Purpose and Scope

The purpose of this report is to compare measurements of T_c and related runoff characteristics to estimates of T_c and related runoff characteristics made using accepted methods. Objectives of this report are to:

Precipitation

The 2018 water year was unusually wet at all sites, ranging from 53 to 59 in., compared to the 1991–2020 normal precipitation of 46 in. at a nearby long-term weather station at Charleston, W. Va., Yeager Airport (table 2; National Centers for Environmental Information, 2021). The 2018 calendar year was the wettest year on record at Charleston, W. Va., with a total of 67 in. of precipitation (National Oceanic and Atmospheric Administration, 2018). Total annual precipitation for the 2019 water year was moderately drier than normal, ranging from 42 to 43 in. at all sites. September 2019, however, was unusually dry, which accounted for most of the precipitation deficit for the year (National Centers for Environmental Information, 2021). Total precipitation for September 2019 ranged from 0.10 in. at Ripley to 0.42 in. at Wallback, in comparison to the 1991–2020 September normal precipitation at Charleston, W. Va., Yeager Airport of 3.46 in. The 2020 water year, like the 2018 water year, was unusually wet at all sites. Rainfall totals for the partial year of 2020 ranged from 39 to 50 in., but some of the differences among sites were caused by differences in the duration of data collection and dates when data collection was discontinued (fig. 3). Data collection ended on June 17, 2020, at Ripley, June 26, 2020, at Fairplain, June 30, 2020, at Hico, and July 2, 2020, at Wallback. Also, some annual and monthly precipitation totals include periods of missing record, which were not adjusted by estimation. The study period’s rainfall totals were normal to wetter than normal overall, except for September 2019.

As would be expected in a generally wet period, many days had storms and substantial total precipitation. Daily precipitation exceeded 2 in. between 1 and 4 times at all sites (table 3). Daily precipitation exceeded 1.5 in. between 9 and 11 times at all sites and exceeded 1 in. between 24 and 36 times at all sites. The number of days with substantial total precipitation varied spatially but was also affected by slight differences in study duration. Even though the closest pair of sites, Ripley and Fairplain, were only about 7.5 miles apart, the number of days with 1-inch or greater rainfall varied—Ripley had 26 days and Fairplain had 35. Four of the days when Fairplain received more than an inch of precipitation were days when the rain gage at Ripley was not operational or had been discontinued, compounding the differences that were to be expected because of spatial variance.

Precipitation totals for selected durations for each site were compared to intensity-duration-frequency values commonly used in design (Bonnin and others, 2006). The storm on June 14, 2020, at Fairplain yielded 0.57 in. of rain in 5 minutes and 1.54 in. of rain in 30 minutes. This storm exceeded the 10-year return interval for durations of 5, 10, 15, and 30 minutes; the largest 15-minute total during that storm was between the 50- and 100-year return interval for that duration (table 4). For the longer durations that were compared (60 minutes and 2, 3, 6, 12, and 24 hours), however, the largest totals were only between the 1- and 2-year and 2- and 5-year recurrence intervals. At Fairplain, five storms exceeded the 1-year recurrence interval for durations of 5, 10, and 15 minutes, and three storms exceeded the 2-year recurrence interval for these same durations.

At Ripley, the storm of July 7, 2019, yielded 0.85 in. of rain in 10 minutes and 1.33 in. of rain in 30 minutes (table 4). This storm was between the 10- and 25-year recurrence intervals for 10-minute durations and the 5- and 10-year recurrence intervals for 5-, 15-, and 30-minute durations. At Ripley, the largest storms for durations of 1, 2, 3, 6, 12, and 24 hours were between the 1- and 2-year and the 2- and 5-year recurrence intervals. Either 2 or 3 storms exceeded the 1-year return interval for 5- through 60-minute durations at Ripley, and either 1 or 2 storms exceeded the 2-year recurrence interval for these same durations.

Hico received some intense storms during the study. At Hico, the storm of June 5, 2020, yielded 0.48 in. of rain in 5 minutes and was between the 5- and 10-year and 25- and 50-year recurrence intervals for the durations of 12 hours or less that were available for comparison (table 4). Hico was the only site where a storm with a duration >30 minutes exceeded the 5-year recurrence interval. Either 3 or 4 storms exceeded the 1-year recurrence interval for durations of 1 hour or less, and 1 to 2 storms exceeded the 2-year recurrence interval for the same durations.

Storms at Wallback were somewhat less intense, or had a lesser sustained intensity, than at the other sites. The maximum storms for all durations that were compared at Wallback were between the 1- and 2-year and 2- and 5-year recurrence intervals. Either 5 or 6 storms exceeded the 1-year recurrence interval for durations of 15 minutes or less. Of these storms, 4 exceeded the 2-year recurrence interval for 5 minutes, 3 exceeded the 2-year recurrence interval for 10 minutes, but only 1 exceeded the 2-year recurrence interval for 15 minutes.

Streamflow

Streamflow data collection began November 1, 2017, at Hico; January 11, 2018, at Ripley; February 6, 2018, at Wallback; and February 7, 2018, at Fairplain. Data collection ended June 25, 2020, at Fairplain; June 17, 2020, at Ripley; June 30, 2020, at Hico; and July 2, 2020, at Wallback.

