Quantitative Assessment of Epistemic Uncertainties in Tsunami Hazard Effects on Building Risk Assessments

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1 geosciences Article Quantitative Assessment of Epistemic Uncertainties in Tsunami Hazard Effects on Building Risk Assessments Yo Fukutani 1, * ID, Anawat Suppasri 2 Fumihiko Imamura 2 1 College of Science Engineering, Kanto Gakuin University, Yokohama , Japan 2 International Research Institute of Disaster Science, Tohoku University, Sendai , Japan; suppasri@irides.tohoku.ac.jp (A.S.); imamura@irides.tohoku.ac.jp (F.I.) * Correspondence: fukutani@kanto-gakuin.ac.jp; Tel.: Received: 30 September 2017; Accepted: 5 January 2018; Published: 10 January 2018 Abstract: Based on definition of risk, we quantitatively evaluated annual expected loss ratio (Tsunami Risk Index) clarified quantitative effects of epistemic uncertainties in hazard assessments on risk of buildings by combining probabilistic information regarding inundation depths at target points fragility assessments of buildings. For risk assessment, we targeted buildings with four different structures (reinforced concrete, steel, brick, wood) located in three different areas (Soma, Sendai, Kesennuma). In conclusion, we demonstrated that expected risk could vary by approximately two orders of magnitude when considering hazard uncertainties between 95th percentile 5th percentile. In addition, we quantitatively clarified fact that we cannot properly underst risk by evaluating fragility alone. For example, analysis results indicate that risk of a wood building located in Kesennuma is lower than that of a reinforced concrete building located in eir Soma or Sendai. Keywords: hazard uncertainty; risk; risk quantification; quantitative effect 1. Introduction In general, risks of natural disasters can be understood by multiplying magnitude of hazard evaluated vulnerability [1]. The magnitude of hazard must include severity of hazard its occurrence probability. Therefore, it is necessary to carry out probabilistic natural hazard assessments in order to properly evaluate such risks. Based on this concept, risks can be understood by multiplying probabilistic hazard assessment fragility assessment, which represents vulnerability of. In or words, neir hazard nor fragility alone can capture risk for an appropriate evaluation. Numerous studies on probabilistic hazard assessment methods fragility evaluation methods have been conducted individually but few studies have quantitatively assessed risks by multiplying m toger. Moreover, probability estimations of inundation depths at inl locations constitute necessary input information for quantitative evaluation of risks; however, not many studies have conducted probabilistic inundation assessments (e.g., [2 8]). González et al. [2] applied a probabilistic hazard assessment methodology to Seaside, Oregon, that combines inundation modeling with probabilistic concepts. Goda et al. [4] developed stochastic rom-field slip models for 2011 Tohoku earthquake conducted a inundation simulation using those developed models, after which y concluded that inundation heights at a coastal location are major sources of uncertainties in prediction of risks. Park Cox [5] demonstrated an approach for Geosciences 2018, 8, 17; doi: /geosciences

2 Geosciences 2018, 8, 17 2 of 27 prediction of risks. Park Cox [5] demonstrated an approach for assessing probabilistic assessing near-field probabilistic hazards near-field within Cascadia hazards Subduction within Zone Cascadia using Subduction logic tree method. Zone using logic The tree logic method. tree method has been widely utilized to conduct probabilistic hazard analysis (e.g., The [5 10]). logic In tree this method method, has been uncertainties widely utilized in to conduct hazard probabilistic assessments are classified hazard as analysis eir (e.g., epistemic [5 10]). uncertainties In this method, or aleatory uncertainties uncertainties. in Historically, hazard Cornell assessments [11] categorized are classified as types eir of epistemic uncertainties uncertainties that should or be aleatory considered uncertainties. during an Historically, evaluation Cornell of strong [11] earthquake categorized motion types into of epistemic uncertainties uncertainties that should aleatory be considered uncertainties. during It an has evaluation since become of strong commonplace earthquake to evaluate motion into epistemic epistemic uncertainties aleatory uncertainties aleatory in probabilistic uncertainties. It has hazard since become assessments. commonplace As shown to in evaluate Figure 1, epistemic epistemic uncertainty aleatory can uncertainties be captured in using probabilistic a logic tree, while hazard aleatory assessments. uncertainty As shown can be in Figure evaluated 1, through epistemic a probability uncertainty distribution can be captured function using of a logic tree, height. while aleatory uncertainty can be In evaluated this study, through we first a demonstrate probability distribution a method for function evaluating of probabilistic height. inundation depths In this at inl study, locations we first demonstrate by employing a method a probabilistic for evaluating hazard probabilistic assessment model inundation using depths logic tree at inl method. locations The demonstrated by employing probabilistic a probabilistic inundation hazard assessment depths are model combined using with logic information tree method. regarding The demonstrated fragility probabilistic of buildings inundation is refore depths capable are of combined quantitatively with information evaluating regarding risk of a specific fragility building of buildings located at a specific is refore location. capable If of quantitatively risk can evaluating be evaluated quantitatively, risk of it a is specific possible building to objectively located compare at a specific location. risks among If buildings with risk can different be evaluated structures quantitatively, in different areas. it is possible Since to objectively method proposed compare in this risks study among is relatively buildings simple, with different its calculation structures cost is in smaller different than areas. those Since of methods method proposed employed in within this study previous is relatively studies. simple, its calculation After cost 2011 is smaller Great East than Japan those Tsunami, of methods many improvements employed within of previous countermeasures studies. have been After made in 2011 both Great structural East Japan measures Tsunami, many nonstructural improvements measures of but large countermeasures future challenges have been remain made such in as both probabilistic structural measures risk prediction nonstructural an advanced measures simulation but large technique future challenges system remain for such real-time as probabilistic hazard etc. risk [12]. prediction The method proposed an advanced in this simulation study will technique be used to produce system for real-time risk hazard index using etc. [12]. objective The method judgment proposed for various in this situations study will in disaster be used prevention to produce scenarios. risk Consequently, index using objective we can also judgment examine for how various evaluation situations of in disaster prevention risk index scenarios. varies depending Consequently, on we structures can also examine of buildings how evaluation or regional of differences. risk The index method varies is depending intended on to quantitatively structures of clarify buildings contribution or regional from differences. epistemic The uncertainty method is intended in to quantitatively hazard evaluation clarify to contribution quantitative from amount epistemic of risk. uncertainty in hazard evaluation to quantitative amount of risk. (a) (b) Figure Figure A diagram diagram of of evaluation evaluation methods methods for for (a) (a) epistemic epistemic uncertainty uncertainty using using a logic logic tree tree [10] [10] (b) (b) aleatory aleatory uncertainty uncertainty using using a probabilistic probabilistic density density function function [13]. [13].

