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Flood frequency analysis using partial series data has been shown to provide better estimates of small to medium magnitude flood events than the annual series, but the annual series is more often employed due to its simplicity. Where partial series average recurrence intervals are required, annual series values are often “converted” to partial series values using the Langbein equation, regardless of whether the statistical assumptions behind the equation are fulfilled. This study uses data from Northern Tasmanian stream-gauging stations to make empirical comparisons between annual series and partial flood frequency estimates and values provided by the Langbein equation. At

Estimates of the size and frequency of floods is important for infrastructure planning and design and in the management of water resources and riparian areas [

Flood frequency analysis is used for making probabilistic estimates of a future flood event based on the historical stream-flow record, with probability often expressed as the average length of time between floods and called the return period or average recurrence interval (

Of the two main choices of data series in flood frequency analysis, the most frequently used is the annual series, which is composed of the single maximum discharge for each year of the record. IEA [

The partial series, also known as Peaks Over Threshold (POT), is composed of all discharges over a chosen threshold for the entire stream gauge record—some years may contribute several floods and other years none. Advantages of the partial series are that insignificant floods are excluded, which can improve magnitude estimates of high frequency floods [

Due to the difficulties in defining the partial series, the estimation of the magnitude-frequency of frequent floods is often made using the easier to define annual series, despite evidence it underestimates their magnitude [

Assuming that the floods in the partial series are independent and distributed according to a Poisson process, Langbein [_{P}_{A}

The objective of this paper is to compare magnitude-frequency estimates of frequent floods determined using the annual series, the Langbein adjusted annual series and the partial series, and to determine whether Langbein’s equation provides a suitable empirical method to convert annual series flood frequency average return intervals to partial series intervals. The purpose of this study is to improve understanding of practical methods available to fluvial geomorphologists and catchment managers for the estimation of high-frequency low-magnitude flood events. This would allow estimation of the frequency or magnitude of geomorphically important flood events such as bankfull discharge.

Tasmania is the southernmost state in Australia, with the main island extending across a latitudinal range of 39°40′–43°20′ S. The North-Eastern Region covers almost one-third of Tasmania’s landmass (

Location of major rivers and stream-flow stations in North-Eastern Tasmania used in this study. State Government stream-gauge codes are used to identify sites.

Stream-flow data for the thirteen gauging stations shown in ^{2} to 3306.4 km^{2}, with a mean of 509.5 km^{2}. The number of years of stream-flow record (

Stream gauging sites and flow records used in the flood frequency analyses.

Site Code | Site Name | Years of record ( |
Catchment area (km^{2}) |
---|---|---|---|

2214 | Ansons River downstream of Big Boggy Creek | 10 | 228.9 |

191 | Break O’Day River at Killymoon | 28 | 186.2 |

19200 | Brid River 2.6 km upstream of tidal limit | 34 | 138.9 |

19201 | Great Forester River 2 km upstream of Forester Road | 41 | 192.0 |

18217 | Macquarie River at Trefusis | 32 | 375.3 |

76 | North Esk River at Ballroom | 85 | 375.9 |

25 | Nile River at Deddington | 10 | 220.9 |

19204 | Pipers River downstream of Yarrow Creek | 39 | 298.4 |

30 | Ringarooma River upstream of Moorina Bridge | 34 | 482.3 |

2217 | Ransom River at Sweet Hills | 28 | 26.4 |

2206 | Scamander River upstream Scamander water intake | 10 | 268.0 |

181 | South Esk River above Macquarie River | 55 | 3306.4 |

18311 | St. Pauls River upstream of South Esk River | 23 | 524.2 |

Note: years of record relates to the period immediately prior to 1 January 2012.

Daily stream-flow data from each site was time-stepped to annual maxima, with checks made to ensure peak events from one year were not included as peak events for the following year. While there are various a-priori theories for choosing particular probability distributions for flood frequency data, in practical applications empirical suitability plays a much larger role in distribution choice [

A peaks-over-threshold (POT) analysis was undertaken on daily stream-flow data from each site [^{2}, 10 days for catchments 45,000–100,000 km^{2}, and 20 days for catchments >100,000 km^{2}. As the largest catchment in this study was 3306 km^{2} 14 days between flood events was used as a criterion to ensure independence. In consideration of the range of values suggested by the literature, four different partial series were defined for each site. Thresholds were adjusted to provide partial series data sets where the number of events (_{1}, PS_{1.5}, PS_{2} and PS_{2.5} respectively). The IEA [

At-a-site flood frequency estimates for _{1}, PS_{1.5}, PS_{2}, PS_{2.5}) using the procedures detailed above. Langbein adjusted flood frequency estimates (LC) were determined from the annual series estimates using Equation (1). It should be noted that Langbein’s equation is used in this study to determine if it provides an empirical method to convert annual series average recurrence intervals to partial series average recurrence intervals, and that consequently the theoretical assumptions behind the equation were not considered in the choice of flood frequency analysis method. The PS_{1} values (also referred to as PS) were chosen for comparison with annual series estimates (AS) and Langbein adjusted annual series estimates (LC), as the data set on which the PS_{1} estimates were based contained the same number of flood events as the annual series. In addition, all partial series magnitude estimates were generally closely clustered, irrelevant of the number of flood events included. The three estimates (AS, PS and LC) were then compared, and the ratio of AS to PS and LC to PS was calculated for each station. Mean ratios averaged across all thirteen stations were also compared.

