# 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 – 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?

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

R functions and example calculation available as a gist.

# Highlights from ARR Book 7

Book 7 of Australian Rainfall and Runoff is titled Application of Catchment Modelling Systems.  It has been written by experienced people and there is some great information. A few, paraphrased, highlights follow.

• Its often challenging to get good calibrations for all the available historical events and there may good reasons why.

Difficulties in calibrating a model to observed flood events of different magnitude should be taken as an indication of the changing role of processes.

In many cases a significant change occurs between floods that are mostly contained within the stream channel and floods in which floodplain storage plays an important role in the routing process.

If the model has only been calibrated to in-bank floods, confidence in its ability to represent larger floods will be lower.

• Calibration needs to focus on what the model is to be used for, not just ensuring past events are well represented.

The focus of model calibration is not just to develop a model that is well calibrated to the available flood data.  Application of the model to the design requirements must be the primary focus.

It is often the case that calibration floods are relatively frequent while design applications require much rarer floods.  In this case, work in refining the model calibration to the frequent floods may not be justified.

Parameter values should account for the expected future design conditions, rather than an unrepresentative calibration event.

Calibration usually works with historic flood events while the design requirements are for probabilistic events.  The parameters calculated for the historic events may not be applicable to the design flood events.

• On using all available data.

Even if the data is of poor quality or incomplete, it is important that the model calibration be at least consistent with the available information.

Even poor quality observations may be sufficient to apply a ‘common sense test’.

…at least ensure that model performance is consistent with minimal data [available]…

• On inconsistent data

Effort should be concentrated on resolving the source of the inconsistency rather than pursing further calibration.

• Dealing with poor calibration.

It is far more important to understand why a model may not be calibrating well at a particular location than to use unrealistic parameter values to ‘force’ the model to calibrate.

• Don’t expect your model to provide a good fit to all data.

It is extremely unlikely that your simple model is perfectly representing the complex real world well, all your data has been collected without error, or is unaffected by local factors.

• The appearance of great calibrations may mean:

The model has been overfitted to the data with unrealistic parameter values, or

Some of the data, that does not fit well, has been ignored or not presented.

Calibration events should be re-run with adopted parameters and results should show at least reasonable performance for all of the calibration events.

• Confirming model suitability for design events

Model performance, for design events, should be confirmed using Flood Frequency Analysis results, if available, or regional flood frequency information.

Book 7 also has worthwhile guidance on uncertainty analysis, model checking and reporting.

# ARR update from the FMA conference

There were several papers related to Australian Rainfall and Runoff at the FMA conference last week.  Once the papers become available on the FMA website, it would be worth checking, at least these three:

• What Do Floodplain Managers Do Now That Australian Rainfall and Runoff Has Been Released? – Monique Retallick, WMAwater.
• Australian Rainfall and Runoff: Case Study on Applying the New Guidelines -Isabelle Testoni, WMAwater.
• Impact of Ensemble and Joint Probability Techniques on Design Flood Levels -David Stephens, Hydrology and Risk Consulting.

There was also a workshop session where software vendors and maintainers discussed how they were updating their products to become compliant with the new ARR.

A few highlights:

1. The ARR team are working on a single temporal pattern that can be used with hydrologic models to get a preliminary and rapid assessment of flood magnitudes for a given frequency. This means an ensemble or Monte Carlo approach won’t be necessary in all cases but is recommended for all but very approximate flood estimates.

2. The main software vendors presented on their efforts to incorporate ARR2016 data and procedures into models. This included: RORB, URBS, WBMN, RAFTS. Drains has also included functionality. All the models use similar approaches but speakers acknowledged further changes were likely as we learn more about the implications of ARR2016. The modelling of spatial rainfall patterns did not seem well advanced as most programs only accept a single pattern so don’t allow for the influence of AEP and duration.

3. WMA Water have developed a guide on how to use ARR2016 for flood studies. This has been done for the NSW Office of Environment and Heritage (OEH) and looks to be very useful as it includes several case studies. The guide is not yet publicly available but will be provided to the NFRAG committee so may released.

4. Hydrologists need to take care when selecting the hydrograph, from the ensemble of hydrographs, to use for hydraulic modelling. A peaked, low-volume hydrograph may end up being attenuated by hydraulic routing. We need to look at the peaks of the ensemble of hydrographs as well as their volumes. The selection of a single design hydrograph from an ensemble of hydrographs was seen as an area requiring further research.