3 Geosciences 2018, 8, 17 3 of Methodology for Tsunami Risk Assessment Quantification of Hazard Uncertainty Figure 2 shows workflow for quantitative assessment of risk posed by a of direct damage of a building located at a specific location in consideration of hazard uncertainties. First, wave height (h) simulated from each fault source exceeding a specific level (h th ) is determined as follows: q ijk (h th) = h th p ijk(h)dh (1) where i is number of assumed fault sources in one earthquake region, j is number of logic tree branches, k is number of assumed earthquake regions p ijk (h) is aleatory uncertainty, which is defined as a lognormal normal distribution. ( Then, annual exceedance frequency of wave height in one earthquake region (Q jk h th)) is calculated as follows using annual exceedance frequency of each fault source µ i : Q jk (h th) = q ijk (h th) µ i (2) i If we assume a Poissonian arrival ( time ( distribution, annual exceedance probability of wave height in one earthquake region f jk h th)) is given using following exponential function: f jk (h th) ( = 1 exp ( Q jk h th)) (3) By considering annual excess probability of wave height created for each logic tree branch weight of each logic tree branch (w j ), annual exceedance probability of wave height for each percentile (perc) in each earthquake region (F k, perc (h th) ) can be obtained as follows: F k, perc ( h th) = f jk ( h th, w j ) After calculating se curves for each earthquake region, we can obtain annual exceedance probability of wave height for each percentile (perc) for entire earthquake region (P perc (h th) ) through following formula (Figure 2a): P perc (h th) ( = 1 (1 F k, perc h th)) (5) k We convert this annual exceedance probability of wave height into annual exceedance probability of inundation depth using results from numerical simulation (Figure 2b). Meanwhile, assuming that -induced damage level of each building is n, damage probability for each damage level (d n (h)) can be written as follows (Figure 2c): ( ) ln(h) λ d n (h) = Φ ζ where Φ( ) n represents stardized normal distribution function for damage levels λ ζ represent mean stard deviation of ln(h), respectively. Then, by using damage probability coefficient for each damage level (a n ), damage probability function (D(h)) can be written as follows: 6 D(h) = a l (d l (h) d l 1 (h)) (7) l=1 n (4) (6)

4 Geosciences 2018, 8, 17 4 of 27 Geosciences 2018, 8, 17 4 of 27 (a) Probabilistic wave height assessment (Section 3.2) 1 Annual Annual Exceedance Exceedance Probability Probability Tsunami height (m) 1 5 percentile 50 percentile Average 95 percentile (b) Probabilistic inundation assessment (Section 3.3) 5 percentile 50 percentile Average 95 percentile Tsunami inundation depth (m) Steel Building Tsunami inundation height (m) (c) Fragility assessment (Section 3.4) (d) Risk assessment (Section 3.5) Minor Moderate Major Complete Collapsed Washed away Reinforced Concrete Steel Brick Wood 5 percentile hazard Annual expected loss ratio (%/year) E-03 E-04 E-05 Wood Steel Brick RC Wood Steel Brick RC Wood Steel RC Brick Soma Soma Soma Soma SendaiSendaiSendaiSendaiKesen Kesen Kesen Kesen numa numa numa numa E-02 E-03 (b) Annual expected loss ratio (5th percentile) (c) Annual expected loss ratio (50th percentile) E-04 Wood Steel Brick Wood RC Steel Brick RC Wood Steel Brick RC Soma Soma SomaSendai SomaSendaiSendaiSendaiKesen Kesen Kesen Kesen numa numa numa numa E-01 E-02 (e) Tsunami Risk Index uncertainty assessment (Section 3.5) In case of using 5 th percentile hazard In case of using 50 th percentile hazard In case of using 95 th percentile hazard (d) Annual expected loss ratio (95th percentile) E-03 Wood Steel Brick RC Wood Steel Brick RC Wood Steel Brick RC Soma Soma Soma Soma SendaiSendaiSendaiSendaiKesen Kesen Kesen Kesen numa numa numa numa Figure 2. Flow of risk quantification. (a) The probabilistic wave height is converted to (b) probabilistic inundation information, Figure 2. Flow of risk quantification. (a) The probabilistic wave height is converted to (b) probabilistic inundation information, (d,e) risk assessment is carried out by combining hazard information with (c) fragility assessment. (d,e) risk assessment is carried out by combining hazard information with (c) fragility assessment.

5 Geosciences 2018, 8, 17 5 of 27 Geosciences 2018, 8, 17 5 of 27 In this study, we consider six damage levels: minor damage, moderate damage, major damage, In this study, we consider six damage levels: minor damage, moderate damage, major damage, complete damage, collapsed damage washed away. Consequently, by using annual exceedance complete damage, collapsed damage washed away. Consequently, by using annual probability of inundation depth (Equation (5)) damage probability (Equation (7)), exceedance probability of inundation depth (Equation (5)) damage probability (Equation (7)), risk curve for each percentile hazard can be written as follows (Figure 2d): risk curve for each percentile hazard can be written as follows (Figure 2d): ( R perc Pperc, D ), (8) We should note that we have not performed a risk estimation that includes variability We should note that we have not performed a risk estimation that includes variability in in building response uncertainties in loss estimates because main objective of this building response uncertainties in loss estimates because main objective of this study is study is to quantify epistemic uncertainties in hazards affecting building risk. Future to quantify epistemic uncertainties in hazards affecting building risk. Future research research will include or uncertainties in addition to those of hazards within risk assessment. will include or uncertainties in addition to those of hazards within risk assessment. Finally, Finally, we can quantitatively evaluate risk by integrating Equation 8 based on we can quantitatively evaluate risk by integrating Equation 8 based on definition of definition of risk. We can consider this index representative of annual expected risk. We can consider this index representative of annual expected loss ratio loss ratio (%/year), that is, a Tsunami Risk Index (TRI) (Figure 2e): (%/year), that is, a Tsunami Risk Index (TRI) (Figure 2e): 1 = (, Tsunami Risk Index perc = R perc Pperc, D ) (9) dd (9) 3. Application to Tohoku Area 3.1. Assessment Targets For targets of risk assessment, we considered wooden, brick, steel reinforced concrete buildings located in in city of Soma in Fukushima Prefecture cities of Sendai Kesennuma in Miyagi Prefecture. Figure 3 Table 1 show information for locations of buildings in each city. Kesennuma is is in in ria coast, Sendai is inside of Sendai bay Soma is directly facing Pacific Ocean. We selected se three regions to tocapture difference of of risk risk due due to to different different geography. The Theelevation distance from fromcoastline of of three cities also indicated in Table Figure 3. Locations of target buildings. Figure 3. Locations of target buildings.

