A practical method for developing future joint probabilities of riverine and coastal flood risk in complex tidal river systems – a case study Open Access

Compound flood events can occur when various flood drivers combine at a location resulting in unusually severe flooding. Typically, compound events are defined as:

• two or more extreme events occurring simultaneously or successively,

• combinations of extreme events with underlying conditions that amplify the impact of the events, or

• combinations of events that are not themselves extremes but lead to an extreme event or impact when combined.

In coastal and tidal river locations, compound flooding is often driven by fluvial and coastal processes. Often an individual driver may not be extreme, but the complex nonlinear interactions between fluvial and coastal processes can intensify the joint impact of the drivers, which may result in significant flood hazards (AghaKouchak et al. 2018; Dykstra & Dzwonkowski 2020).

Heinrich et al. (2023) wrote that coastal flooding is one of the most frequent, expensive, and fatal natural disasters. With more than 600 million people living in coastal areas that are less than 10 m above sea level, the urgency of this problem is clear (United Nations 2017).

Flooding in coastal areas primarily arises from four main physical factors or source mechanisms (Latif & Simonovic 2023):

• a combination of storm surge with high predicted astronomical tides or high normal tides (also called storm tides),

• locally or remotely generated waves (swell),

• river discharges at the lower estuary or fluvial flooding, and

• Direct surface runoff or pluvial flooding.

Bivariate or multivariate analyses are often used to measure compound flood hazard in terms of the joint occurrence of event extremes (Zscheischler et al. 2020). The co-occurrence rate may be calculated as the joint exceedance probability when a single driver or multiple drivers are above their predefined thresholds (Moftakhari et al. 2017). These can be defined as an ‘OR’ hazard scenario, where the probability is that at least one driver exceeds their given flood thresholds, or an ‘AND’ hazard scenario, where the probability is that both drivers exceed their flood thresholds simultaneously (Salvadori et al. 2016).

Multiple statistical methods have been suggested to study compound flooding in coastal regions (Hao et al. 2018; Tilloy et al. 2019). These methods estimate the joint extremes or the statistical dependance between a variety of flood variables such as rainfall and storm surge, river flow and storm surge, and river flow and sea level (Wahl et al. 2015; Sadegh et al. 2018; Bevacqua et al. 2019; Hendry et al. 2019).

In the past decades, several flooding studies have used copula theory to estimate the bivariate joint distribution and assess complex dependance structures between flood drivers (Paprotny et al. 2018; Ward et al. 2018; Bevacqua et al. 2019). For example, an earlier study in the Lower Rhine Delta (Van der Made 1969) applied a joint probability approach considering three individual categories: high discharges and high sea levels, high discharges and normal sea levels, and normal discharges and high sea levels.

Recently, copula-based methodologies have been used to model extreme flood and meteorological events (Kong et al. 2015; Latif & Mustafa 2020). Wahl et al. (2015) modeled the joint impacts of storm surge and rainfall. Moftakhari et al. (2017) proposed a copula-based bivariate flood hazard framework for the joint impacts of coastal sea levels (with future sea level rise (SLR)) and riverine flooding. Couasnon et al. (2020) modeled bivariate joint distribution analyses of storm surge and daily river discharge observations to identify hot spots vulnerable to compound flooding in river mouths around the world.

For this study, the location of a tidal section of a tributary river (Schuylkill River) to the main tidal river (Delaware River) results in an additional complexity to the more common approaches encountered at the mouth of the tidal river.

Joint probability studies have been carried out for many flood-related studies, however, the ones found in the literature primarily focus on combinations of fluvial and coastal flooding, fluvial and pluvial flooding, or coastal and pluvial flooding. Other studies, including an earlier study for this particular project, simulate riverine flooding in coastal areas by incorporating coastal conditions as boundary conditions in a river model. The subject of this paper was prompted by the need to examine

• possible differences in flood elevation by directly incorporating coastal flood elevations in a riverine flood study instead of the use of estimated boundary conditions; and

• The need to study the unique conditions of a tidal river discharging to another tidal river.