5. Critical duration – The identification of a single critical duration is often much less obvious now we are using ensemble rainfall patterns. It seems that many durations produce similar flood magnitudes. The implications of this are not yet clear. Perhaps if the peaks are similar, we should consider hydrographs with more volume as they will be subject to less attenuation from further routing.

6. There was lots of discussion around whether we should use the mean or median of an ensemble of events.  The take away message was that in general we should be using the median of inputs and mean of outputs.

7. When determining the flood risk at many points is a large catchment, different points will have different critical durations. There was talk of “enveloping” the results. This is likely to be an envelope of means rather than extremes.

8. The probabilistic rational method, previously used for rural flood estimates in ungauged catchments, is no longer supported. The RFFE is now recommended.

9. The urban rational method will only be recommended for small catchments such as a “two lot subdivision”.

10. There was no update on when a complete draft of ARR Book 9 would be released.

11. Losses should be based on local data if there is any available. This includes estimating losses by calibration to a flood frequency curve. Only use data hub losses if there is no better information. In one case study that was presented, the initial loss was taken from the data hub and the continuing loss was determined by calibration to a flood frequency curve.

12. NSW will not be adopting the ARR2016 approach to the interaction of coastal and riverine flooding. Apparently their current approaches are better and have an allowance for entrance conditions that are not embedded in the ARR approach.

13. NSW will not be using ARR approaches to estimate the impacts of climate change on flooding. Instead they will use NARCLIM.

14. NSW have mapped the difference between the 1987 IFD and the 2016 IFD rainfalls and use this to assist in setting priorities for undertaking flood studies.

15. A case study was presented for a highly urbanized catchment in Woolloomooloo. There was quite an involved procedure to determine the critical duration for all points in the catchment and the temporal patterns that led to the critical cases. Results using all 10 patterns were mapped, gridded and averaged. I didn’t fully understand the approach as presented but there may be more information in the published version of Isabelle Testoni’s paper once it becomes available.

There is still much to learn about the new Australian Rainfall and Runoff and much to be decided.  The papers at the FMA conference were a big help in understanding how people are interpreting and responding to the new guideline.

# Time of concentration: Pilgrim McDermott formula

There are many formulas for the time of concentration.  A previous post discussed the Bransby Williams approach. Here I look at the Pilgrim McDermott formula, which is another method commonly used in Australia and relates time of concentration to catchment area (A):

$t_c = 0.76A^{0.38}$    (hours)                                                     (equation 1)

where A is measured in km2.

This formula is a component of the Probabilistic Rational Method as discussed in Australian Rainfall and Runoff 1987 (ARR1987) Book IV and is recommended for use in:

• Eastern New South Wales
• Victoria (as developed by Adams, 1987)
• Western Australia – wheatbelt region

McDermott and Pilgrim (1982) needed a formula for the time of concentration to develop their probabilistic rational method approach which was ultimately adopted in ARR1987.  They make the point that, for their statistical method, it is not necessary that the time of concentration closely matches the time for water to traverse a catchment, rather a characteristic time is required for a catchment to determine the duration of the design rainfall.  This characteristic time must be able to be determined directly by designers and lead to consistent values of the runoff coefficient and design flood values.

The basic formula for the probabilistic rational method is:

$Q_y = C_y I_{(y,t_c)} A$                                                                  (equation 2)

Where:

• $Q_y$ is the flood of $y$ years average recurrence interval.
• $C_y$ is the runoff coefficient for a particular average recurrence interval.
• $I$ is the rainfall intensity which is a function of $t_c$ (time of concentration) and $y$.
• $A$ is the catchment area.

For a catchment with a stream gauge, where flood frequency analysis can be undertaken, this will provide the $Q_y$ values on the left hand side of equation 2. We also know the catchment area ($A$). If $t_c$ can be estimated via a time of concentration formula, then the rainfall intensity can be looked up in an IFD table for the location and the only unknown is $C_y$.

$C_y = \frac{Q_y}{I_{(y,t_c)} A}$                                                              (equation 3)

This was the approach used in ARR1987. A large number of gauges were selected and $C_y$ values calculated. Ultimately $C_{10}$ values were mapped in Volume 2 of Australian Rainfall and Runoff.   For floods other than those with a 10 year average recurrence interval, frequency factors were provided to calculate the required runoff coefficient values.  This meant design floods could be estimated for ungauged catchments given information on design rainfall intensity which is available everywhere in Australia.