Dry channels and zero flows were observed at all four streamgages during field visits. Only one streamgage, Hico, was able to accurately record values of zero flow (table 5). At the other three streamgages, stage could be too low to reach either stage sensor in use at a given time. In this report, the minimum flow that could be measured accurately in each stream is referred to as the “detection level” for that streamgage. The minimum flows recorded at the other three streamgages were 0.02 cubic foot per second (ft³/s) at Wallback and Ripley and 0.15 ft³/s at Fairplain. The number of days when flow was above the detection level for part of the day ranged from 117 days at Fairplain to 851 days at Hico, whereas the number of days when flow remained above the detection level all day ranged from 7 days at Fairplain to 708 days at Hico (table 5). The low number of measured flow values at Fairplain was directly related to the high detection level of flow (0.15 ft³/s) needed to submerse the AV sensor deeply enough for measurements to be accurate.

When flows were high for several days at one streamgage, they generally were high at the other streamgages. However, the maximum flows for the study period were on different days for all four streamgages (fig. 3). The maximum flow for Wallback resulted from a series of storms in September 2018 that produced a peak on September 25 with a flow of 16.6 ft³/s and another peak on September 27 with a flow of 16.7 ft³/s. Thunderstorms in spring of 2020 produced the maximum flow for the other three streamgages. The maximum flow at Ripley (16.5 ft³/s), on April 8, 2020, was nearly twice the magnitude of the second largest peak 8.43 ft³/s recorded on May 29, 2020. The largest peaks at Fairplain were produced by different storms than at Ripley and were closer to each other in magnitude than the largest peaks at the other streamgages: 12.7 ft³/s on June 14, 2020, and 8.93 ft³/s, 9.74 ft³/s, and 10.2 ft³/s on June 22, 2018, February 16, 2018, and December 12, 2018, respectively. At Hico, the maximum flow (17.4 ft³/s), on June 14, 2020, was more than twice the magnitude of the second largest peak, 7.71 ft³/s on September 27, 2018.

The longest period with uninterrupted low flows at all streamgages was in August–September 2019. A period between August 23 and October 5 when no rain gage received more than 0.67 in. of rain resulted in sustained zero flows at Hico and flows below the detection level at the other three streamgages during September and early October 2019 (fig. 3). Flows increased at Hico and Wallback after rain in October 2019. Flows in the smaller streams at Ripley and Fairplain receded below the detection level earlier in August than in the two larger streams and also remained generally dry later until December.

At Hico, a recorder malfunction caused data recorded at 1-minute and 5-minute intervals on the backup recorder between August 31 and October 3, 2018, to be lost, so that only the subset of data transmitted through the GOES system was available for this period. Satellite-transmitted data were only available at 15-minute intervals, instead of 1-minute or 5-minute intervals. This September 2018 period included two large, extended storms and the second highest peak streamflow recorded at Hico. For precipitation, the 15-minute totals were accurate because the rain gage recorded tips throughout the interval and transmitted total precipitation every 15 minutes. The 1-minute precipitation subtotals, however, were lost, and information on precipitation timing was less precise than for any of the other data discussed in this report. For streamflow, not only was information on timing limited to 15-minute intervals, but the recorded value was the instantaneous stage reading from the NTRAN at the 15-minute reading, so that information on rises or falls in stage between those measurements was lost.

Hydrograph Time Metrics

Three hydrograph time metrics were used in this study to compute T_c: time to rise, time to recede to an inflection on the recession, and the time between the inflection on the hyetograph at the end of major precipitation (the precipitation inflection) and the inflection on the falling limb of the hydrograph (the recession inflection). The time from the precipitation inflection to the recession inflection is abbreviated in this report as PI-to-RI. These three time metrics represent the two primary conceptual definitions of T_c used in this report.

Time to rise is intended to represent the time needed for a particle of water to travel to the outlet from the most hydraulically remote part of the watershed (Hayes and Young, 2005). Time to rise is most likely to give a good estimate of T_c when the assumptions of the Rational Method are closest to being met in a small basin. Among these assumptions are that the basin area is <200 acres, streamflow peaks when the entire basin contributes stormflow, a storm that has a duration equal to T_c produces the highest peak discharge for a given precipitation intensity, and precipitation intensity is uniform for a storm that has a duration equal to T_c (McCuen, 2002).

The other two time metrics (as previously mentioned) can also be used to compute T_c when T_c is defined as the end of excess precipitation until the end of direct runoff or stormflow (McCuen, 2009). The two primary conceptual definitions of T_c used in this study are related if the rain in the most remote part of a basin when excess precipitation ends is also the last stormflow to leave the basin. In hydrograph analysis, the end of stormflow is taken to be the inflection on the falling limb of the hydrograph from steeply receding flows, representing stormflow, and gradually receding flows, representing mixed base flow and interflow (Natural Resources Conservation Service, 2010). PI-to-RI is a computational definition of T_c that directly corresponds to this conceptual definition (Hayes and Young, 2005; McCuen, 2009).

