USGS hydrologists are often called upon to comment on the rarity of an observed flood.  In many cases the observed flood peak is compared to flood quantiles from either a published or a newly derived flood-frequency distribution and the Annual Exceedance Probability (AEP) is inferred from the comparison.  There are pitfalls associated with this practice and judgments are required to provide a reasonable and consistent result.  The need for careful judgment is even greater for streams that are heavily regulated to control or moderate flood flows.  The purpose of this memo is to provide guidance for the computations and judgments associated with estimating the AEPs for observed flood peaks on both unregulated and regulated streams.

Flood-Frequency Computations for Unregulated Streams

The starting point for estimating the AEP for an observed flood peak is the floodfrequency computation based on Bulletin 17B (U.S. Interagency Advisory Committee on Water Data, 1982, Guidelines for determining flood flow frequency, Bulletin 17-B of the Hydrology Subcommittee).  Bulletin 17B procedures were designed to incorporate relevant regional and historical information into the flood-frequency computation process to stabilize and improve the accuracy of flood-frequency estimates for unregulated streams.  The increase in accuracy is accomplished first by determination of a weighted skew based on both the regional and the atsite skew estimates.  Next, the weighted skew is used to determine the at-site flood frequencies.  Finally, the at-site estimates are weighted with independent estimates of the flood frequencies for that site, typically determined using regional regression techniques.  Office of Surface Water Technical Memorandum 2010.05 (SW 2010.05) describes the method for weighting the two estimates.

Flood-Frequency Computations for Regulated Streams

In contrast to unregulated streams, computations of flood-frequency estimates for regulated streams depend on the unique details of the regulation, as well as at-site flood history.  Accordingly, here are some general guidelines for flood-frequency computations for regulated systems:

• Select the period of record (POR) that is appropriate for the purpose of the computations.
  1. If the purpose is to ascertain the AEP of an observed flood under regulated conditions, the POR associated with the regulation is usually the appropriate base period for both the computations and comparisons. However, in some cases, the flood-control reservoir may be full during the flood such that the system is in a “runof-the-river” situation.  In such cases, the regulation may not significantly affect the magnitude of the peak flows.  If an observed flood occurred during such conditions, the appropriate reference period would likely include the pre-regulated period, the observed current flood, along with any other floods during the regulated period which occurred under “run-of-the-river” conditions.  Ultimately, the selection of the appropriate POR is a matter of judgment that should be documented with the computations and release of the AEPs.
  2. Use data drawn from a consistent POR. Except for run-of-the-river situations as described in item “a” above, flood-frequency estimates generally should be computed separately for the regulated and unregulated periods.  Combining the records would inappropriately mix together two different populations.  In addition, the analyst should determine that there were no major changes in operating rules during the regulated period.  If major changes did occur, the POR for the regulated period would likely be further reduced and may not be long enough to support flood frequency estimates.
  3. Explain and document the selection of the flood data and similarity or differences in flood conditions, especially the effects of regulation on individual floods, including the observed flood for which the AEP estimate is sought. Include commentary and documentation of discussions with personnel from agencies responsible for the regulation.
• Use only the station skew. The regional (generalized) skew (for example, Bulletin 17B, plate 1) and the weighted skew (which is calculated using the regional skew) are based on skews observed at streamgages with unregulated flow.  As a result they are not applicable for a streamgage with regulated flow.
• Use the “systematic record” output from PeakFQ (figure 1). In almost all cases involving regulated flow, the systematic results from PeakFQ are the estimates that should be used.  Since the Bulletin 17B estimates include the weighted skew, the “systematic record estimate” should be used.
• For a final estimate of the flood frequency relation, do not weight the at-site floodfrequency estimates with flood-frequency estimates from regional regression equations. Weighting techniques should be used only when the site is a valid location where the regional regression equations apply.  Since the equations do not generally apply to regulated streams, use of the weighted estimates, as described in SW Technical Memorandum 2010.05, is not appropriate.

