Developing Flood Vulnerability Functions through Questionnaire Survey for Flood Risk Assessments in the Meghna Basin, Bangladesh

: Flood vulnerability is estimated by Flood Damage Functions (FDFs), which are crucial for integrated ﬂood risk assessment for developing sustainable ﬂood management, mitigation, and adaptation strategies under global change. However, the FDFs, either empirical or synthetic, are not available in Bangladesh. Therefore, this paper focused on developing the synthetic type of FDFs for agriculture and rural households through the data of a well–structured questionnaire survey conducted in two pilot sub–districts of northeastern Bangladesh in the Meghna River basin. Multiple regression analyses were performed on the collected data, and the best performing models were selected to establish FDFs. The FDF for agriculture (~196 samples) was developed concerning damage to Boro rice, whereas the FDFs for households (~165 samples) were developed concerning damage to the buildings and household property of three house types (Mud, Brick, and Concrete), separately. The results revealed that there were no yield losses when the water levels were lower than 25 cm (~rice tiller height), and the yield losses were ~100% when the water levels were 70–75 cm deep (~rice grain height). Mud houses and their household property were found the most ﬂood–vulnerable and likely to experience total damage when the water levels exceeded 150 cm above the plinth level, whereas the damage to Brick and Concrete houses and their household property was found likely to remain partial even when the water levels exceeded 150 cm above the plinth level. The developed FDFs can be used to assess potential ﬂood risk in the study area for sustainable and effective management of ﬂood disasters and build back better under global change in the future.


Introduction
Bangladesh is a deltaic country located at the confluence of three large transboundary rivers, the Ganges, Brahmaputra, and Meghna; however, about 92-93% of their total catchment area is situated outside Bangladesh [1,2]. Due to this complex geographical location, flooding is the major and frequent natural hazard in Bangladesh, resulting in devastating damage to the people and socio-economic sectors of the country [3,4]. The northeastern part of the country, located downstream of the Meghna River basin, is suitable for agricultural and aquaculture activities. The area contributes to producing more than 18% of the nation's total rice grains and providing livelihoods for about 20 million people, thus supporting the country and its economy significantly [5][6][7][8]. However, this important area of the country is adversely affected by pre-monsoon (April-May) and monsoon (June-September) floods every year. Particularly, frequent pre-monsoon flash floods rule-based computerized decision support system to simulate rice yield loss through a crop simulation model for monsoon flooding (June-September) in Bangladesh. The main constraint of the study is that the model was run with site-specific input data, and, thus, a national minimum dataset is required to support decision making. Another limitation is that it only considered rice crops cultivated during the monsoon period in the region of Bangladesh different from the rice crops and region considered in this study. In addition, the validation of the simulated rice yield loss was not performed and, thus, the results are likely to have uncertainties. Two recent studies [44,45] conducted for flood risk assessment in the eastern part of Dhaka also estimated vulnerability by assuming land use-based weight factors or using stage-damage relationships obtained from several other studies. The fact is, in Bangladesh, no regional-scale damage functions with identical content characteristics exist, and, thus, establishing such functions on a local scale is necessary. Therefore, as a first attempt, this study established a synthetic type of flood damage functions for agriculture and rural households through the data of a structured questionnaire survey conducted in the two sub-districts of the northeastern region of Bangladesh for assessing potential flood risk in the region.

Study Area
The northeastern part of Bangladesh, located downstream of the Meghna River basin, was chosen for this study. The Meghna River originates in the Meghalaya and Manipur Mountain chains of India and flows down to agriculture-dominated northeastern Bangladesh (Figure 1a). The basin topography in Bangladesh is characterized by relatively flat plains comprising mostly cropland and paddy fields and gentle hills [46,47]. The Barak River, the main source of the Meghna River, bifurcates into the Surma and Kushiyara Rivers at the Amalshid point near the Bangladesh-India border (Figure 1a). Discharges from steep and highly flashy rivers (e.g., the Lubachara, Jadukata, and Chela Rivers) originating in the Khasi and Jaintia Hills, the wettest places in the world with an annual average rainfall of about 12,000 mm [9,10], contribute to the Surma River through its right bank. On the other hand, the Kushiyara River receives water from the rivers that originate in the Tripura Hills (e.g., the Manu and Khowai Rivers).
Water 2022, 13, x FOR PEER REVIEW 3 of 25 density or by calculating vulnerability as indices assuming several weight factors according to land use classes [40][41][42]. One study [43] developed a rule-based computerized decision support system to simulate rice yield loss through a crop simulation model for monsoon flooding (June-September) in Bangladesh. The main constraint of the study is that the model was run with site-specific input data, and, thus, a national minimum dataset is required to support decision making. Another limitation is that it only considered rice crops cultivated during the monsoon period in the region of Bangladesh different from the rice crops and region considered in this study. In addition, the validation of the simulated rice yield loss was not performed and, thus, the results are likely to have uncertainties. Two recent studies [44,45] conducted for flood risk assessment in the eastern part of Dhaka also estimated vulnerability by assuming land use-based weight factors or using stage-damage relationships obtained from several other studies. The fact is, in Bangladesh, no regional-scale damage functions with identical content characteristics exist, and, thus, establishing such functions on a local scale is necessary. Therefore, as a first attempt, this study established a synthetic type of flood damage functions for agriculture and rural households through the data of a structured questionnaire survey conducted in the two sub-districts of the northeastern region of Bangladesh for assessing potential flood risk in the region.

