# Converting between EY, AEP and ARI

The latest version of Australian Rainfall and Runoff (ARR2016) proposes new terminology for flood risk (see Book 1, Chapter 2.2.5).  Preferred terminology is provided in Figure 1.2.1 which is reproduced below.

Definitions:

• EY – Average number of exceedances per year
• AEP – Annual exceedance probability
• AEP (1 in x) – 1/AEP
• ARI – Average Recurrence Interval (years)

Australian Rainfall and Runoff preferred terminology

For floods rarer than 5%, the relationship between the various frequency descriptors can be estimated by the following straightforward equations.

$\mathrm{EY} = \frac{1}{\mathrm{ARI}}$
$\mathrm{EY} = \mathrm{AEP}$
$\mathrm{AEP(1\; in\; x \;Years)} = \frac{1}{\mathrm{AEP}}$
$\mathrm{ARI} = \mathrm{AEP(1\; in \; x \; Years)}$
$\mathrm{AEP} = \frac{1}{\mathrm{ARI}}$

For common events, more complex equations are required (these will also work for any frequency):

$\mathrm{EY} = \frac{1}{\mathrm{ARI}}$
$\mathrm{AEP(1\; in\; x \;Years)} = \frac{1}{\mathrm{AEP}}$
$\mathrm{AEP(1\; in\; x \;Years)} = \frac{\exp(\mathrm{EY})}{\left( \exp(\mathrm{EY}) - 1 \right)}$
$\mathrm{ARI} =\frac{1}{-\log_e(1-AEP)}$
$\mathrm{AEP} = \frac{\exp(\frac{1}{\mathrm{ARI}}) - 1}{\exp(\frac{1}{\mathrm{ARI}})}$

A key result is that we can’t use the simple relationship ARI = 1/AEP for frequent events.  So, for example, the 50% AEP event is not the same as the 2-year ARI event.

### Example calculations

For an ARI of 5 years, what is the AEP:

$\mathrm{AEP} = \frac{\exp(\frac{1}{\mathrm{5}}) - 1}{\exp(\frac{1}{\mathrm{5}})} = 0.1813$

For an AEP of 50%, what is the ARI?

$\mathrm{ARI} =\frac{1}{-\log_e(1-0.5)} = 1.443$

For an EY of 12 what is the AEP (this is missing from the table in Australian Rainfall and Runoff)

$\mathrm{AEP(1\; in\; x \;Years)} = \frac{\exp(\mathrm{12})}{\left( \exp(\mathrm{12}) - 1 \right)}$

1 in 1.000006 years or 99.99939%

R functions and example calculation available as a gist.

## 1 thought on “Converting between EY, AEP and ARI”

1. Scottish Sceptic

Thanks for the article. Useful definitions, but I still don’t understand why number of exceedances per year starts at 1 and drops, but probability of the same flow being exceeded doesn’t start at 1.