Both rivers have caused flooding in and near Philadelphia in the past, driven by a variety of factors such as tides, storm surges, and rainfall–runoff-induced riverine flooding. A significant event highlighting these complexities was Hurricane Ida, which brought up to 25 cm of rain to the area on 2nd and 3rd September 2021, resulting in an increase in Schuylkill River flow from an average of 100 m³/s to a flood flow of 3,540 m³/s. The convergence of storm surge from the Atlantic Ocean and intense rainfall caused unprecedented flooding along the Schuylkill River and its tributaries, inundating homes, businesses, and infrastructure. This event underscored the need for more advanced methods, such as copula-based models, which can better capture the joint behavior of multiple drivers providing a more accurate estimation of extreme flood risks.

The Vine Street Expressway, a section of I-676, is located in northeast Philadelphia. It is depressed below street level as it approaches the Schuylkill River. In the past, there have been multiple events affecting the project location with Hurricane Ida being the largest event. During Hurricane Ida, the depressed section of the highway flooded. With no avenue to flow back to the river, the section suffered damage and had to be pumped out before the highway could be reopened. As a result, the Pennsylvania Department of Transportation wanted to explore both the current and future flood risk to the highway to inform the development of flood mitigation measures. As sea levels continue to rise and storms intensify, more events of this magnitude or greater are expected.

The location presents a challenge as Vine Street is located in a downstream tidal section of the Schuylkill River which discharges to the tidal Delaware River. This means that elevations at Vine Street will be influenced by flows on the Schuylkill River, along with extreme flows, tides, and storm surges on the Delaware River. The method had to account for all these influences to develop both current and future flood elevation probabilities. With gages located in both tidal rivers (water elevations) as well as upstream of the dam (flow) in the non-tidal Schuylkill River, the analysis of the past data concurrent at all three gages provided an opportunity to calculate extreme water level probability using past data. Because the next phase of the study will develop flood protection measures, estimates of future changes to these flood elevations due to climate change were also required.

The overall approach to the flood analysis is discussed in the following section. Each aspect of the approach is then presented in greater detail in subsequent sections.

The datasets used for this analysis were taken from several sources and include hourly data for Schuylkill River discharge and tidal elevations for the Schuylkill and Delaware Rivers (Figure 1).

• i. United States Geologic Survey (USGS) 01474500 Schuylkill River upstream gage (non-tidal): Hourly discharge data in cubic feet per second was downloaded for the period 1986 through 2023. These data were used to determine extreme discharge values for the 2, 5, 10, 25, 50, and 100-year return intervals.

• ii. USGS 01474501 Schuylkill River downstream gage (tidal): The USGS gage provided hourly elevation data for the period from September 2016 through December 2023. This gage was used for tidal decomposition.

• iii. National Oceanic and Atmospheric Administration (NOAA) Philadelphia tidal gage (Delaware River) 8545240: The NOAA gage provided hourly elevation data from 1901 through 2023. The elevations represent the combined effects of tide, storm surges from the Atlantic that impact the Delaware River in Philadelphia, and Delaware River watershed runoff events and baseflows.

• iv. Downscaled Schuylkill River Watershed Runoff Projections: Statistically downscaled global climate model (GCM) output provided by the US Bureau of Reclamation's downscaled CMIP3 and CMIP5 climate and hydrology projections website was used for this study (https://gdo-dcp.ucllnl.org/). The data portal was used to access downscaled runoff projections for the Schuylkill Watershed (upstream of the project location). The runoff projections available are provided by a variable infiltration capacity (VIC) hydrology model (Liang et al. 1994; Hamman et al. 2018). The VIC model is a spatially distributed hydrologic model that solves the water balance at each model grid cell. The VIC model runoff values were downloaded at nodes located within the Schuylkill River watershed (Figure 5).