Study Area
The northeastern part of Bangladesh, located downstream of the Meghna River basin, was chosen for this study. The Meghna River originates in the Meghalaya and Manipur Mountain chains of India and flows down to agriculture-dominated northeastern Bangladesh (Figure 1a). The basin topography in Bangladesh is characterized by relatively flat plains comprising mostly cropland and paddy fields and gentle hills [46,47]. The Barak River, the main source of the Meghna River, bifurcates into the Surma and Kushiyara Rivers at the Amalshid point near the Bangladesh-India border (Figure 1a). Discharges from steep and highly flashy rivers (e.g., the Lubachara, Jadukata, and Chela Rivers) originating in the Khasi and Jaintia Hills, the wettest places in the world with an annual average rainfall of about 12,000 mm [9,10], contribute to the Surma River through its right bank. On the other hand, the Kushiyara River receives water from the rivers that originate in the Tripura Hills (e.g., the Manu and Khowai Rivers).  The study area, consisting of six administrative districts, mainly lies in the Surma-Kushiyara river system (Figure 1b). Between the Surma and Kushiyara Rivers there are many low-land regions, internal draining depressions (locally called Haors), meandering flood channels, and abandoned river courses, which get flooded in every pre-monsoon (April-May) and monsoon (June-September) season. Many Haors (about 423), which are typical wet-and low-lands in Bangladesh, are located in this region, and most of them remain inundated for about 4-7 months a year. Thus, people of this area can cultivate one crop a year, which is a type of rice locally called Boro rice. Unfortunately, almost every year, the rice fields are affected just about a month before the harvesting time (at the time of the rice's flowering through maturity stage) by pre-monsoon floods, causing immense socio-economic losses to the nation. In 2017, a massive pre-monsoon flash flood devastated the study area, affecting about 1 million people and one-third of the households. The flood exceeded the danger levels at various locations of both the Surma and Kushiyara Rivers, inundating vast areas of rice fields. The Department of Disaster Management, Bangladesh reported that 219,840 hectares of crops, mainly Boro rice at the nearly-readyfor-harvesting stage, were damaged. Moreover, nearly 776,579 households were affected, with approximately 30,000 of them fully or partially destroyed [48,49].
Two administrative sub-districts of the study area (namely, Dakkhin Sunamganj and Balaganj in Figure 1b) were selected to conduct the aforementioned questionnaire survey to collect flood damage data of three damage classes (i.e., Boro rice, house building, and household property). The main reasons for selecting the two sub-districts are: (1) Boro rice and households of the sub-districts were affected and damaged in a larger quantity during the recent 2017 flood [48]; (2) since the study area is surrounded by two independent rivers (i.e., Surma and Kushiyara), one sub-district each was chosen from these two river basins; and (3) three types of house buildings are seen in the study area [50]. However, the houses in the Dakkhin Sunamganj sub-district are mainly a mud type (~81%), whereas those in the Balaganj sub-district consist of all three types proportionately [50]. Therefore, to get more generalized results, these two sub-districts were selected. Figure 2 shows the flowchart of the methodology used in this study. The methodology consisted of the following steps: (1) Conducting a well-structured questionnaire survey in the chosen sub-districts to collect flood damage data based on the calculated minimum sample size for the three damage classes (i.e., Boro rice, house building, and household property); (2) Investigating the similarity and variations of the collected datasets between the two sub-districts for each damage category; (3) Evaluating the consistency of the collected data statistically for each damage category; (4) Performing multiple regression analyses with the data to fit the depth-damage relationship for each damage category; and (5) Identifying the best regression function based on several performance indicators to establish damage curves for the mentioned damage classes.

Outline of the Questionnaire Survey
During flooding periods, the people of the study area mainly experience losses of their nearly-ready-for-harvesting Boro rice and their houses with household property [48,49]. The Bangladesh Bureau of Statistics and the Department of Disaster Management, Bangladesh, reported that only Boro rice fields, households, and livestock were affected by the 2017 flood [48,50]. No other private and public property damage (e.g., local road damage, embankment damage) was stated in the report. No casualties were reported as well. Additionally, during the past floods of 2004, 2010, and 2016, damage to Boro rice and households was only reported by the Bangladesh Bureau of Statistics [50]. Given those reports, the damage to Boro rice and households was found as a major concern for the government and policy makers to draw a roadmap for sustainable development in the study area. Therefore, the three damage classes of Boro rice, house building, and in-house property were considered in this study to conduct the aforesaid questionnaire survey. Since Boro paddies at their flowering through maturity stage are usually damaged by pre-monsoon flash floods in the month of April, just 20-30 days before harvesting time, the maturity stage ('flowering through maturity stage' is termed as 'maturity stage') of the rice was only considered for the survey.

