Comparison of Flood Vulnerability Assessments to Climate Change by Construction Frameworks for a Composite Indicator

As extreme weather conditions due to climate change can cause deadly flood damages all around the world, a role of the flood vulnerability assessment has become recognized as one of the preemptive measures in nonstructural flood mitigation strategies. Although the flood vulnerability is most commonly assessed by a composite indicator compiled from multidimensional phenomena and multiple conflicting criteria associated with floods, directly or indirectly, it has been often overlooked that the construction frameworks and processes can have a significant influence on the flood vulnerability indicator outcomes. This study has, therefore, compared the flood vulnerability ranking orders for the 54 administrative districts in the Nakdong River Watershed of the Korean Peninsula, ranked from composite indicators by different frameworks and multi-attribute utility functions for combining the three assessment components, such as exposure, sensitivity, and coping, presented in the IPCC Third Assessment Report. The results show that the different aggregation components and utility functions under the same proxy variable system can lead to larger volatility of flood vulnerability rankings than expected. It is concluded that the vulnerability indicator needs to be derived from all three assessment components by a multiplicative utility function for a desirable flood vulnerability assessment to climate change.


Introduction
As global and regional climate change has been irrefutably occurring due to global warming [1], extreme weather conditions, such as severe torrential storms and super typhoons, recently and frequently occur at, and cause damage to, the Korean peninsula.Since it is geographically located on the east coast of the Eurasian Continent and also adjacent to the Western Pacific, it has much more precipitation than the continental average because of typhoons in summer and a rainy period from the East-Asian Monsoon.Such complex geographical and climatological circumstances require studies on the assessment of the regional flood vulnerability for each administrative district to establish the customized mitigation plans.Furthermore, as the occurrence of heavy floods increases due to climate change, the flood vulnerability assessment is recognized as an essential tool to provide the preventative information in nonstructural flood mitigation measures, especially concerning climate variability, changes, and impacts.One of the most well-known definitions on the vulnerability to climate change is presented in the Intergovernmental Panel on Climate Change (IPCC) Third Assessment Report [2] as a degree to which a system is exposed to, sensitive to, and/or unable to cope with adverse effects of climate change.The exposure denotes the degree of extrinsic factors exposed to significant variations.The sensitivity means the degree of intrinsic factors affected by dangerous phenomena likely to suffer harm.The lack of coping indicates the degree of limitations and incapacity of a system to adapt and reduce adverse effects of experienced hazards.The IPCC [2] framework for the vulnerability assessment to climate change has been used to interpret elusive issues in a wide range of fields, science, environment, economy, society, etc. [3][4][5][6][7][8][9][10][11].
In case that it is almost impossible in decision making to represent a multi-dimensional issue by individual attribute variables especially under multiple conflicting criteria, they can be aggregated into a composite indicator to measure the overall performance of the multi-dimensional concept [12].Accordingly, a composite indicator are derived from individual constituent components most commonly by multi-attribute utility functions, which can provide the significant information on the prioritization of alternatives for policy and decision problems.However, the vulnerability composite indicators have been compiled by the conventional aggregation methods without concerns for any effect of frameworks or methods.Some previous studies [13][14][15] generated the vulnerability indicators from the constituent attribute components by additive multi-attribute utility functions while the composite indicators were combined by multiplicative multi-attribute utility functions in some other studies [16][17][18].In addition, the potential impact component were also used as a function of the exposure and sensitivity components to compile with the coping (or lack of coping) component into a composite indicator by additive or multiplicative aggregations in some literature [19][20][21][22][23].