• v. Sea Level Rise Projections: NOAA 2022 Sea Level Rise Projections were used for this project. NOAA made locally relevant results by downscaling the Global Mean Sea Level (GMSL) for the US coastline and providing data to the public via NOAA's Sea Level Rise Viewer (https://coast.noaa.gov/slr/).

Each of these has its own probabilities, ranging from the daily tidal swings to the differing probabilities of extreme rainfall events in both the Schuylkill and Delaware River watersheds and the low probability storm surges from Atlantic Ocean hurricanes or large-scale extratropical cyclones often called nor'easters.

Prior estimates of flood elevation probabilities at Vine Street focused only on extreme discharge events from the non-tidal Schuylkill River based on USGS river discharge probabilities and coupled with a Hydrologic Engineering Center's River Analysis System (HEC–RAS) model from Newell Tereska & MacKay (NTM 2023). This univariate analysis ignores the possibility that the other three flood factors associated with the Delaware River could influence the probabilities. Because the tide gauge in the Delaware River's long-term hourly record included past storm surges, extreme upstream discharge events and the daily tidal cycle, it was not necessary to assess the probabilities of all three separately. Thus, a joint probability approach that included the calculated probabilities associated with Schuylkill River extreme flows using the tidal Schuylkill River gage data and the probabilities of high-water levels on the Delaware River, no matter the cause, would suffice to get a more accurate picture of flood probabilities at Vine Street.

Before the analysis could take place, it had to be established that the elevations in the tidal Schuylkill River were essentially identical to the elevations in the Delaware River, with differences only attributable to the Schuylkill River discharge over the Fairmount dam, the location of the transition from the non-tidal to tidal portion of the Schuylkill River. To do this, elevations at the Schuylkill River tidal gage and simultaneous elevations at the Delaware tidal gage were examined under conditions of normal Schuylkill River discharge conditions, conditions of high Schuylkill River discharge, and conditions of elevated Delaware River water levels under normal Schuylkill River discharge. The results confirmed that elevations at both the Schuylkill and Delaware tidal gages responded the same to tidal cycles and to changes in the Delaware River elevation and were only different due to Schuylkill River discharges.

Having established that the water levels in the Delaware River and the Schuylkill River are the two main contributing factors to flood elevations at Vine Street, the overall approach to estimating current flood elevation probabilities can be summarized in the following steps.

• Tidal cycle removal: Because the joint probability analysis is based on data with return intervals of 1 year or greater, the tidal cycle had to be removed from both the Schuylkill River tidal elevations and the Delaware River water levels. This was done through tidal harmonic analysis. This resulted in residual elevations for both rivers representing only the influence of the extreme events on baseflow conditions.

• Schuylkill River discharge probability: The probability of increased elevations due to extreme discharge could be calculated by examining the tidal gage elevation data for high flow events. To do this, a regression equation was developed relating the Schuylkill River residual elevations at Vine Street to the USGS estimates of extreme discharge and associated univariate probabilities.

• Delaware River water level probability: The univariate probabilities of Delaware River extreme water levels were calculated using the Delaware River tidal gage hourly data. These extreme high-water levels include extreme upstream discharge events and/or storm surges.

• Joint probability: A bivariate joint probability analysis was used to estimate the joint probability of flooding that accounts for the compounding effect of extreme Schuylkill River discharges and water levels in the Delaware River transmitted to the tidal portion of the Schuylkill River.

To account for future changes to flood elevations at Vine Street, the joint probability of flood elevations under current conditions was adjusted for both sea level rise and changes to extreme rainfall in the Schuylkill River watershed. Although theoretically, changes in extreme rainfall in the Delaware River watershed could also affect future flood elevations at Vine Street, the dominance of Schuylkill River discharge in explaining Vine Street flood elevations suggested that changes in river discharges in the Delaware River would not affect the results to any significant degree.