Time to recede has been used previously as a surrogate for time from the end of excess precipitation to the end of stormflow (Hayes and Young, 2005). The assumption underlying this surrogacy is that in small basins, peaks closely follow the end of excess precipitation. One benefit of using this computational method is that it relies only on the hydrograph and, therefore, one environmental sensor. This was most important in studies done when the best available precipitation and stage recorders used separate clocks that sometimes became unsynchronized, causing errors in time measurement to be an important source of overall error. In this study, the rain gages and stage sensors shared a common recorder that was synchronized with GPS satellites, which allowed a comparison between time to recede and PI-to-RI that was not affected by possible errors in time measurement. Hydrograph time metrics were computed using inflections that were identified on graphs that represent the beginning and end of hydrograph events. In this report, the phrase hydrograph event describes an individual rise, recession, or PI-to-RI that was identified as part of a storm graph. Graphs were plotted in R (R Core Team, 2021) using the package “lattice” (Sarkar, 2021). For most storms, a fixed, site-specific flow range was assigned as a common scale for graphs. The largest storms from all streamgages, those that would not plot properly on the fixed scale for the streamgage in question, were analyzed as a group and plotted using the lattice autoscaling algorithm. The maximum discharge from the 50th ranked storm at a streamgage was plotted on the graphs as a reference line. Inflections were identified onscreen using “lattice,” and codes developed as unique identifiers for the inflections were copied and pasted into a Microsoft Excel spreadsheet to associate them with time and other relevant information. The time at the beginning of each hydrograph event was subtracted from the time at the end of the event in Microsoft Excel.

All three types of hydrograph events were identified at and included in analyses for all four streamgages. However, hydrograph events were not identified for all storms. When a complex storm had multiple rises and recessions in precipitation that met the criteria for a continuous storm, multiple hydrograph events were identified, and hydrograph time metrics were computed multiple times. Hydrograph events from different times during a long, complex storm were judged to represent different antecedent conditions. The amount of water in storage in the basins at the beginning of stormflow is the most important consideration concerning antecedence, not the amount of water in storage hours or days earlier.

In general, 7-minute running averages were used to compute time metrics for Fairplain, Ripley, and Wallback—the streamgages with flow data collected at 1-minute intervals. Running averages were computed as the unit value at the nominal measurement time, the three unit values before the nominal measurement time, and the three unit values following the nominal measurement time. High flows inside culverts were turbulent, and standing waves moved back and forth across the AV sensors. The 1-minute interval data were affected by the turbulence and therefore noisy, which masked patterns. Running averages of several durations, from 3 to 9 minutes, were examined, but 7-minute running averages were determined to be the best compromise among resolution, responsiveness, and smoothing.

At Hico, the flow data collected at 5-minute intervals from the NTRAN were not smoothed and were used as the flow record throughout the study period at this streamgage. At Wallback, when 1-minute interval data from the AV sensor were available, they were used with smoothing. Periods of missing data from the AV sensor, however, used 5-minute interval data from the NTRAN, which was used without smoothing. At both streamgages, the use of 5-minute interval data instead of 1-minute interval data introduced some uncertainty when determining time metrics because the greater the period between measurements meant a greater chance of missing the true peak or inflection. At Wallback, the 5-minute interval measurements collected by the NTRAN had many of the same problems seen in the 1-minute interval measurements from the AV sensor, but without the dense cloud of points to smooth that increased uncertainty in the timing of inflections in hydrographs from this streamgage.

There were 14 exceptions to the 7-minute averaging rule when 1-minute data were available. In 6 storms at Ripley and 8 storms at Fairplain, intense precipitation at the beginning of a storm caused a rapid initial rise in flow less than 7 minutes after precipitation began and, therefore, before enough values were available to compute a running average. For these storms, three 1-minute observed values were added at the beginning of stormflow so that time-to-rise calculations would be accurate (Messinger and Scott, 2024; table 1).

A few times to rise at Fairplain might be slightly, but incorrectly, faster than at other streamgages because the streamgage had a high threshold for minimum flow, and the time needed to bring flow above the detection level is not included in events from this streamgage. Times to rise at Hico might be slightly slower than the other streamgages because flows went all the way to zero, and a few times to rise might include a measurement at the beginning of a hydrograph that would not have been recorded at the other streamgages. Computation of times to rise began at the absolute beginning of stormflow only a few times; notable exceptions were the storms for which the 7-minute averages were supplemented with unit flow values at the beginning of the storm.

The rise and recession metrics were unreasonably long for nearly all storms if they were computed from either the initial increase in flow at the beginning of a storm to the absolute peak or from the absolute peak to pre-stormflow magnitudes. The time from initial rise to absolute peak was not determined for all storms but was more frequently measured in hours or even days than in minutes. Time to rise was highly inconsistent among storms because the absolute peak of a storm could be delayed depending on several factors including how many spates of rain came, time between the spates, and variations in precipitation intensity. Times to rise measured in hours or days were judged to be unrepresentative of how long water took to move through the small basins of interest in this study.