Outline of the Questionnaire Survey
During flooding periods, the people of the study area mainly experience losses of their nearly-ready-for-harvesting Boro rice and their houses with household property [48,49]. The Bangladesh Bureau of Statistics and the Department of Disaster Management, Bangladesh, reported that only Boro rice fields, households, and livestock were affected by the 2017 flood [48,50]. No other private and public property damage (e.g., local road damage, embankment damage) was stated in the report. No casualties were reported as well. Additionally, during the past floods of 2004, 2010, and 2016, damage to Boro rice and households was only reported by the Bangladesh Bureau of Statistics [50]. Given those reports, the damage to Boro rice and households was found as a major concern for the government and policy makers to draw a roadmap for sustainable development in the study area. Therefore, the three damage classes of Boro rice, house building, and in-house property were considered in this study to conduct the aforesaid questionnaire survey. Since Boro paddies at their flowering through maturity stage are usually damaged by premonsoon flash floods in the month of April, just 20-30 days before harvesting time, the maturity stage ('flowering through maturity stage' is termed as 'maturity stage') of the rice was only considered for the survey.
Before the ground survey, the minimum representative sample size for each damage category (i.e., samples of farmers for Boro rice damage and households for house and property damage) was calculated using a probability sampling technique [51]. Using this technique, a population size of farmers with more than 0.05 acre of operating land and that of households comprised of at least two members were obtained from the Bangladesh Bureau of Statistics for the two selected sub-districts. The obtained population sizes were 39,941 for farmers and 86,279 for households, respectively [50]. Using a confidence level of 90%, a margin of error of 6% for farmers and 6.5% for households, and considering a response rate of 50%, the targeted minimum sample sizes were calculated. They were 188 for farmers and 160 for households, respectively. The actual number of samples collected Before the ground survey, the minimum representative sample size for each damage category (i.e., samples of farmers for Boro rice damage and households for house and property damage) was calculated using a probability sampling technique [51]. Using this technique, a population size of farmers with more than 0.05 acre of operating land and that of households comprised of at least two members were obtained from the Bangladesh Bureau of Statistics for the two selected sub-districts. The obtained population sizes were 39,941 for farmers and 86,279 for households, respectively [50]. Using a confidence level of 90%, a margin of error of 6% for farmers and 6.5% for households, and considering a response rate of 50%, the targeted minimum sample sizes were calculated. They were 188 for farmers and 160 for households, respectively. The actual number of samples collected in the field during the questionnaire survey were 196 for farmers and 165 for households, respectively. Therefore, the surveyed dataset for this study can be considered sufficient and reliable. Table 1 shows the total population size and calculated and collected sample sizes by sub-district. The samples of the questionnaires designed to collect Boro-rice and household damage data are presented in Appendices A and B, respectively. The survey to collect flood damage data for each of the target classes was conducted by the first author for 12 days (25 November to 6 December 2018); 7 days in the Dakkhin Sunamganj sub-district of the Sunamganj district and 5 days in the Balaganj sub-district of the Sylhet district. As depicted in Table 1, Boro-rice and household damage data were collected, respectively, from 131 farmers and 67 households in the Dakkhin Sunamganj sub-district. On the other hand, the total samples collected in the Balaganj sub-district for Boro rice and household damage were 65 and 98, respectively. An average of 30 samples was collected each day from dawn to dusk by spending an average of 15-20 min per sample. An average of 5 min was spent to make the questionnaire understandable for the respondents, and the remaining time was used to collect answers to the questions.
Three employees (sub assistant engineers and surveyors) from the Bangladesh Water Development Board (BWDB; https://www.bwdb.gov.bd/, accessed on 10 January 2022) constantly supported the first author in conducting the survey. Every morning, the Boro damage survey was first conducted at a local village Bazaar (marketplace) by gathering 5 to 10 farmers. The farmers were notified beforehand to gather in such a place with the help of the BWDB staff. An average of~30% Boro damage samples were collected from this kind of gathering. The remaining~70% samples were collected from individual farmers by moving from one village to another. While collecting Boro damage data from individual farmers and households in a village and paddy fields, household damage data (damage to house buildings and in-house property) were also collected at the same time. Sometimes the same individuals answered questions about both Boro rice damage and household damage. All of the respondents who answered the questionnaires of Boro damage were male; only about 5% of the respondents who answered the household damage questionnaires were female. Almost all of the respondents were of 30-50 years of age.
It is worth mentioning that about 2 months before the actual survey, draft questionnaires were prepared by the authors and sent to the BWDB employees to carry out discussions with farmers and households as to whether the questions in the draft questionnaires were understandable to the respondents. The BWDB staff went to sample areas, where 60 potential farmers and households (30 for each sub-district) were randomly selected and directly asked to discuss the draft questions to examine actual responses. From such discussions, the recommendations and corrections made by the respondents over the draft questionnaires were collected by the BWDB staff and sent back to the authors to consider them for improvements. Based on the obtained feedback, the design of the sample questionnaires was finalized by making them easily understandable for respondents and accordingly used in the actual questionnaire survey. During the actual survey, all the questions in the questionnaires were asked to the respondents by the first author with the help of the BWDB staff, and the answers were written down mostly by the first author. No questionnaire was given to the respondents to answer the questions by themselves. In such a way, the subjectivity of the respondents was significantly avoided in the questionnaire survey, which increased the accuracy of the survey [52].
In this study, the questionnaire survey was mainly conducted considering the recent 2017 flood. However, the sample questionnaires also included some comprehensive questions for other floods, such as ones in 2004, 2010, 2016, and 2018 (refer to Appendices A and B). However, during the survey, the respondents seemed to have difficulty answering the questions related to flood depths, duration, and damage for the flood years of 2004, 2010, and 2016. Due to their forgotten memory, most of the respondents could not provide adequate information on flood depths, duration, and damage to Boro rice and households for the floods of 2004, 2010, and 2016. However, the respondents were able to recall such information about the recent 2017 flood and answer the questions. Therefore, the damage data (considering flood depths and duration) collected for each damage class via the questionnaire survey in this study were based on the 2017 flood.

Boro Rice Damage Survey
Boro rice is the major crop of the study area, usually planted in mid-December to early January and harvested by late April [53,54]. Flood damage to Boro rice usually takes place in its flowering through maturity stages, just about a month before the harvesting time. Therefore, in the questionnaire survey, other growth stages of the rice were not included and only the maturity stage of the rice was considered. In this study, a total number of 196 flood-affected farmers were interviewed to collect Boro damage data. Each interviewee was requested to answer the reduced amount of rice yield in terms of varying flood depth and duration, particularly during the 2017 flood. During the survey period, many other data were also collected, including the standard height of Boro paddies, the average expected yield per unit area, and the value per unit weight. Refer to Appendix A for the sample questionnaire.

Household Damage Survey
The household damage survey was conducted for house building and household property damage. Classifying households by type, age, and social status is the primary step of undertaking such a survey [25]. The Bangladesh Bureau of Statistics groups the households of the study area into three classes according to their building materials [50]. They are shown in Figure 3a-c and are called (a) Mud houses (locally called Kaccha), (b) Brick houses (locally called Semi-Packa), and (c) Concrete houses (locally called Packa). It is worth mentioning that the majority of houses observed in the study area were Mud-type, one-story houses.
respondents could not provide adequate information on flood depths, duration, and damage to Boro rice and households for the floods of 2004, 2010, and 2016. However, the respondents were able to recall such information about the recent 2017 flood and answer the questions. Therefore, the damage data (considering flood depths and duration) collected for each damage class via the questionnaire survey in this study were based on the 2017 flood.

Boro Rice Damage Survey
Boro rice is the major crop of the study area, usually planted in mid-December to early January and harvested by late April [53,54]. Flood damage to Boro rice usually takes place in its flowering through maturity stages, just about a month before the harvesting time. Therefore, in the questionnaire survey, other growth stages of the rice were not included and only the maturity stage of the rice was considered. In this study, a total number of 196 flood-affected farmers were interviewed to collect Boro damage data. Each interviewee was requested to answer the reduced amount of rice yield in terms of varying flood depth and duration, particularly during the 2017 flood. During the survey period, many other data were also collected, including the standard height of Boro paddies, the average expected yield per unit area, and the value per unit weight. Refer to Appendix A for the sample questionnaire.

