The input data (e.g., Supplementary Table 1) used in the AAL analysis are the number of stories (1 or 2+), basement existence (0 = No, 1= Yes), area in square feet (A), unit area repair cost in USD per square foot (C_R), and FFH₀. FFH₀ is a general term that may represent either the actual FFH for existing buildings, the minimum regulatory standard for FFH for a particular area, a hypothetical beginning FFH for consideration of additional elevation, or a practical minimum FFH based on foundation type. Also, the flood depth above the ground for multiple return periods at each home and the corresponding AEP values for those flood depths are input.

The DDF selection is a critical decision for flood loss, and therefore risk assessment (Mostafiz et al., 2021b). The most effective DDF will characterize the relationship between the water depth in the structure (dh) and the percent of damage most appropriately while also identifying properly the flood damage initiation point with respect to dh. Several sources examine the dh-damage relationship, including United States Army Corps of Engineers (2000), United States Army Corps of Engineers (2006), and Nofal et al. (2020) who provided loss functions for multiple models, and Wing et al. (2020) who used NFIP flood damage claims to represent the dh-damage relationships.

In this study, the United States Army Corps of Engineers (2000) DDFs are chosen to determine the relationship between building (or contents) loss L_B (and separately, L_C) and dh. These DDFs are selected in this study to demonstrate the methodology because they are based on generic data (1996–2000). DDFs such as United States Army Corps of Engineers (2006) include flood duration and foundation type and could be substituted in cases for which more descriptive input data are available. Four of the United States Army Corps of Engineers (2000) home types are incorporated into the numerically integrated computational framework here: one story with no basement (DDF1), two or more stories with no basement (DDF2), one story with basement (DDF3), and two or more stories with basement (DDF4). Within the DDF, the mean and standard deviation of building and contents loss are expressed as a proportion of home replacement cost value (V_R) for each one-foot increment of dh beginning at the damage initiation point of −2 feet, up to 16 feet (United States Army Corps of Engineers, 2000). Interpolation is used here to determine the percentage of building and contents losses at each 0.5-foot increment of dh. Finally, restoration time (in months) is taken from Federal Emergency Management Agency (2013) as a surrogate for use loss (L_U), at each flood depth. Because restoration time tables in Federal Emergency Management Agency (2013) define L_U at four-foot increments, interpolation is employed to estimate L_U at 0.5-foot increments of dh. The four utilized DDFs are shown in Supplementary Tables 2–5. The DDF selected for each home corresponds to the number of stories and presence/absence of a basement.

The Gumbel distribution function has been shown to be effective for modeling flood frequency (e.g., Singh et al., 2018; Patel, 2020; Rahim et al., 2021). Therefore, the two-parameter Gumbel distribution function is used to model the probability of exceedance for the expected flood depths. The cumulative distribution function (CDF) of the Gumbel distribution is the probability (p) that a random variable X has a value less than or equal to a threshold d, also known as the probability of non-exceedance. That value is generally expressed in Equation 1 (Patel, 2020), with u and a representing the Gumbel location and scale parameter, respectively.

The complement of the CDF represents the AEP of a potential flood event with depth above the ground d, as shown in Equation 2.

where P = p(X>d)

Solving Equation 1 for d yields Equation 3, which represents the relationship between d and the double natural logarithm of p, where a and u are the regression coefficients (slope and y-intercept, respectively). Using the flood depth (d) data and corresponding double natural logarithm of p for each return period, the Gumbel distribution is fit and a unique regression line is generated for each home location to yield its distinctive a and u hazard parameters (Mostafiz et al., 2021c, 2022d). The flood depth within the house, represented in the DDFs as dh, is expressed as the difference between d and FFH₀ (Equation 4).

For each 0.5-foot water depth increment of dh, the corresponding d is calculated using Equation 4. The generated flood parameters and the calculated d are used to calculate the corresponding P using Equation 3. For each home, the selected DDFs are transformed into a function of P using the relationships in Equations 2, 3. The functions L_B(P) and L_C(P) represent building and contents losses as a function of P, expressed as a proportion of the V_R. Similarly, L_U(P) represents use loss in months as a function of P. AAL is calculated as the integral of loss as a function of flood probability, where P_min represents the lowest exceedance probability value and P_max is the highest exceedance probability value. Equations 5–7 describe the theoretical formulation of the AAL for building, contents, and use (AAL_B/VR, AAL_C/VR, and AAL_{U, months}, respectively) for each home. AAL_B/VR and AAL_C/VR represent the annual building and contents flood risk as a proportion of the V_R, while AAL_{U, months} represents annual use flood risk in months.

