Introduction

The Hydrologic Sampler (HS) plugin generates flood events based on a flow or precipitation-frequency curve or from a set of time-series data. This analysis uses the Precipitation-Frequency sampling method to select rainfall depths from a precipitation-frequency curve and scale various hyetograph patterns to the selected depth. Additionally, the HS plugin can define flood seasons, representing the likelihood a flood event would occur for a particular month.

This section explains how to:

  • Set up NOAA Atlas 14 depths as a precipitation-frequency curve using a Generalized Extreme Value (GEV) distribution and estimate an Equivalent Record Length (ERL) to characterize uncertainty. 
  • Estimate Flood Season and incorporate temporal distribution.

1. Estimating Frequency Curve Moments and Equivalent Record Length for NOAA Atlas 14


The Hydrologic Sampler can incorporate precipitation frequency data into modeling workflows by accepting both the distribution type and statistical moments as inputs. For California (Volume 6) at the San Lorenzo watershed, the NOAA Atlas 14 documentation does not provide the exact statistical parameters used to fit the frequency curves, nor does it specify the equivalent record length (ERL) needed to quantify uncertainty. However, the documentation does indicate that the Generalized Extreme Value distribution provides an acceptable fit to the annual maximum series (AMS) data [citation]. For this analysis, the GEV distribution was assumed, and its moments were estimated through a root mean square error (RMSE) minimization process.

The procedure for estimating the precipitation frequency curve was as follows:

  1. Compute Basin-Average Depths: For each frequency (50% to 0.1% AEP) at the 48 hour duration, compute basin-average precipitation frequency depths corresponding to the median, upper (5%), and lower (95%) confidence limits.
  2. Fit GEV Parameters: In Excel, iteratively adjust the GEV moments (location, scale, and shape parameters) for each duration to best fit the computed basin-average median frequency depths. The optimal set of parameters minimizes the RMSE.
  3. Apply Importance Sampling: To efficiently sample rare events (frequencies as low as 10-4 annual exceedance probability), importance sampling was enabled in the Hydrologic Sampler's stratified sampling settings (set to "Exponential, 10-4"), reducing computation time while adequately representing low-probability depths.
  4. Input Parameters and Estimate Uncertainty: Input the best-fit GEV moments into HEC-WAT's Hydrologic Sampler plugin. Use a trial-and-error approach on the Equivalent Record Length (ERL), running a bootstrap routine to generate 500 Monte Carlo realizations for each candidate ERL value. Compare the resulting 5% and 95% confidence limits to those provided by NOAA Atlas 14, adjusting the ERL until a close match is achieved.

GEV moments moments estimate: GEV_frequencyestimate.xlsx

The Hydrologic Sampler (HS) offers an option for stratified sampling, which samples broadly across the frequency curve. This approach reduces the total number of events required to adequately represent the upper tail of the curve. For example, instead of needing to sample 1,000 events to capture an event as rare as the 0.1% Annual Exceedance Probability (AEP), stratified sampling can target rarer events with fewer samples by dividing the curve into “bins” and assigning weights to each bin. Several methods are available for defining bin sizes, including the Distribution option and the Smallest Exceedance Probability of Interest setting. These bins correspond to exceedance probability ranges from which samples are drawn. Testing has shown that using an exponential function to define bin sizes tends to work well for probabilities above the 50% AEP.  Within the Flood Risk Analysis Simulation Editor, 200 events were selected over an analysis period of 50 years.  Given these parameters and the stratification settings, the number of bins created equals 20 (calculated as 200 events divided by 50 years). In this analysis, the Smallest Exceedance Probability of Interest was set to 10⁻⁴, which creates the last bin extending beyond this probability threshold. 

Stratification SettingOption
DistributionExponential
Location to StratifyWatershed 1
Smallest Exceedance Probability of Interest10⁻⁴

The ERL influences the width of the uncertainty bounds around the frequency curve: higher ERL values yield tighter bounds, while lower ERL values result in wider uncertainty intervals. Because NOAA Atlas 14 does not report ERL values but does provide upper and lower bound estimates for each point depth, ERL was determined iteratively by matching bootstrap-generated bounds with those reported by NOAA Atlas 14 at each quantile. 

The following settings are used for testing the ERL lengths. 

SettingsSelection
Duration Sampling OptionsScale to single duration, by key AEP
Maximum Time-Window OptionBasin Average
Duration Selection2 Days
Distribution TypeGEV
UncertaintyEquiv. years of record

image-2024-11-12_11-59-37.png

The ERL lengths that generated the lowest RMSE when comparing 95/5 percent confidence limits to NOAA Atlas ranges were chosen. ERL values between 90 - 160 were tested. An ERL length of 140 generated the lowest RMSE. The 2, 5, 10, 25, 50, 100, 200, 500, and 1000 year depths were extracted from the 500 realizations precipitation frequency depths and shown in the table below.  

Return PeriodAtlas 14 90% Lower ConfidenceAtlas 14 90% Upper ConfidenceAtlas 14 MedianBootstrap 90% Lower ConfidenceBootstrap 90% Upper ConfidenceBootstrap Median
100019.132.924.818.833.224.6
50017.929.122.617.728.922.5
20016.224.619.816.124.219.7
10014.921.517.715.121.017.7
5013.518.715.813.818.115.7
2512.116.113.812.415.413.8
1010.212.811.310.412.211.3
58.510.69.418.710.09.3
25.97.36.56.17.06.5

2. Estimating Flood Seasons and Temporal Distributions

The Flood Season refers to the likelihood of a flood occurring in any given month. In San Lorenzo, the period of heavy rainfall—and consequently higher flood risk—typically occurs during winter, from December through March. The Flood Season option enables the Hydrologic Sampler (HS) to randomly select a Flood Date, which is then passed to the HEC-HMS model. Within HEC-HMS, probabilistic distributions are established for each month to define initial conditions, specifically the initial deficit and initial baseflow values. Once the Flood Date is obtained from the Hydrologic Sampler, HEC-HMS uses this date to select corresponding initial deficit and baseflow values based on the probability distributions assigned for that month.

Users have the flexibility to define a flood season using various probability distributions or by creating an Empirical Probability Distribution. For this analysis, an Empirical Probability Distribution was chosen. This distribution was developed by analyzing daily flow records and setting a flow threshold of 1,000 cfs. The number of flood events exceeding this threshold was counted for each month. These monthly counts were then divided by the total number of floods over the entire period of record to estimate the probability of flooding in each month. Finally, the cumulative probabilities derived from this analysis were input into the Hydrologic Sampler Plugin to guide the selection of Flood Dates consistent with observed flood seasonality.

Six temporal distributions were added to the Hydrologic Sampler as Shape Sets obtained from the analysis from the HEC-HMS calibration effort.  These storms were given equal weight and were applied to the entire probability range.  A Data Check was performed to validate all inputs were applied correctly.