Estimating the AEP of Observed Flood Peak on Unregulated and Regulated Streams

When estimating the AEP of a recent flood, a new flow frequency analysis should be conducted that includes the observed flood peak for which the AEP is sought.  The AEP can be determined through log-linear interpolation of PeakFQ flood-frequency output using equation 1:

= 1 + (2−1)×10(102−10(1)−10(1)),                                        (1)

 

where:

 

AEP₁ and AEP₂ are taken from column of AEP values associated with the flows Q₁ and Q₂ that bracket the observed flood.  For example, given an observed peak flow of 20,000 cubic feet per second (cfs) and the flood-frequency quantiles listed in the PeakFQ results shown in figure 1 (columns A and B for unregulated streams and columns A and C for regulated streams), the AEP is computed as:

 

AEP = 0.02 ₊ (.01−.02₁₀)∗(10_(22,810(20₎₋,000)₁₀−₍₁₈10_,760(18₎,760))

= 0.017

 

If the final flood frequency estimates are obtained by weighting the at-site estimates with regional regression estimates, then the interpolation should be performed on the weighted estimated (from WIE).

Reporting and presenting the AEP of Observed Flood Peaks

Flood-frequency concepts including those found in Bulletin 17B were developed to provide reasonably precise estimates of the flood magnitudes associated with a set of AEPs.  However, the converse of that operation, assigning an AEP to an observed flood, is associated with more uncertainty.  To communicate the uncertainty, AEP estimates must be accompanied with a confidence internal likely to include the true AEP.

The computation of a confidence interval for the AEP could be performed by interpolation of the upper and lower confidence interval information contained in PeakFQ output in a manner similar to interpolation process described above.  However, the flood-frequency distribution computed by PeakFQ is based on the computed mean, standard deviation, and skew and an assumption that these statistics represent their true values and the true flood distribution, assumptions that may be flawed.  While the interpolation process is reasonable for the estimation of the AEP, it fails to account for the uncertainty of the distributional assumptions and the increased uncertainty arising for the lack of reference data for extreme flood being assessed.  AEP confidence intervals that are less susceptible to such assumptions can be computed using nonparametric methods.  OSW recommends that the confidence interval of the AEP be calculated using the procedure based on the Beta distribution, which can be implemented in Excel using the cell formula:

 

AEP_CL = 1-BETAINV(Prob, Alpha, Beta, lowerbound, upperbound)

where:

AEP_CL= the confidence interval (upper or lower range) of the AEP;

Prob = the probability associated with the upper or lower confidence limit (0.167 or

0.833, based on the 66.7% confidence level);

Alpha = the number of floods minus the rank plus 1 (i.e., N-Rank+1);

Beta = the rank of the observed flood; lowerbound = 0; and, upperbound = 1.

 

While the use of the 66.7-percent confidence level is unusual, the use of confidence intervals based on the 90- or 95-precent level (which are more typical) result in ranges that are extremely large and tend to be misunderstood by lay audiences.  There is some precedent for use of the 66.7-percent confidence level.  The Intergovernmental Panel on Climate Change uses it to describe events it considers “likely”.

As an example, a streamgage with 19 years of record experiences a peak of record in the current (20^th) year of record.  Using Excel and the BETA.INV statistical function, the formula arguments for the upper confidence limit would be:

 

AEP_CL (upper) = 1 - BETA.INV(0.167, 20, 1, 0, 1) = 1 - 0.914

 

producing a result of 0.086.  This corresponds to the upper CL value in Table 1 for a streamgage with 20 years of record and an observed flood rank of 1, and a confidence interval of 66.7 percent.

Table 1 provides upper and lower confidence limits at the 66.7-percent level computed from the Excel Beta function for selected combinations of lengths of POR and flood ranks.  Table 1 is the preferred source for ranges of the AEP of an observed flood.  In order to use it, an analyst need only determine the number of floods for which the conditions (regulated, unregulated, or ‘run-of-the-river”) are comparable to the observed flood and the rank of the observed flood relative to those data.