Riemann summation is a computational approach to approximate an exact integration solution as the sum of trapezoidal areas under a curve. To evaluate Equations 5–7 computationally, the area of each trapezoid under the L(P) functions is estimated as the difference in exceedance probabilities multiplied by the average loss (building, contents, or use) for the corresponding probabilities. The trapezoidal Riemann sums approach is used to aggregate the product results across all probabilities to yield AAL (Meyer et al., 2009; Gnan et al., 2022a) as generally shown in Equation 8.

Here, K is the number of trapezoids under the L(P) curve.

The total AAL calculations vary based on the owner/occupant type (i.e., homeowner, landlord, or tenant). Any building risk will always be incurred by the owner (i.e., homeowner or landlord), regardless of occupancy. Homeowners and tenants, but not landlords, will incur contents risk. For this study, AALs for building and contents are initially expressed as a proportion of the V_R (Equations 5, 6). However, because monetary values are understood and appreciated more readily when communicating risk to the public, the AAL variables are then converted to dollar figures for building (AAL_B$) and contents (AAL_C$) via V_R, which is the unit building replacement cost per square foot (C_R) multiplied by the home area (A), as shown in Equation 9:

Then, AAL_B$ and AAL_C$ are calculated as shown in Equations 10, 11.

The AAL for use, or restoration time, is expressed in units of months (AAL_{U, months}) and is partitioned into that assumed by the homeowner (AAL_{UH, months}), landlord (AAL_{UL, months}), and tenant (AAL_{UT, months}). These three variables are then converted to economic value, as shown in Equations 12–14, where R_l is the monthly rent incurred (for the homeowner) or lost (for the landlord), and H_R is the nightly hotel rent. The total annual flood risk (AAL_T$) calculated as the sum of building, contents, and use annual risk by owner/occupant type (Equation 15). AAL_U$ is a generalized representation of use risk for which the applicable equation should be selected based on owner/occupant type. Table 1 shows the equation numbers to use when calculating AAL_B$, AAL_C$, and AAL_U$ for each owner/occupant type.

To calculate the AAL for FFH above FFH₀, an increment (j) is added to FFH₀ and all calculations represented in Equations 3–15 are repeated using the new FFH value. The relative AAL reduction with each increment is calculated using Equation 16.

Flood depth grids at 10-, 25-, 50-, 100-, and 500-year return periods are required to calculate “average annualized loss.” Hazus-MH represents AAL via summed calculations of the product of difference in return period (RP) flood frequency (f_RP; analogous to AEP) and the corresponding mean loss (L_RP), as shown in Equation 17.

The process begins when “create a new region” is selected, with “riverine only” clicked in the study region flood hazard type. Then, the 10-, 25-, 50-, 100-, and 500-year flood depth grids under “user data” are uploaded. Next, the building input data are imported under “user defined facilities (UDF).” The proper DDFs are then selected for the flood loss calculation. Then, a new hazard scenario is created with the 10-, 25-, 50-, 100-, and 500-year flood depth grids, and the riverine floodplain is delineated for “full suite of return periods” using the default cell size (3.048 × 3.048 m). Then, AAL analysis is run with the UDF. The result shows AAL_B$ and AAL_C$ for each building separately, but Hazus-MH does not provide AAL_U$.

The dh is an important parameter for all flood loss models (Apel et al., 2009), in part because it is used in the DDFs to estimate building loss (L_B) and contents losses (L_C); and therefore total AAL (Pistrika et al., 2014). More recent research has highlighted the importance of dh where damage is assumed to initiate. Gnan et al. (2022a) demonstrated that AAL derived from USACE functions that initiate damage at a dh of −2 feet (i.e., 2 feet below the top of the first floor) were much higher than AAL calculated from the same function with damage initiation assumed to occur at dh = 0 (i.e., top of first floor; Figure 2). Consideration of flood risk reduction through optimal home elevation and the sensitivity of risk assessment to DDF selection remain as fundamental issues in the development of consistent flood risk assessment procedures.

