Estimation of Historical-Annual and Historical-Monthly Scale-Invariant Flow Duration Curves with Implementation for Iowa

The flow duration curve (FDC) is a plot that shows the percentage of time when streamflow is likely to exceed or equal a given discharge value over a specific period of observation. The FDC provides a depiction of streamflow temporal variability for a specific location. Its interpretation in terms of recurrence depends on the time scale of flow averages (days, weeks, months, or years), and the period of streamflow records. The most conventional FDCs are the historical and the median-annual. Researchers calculate the historical FDC using the entire period of record utilizing historical daily flows. On the other hand, the median-annual FDC is calculated as the median of FDCs for independent water year records of daily flows. These FDCs have a wide application range in water resources, including hydropower design, water-quality management, sediment load estimates, and infrastructure design (Hickox and Wessenauer 1933; Mitchell 1957; Vogel and Fennessey 1995). A deficiency of the historical FDC and the median-annual FDC is that they do not describe changes in flow likelihood due to the seasonal variability of the streamflow throughout the year. For example, low flows caused by droughts during the month of June can be comparable in value to high streamflow values typical for the month of December. Alternatively, we can describe this type of variability by constructing FDCs from daily flow records associated with specific months. For instance, if the historical daily flow records are separated by month, then an FDC can be estimated for every month (e.g., January’s FDC is constructed with all the daily records observed that month), and these 12 monthly FDCs can more accurately represent the seasonal variability in probability of occurrence month by month.

When streamflow data are available for a specific site, the FDC can be straightforwardly estimated using parametric or nonparametric fitting procedures. However, streamflow records are limited around the world, and the prediction of FDC at ungauged locations is one of the challenges for the hydrologic community (Sivapalan et al. 2003). In response to this problem, researchers have proposed and implemented regionalization methods to estimate FDCs. The effectiveness of these regionalization methods depends on their ability to capture the spatial and temporal hydrological variability in a region using independent explanatory variables such as precipitation, land cover, evapotranspiration, and geometric properties of the basin and drainage network. In cases where hydrological heterogeneity cannot be fully captured or described by independent variables, the region is partitioned into hydrologic subregions. The presence of these sub-regions creates a disadvantage in the estimations of FDC for large domains, both in terms of the development of regional equations, as the primary explanatory variables need to be identified independently, and in terms of applications, because in many cases a catchment of interest can include many subregions.

One of the most recent motivations for the estimation of FDC at ungauged sites comes from the need to classify streamflow outputs from distributed hydrological models (DHM) that provide streamflow estimations at local, regional, and national scales. To communicate current daily discharge to the public using a broader hydrologic context, the USGS provides classified streamflow maps for the nation using observed streamflow values and historical FDCs for sites with long-term records (USGS 2016). USGS scientists classify the flow condition as a function of the percentile of the streamflow for a particular day of the year with a total of 365 references. However, this classification, based on historical observations, is for gauged sites only. An alternative is to estimate the FDC based on the model outputs; however, this requires an extensive simulation of long periods (more than 30 years), which is a difficult task for large regions. The USGS maps with classified streamflow are useful to the public because they provide an immediate visual assessment of the state of rivers in different regions of the country. However, when there is a need to analyze a region or state in more detail, the sparsity of information makes the picture incomplete (Fig. 1). In contrast, DHMs provide current and future streamflow estimates across the entire drainage network in the domain, including ungauged locations. Yet, model-based streamflow values shown on a map are difficult to interpret because of the multiple scales and regional hydroclimatic heterogeneity in space and time. A value of $100{\mathrm{m}}^3/\mathrm{s}$ on a map would be difficult to interpret correctly: is it a high flow for a small basin or a low flow for a large basin? The correct interpretation of streamflow outputs from hydrologic models is crucial for water-use decision making. The potential use of these streamflow outputs requires understanding of the flow conditions relative to flow extent. Without such tools, providing discharge estimates at locations where there is no reference data is not very meaningful. The simplest interpretation tool is the translation of flow magnitudes into flow frequencies using an FDC. Researchers can establish flow frequency thresholds to determine if the streamflow is in very low, low, normal, or high condition at an arbitrary location. This hydrologic information is not only relevant for the scientific community, but also for the general public because local communities can better understand streamflow conditions in a stream channel of interest, identifying impacts to the community either by droughts, floods, agricultural practices, recreational uses, or water supply activities.