To isolate the non-tidal influence in water level elevations in the tidal portion of the Schuylkill River and to be able to determine joint probabilities between riverine flow and storm surge, it was important to extract the tidal signal in water elevations of the Delaware and Schuylkill Rivers using a harmonic tidal analysis, which allows decomposition of the water level data and removal of tidal effects from the observational data. The results are residual elevations for both the Schuylkill and Delaware Rivers tidal gage data. This was needed to eliminate Delaware River residuals (storm surge and Delaware River discharge effects only) from the calculation of Schuylkill River residuals (the effect of high flows from the Schuylkill River). The Schuylkill River residuals were calculated as the elevations at Vine Street with the tides removed minus the simultaneous Delaware River residuals. The estimated Schuylkill River residual elevations, along with Delaware River water levels, provided the data needed for the joint probability.

A joint probability is needed if the probabilities under consideration are not statistically independent of each other. The significance of statistical dependance between the paired data was assessed using Kendall's rank correlation coefficient, which provides a nonparametric measure of association between paired high Schuylkill River discharge and Delaware River water levels (Kendall 1938). If the Kendall rank correlation coefficient indicates a dependency between high river discharge and tidal water levels, the copula method should be utilized to build the joint distribution of pairs data. Kendall's rank correlation coefficient indicated a significant correlation between high Schuylkill River discharge and Delaware River water levels with a coefficient of 0.45 and a p-value close to zero (cutoff <0.05). Thus, it was expected that the joint probability approach would result in a different probability of flood elevations due to both Delaware River water levels and extreme discharge of the Schuylkill River than if independently assessed through univariate analysis.

Through the isolation of Schuylkill River residuals (no tidal or Delaware River effects), the flooding at Vine Street could now be defined by the joint probability of extreme water levels on the tidal Delaware River and the probability of extreme discharge and related flood elevations originating from the non-tidal Schuylkill River. The analysis used a copula approach. Copulas are special forms of bivariate and multivariate functions, and they are an elegant way to express those multivariate distributions. Sklar (1959) showed that a copula exists for any multivariate distribution, such that the joint distribution equals the copula applied to the marginal distributions. This is very advantageous when estimating multivariate distributions because the problem now reduces to estimating the univariate distributions and their dependency structure.

For this analysis, the largest annual Schuylkill River discharge/elevations and the corresponding largest Delaware River water levels were selected within ±1 day of each other. This procedure yields pairs of interest for each year (annual bivariate vectors), which can be assumed to be independent for physical reasons. The best copula fit among 26 copulas was selected for bivariate dependance analysis based on akaike information criterion and Bayesian information criterion to describe the dependance structure between the paired data (Multivariate Copula Analysis Toolbox, Sadegh et al. 2018). The approach followed a published methodology presented by Ghanbari et al. (2021).

Unlike univariate flood frequency analysis, hazard scenarios in bivariate analysis are not uniquely defined. Each definition provides specific information, and the choice of definition should align with the study's objectives (Salvadori & De Michele 2004; Salvadori et al. 2016). In this analysis, the ‘AND’ hazard scenario was considered, as the focus is on estimating the joint probability of exceeding given thresholds on both the tidal Delaware River and the non-tidal Schuylkill River.

A critical part of the analysis was to develop projections of future flood elevations, as the flood elevations at the project location are affected both by sea level rise and the expected intensification of precipitation in the Schuylkill River watershed. Sea level rise requires consideration because the project location is in a tidal river section, and sea level rise will elevate the daily base elevation above which flood elevations due to high Schuylkill River discharge and the Delaware River tidal elevations occur. Increases in precipitation and subsequent peak watershed runoff require consideration, as it will increase the Schuylkill River's extreme discharge events. The increased discharges were calculated for both Representative Concentration Pathways (RCP) RCP4.5 and RCP8.5 for mid-century and end-of-century.