Household Damage Survey
The household damage survey was conducted for house building and household property damage. Classifying households by type, age, and social status is the primary step of undertaking such a survey [25]. The Bangladesh Bureau of Statistics groups the households of the study area into three classes according to their building materials [50]. They are shown in Figure 3a-c and are called (a) Mud houses (locally called Kaccha), (b) Brick houses (locally called Semi-Packa), and (c) Concrete houses (locally called Packa). It is worth mentioning that the majority of houses observed in the study area were Mudtype, one-story houses. During the survey, the damage to house buildings was calculated considering the repairing and/or reconstruction costs of flood-damaged or destroyed houses. On the other hand, the damage to rice stocked up inside the house, furniture, kitchen items, appliances, clothes, radios, televisions, and other household goods was considered for household property damage estimation. A total of 165 flood-affected households were interviewed to gather information on the damage to houses and household property according to varying flood inundation depths, considering mainly the duration of both pre-monsoon and monsoon floods. Several other questions, such as the average building value of each type of house, the plinth levels (floor height from the ground level) of houses, and the accumulated value of the household property, were also asked at the time of the household damage survey. Since the collected data were based on the recent past floods, it was assumed that the respondents were like to have clear picture on the damage data of the house and household property. Refer to Appendix B for the sample questionnaire.

Development of Flood Damage Functions
The collected data were used to establish damage functions for the target damage classes. Damage data on Boro yield were collected as per the local measurement unit called Maund per Bigha (equivalent to 330 kg per hectare). As suggested by several researchers [21,27,31,[37][38][39], Boro yield damage data were converted into relative terms as percent damage using Equation (1), to make them clear for all readers and applicable to different times and spaces.
where PRD i is the percent rice yield damage for sample i, RRY i is the reduced rice yield due to flood inundation for sample i, and MEY is the maximum expected rice yield with no inundation. The house and household property damage data, on the other hand, were collected in thousand Bangladesh currency (Bangladesh Taka = BDT). In this study, to make it independent of changes in market values and make it applicable to different times and areas, the damage to houses and household property was normalized and expressed in percent damage using the following equations, as also suggested by [20,55]: where PHD i,j and PHPD i,j , respectively, are the percent house damage and percent household property damage for house type i and sample j, HDV i,j is the house building damage value, HV i,j is the house building value, HPDV i,j is the in-house property damage value, and HPV i,j is the total in-house property value for house type i and sample j.
With the collected data, regression analyses were performed to establish flood damage functions (or depth-damage relationships), as suggested by many researchers [21,27,35,56]. Five regression models were used in this regard, as shown in Table 2. Table 2. Names and equations of the regression models used in this study.

Regression Models Equations of Functions
Linear To evaluate the performance of the models and identify the best one, four indicators, i.e., the Coefficient of Determination (CD), Correlation Coefficient (CC), Standard Error (SE), and Akaike's Information Criterion (AICC), were used. A perfect model should have the CC and CD indices of 1. In addition, the lower the SE and AICC indices, the better the model performance.