Examples of the extensive use of regional hydrologic distributed models include the Iowa Flood Center (IFC) Hillslope Link Model (HLM) for the state of Iowa (Mantilla and Gupta 2005; Quintero et al. 2016; Krajewski et al. 2017) and the National Water Model (NWM) for the entire continental United States (Maidment 2017). The HLM estimates the current and future streamflow for 420,000 stream channels embedded in Iowa’s drainage domain. The Iowa Flood Information System web platform (Krajewski et al. 2017; Demir and Krajewski 2013) disseminates this information, providing access for local authorities and the public. At a national scale, the National Oceanic and Atmospheric Administration (NOAA) launched the National Water Model, which provides streamflow forecasting for 2.7 million locations in the United States based on the National Hydrography Dataset NHDPlusV2. These hydrologic models make important contributions to support water-use decision making for current and future scenarios.

The literature documents a broad range of methodologies to estimate FDCs at ungauged locations. These methodologies can be classified into two broad categories. The first approach estimates statistical distributions of streamflow, where the distribution parameters are approximated from explanatory variables that enable estimation at ungauged sites (Doulatyari et al. 2015; Fennessey and Vogel 1990; Li et al. 2010; Longobardi and Villani 2013). The second approach uses regression analyses performed between specific streamflow quantiles or distribution-moments with respect to a set of explanatory variables. This method is described by Farmer et al. (2015); Flynn (2003); Singh (1971); and Yue and Yew Gan (2004), among others.

The USGS has developed a methodology to estimate streamflow quantiles for the construction of FDCs at ungauged sites in Iowa (Linhart et al. 2012). They used independent streamflow quantile regressions employing 113 streamflow gauges with at least 10 complete years of daily mean streamflow, without regulation or diversion. The method uses a regression analysis between discharge for a particular quantile and independent variables that describe physical and climatic basin characteristics. In total, they explored 57 physical and climate basin characteristics as explanatory variables. They used the stepwise selection method to identify the significant independent variables in the regression analysis. After filtering the independent variables, they performed an ordinary least square (OLS) regression analysis and weighted least squares (WLS) to determine the best set of regression equations. They conducted this regression analysis with an imposed limit of three independent variables for each equation to minimize overfitting of the regression models. The WLS multiple linear regression’s advantage is that it accounts for differences in record length. In addition, the study uses left-censored regression for streamflows with a 99% probability of exceedance (low flow), with a censoring threshold of $0.00283{\mathrm{m}}^3/\mathrm{s}$ ($0.1{\mathrm{ft}}^3/\mathrm{s}$), and the goal of reducing the standard error in the regression analysis. They fitted 15 equations for 15 quantiles: 0.01, 0.05, 0.10, 0.15, 0.20, 0.30, 0.50, 0.60, 0.70, 0.80, 0.85, 0.90, 0.95, and 0.99. Each equation includes up to three independent variables, accounting for the following: drainage area; mean annual precipitation; percent area underlain by hydrologic soil type B, C, and D; relative stream density; hydrograph separation; streamflow-variability index; and a measure of the steepness of the slope of a duration curve. The web-based tool StreamStats (Ries et al. 2009) implements this method.