Flood waters can damage subfloor assemblies and utilities of elevated homes, which may require consideration of losses for dh < 0. However, the extent to which these losses may occur is unclear. A sensitivity analysis is therefore conducted to compare AAL values estimated using United States Army Corps of Engineers (2000) considering damage initiation points of dh = −1 foot, −0.5 foot, and 0 feet, holding all input parameters and the DDF constant. While this sensitivity analysis does not identify which damage initiation point is “correct,” it does shine light on the effect of DDF damage initiation point variability on flood risk assessment results.

A pairwise t-test is used to assess whether a statistically significant difference (α = 0.05) exists in population mean (μ) values between AAL_B$ (and in a separate analysis, AAL_C$) calculated by the AAL approach presented in this paper vs. Hazus-MH output. Because Hazus-MH ignores use risk, AAL_U$ is not considered in this analysis. Thus, the null and alternative hypotheses for both AAL_B$ and AAL_C$ are provided in Equations 18, 19, respectively. The intent of this test is to evaluate whether the refined approach presented in this paper differs significantly from the results of the existing, widely used Hazus-MH tool.

Flood risk mean values for the 29 homes in this case study represent neighborhood-level values. For homes with FFH equal to the 100-year flood elevation, the mean (i.e., neighborhood-level) AAL_B$ and AAL_C$ is $2,300 and $1,500 per home, respectively, with a range of ~$600–$7,000 and $300–$4,700 per home, respectively (Figure 4). Table 2 shows that the building attributes, V_R, and regression parameters differ substantially for the homes with the largest (i.e., $11,636) and smallest (i.e., $894) AAL values.

Because the AAL_U$ calculation depends on owner/occupant type (i.e., homeowner, landlord, and tenant), it is not shown in aggregate form. As owner/occupant type is unknown, the sum of AAL_B$, AAL_C$, and AAL_U$ for homes at E₁₀₀ assumes that all 29 homes are homeowner-occupied (and then landlord-owned and tenant-occupied), yielding a mean AAL_T$ of $4,390, $2,960, and $1,590 per home, respectively (Figure 5). The complete set of output values for each home in the study area is shown in Supplementary Table 6.

The mean AAL_B/VR (and separately, AAL_C/VR) for the 29 homes for each increase in FFH above E₁₀₀ is shown in Figure 6. The calculated AAL_B$, AAL_C$, and AAL_U$ by owner/occupant type and for each increase in FFH above E₁₀₀ appear in Table 3. Substantial decreases in all components of AAL are realized with a 1-foot increase in FFH above E₁₀₀, and more modest decreases in AAL occur with additional increases above E₁₀₀, for all owner/occupant types (Table 4).

The means of AAL_B$ and sum of AAL_B$ and AAL_C$ increase ~6-fold, and AAL_C$ increases 7-fold, when the damage initiation point is considered at dh = −1.0 foot vs. 0 feet (Table 5). Similarly, use of DDFs starting at dh = −1.0 foot causes a 5-fold increase in mean AAL_T (i.e., building, contents, and use) for homeowner and landlord, and a 6-fold increase for tenant, compared with AAL_T at dh = 0 feet (Table 6).

A test of the null hypotheses equating mean AAL with Hazus-MH output results in p-values of 0.39 and 0.23 for building and contents, respectively. These results indicate that no statistically significant difference exists between the mean AAL value from the presented approach vs. results from Hazus for both building and contents risk. The mean AAL_B$ and AAL_C$ are $2,330 and $1,513 per home, respectively, for the numerically integrated approach. In contrast, these values are $2,069 and $1,279 per home based on Hazus output.

The wide variation in AAL across the case study area, with homes 21 and 9 having the highest and lowest AAL_B$ and AAL_C$ (Figure 4), demonstrates the value of performing neighborhood-level risk assessment. Many factors affect AAL results. Further analysis to interpret the results presented in Supplementary Table 6 demonstrates that unique DDFs based on number of stories result in statistically significant differences in AAL_C$ to AAL_B$ ratios (p-value of 0.0004), with mean values of 67 and 49% for one- and two-story homes, respectively. Other factors, such as a and u parameters, FFH₀, and V_R also affect AAL (Table 2). The finding that AAL_T$ for homeowner-occupied homes exceeds that for the other owner/occupant types (Figure 5) is reasonable because the homeowner bears the risk for building, contents, and use, while the smallest AAL_T$ is borne by tenants because they incur no AAL_B$.