Development of Agricultural Damage Curves
The agricultural damage curve for Boro rice was developed in this study using the survey data. The data revealed that the average height of the Boro paddies is about 85 cm, and its initial flood damage usually starts after 2-3 days of inundation at a depth of about 30 cm. Some characteristics of Boro rice revealed from the survey are depicted in Table 3. During the survey, information on Boro yield damage was collected in the two subdistricts, considering varying flood depths of 30, 45, 60, 75, and 90 cm and flood durations of 1-3, 4-7, and more than 7 days. To investigate data variability between the sub-districts, the damage data collected for each sub-district (i.e., 131 samples for Dakkhin Sunamganj and 65 for Balaganj) were compared using four statistical indices (mean, median, mode, and standard deviation). This comparison was performed for each set of flood duration and is shown in Figure 4a-c. The results showed only slight variability between the datasets of the sub-districts according to the four indices. This might be because the topography, land use, households, and people's livelihoods in the study area are very similar. It, thus, suggests that the damage curves that this study attempted to develop for Boro rice, can be used for flood damage and risk assessment not only in the pilot sub-districts but also in other regions of similar characteristics.   To depict the relationship between Boro yield loss and flood depths, box and whisker plots were plotted with the aggregated datasets of the two sub-districts (i.e., 196 samples) for each set of the selected flood durations. The plots are shown in Figure 5a-c. Several statistics of the datasets, such as the mean, median, mode, and standard deviation, are also drawn in the figure.
The Boro yield loss at depths of 45, 60, and 75 cm showed larger variability in the data for the flood duration of 1-3 days (Figure 5a). The same variability for the depths of 45 and 60 cm was also shown in the case of a flood that lasted more than 7 days (Figure 5c). However, the data for a 4-7 day flood showed minimum variations (an average of 8%) in yield loss at all depths of inundation (Figure 5b). The variability shown in the data might be due to the effect of selecting several days on a single set of flood duration. This might also be caused by the interviewee's responses that are subject to their individual experiences and memories related to flood damage. Although variability and inconsistency were somewhat observed in the collected data, their standard deviations (σ) were obtained as reasonably small in quantity (an average of 9%) for each range of duration (Figure 5a-c, purple-colored line), which indicates satisfactory reliability of the collected data. Regression analyses were, thus, performed with the data using five models to establish a best-fit relationship between Boro damage and flood depths for each range of flood duration. The results of the regression analyses are provided in Table 4. Figure 4. Comparison between the collected datasets of the two sub-districts using four statistical indices for the selected flood durations of (a) 1-3 days, (b) 4-7 days, and (c) more than 7 days.
To depict the relationship between Boro yield loss and flood depths, box and whisker plots were plotted with the aggregated datasets of the two sub-districts (i.e., 196 samples) for each set of the selected flood durations. The plots are shown in Figure 5a-c. Several statistics of the datasets, such as the mean, median, mode, and standard deviation, are also drawn in the figure. The Boro yield loss at depths of 45, 60, and 75 cm showed larger variability in the data for the flood duration of 1-3 days (Figure 5a). The same variability for the depths of 45 and 60 cm was also shown in the case of a flood that lasted more than 7 days ( Figure  5c). However, the data for a 4-7 day flood showed minimum variations (an average of 8%) in yield loss at all depths of inundation ( Figure 5b). The variability shown in the data might be due to the effect of selecting several days on a single set of flood duration. This might also be caused by the interviewee's responses that are subject to their individual experiences and memories related to flood damage. Although variability and inconsistency were somewhat observed in the collected data, their standard deviations (σ)  Following the equations of their functions, each regression model could predict a relationship of Boro yield loss with inundation depths. Figure 6a-c shows the depthdamage relationships of Boro rice predicted by the five models for each set of flood duration. The calculated performance indicators are shown in Table 4.
The results depicted in Table 4 and Figure 6a-c indicate that each regression model for each range of flood duration performed reasonably well with satisfactory indices. Even the simplest linear-type model showed good performance, with CC and CD values ranging from 0.89 to 0.95 and from 0.80 to 0.90, respectively. However, the polynomial regression model performed the best for 1-3 days of flood duration. It showed a higher magnitude of CC (0.96) and CD (0.92) values together with the lower indices of SE (11.54) and AICC (4796.84). For 4-7 days and more than 7 days of inundation, on the other hand, the logistic model overtopped the other models and resulted with higher indices of CC and CD (ranging between 0.96 and 0.97 and between 0.92 and 0.93, respectively) together with a lower range of SE (from 10.62 to 11.52) and AICC (from 4631.30 to 4791.76) indices. Therefore, by considering their best performance among the selected models, the thirdorder polynomial type for 1-3 days and the logistic type for 4-7 days and more than 7 days of inundation were chosen to develop the damage function of Boro rice for this study. Following the equations of their functions, each regression model could predict a relationship of Boro yield loss with inundation depths. Figure 6a-c shows the depthdamage relationships of Boro rice predicted by the five models for each set of flood duration. The calculated performance indicators are shown in Table 4. The results depicted in Table 4 and Figure 6a-c indicate that each regression model for each range of flood duration performed reasonably well with satisfactory indices. Even the simplest linear-type model showed good performance, with CC and CD values ranging from 0.89 to 0.95 and from 0.80 to 0.90, respectively. However, the polynomial regression model performed the best for 1-3 days of flood duration. It showed a higher To further verify the best selected three regression models, a validation approach was conducted in this study by dividing the survey data of rice damage (196 samples) into two groups. First, 100 samples were used as training data to set up the three models and the next 96 samples were used as testing data to check the models. First, regression analysis with the 100 samples was performed with the best three models and the damage functions were derived for the flood durations of 1-3 days, 4-7 days, and more than 7 days. Then, an average yield loss from the 96 sample was calculated according to the flood depths of 30, 45, 60, 75, and 90 cm and for the three sets of flood durations. Finally, a comparison between the yield loss estimated from the 96 samples and the yield loss calculated from the damage functions derived from the 100 samples is shown in Figure 7a Therefore, by considering their best performance among the selected models, the thirdorder polynomial type for 1-3 days and the logistic type for 4-7 days and more than 7 days of inundation were chosen to develop the damage function of Boro rice for this study.
To further verify the best selected three regression models, a validation approach was conducted in this study by dividing the survey data of rice damage (196 samples) into two groups. First, 100 samples were used as training data to set up the three models and the next 96 samples were used as testing data to check the models. First, regression analysis with the 100 samples was performed with the best three models and the damage functions were derived for the flood durations of 1-3 days, 4-7 days, and more than 7 days. Then, an average yield loss from the 96 sample was calculated according to the flood depths of 30, 45, 60, 75, and 90 cm and for the three sets of flood durations. Finally, a comparison between the yield loss estimated from the 96 samples and the yield loss calculated from the damage functions derived from the 100 samples is shown in Figure 7a-c. The results shown in Figure 7a-c indicate that the best performing models selected for developing Boro rice damage curves produced the satisfactory results with a CD (Rsquared) values ranging from 0.98 to 0.99. Therefore, it is evident that the best selected regression models are likely to develop flood damage functions of Boro rice in the study area. Figure 8a-c shows the developed damage curves of Boro rice for three sets of flood duration. To incorporate the uncertainty of the collected data, an uncertainty band of one standard deviation (±1σ) was integrated with the damage curves. As pointed out earlier, The results shown in Figure 7a-c indicate that the best performing models selected for developing Boro rice damage curves produced the satisfactory results with a CD (R-squared) values ranging from 0.98 to 0.99. Therefore, it is evident that the best selected regression models are likely to develop flood damage functions of Boro rice in the study area. Figure 8a-c shows the developed damage curves of Boro rice for three sets of flood duration. To incorporate the uncertainty of the collected data, an uncertainty band of one standard deviation (±1σ) was integrated with the damage curves. As pointed out earlier, this study considered only the maturity stage of Boro rice for damage estimation. The damage curves developed at this stage of the rice showed the difference in the shape, thresholds for damage occurrence, and range of expected damage ratios with the other existing damage curves developed by various studies [31,[37][38][39]. This might be due to the fact of containing regional differences in various contexts such as crop type, flood timing and duration, potential yield, cultivation practices, etc. The developed curves revealed that the rice yield became nearly zero when the flood height exceeded 75 cm (the depth at which rice grains start to flourish), regardless of the inundation duration. One finding revealed from the developed curves was that there was no damage to Boro paddies when the water level remained up to an average depth of 25 cm (at which the tiller of rice evolves). This finding can effectively be used to facilitate excess water during the flood period to reduce agricultural risk, as a huge volume of floodwater can be diverted into around 423 Haors (in which Boro rice is cultivated) up to a 25 cm depth. However, flood inundation up to such a depth in a Boro field usually delays the maturity of rice by about 15-20 days.

Development of Household Damage Curves
Household damage curves were developed for house and household property damage. Of 165 flood-affected households surveyed in this study, 91 were categorized as Mud, 41 as Brick, and 33 as Concrete. The characteristics of these households observed during the survey are shown in Table 5.

House Building Damage Curves
House building damage is mainly dominated by flood heights above the plinth level (PL). Considering varying flood heights above PL of 30, 60, 90, 120, 150, 180, and 215 cm, damage data to house buildings were collected in a monetary value expressed in thousand BDT. To be used for different times and space, they were normalized and expressed in percent damage using Equation (2). The variability of the building damage data between the sub-districts was also investigated and is presented in Figure 9a-c.

Development of Household Damage Curves
Household damage curves were developed for house and household property damage. Of 165 flood-affected households surveyed in this study, 91 were categorized as Mud, 41 as Brick, and 33 as Concrete. The characteristics of these households observed during the survey are shown in Table 5.