Although the rigorous statistical analysis presented in Linhart et al. (2012) provides convenient estimations of 15 streamflow quantiles that allow us to reconstruct the FDC, the estimation of FDC could be improved and compacted for easier implementation in larger domains. We identified three main drawbacks in Linhart et al. (2012). First, the changes to the physical and climatic variables used to describe different streamflow quantiles make it difficult to track the overall role of an explanatory variable in the entire streamflow distribution. Second, the fitting of independent equations for specific streamflow quantiles ($Q_p$) with different explanatory variables does not guarantee a monotonically decreasing trend in the FDC ($Q_{p_i}>Q_{p_j}\text{with}{p}_i<p_j$), with respect to the probability of exceedance $p$, which implies that $Q_{p_i}<Q_{p_j}$ for $p_i<p_j$ could occur. Third, the method does not capture the streamflow seasonality described at a monthly scale. Addressing these three shortcomings is the main contribution of our work. The FDC we develop below enables meaningful communication of streamflow conditions to the public in Iowa.

This paper is organized as follows: Section “Research Objectives” describes the specific research objectives of this study. Section “Materials and Methods” describes the information and procedures used to estimate FDCs at ungauged sites, including the regression analysis, the selection of the explanatory variables, the validation test, and the final construction of a piecewise continuous monotonic FDC. In section “Results and Discussion,” we discuss our results and examine the outcomes of our FDC estimations for the state of Iowa at ungauged sites. Finally, we synthesize our main results in the closing comments.

Our primary objective is to estimate historical-annual and historical-monthly FDCs at ungauged sites in Iowa. We define the following specific objectives in the construction of FDCs: (1) estimate FDCs with a single explanatory variable for an easy implementation; (2) fit a continuous and monotonic FDC for a representative range of quantiles; and (3) estimate FDCs for both historical and monthly scales to capture the seasonal streamflow variability. To pursue our objectives, we hypothesize that the long-term mean annual streamflow ($\overline{Q}$) captures most of the hydrological heterogeneities of streamflow quantiles across space and watershed scales. We explore this hypothesis recognizing that the mean annual flow integrates the spatial variability of precipitation and evapotranspiration across the watershed through the long-term water balance relationship. We follow the work by Poveda et al. (2007), who uncovered the scaling connections between peak flow quantiles and mean annual flow. While Poveda et al. (2007) made the connections with peak flow quantiles rather than streamflow quantiles, their results suggest that the mean annual flow contains valuable information to describe different streamflow frequencies.

We calculate the empirical nonparametric historical-annual FDCs for gauged sites by sorting the daily streamflow record of $n$ values from the largest to the smallest and assigning each position a rank ($R$). Each ranked position is associated with a probability ($p$), which represents the chance that the sample exceeds or equals the corresponding streamflow. The cumulative probability $p$ associated with $R$ can be calculated with different approaches (Beard 1943; Blom 1958). We calculated $p$ using the Weibull plotting position (Weibull 1939), which is approximately unbiased for quantiles:

This study uses daily mean streamflow records from 114 streamflow gauges not affected by regulation or diversions (Fig. 2) and located across Iowa up to September 2014, employing only complete water years. Data flags provided by the USGS help identify the sites with regulation or diversion. We grouped these gauges into two different data sets for different analyses. The first, called the calibration set, included 74 streamflow gauges with more than 30 years of records; we used these in the regression analysis section. The remaining 40 streamflow gauges were used for a validation test and have streamflow record durations ranging from 10 to 30 years. (See Supplemental Data for some climatologic features and duration of records for the calibration and validation gauge sets.)

To construct the historical-annual FDC, we estimate the streamflow quantiles with the complete daily streamflow record for each streamflow gauge using Eq. (1). For the historical-monthly FDC, we divided the streamflow records by months and again used Eq. (1). The streamflow quantiles are defined for probabilities of exceedance ranging from 0.01 to 0.99 with a step of 0.001, for a total of 981 quantiles. By selecting streamflow gauges with more than 30 years, we were able to use the small step (0.001) between frequencies. In the case of monthly FDCs, we grouped the data by months; therefore, there is a significant reduction in the number of streamflow observations. Thus, the streamflow gauges in the calibration set must have more than 30 years of records to guarantee at least 980 daily streamflow observations per month. In cases where the data are not sufficient, Eq. (1) does not give frequency values for the exact quantile in the range of 0.01–0.99 with step of 0.001. Therefore, we use a linear interpolation between frequencies obtained from Eq. (1) to get the desired streamflow quantiles estimate for every streamflow gauge. We censored streamflow records with zero flow values at $0.00028{\mathrm{m}}^3/\mathrm{s}$ ($\approx 0.01{\mathrm{ft}}^3/\mathrm{s}$) to avoid problems with the logarithmic transformation in the regression equations described in the next section.