This study has, therefore, compared and investigated the flood vulnerability assessment outcomes derived by various construction frameworks and processes that have influence on the composite indicators for the use in the process of prioritizing and selecting flood mitigation strategies.In this study, the flood vulnerability indicator (FV I) to climate change is derived by the following construction processes; (1) a theoretical framework is selected from the IPCC [2] vulnerability assessment to climate change, comprising the three assessment components, exposure, sensitivity (or susceptibility or fragility), and coping (or adaptive capacity or resilience); (2) proxy variables for the three individual components are selected in terms of measurability, representatives, soundness, etc.; (3) all proxy variables at different scales and units are normalized into ratios to the maximum value of each variable for a common scale 1 to 0; (4) an equal weighting is assumed and assigned to the proxy variables and the three assessment components; (5)  factors exposed to significant variations.The sensitivity means the degree of intrinsic factors affected by dangerous phenomena likely to suffer harm.The lack of coping indicates the degree of limitations and incapacity of a system to adapt and reduce adverse effects of experienced hazards.The IPCC [2] framework for the vulnerability assessment to climate change has been used to interpret elusive issues in a wide range of fields, science, environment, economy, society, etc. [3][4][5][6][7][8][9][10][11].
In case that it is almost impossible in decision making to represent a multi-dimensional issue by individual attribute variables especially under multiple conflicting criteria, they can be aggregated into a composite indicator to measure the overall performance of the multi-dimensional concept [12].Accordingly, a composite indicator are derived from individual constituent components most commonly by multi-attribute utility functions, which can provide the significant information on the prioritization of alternatives for policy and decision problems.However, the vulnerability composite indicators have been compiled by the conventional aggregation methods without concerns for any effect of frameworks or methods.Some previous studies [13][14][15] generated the vulnerability indicators from the constituent attribute components by additive multi-attribute utility functions while the composite indicators were combined by multiplicative multi-attribute utility functions in some other studies [16][17][18].In addition, the potential impact component were also used as a function of the exposure and sensitivity components to compile with the coping (or lack of coping) component into a composite indicator by additive or multiplicative aggregations in some literature [19][20][21][22][23].
This study has, therefore, compared and investigated the flood vulnerability assessment outcomes derived by various construction frameworks and processes that have influence on the composite indicators for the use in the process of prioritizing and selecting flood mitigation strategies.In this study, the flood vulnerability indicator () to climate change is derived by the following construction processes; (1) a theoretical framework is selected from the IPCC [2] vulnerability assessment to climate change, comprising the three assessment components, exposure, sensitivity (or susceptibility or fragility), and coping (or adaptive capacity or resilience); (2) proxy variables for the three individual components are selected in terms of measurability, representatives, soundness, etc.; (3) all proxy variables at different scales and units are normalized into ratios to the maximum value of each variable for a common scale 1 to 0; (4) an equal weighting is assumed and assigned to the proxy variables and the three assessment components; (5) an additive multi-attribute utility function is employed to combine individual proxy variables into each component indicator; and (6) component indicators are aggregated into a composite indicator by either additive or multiplicative utility functions.The flood vulnerability ranking orders are then placed and compared for each administrative district under study from the composite indicator outcomes derived by different construction frameworks.The flood vulnerability assessment results for the 54 administrative districts are classified into the three levels, and the 18 administrative districts ranked at the high level are finally compared with flood hazard areas managed by a government office.The analysis procedure of this study is summarized in Figure 1.

