedsource fault model

Goal of tsunami evaluation

Eq. (18)

Seismic moment of the
backgrournd areas M0B’

Eq(19)

Average slip of the
background areas DB’

Area of the background areas SB’

Eq. (15)

Characteristics of the background area

Figure F2. Flowchart used for configuration of source fault characteristic parameters for interplate earthquakes (large slip zones and super large slip zones configured)

No

Seismic moment of the superlarge slip area M0S

Eq(16)

Average slip of the superlarge slip area DS

Eq(11)

Eq(13)

Area of the super large slip area SS

Characteristics of the super large slip area

Verification witha past records

Characterized source fault model

When a past tsunami record is available

Eq. (12)Eq. (14)

Tsunami simulation with a characterized
source fault model

Rigidity μ

Average slip D

Eq. (2)

Seismic moment M0

Eq. (1)

Area of the source fault S

Shape of a source fault

Macroscopic source fault
characteristics

Long-term evaluation, Source faults of past events