Fig. 3 shows the $R^2$, MAPE, SMAPE, and MSAR for the 981 independent streamflow quantile regressions for the historical-annual FDC using the SSA, SSQ, MSA, and MSQ models. The model selection criteria $R^2$ suggest that the models based on the $\overline{Q}$ (SSQ and MSQ) explain more the total variation of the streamflow quantiles than the models based on the $A$ (SSA and MSA). Although the SSQ and MSQ models have similar values of $R^2$, the MAPE, SMAPE, and MSAR display lower errors for the MSQ model than the SSQ model; hence, we select the MSQ model as the best basis to construct the historical-annual FDC.

In the four models, the streamflow estimations for low probabilities of exceedance (high flows) fit better in comparison with the streamflow with higher probabilities (low flows). Here, it is interesting to observe that the $p$ of the $\overline{Q}$ varies between 0.2 and 0.3 (see Supplemental Data), which is close to the range where the regression models show better performance. However, moving away from this range toward high frequencies (low flows), the performance decreases. In general, we can argue that the decreasing in performance for low flows can be attributed to the fact that $A$ or $\overline{Q}$ do not fully capture the streamflow variability in this range of frequencies. We found similar results for the historical-monthly FDCs, which show a better performance for the MSQ model. Fig. 4 shows the statistic for the historical-monthly FDCs using the MSQ model. The error metrics change from month to month. Based on the $R^2$, the months with lower performance correspond to the winter season. Similarly, the other error metrics show that the months in fall show lower performance, with higher MAPE, SMAPE, and MSAR values for the months of September and October. Certainly, the MSQ model fits better than the other models explored in this study. Fig. 5 shows the mean error and the root mean square error (RMSE) for the MSQ model for the historical-annual FDC and the 12 historical-monthly FDCs.

The MSQ model works much better than the MSA model because the MSQ model is region independent in the domain of the state of Iowa, while the model MSA is regionally dependent. For an example of this idea, we normalized the $Q_p$ for the 981 quantiles with respect to its $A$ or $\overline{Q}$ [Fig. 6(a)]. This plot allowed us to visualize the presence of regional dependence. A visual inspection grouping the data by 4-digit hydrologic unit codes (HUCs) reveals spatial clusters between the normalized variable and the $A$. Fig. 6(b) shows an example: we fixed $p=0.1$ and plotted the normalized variables against the $A$. Here we identified a clear regional dependency in $Q_{0.1}/A$ by observing that most of the rescaled streamflow values for Region 1 (black dots) were above 0.016; values for region 2 (gray squares) are between 0.011 and 0.016; and values for region 3 are below 0.011. Note that these clusters are not evident in the $Q_{0.1}/\overline{Q}$ plot. Fig. 7 shows the location of the three regions based on the clusters found in the relation $Q_p/A$ and $A$. Although the identification of these clusters is a function of the frequency evaluated and its delineation can be redefined with clustering techniques, we are satisfied with the visual identification of the three regions shown in Fig. 7 because they illustrate how $Q_p/A$ depends on the region, while $Q_p/\overline{Q}$ does not.