6 Geosciences 2018, 8, 17 6 of 27 Table 1. Location information for target buildings in each city. Target Location Latitude ( ) Longitude ( ) Elevation (m) Distance from Coastline (m) Soma Sendai Kesennuma Probabilistic Tsunami Wave Height Construction of Logic Trees In this section, we aim to evaluate probabilistic coastal height in Tohoku region adjacent to Pacific Ocean. First, we need to select an appropriate earthquake-generating fault to produce a. We select ten regions one region of interlocking earthquakes from among occurrence areas of trench-type earthquakes along Japan Trench used in probabilistic earthquake prediction map released by NIED [14], all of which are shown in Figure 4 Table 2 as earthquake faults that could generate s. These selections exclude both earthquakes with moment magnitudes (Mw) reaching 7.4 or less in consideration of a variation of ± Mw earthquakes for which source fault is unlikely to be predicted beforeh. To evaluate epistemic uncertainties for se eleven fault regions, we use logic tree method proposed in Annaka s study [10]. Figure 5 shows logic tree constructed for se regions. We establish five branches within logic tree: The Mw range of earthquake, asperity position of earthquake fault, average occurrence interval (return period) of earthquake, stard deviation of lognormal distribution followed by error of wave height truncation range of lognormal distribution. Except for asperity position of earthquake fault, or four branches follow branches shown in Annaka s study [10]. The total number of branches in logic tree constructed using this approach is 3384 branches. The outline of setting for each branch is as follows. Table 2. Abbreviation explanations. Abbreviation JTN1-1 JTN1-2 JTN2 JTN3-1 JTN3-2 JTN2 + JTN3 TOHOKU JTT JTNR JTS1 IBRK Name Large interplate earthquakes in Norrn Sanriku-Oki (repeating earthquakes) Large interplate earthquakes in Norrn Sanriku-Oki (or than repeating earthquakes) Miyagi-ken-Oki earthquake (repeating earthquakes) s close to offshore trenches in Sourn Sanriku-Oki (repeating earthquakes) s close to offshore trenches in Sourn Sanriku-Oki (or than repeating earthquakes) Miyagi-ken-Oki, earthquakes close to offshore trenches in Sourn Sanriku-Oki consolidated-type-earthquake Great East Japan (2011 Tohoku-type earthquake) Large interplate earthquakes close to offshore trenches in Sanriku-Oki to Boso-Oki regions ( earthquakes) Large intraplate earthquakes close to offshore trenches in Sanriku-Oki to Boso-Oki regions (normal fault-type) Interplate earthquakes in Fukushima-ken-Oki Interplate earthquakes in Ibaraki-ken-Oki (or than repeating earthquakes)

7 Geosciences 2018, 8, 17 7 of 27 Geosciences 2018, 8, 17 Geosciences 2018, 8, 17 7 of 27 7 of 27 Figure 4. The selected earthquakes that could generate a based on occurrence region of Figure 4. The selected earthquakes that could generate based on occurrence region of Figure 4. The selected earthquakes that could generate aa occurrence region of trench-type earthquake along Japan Trench, which is used based in on probabilistic seismic motion trench-type earthquake along whichisisused used probabilistic seismic motion trench-type along Japan Japan Trench, Trench, which in in probabilistic seismic motion prediction mapearthquake [14]. prediction mapmap [14].[14]. prediction Source area Source area JTN1-1 JTN1-1 Mw=8.2 North α= years Mw=8.2 Mw=8.3 North α=0.08 α=8 89 years 97 years Mw=8.3 Mw=8.4 Mw=7.4 Mw=7.4 Mw=7.5 Mw=7.5 Mw=7.6 Mw=7.3 Mw=7.3 Mw=7.4 South South α=8 α=0.28 α=0.28 Mw=7.4 Mw=7.5 Mw=7.6 JTN3-1 JTN3-1 Mw=8.4 JTN2 JTN2 Log normal Return period stard deviation Log normal Return period stard deviation α value of BPT distribution α value of BPT distribution JTN1-2 JTN1-2 Asperity position Asperity position Moment magnitude Moment magnitude Mw= years 106 years 106 years 10 years 10 years 14 years 14 years 21 years 21 years 15 years 15 years 38 years 38 years 53 years 53 years Mw=7.8 North α=2 96 years Mw=7.8 Mw=7.9 North α=2 α= years 109 years Mw=7.9 Mw=8.0 Mw=8.0 South South α=0.22 α=0.32 α=0.32 Figure 5. Cont. 109 years 124 years 124 years

8 Geosciences 2018, 8, 17 8 of 27 Geosciences 2018, 8, 17 8 of 27 Asperity α value of BPT Source area Moment magnitude position Return period distribution Log normal stard deviation Truncation of log normal distribution JTN3-2 Mw=7.4 Mw=7.5 Mw= years 42 years 93 years JTN2 + JTN3 Mw=8.0 Mw=8.1 Mw=8.2 North South 0 66 years 218 years 1260 years JTT Mw=7.9 Mw=8.0 Mw=8.1 North South 0 56 years 103 years 191 years JTNR Mw=8.1 Mw=8.2 Mw=8.3 North South years 575 years 3324 years JTS1 Mw=7.3 Mw=7.4 Mw= years 206 years 582 years IBRK Mw=7.4 Mw=7.5 Mw= years 26 years 45 years TOHOKU Mw=8.9 Mw=9.0 Mw=9.1 1/5 1/5 1/5 1/5 1/5 North North- South- South α=4 α=0.24 α= years 600 years 676 years Figure 5. Logic trees constructed for eleven fault regions. The numbers appended onto Figurebranches 5. Logicof trees logic constructed trees are for weights eleven of fault branches. regions. The numbers appended onto branches of logic trees are weights of branches. The Mw range of each earthquake is varied by ±. This Mw variation is accomplished by changing average slip amount along entire fault. For asperity position of fault, three The types Mw of branches range of with each asperities earthquake located is at varied center by ±. of fault This Mw near variation both ends is of accomplished fault are by changing established average when slipfault amount length along is 150 km or entire more. fault. Since For only Tohoku-type asperity position earthquakes of have fault, long three typeslengths of branches (approximately with asperities 500 km), located we set atfive center branches of by adding fault two near cases both where ends an of additional fault are established asperity when is located fault between length three is 150 asperities. km or more. The method Since utilized only Tohoku-type to establish earthquakes asperity positions have long lengths along (approximately fault is detailed 500 km), in wefukutani set five branches et al. [15]. byregarding adding two cases occurrence where an probability additional of asperity is located between three asperities. The method utilized to establish asperity positions along fault is detailed in Fukutani et al. [15]. Regarding occurrence probability of earthquake, we construct three types of branches that take into consideration confidence interval of