Study Site
To compare the flood vulnerability assessment outcomes to climate change by a variety of construction frameworks for the vulnerability composite indicators, the administrative districts in the Nakdong River Watershed of the Korean peninsula were chosen for study because this study site, located between 127 • 29 ~129 • 18 E and 35 • 03 ~37 • 13 N, has various characteristics in geophysical, climatological, and socio-economic aspects for each administrative district.The size of the study watershed is 23,690.32km 2 , the second largest one in Korea, and the main channel length is 511.92 km, the nation's longest river that flows from the Taebaek Mountains (1945.15EL.m) over the north region to the South Sea, as shown Figure 2. The mean elevation is 290.51EL.m and the averaged slope is 37% over the Nakdong River Watershed.The annual mean rainfall depth is 1337.4mm over the study site for the past decade during 2007 to 2016 years, and there is difference between 1247.8 mm for the upstream and 1430.2 mm near the Nakdong River estuary area.The study site is composed of a total of 54 administrative districts, such as the three metropolitan cities comprising 14 districts (Gu in Korean), along with 19 cities (Si in Korean) and 21 counties (Gun in Korean) in five provinces (Do in Korean).

Study Site
To compare the flood vulnerability assessment outcomes to climate change by a variety of construction frameworks for the vulnerability composite indicators, the administrative districts in the Nakdong River Watershed of the Korean peninsula were chosen for study because this study site, located between 127°29′~129°18′ E and 35°03′~37°13′ N, has various characteristics in geophysical, climatological, and socio-economic aspects for each administrative district.The size of the study watershed is 23,690.32km 2 , the second largest one in Korea, and the main channel length is 511.92 km, the nation's longest river that flows from the Taebaek Mountains (1945.15EL.m) over the north region to the South Sea, as shown Figure 2. The mean elevation is 290.51EL.m and the averaged slope is 37% over the Nakdong River Watershed.The annual mean rainfall depth is 1337.4mm over the study site for the past decade during 2007 to 2016 years, and there is difference between 1247.8 mm for the upstream and 1430.2 mm near the Nakdong River estuary area.The study site is composed of a total of 54 administrative districts, such as the three metropolitan cities comprising 14 districts (Gu in Korean), along with 19 cities (Si in Korean) and 21 counties (Gun in Korean) in five provinces (Do in Korean).

Framework and Variables
The selection of a sound theoretical framework is the most important process in constructing the flood vulnerability composite indicator.This study adopts a conceptual framework of vulnerability to climate change in the IPCC [2] that comprises the three assessment components, such as exposure, sensitivity, and coping, or the two assessment components, such as potential impact and coping, as shown in Table 1.

Framework and Variables
The selection of a sound theoretical framework is the most important process in constructing the flood vulnerability composite indicator.This study adopts a conceptual framework of vulnerability to climate change in the IPCC [2] that comprises the three assessment components, such as exposure, sensitivity, and coping, or the two assessment components, such as potential impact and coping, as shown in Table 1.
To measure the three constituent components of the flood vulnerability indicator to climate change, the three proxy variables are selected for each assessment component as shown in Table 1 in the light of previous studies on flood vulnerability assessments [15][16][17]23,24].The selection of the nine total proxy variables is based on the simplicity of data acquisition and construction, as well as the principles of measurability, representatives, and integrity.civil servant ratio people/inhabitants 1 The exposure and sensitivity components can be combined into the potential impact component.
Exposure is defined as the nature and degree to which a system is exposed to significant perturbations [2].The proxy variables for the exposure component are composed of the maximum daily rainfall (E1) (mm/day), rainfall intensity (E2) (mm/hr), and days of heavy rainfall greater than 80 mm per day (E3) (days).For the consideration of climate change impacts the three proxy variables are temporally averaged for the past decade during 2007 to 2016 years, and then spatially averaged for 54 administrative districts by the inverse distance weighted interpolation from observations at 75 gauge stations managed by the Korea Meteorological Administration (KMA) in the Korean Peninsula.
Sensitivity is defined as the degree to which a system is affected by disturbances [2].The sensitivity component is measured by the number of the population per district area (S1) (inhabitants/km 2 ) for the human feature, a ratio of the houses, buildings, and infrastructure area to the district area (S2) (km 2 /km 2 ) for the property feature, and the averaged terrain slope over a district (S3) (m/m) for the geographical feature.
Coping is defined as the ability of a system to adjust to disturbance, moderate damage, or cope with consequences [2].The coping component comprises the three proxy variables such as a ratio of the river improvement length to the total river length in district (km/km) (C1) representative for coping with river flooding, drainage rate capacity of stormwater pumping stations in district (C2) (m 3 /min) representative for coping with inland flooding, and the number of civil servants working for water management per inhabitants in a district (people/inhabitants) representative for nonstructural flood countermeasures.

Normalizations
Since variables with different characteristics from different sources are usually measured at different scales and units, they need to be transformed into a commensurate scale by a proper normalization method.This study divides all data by each maximum value for normalized scores in consideration of the functional relationship between variables and vulnerability.As previous studies computed the normalized values based on each variable's positive or negative contribution to vulnerability [18,24,25], the variable that has a positive elasticity to vulnerability is normalized by Equation ( 1) and the variable that has a negative elasticity to vulnerability is normalized by Equation (2) as follows: where u i is the normalized score, x i is the actual value of the proxy variable i, and max(x i ) and min(x i ) are, respectively, the maximum and minimum values of the proxy variable i for the vulnerability assessment components.The two opposite values can correspond bilaterally and symmetrically to each other by the above two equations.Note that this method can generate normalized scores while preserving the distribution of the original data values without distorting outliers.
The variables E1, E2, E3, S1, and S2 are normalized by Equation (1) because they have the positive (increasing vulnerability) functional relationship between variables and vulnerability, while Equation ( 2) is employed to normalize the variables S3, C1, C2, and C3 whose higher values decrease the vulnerability due to the negative functional relationship with vulnerability.Thus, all the proxy variables are transformed into the normalized scores with the positive elasticity to vulnerability.