Instead of treating the entire time that flow rose during a complex storm or series of storms as representative of T_c, time to rise and time to recede were determined from inflections on the rising and falling limbs, respectively, of hydrographs (fig. 5; table 3 in Messinger and Scott, 2024). A rise, for purposes of delineating a hydrograph event, was counted as the time where flow clearly inflected upward in response to a spate of intense rain until a point where the slope of that rise caused by that spate of rain levelled or decreased. To be included in analyses, the rise needed to be (1) steady, (2) clearly the result of a specific spate of rain, (3) visually distinct, and (4) representative of a meaningful change in flow magnitude. A distinct rise as a section of a rising limb of a hydrograph should represent T_c because the initial rise represents the arrival of the first stormflow and the slope changes at the end of the rise because water from that spate has traveled from the most remote part of the basin to the outlet. Time to recede was assessed like time to rise, with the additional constraint that when repeated or sustained rain fell during the recession and caused flow to rise or remain stable, the event was excluded. The time between PI-to-RI was defined as the difference between a (1) final inflection on the hyetograph before a visually distinct storm peak and (2) the inflection on the receding limb of the hydrograph from a steep decrease in stormflow to a part of the hydrograph with a flatter slope. The inflection in the recession represents the point where stormflow from the most hydraulically remote point in the basin passes through the outlet. Generally, this recession inflection was selected at the first unit flow value with zero change in flow that followed the peak. Exceptions included extended hydrographs, when flow remained near peak due to continued steady rain; some complex hydrographs, when the basin received rain before the hydrograph appeared to reach an inflection; hydrographs when excessive turbulence or surges in flow at the recorder masked changes in flow, even after smoothing; and peaks that represented generally minor rises that exceeded the flow threshold because of high initial base flow at the beginning of stormflow. All precipitation inflections were chosen visually. The inflection in the plot of cumulative precipitation, referred to as “precipitation inflection” on figure 5, marks a decrease in storm intensity and is taken to represent the end of excess precipitation (fig. 5). Excess precipitation is defined as precipitation that exceeds available detention storage in the basin and, therefore, results in stormflow (Black, 1991). The primary quality-assurance metric for precipitation inflections was the amount of precipitation received during the storm segment after the inflection. Values for post-inflection precipitation ranged from 0 to 0.28 in. (table 4 in Messinger and Scott, 2024). Hydrograph events that included post-inflection precipitation exceeding 0.10 in. were identified and ultimately included in analyses.

Hydrograph events were included in analyses if (1) their maximum values were large enough to be among the 50 largest peaks at each streamgage; (2) the event was not affected by snow; (3) the slope of the rise or recession was consistent and steady during a relevant part of an event; (4) the flow record during the hydrograph event was complete without estimated values; (5) the event in question represented a meaningful change in flow; (6) rain that continued while flow receded was minimal and did not appear to interrupt recessions with secondary rises in flow; and (7) the changes between successive unit flow measurements during the hydrograph event were either primarily increases or decreases, indicating that it was primarily stormflow, in contrast to events that were largely stable, which characterized the minor rises and recessions that exceeded flow thresholds only because they began when base flow was already high. Quality-assurance metrics were developed and computed to show how well the hydrograph events met these criteria (Messinger and Scott, 2024). Hydrograph events provisionally identified through inspection of graphs, but that failed to meet quality-assurance criteria, were not included in summary statistics or further analyses.

The quality-assurance metrics that confirmed that hydrograph events had large enough maximum flow values, were affected by snow, or had a complete flow record applied to all three types of hydrograph time metrics. Thresholds for flows were used because the purpose of the study was to characterize high flows. Snow-affected storms were excluded because limited information was available for when the snow melted in the basins or for how much an effect the melted snow had on runoff timing. Days when snow was recorded by the National Oceanic and Atmospheric Administration climate station nearest each streamgage (either Charleston, Beckley, or Gassaway) were identified. Precipitation records for the next storms after snow was recorded were assumed to include snowmelt and excluded from analysis. A complete or mostly complete flow record was needed to identify inflections on hydrographs because estimated values were made by linear interpolation, making them unsuitable for identifying inflections.

Quality-assurance metrics for rises and recessions included the number of increases in unit flow values during a recession, the number of decreases in unit flow values during a rise, and the number of unit flow values with zero change during both types of events. For computing these quality-assurance metrics, a unit flow value was an individual unit of flow either at a 1-minute increment in the 7-minute running average for streamgages and periods where flow data were available at 1-minute increments or a 5-minute increment for the streamgages and periods for which flow data were available at 5-minute increments. Recessions could have no more than three unit rises, rises could have no more than three unit decreases, and no more than 25 percent of unit value changes in flow equal to zero.

A meaningful change in flow was evaluated by comparing the initial and final unit value of flow for each event that was delineated (Messinger and Scott, 2024). This metric was also used to determine if the change in flow was sufficient for inflections to be distinct throughout the hydrograph event. Meaningful changes in flows increased by 25 percent or more during a rise, decreased by 10 percent or more during a recession, and had to equal or exceed 25 percent of the maximum flow during a PI-to-RI event.

In general, hydrograph events were subjectively easier to delineate from simple hydrographs than from extended or complex hydrographs. Events from simple hydrographs were more consistent with delineation criteria than events delineated from extended hydrographs, but about the same in consistency as complex hydrographs (Messinger and Scott, 2024). For simple hydrographs, 13 of 51 provisionally identified events were inconsistent with delineation criteria. For extended hydrographs, 24 of 51 provisionally identified events were inconsistent with delineation criteria, whereas 38 of 148 provisionally identified events from complex hydrographs were inconsistent with delineation criteria.