9 Geosciences 2018, 8, 17 9 of 27 occurrence probability determined by probabilistic seismic motion prediction evaluation published by NIED [14]. Table 3 shows model name for generation interval of earthquake, α value of BPT distribution, average return period, sample period, earthquake generation time within period used to determine average occurrence interval lower upper limits of confidence interval for each earthquake fault, which are shown in Headquarters for Research Promotion [16]. See Appendix A for method used to establish confidence intervals. Although probability of occurrence of an earthquake with a relatively small magnitude can be evaluated with high accuracy using general Gutenberg-Richter rule, it is known that probability of occurrence of a relatively high-magnitude earthquake that causes a large cannot be perfectly evaluated using Gutenberg-Richter rule. Based upon this background, in this study, we note that data of return period collected as result of a detailed examination of historical earthquake record are used. Regarding stard deviation of lognormal distribution (i.e., aleatory uncertainty) followed by error of wave height, we use modeling error in numerical simulation results observation records, that is, geometrical stard deviation κ of Aida [17], for past eleven historical earthquakes represented by following expression based on Annaka s study [10]: logβ = logβ i (10) i logκ = 1 n i (logβ i ) 2 (logβ) 2 (11) where n is number of observation points, i is observation point, βi = (Ri/Hi), Ri is observed height at i-th point Hi is simulated value at i-th point. The values of κ are evaluated from past eleven historical earthquakes. The minimum value is result for 1707 Hoei earthquake, where κ = 1.35 (σ = log κ = 0.300) maximum value is result for 1946 Nankai earthquake, where κ = 1.60 (σ = log κ = 0.470). Finally, two types of branches with are established as truncation values at both ends of lognormal distribution. The numbers attached to branches of logic trees are weights of each branch sum of all of weights is set to. The weights of branches containing Mw range asperity positions along fault are set by equally dividing ir weights. Regarding branch consisting of occurrence probability of earthquakes, we use a weight of 0 for central branch for branches at both ends to consider confidence interval. The weight value adopted in Annaka s study [10] is also adopted for weights of branches for stard deviation of lognormal distribution truncation range. Table 3. Model names for generation intervals of earthquakes, α values of BPT distribution, average return periods, sample periods, earthquake generation times within periods used to determine average occurrence intervals lower upper limits of confidence intervals for eleven earthquakes, which are shown in Headquarters for Research Promotion [16]. Name (Abbreviation) JTN1-1 JTN2 JTN3-1 Model for Generation Interval of BPT Poisson process BPT α Value of BPT Distribution Average Return Period (Year) Sample Period (Year) Generation Time Lower Limit of Confidence Interval for Return Period Upper Limit of Confidence Interval for Return Period

10 Geosciences 2018, 8, of 27 Table 3. Cont. Name (Abbreviation) JTN2 + JTN3 JTT JTNR JTN1-2 JTN3-2 JTS1 IBRK TOHOKU Model for Generation Interval of Poisson process Poisson process Poisson process Poisson process Poisson process Poisson process Poisson process BPT α Value of BPT Distribution Average Return Period (Year) Sample Period (Year) Generation Time Lower Limit of Confidence Interval for Return Period Upper Limit of Confidence Interval for Return Period Tsunami Numerical Simulation For each branch comprising eir Mw range or asperity positions in logic trees constructed in previous section, fault parameters for numerical simulation are determined. The fault parameters for reference magnitudes of each fault are shown in Table 4. These data, which are related to position of fault, were published by NIED [14]. To provide heterogeneous asperities along fault planes, we generated 10 km mesh points covering Japan Trench area shown in Figure 6 while assuming presence of small faults with lengths widths of 10 km setting a slip amount to each fault. As described above, method used to establish asperity along 3.11 Tohoku earthquake-type fault is described in Fukutani et al. [15]. The depth of each small fault was set considering its length, width dip estimated from depth along entire fault. The strike, dip rake of each small fault were set to same values as those along entire fault. The calculation conditions of numerical simulation are shown in Table 5 terrain data used for simulation are shown in Figure 7. The terrain data were generated from 30 s gridded depth data (J-TOPO 30, Japan Hydrographic Association) near isls of Japan a mesh of 500 m bathymetry data (J-EGG 500, Japan Oceanographic Data Center). We obtained initial displaced water height using formula of Okada [18] from se fault parameters. Using calculated initially displaced water heights as input values, time integration was performed in each mesh using continuous equation (Equation (12)) equations of motion (Equations (13) (14)) based on a nonlinear longwave equation using TUNAMI model (Tohoku University Numerical Analysis Modeling for Inundation) [19]: M t + ( M 2 x D ( MN D N t + x ) + y ) + y η t + M x + N y = 0 (12) ( MN D ( N 2 D ) + gd η x + gn2 M D 7/3 M 2 + N 2 = 0 (13) ) + gd η y + gn2 N D 7/3 M 2 + N 2 = 0 (14) where η is water level, D is total water depth, g is gravitational acceleration, n is Manning s roughness coefficient M N are flow fluxes in x y directions. Taking into consideration number of branches comprising Mw range asperity positions along earthquake faults, re are 258 cases in which numerical simulation can be performed.

11 Geosciences 2018, 8, of 27 Table 4. Fault parameters for reference magnitude of each assumed fault. Name (Abbreviation) Moment Number of Magnitude (Mw) Faults Longitude ( ) Lattitude ( ) Fault Parameter Depth (km) Length (km) Width (km) Strike ( ) Rake ( ) Dip ( ) Moment Mo (Nm) Shear Modulus µ (N/m2) Average Slip (m) JTN JTN JTN JTN2+JTN The fault parameters were assumed as JTN2, JTN3-1 Consolidated Type TOHOKU , Deep part: Shallow part: JTT 8.0 JTNR 8.3 JTN JTN JTS1 7.4 IBRK 7.5 Shallow part: 18.1, Deep part: Table 5. Calculation conditions for numerical simulation. Item Calculation Condition Governing equation 2D non-linear shallow water equation (Tohoku University TUNAMI model) [19] Numerical integration method Staggered leap-frog differential method Initial condition Okada equation [18] Boundary condition Open boundary Coordination system Spherical coordinate system Tidal setting T.P m Mesh size 450 m Time step 0.9 s Calculation time 3 h

12 Geosciences 2018, 8, of 27 Geosciences 2018, 8, 17 Geosciences 2018, 8, 17 Figure 6. Mesh consisting of 10 km points encompassing Japan Trench area (black dots). 12 of of 27 Figure 6. Mesh consisting of 10 km points encompassing Japan Trench area (black dots). Figure 6. Mesh consisting of 10 km points encompassing Japan Trench area (black dots). Figure 7. Calculation area for numerical simulation (450 m mesh). Figure 7. Calculation area for numerical simulation (450 m mesh). Figure 7. Calculation area for numerical simulation (450 m mesh).