Aggregations
The next process is to aggregate the normalized proxy variables for each assessment component j into each indicator I j as: where I j denotes EI, SI, and CI for the exposure indicator, sensitivity indicator, and coping indicator, respectively, w i is the weight of the normalized score of the proxy variable i for which the value of 1/3 is assigned equally to the proxy variables by adopting an equal weighting method, and k is the number of variables for each component E, S, and C.
The three component indicators EI, SI, and CI are then aggregated into a composite indicator for the flood vulnerability to climate change using either an additive or a multiplicative multi-attribute utility function as: or: where FV I a (3) and FV I m (3) are the flood vulnerability composite indicators from the three components by additive and multiplicative forms, respectively. or: where PI a or PI m is the potential impact indicator by an additive form or a multiplicative form.Then, the flood vulnerability indicator can be compiled from the two component indicators by an additive utility form FV I a (2) or a multiplicative utility form FV I m (2) as: or: Thus, the flood vulnerability indicators FV I a (2) and FV I m (2) are aggregated from the coping component weighted by 1/2 and the exposure and sensitivity components with 1/4 weight for each.
Note that all the component indicators EI, SI, CI, PI a , and PI m aggregated from proxy variables need to be normalized with the same maximum value of 1 to affect a composite indicator equally, which can prevent any unintended weighting effects for further comparison of assessment components in the aggregation process by Equations ( 4), ( 5), (8), or (9).The computational framework for the FV Is is presented in Figure 3.
Finally, the ranking orders can be assigned to the composite indicator outcomes in the process of ordering priority or selecting alternatives for policy and decision making. or: Thus, the flood vulnerability indicators   (2) and   (2) are aggregated from the coping component weighted by 1/2 and the exposure and sensitivity components with 1/4 weight for each.
Note that all the component indicators , ,  ,   , and   aggregated from proxy variables need to be normalized with the same maximum value of 1 to affect a composite indicator equally, which can prevent any unintended weighting effects for further comparison of assessment components in the aggregation process by Equations ( 4), ( 5), (8), or (9).The computational framework for the  is presented in Figure 3.
Finally, the ranking orders can be assigned to the composite indicator outcomes in the process of ordering priority or selecting alternatives for policy and decision making.3) Equation ( 4) or ( 5) Equation ( 6) or ( 7) Equation ( 8) or ( 9)

Comparison of Indicators
Figure 4 shows the spatial distribution of three constituent component indicators, EI, SI, and CI, along with the two kinds of potential impact indicators, PI a and PI m .The value of the exposure indicator EI is higher in the south part close to the coastline where typhoons and the summer monsoon effects are more dominant.The value of the sensitive indicator SI is higher in the middle and the right bottom parts where the large cities with higher population densities and urbanization rates are located.The value of the coping (lack of coping) indicator CI is higher in the southwestern part of the study site under relatively poor conditions of structural and nonstructural countermeasures for flood disasters.Such a greater vulnerable condition in the coping component can lead to higher flood vulnerability outcomes despite relatively lower sensitivity conditions, particularity for additive composite indicators, as a result of higher compensability in aggregation schemes.In the potential impact indicators PI a and PI m combined from the two component indicators EI and SI, the sensitivity indicator SI with large variations is more influential one generating higher values for the large metropolitan cities.A peculiar feature in the component indicator outcomes is that the values of exposure and coping indicators EI and CI are highly distributed over this study site with accordingly higher minimum values.
Figure 5 shows the spatial distribution of the flood vulnerability ranks based on the four different composite indicator outcomes FV I a (3) and FV I m (3) from the three components, and FV I a (2) and FV I m (2) from the two components by additive and multiplicative utility functions.Regardless of the construction frameworks for the flood vulnerability indicator, higher flood vulnerability to climate change is evaluated for most of districts in the three metropolitan cities, Busan, Ulsan (right lower), and Daegu (around center) that have relatively higher values of SI, and for the administrative districts in the southwestern part of the study site that have relatively higher values of EI and CI.Meanwhile, lower flood vulnerability to climate change is shown in the mid-northern administrative districts that have relatively lower or moderate values of all the three components.It is also found that the volatility of rankings and rank reversals occur in the four flood vulnerability results for some administrative districts.A greater interpretation on the differences in flood vulnerability results is presented in the following sections.
can lead to higher flood vulnerability outcomes despite relatively lower sensitivity conditions, particularity for additive composite indicators, as a result of higher compensability in aggregation schemes.In the potential impact indicators   and   combined from the two component indicators  and  , the sensitivity indicator  with large variations is more influential one generating higher values for the large metropolitan cities.A peculiar feature in the component indicator outcomes is that the values of exposure and coping indicators  and  are highly distributed over this study site with accordingly higher minimum values.4) and ( 5).Table 2 denotes the ten administrative districts that have large

