Abstract

This study proposes a flood potential index suitable for use in streamflow forecasting at any location in a drainage network. We obtained the index by comparing the discharge magnitude derived from a hydrologic model and the expected mean annual peak flow at the spatial scale of the basin. We use the term “flood potential” to indicate that uncertainty is associated with this information. The index helps communicate flood potential alerts to communities near rivers where there are no quantitative records of historical floods to provide a reference. This method establishes a reference that we can compare to forecasted hydrographs and that facilitates communication of their relative importance. As a proof of concept, the authors present an assessment of the index as applied to the peak flows that caused severe floods in Iowa in June 2008. The Iowa Flood Center uses the proposed approach operationally as part of its real-time hydrologic forecasting system.

1. Introduction

In principle, distributed hydrological models allow river discharge forecasts at every channel in a drainage network. The spatial and temporal distribution of peak flows across the channels is a reflection of the interactions among the spatial and temporal variability of the model input (e.g., rainfall, snowmelt), the soil properties controlling the generation of overland flow, and the structure of the drainage network (e.g., Lu et al. 2017; Mantilla et al. 2006; Ayalew et al. 2014b; Mandapaka et al. 2009; Ayalew et al. 2014a). To properly judge the magnitude of the forecast discharge derived from hydrologic models, the general public needs adequate reference for a particular location of interest. For locations where streamflow gauges are installed, quantiles of observed past discharges provide this reference. Difficulties occur when it is necessary to provide guidance on forecasted streamflow for river communities at ungauged locations. The situations where flooding is likely to affect communities are of particular importance. Alerts issued with enough lead time can help the community members prepare and thus lessen the flood consequences.

In this study, we address this problem by proposing a methodology that establishes a discharge reference at any point in the drainage network. The goal is to provide guidance to determine the relative importance of forecasted streamflow, given a historically observed hydrological regime at gauged locations. The literature includes efforts to address this problem using references based on the 10-yr flood discharge magnitude. Villarini and Smith (2010) proposed a flood index obtained from the ratio of flood peak discharge at a particular location along a river network to the 10-yr flood discharge at the same location, providing a dimensionless representation of flood magnitude. To represent the spatial distribution of flooding, they created a continuous spatial field of flood index values obtained by inverse distance (in Euclidean space) weighted interpolation of discrete point observations. More recently, Lu et al. (2017) adopted a more physically based representation of the index, implementing a distributed hydrologic model that allowed reporting the 10-yr flood magnitude continuously over a river network. For our study, one limitation of this method is that the 10-yr flood magnitude is a good reference for analyzing rare events, but not very useful for smaller flow events. Also, it is not easy to interpret a 10-yr event in terms of the corresponding water stage. To avoid this limitation, we propose as a reference the mean annual peak flow.

Based on earlier studies (Wolman and Leopold 1957; Dury 1973; Leopold 1997), mean annual peak flow corresponds closely to the bankfull discharge, that is, the discharge where flow just fills the stream channel and begins to inundate the floodplain. Mean annual peak flow is easy to obtain, and the literature shows that it can be used as a good proxy to estimate bankfull discharge. Wolman and Leopold (1957) suggested that bankfull discharge has a recurrence interval of 1–2 years. Dury (1973) concluded that bankfull discharge is approximately 97% of the 1.58-yr discharge or the most probable annual flood. Richards (1982) suggested that in a partial duration series, bankfull discharge equals the most probable annual flood, which has a 1-yr return period. Leopold (1994) stated that most investigations have concluded that the bankfull discharge recurrence intervals range from 1.0 to 2.5 years.

We propose using an index based on the mean annual peak flow value at a particular location that can be represented over a river network. The approach can be implemented with any distributed hydrologic model that includes a faithful representation of the drainage network. As an illustration, we discuss the methodology as implemented in the hydrologic forecasting system developed by the Iowa Flood Center (IFC), which uses the Hillslope-Link Model (HLM; Krajewski et al. 2017).