We used the Potthoff method explained in Section 3.2 to evaluate whether the scaling parameters of the MSA and MSQ models were changing significantly in the three regions identified above. For the MSA model, we found that the paired regions 1–3 and 2–3 differ in the ${\theta }_p$, with p-values of less than 0.05 for low and high frequencies ($p<0.1$ and $p>0.9$). For the regions 1–2, we found p-values of less than 0.05 in probabilities of exceedance less than 0.1. On the other hand, the same analysis for $\overline{Q}$ as explanatory variable (MSQ model) indicated that for all the frequencies, the p-value is greater than 0.05; therefore, for the MSQ model we cannot reject the hypothesis that the ${\theta }_p$ is the same along the three regions identified in Iowa. Regarding the ${\alpha }_p$, we found that the p-values for the MSA model in the regions 1–2 have a wide range ($0.2<p<0.7$), with p-values of less than 0.05. The sets of regions 1–3 and 2–3 exhibits p-values of less than 0.05 in low and high frequencies ($p<0.1$ and $p>0.9$); thus, we concluded that the ${\alpha }_p$ are significantly different in these frequencies. On the other hand, the test for the ${\alpha }_p$ using the $\overline{Q}$ as explanatory variable exposed p-values above 0.05 for the three sets of regions. Therefore, the hypothesis that the MSQ model is region independent cannot be rejected. In conclusion, the MSA model with the $A$ as explanatory variable exhibited a significant regional dependency in Iowa.

We replicated the hypothesis test for monthly scale FDCs, with a total of three pairs of regions, 12 months, and 981 quantiles. We found that there is not a significant signature of regional dependency when the $\overline{Q}$ is used as explanatory variable for the regression analysis at monthly scale. However, for the MSA model we found large ranges for diverse months in which the p-value is less than 0.05. Hence, we can determine that the MSA model also presents a regional dependency at monthly scale, with a prevalence in low and high flows. The MSQ model does not exhibit such dependence (see Supplemental Data for the p-values at annual and monthly scales).

Regarding the validation test, Fig. 8 shows the error metrics for quantile estimations at 40 streamflow gauges for the historical-annual FDCs using the four models explored in this study and the standard method obtained from StreamStats based on Linhart et al. (2012). The MAPE results show that the MSQ model performed best in estimating streamflow quantiles among the four proposed models. StreamStats outperforms the MSQ model in low streamflows. The MAPE results show that the MSQ model performs in a similar manner in the range of frequencies from 0.01 to 0.6, and the MSAR indicates that the performance is comparable for the ranges 0.01 to 0.4. This result is consistent with the statistics shown in the regression analysis in which lower performances were obtained at low flows, with an increase in performance near the frequency range of 0.2 to 0.3. Of the four methods explored in this study, we observed that the MSQ model with multiscaling approach and explanatory variable $\overline{Q}$ is the best option to replicate some quantiles of the FDC. We replicated the same validation test for FDC at monthly scales; however, in this case, it was not possible to compare the FDC with StreamStats results because the USGS method only provides estimates for the historical-annual FDC. Fig. 9 shows the MAPE, SMAPE, and MSAR statistics at a monthly scale for the MSQ model. The results indicate that the historical-monthly FDC is well described in all the months with decreasing performance for low flows. The months of February, March, and April show a better model performance for most frequencies. On the other hand, the months with lower performance (July, August, and September) show the largest errors over the range of low flows. These results show the variability in time (months) and frequency (percentage of time exceeded) in the model performance.

With the MSQ model selected as the best model for the estimation of the historical-annual and historical-monthly FDCs, we fit the ${\alpha }_p$ and ${\theta }_p$ of the MSQ model to a piecewise continuous function with respect to the $p$. The results show that the sum of squared errors and the RMSE in these regressions are small (Table 1). Therefore, the estimated scaling parameters ($\widehat{\theta }$ and $\widehat{\alpha }$) from the piecewise continuous function can be consider an accurate representation of ${\theta }_p$ and ${\alpha }_p$ (Fig. 10). We replicated this analysis at the monthly scale and again had the low sum of squared errors and RMSE (Table 1), which allowed us to calculate piecewise continuous functions for the estimation of $\widehat{\theta }$ and $\widehat{\alpha }$ for each month. We synthesized the final historical-annual and historical-monthly FDCs based on these piecewise continuous functions as a set of equations for the three different ranges of $p$ (Tables 2–4). Based on the streamflow gauges used in this study, reasonable values of $\overline{Q}$ for the state of Iowa range from 0 to $200{\mathrm{m}}^3/\mathrm{s}$. We verified the monotonic property of these piecewise continuous FDCs in the range of $\overline{Q}$ between 0 and $200{\mathrm{m}}^3/\mathrm{s}$, checking the existence of negative derivatives on the final equation with respect to the probability of exceedance $p$. We confirmed that the monotonic property emerges because of the small steps between frequencies in the streamflow regression, and because of the use of a single explanatory variable in the regression analysis.