Comparision of FVIs by Aggregation Functions
Figure 5a,b compares the rank ordering results of the two flood vulnerability indicators FV I a (3) and FV I m (3) aggregated from the three components by additive and multiplicative utility functions, respectively in Equations ( 4) and ( 5).Table 2 denotes the ten administrative districts that have large differences between the two flood vulnerability rank results from FV I a (3) and FV I m (3).The ranking orders from FV I a (3) are much higher (with smaller numbers) than those from FV I m (3) when any two of the three component indicators have very large values and the value of the other component indicator is quite low.Meanwhile, the ranking orders from FV I m (3) are much higher than those from FV I a (3) when all the three component indicators have moderate or relatively higher level values.Such features of differences between the two rank results are due to higher compensability in the additive aggregation function and lower compensability in the multiplicative aggregation function.Thus, the additive composite indicator FV I a (3) is influenced by the three component values altogether, while the multiplicative composite indicator FV I m (3) is dramatically swayed by a lower value of one of the three components.
Table 2.The comparison of flood vulnerability ranks from the two composite indicators FV I a (3) and FV I m (3) along with the three component values for the top ten administrative districts with large differences between the two rank results.

Comparision of FVIs by Assessment Components
The flood vulnerability rank ordering results of the additive and multiplicative composite indicators FV I a (3) and FV I m (3) aggregated from the three components in Figure 5a,b are, respectively, compared with those of the additive and multiplicative composite indicators FV I a (2) and FV I m (2) aggregated from the two components in Figure 5c,d.The ten administrative districts that have large differences between the two flood vulnerability rank results from FV I a (3) and FV I a (2) are presented in Table 3, and those from FV I m (3) and FV I m (2) are listed in Table 4.The raking orders from FV I a (2) and FV I m (2) are higher than those from FV I a (3) and FV I m (3) when the value of the component indicator CI is very large and one or both of the two component indicators EI and SI have low values.On the contrary, it is expected that the ranking orders from FV I a (3) and FV I m (3) are higher than those from FV I a (2) and FV I m (2) when the component indicator CI is quite low and one, or both, of the two component indicators EI and SI have large values.The ranking orders from FV I a (2) and FV I m (2) become lower for the administrative districts where the component CI is around the minimum value of 0.472 and one of the two component indicators EI and SI has a relatively large value.Such different vulnerability ranking results are induced by the framework using another indicator PI combining the two components EI and SI.The vulnerability ranks from the composite indicators FV I a (2) and FV I m (2) are influenced more by the coping indicator CI weighted one-half than by exposure and sensitivity indicators EI and SI with one-quarter weight for each, when compared with rank results from the composite indicators FV I a (3) and FV I m (3) compiled from the three equally weighted components.
Table 3.The comparison of flood vulnerability ranks from the two composite indicators FV I a (3) and FV I a (2) along with the constituent component values for the top ten administrative districts with large differences between the two rank results.Table 4.The comparison of flood vulnerability ranks from the two composite indicators FV I m (3) and FV I m (2) along with the constituent component values for the top ten administrative districts with large differences between the two rank results.