Data generally conformed well to statistical models, with both annual series and partial series flood frequency curves created from the probability distributions providing a fairly good fit (

Original stream-flow data compared to fitted probability distributions (solid line) for the South Esk River above Macquarie River (Site 181); (

The final partial and annual series flood frequency estimates are presented in _{1}, PS_{1.5}, PS_{2} and PS_{2.5}). Partial series CV was generally larger for sites with shorter stream-flow records, and displayed a general increase at and above

Estimated discharge (m^{3}s^{−1}) for 1.1, 1.5, 2, 3, 4, 5 and 10 year average recurrence interval floods using partial (PS) and annual series (AS) data for Northern Tasmanian stream gauging stations (Coefficient of variation for partial series estimates PS_{1}, PS_{1.5}, PS_{2} and PS_{2.5} shown in parentheses).

Site | Series | Average Recurrence Interval (years) | ||||||
---|---|---|---|---|---|---|---|---|

1.1 | 1.5 | 2 | 3 | 4 | 5 | 10 | ||

2214 | PS | 62.28 | 76.63 | 112.68 | 199.14 | 206.44 | 205.83 | 218.03 |

(3.52) | (3.91) | (3.82) | (5.17) | (2.59) | (7.99) | (20.88) | ||

AS | 7.10 | 26.09 | 48.37 | 90.66 | 125.60 | 156.90 | 292.55 | |

191 | PS | 77.93 | 90.70 | 116.90 | 154.97 | 166.99 | 180.77 | 244.44 |

(0.74) | (1.90) | (2.50) | (2.8) | (0.83) | (2.57) | (6.35) | ||

AS | 14.92 | 40.89 | 67.07 | 107.87 | 142.52 | 173.00 | 287.26 | |

19200 | PS | 10.01 | 11.30 | 12.21 | 14.00 | 15.11 | 16.14 | 20.21 |

(0.93) | (1.50) | (1.50) | (1.60) | (0.69) | (0.67) | (3.69) | ||

AS | 3.91 | 7.19 | 9.51 | 12.52 | 14.62 | 16.25 | 21.55 | |

19201 | PS | 21.42 | 24.78 | 27.54 | 30.56 | 32.37 | 34.79 | 52.57 |

(1.95) | (1.19) | (2.05) | (0.67) | (1.07) | (2.00) | (3.14) | ||

AS | 10.72 | 17.63 | 22.26 | 28.43 | 32.53 | 35.61 | 45.94 | |

18217 | PS | 63.24 | 72.44 | 85.20 | 102.55 | 112.18 | 120.62 | 173.35 |

(1.28) | (0.52) | (1.89) | (1.33) | (0.54) | (1.79) | (1.01) | ||

AS | 6.25 | 22.79 | 42.27 | 78.58 | 110.82 | 138.55 | 264.63 | |

76 | PS | 51.41 | 59.53 | 66.48 | 73.36 | 77.68 | 80.60 | 92.67 |

(1.86) | (1.21) | (2.35) | (0.90) | (0.37) | (1.13) | (1.95) | ||

AS | 26.78 | 41.29 | 50.24 | 61.77 | 69.70 | 75.51 | 92.31 | |

25 | PS | 68.62 | 79.06 | 93.89 | 108.64 | 109.11 | 109.33 | 111.01 |

(2.30) | (0.83) | (2.69) | (1.38) | (2.38) | (4.93) | (10.47) | ||

AS | 32.97 | 50.69 | 62.11 | 75.71 | 84.29 | 91.07 | 110.62 | |

19204 | PS | 49.71 | 56.93 | 67.09 | 73.77 | 79.44 | 92.76 | 132.43 |

(1.53) | (0.51) | (2.94) | (0.52) | (2.44) | (0.93) | (3.03) | ||

AS | 18.46 | 35.77 | 49.40 | 68.38 | 82.17 | 92.24 | 129.21 | |

30 | PS | 72.36 | 81.49 | 86.79 | 102.49 | 109.26 | 112.21 | 127.78 |

(1.17) | (1.02) | (1.13) | (1.24) | (0.30) | (1.95) | (9.29) | ||

AS | 43.01 | 65.86 | 79.92 | 97.30 | 108.25 | 117.08 | 145.20 | |

2217 | PS | 4.53 | 5.31 | 7.09 | 9.45 | 11.25 | 12.80 | 20.28 |

(0.93) | (1.71) | (2.93) | (1.72) | (0.48) | (1.16) | (1.43) | ||

AS | 2.01 | 3.95 | 5.41 | 7.45 | 8.99 | 10.20 | 14.26 | |

2206 | PS | 119.93 | 159.67 | 263.43 | 292.83 | 298.75 | 303.30 | 295.09 |

(7.64) | (4.37) | (8.18) | (2.31) | (4.77) | (8.06) | (16.17) | ||

AS | 9.50 | 39.97 | 79.52 | 157.04 | 228.05 | 298.17 | 601.73 | |

181 | PS | 310.78 | 406.08 | 512.45 | 663.32 | 798.91 | 896.20 | 1162.37 |

(3.34) | (1.10) | (3.08) | (1.38) | (0.84) | (0.73) | (7.12) | ||

AS | 102.60 | 228.37 | 337.54 | 491.06 | 609.74 | 716.14 | 1069.25 | |

18311 | PS | 106.63 | 144.54 | 165.07 | 207.46 | 284.52 | 334.64 | 419.42 |

(4.11) | (3.14) | (1.65) | (2.39) | (1.75) | (1.98) | (6.91) | ||

AS | 20.94 | 59.48 | 95.28 | 152.71 | 198.40 | 240.11 | 394.74 |