House Building Damage Curves
House building damage is mainly dominated by flood heights above the plinth level (PL). Considering varying flood heights above PL of 30, 60, 90, 120, 150, 180, and 215 cm, damage data to house buildings were collected in a monetary value expressed in thousand BDT. To be used for different times and space, they were normalized and expressed in percent damage using Equation (2). The variability of the building damage data between the sub-districts was also investigated and is presented in Figure 9a-c. Because of the similar topography, land use, households, and people's livelihoods in the study area, an insignificant variation of the house damage data was observed between the sub-districts (Figure 9a-c). Therefore, by considering the datasets of two sub-districts as one, the depth-damage relationships were depicted for all three types of houses in box and whisker plots and are presented in Figure 10a-c. The mean, median, mode, and standard deviation of the dataset are also shown in the figure. A Larger variation was observed in Mud house damage, particularly for 90 and 120 cm of flood heights. This might be due to the effect of household responses that were dependent on the experience, knowledge, fatigue, and memories of the respondents [52]. Another reason could be the variety of building materials used for constructing Mud houses. The roofs of some Mud houses are made of iron sheets, whereas some are of straws. The walls are sometimes built with straws and sometimes with jute sticks or bamboo sticks. The Brick and Concrete houses showed a mild variability within the damage data, which indicated the consistency of the collected data regardless of the regions. Although the house damage data resulted in larger to mild variability for all three types of houses, the standard deviations (σ) of the data were found satisfactorily low (an average of 7%) for each type of house (Figure 10a-c, purple-colored line), which indicates that the collected date were reliable. Hence, with these data, regression analyses with five Because of the similar topography, land use, households, and people's livelihoods in the study area, an insignificant variation of the house damage data was observed between the sub-districts (Figure 9a-c). Therefore, by considering the datasets of two sub-districts as one, the depth-damage relationships were depicted for all three types of houses in box and whisker plots and are presented in Figure 10a-c. The mean, median, mode, and standard deviation of the dataset are also shown in the figure. Because of the similar topography, land use, households, and people's livelihoods in the study area, an insignificant variation of the house damage data was observed between the sub-districts (Figure 9a-c). Therefore, by considering the datasets of two sub-districts as one, the depth-damage relationships were depicted for all three types of houses in box and whisker plots and are presented in Figure 10a-c. The mean, median, mode, and standard deviation of the dataset are also shown in the figure. A Larger variation was observed in Mud house damage, particularly for 90 and 120 cm of flood heights. This might be due to the effect of household responses that were dependent on the experience, knowledge, fatigue, and memories of the respondents [52]. Another reason could be the variety of building materials used for constructing Mud houses. The roofs of some Mud houses are made of iron sheets, whereas some are of straws. The walls are sometimes built with straws and sometimes with jute sticks or bamboo sticks. The Brick and Concrete houses showed a mild variability within the damage data, which indicated the consistency of the collected data regardless of the regions. Although the house damage data resulted in larger to mild variability for all three types of houses, the standard deviations (σ) of the data were found satisfactorily low (an average of 7%) for each type of house (Figure 10a-c, purple-colored line), which indicates that the collected date were reliable. Hence, with these data, regression analyses with five A Larger variation was observed in Mud house damage, particularly for 90 and 120 cm of flood heights. This might be due to the effect of household responses that were dependent on the experience, knowledge, fatigue, and memories of the respondents [52]. Another reason could be the variety of building materials used for constructing Mud houses. The roofs of some Mud houses are made of iron sheets, whereas some are of straws. The walls are sometimes built with straws and sometimes with jute sticks or bamboo sticks. The Brick and Concrete houses showed a mild variability within the damage data, which indicated the consistency of the collected data regardless of the regions. Although the house damage data resulted in larger to mild variability for all three types of houses, the standard deviations (σ) of the data were found satisfactorily low (an average of 7%) for each type of house (Figure 10a-c, purple-colored line), which indicates that the collected date were reliable. Hence, with these data, regression analyses with five models were performed to identify the best depth-damage relationships for the three types of houses, and the results are presented in Table 6 and Figure 11a-c. models were performed to identify the best depth-damage relationships for the three types of houses, and the results are presented in Table 6 and Figure 11a-c.  The results, shown in Table 6 and Figure 11a-c, revealed the satisfactory performance of each regression model to establish depth-damage relationships for all three types of houses. However, the logistic model performed the best for Mud and Brick houses, producing larger CC and CD values with smaller SE and AICC values (Table 6 and Figure  11a,b). For Concrete-type houses, on the other hand, the power, logistic, and polynomial models produced identical CC and CD indices; however, the other two indices (SE and AICC) showed better values for the power model than the logistic and polynomial models ( Table 6 and Figure 11c). Therefore, flood damage curves for all the three types of houses were developed using the results of these best-performing models (i.e., the logistic model for Mud and Brick houses and the power model for Concrete houses), as shown in Figure  12a-c. To incorporate the uncertainty in the collected building damage dataset, an The results, shown in Table 6 and Figure 11a-c, revealed the satisfactory performance of each regression model to establish depth-damage relationships for all three types of houses. However, the logistic model performed the best for Mud and Brick houses, producing larger CC and CD values with smaller SE and AICC values (Table 6 and Figure 11a,b). For Concrete-type houses, on the other hand, the power, logistic, and polynomial models produced identical CC and CD indices; however, the other two indices (SE and AICC) showed better values for the power model than the logistic and polynomial models (Table 6 and Figure 11c). Therefore, flood damage curves for all the three types of houses were developed using the results of these best-performing models (i.e., the logistic model for Mud and Brick houses and the power model for Concrete houses), as shown in Figure 12a-c. To incorporate the uncertainty in the collected building damage dataset, an uncertainty band of one standard deviation (±1σ) was integrated with the developed damage curves. The curves revealed that the flood damage to Brick and Concrete houses did not reach 100% for any selected flood heights because of their building materials and construction methods. On the other hand, the construction method and materials of Mud houses make them most vulnerable to floods, compared to the other two types of houses.

Household Property Damage Curves
Flood damage data for household property were collected in thousand BDT for all 165 samples. The data were converted into percent damage using Equation (3). This percent damage data for each sub-district were statistically compared based on four indices, as shown in Figure 13a-c. No significant variability was observed; hence, to see the flood depth-damage relationship of household property, box and whisker plots were drawn for the entire dataset and are presented in Figure 14a-c with the data's mean, median, mode, and standard deviation.