Judgments made about how to treat quality-assurance metrics for hydrograph events affected T_c computation at all streamgages. Metrics were developed for this project after initial graphical delineation of hydrograph events. The metrics were developed to quantify the criteria used to delineate events and used to eliminate events that analysts were not fully confident met delineation criteria. Several of the events that were provisionally delineated but later excluded were inconsistent with multiple criteria.

Time of Concentration

All the time metrics used to compute T_c varied substantially among storms at all streamgages (table 6). All three metrics were slower at Hico than they were at any of the other streamgages. At Hico, average time to rise and time to recede were 34 and 32 minutes, respectively. The average time from PI-to-RI at Hico, 38 minutes, was the longest for any of the characteristics at any of the streamgages. Hico had the most variation of any streamgage in its time to rise and PI-to-RI values; PI-to-RI had the greatest range of values of any time metrics at any streamgage (fig. 6).

Wallback had the second slowest time metrics of all the streamgages. Average time to rise and time to recede were similar, 23 and 25 minutes, respectively. As at Hico, the average time from PI-to-RI for Wallback was greater than the time to rise or time to recede, 32 minutes. Time metrics at Wallback varied less than Hico, but more than Fairplain and Ripley.

Time metrics were similar at Fairplain and Ripley. At these two streamgages, average time to rise was 18 and 19 minutes, average time to recede was 14 and 16 minutes, and average time from PI-to-RI was 22 and 27 minutes, respectively. Ripley had the smallest standard deviation in time to rise and time from PI-to-RI among all streamgages, whereas Fairplain had the smallest standard deviation in time to recede.

Hydrograph events that had been provisionally delineated but failed to meet the quality-assurance criteria were examined to see if they were systematically different from those that met the criteria (table 6). Some differences were negligible or slight. At Fairplain, average time to recede was identical, and PI-to-RI was 3 minutes shorter for hydrograph events that failed to meet quality assurance criteria than for those that met criteria. At Ripley, average time to rise was 1 minute longer for hydrograph events that failed to meet quality-assurance criteria than for those that met criteria. At Wallback, average time to recede was 1 minute shorter for hydrograph events that failed to meet quality-assurance criteria than for those that met criteria. Other differences were greater, and for them, the differences between the two groups of hydrograph events were inconsistent in direction. At Fairplain, time to rise was 8 minutes longer for hydrograph events that did not meet quality-assurance criteria than for those that met criteria. At Ripley, PI-to-RI was 6 minutes shorter, but time to recede was 9 minutes longer for hydrograph events that did not meet quality-assurance criteria than for those that met criteria. Time to rise (13-minutes difference) and PI-to-RI (27-minutes difference) at Wallback, however, were both substantially longer for storms that failed to meet criteria, and the storms that caused most of the difference between the categories were unrepresentative of the largest storms at this streamgage. The difference in time to rise at Wallback was primarily because of a 90-minute rise that was affected by snow during a storm that began on January 23, 2019, and the difference in PI-to-RI was because of a 65-minute event that included more than 25-percent unit flows with zero change and little overall change in flow, during a storm that began on December 16, 2019.

At Hico, storms that were excluded from the T_c summaries were generally longer than those that met all criteria for inclusion. These storms did not meet the criterion that fewer than 25 percent of unit flows had zero change (10 of 57 provisionally identified events) and included some major storms. Among excluded storms, average time to recede was only 3 minutes longer than among included storms, but time to rise was 17 minutes longer and PI-to-RI was 127 minutes longer. Storm hydrographs at Hico could be separated into two distinct groups, those that rose quickly and those that rose slowly. Many of the slow rises started with either high base flow or were caused by steady light or moderate rain. Some of them, though, were generated by sustained hard rain, including the second largest peak flow measured at this streamgage, 7.71 ft³/s on September 27, 2018. Slow rises frequently rose and receded by less than the resolution of the water-level recorder during a time step between measurements, which was expressed as successive unit flow values with zero difference. The fast rises resulted from intense rain, and sometimes they were followed by a smaller, broader secondary peak that resembled the slow rises. Two storms from September 2018, for which only 15-minute data were available, were examples of storms with intense early rain (0.36 in. or 1.44 inches per hour (in/hr) for a storm that began on September 8, 2018, just before midnight, and 0.40 in. or 1.60 in/hr for a storm that began on September 10, 2018) that resulted in a small initial rise followed by broad, sustained secondary peaks that exceeded the initial peak (fig. 7). These major storms, and two other major storms from the same period (a second storm segment on September 9 and a storm on September 27), were not included in the T_c computations because the 15-minute precipitation data lacked enough resolution to delineate hydrograph events that were comparable to others in the study. If the events from these storms that met the other criteria were included, however, they would change the average times for PI-to-RI from 38 to 64 minutes and time to rise from 34 to 44 minutes. Time to recede would be unchanged, because no events were delineated for these storms that met other criteria. The largest storm at Hico, on June 5, 2020, was a quick storm with intense rain that caused a large immediate peak; the peak had a time to rise of 10 minutes, a time to recede of 15 minutes, and a PI-to-RI of 6 minutes. Either a quick, intense storm or a protracted storm could cause a large flood in this basin.

