Peaks Over Threshold Filtering

The most challenging part of a peaks over threshold analysis is selecting an appropriate threshold value. A high threshold ensures that the sample is comprised of extreme events. However, selecting too high a threshold results in a small sample size and may violate the assumption that the model is representative of the population. On the other hand, selecting too low of a threshold value means that events that are not truly extreme are included in the sample. 

In addition to using a representative sample, the sample values should be independent of each other. To ensure independence, time separation and magnitude differential criteria can be applied. Time separation criteria ensure that enough time has passed between floods. Magnitude differential criteria ensures that the hydrograph has sufficiently receded between floods. Magnitude differential criteria are less applicable to other variables that do not exhibit serial correlation, such as precipitation.

Users can select a previously extracted partial duration series within the Peaks Over Threshold Analysis.

When changes are made to the filtering criteria (Selected Threshold, Time Separation, and/or Magnitude Differential), the Refresh Plot button should be used to update the time series plot.

The Save Filtered Data Set As... button can be used to apply the filtering criteria to the input dataset and extract a partial duration series dataset to the study.

Threshold Selection

If the selected data set is a partial duration series, the user should select the Data set is already a partial duration series checkbox. If the user is extracting a partial duration series, the checkbox should be left unchecked.

The user must enter the Selected Threshold. If the dataset is a regular time series, the threshold will be applied to extract the partial duration series. If the dataset is a partial duration series, the threshold will be used in the plot. 

The user has the option to enter a range of threshold values to evaluate. When the Investigate Partial Duration Threshold Value checkbox is select, the user can enter a Minimum Threshold, Maximum Threshold, and Number of Thresholds. HEC-SSP will generate a vector of equally spaced threshold values between the Minimum Threshold and Maximum Threshold values (inclusive).

The user can elect to compute diagnostic quantities for 10 to 100 threshold values (Number of Thresholds). If the user enters a Number of Thresholds value less than 10 or greater than 100, the program will throw a warning message. The Minimum Threshold and Maximum Threshold values should be set such that 10 to 100 thresholds and the corresponding diagnostic data provides enough information for selection of a single threshold. For example, spanning the entire range of the dataset is not appropriate. 

The diagnostic plot quantities (discussed later) will then be computed for each threshold value. 

Users can still extract a partial duration series using the Data Filter. The ability to investigate threshold selection is only available within the Peaks Over Threshold Analysis.

Another method for determining an appropriate threshold value for flow frequency analysis is through the use of bankfull discharge. Bankfull discharge refers to the flow which just fills the channel, without overtopping the banks. Bankfull discharge can be determined from:

• Channel cross-section and Manning's equation,
• A HEC-RAS model, 
• Inflection point(s) on a rating curve, and/or
• From sediment data (HEC-RAS Sediment Rating Curve Analysis Tool).

However, using bankfull discharge to determine a flow threshold value is not recommended in an incised channel as this will result in an artificially high bankfull discharge.

Time Separation

The user can apply a time separation criterion, meaning that a certain amount of time must elapse before another peak is identified. Two time separation method choices are available for user: User-specified and ceil[5 + ln(DA)].

The user-specified option allows the user to enter a time duration, in days, between peaks.

The second option uses an equation, provided in Bulletin 17, that is used as a time separation criterion to ensure independence between partial duration series flood events (U.S. Water Resources Council, 1976). The equation is:

$//<![CDATA[ \begin{array}{l}t = ceil[5 + ln(DA)]\end{array} //]]>$

where t is the time duration (days), ceil is the ceiling function (returns the smallest integer greater than or equal to the argument), and DA is the drainage area (square miles). 

This equation is relevant when applying a Peaks Over Threshold Analysis to streamflow events. A streamflow gage with a drainage area of 100 square miles would have a time separation of ceil[5 + ln(100)] = ceil(9.6) = 10 days.

Magnitude Differential

Navigate to the Filtering Data section of the User's Manual. Read the Peaks Tab section and then return to this page.

Diagnostic Plots

Five diagnostic plots are plotted when the Plot Threshold Selection Diagnostic Plots button is selected: number of exceedances, rate of exceedances, Generalized Pareto distribution shape parameter, mean excess, and the Anderson-Darling right tail weighted test statistic. Each of these quantities is plotted against threshold value. In addition, users can view the diagnostic data values in tabular format by selecting File | Tabulate in the plot dialog.

Number of Exceedances

The number of exceedances is the sample size of the partial duration series. For example, 50 exceedances of the threshold means that there are 50 events in the PDS. 

Rate of Exceedances

Using an average rate of 1 event per year as a criterion ensures that the partial duration series and annual maximum series have the same sample size. There is no special meaning beyond that but this criterion can be useful in the absence of other information.

Generalized Pareto Distribution Shape Parameter

The smallest threshold value corresponding to a stable shape parameter is selected (Naghettini, 2017). The Generalized Pareto distribution shape parameter is fitted using the Distribution Fitting Method selected on the General tab.

Mean Excess

Typically, the threshold value is selected when the mean residual life plot’s behavior becomes nonlinear, or there is a break in the slope (Naghettini, 2017).

Anderson-Darling Right Tail Weighted Test Statistic

Solari et al. (2017) proposed a method for automated threshold selection, based on an empirical distribution function (EDF)-statistic and a goodness-of-fit test to test the null hypothesis that exceedances of a high threshold come from a Generalized Pareto distribution. The Anderson Darling A² statistic is an EDF-statistic, which assigns more weight to the tails of the data than similar measures. Sinclair et al. (1990) proposed the right-tail weighted Anderson Darling statistic (ADRTW) A_R² which allocates more weight to the upper tail and less to the lower tail of the distribution than A² and is given by:

$//<![CDATA[ \begin{array}{l}A_R^2=\frac{n}{2}-\[ \sum_{i=1}^{n} [(2 - \frac{2i - 1}{n}) ln(1 - z_i) + 2z_i] \]\end{array} //]]>$

where z = F(x) and n is the sample size.

When an inappropriately low threshold value is selected, the ADRTW test statistic value will be very large and can approach infinity. When this occurs, the computed ADRTW test statistic values will not appear on the Threshold Selection Diagnostic Plots. Users can view the ADRTW test statistic values in tabular format by selecting File | Tabulate at the top of the Threshold Selection Diagnostic Plots dialog.