Storm surge risk mitigation under ambiguity by coupled tropical cyclone and decision model

stated equal indicates cost-benefit suggests that Inundation from storm surges. This conducts simulations by using GSTCM under the current climate conditions and selects the worst four simulated satisfying the following three conditions: (i) the minimum distance to Osaka is less than 200 km, (ii) the minimum central pressure is less than 950 hPa, and (iii) the velocity of the typhoon at landfall is higher than 20 km/h. Then, by adding inputs like the simulated properties (track, speed, and central pressure) of the four selected typhoons, storm surges in Osaka are simulated using the SuWAT. The model simulates storm surge heights as well as maximum wind wave heights using the nonlinear shallow water equation and spectral wave model Waves Nearshore), respectively. Entering the properties of the simulated storm surge, we each 30m in using an inundation model the 2D shallow water equation fifth worst four first, second, third, and fourth worst inundation with return periods once 200, 100, 70, and 50 years. We PDF storm surge risk. To explore the process from simulation 25 PDFs storm surge inundation each mesh. the simulated inundation depths of 30m meshes each average and

Even as storm surge risks are increasing, the projections of such risks have an element of ambiguity. Consequently, policymakers find it extremely difficult to design policies to deal with storm surge risks. Therefore, in this study, we have linked the tropical cyclone models and stated preference experiments with decision models to provide a fresh perspective on households' preferences for storm surge risk mitigation under ambiguity. We have validated households' choices under the average and worst projections of storm surges and estimated the expected loss reduction, risk premium, and ambiguity premium for storm surge risk mitigation. Our study reveals that households pay disproportionately more attention to the worst case and that the ambiguity premium is not negligible. This leads to an important conclusion that policymakers should factor in the ambiguity premium to formulate risk mitigation policies.
According to IPCC's recent report, in coastal cities, the combination of more frequent extreme sea level events (due to sea level rise and storm surge) and extreme rainfall/river flow events will make flooding more probable. 1 Indeed, climate change and global warming pose an increasing threat to coastal flood risks from tropical cyclone and storm surge. [2][3][4][5][6][7][8][9][10] Moreover, the projected upsurge intensity and frequency of extreme sea levels for the coming decades increases the risk of catastrophic losses, 11 hence, both structural for example dike construction, and non-structural (education and awareness, insurance, social safety net etc.) measures are essential as coastal defense.
Instead, the scientific projection of storm surge risks inevitably involves uncertainty due to the stochastic nature of extreme events and unknowns or limited knowledge about the mechanism of tropical cyclones or typhoons, the rates and magnitudes of sea level rise, and model parameter values. 8,[12][13][14][15] The projections of storm surge risks include ambiguity (uncertainty about probabilities), posing significant challenges for risk reduction or mitigation policies, such as rising dikes, building code restrictions, land use restrictions, and flood insurances. Uncertainty can be classified into "aleatory uncertainty" and "epistemic or deep uncertainty." 16,17 The aleatory uncertainty refers to natural randomness of the relevant event, whereas epistemic uncertainty is the situation where there is hardly any information to specify a single probability density function (PDF). In the context of storm surge, epistemic uncertainty may arise due to a lack of sufficient knowledge and data on typhoon systems such as cyclogenesis factors, developmental processes, and their movement. In economics, aleatory uncertainty is classified as "risk" and epistemic uncertainty corresponds to "ambiguity." 18,19 While risk is represented by a single well-defined PDF, ambiguity often denotes multiple PDFs derived from different models with various parameter settings. 17,20 Ambiguity (here, multiple projected PDFs) of storm surge risk makes it difficult to formulate risk mitigation policies, 21,22 which motivates to conduct our unique analysis of coupled tropical cyclone and decision models. The common approach using average projection of multiple predictions [23][24][25][26][27][28] can be straightforward, but it may lead to ignoring the possibility of worst-case occurrences. Instead, the worst projections, such as probable maximum flood 29 and plausible worstcase estimate, 30,31 are often considered in flood risk management. Although a risk mitigation policy designed for the worst projection performs to tackle most inundation risks, the potential cost of implementing the policy remains a challenge. 7,11 Thus, policymakers face difficulty on how to use the multiple projections for risk mitigation policies. This problem is particularly acute when designing structural flood mitigation measures, such as dikes or sea walls to prevent flood damage only from storm surges with predefined design levels (e.g., a 100-year storm surge) or less, so that their effect might be quite different between average and worst projections. 20 Suppose that areas A and B face 1m (meter) and 2m storm surges in the average projection, and 3m and 4m storm surges in the worst projection, respectively. A dike with a 3m height can protect both areas in the average projection but cannot protect area B in the worst projection. Therefore, it is crucial to emphasize the importance of geographical conditions and distributions in evaluating the risk mitigation effects of the structural measures under multiple projections.