The statistically downscaled runoff projections produced by the VIC model were downloaded at nodes located within the Schuylkill River watershed. The spatial distribution of nodes is shown in Figure 5.

Using a published method (Maimone & Adams 2023), the VIC Model output for runoff was then translated into changes in extreme river flow for the watershed at the project location.

The NOAA 2022 Intermediate and Intermediate-High Scenarios were used to assess sea level rise impacts for two future periods: mid- and end-century.

Flood elevations at Vine Street were assessed for four future scenarios related to changes in runoff in the Schuylkill River watershed and changes to sea level impacting the Delaware River elevations.

• Mid-century (2040–2070): RCP4.5 combined with the Intermediate Sea Level Rise

• Mid-century (2040–2070): RCP8.5 combined with the Intermediate-High Sea Level Rise

• End-of-century (2070–2099): RCP4.5 combined with the Intermediate Sea Level Rise

• End-of-century (2070–2099): RCP8.5 combined with the Intermediate-High Sea Level Rise

The intent of the analysis was to provide a series of current and future probability estimates of flooding at Vine Street for a range of flood elevations to support decisions on what measures to take to prevent flooding now and in the future. The elevation at which Vine Street currently floods (4.3 m NAVD88) can be caused by high Schuylkill River discharge elevations occurring simultaneously with high Delaware River tidal surges. In essence, it is the probability of the sum of the two elevations, which can have many permutations. Only pairs that occur with a 1-year or more return interval were used as the probabilities were calculated using annual maximum values. If considered in 1-foot (0.3 m) increments, a 14-ft (4.3 m) floodwater elevation at Vine Street could result from seven different elevation pairs. Pairs vary from the pair 4 ft (1.2 m) in the Delaware River and 10 ft (3 m) in the Schuylkill River through the pair 10 ft (3 m) in the Delaware River and 4 ft (1.2 m) in the Schuylkill River.

The joint probability for all combinations resulting in elevations of 9 (2.7 m)–22 feet (6.7 m) was calculated using a copula function with a result matrix of almost 1 million values. The highest probability (lowest return period) of all the cells for each flood elevation was reported as the joint probability for that specific floodwater elevation.

For developing a basis of design for any flood protection measure at Vine Street, the return intervals for a range of flood elevations are provided in Table 1. Table 1 also provides a comparison between an estimation of return intervals using a univariate estimate and the joint probability approach presented here. The first estimation (column 3) is based on using the USGS Schuylkill River discharge probabilities coupled with modeled elevations measured at the tidal Schuylkill River gage (NTM 2023). This is the univariate estimate of flood elevations based only on Schuylkill River discharge. The second estimation (column 4) is the joint probability of the same range of flood elevations at Vine Street using gage data and a copula joint value analysis that accounts for Schuylkill River discharge and water levels in the Delaware River that include tides, high upstream Delaware River discharges, and storm surges coming from the coast. It illustrates the clear need for the joint probability approach, as the joint probability return intervals are consistently lower than those using only Schuylkill River discharge. This is important for the current critical flood elevation at Vine Street. For example, the return interval estimate for elevation 4.30 m is 21 years due to the univariate estimate of Schuylkill River extreme flood probabilities, but only about 10 years if the Delaware water levels are also accounted for.

Table 1

Comparison of return intervals for Schuylkill River discharge only with joint probability estimate using Schuylkill River discharge and Delaware River water levels

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Designing a flood protection measure is a significant investment and it must be designed to protect against both current and future flood elevations. Because Vine Street is subject both to impacts related to storm surge from the coast as well as high river discharges from the upstream Schuylkill River watershed, climate impacts to Schuylkill River flows and changes to sea level rise must be considered. Like all climate-related projections, sea level rise and changes in extreme rainfall-related runoff can only be estimated as a range, depending on the selected emission scenario.