Comparision of FVIs and FHAs
As a means of evaluating the four FV Is to find the flood vulnerability assessment result suitable to the study site, this study compares the FV I outcomes with one of official flood vulnerability maps, the flood hazard areas (FHA) in 2010 managed by the Ministry of the Interior and Safety (MOIS) [26].For the comparison with the flood hazard map, the 54 administrative districts are classified into the three levels, such as high (H), middle (M), and low (L), with respect to the flood vulnerability by the four FV Is.Thus, the 18 administrative districts ranked at Level H are compared with the 10 administrative districts including FHA rated at Class 1 by the MOIS as shown in Figure 6.The total 4, 9, 3, and 7 administrative districts in Level H, respectively, from FV I a (3), FV I m (3), FV I a (2), and FV I m (2) match the FHA Class 1.This implies that the flood vulnerability assessment using FV I m (3) can provide better information on flood mitigation strategies for this study site than vulnerability results from other aggregation schemes.It is also found that the actual flood hazard feature is relatively different from the additive composite indicators FV I a (3) and FV I a (2) derived with higher compensability of attribute components.It is, therefore, expected that for the southwestern part of the study site that shows higher FV I a (3) and FV I a (2) by lower SI and higher EI and CI, the future flood mitigation strategies can reduce the flood vulnerability by additive composite indicators as well.

Discussion and Limitations
Many composite indicators have been proposed and used to illustrate multidimensional and complex issues in a variety of fields, such as science, environment, economy, society, etc.The composite indicators need to be properly constructed and interpreted not to provide misleading information for policy and decision making.There have been, however, limited studies conducted on the influence of aggregation frameworks and methods on the composite indicator outcomes.As disastrous floods occur recently and frequently due to extreme weather conditions, such as severe torrential storms and super typhoons, the flood vulnerability assessment becomes recognized as an essential tool to provide the preemptive and preventative information on nonstructural flood mitigation measures to climate change.One of remarkable frameworks to measure the vulnerability to climate change comprises the three contributing components, such as exposure, sensitivity, and coping presented in IPCC [2], and the vulnerability composite indicators have been compiled by various multi-attribute utility functions, representatively additive or multiplicative forms.This study has, therefore, investigated the influence of aggregation processes for constructing flood vulnerability composite indicators based on the framework for the vulnerability measurement to climate change in IPCC [2].The comparison of flood vulnerability composite indicator outcomes by different aggregation frameworks and methods are implemented for the 54 administrative districts of the Nakdong River Watershed in the Korean peninsula where severe torrential storms and super typhoons occur and districts with high population density and urbanization ratio exist.This study has focused on the comparison of the flood vulnerability ranking orders from different composite indicator outcomes by various aggregation schemes, and demonstrated that a selection of the better aggregation framework for constructing a vulnerability composite indicator is required because the different frameworks and methods can lead to quite different vulnerability ranking results than expected.In comparison of the two vulnerability rank results based on the additive and the