2. Methodology and data

The IFC uses the HLM in its real-time streamflow forecasting system (Krajewski et al. 2017; Quintero et al. 2016; ElSaadani et al. 2018) for all the communities in Iowa. The model uses channel links and hillslopes as the primary units of landscape decomposition where the hydrologic processes are modeled. Conversion of rainfall to runoff is modeled through soil moisture accounting at hillslopes; a hillslope drains to a single link of the river network. The model bases channel routing on a nonlinear representation of water velocity that considers the discharge amount as well as the upstream drainage area (Ghimire et al. 2018; Gupta et al. 2010). The model is basically a large system of ordinary differential equations organized to correspond with the river network topology. HLM solves the equations using an efficient numerical solver suitable for high-performance computing architecture (Small et al. 2013). A recent comprehensive study used seven years of data and two radar rainfall products across 140 U.S. Geological Survey (USGS) gauges in Iowa with available streamflow observations to exhaustively examine HLM’s streamflow prediction performance (Quintero et al. 2020). HLM has average peak estimation errors below 5% at large basins, and below 20% at small basins; its mean absolute error for river stage prediction is below 1 ft over 65% of the simulations. Quintero et al. (2020) pointed out the model’s difficulties in reproducing the streamflow behavior at landforms in western Iowa with geomorphologic characteristics that differ from the rest of the state; the authors also found that the model sometimes overestimates the hydrographs’ falling limb rate.

The HLM uses a digital representation of Iowa’s river network based on the processing of a digital elevation model with 90-m squared cells. The landscape is decomposed into hillslopes and channels derived from digital elevation models. The Iowa domain is split into about 420 000 hillslope–link pairs with an average size of approximately 0.1 km2; the average length of the channels is about 400 m (Quintero and Krajewski 2018). The network description includes multiple geomorphological characteristics of the hillslopes and links, as well as the upstream drainage area of each channel.

We obtained the mean annual peak flows as observed at 187 USGS river gauges in Iowa during the period between 1900 and 2018 (Fig. 1). Mean annual peak flow is defined here as the average of the annual maxima observed at a streamflow gauge. Note than mean annual peak flow does not necessarily cause flood conditions at a given location. Evidence that peak flows for flood events shows power-law scaling with the area of the basin supports our approach for obtaining a flood index that can be represented over the river network (e.g., Ayalew et al. 2014a,b; Mantilla et al. 2006; Ogden and Dawdy 2003). The literature has reported that scaling of peak flows with drainage area functions across multiple spatial scales, including the range of scales of the river network of Iowa and over watersheds with different climate characteristics (Perez et al. 2018, 2019).

Fig. 1.

Location of 187 USGS gauges in Iowa that were used to calculate the relation between mean annual peak flow and drainage area. The thick white lines show the segments of the river network located downstream of reservoirs.

Fig. 1.

Location of 187 USGS gauges in Iowa that were used to calculate the relation between mean annual peak flow and drainage area. The thick white lines show the segments of the river network located downstream of reservoirs.

We compared the relationship between the mean annual peak discharges observed at each gauge and its drainage area. Figure 2 shows a power-law relationship between the drainage area and the mean annual peak discharge. The mean annual peak flow scales with the drainage area following a function of the form

 
Q=αAθ,
(1)

where A is the area of the basin and α and θ are the coefficient and exponent of the equation. We included gauge data downstream of reservoirs because during very large flood events, reservoirs can lose control of discharge regulation; communities downstream of the reservoirs can benefit of the information provided by the flood potential index. Also, the effects of flow regulation decrease downstream of the reservoirs and, given all the uncertainty sources, it is difficult to decide where it becomes negligible.

Fig. 2.

Scatter data compare the basin upstream area and the mean annual peak flows at gauges with observed data. In black is shown the power-law fit when not including gauges downstream of reservoirs, and in red the fit when the gauges downstream of reservoirs are included.

Fig. 2.

Scatter data compare the basin upstream area and the mean annual peak flows at gauges with observed data. In black is shown the power-law fit when not including gauges downstream of reservoirs, and in red the fit when the gauges downstream of reservoirs are included.

