1. Basic Hydrology: Flood Frequency

2. Probability and Discharge
Discharge is the flow rate (cubic feet per second) in a conduit (stream, pipe, overland, etc.)
Probability is the chance of observing a particular value of discharge or greater in a given period, typically a year.
These exceedance probabilities are sometimes expressed for stage (depth), or hydraulic structure capacity.
Chapter 4, HDM

3. Annual Exceedance Probabiltiy
In TxDOT HDM, the preferred terminology is Annual Exceedence Probability (AEP)
In other contexts recurrence intervals are used interchangeably
1-percent chance, 0.01 chance, and 100-year recurrence interval all represent the same “amount” of probability.
In recent years, the use of T-year designation is discouraged because it is easy to misinterpret!.

4. Annual return interval
An annual return interval is an alternative way to express the AEP.
The abbreviation is ARI.
The ARI is the average number of periods (years) between periods containing one or more events (discharges) exceeding a prescribed magnitude.

5. Annual Exceedance Probabiltiy
Typical plot of discharge and probability (from HDM)

6. Annual Exceedance Probabiltiy
Probability of observing 20,000 cfs or greater in any year is 50% (0.05) (2-year).
Exceedance
Non-exccedance

7. Annual Exceedance Probabiltiy
Probability of observing 150,000 cfs or greater in any year is 1% (0.01) (100-year)

8. Annual Exceedance Probabiltiy
The “magnitude” axis does not always have to be discharge, it can be stage, volume, or other hydrologic variable.
Magnitude (stage, elevation, discharge, etc.)

9. Estimating Probability
Subjective assessment – probability you will be bored in the next 10 minutes (hard to judge, depends on my “entertainment value”, time of day, how well you slept, interest, etc.)
Fault-tree analysis – probability that a system (computer) will fail but linking the failure probabilities of individual components (transistors, capacitors, etc.)
Historical outcome analysis – estimate probability on past system behavior (this is the method used in hydrology most of the time)
Reference: Engineering Statistics Handbook; Section ; NIST – US Commerce Department.

10. Estimating Probability
Historical outcome analysis – estimate probability on past system behavior (this is the method used in hydrology most of the time)
Time-series – e.g. annual peak discharge versus time
No anticipation the peak comes on the same day each year
Anticipate that the annual peaks are sort of caused by similar, random, processes

11. Estimating Probability
Time-series – e.g. annual peak discharge versus time
Appeal to the concept of “relative frequency” as a model to explain the time-series behavior.
Each year is a roll of “dice”, we record the result, and use the result to postulate the long-term average, anticipated behavior

12. Probability plots
The probability plot is a graphical technique for assessing whether or not a data set follows a given distribution such as the normal or Weibull.
The distribution is the model of the observations, hence it is kind of important to be comfortable we are choosing the most appropriate model from our tool kit.
Perfect agreement is impossible! If the model exactly fits, we probably made an error (i.e. plotted model vs. model, instead of data vs. model)
Reference: Engineering Statistics Handbook; Section ; NIST – US Commerce Department.

13. Example Time-series: (YYYY,Peak Q)
Illustrates concepts related to probability, magnitude, and the underlying mechanics of assessing such behavior.
Time-series: (YYYY,Peak Q)

14. Example
Generally, rank series (small to big, big to small – analyst preference).
Assign a relative frequency to each year assuming each year is a dice roll (independent, identically distributed)
Typical “relative frequency” is the Weibul plotting position (there are others, next lesson)
Plot Magnitude and Cum.Freq

15. Example Ranked, relative frequency, and plotted QPEAK
Cumulative Relative Frequency

16. Example
So at this step, we have an “empirical” probability-discharge plot.
Sometimes can use as-is, but usually we fit a distribution model to the plot, and make inferences FROM THE MODEL!

17. Example
As an illustration, we can fit a normal distribution to the time series
Normal Distribution using the
Time-Series Mean and Variance as fitting parameters
Fit is not all that great
Point here is to illustrate how AEP models are
constructed from observations.

18. (like a key on a calculator
Example
Assume we “like” this fit, then one can interpolate/extrapolate from the distribution model (and dispense with underlying data)
AEP
Magnitude
Error function
(like a key on a calculator
e.g. log(), ln(), etc.)
Distribution Parameters

19. Example
Naturally we would prefer to supply a “F” and recover the “x” directly – not always possible, but in a lot of cases it is.
More importantly, is when we extrapolate – the participant should observe the 1% chance value is NOT contained in the observation record.
To estimate from the model, we simply find the value “x” that makes “F” equal 0.01 (about 3920 cfs in this example)
AEP
Magnitude

20. Summary Probability and Magnitude are Related via a Frequency Curve.
Typically on a log-log plot but not always

21. Summary
The probability is called the Annual Exceedance Probability (AEP) or Annual Recurrence Interval (ARI). AEP is the preferred terminology.
Historical observations are examined to construct “models” of the probability and discharge relationship.
These models are used to extrapolate/interpolate to recover magnitudes at prescribed values of AEP