13 Geosciences 2018, 8, of 27 Geosciences 2018, 8, of Tsunami Hazard Curves at Offshore Points For Tsunami each branch Hazard of Curves constructed at Offshore logic Points tree, maximum wave height in each mesh is determined For each according branch to of results constructed of logic tree, numerical maximum simulation. wave If we height assume in each simulated mesh is maximum determined wave according heightto is results medianof value µ ifnumerical we use simulation. lognormalif stard we assume deviation simulated σ in logic maximum tree, we wave canheight obtain is a probability median density value μ function if we of use lognormal wave height stard represented deviation byσ in following logic tree, (Equation we can (15)): obtain a probability density function of wave height represented by following (Equation (15)): { } 1 (logx µ) 2 f (x) = exp 2πσx 2σ 2, 0 x (15) ( ) = 1 ) ( 2 2,0 (15) Next, by converting probability density function into exceedance probability distribution under an ergodic assumption, it it is possible to obtain hazard curve expressed by relationship between wave height annual exceedance probability for each branch of logic logictree. The The ergodic ergodic assumption assumption is a isstatistical a statistical assumption assumption that thatspatial spatial variation variation is equal is equal to to temporal temporal variation. variation. By evaluating By evaluating annual annual exceedance exceedance probability probability distribution distribution within within an an earthquake earthquake area area drawing drawing curves curves along along percentile percentile paths paths in consideration in consideration of of weights weights of of logic logic tree tree branches, branches, it is it possible is possible to estimate to estimate hazard hazard curve curve (i.e. (i.e. a fractile a fractile curve) curve) for that for that earthquake earthquake area. area. Finally, Finally, we integrate we integrate each each hazard hazard curve curve for each for each earthquake earthquake area. area. The The evaluation evaluation results results for 10 form 10water m water depths depths off Soma, off Soma, Sendai Sendai Kesennuma Kesennuma are are shown shown in Figure in Figure (a) Annual exceedance probability 1 5 percentile 50 percentile Average 95 percentile Tsunami height (m) 1 (c) Annual exceedance probability (b) Tsunami height (m) Figure Relationships between annual exceedance probability wave height at (a) a water depth of 10 m ( N, E) E) off port of Soma; (b) a water depth of 10 m ( N, E) E) off off coast of Sendai Plain; (c) a water depth of 10 m ( N, E) off coast of city of Kesennuma. Annual exceedance probability 1 5 percentile 50 percentile Average 95 percentile Tsunami height (m) 5 percentile 50 percentile Average 95 percentile

14 Geosciences 2018, 8, of Tsunami Inundation Assessment Probabilistic Tsunami Hazard Map The results shown in Table 6 for Soma, Table 7 for Sendai Table 8 for Kesennuma represent values converted from wave heights at water depths of 10 m generated for each earthquake to annual exceedance probability return period using each hazard curve. It should be noted that we used average values of hazard curves in this section. Using se tables, we can consider possibility of specifying return period of an earthquake-generating fault by focusing on height in a coastal area. Then, if a run-up simulation is carried out using parameters of fault specifying return period, a inundation area a inundation depth for every return period on l are obtained by running a simulation. The run-up simulations were conducted using nonlinear longwave equations under calculation conditions shown in Table 9 (Soma) Table 10 (Sendai Kesennuma). The calculation regions for each area are shown in Figure 9 (Soma) Figure 10 (Sendai Kesennuma). The inundation areas for each return period simulated using above method are shown in Figure 11 inundation heights inundation depths at each target point within risk assessment are shown in Table 6 through Table 8 (g), (h). This method of calculating inundation area for each return period is advantageous because it is possible to easily perform inundation calculations by appropriately changing information regarding initial tide levels or artificial structures (e.g., dikes buildings). Table 6. Calculation results for Soma. (a) source name; (b) moment magnitude of earthquake; (c) position of asperity; (d) wave height at a water depth of 10 m; (e) annual exceedance probability estimated using hazard curve; (f) return period calculated from annual exceedance probability; (g) inundation height at risk assessment point simulated using a nonlinear longwave equation with fault parameters. (a) (b) (c) (d) (e) (f) (g) (h) Source TOHOKU Moment Magnitude (Mw) Position of Asperity Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) 9.1 Center Between south South Center Between north South North Between south Between north Center North South Between south Between north North

15 Geosciences 2018, 8, of 27 Table 6. Cont. (a) (b) (c) (d) (e) (f) (g) (h) Source JTNR JTN2 + JTN3 JTT JTNR JTT Moment Magnitude (Mw) Position of Asperity Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) 8.4 North Center South South Center Center North Center South North Center North Center South Center Table 7. Calculation results for Sendai. (a) source name; (b) moment magnitude of earthquake; (c) position of asperity; (d) wave height at a water depth of 10 m; (e) annual exceedance probability estimated using hazard curve; (f) return period calculated from annual exceedance probability; (g) inundation height at risk assessment point simulated using a nonlinear longwave equation with fault parameters. (a) (b) (c) (d) (e) (f) (g) (h) Source TOHOKU Moment Magnitude (Mw) Position of Asperity Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) 9.1 Center Center Between north South Center Between north Between south South North Between north Between south North South North Between south

16 Geosciences 2018, 8, of 27 Table 7. Cont. (a) (b) (c) (d) (e) (f) (g) (h) Source JTN2 + JTN3 JTN3-1 JTN2 + JTN3 Moment Magnitude (Mw) Position of Asperity Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) 8.2 South Center North South Center Center JTN North JTN2 + JTN3 8.1 South JTT 8.1 South JTN2 + JTN3 8.1 North JTT 8.1 Center JTN Center JTNR 8.4 South North Center Table 8. Calculation results for Kesennuma. (a) source name; (b) moment magnitude of earthquake; (c) position of asperity; (d) wave height at a water depth of 10 m; (e) annual exceedance probability estimated using hazard curve; (f) return period calculated from annual exceedance probability; (g) inundation height at risk assessment point simulated using a nonlinear longwave equation with fault parameters. (a) (b) (c) (d) (e) (f) (g) (h) Source TOHOKU Moment Magnitude (Mw) 9.1 Position of Asperity Between north Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) Center Between north Center Between south North Between north Center North Between south South North South Between south South