Note that use of the non-parametric confidence intervals calculated using either the Excel Beta function or Table 1 provide the AEP range which is expected for a flood of a given rank within a specified period of record and that these confidence intervals are not centered around the interpolated AEP estimate.  Consequently, the AEP estimate will occasionally be outside of the range of the confidence interval.  To explain this result, we suggest the following language:

 

We estimate the exceedance probability corresponding to this particular flood at 2%.  However, the uncertainty in this statistic is large, and, in general, a 66.7% confidence interval for the true exceedance probability of the largest flood in 47 years extends from 0.4% to 4%. Finally,  in an attempt to make the end user of USGS AEP estimates fully aware of the uncertainty, it is recommended that the flood quantiles and associated 95% confidence intervals for selected AEP’s (for example, 4%, 2%, 1%, and 0.2%) also be presented.  The 95% confidence intervals are basic output in the analysis methods used by USGS for unregulated streams and regulated streams.  An example of this information in table form is presented in table 2.  Note that if the AEP estimate for a flood is less than 0.2% (500-year flood), the AEP estimate should be listed simply as being less than 0.2%, without specifying the actual calculated value.  The confidence intervals should still be listed.

Figure 1.  Output from PeakFQ showing "Bulletin 17B" (column B) and "Systematic" (column

66. C) flood-frequency estimates, upper confidence limit for 66.7-percent confidence level (column F) and lower confidence limit (column G). The box encloses the input data for a hypothetical observed flood of 20,000 cubic feet per second. If the computation is for a regulated site, the “systematic record” estimates should be used.

 

Table 1.  Lower and Upper Nonparametric AEP Confidence Limits as a Function of the Rank of Observed Flood and Length of Record for the 66.7-percent level.

Lower and Upper Nonparametric AEP Confidence Limits as a Function of the Rank of Observed Flood and Length of Record

	Rank of Observed Flood
	1		2		3		5		10
Length of Record (in Years)	Lower  CL	Upper  CL	Lower CL	Upper CL	Lower CL	Upper CL	Lower CL	Upper CL	Lower CL	Upper CI
10	0.018	0.164	0.074	0.289	0.145	0.400	0.309	0.600	0.836	0.982
11	0.016	0.150	0.067	0.266	0.131	0.368	0.279	0.554	0.734	0.933
12	0.015	0.139	0.062	0.246	0.120	0.341	0.254	0.515	0.659	0.880
13	0.014	0.129	0.057	0.228	0.110	0.318	0.234	0.480	0.599	0.830
14	0.013	0.120	0.053	0.213	0.102	0.297	0.216	0.450	0.550	0.784
15	0.012	0.113	0.049	0.200	0.095	0.279	0.201	0.424	0.508	0.742
16	0.011	0.106	0.046	0.189	0.089	0.263	0.188	0.400	0.473	0.704
17	0.011	0.100	0.043	0.178	0.084	0.249	0.176	0.379	0.442	0.669
18	0.010	0.095	0.041	0.169	0.079	0.236	0.166	0.360	0.415	0.637
19	0.010	0.090	0.039	0.161	0.075	0.224	0.157	0.342	0.392	0.608
20	0.009	0.086	0.037	0.153	0.071	0.214	0.149	0.327	0.371	0.582
25	0.007	0.069	0.029	0.124	0.057	0.173	0.118	0.266	0.292	0.477
30	0.006	0.058	0.024	0.104	0.047	0.146	0.098	0.224	0.241	0.403
35	0.005	0.050	0.021	0.090	0.040	0.126	0.084	0.193	0.206	0.349
40	0.005	0.044	0.018	0.079	0.035	0.110	0.073	0.170	0.179	0.308
45	0.004	0.039	0.016	0.070	0.031	0.099	0.065	0.152	0.159	0.276
50	0.004	0.035	0.015	0.063	0.028	0.089	0.058	0.137	0.142	0.249
55	0.0033	0.032	0.013	0.058	0.026	0.081	0.053	0.125	0.129	0.227
60	0.0030	0.029	0.012	0.053	0.023	0.074	0.049	0.115	0.118	0.209
65	0.0028	0.027	0.011	0.049	0.022	0.069	0.045	0.106	0.109	0.194
70	0.0026	0.025	0.010	0.045	0.020	0.064	0.042	0.099	0.101	0.180
75	0.0024	0.024	0.010	0.042	0.019	0.060	0.039	0.092	0.094	0.169
80	0.0023	0.022	0.009	0.040	0.018	0.056	0.036	0.087	0.088	0.158
85	0.0021	0.021	0.009	0.038	0.017	0.053	0.034	0.082	0.083	0.149
90	0.0020	0.020	0.008	0.036	0.016	0.050	0.032	0.077	0.078	0.141
95	0.0019	0.019	0.008	0.034	0.015	0.047	0.031	0.073	0.074	0.134
100	0.0018	0.018	0.007	0.032	0.014	0.045	0.029	0.070	0.070	0.127
105	0.0017	0.017	0.007	0.030	0.013	0.043	0.028	0.066	0.067	0.121
110	0.0017	0.016	0.007	0.029	0.013	0.041	0.026	0.063	0.064	0.116
115	0.0016	0.015	0.006	0.028	0.012	0.039	0.025	0.061	0.061	0.111
120	0.0015	0.015	0.006	0.027	0.012	0.038	0.024	0.058	0.059	0.106