Household Property Damage Curves
Flood damage data for household property were collected in thousand BDT for all 165 samples. The data were converted into percent damage using Equation (3). This percent damage data for each sub-district were statistically compared based on four indices, as shown in Figure 13a-c. No significant variability was observed; hence, to see the flood depth-damage relationship of household property, box and whisker plots were drawn for the entire dataset and are presented in Figure 14a-c with the data's mean, median, mode, and standard deviation.

Household Property Damage Curves
Flood damage data for household property were collected in thousand BDT for all 165 samples. The data were converted into percent damage using Equation (3). This percent damage data for each sub-district were statistically compared based on four indices, as shown in Figure 13a-c. No significant variability was observed; hence, to see the flood depth-damage relationship of household property, box and whisker plots were drawn for the entire dataset and are presented in Figure 14a-c with the data's mean, median, mode, and standard deviation.  The distribution of the in-house property damage dataset (Figure 14a-c) also showed inconsistency and variability, as observed in Figure 10a-c for the building damage dataset, and that may be due to the same reasons described in Section 4.2.1. However, the property damage data produced an average of only 8% standard deviations. With such a minimum value, regression analyses with the five models were also conducted using the property damage data for all three types of houses to predict flood depth and property damage relationships. The model results are presented in Table 7, and the damage curves predicted by each model for the three types of households are shown in Figure 15a-c.  The distribution of the in-house property damage dataset (Figure 14a-c) also showed inconsistency and variability, as observed in Figure 10a-c for the building damage dataset, and that may be due to the same reasons described in Section 4.2.1. However, the property damage data produced an average of only 8% standard deviations. With such a minimum value, regression analyses with the five models were also conducted using the property damage data for all three types of houses to predict flood depth and property damage relationships. The model results are presented in Table 7, and the damage curves predicted by each model for the three types of households are shown in Figure 15a-c.  Among all performance indices stated in Table 7, the logistic and polynomial models performed better than the other three models. However, the less complicated logistic model was selected as the best model over the more complicated polynomial model through the SE-and AICC-Test. Therefore, using the results of this best-performing logistic model, flood damage curves for the household property of the three house types were developed and are shown in Figure 16a-c. To incorporate the uncertainty of the collected data, an uncertainty band of one standard deviation (±1σ) was integrated with the developed damage curves. The damage curves revealed that the owners of Mud houses are unlikely to save household goods when the flood height exceeds 150 cm, whereas the goods of Brick and Concrete houses can be saved to some extent since they are usually constructed with an attic (loft) and/or half ceiling at lintel height. It is, thus, evident that the construction method and materials of Mud houses make the buildings and in-house property most vulnerable to floods.

Validation of the Damage Curves
This study verified the developed damage curves of Boro rice by conducting a damage assessment of the 2017 flood. In doing this, a hazard assessment and an exposure assessment were also performed. Hazard parameters such as inundation depths and duration were simulated by a 2D hydrological Rainfall-Runoff-Inundation (RRI) model [57,58] forced with distributed rainfall data previously developed by the authors [28]. Figure 17a shows a flood inundation map generated at a 500 m grid resolution for the 2017 flood of the study area with a duration range of 28 days (from 1 April to 28 April). The Among all performance indices stated in Table 7, the logistic and polynomial models performed better than the other three models. However, the less complicated logistic model was selected as the best model over the more complicated polynomial model through the SE-and AICC-Test. Therefore, using the results of this best-performing logistic model, flood damage curves for the household property of the three house types were developed and are shown in Figure 16a-c. To incorporate the uncertainty of the collected data, an uncertainty band of one standard deviation (±1σ) was integrated with the developed damage curves. The damage curves revealed that the owners of Mud houses are unlikely to save household goods when the flood height exceeds 150 cm, whereas the goods of Brick and Concrete houses can be saved to some extent since they are usually constructed with an attic (loft) and/or half ceiling at lintel height. It is, thus, evident that the construction method and materials of Mud houses make the buildings and in-house property most vulnerable to floods. Among all performance indices stated in Table 7, the logistic and polynomial models performed better than the other three models. However, the less complicated logistic model was selected as the best model over the more complicated polynomial model through the SE-and AICC-Test. Therefore, using the results of this best-performing logistic model, flood damage curves for the household property of the three house types were developed and are shown in Figure 16a-c. To incorporate the uncertainty of the collected data, an uncertainty band of one standard deviation (±1σ) was integrated with the developed damage curves. The damage curves revealed that the owners of Mud houses are unlikely to save household goods when the flood height exceeds 150 cm, whereas the goods of Brick and Concrete houses can be saved to some extent since they are usually constructed with an attic (loft) and/or half ceiling at lintel height. It is, thus, evident that the construction method and materials of Mud houses make the buildings and in-house property most vulnerable to floods.

Validation of the Damage Curves
This study verified the developed damage curves of Boro rice by conducting a damage assessment of the 2017 flood. In doing this, a hazard assessment and an exposure assessment were also performed. Hazard parameters such as inundation depths and duration were simulated by a 2D hydrological Rainfall-Runoff-Inundation (RRI) model [57,58] forced with distributed rainfall data previously developed by the authors [28]. Figure 17a shows a flood inundation map generated at a 500 m grid resolution for the 2017 flood of the study area with a duration range of 28 days (from 1 April to 28 April). The map indicated that the study area mostly experienced a range of flood inundation depth

Validation of the Damage Curves
This study verified the developed damage curves of Boro rice by conducting a damage assessment of the 2017 flood. In doing this, a hazard assessment and an exposure assessment were also performed. Hazard parameters such as inundation depths and duration were simulated by a 2D hydrological Rainfall-Runoff-Inundation (RRI) model [57,58] forced with distributed rainfall data previously developed by the authors [28]. Figure 17a shows a flood inundation map generated at a 500 m grid resolution for the 2017 flood of the study area with a duration range of 28 days (from 1 April to 28 April). The map indicated that the study area mostly experienced a range of flood inundation depth (FID) between 0.5 m and 4 m during the 2017 flood. Figure 17b shows an exposure map of Boro rice fields at a 500 m grid resolution by administrative districts of the study area. The raster data of the exposure map was prepared by using a previous study [59] that mapped rice area of the South Asia using MODIS Multitemporal data.  Figure 17b shows an exposure map of Boro rice fields at a 500 m grid resolution by administrative districts of the study area.
The raster data of the exposure map was prepared by using a previous study [59] that mapped rice area of the South Asia using MODIS Multitemporal data. By combining the flood inundation map simulated by the RRI model (Figure 17a), the depth-duration-damage relationship obtained from the developed Boro damage curves (Figure 8a-c) and the Boro extent map (Figure 17b), a distributed flood damage map at a 500 m resolution for Boro rice was produced for the study area. As depicted in Table 3, the flood depth at which the damage to rice starts is 30 cm. Therefore, the marginal flooding areas considered in this study are the areas that experience less than 30 cm of flood inundation. These types of marginal flooding areas in a hazard map were filtered and left out for flood damage assessment by many other studies [60][61][62]. Therefore, by leaving out the grid cells experiencing a marginal flooding of less than a 30 cm FID of the simulated hazard map, the maximum, minimum, and an average total of rice damage were estimated and compared with those of the reported damage obtained from three different sources. Table 8 shows the comparison. The damage to Boro rice estimated by this study for each case (maximum, minimum, and average total) was found to satisfactorily agree with the reported damage. The developed damage curves of house and household property were unlikely to be validated due to the lack of reported damage data.