Estimated increases in flood elevations due to sea level rise and increased Schuylkill River discharge at the project location are shown in Table 2. RCP4.5 is the lower emission scenario, and RCP8.5 is the higher emission scenario. Projected changes in mid- and end-of-century due to increased discharges of the Schuylkill River range from 0.3 to 0.9 m. Sea level rise projections are not provided directly by RCP but by a range of low to high estimates. The selected sea level rise increase ranges from 0.5 to 1.2 m associated with the intermediate and intermediate-high projections. The resulting combined increases in flood elevation range from 1 to over 2 m by the end of the century, depending on the selected RCP and sea level rise scenario.

Table 3 is designed to estimate the effects of climate change on the current flood elevations. Column 1 shows the elevations of interest for flooding at Vine Street (see Table 1). Climate change impacts are a combination of sea level rise and increased flows in the Schuylkill River, as shown in Table 2. Elevations related to each combination in Table 2 were added to the joint probability estimates of flood elevation for the associated return intervals shown in Column 2. Columns 3 through 6 show the climate change impacts, increasing the current flood elevations in Column 1 with the combined effects of increased Schuylkill River discharges and projected sea level rise. The combinations shown are for RCP4.5 plus intermediate sea level rise and RCP8.5 plus intermediate-high sea level rise for mid- and end-of-century.

Table 3

Estimates of current and future flood elevations at Vine Street

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Table 3 shows that the estimated difference between the low emission scenario RCP4.5/intermediate sea level rise and the high emission scenario RCP8.5/intermediate-high sea level rise is over 0.5 m by mid-century and more than 1 m by the end of the century for all the return intervals. Note that the critical flood elevation of 4.30 m, shaded in green, is the elevation at which Vine Street floods. A 4.30 m flood elevation currently has a return interval of about 10 years. The flood elevation for a 10-year return interval is projected to increase from the current 4.30 m to about 6.4 m under the worst-case RCP8.5 and intermediate-high sea level rise near the end of the century.

Table 4 presents the same information as in Table 3, but this time with the focus on return intervals. Return intervals between 1 and 100 years are commonly used in flood studies to provide information necessary for risk studies and are shown in column 1. Columns 2 through 6 provide the estimated flood elevations associated with each return interval under current conditions and the combinations of sea level rise and emission scenarios shown in Table 2. This table provides vital information for making probability-based decisions about the eventual height of flood protection measures. The future frequency of the critical flood elevation of 4.30 m at Vine Street is projected to change from a current return interval value of 10 years to less than a 5-year return interval by mid-century, even under the less conservative assumptions of RCP4.5 and intermediate sea level rise. Under the high emission and intermediate-high sea level rise scenario, flooding of Vine Street is projected to occur annually.

Table 4

Return intervals between 1 and 100 years and associated flood elevations at Vine Street

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Floods in coastal and riverine environments are frequently subject to dual causes. In rivers discharging to coastal areas, storm surge and sea levels can combine with high river discharge to result in coastal flooding of low-lying areas, but also to prevent storm sewers from discharging, resulting in pluvial flooding in more inland, higher-elevation areas. In areas adjacent to rivers away from the coast, the same dual effect can be seen due to riverine flooding and exacerbated pluvial flooding. The dual cause examined in this paper is the extreme upstream discharge of a tributary river and water elevations on the receiving tidal river. This situation is less commonly encountered, and we found no studies of a similar nature.

This study has shown that assuming flooding is caused by only one factor can result in a significant underestimation of current flood risk (Wu et al. 2021). An initial estimate of the return interval for the critical flood elevation at Vine Street was made using upstream discharge probabilities and an HEC–RAS model to estimate resulting flood elevations in the tidal section of the river. The resulting estimated probability for the critical flood elevation at Vine Street underestimated the probability of flooding by a factor of two when compared to a joint probability approach. This highlights the significant advantage of using joint probability as opposed to making a simple conservative assumption about the downstream model boundary.