In this regard, policy makers' judgement should reflect the preferences of stakeholders for risk and ambiguity and the extent to which they plan to allocate resources for protection against the worst projection. 32,33 This is vital for the residential or household sectors in the coastal areas, as people's decisions are often influenced by ambiguity. 18,19 Even though some studies have explored the ambiguity of storm surge risks, 15,28,34,35 proposed decision support methods under ambiguity, 21,31,36,37 and estimated residents' preferences to mitigate flood risks from storm surges, 38,39,40 the scholars have not paid sufficient attention to ambiguity and estimation of both risk and ambiguity premiums for storm surge risk mitigation. To our knowledge, no study have investigated households' preferences in mitigating storm-surge inundation risks under ambiguity; therefore, in this study, we attempt to fill this gap. Increased mean and extreme sea level, alongside ocean warming and acidification, are projected to exacerbate risks for human communities in low-lying coastal areas. 41 Japan experiences huge typhoons, causing serious storm surge damages. Osaka Bay is one of the most vulnerable areas in terms of storm surge risks in Japan and is ranked fifth among the world's 120 cities in terms of expected annual losses due to storm surges in 2050. 42 Recently, in 2018, Typhoon Jebi caused 14 deaths, 1014 injuries and destroyed 686 houses in Osaka Bay. 43 Thus, we assess storm surge risks and ambiguity in the coastal areas of Osaka Bay in this study. Fig. S1 and Fig. S2 show our study areas and the flow of methodological framework and analysis. Here, we briefly illustrate the steps followed for analysis (see Methods for details): i) we conduct a simulation of typhoon generations for 200 years using global stochastic tropical cyclone model (GSTCM); 44 ii) the four worst typhoons are selected (after fulfilling the conditions), and the storm surges of Osaka Bay are simulated by a full-coupled surge-wave-tide coupled model (SuWAT); 45 iii) we predict the inundation depth due to storm surges using the inundation simulation model; 46,47 iv) by repeating step i) to iii), we get 25 PDFs of the inundation risk for each dike level: current level and rising by 0.5m, 1.0m, 1.5m, and 2.0m (25 × 5); v) the average and worst projections of the inundation risk (PDFs) are specified by each zip-code in the web-based survey; vi) we estimate households' preferences by asking them to choose whether to buy a hypothetical insurance covering all the losses due to storm surges by presenting the average and worst projections of inundation risks to their houses; vii) by using the choice experiment data, a decision model is applied for estimating risk and ambiguity premiums; and viii) finally, we analyze the geographical distribution of risk and ambiguity premiums by geographic information system (GIS). Fig.1 shows the average and worst projections of storm surge inundation risks (based on exceedance probability) with different dike levels in each zip-code area.