At most streamgages, differences among the time metrics were not statistically significant. The Kruskal-Wallis test was used to explore possible differences among the three time metrics within streamgages. Probability values (p-values) were 0.281, 0.012, 0.102, and 0.899 for Fairplain, Ripley, Wallback, and Hico, respectively; these values indicate that differences among the time metrics were significant for Ripley (values <0.05 being significant and values >0.05 being not significant), but not for the other streamgages.

Further testing for significant differences within streamgages was done with the paired Wilcoxon rank-sum test and Dunn’s all-pairs test, using Benjamini and Hochberg (BH) and Bonferroni false-discovery corrections to adjust p-values for the possibility of finding a falsely significant difference in repeated post-hoc hypothesis testing (Helsel and others, 2020). Most pairings of time metrics from the same streamgage were not significantly different from each other. The paired Wilcoxon rank-sum test with BH false-discovery correction was used to test for differences in all pairwise combinations of time metrics within streamgages. Two p-values were <0.05, both at Ripley, between time to rise and PI-to-RI (p=0.025) and between time to recede and PI-to-RI (p=0.011). P-values ranged from 0.15 to 0.87 for the other within-streamgage pairings of time metrics. Dunn’s all-pairs test with Bonferroni false-discovery correction was used to test if one time metric was different from the other time metrics at each streamgage. The same pairings of time metrics had p-values <0.05 as for the Wilcoxon rank-sum test: between time to rise and PI-to-RI (p=0.040) and between time to recede and PI-to-RI (p=0.012) for Ripley. For the other within-streamgage combinations of time metrics, p-values ranged from 0.11 to 1.00.

The same analytical approaches used to determine differences within streamgages were taken to explore possible differences among streamgages for the three time metrics. The Kruskal-Wallis test found that p-values were 0.017, 0.057, and 0.238 for time to recede, time to rise, and PI-to-RI, respectively. These values indicate that differences among streamgages were likely for time to recede, and possible for time to rise, but that they were unlikely for PI-to-RI.

Further testing confirmed that most pairings of streamgages were not significantly different from each other for any of the three time metrics, indicating that variance within time metrics at each streamgage exceeded differences among streamgages. According to paired Wilcoxon rank-sum tests with BH false-discovery corrections, p-values ranged from 0.14 to 0.85 for pairwise combinations of sites. Dunn’s all-pairs test with the Bonferroni false-discovery correction showed that time to recede was significantly different for one pairing of streamgages, Hico and Fairplain (p=0.036), but for all the other combinations of streamgages, p-values ranged from 0.077 to 1.000.

Hydrograph Time Metrics Compared to Storm Characteristics

The time metrics used to compute T_c were compared to total storm and storm segment precipitation, maximum streamflow during the storm segment, and precipitation intensity terms representing 1-minute, 5-minute, 15-minute, and 60-minute durations. Relations among time metrics and other storm characteristics were inconsistent within and among streamgages (figs. 8, 9, 10, 11; table 7). Precipitation intensity was the storm characteristic most often significantly correlated (p<0.05) with T_c. At least one of the precipitation intensity terms was significantly correlated with at least one time metric at three of the streamgages (table 7). All the significant correlations were negative, indicating that T_c was faster with the increasing values of storm characteristics.

At Hico, all three time metrics were significantly correlated (p<0.05) with one or more precipitation intensity terms; the significant correlation coefficients ranged from −0.570 to −0.985. At Wallback, PI-to-RI and time to rise were significantly correlated to all the precipitation intensity terms; the significant correlation coefficients ranged in value from −0.454 to −0.598. None of the other storm characteristics (total storm precipitation, storm segment precipitation, and streamflow) were significantly correlated with T_c at either Hico or Wallback, and the correlation coefficients included both positive and negative values (table 7).

At Fairplain, PI-to-RI was significantly correlated (p<0.05) with total storm precipitation and storm segment precipitation and all the precipitation intensity terms. At Ripley, PI-to-RI was significantly correlated with total storm precipitation and maximum streamflow for the storm segment. Correlations between time to recede and time to rise for Fairplain and Ripley included both positive and negative values and were generally weak; none were significant.

Stronger relations between time metrics and precipitation intensity at Hico than at the other streamgages may result from a greater range in T_c values computed at this streamgage, which is related to the differences in the two populations of hydrographs and how the basin responds differently to high-intensity and low-intensity precipitation. Wallback also had some short T_cs for some intense storms that resulted in large peaks. Ripley and Fairplain generally had a smaller range of T_c values than the two larger basins, though some of the shortest times at these two streamgages were associated with some of the most intense storms. Separate processes may have generated flows at Ripley and Fairplain in response to storms with distinct characteristics, but there was little variability in T_c between flows generated by different processes.