 

 

Table 2.  Example presentation of observed flood peak data along with AEP estimates

 

Flood data                   AEP for observed 2011 flood # of Annual                 Peak            Peak            66.7% Confidence Interval Expected peak streamflows for selected AEP with 95-percent confidence intervals (ft³/s)

10-percent AEP                                              1-percent AEP                                      0.2-percent AEP

(10-year recurrence)                                 (100-year recurrence)                            (500-year recurrence)

95% Confidence                                        95% Confidence                                     95% Confidence

 

Estimate

Lower           Upper

Estimate

Lower           Upper

02473460 02473480 02477330 02483000 02484600 03291780 03303280 03381500 03333333

4 1 2 2 3 1 2 1 1

44 47 47 75 39 42 33 72 111

3/9/2011 3/9/2011 3/9/2011 9/6/2011 9/6/2011 4/19/2011 4/28/2011 5/3/2011 5/28/2011

22.69 18.15 22.90 31.10 16.59 11.60 49.30 36.42 15.20

12,700 8,000 9,850 39,500 8,460 8,300 648,000 55,300 151,000

8.5% 1.7% 4.6% 1.6% 7.4% 2.2% 5.8% 0.7% < 0.2%

4.9% 0.4% 1.6% 1.0% 3.6% 0.4% 2.2% 0.3% 0.16%

12.8% 3.7% 6.7% 4.2% 11.3% 4.2% 9.5% 2.5% 1.6%

11,900 4,440 7,240 22,600 7,430 6,290 610,000 29,800 83,800

9,480 3,400 5,560 18,900 5,820 5,470 568,000 25,800 75,600

14,900 5,790 9,450 27,000 9,490 7,230 698,000 36,100 93,000

21,000 8,940 14,900 43,600 14,100 9,350 742,000 49,400 120,000

14,600 5,970 9,820 32,200 9,490 7,550 664,000 39,400 99,800

30,100 13,400 22,500 59,100 21,000 11,600 1,070,000 79,500 145,000

28,200 12,400 21,100 61,000 19,600 11,300 827,000 64,200 144,000

17,600 7,380 12,400 40,600 11,700 8,760 713,000 46,800 112,000

45,200 20,900 36,100 91,700 32,900 15,500 1,370,000 128,000 186,000

Station Rank Peaks in Date of Peak Stage Streamflow Interval Interval Interval number analysis Streamflow (ft ) (ft³/s) Estimate Lower UpperEstimate Lower Upper

 

 

 

Robert R. Mason, Jr.

Acting Chief, Office of Surface Water

 

Distribution:  GS-W All

 

 

Additional Claification and Guidance for this memo can be found at:  https://xcollaboration.usgs.gov/wg/osw/OSWNotes/Shared%20Documents/OSW%20Note%202013.21%20Additional%20Clarification [link or image removed]Guidance%20OSW%20TM%202013.01.pdf [link or image removed]