Several studies have applied the stated preference method to estimate households' preferences to reduce storm surge risks, 38-40 however, our study is the first attempt to estimate households' preferences under ambiguity of storm surge risk mitigation (Fig.2). It is worth noting that the risk premium is the additional payment for avoiding random variability of the economic loss from storm surge risks. For instance, the risk premium is zero if the economic loss has no random variability (fixed amount as expected loss) or an individual's risk attitude is neutral (i.e. his/her utility function is linear). Instead, the ambiguity premium is the additional payment for avoiding multiple projections, that is, zero if a unique projection of storm surge risks is specified. We used limited degree of confidence (LDC) model to deal with the ambiguity of storm surge risks because this model is useful in designing robust strategies for dealing with the uncertainty of climate change. [48][49][50] In addition to resolving the ambiguity of the inundation risks, this study attests to the importance of coupled stochastic tropical cyclone model and stated preference method. The details of storm surge and inundation simulation, survey procedures, and decision models are explained in the Methods section. Using the estimated parameters of Model 1 and the sample average of household assets (34.1 million yen), we calculate the expected loss reduction, risk premium, and ambiguity premium for full protection from the storm surge risk (for details, see Methods section). Then, each value is multiplied by the number of households in each zip-code area ( Supplementary Fig. S3) and aggregated for the entire target area, as presented in Table 2. The total economic value or WTP is 1,743 million yen (about 17 million USD) for full protection insurance policy from the storm surge * 1 Technically, the worst projection is constructed as the worst envelope curve of 25 projected distribution on storm surge inundation depth. Therefore, the weight of the worst projection should be less than 4% even though we give equal weight to each of the 25 projections. risk under ambiguity in the target area. It consists of the expected loss reduction, risk premium, and ambiguity premium, which are 821, 65, and 849 million yen, respectively. The risk premium and ambiguity premium are 8% and 103%, respectively, compared with the value of expected loss reduction for eliminating storm surge risks in the target area. Thus, the total economic value is 211% of the expected loss reduction.
In LDC model, the expected loss reduction and risk premium are calculated using the average prediction, while ambiguity premium is estimated with the worst prediction. The estimated LDC model indicates that a statistically representative household weighs 90% on the average projection and 10% on the worst projection, which apparently imply small ambiguity premium compared to the value of expected loss reduction and risk premium. However, the results demonstrate that the ambiguity premium is much larger than risk premium and almost equivalent to the expected loss reduction. This is because the asset loss based on the worst projection is much larger than the loss based on the average projection. As can be seen in Fig. 1, there are many areas where houses will be submerged or the flooding on the floor will be occurred in the worst projection, while the average prediction shows no areas of submerged houses and very small area of flooding on the floor will occur. Due to this relationship between the geographic conditions and inundation levels, the asset loss and its dispersion in the average and worst projections are very different, as shown numerically in Table S2.
Estimating the economic value of dike rise is important for the storm surge risk mitigation policies. Our study considers four dike rising policies assuming all dikes in the target area are uniformly raised by 0.5m, 1.0m, 1.5m and 2.0m, respectively. Table 2 indicates that the total economic value of dike rising by 2.0m is 1,682 million yen, which is quite close to the benefit of full protection (1,743 million yen). This implies that dike rising by 2.0m can prevent almost all damages from storm surges. Further, we divided the dike rising by 2m into four steps: "0 to 0.5m," "0.5 to 1.0m," "1.0 to 1.5m," and "1.5 to 2.0m" and calculate their values. The results demonstrate that the marginal values of dike rising decrease with the higher level of dike rising. The expected loss reduction and risk premium for dike rising of "0 to 0.5m" are 581 and 64 million yen, which drop sharply to 110 and 1 million yen, respectively, for dike rising of "0.5 to 1.0m." The ambiguity premium is about 390 million yen for dike rising of both "0 to 0.5m" and "0.5 to 1.0m" and then largely falls to 55 million yen for dike rising of "1.0 to 1.5m." This is because the first dike rising by 0.5m to the current level may protect the target area mostly from storm surges in the average prediction but cannot protect in the worst prediction. Additional dike rising by 0.5m (or +1.0m to the current level) can prevent storm surge inundation largely, even in the worst prediction. Thus, the marginal ambiguity premium decreases more slowly than the marginal value of the expected loss reduction and the marginal risk premium as dike rising increases.
The geographical distributions of the expected loss reduction, risk premium, and ambiguity premium provide useful information for designing effective storm surge risk mitigation policies.  Our results provide several policy implications for mitigating storm surge risks. First, ignoring the ambiguity premium causes significant undervaluation of storm surge risk mitigation, in this case, dike rising. Our decision model of LDC finds that a statistically representative household weighs 90% on the average projection and 10% on the worst projection. Although the weight on the worst projection is not very high, it causes a large ambiguity premium that is almost equivalent to the value of the expected loss reduction for eliminating storm surge risks. This is because the economic loss due to inundation under the worst projection is much larger than the loss under the average projection. Thus, the total economic value of eliminating storm surge risk is almost half if the ambiguity premium is ignored.