The general patterns between time metrics, precipitation, and peak flows may be unclear because of this study’s short period of record and the small number of large storms available for analysis. Peak flow magnitudes were weakly related to total precipitation and precipitation intensity among the storm segments (storm segments were created to help delineate hydrograph events; fig. 12). This weak relation is likely because total storm precipitation and precipitation intensity were only weakly related to each other, especially during moderate storms. At three streamgages, the largest peak streamflows were measured during the storms with the greatest precipitation intensity. If the storms with the greatest precipitation intensity were excluded from analysis, correlations between peak streamflow and precipitation intensity were weak. Among storms with moderate total precipitation, the most intense storms did not necessarily produce the greatest total precipitation, and either a short, intense storm or a long storm with moderate intensity might produce similar peak magnitudes. More moderate storms than large storms were included in the analysis. Only a few storms had spates of rain that exceeded the 2-year or 5-year recurrence interval for a given duration (table 4). Although these particular storms generally caused the largest peak streamflows at each streamgage, they may or may not have been representative of the full range of large storms.

Measured and Estimated Time of Concentration

Times of concentration that were estimated during site selection were compared to the measurements made during the study. The estimates are compared to time to rise, time to recede, and PI-to-RI. Comparisons are made among time metrics at the same streamgage and among streamgages.

Comparison Between Estimated and Measured Time of Concentration

All T_c estimates for all design flows for both estimation procedures were within the range of observations for their respective streamgage; design flows for only the 10-year flood are shown for clarity (fig. 13). All estimates were also within one standard deviation of the overall average for their streamgage, computed as the average of the averages of the three time metrics. Overall averages and averaged standard deviations of the T_c values for the three time metrics were 18 and 10 minutes for Fairplain, 21 and 7 minutes for Ripley, 35 and 24 minutes for Hico, and 27 and 11 minutes for Wallback, respectively. The general pattern contains a great deal of variation, however.

Fairplain had an average T_c value that matched T_c estimates from the initial procedure for frequent, moderate storms, whereas the T_c value for Ripley and Wallback had the closest match with estimates from the initial procedure for infrequent, large storms. Adjusting the initial estimate with regional bankfull-channel characteristics provided shorter and less accurate T_c estimates for Fairplain, Ripley, and Wallback. For Hico, however, adjusting the T_c estimates made them longer and closer to the overall measured averages.

None of the three time metrics had a consistent relationship with the estimates. Time to rise and time to recede were generally similar, both among streamgages and to each other, as demonstrated by the testing discussed in the “Time of Concentration” section. Averages for time to rise and time to recede were shorter than PI-to-RI for Fairplain, Ripley, and Wallback (fig. 13). For Fairplain and Ripley, average PI-to-RI was longer than both estimates; for Hico, average PI-to-RI was slightly shorter than both estimates; and for Wallback, average PI-to-RI was longer than the initial estimate but slightly shorter than the adjusted estimate. All T_c estimates were within one standard deviation of the mean for time to rise. Most estimates for time to recede were within one standard deviation of the mean, except for regionally adjusted estimates for Fairplain and initial estimates for Hico. Hico was different from the other basins. Average time from the PI-to-RI, which was longer than time to rise and time to recede, had the best match with estimates. The range of initial estimates for Hico were more than one standard deviation from the mean of both time to rise and time to recede. At Hico, however, the incomplete storms in September 2018 appeared to include hydrograph events with long enough times that, if the record had been complete and the events were included in analyses, could have lengthened the average, median, and interquartile range (IQR) for PI-to-RI and time to rise.

Average T_cs were largely derived from storms that had magnitudes less than those for the 1-year recurrence-interval storm (table 4). Correlations between individual hydrograph time metrics and precipitation intensity were weak everywhere but Hico and, to a lesser extent, at Wallback. At all streamgages, some of the shortest T_cs were for some of the most intense storms, which included some of the largest peak flows in the study (fig. 12). Estimated T_c values decreased slightly with increasing storm magnitude and intensity, but all estimates were developed for storm intensities that exceeded the 1-year recurrence interval.

Averages and standard deviations, both overall and of the separate time metrics, are useful analytical tools to compare broad patterns. However, these measures of variance and central tendency can be affected by outliers (Helsel and others, 2020). Some of the combinations of time metrics and streamgages have apparent outliers among small numbers of observations. The small number of observations make it difficult to assess whether the apparent outliers are truly rare or are rare only among observations from a short study period. The interquartile range (IQR) represents 50 percent of the observations in a group, the range between the 25th and 75th percentiles, and is a commonly used measure of variance that is resistant to outliers.

Predicted values for the 10-year storm, one of the key recurrence intervals in design, were within the IQR for 4 of 12 combinations of streamgages and time metrics (fig. 13). Two of these observations were for observed time metrics for Fairplain, one was for Wallback, and the other was for Hico. At Ripley, which was the streamgage where time metrics were significantly different from each other, the adjusted prediction was not within the IQR for any of the time metrics, but it was shorter than one IQR and longer than the other two. The adjusted prediction was shorter than the IQR for one time metric at one streamgage (PI-to-RI at Ripley), and it was longer than the IQR for seven combinations of streamgages and time metrics.

Comparisons between predictions and individual time metrics showed differences. Adjusted predictions of T_c were within the IQR of PI-to-RI for Fairplain and Wallback, longer than the IQR of observed PI-to-RI at Hico, and shorter than the IQR of observed PI-to-RI at Ripley. The adjusted predictions of T_c were within the IQR of time to rise for Hico and Fairplain but were longer than the IQR of time to rise for Ripley and Wallback. The IQR for time to recede was shorter than the adjusted prediction for each streamgage.