Discussion and Limitations
Many composite indicators have been proposed and used to illustrate multidimensional and complex issues in a variety of fields, such as science, environment, economy, society, etc.The composite indicators need to be properly constructed and interpreted not to provide misleading information for policy and decision making.There have been, however, limited studies conducted on the influence of aggregation frameworks and methods on the composite indicator outcomes.As disastrous floods occur recently and frequently due to extreme weather conditions, such as severe torrential storms and super typhoons, the flood vulnerability assessment becomes recognized as an essential tool to provide the preemptive and preventative information on nonstructural flood mitigation measures to climate change.One of remarkable frameworks to measure the vulnerability to climate change comprises the three contributing components, such as exposure, sensitivity, and coping presented in IPCC [2], and the vulnerability composite indicators have been compiled by various multi-attribute utility functions, representatively additive or multiplicative forms.This study has, therefore, investigated the influence of aggregation processes for constructing flood vulnerability composite indicators based on the framework for the vulnerability measurement to climate change in IPCC [2].The comparison of flood vulnerability composite indicator outcomes by different aggregation frameworks and methods are implemented for the 54 administrative districts of the Nakdong River Watershed in the Korean peninsula where severe torrential storms and super typhoons occur and districts with high population density and urbanization ratio exist.This study has focused on the comparison of the flood vulnerability ranking orders from different composite indicator outcomes by various aggregation schemes, and demonstrated that a selection of the better aggregation framework for constructing a vulnerability composite indicator is required because the different frameworks and methods can lead to quite different vulnerability ranking results than expected.In comparison of the two vulnerability rank results based on the additive and the multiplicative composite indicators FV I a (3) and FV I m (3) from the three components EI, SI, and CI, some administrative districts present the considerable ranking changes around 20 levels higher or lower in the total 54 ranking levels.It can be attributed to the functional feature differences in aggregation schemes.The additive aggregation function has higher compensability that the composite indicator is influenced by the three component values altogether, while the multiplicative aggregation function has lower compensability that decreases the indicator outcome by a low value of one of the three components.The vulnerability ranks based on FV I a (2) and FV I m (2) from the framework comprising the two assessment components CI and PI (combined by EI and SI) are also compared with those based on FV I a (3) and FV I m (3) from the three assessment components.It is expected that the ranking results from FV I a (2) and FV I m (2) are more sensitive to the component indicator CI weighted by one-half than the other indicators EI and SI with one-quarter weight for each.The ranking changes can occur especially for the administrative districts with very high or low values of CI, and the considerable ranking changes are around 10 levels in the total 54 ranking levels in comparison of the two results by different assessment component frameworks.Such underestimated or overestimated vulnerability results by aggregation schemes may provide misleading information on decision making.When the four FV I outcomes for the 54 districts are classified into the three levels, such as high, middle, and low, and compared with an official hazard map by a government office, the 18 administrative districts ranked at Level H by the FV I m (3) shows the best agreement with the FHA ranked at Class 1, whereas there is worse agreement for the additive composite indicators aggregated by higher compensability.It is more reasonable in the IPCC framework [2] that the region where all the assessment components increase in vulnerability can have a higher flood vulnerability rather than the flood vulnerability can be compensated for by any surplus values of some components.Additionally, the flood vulnerability indicators compiled from a more weighted component CI can be less robust in the concern that the assessment of coping is on the nature of more uncertainty since it is usually measured by behavior and politics rather than by data and observations [27].In conclusion, a composite indicator aggregated from the three assessment components by the multiplicative utility function is recommended for the flood vulnerability assessment to climate change for this study site.It is expected that the proposed framework for constructing the flood vulnerability indicator can provide basis and precedence information on mitigation or adaptation policies to climate change.
As in previous studies [28][29][30][31][32][33], this study has evaluated the influence of construction frameworks on the composite indicator outcomes and additionally proposed the proper scheme by comparison with a flood hazard map for the study site.However, this study has the following limitations.Although a composite indicator is useful in decision making for multidimensional phenomena and multiple conflicting issues, it is difficult, or almost impossible, to have a precise mathematical scheme for the vulnerability assessment by a composite indicator because the vulnerability indicators might be technically problematic and have limitations and drawbacks in construction processes.In addition to the construction frameworks for the vulnerability indicators, the selection of proxy variables for assessment components has a large influence on the vulnerability outcomes.The proxy variables relevant to individual components need to be selected and identified by concerns of mutual independency, as well as representatives based on a complete understanding of the analysis system, since the preference independence among constituent assessment components is a necessary and sufficient condition for the existence of a proper composite indicator aggregated from them.There are a number of methods in normalization for data types and distribution, and this study uses a method that can conserve the value distribution of the original data.This study also adopts an equal weighting assumption for the proxy variables and assessment components for which the proper weighting coefficients need to be determined and assigned in a logical process as following some studies in the Korean Peninsula [34][35][36].Differences and uncertainties in the normalization and weight determination methods can have another impact on composite indicator outcomes.The upcoming study is, therefore, required to scrutinize these important issues affecting composite indicator outcomes, particularly for the selection of representative proxy variables through the multivariate analysis, and the uncertainty in normalization and weighting methods for each variable and component contributing to the analysis.
an additive multi-attribute utility function is employed to combine individual proxy variables into each component indicator; and (6) component indicators are aggregated into a composite indicator by either additive or multiplicative utility functions.The flood vulnerability ranking orders are then placed and compared for each administrative district under study from the composite indicator outcomes derived by different construction frameworks.The flood vulnerability assessment results for the 54 administrative districts are classified into the three levels, and the 18 administrative districts ranked at the high level are finally compared with flood hazard areas managed by a government office.The analysis procedure of this study is summarized in Figure 1.Sustainability 2018, 10, x FOR PEER REVIEW 2 of 14

Figure 1 .
Figure 1.The analysis procedure diagram for the  (flood vulnerability indicator) construction and comparison.