Our method for estimating FDCs can be characterized as follows: (1) the FDC is described by a scale-invariant power law; (2) we construct the power law using the multiscaling framework, with variation in the scaling exponent and intercept across frequencies; (3) the FDC captures regional nonhomogeneities in Iowa via the $\overline{Q}$, which captures changes in precipitation and evapotranspiration across the domain; (4) the FDC is region-independent in the sense that we can implement the method anywhere in the state of Iowa; (5) we can calculate the FDC using a piecewise continuous function based on the $\overline{Q}$ and $p$; and (6) we can estimate the FDC for historical annual and monthly scales, which allows us to represent more accurately the seasonal variability in the streamflow distribution.

The proposed methodology provides reasonable FDC estimates with respect to estimates from streamflow observations at historical annual and monthly scales. Our method represents an engineering solution to the problem of communicating to the public streamflow values from arbitrary locations in a hydrologically meaningful context. The historical-annual FDC comparison with respect to the FDC estimated with StreamStats shows that our approach demonstrates a similar performance in streamflow quantiles for probabilities of exceedance less than 0.6. We found that the errors in our FDC estimations are larger for low flows with $p>0.6$, but well within the margin of error of many engineering applications. This methodology is valid for unregulated rivers within the state of Iowa, excluding the Mississippi River and Missouri River, which are heavily regulated. Reasonable values of $\overline{Q}$ for Iowa range from 0 to $200{\mathrm{m}}^3/\mathrm{s}$ across all watershed scales.

A few caveats on our analysis include the following: First, the results of performance in the regression estimates for the calibration and the validation sets suggest that the variance in our estimates is higher in low flows, and the StreamStats methodology shows better performance at the annual scale for these low flows. Second, the proposed method uses the mean annual flow and therefore, the performance of the FDC estimation depends on the accuracy of its estimation. Third, other basin characteristics or hydrological processes not captured by the mean annual flow drive the description of variability in low flow quantiles. This can be explained by noting that the mean annual flow has a probability of exceedance between 0.2 and 0.3; therefore, the information contained in $\overline{Q}$ in the representation of the streamflow distribution decreases toward frequencies closer to 1 (low flows). Finally, The suitability of translating streamflow predictions from real-time hydrologic distributed models to streamflow frequencies relies on the performance of the hydrologic model to represent the observed streamflow distribution. If the outputs of the hydrologic model are biased, the streamflow classification based on the FDC will be biased as well.

Although we developed this study for Iowa, our methodology can easily be extended to other regions. This makes it an attractive approach for the classification of streamflow outputs from regional or national hydrologic distributed models. Mapping streamflow frequencies in a basin or a region helps to interpret streamflow conditions at ungauged sites.

Tables with the description of the streamflow gauges used on the regression analysis and validation test for the SSA, SSQ, MSA, and MSQ models, and tables with p-value for testing of the null hypothesis to determine whether the scaling parameters (${\alpha }_p$ and ${\theta }_p$) for different regions differ significantly are available online in the ASCE Library (www.ascelibrary.org).

The authors thank the Iowa Flood Center and the Iowa Energy Center for supporting this study. We also wish to thank the anonymous reviewers and editors of our manuscript for their insightful comments and corrections.