Five of the eight adjusted predictions of T_c that were outside the IQR for their combination of streamgage and time metric were for combinations that had fewer than 10 observations, whereas all four of the adjusted predictions that were within the IQR for a time metric at a streamgage were for combinations with more than 10 observations. The small number of observations for some of the metrics may indicate that the study did not run long enough to provide reliable estimates of median and IQR. The study ran long enough, though, to confirm that the time metrics generally have high variance and wide range.

As a metric, PI-to-RI seems to match one of the most used conceptual definitions of T_c, the time between the end of excess precipitation and the end of stormflow. Concerns about PI-to-RI include whether timing of precipitation can be adequately measured for a basin with a single rain gage at the outlet, whether inflections in the hyetograph truly represent the end of excess precipitation, and whether using two different datasets results in unacceptable increases in measurement error, particularly in time measurement. The sensors used in this study were synchronized with GPS clocks so that errors in time measurement were negligible, addressing a concern in studies that used older technology. However, use of a single rain gage might be a source of error in timing of precipitation throughout the basins relative to changes in streamflow.

Time to rise and time to recede avoid some of the questions about differences in precipitation timing across basins. Although basins in this study were small, precipitation across them was not uniform in space and time. To accept the timing of storms measured with a single rain gage at the basin outlet, it is necessary to assume that differences in timing across the basins were inconsequential. Even these hydrograph-derived metrics incorporated precipitation data, however. Delineation of hydrograph events for time to rise and time to recede included examining hyetographs to exclude recessions complicated by excessive rain and rises that were interrupted or extended by breaks in rain.

Time to recede is intended to quantify the same conceptual definition as PI-to-RI, the time between the end of excess precipitation and the end of stormflow, without requiring detailed precipitation data. It is to be expected that time to recede was faster than PI-to-RI; as an individual hydrograph event in a single storm, it could only be longer than PI-to-RI for that storm if the precipitation inflection follows peak streamflow. Time to recede may represent T_c well if it meets the assumption that in small basins, peak flow is a good surrogate for the end of excess precipitation. For storms measured in this study, it was an imperfect surrogate. For all storms analyzed in this study, precipitation intensity decreased from storm maxima before flows peaked, although in many of the most intense storms, only a minute or two before peaks. Decreases in precipitation intensity may not be ideal measures of the end of excess precipitation in storms when rain continues, but they would seem to be better measures than the timing of peak flow. Additionally, the number of observations for time to recede was generally small. Only 34 such events were identified for the entire study. Time to recede was the least frequently delineated time metric for Fairplain, Hico, and Wallback. This metric is useful to consider, but it probably represents a secondary line of evidence, rather than strong evidence that the true T_c values for these basins are shorter than estimates.

The predictions of peak flow for selected precipitation intensities and durations did not match observed peak flows as accurately as predictions matched observations for T_c. Maximum peak flows during the study period were 12.7, 16.5, 17.4, and 16.7 ft³/s for Fairplain, Ripley, Hico, and Wallback, respectively. Maximum precipitation totals at all streamgages reached the range of 2- to 5-year recurrence intervals for some durations, and maximum storm intensity exceeded the 50-year recurrence interval for a 15-minute duration at Fairplain. The maximum peaks recorded at Fairplain, Ripley, and Hico coincided with the most intense storms at each of those streamgages, and the maximum peak at Wallback coincided with one of the most intense storms. The only streamgage where the maximum observed stormflow for a streamgage reached 50 percent of the smallest prediction was at Ripley, during the April 8, 2020, storm. Although relating peak streamflow to channel and floodplain elevations was not a focus of the study, field inspections included checks for evidence of out-of-bank flow, which was noted only following the two largest storms at Hico and Wallback. Maximum depths at stage sensors during the study were 0.44 ft at Fairplain and 1.17 ft at Ripley, for the AV sensor inside the culverts, and 1.81 ft at Hico and 1.51 ft at Wallback, for the NTRAN upstream from the culverts; these depths are not indicative of major floods, even for such small streams.

More of the predictions seemed too slow than too fast, but that is in comparison to a high-variance set of data. The variance is demonstrated not just by values of standard deviations and IQR, but also by the absence of a metric that was significantly different among streamgages, even though the streamgages were selected to represent different basin slopes and shapes. Differences among streamgages were expected, yet for all time metrics, the differences within streamgages were greater than the differences among streamgages. Time to recede accounted for four of the seven combinations of time metrics and streamgages for which the IQR of observations was shorter than the adjusted estimates for the 10-year storm; however, PI-to-RI better represents assumptions of the same conceptual definition of T_c. If time to recede is considered a secondary line of evidence, then the IQR for three combinations of time metrics and streamgages is shorter than estimates, compared with one that was longer, and the IQR for four combinations included the estimates. All predictions were within one average standard deviation of the overall mean for the streamgage in question, and within the range of observations for all time metrics at all streamgages. These lines of evidence do not indicate large, systematic errors in T_c estimates.