Figure 1 .
Figure 1.The analysis procedure diagram for the FV I (flood vulnerability indicator) construction and comparison.

Figure 2 .
Figure 2. The localization of 54 administrative districts (yellow) under study along with 75 gauge stations (green) managed by the Korea Meteorological Administration and the main streamlines (blue) in the Nakdong River Watershed (red).

Figure 2 .
Figure 2. The localization of 54 administrative districts (yellow) under study along with 75 gauge stations (green) managed by the Korea Meteorological Administration and the main streamlines (blue) in the Nakdong River Watershed (red).
w j is the weight of the component indicator I j under the equal weighting value of 1/3 for each component.n is the number of component indicators.Note that all the three component indicators from Equation (3) have the positive elasticity to the flood vulnerability because all the proxy variables are normalized with the functional relationship for increasing vulnerability as mentioned in Section 3.2.A mixed use of both positive and negative functional relationships between the component indicators and the vulnerability indicator can cause undesirable vulnerability outcomes especially when quite low values of a component indicator employed as a divisor (for the negative contribution in the multiplicative utility form) may generate abnormally large composite indicator outcomes numerically.When the potential impact indicator PI is used as a function of the exposure indicator EI and the sensitivity indicator SI as:

Figure 3 .FVI
Figure 3.The process diagram for the  (flood vulnerability indicator) computations.

Figure 3 .
Figure 3.The process diagram for the FV I (flood vulnerability indicator) computations.

Figure 4 .
Figure 4. Spatial distribution of the vulnerability assessment component indicators for the study site: (a) exposure indicator ; (b) sensitivity indicator S; (c) coping indicator ; (d) additive potential impact indicator   ; and (e) Multiplicative potential impact indicator   .

Figure 4 . 14 Figure 5 Figure 5 .
Figure 4. Spatial distribution of the vulnerability assessment component indicators for the study site: (a) exposure indicator EI; (b) sensitivity indicator SI; (c) coping indicator CI; (d) additive potential impact indicator PI a ; and (e) Multiplicative potential impact indicator PI m .

Figure 5a ,
Figure 5a,b compares the rank ordering results of the two flood vulnerability indicators   (3) and   (3) aggregated from the three components by additive and multiplicative utility functions, respectively in Equations (4) and (5).Table2denotes the ten administrative districts that have large

Figure 5 .
Figure 5. Spatial distribution of the flood vulnerability ranking results based on the four composite indicators for the study site: (a) ranking orders from FV I a (3); (b) ranking orders from FV I m (3); (c) ranking orders from FV I a (2); and (d) ranking orders from FV I m (2).

Figure 6 .
Figure 6.Spatial distribution of the 18 administrative districts with high vulnerability level (H) based on the four composite indicators for the study site: (a) districts rated H from   (3); (b) districts rated H from   (3); (c) districts rated H from   (2); and (d) districts rated H from   (2) along with (e) the 10 administrative districts including flood hazard areas (FHA) rated Class 1 by the MOIS in 2010.

Figure 6 .
Figure 6.Spatial distribution of the 18 administrative districts with high vulnerability level (H) based on the four composite indicators for the study site: (a) districts rated H from FV I a (3); (b) districts rated H from FV I m (3); (c) districts rated H from FV I a (2); and (d) districts rated H from FV I m (2) along with (e) the 10 administrative districts including flood hazard areas (FHA) rated Class 1 by the MOIS in 2010.

Table 1 .
[2] assessment components and selected proxy variables for the FV I (flood vulnerability indicator) based on the vulnerability assessment concept in the IPCC[2].