We used the parameters of the mean annual peak flow regression [Eq. (1)] to extrapolate the expected annual peaks to all the channels in the representation of the drainage network. Figure 3 shows the results of the spatially distributed mean annual flood. The spatially distributed mean annual peak flow can be used as a reference to compare with the magnitude of streamflow at any location in the drainage network for any period of time, including the forecast lead time. The streamflow simulated with HLM can be transformed into a nondimensional flood potential index I defined as

 
I=QQmaf,
(2)

where Q is the streamflow forecast obtained with the hydrologic model at a specific location and time, and Qmaf is the average of the annual peak flows at the same location. We coined the term “flood potential” for the index to indicate that there is uncertainty associated with this measure. The error in the estimation of flood potential index at a given location is subject to the accuracy of the streamflow forecasts and the errors in the estimation of mean annual peak flow using Eq. (1).

Fig. 3.

Mean annual peak value estimated with the regression model.

Fig. 3.

Mean annual peak value estimated with the regression model.

We illustrate the proposed methodology by estimating the flood potential index caused by the devastating flood events that took place in Iowa in June 2008. This is the most damaging flood event in Iowa’s recorded history. Even though the 1993 floods were larger in scope and damages in the Upper Midwest, the 2008 flood appears to have caused more damage and losses for Iowa communities and individuals (e.g., Mutel 2010; Smith et al. 2013). Estimates place the total damages to private property and public infrastructure at more than $3 billion, without accounting for lost sales by businesses and industries in flooded areas (Mutel 2010). Of the 99 counties in Iowa, 85 were included in the state’s Presidential Disaster Declaration. Most of the devastation occurred in the city of Cedar Rapids, where the flow of the Cedar River reached 3964 m3 s−1 (e.g., Budikova et al. 2010; Krajewski and Mantilla 2010). We used HLM to estimate peak flows for the 2008 event across the river network for the entire state. We forced HLM with the observed rainfall data obtained from the Stage IV radar rainfall products over the continental United States (CONUS), provided by the National Centers for Environmental Prediction (Lin and Mitchell 2005) We obtained the discharge data from the USGS.

3. Results

a. Regression parameters and mean annual peak flow map

We obtained the upstream area and the average of the annual maxima values for the 187 analyzed stations (Fig. 2), including gauges downstream of reservoirs. We estimated the parameters of the regression function proposed in Eq. (1). We found the values of α = 3.12 and θ = 0.57, respectively. Figure 2 shows the fit of the power law as a solid red line. The figure shows that the difference between including or excluding gauge data downstream of reservoirs is minimal, that is, within the scatter of the relationship. With these parameters of the regression, we were able to estimate the mean annual peak flow at any location in the drainage network, based only on the upstream area information of each link. These results look as expected and are shown in Fig. 3. The map shows results for streams of Horton order larger than three, but we have the results available for finer spatial scales as well. Certainly, this analysis could be conducted in a basin-by-basin manner to account for differences in the spatial distribution of peak flows across the state. The authors limited their analysis by assuming that the distribution of peak flows in the upstream area is similar across the state.

b. Hydrologic simulation of the June 2008 event in Iowa

We used HLM to simulate the streamflow that occurred in Iowa during the floods of June 2008. We initialized the model with the hydrologic states observed in April 2008. The snow accumulation during the winter months of the 2007–08 produced wet conditions in the ground that lasted into early summer; the saturated soils were unable to absorb rainfall produced by the intense storms that took place in Eastern Iowa during April and later in the first days of June 2008. In Fig. 4, we compare the outputs of HLM with observed streamflow at six gauges in major basins in eastern Iowa, where the flooding occurred. HLM accurately predicts the magnitude of peak flows, as well as other characteristics of the hydrographs, such as the rising limb and volume. Certain locations have a problem with peak timing (e.g., the Wapsipinicon River near Tripoli and the Cedar River at Cedar Rapids), where the model predicted peak flows a few hours earlier than observed. Figure 5 compares the streamflow peaks at gauges where observations were available with the corresponding peaks derived from the hydrological simulation. For this example, we ensured that the analysis includes only observed and simulated peaks caused by the storms of June 2008. The coefficient of determination of the regression is R2 = 0.81; the dispersion of the regression is larger for lower peak values. The regression shows that the hydrologic model accurately predicts peak flows for large streamflow values, which are expected to occur at larger basins.

Fig. 4.

Comparison of observed streamflow (black lines) and flow simulations (red lines) obtained with HLM during the flood events of 2008 at six USGS gauges in eastern Iowa.

Fig. 4.