Second, the areas that benefit from the structural mitigation measures, such as dike rising, are differently distributed depending on the geographical heterogeneity of the areas and the current level of dikes. The projection of storm surge risks and ambiguity can be largely affected by geographical conditions such as land elevation and distance from the sea or river. To ensure a predetermined safety level for all areas, policymakers should capture the geographical distribution of benefited areas. In particular, the ambiguity premiums need special attention because they are distributed differently from the expected loss reduction and risk premium. Thus, the ambiguity premium emphasizes the importance of considering geographical heterogeneity for effective risk mitigation policies.
Third, the ambiguity premium stems from households' over-concern of the worst projection, which may lead to an over-allocation of resources to prevent storm surge damage before a disaster.
However, compensation for economic loss after disaster is never over-valued because this amount is independent of households' preferences for ambiguity. Hence, this study suggests that post-disaster compensation, such as public insurance, is economically effective in coping with ambiguity while continuing with pre-disaster prevention measures, such as dike rising, land use regulation, and building code. Thus, a risk mitigation policy combining public insurance for the worst projection and pre-disaster prevention measures, such as dike construction for average projection, can be effective and efficient approach.
This study provides a new perspective on households' preferences for storm surge risk where 0 , 1 , 2 , and 3 are the average probabilities of "no flooding," "flooding under floor," "flooding on floor," "house submerged," respectively. Similarly, 0 , 1 , 2 , and 3 are the worst probabilities. The house value of household is , which becomes 1 , 2 , and 3 when it gets "flooding under floor," "flooding on floor," "house submerged," respectively.
The econometric model is as follows. The LDC expected utility without insurance can be written as: where is the CRRA utility function with coefficient as follows: The LDC expected utility with insurance at cost of is given as: where is the insurance fare rate such that is a household's payment for buying an insurance.
A household chooses to buy insurance if the following inequality holds: where 1 and 2 are random components of utilities and = 1 − 2~( 0,1). The probability of buying insurance can be written as: where is the standard normal distribution function. Thus, the log-likelihood to be maximized is written as: where is a dummy variable taking a value of 1 if household chooses to buy the insurance and 0 otherwise, and is the number of respondents.
Estimation of expected loss, risk premium, and ambiguity premium. Based on the LDC model, the willingness to pay of household for the full protection of insurance policy ( ) is calculated as follows: where −1 is an inverse function of . The expected loss, risk premium, and ambiguity premium are defined by the following equations. The expected loss, is defined as the full asset value, , minus the expected value, , with the average projection of storm surge risk as: The certainty equivalent for risk is a certain value that is equally attractive to the household's asset facing storm surge risks in the average projection and can be written as follows: The risk premium, , is defined as the expected value minus the certainty equivalent for risk.

= −
There is no widely accepted definition of the ambiguity premium, while the expected loss and risk premium are clearly defined. However, we follow the definition proposed by Cubit, van de Kuilen, and Mukerji, 62 as it can be applied to any decision model. By an analogous notion to certainty equivalent for risk, we consider the certainty equivalent for ambiguity . It is a certain value that is equally attractive to the household's asset facing storm surge risks under ambiguity and can be written as follows: The ambiguity premium is defined as certainty equivalent for risk minus certainty equivalent for ambiguity. 62 Note that the following equation holds because is equivalent to − .

= + +
Finally, these values are calculated by using the estimated econometric model of the LDC model.

Experimental design of web survey. We conducted the choice experiments on households'
preferences for purchasing the hypothetical insurance through a web survey. The steps are as follows: (1) Explain the impact of disasters on daily life and past damages due to storm surge disasters and inundation (abstracted from the leaflet by the Japanese government).
(2) Ask respondents about the estimated values of their house and furniture. These data were used in the latter part of the survey.
(3) Explain that projections of storm surge inundation risks inevitably include ambiguity and that 25 PDFs of storm surge inundation risks were projected as ambiguity.