17 Geosciences 2018, 8, of 27 Table 8. Cont. (a) (b) (c) (d) (e) (f) (g) (h) Source JTT JTNR Moment Magnitude (Mw) Position of Asperity Tsunami Height (m) (10 m Water Depth Point) Annual Exceedance Probability Return Period (year) Tsunami Inundation Height (m) (Risk Assessment Point) Tsunami Inundation Depth (m) (Risk Assessment Point) 8.1 South South Center North Center South North JTT 8.1 North JTNR 8.4 Center JTN2 + JTN3 JTNR JTT 8.2 North South Center South Center South Table 9. Calculation conditions for nonlinear longwave equation used in numerical simulation for Soma. Item Calculation Condition Governing equation 2D non-linear shallow water equation (Tohoku University TUNAMI model) [19] Numerical integration method Staggered leap-frog differential method Initial condition Okada equation [18] Boundary condition Run-up boundary Coordinate system Plane rectangular coordinate system IX Tidal setting T.P m Mesh size 810 m, 270 m, 90 m, 30 m, 10 m Time step 0.9 s, 0.3 s, s, 0.03 s, s Calculation time 3 h Table 10. Calculation conditions for nonlinear longwave equation used in numerical simulations for Sendai Kesennuma. Item Calculation Condition Governing equation 2D non-linear shallow water equation (Tohoku University TUNAMI model) [19] Numerical integration method Staggered leap-frog differential method Initial condition Okada equation [18] Boundary condition Run-up boundary Coordinate system Plane rectangular coordinate system X Tidal setting T.P m Mesh size 1215 m, 405 m, 135 m, 45 m, 15 m Time step 0.9 s, 0.3 s, s, 0.03 s, s Calculation time 3 h

18 Numerical integration method Staggered leap-frog differential method Initial condition Okada equation [18] Boundary condition Run-up boundary Coordinate system Plane rectangular coordinate system IX Tidal setting T.P m Mesh size 810 m, 270 m, 90 m, 30 m, 10 m Geosciences 2018, 8, 17 Time step 0.9 s, 0.3 s, s, 0.03 s, s 18 of 27 Calculation time 3 h Figure 9. Five regions of terrain data used for numerical run-up simulations in Soma Figure 9. Five regions of terrain data used for numerical run-up simulations in Soma area (Domain 1, Domain 2, Domain 3, Domain Domain 5). The numbers in parenses are area (Domain 1, Domain 2, Domain 3, Domain 4 Domain 5). The numbers in parenses are east-west north-south mesh dimensions. east-west north-south mesh dimensions. Table 10. Calculation conditions for nonlinear longwave equation used in numerical simulations for Sendai Kesennuma. Domain3: 135m 39.1 N ( ) Domain1: 1215m Item Calculation Condition 43.9 N ( ) Governing equation 2D non-linear shallow water equation (Tohoku University TUNAMI model) [19] Domain4: 45m Numerical integration method Staggered leap-frog differential method Initial condition Okada Risk リスク評価地点 assessment equation point [18] ( ) Boundary condition Run-up (Kesennuma) ( 気仙沼 boundary ) Coordinate system Plane rectangular coordinate system X Domain5: 15m Tidal setting T.P m ( ) Mesh size Domain2: 405m 1215 m, 405 m, 135 m, 45 m, 15 m Time step ( ) 0.9 s, 0.3 s, s, 0.03 s, s Calculation time 3 h Domain4: 45m ( ) Domain3: 135m ( ) Risk リスク評価地点 assessment point ( 仙台 (Sendai) ) Domain5: 15m ( ) Elevation 1530m 0m E 32.9 N E (east-west mesh number north-south mesh number) E ( )(east-west 内は ( 東西方向のメッシュ数 mesh number north-south 南北方向のメッシュ数 mesh number) ) を表す 37.7 N E Figure 10. Five regions of terrain data used for numerical run-up simulations in Sendai Kesennuma areas (Domain 1, Domain 2, Domain 3, Domain 4 Domain 5). The numbers in parenses are east-west north-south mesh dimensions Tsunami Hazard Curves at Inl Points Based on annual exceedance probability of wave height numerical simulation results of inundation depths, we evaluate annual exceedance probability curves of inl inundation depths. First, in each case of numerical simulation, we assumed that probability density of wave height was equal to probability density of inundation depth. Then, after normalizing probability density of inundation depth so that maximum value of annual exceedance probability of inundation depth represents annual exceedance probability of wave height in case where inundation depth is zero, we calculated annual

19 Geosciences 2018, 8, of 27 exceedance probability of inundation depth from corresponding probability density data (see Figure 12). From Tables 6 8, wave height (in case where inundation depth at each evaluation point is zero) annual exceedance probability are respectively 2.01 m (average value) for Soma, 4.30 m (average value) for Sendai 3.63 m (average value) for Kesennuma. In this study, although procedure is not entirely probabilistic because numerical simulations are performed only discretely with a limited number, we constructed annual exceedance probability curve of inundation depth by regressing each plot with a straight line. The average values in addition to 5th percentile, 50th percentile 95th percentile curves created using same method are also shown in Figure 12. Geosciences 2018, 8, of 27 Figure 11. Results of inundation assessments for (a) Soma; (b) Sendai (c) Kesennuma at Figure 11. Results of inundation assessments for (a) Soma; (b) Sendai (c) Kesennuma at return periods of approximately 200, years. return periods of approximately 200, years. 1 5 percentile 50 percentile Average 95 percentile (b) (a) 1 5 percentile 50 percentile Average 95 percentile