Comparison of observed streamflow (black lines) and flow simulations (red lines) obtained with HLM during the flood events of 2008 at six USGS gauges in eastern Iowa.

Fig. 5.

The scatter compares the observed peaks in the floods of 2008 with HLM estimates at gauges where peak flows were caused by the June event.

Fig. 5.

The scatter compares the observed peaks in the floods of 2008 with HLM estimates at gauges where peak flows were caused by the June event.

We obtained the peak flow at every channel in the Iowa river network, based on HLM’s distributed streamflow simulations. The results are shown in Fig. 6. The storm events generated high values of discharge in all the watersheds of central and eastern Iowa. The values are especially high for the Cedar River after its junction with the Iowa River. The communities located in these areas were the ones most affected by the flooding. Note that even though the model can provide accurate estimates of discharge for every channel in the network, this is not very informative for general public. Only people with prior experience and knowledge of the flood magnitude or river stage that could cause problems for their specific location of interest (e.g., home, school, or business), could benefit from this information. For others, the discharge values are meaningless. However, it is important to alert the general public about potentially dangerous conditions at bridge crossings, water infrastructure, and homes in the floodplain. In the next section, we will address this problem by reporting the flood potential index caused by the storm event.

Fig. 6.

Peak discharge values simulated with HLM at every channel of Iowa’s river network after the floods of 2008.

Fig. 6.

Peak discharge values simulated with HLM at every channel of Iowa’s river network after the floods of 2008.

c. Assessment of the flood potential index

We used Eq. (2) to estimate the flood potential index using the distributed simulated peak flows (Fig. 6) and the distributed mean annual peak obtained with the regression model (Fig. 3). The estimated index for streams and rivers in Iowa is shown in Fig. 7. According to the proposed index, the peaks of 2008 exceeded the mean annual peak flow in the Cedar River basin downstream of Cedar Rapids by more than 5 times, including the streams after the junction with the Iowa River. The mean annual peak in some streams was exceeded by up to 4 times because of heavy localized convective rainfall. We saw this in the upper Iowa River, the Turkey River basin, the upper part of the Cedar and Wapsipinicon Rivers, the upper part of the Maquoketa River, and the Des Moines River after the junction with Raccoon River. Several small tributaries of the main rivers also reached values of up to 3 times the mean annual peak flow.

Fig. 7.

Flood potential index obtained for the peak flow at every channel of Iowa’s river network after the floods of 2008.

Fig. 7.

Flood potential index obtained for the peak flow at every channel of Iowa’s river network after the floods of 2008.

The index map (Fig. 7) provides more information than the simple discharge map (Fig. 6) on the relative magnitude of the flow. Emergency responders can use the index map to help them to prioritize their efforts during flood response by identifying the communities that will be more affected by floods. The index map makes it easier to communicate awareness of the flood risk at the communities. However, the flood potential index could be simplified to further facilitate communication of the risk to the general public by using flood threat categories rather than numerical index values to express the flood threat. We believe these will be easier for the general public to interpret. We elaborate below.

d. Implementation in the Iowa Flood Information System

The Iowa Flood Center uses HLM to make real-time streamflow predictions every 15 min for well over 2000 locations, including about 1000 small communities and 400 stream gauge locations. These streamflow forecasts are communicated to general public through the Iowa Flood Information System (IFIS) (Demir and Krajewski 2013; Krajewski et al. 2017), a browser-based, comprehensive, interactive platform. Streamflow forecasts at gauged locations (e.g., USGS stream gauges) are converted into river stages using the rating curves for the gauge location. Small communities, however, lack the reference needed to interpret streamflow forecasts; therefore, these communities can benefit from use of a flood potential index. We implemented the methodology in IFIS to improve the communication of discharge forecasts for river communities.