The percentage differences between partial series (PS) and annual series estimates (AS) and between the partial series and Langbein adjusted annual series estimates (LC) averaged across all 13 sites are listed in

Ratio of annual series estimates (AS) and Langbein adjusted annual series estimates (LC) to partial series (PS) estimates averaged across 13 North-Eastern Tasmanian stream gauging stations.

Ratio | Average Recurrence Interval (years) | ||||||
---|---|---|---|---|---|---|---|

1.1 | 1.5 | 2 | 3 | 4 | 5 | 10 | |

AS/PS | 0.33 | 0.53 | 0.65 | 0.79 | 0.88 | 0.95 | 1.19 |

LC/PS | 0.75 | 0.78 | 0.81 | 0.89 | 0.95 | 1.01 | 1.22 |

Comparison of annual and partial series average recurrence intervals (

Annual series average recurrence intervals (in years) equivalent to partial series as predicted by Langbein’s function and estimated from mean values across 13 North-Eastern Tasmanian stream-flow stations.

Partial Series | Annual Series | |
---|---|---|

Langbein function | North-Eastern Tasmanian data | |

1.1 | 1.67 | 2.17 |

1.5 | 2.06 | 2.71 |

2.0 | 2.54 | 3.28 |

3.0 | 3.53 | 4.22 |

4.0 | 4.52 | 5.01 |

5.0 | 5.52 | 5.71 |

10.0 | 10.51 | 8.63 |

Differences between mean Langbein adjusted annual series average recurrence intervals and partial series intervals were smaller than differences between annual series and partial series intervals, but remained significant at low average recurrence intervals. Mean partial series estimated discharge at

Annual series flood frequency estimates made using data from Northern Tasmanian stream gauging stations differed from partial series estimates at most average recurrence intervals. Differences were largest for the most frequent floods, with annual series estimates only 33 percent of partial series estimates at

Differences between the two series reflect the different data sets used. The partial series, which uses all floods above a threshold, is likely to include more medium sized flood events than the annual series, which only uses the largest flood event of each year. As more flood events are included the average recurrence interval between peaks of a given magnitude automatically declines [

The differences between Langbein adjusted values and partial series values reflect the differences between the annual and partial series. The largest differences between the two values occurred at the smallest average recurrence intervals, and decreased as

Previous studies [

The variation between the different partial series estimates at a site was generally small at low average recurrence intervals (

This study found large differences between annual and partial series flood frequency estimates made using Northern Tasmanian stream-flow data for average recurrence intervals of less than five years, similar to other studies finding such significant deviations [

In addition, this study found that Langbein’s equation did not provide a suitable empirical method to convert annual series flood frequency estimates to partial series estimates at average recurrence intervals of less than five years. Langbein adjusted annual series estimates were three quarters the magnitude of partial series estimates at

These results suggest that both the annual series and the Langbein adjusted annual series significantly underestimate the magnitude of frequent floods and should not be used at average recurrence intervals of less than five years. Rather, the partial series should be used for estimates of high frequency-low magnitude floods (

The paper was greatly improved following the insightful comments of four Reviewers and the External Editor.

The authors declare no conflict of interest.