20 Geosciences 17 Figure2018, 11. 8, Results of inundation assessments for (a) Soma; (b) Sendai (c) Kesennuma at20 of 27 return periods of approximately 200, years Geosciences 2018, 8, 17 Tsunami inundation depth (m) (c) 1 (b) 5 percentile 50 percentile Average 95 percentile (a) percentile 50 percentile Average 95 percentile Tsunami inundation depth (m) p ( ) 5 percentile 50 percentile Average 95 percentile Tsunami inundation height (m) 10.0 Figure 12. between annual exceedance probability height inundation heigh Figure 12. Relationships between Relationships annual exceedance probability inundation for (a) Soma; (b) Sendai (c) Kesennuma. for (a) Soma; (b) Sendai (c) Kesennuma Fragility Assessment 3.4. Fragility Assessment To evaluate fragilities of buildings with respect to s, we used fragility To evaluate fragilities of buildings with respect to s, we used fragility curve studied by Suppasri et al. [20], who performed regression analyses using damage data from buil studied by Suppasri et al. [20], who performed regression analyses using damage data from buildings that suffered from inundation during 3.11 Tohoku earthquake. Their study prop that suffered from inundation during 3.11 Tohoku earthquake. Their study proposed various fragility curves for different building structures (i.e. reinforced concrete, steel, brick various fragility curves for different building structures (i.e. reinforced concrete, steel, brick wood) wood) according to six different damage levels: minor damage, moderate damage, major dam according to six different damage levels: minor damage, moderate damage, major damage, complete complete damage, collapsed damage washed away (Figure 13). Although re are num damage, collapsed damage washed away (Figure 13). Although re are numerous studies about studies about fragility functions [21], we selected before-mentioned curves in this fragility functions [21], we selected before-mentioned curves in this time. That is, why That is, why we also need to investigate uncertainties of risk due to difference of tsu we also need to investigate uncertainties of risk due to difference of fragility fragility functions in future study. functions in future study Minor 0.7 Moderate 0.6 Major Complete Collapsed 0.2 (a) Tsunami inundation depth (m) Washed away Major Comp Collap (b) Tsunami inundation depth (m) Wash away Minor Moderate Major Complete ge Probability ge Probability 0.6 Mode Minor Minor 0.7 Mode 0.6 Major

21 various fragility curves for different building structures (i.e. reinforced concrete, steel, brick wood) according to six different damage levels: minor damage, moderate damage, major damage, complete damage, collapsed damage washed away (Figure 13). Although re are numerous studies about fragility functions [21], we selected before-mentioned curves in this time. Geosciences That is, 2018, why 8, we 17also need to investigate uncertainties of risk due to difference of 21 of 27 fragility functions in future study Tsunami inundation depth (m) (a) (c) Tsunami inundation depth (m) Minor Moderate Major Complete Collapsed Washed away Minor Moderate Major Complete Collapsed Washed away (b) Tsunami inundation depth (m) (d) Tsunami inundation depth (m) Minor Moderate Major Complete Collapsed Washed away Minor Moderate Major Complete Collapsed Washed away Figure 13. Tsunami fragility curves for different building structures ((a) Reinforced concrete; (b) Steel; Figure 13. Tsunami fragility curves for different building structures ((a) Reinforced concrete; (b) Steel; (c) Brick (d) Wood) different damage levels, which were created by regressing damage (c) Brick (d) Wood) different damage levels, which were created by regressing damage data from 3.11 Tohoku earthquake [20]. data from 3.11 Tohoku earthquake [20] Risk Assessment Quantitative Effects of Hazard Assessment To quantify risk with regard to direct damage of a building located at a risk assessment point, we eliminated inundation depth axis from Figures obtained risk curve represented by relationship between damage probability of a building annual exceedance probability. The risk curve for each type of building structure is shown in Figure 14. In calculating damage probability of a building, six different types of destruction were united using each damage probability. We set damage probabilities of minor damage to, moderate damage to 0.3, major damage to, complete damage to 0.8, collapsed damage to 0.9 washed away to. As indicated in abovementioned methodology, we note that we have not performed a risk estimation that includes variability in building response uncertainties in loss estimates because main objective of this study is to quantify epistemic uncertainties in hazards affecting building risk. Generally, risk is expressed by product of degree of loss with its occurrence probability. Applying this idea, if we consider that risk for direct damage to a building is product of probability of destruction of a building by a its generation probability, we can estimate risk using product of horizontal axis vertical axis of risk curve. Therefore, we can quantify risk by integrating derived risk curve, calculating area under curve estimating annual expected loss ratio (i.e. Tsunami Risk Index, or TRI). Figure 15 shows calculation results obtained by integrating area under regressed exponential function for (a) average value of hazard for each percentile value ((b) 5th percentile value, (c) 50th percentile value (d) 95th percentile value). Figure 16 shows calculation results for each area.

22 Geosciences 2018, 8, of 27 Geosciences 2018, 8, of 27 5 percentile 50 percentile Average 95 percentile Soma Sendai Kesennuma Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Reinforced Concrete Steel Brick Wood Figure Figure Tsunami risk risk curves curves for for different different types types of of building building construction construction with with different different percentile percentile of of hazard hazard for for three three target target points. points.

23 Geosciences 2018, 8, of 27 Geosciences 2018, 8, of 27 (%/year) E-02 (a) Annual expected loss ratio (Average) E-03 E-04 (%/year) E-03 Wood Steel Wood Brick RC Steel Brick RC Wood Steel Brick RC Soma Soma Sendai Soma Soma SendaiSendaiSendaiKesen numa Kesen numa Kesen Kesen numa numa (b) Annual expected loss ratio (5th percentile) E-04 E-05 (%/year) E-02 Wood Steel Brick RC Wood Steel Brick RC Wood Steel RC Brick Soma Soma Soma Soma SendaiSendaiSendaiSendaiKesen numa Kesen numa Kesen Kesen numa numa (c) Annual expected loss ratio (50th percentile) E-03 E-04 (%/year) E-01 Wood Steel Brick Wood RC Steel Brick RC Wood Steel Brick RC Soma Soma Soma Sendai Soma SendaiSendaiSendaiKesen numa Kesen numa Kesen Kesen numa numa (d) Annual expected loss ratio (95th percentile) E-02 E-03 Wood Steel Brick RC Wood Steel Brick RC Wood Steel Brick RC Soma Soma Soma Soma SendaiSendaiSendaiSendaiKesen numa Kesen numa Kesen Kesen numa numa Figure 15. Expected value of risk using each hazard value ((a) average value; (b) Figure 15. Expected value of risk using each hazard value ((a) average value; (b) 5th 5th percentile percentile value; value; (c) (c) 50th 50th percentile percentile value value (d) (d) 95th 95th percentile percentile value) value) indicated indicated according according to to structure structure (upper (upper horizontal horizontal axis) axis) region region (lower (lower horizontal horizontal axis). axis).