Because of the relatively high current probability of a reoccurrence of flooding at Vine Street, protective measures are being investigated. With climate change affecting both the Delaware River water levels as well as the extreme discharges from the upstream Schuylkill River, ignoring the trends related to climate change can result in an increasing underestimation of flood risk over time. Flood protection measures built to handle current conditions are likely to fail with increasing frequency in the future.

Joint probability estimates of flooding are important for the purposes of designing flood protection measures. It can also be critical in deciding the degree of protection that can be practically implemented through the use of techniques to assess the benefit/cost ratio of flood protection measures over time. Similarly, insurance rates, which are a reflection of relative risk, are also served by the more accurate estimation of flood probability in those situations where two factors can contribute to the final flood elevation. In this particular study, the location required additional analyses above the simpler univariate or joint probability of pluvial/riverine or pluvial/coastal flood impacts due to its unique location of a tidal tributary to a tidal river. To ensure that any flood protection measures being considered continue to be protective in the future, climate change impacts must also be considered.

Any statistical model should be evaluated to ensure its accuracy and reliability, particularly when accounting for complex hydrologic interactions and uncertainties. Various methodologies exist for assessing model performance, encompassing statistical techniques as well as comprehensive performance evaluation frameworks. While statistical methods focus on measuring predictive accuracy and model fit, comprehensive frameworks provide structured ways to assess a model's applicability, robustness, and effectiveness in real-world scenarios, considering uncertainties, risks, and overall system performance (Lopes et al. 2017; Ugural & Burgan 2021). These methodologies help improve decision-making by ensuring models are properly assessed and their outputs are reliable for flood risk analysis and mitigation planning. Integrating multiple evaluation approaches can strengthen the overall assessment of flood hazards and enhance the effectiveness of management strategies.

Although we believe that joint probabilities should be considered in all cases where there are two significant drivers of flood elevation, every analysis has limitations, and results must be viewed with the following limitations in mind.

GCMs are currently the most credible source available for providing information about the response of global climate systems to increasing greenhouse gas concentrations. Uncertainties include unknowable amounts of future GHG Greenhouse Gas (GHG) emissions as well as complex natural and physical processes and how they are parameterized by the GCMs. Furthermore, extreme rainfall events are often underestimated by climate models, which can lead to an underrepresentation of flood risks and potential damage.

The data for both of the variables in the joint probability analysis must be available with a sufficient level of detail and duration to capture a range of extreme events.

It is assumed that changes in runoff can be directly related to changes in discharge, which is a common assumption in the field of water resources.

The presence of a gage in the tidal section of the Schuylkill River introduced complexities in ensuring that elevations at the Delaware River are accurately reflected in the Schuylkill River. Tidal influences can cause fluctuations that may not be entirely synchronous with upstream conditions. It is important to make sure that any assumptions about the relationship between the downstream water levels and the flood elevations hold for the range of flood elevations under consideration.

This method is specifically tailored for analyzing extreme events with return periods greater than one year. Any extreme value analysis is subject to uncertainty, and values are actually representative of a possible range.

This study presented a joint probability analysis of flooding in a complex location where tidal influences, storm surge, and discharge in both the tributary and main river all contribute to the flood potential at the location in question. It showed the importance of considering all the potential contributors to flooding and the likelihood that flood probabilities based on the joint probability of two factors are likely to be higher than the more typical approach of calculating the probability of a single cause. Although the method was specific to this particular location, it provides an illustration of a practical method in a complex tidal tributary location.

Flood probability estimates are needed to make key decisions about the current and future risk of a location to flooding and to plan and design flood protection measures for critical infrastructure. The results illustrate the importance of considering the changing probabilities of flood elevations as both sea level rise and intensifying extreme rainfall are projected due to climate change. These estimates provide a good basis to calculate depth-damage functions and annualized losses over time in a later phase of the study to assess the benefit–cost ratio of potential protection measures to avoid flooding of Vine Street in the future.

All relevant data are included in the paper.

The authors declare there is no conflict.