When a user selects a river community in IFIS (as a point location), a widget displays the time sequence of the flood potential index, derived from the streamflow prediction of HLM for the next five days (see Fig. 8). Instead of using numerical values of the flood potential index, the IFC simplified the index for easier interpretation by the general public. Rather than reporting its numerical values, we assign categories expressed by a color scale. The categories range from low flood potential to high flood potential. The widget’s green color indicates the time period when the streamflow prediction is below the estimated mean annual peak flow according to the model and therefore the flood potential is low. The yellow color reflects a flood potential index value of 0.6, which is significantly less than the approximate bankfull flow (index of 1). However, because the predictive model has substantial scatter and prediction error, we decided to be conservative and alert the public with the yellow color. For example, several locations exist where the actual value (based on the USGS annual maxima data) is below the predictive model. This corresponds to a value of the index less than 1.0. While the threshold of 0.6 is somewhat arbitrary, 80% of the index distribution is above this value and approximately one standard deviation is below the predictive model. A good analogy here is the National Weather Service’s “action level,” where the water is still well within the banks but the agency is alert and begins to keep close eye on the situation. When the model predicts streamflow that exceeds the mean annual peak flow, the widget appears orange to red, depending on the degree of exceedance.

Fig. 8.

Flood potential index as reported in IFIS.

Fig. 8.

Flood potential index as reported in IFIS.

The colored river network map also indicates the flood potential for a given moment in time. The time controls in the bottom right panel allow the user to track how the flood potential index evolves over time as the flood wave moves through the stream network. Blue tones in the flow maps indicate conditions below the estimated mean annual flood. We chose to use different shades of blue, even though they all indicate low flood potential, to increase the dynamic range of the information about the flow. As the user selects a time frame, the visible features (i.e., the shades of blue) move through the river network.

4. Discussion and conclusions

We proposed a method to estimate a flood potential index that communicates the relative importance of forecasted streamflow. This method can be implemented with any distributed hydrologic model that makes forecasts over a river network. The index relies on historical peak flow data and is intended to provide information at every stream of the river network, serving as a reference for nongauged river communities. We implemented the flood potential index approach as part of the forecasts disseminated through the Iowa Flood Information System. Effective communication of potential floods is crucial for members of the public, who consume the early warning system information. Communicating direct discharge values is not very informative for general public. We proposed an index that provides context of the relative importance of the forecasted discharge.

Based on our results, we found that the function Q = 3.12A0.57 can be used as a predictor of mean annual peak flows at ungauged sites based on drainage area in Iowa for the purposes of reporting importance of streamflow values. One could argue that we could have used existing regression equations for low quantiles (e.g., a 2-yr return period) obtained from regionalization studies in Iowa (Eash et al. 2013) that also use drainage area as a predictor rather than the relationship we found between mean annual peak flow and the upstream area. We compared the relationship we obtained in our study with the regionalization equations of the percentile Q50% and found, as expected, some similarity with the regressions. We decided to use the mean annual maximum because the existing regionalization equations for Iowa are obtained for three different regions, the boundaries of which split many watersheds. Also, calculating the index separately for different basins reduces the amount of gauge data available and thus increases the scatter.

The accuracy of the flood potential index is subject to the quality of the mean annual peak flow reference. In this study, we included observed peak flows between 1900 and 2018 to obtain the mean annual peak flow at each gauge. Therefore, our reference accounts for Iowa’s major floods in 1993, 2008, and 2016. If our reference did not include the observed peaks of 2008 and 2016, for example, the results we present would show a much higher flood potential index. The Iowa Flood Center continues to update the mean annual peak flow reference as new annual maxima become available.

Clearly communicating predicted streamflow in a dynamic and spatially explicit way using inundation maps showing the extent, water depth, and velocity as well as potential damages is the “holy grail” of flood forecasting. Although our technological and computational capabilities are quickly advancing, at the Iowa Flood Center we opt to present streamflow forecasts at ungauged communities and stream segments using the flood potential index. This allows us to acknowledge that streamflow forecasts are subject to major uncertainty. Forecasting maps that fail to acknowledge their associated uncertainty would quickly expose inaccuracies and thus erode public trust in our forecast. Communicating streamflow forecast uncertainty to general public is a difficult problem for social and hydrometeorological research. As we continue to reduce this uncertainty, the flood potential index provides a way to collapse the streamflow forecast uncertainty in terms of flood potential categories that are easier for the public to interpret.

Acknowledgments

The Iowa Flood Center funded this study. WFK was partially funded by the Rose and Joseph Summers endowment. The authors acknowledge the contributions of their colleague Ricardo Mantilla, who first proposed use of the index in the IFC’s operational system.

Data availability statement: Some or all data, models, or code generated or used during the study are available from the corresponding author upon request.

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