24 Geosciences 2018, 8, of 27 Geosciences 2018, 8, of 27 (%/year) E-01 (a) Annual expected tsuami loss ratio (Soma) E-02 E-03 E-04 E-05 Wood Steel Brick RC Wood Steel Brick RC Wood Steel Brick RC (%/year) E-01 (b) Annual expected tsuami loss ratio (Sendai) E-02 E-03 E-04 E-05 (%/year) E-01 Wood Steel Brick RC Wood Steel Brick RC Wood Steel Brick RC (c) Annual expected loss ratio (Kesennuma) E-02 E-03 E-04 E-05 Wood Steel Brick RC Wood Steel Brick RC Wood Steel RC Brick Figure 16. Expected valueof of risk riskat at each each risk risk assessment point point ((a) ((a) Soma; Soma; (b) Sendai; (b) Sendai; (c) (c) Kesennuma) according to to structure (upper horizontal axis) percentile value of hazard (lower horizontal axis) Discussion From results shown in Figure 16, we can compare results from evaluating Tsunami Risk Index As mentioned previously, risk changes dramatically depending on Based on results shown in Figure 15, we can quantitatively underst risk percentile hazard that is considered. Among m, difference between risk using imposed on different types of buildings, that is, a reinforced concrete building located in Kesennuma 50th percentile hazard 5th percentile hazard is much greater than difference between is safest ( lowest risk) while a wooden building located in Soma is insecure ( highest risk) risk using 95th hazard 50th percentile hazard. As a matter of course, se results except for case when average hazard values are used. show that it is necessary to use hazard information from same percentile when comparing Focusing on magnitude, we can see that results using 5th percentile hazard values risks among different regions. results using 50th percentile hazard values differ by approximately one order of magnitude In this way, we can stochastically interpret compare evaluation results of Tsunami Risk Index among several target regions.

25 Geosciences 2018, 8, of 27 that results using 50th percentile values results using 95th percentile values also differ by approximately one order of magnitude. A difference of approximately two orders of magnitude is observed between 5th percentile values 95th percentile values. These results clearly show that risks have substantial hazard uncertainties. Overall, risk of a tends to increase successively from Soma to Sendai n to Kesennuma because impact of hazard is large in that same order. In addition, a wood building located in Kesennuma has a lower risk than a reinforced concrete building located in eir Soma or Sendai a steel building located in Soma is at a higher risk than a wood building located in Kesennuma. We underst that we cannot properly comprehend extent of risk by evaluating fragility only. From results shown in Figure 16, we can compare results from evaluating Tsunami Risk Index As mentioned previously, risk changes dramatically depending on percentile hazard that is considered. Among m, difference between risk using 50th percentile hazard 5th percentile hazard is much greater than difference between risk using 95th hazard 50th percentile hazard. As a matter of course, se results show that it is necessary to use hazard information from same percentile when comparing risks among different regions. In this way, we can stochastically interpret compare evaluation results of Tsunami Risk Index among several target regions. 4. Conclusions In this study, through process of quantifying uncertainty in hazard effecting on building risk assessment, we proposed two new evaluation methods that are essential to implement risk assessment. We first proposed a method that can be used to comprehensively address uncertainties (epistemic uncertainty aleatory uncertainty) in hazard assessments by probabilistically evaluating inundation area inundation depth using hazard curves. In this method, we can estimate inundation area with each return period by performing numerical simulations using fault parameters after specifying return period of fault from coastal hazard curve. This calculation method is advantageous because it is possible to easily perform calculations by appropriately changing information regarding initial tide levels or artificial structures. In addition, we visualized uncertainty in hazard assessment in an easy-to-underst manner by indicating several inundation areas corresponding to return periods of several earthquakes rar than using conventional hazard maps generated from single earthquakes. Next, we proposed a method to evaluate annual expected loss ratio (i.e. risk index, or TRI) targeting buildings located in Soma, Sendai Kesennuma by combining probabilistic inundation information building fragility information. We used fragility curves of different construction building derived from using damaged data of buildings during 3.11 Tohoku earthquake, which were regressed by log-normal distribution. In addition, n, we derived various risk curves, which represent relationship between damage probability of building annual exceedance probability. Based on definition of risk, we considered risk for direct damage to buildings as product of damage probability of a building destroyed by its generation probability. Therefore, we can quantify risk by integrating derived risk curve, that is, calculating area under curve estimating annual expected loss ratio (Tsunami Risk Index: TRI). Focusing on magnitude of calculated TRI, we clearly showed that results using 5th percentile hazard values results using 95th percentile hazard values differ by approximately two orders of magnitude. Furrmore, for example, based on results using average values, A wood building located in Kesennuma is at a lower risk than a reinforced

26 Geosciences 2018, 8, of 27 concrete building located in Soma Sendai. We cannot appropriately underst extent of risk if information regarding hazards fragility evaluations are individually captured. Through this study, we quantitatively showed for that we can underst risk by combining hazard fragility information. The proposed method in this study can be applied to various regions regardless of area refore, Tsunami Risk Index can be an effective index for ranking priority investments in disaster prevention endeavors by comparing magnitudes of risks of several targets located across several regions. Acknowledgments: This research was supported by Specific Project Research from International Research Institute of Disaster Science (IRIDeS) at Tohoku University. This research was also supported by funding from Tokio Marine & Nichido Fire Insurance Co., Ltd. through IRIDeS. The authors also greatly appreciate questions valuable comments of anonymous reviewers that helped improve manuscript. Author Contributions: Yo Fukutani conceived designed experiments, analyzed data wrote paper; Anawat Suppasri Fumihiko Imamura advised experimental design. Conflicts of Interest: The authors declare no conflict of interest. Appendix If we assume a Poisson process, confidence interval of earthquake occurrence interval is established based on study of Weichert [22], as shown in Table A1. That is, if number of records of earthquake occurrence is N (N in table), µ U µ L are determined based on information in Table A1. If sample period is T, confidence interval is as follows: T µ U T µ L (A1) Meanwhile, confidence interval of earthquake occurrence interval assuming an updating process using a BPT distribution is as follows: exp ( n α ) exp (+ n α ) (A2) where variation coefficient of earthquake occurrence interval is α number of records of earthquake occurrence interval is n. Table A1. Lower upper ± stard deviation confidence intervals for a Poisson variable [22]. U N L References 1. United Nations Department of Humanitarian Affairs. Glossary: Internationally Agreed Glossary of Basic Terms Related to Disaster Management, Geneva, Switzerl. Available online: reliefweb.int/files/resources/004dfd3e15b69a67c1256c4c006225c2-dha-glossary-1992.pdf (accessed on 28 September 2017).

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