Quantification of the flood risk is important to overcome the resistance to mitigate the flood hazard (Hollar, 2017). Results from assessing flood risk by owner/occupant type and the effect of increasing FFH on its reduction can enhance awareness and action to mitigate flood effects. For example, the substantial decrease in AAL_B$ and AAL_C$ by increasing the FFH 1 foot above E₁₀₀ (Figure 6 and Table 3) and the virtual elimination of building and contents flood risk through elevation of 4 feet, shown in this case study, demonstrate the economic advantages of mitigation measures and flood risk reduction. The reduction of AAL by ~93% with the first foot of increase in FFH above E₁₀₀, with an additional 6, 0.48, and 0.05% with two through four feet of additional elevation (Table 4), regardless of owner/occupant type, provides the public with specific information for decision-making. These values suggest that adding two feet above E₁₀₀ may provide the most effective flood risk reduction for this study area. However, it is noted that the substantial flood risk prevented by adding one foot above the E₁₀₀ is achieved by the minimum elevation requirement for residential buildings in the U.S.A. specified in the ASCE (2014) technical standard.

These results vary by DDF damage initiation point and completely depend on the accuracy of flood maps, particularly regarding the flood depth information for multiple return periods. Error can also be incurred if the modeling of these data excludes recent changes in development that affect rainfall runoff conditions or climatic data not reflective of the true nature of precipitation events. Future work is needed to understand the extent to which the finding of 93% reduction with first foot of increase in FFH above E₁₀₀ and an additional 6% with two feet elevation is consistent across space and time. Regardless, however, presentation of flood risk in terms of AAL and the corresponding reduction in relative flood risk with additional building information provides new opportunities to validate flood model data and DDFs through observation of future events and losses.

Consideration of the damage initiation point at dh = −1 and −0.5 foot increases the quantified risk substantially, as suggested by Tables 5, 6. Given the 5- to 7-fold increase in AAL calculation based solely on the DDF damage initiation point, this concept warrants considerable future research, as values of dh = −2 (United States Army Corps of Engineers, 2000, 2003), dh = −1 (FIA, 1974), and dh = 0 (Federal Emergency Management Agency, 2003) are employed in the literature. DDFs are available that differentiate foundation type and floodwater characteristics and duration [e.g., (Gulf Engineers and Consultants (GEC), 1997)], and standard deviation values are provided for generalized United States Army Corps of Engineers (2000) DDFs. As micro-scale analysis gains more traction, the use of component-based functions (Matthews et al., 2021) may become viable to generate building-specific DDFs. However, while the current research represents a substantial step forward, without more research in this area it is likely that flood risk estimates continue to underestimate or overestimate the true risk.

Despite the fact that the Hazus-MH does not differ significantly from the refined numerical integration approach-generated mean AALs, an 11 and 15% on average underestimation of AAL_B$ and AAL_C$, respectively, by Hazus in the case study results may be crucial for an individual homeowner. The underestimation can be attributed to two factors. First, Hazus-MH does not consider flooding at shorter than 10-year and longer than 500-year return periods (Equation 17), which leaves damage from the most common, minor floods, and the rarest, major floods, unconsidered. While at first glance such floods may be negligible either because they produce little damage or because they are unlikely to occur during the home's useful life cycle, the frequency with which nuisance floods occur and the devastating impacts of exceedingly large floods, if they do happen to occur during the home's useful life cycle, make consideration of flood risk only for return periods ranging from 10 to 500 years incomplete. A second factor contributing to the Hazus-MH underestimate is that it only considers five return periods, which generates a coarser approximation using Riemann sums of areas under the loss-exceedance curve (Figure 7A) vs. the fine trapezoid definition used in this AAL approach, where 260 trapezoids are used (Figure 7B).

In addition to the building and contents underestimates, other factors point to further inaccuracies by Hazus-MH. The neglect of use risk can generate inaccuracies, especially for longer-term recoveries. Moreover, the final term in Equation 17 (from Hazus-MH), unrepresented in Figure 7A, is an additional factor without theoretical basis that may ostensibly compensate for the underestimation, perhaps leading to the finding that the AAL approach differs insignificantly from Hazus-MH.