Conclusions
The northeastern part of Bangladesh, which is located downstream of the Meghna River basin, is frequently affected by flood disasters. To reduce such disasters and build back better, flood risk assessment is an important tool for the policy makers to identify effective counter-measures and formulate future adaptation plans. Of the three main components of a flood risk assessment model, development of flood damage functions for the estimation of flood vulnerability was the main focus of this study. Of the two types of flood damage functions, the empirical type has yet to be developed in Bangladesh due to By combining the flood inundation map simulated by the RRI model (Figure 17a), the depth-duration-damage relationship obtained from the developed Boro damage curves (Figure 8a-c) and the Boro extent map (Figure 17b), a distributed flood damage map at a 500 m resolution for Boro rice was produced for the study area. As depicted in Table 3, the flood depth at which the damage to rice starts is 30 cm. Therefore, the marginal flooding areas considered in this study are the areas that experience less than 30 cm of flood inundation. These types of marginal flooding areas in a hazard map were filtered and left out for flood damage assessment by many other studies [60][61][62]. Therefore, by leaving out the grid cells experiencing a marginal flooding of less than a 30 cm FID of the simulated hazard map, the maximum, minimum, and an average total of rice damage were estimated and compared with those of the reported damage obtained from three different sources. Table 8 shows the comparison. The damage to Boro rice estimated by this study for each case (maximum, minimum, and average total) was found to satisfactorily agree with the reported damage. The developed damage curves of house and household property were unlikely to be validated due to the lack of reported damage data.

Conclusions
The northeastern part of Bangladesh, which is located downstream of the Meghna River basin, is frequently affected by flood disasters. To reduce such disasters and build back better, flood risk assessment is an important tool for the policy makers to identify effective counter-measures and formulate future adaptation plans. Of the three main components of a flood risk assessment model, development of flood damage functions for the estimation of flood vulnerability was the main focus of this study. Of the two types of flood damage functions, the empirical type has yet to be developed in Bangladesh due to the lack of an integrated and consistent damage data bank. Therefore, development of the other type, the synthetic type of damage functions, which are based on based on questionnaire survey data, was the only alternative. However, to the authors' best knowledge, the synthetic type of flood damage functions has not been established yet in Bangladesh. Therefore, as a first attempt, this paper developed a synthetic type of reliable flood damage functions for agriculture and rural households using the data collected from a well-structured questionnaire survey conducted in northeastern Bangladesh. The survey data included a total of 196 and 165 samples, respectively, for agriculture and household damage due to a flood. Agriculture damage considered the damage to Boro rice (the main crop of the study area), and household damage included the damage to buildings and household property for three types of houses (Mud, Brick, and Concrete). Variability and inconsistency were observed to some extent in the collected data for each damage class (i.e., Boro rice, house building, and in-house property). However, the standard deviations (σ) of each class were obtained as reasonably small in quantity (an average of 7-9%), indicating satisfactory reliability of the collected data. With the data, five regression models were performed to establish reliable depth-damage relationships. The best model based on several performance indicators was used to draw damage curves of the three damage classes.
The Boro damage curves revealed that the damage was mainly governed by flood depths and duration. The yield of Boro rice became near zero at a flood depth of about 70-75 cm (~rice grain height) and a flood duration of 7 days or longer. The Boro damage curves also revealed that the Boro paddies can tolerate water without any damage up tõ 25 cm depth (~rice tiller height) at their flowering through maturity stage. However, at this water depth, the maturity of the rice might be delayed by about 15-20 days. This finding can effectively be used to facilitate a huge volume of excess water during a flood time through diversion into Boro fields inside~423 Haors up to 25 cm depth.
The house and in-house property damage functions revealed that the household damage is mainly dependent on inundation height above the floor level and the type of houses. Due to their construction method and building materials, Mud houses were found to be the most vulnerable to floods, and the damage became 100% when the flood height exceeded 150 cm. The damage to Brick and Concrete houses, on the other hand, never reached 100%. Therefore, by constructing Mud houses (construction cost:~3250 USD) in an elevated place or upgrading them to Brick houses (construction cost:~8060 USD) or Concrete houses (construction cost:~16,895 USD), the flood risk to Mud houses can be reduced significantly.
The FDFs of Boro rice were validated to damage assessment caused by the 2017 flood. The total maximum, minimum, and average damage to Boro rice was found to be satisfactorily agreeable to those calculated by the developed FDFs. The FDFs of households were unlikely to be validated due to the lack of observed damage data.
This study has two limitations, which are likely to be addressed in future studies. First is the building size of the houses that was not considered in the questionnaire survey conducted by this study. Secondly, the changes in production costs of Boro rice due to flood were not assessed, as done by several other studies [65][66][67]. Since the changes in production costs due to flood are dependent on the growth stages of the rice and since the maturity stage of the rice was only considered in this study, it is, therefore, likely that the changes in production costs of rice may not play an important role in flood risk assessment. However, this can be explored and addressed in the future studies.
Despite having few limitations, the flood damage functions/curves (FDFs) developed in this study revealed several findings that can help to reduce flood risk in the study area. In addition, the FDFs can be useful for pre-and post-disaster flood damage assessments and potential flood risk assessment under global change for road-mapping sustainable and effective management of flood disasters. In a future study, the FDFs will be further investigated for the applicability to potential flood risk assessment under global change in the study area.