(4) Explain that the damage to house from storm surge inundation is classified into four categories according to the inundation depth: "no flooding," "flooding under floor," "flooding on floor," and "house submerged" when the inundation depth is less than 0.01m, 0.01m-0.5m, 0.5m-2.0m and 2.0m or over, respectively.
(5) Two projections of storm surge inundation risks were randomly chosen from a set of five inundation projections that are typical in the target area (Fig. S4). The more serious one was set as the worst projection and the other was set as the average projection. Then these projections were presented to the respondents.
(6) When "flooding under floor," "flooding on floor," and "house submerged" occur due to storm surge floods, we assume 1%, 50%, and 100% loss of the households' assets (house and furniture) value as responded in step (2). The amount of economic loss due to storm surge inundation was automatically calculated on the web system and presented to the respondents.
(7) To evaluate the economic value that completely eliminates the risk of damage due to storm surge flooding, respondents were asked whether or not to buy a hypothetical insurance that fully compensates them for the storm surge damages (Fig.2). The question was as follows: "Suppose that the government sells a new insurance, which covers all losses due to storm surge floods with full compensation for house restoring cost and household assets, and that the insurance will be contracted once in a year, would you like to buy this new insurance policy with the annual fee of XXX yen?" The amount of insurance fee (XXX yen/year) was automatically calculated and presented to the respondents according to the following formula: their estimated household asset value×0.001%, 0.005%, 0.01%, 0.05%, 0.1%, 0.5%, and 1% (one of these amounts was randomly chosen and presented to the respondent in the web system). (8) We estimated the LDC model using the survey data after excluding "protest responses." 54 The protest responses were obtained from the respondents who chose not to buy the hypothetical insurance due to the following two reasons: "I cannot understand the question" and "I cannot accept the hypothetical scenarios." The number of protest responses is 340 (34%). As a result, the effective number of observations is 660.
Outline of survey attributes. The respondents of the web surveys were recruited from1.9 million panel members registered with Cross Marketing, a Japanese company. The web survey was conducted with 1,000 households who responded that they live at an altitude of 5m or less among the detached houses in Osaka Bay. This survey was conducted from December 16 to December 22 in 2016. Table S1 shows the means and standard deviations of the socioeconomic attributes of the respondents.

Data availability
The data that support the findings of this study are available online at https://drive.google.com/drive/u/1/folders/1XHlMFZuCATM0mtpMsQ4-mzwATEiaKQiH  This also shows an example of scenarios for choice experiment and the key parts of the questionnaire. Economic loss of AAA, BBB, and CCC were calculated with 1%, 50%, and 100% of the respondent's asset value. XXX were randomly chosen to the respondents based on the given asset value: their estimated household asset value×0.001%, 0.005%, 0.01%, 0.05%, 0.1%, 0.5%, and 1%. The amounts of AAA, BBB, CCC, and XXX were automatically calculated in the web survey system.
In the average one among 25 projections of inundation risk from typhoon simulation, the inundation risk of your house was projected as the following figure. Average projection In the worst one among 25 projections of inundation risk from typhoon simulation, the inundation risk of your house was projected as the following figure. Worst projection Scenario Assumption.
• If your house gets flooded under floor (inundation depth is below 0.5m) from storm surge, your economic loss is AAA yen (1% of your asset value).
• If your house gets flooded on floor (inundation depth is 0.5m-2.0m) from storm surge, your economic loss is BBB yen (50% of your asset value).
• If your house gets submerged (inundation depth is over 2.0m) from storm surge, your economic loss is CCC yen (100% of your asset value).
• You do not have any insurance for compensating economic loss from storm surge.
Suppose that the government sells a new insurance which covers all losses due to storm surge floods with full compensation of house restoring cost and households assets back, and that the insurance will be contracted once in a year. Do you like to buy this new insurance policy with the annual fee of XXX yen?
◼ Yes, I'd like to buy the insurance with annual fee of XXX yen.
◼ No, I'd not like to buy the insurance.
Why did you chose not to buy the insurance? Please choose one of the following options as your reason.