For this approach to work, some relationship is required between $t_c$ and catchment characteristics i.e. we need a time of concentration formula. McDermott and Pilgrim (1982) began their development of such a formula by testing the Bransby Williams approach because that had been shown to be the best of 8 methods examined by French et al. (1974). McDermott and Pilgrim found that Bransby Williams wasn’t suitable for their purposes because it often resulted in runoff coefficients greater than 1 and they thought the use of such large values would be resisted by practising engineers. Equation 2 doesn’t preclude runoff coefficient values greater than 1 but the intuitive definition of $C$ as being “the proportion of rainfall that runs off” requires it.

An alternative time of concentration formula was developed by considering the ‘minimum time of rise of the flood hydrograph’ which McDermott and Pilgrim collected or collated for 96 catchments. This is the time from when storm rainfall starts until stream discharge begins to increase. McDermott and Pilgrim adopted this as their definition of the time of concentration.

The measured times of concentration were regressed against catchment characteristics that included:

• Catchment area
• Main stream length
• Main stream equal area slope
• Main stream average slope
• Catchment shape factor
• Stream slope non-uniformity index
• Vegetation cover
• Median annual rainfall
• Soil type.

Three formulas provided a similar fit to the data with the simple relationship with catchment area ultimately adopted (equation 1).

One of the important implications of the probabilistic rational method approach is that the time of concentration used for design must be calculated using the same formula that was used in the derivation of the runoff coefficients (equation 3).   So, in Victoria (and Eastern NSW and the Wheatbelt of WA), when using the probabilistic rational method to estimate floods in ungauged catchments, it is important to adopt the Pilgrim McDermott formula for the time of concentration and not use any of the many other approaches.

### References

Adams, C. A. (1987) Design flood estimation for ungauged rural catchments in Victoria.  Road Construction Authority, Victoria. (link)

French, R., Pilgrim, D. H. and Laurenson, E. M. (1974) Experimental examination of the rational method for small rural catchments. Civil Engineering Transactions CE16: 95-102.

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Department of National Development and Energy.  Australian Water Resources Council Technical Paper No. 73, pp. 233. (link)

Pilgrim, D. H. and McDermott, G. E. (1982) Design floods for small rural catchments in eastern New South Wales. Civil Engineering Transactions.  Institution of Engineers CE24:226-234.

# Where is ARR?

[Edit 13 Oct 2016]

RFFE software is now available at http://rffe.arr-software.org/

ARR project reports are available at:

The main ARR page is http://arr.ga.gov.au/

The new version of Australian Rainfall and Runoff, plus supporting documents and software, was all available via www.arr.org.au.

This general address has recently been changed to http://arr.ga.gov.au/.

The web-based version of the draft ARR guideline is here or via this direct link.  Flike is still available here.

It seems that all the project reports have been removed, however, Way  Back Machine, a web archiving service, has copies of old ARR sites.

If you go to www.waybackmachine.org and then enter arr.org.au it will bring up a calendar that shows when archives were made.  The most recent version is available via this link
http://web.archive.org/web/20160501050605/http://www.arr.org.au/.  All the project reports seem to be available.

I’ve also made a link to a dropbox folder with those reports that were distributed at the Hydrology and Water Resources Symposium in Hobart in Dec 2015.

The RFFE software does not seem to be available yet on the new site but I know WMA Water are working to get this back up.   Lets hope this is sorted out soon.

# New Draft of Australian Rainfall and Runoff

[Edit 17 Oct 2016: Some web links have changed.  See Where is ARR?]

A new draft version of Australian Rainfall and Runoff has just been released and is available for download here.  The epub file is dated 2016-07-07.

So, what has changed?  After skimming through the document, my preliminary assessment is as follows:

• There is now more than one editor.  The previous version listed James Ball as the editor while this one lists James along with: Mark Babister, Rory Nathan., Bill Weeks, Erwin Weinmann, Monique Retalick and Isabelle Testoni, with Peter Coombes and Steve Roso associate editors for Book 9.
• Industry consultation on the current draft will take place until October 2016 when the editorial team will meet to consider feedback and decide on when the next update will be published.
• Referencing has been improved and many in-text citations are hyperlinked to their location in a reference list.
• Book 2 Rainfall Estimation, Chapter 1, Introduction, has been re-written.
• There is a new chapter on climate change impacts on rainfall (Book 2, Chapter 2.7).
• Book 2, Chapter 4, has a name change, from Spatial Patterns to Areal Reduction Factors.  Alan Seed is no longer listed as an author.  Spatial patterns are now addressed in a new chapter (Book 2, Chapter 6).
• Book 2, Chapter 4, Areal Reduction Factors: the equations used to calculate areal reduction factors have changed (but are still difficult to interpret, at least on the epub version).
• Book 2, Chapter 4, Figure 2.4.1: The ARF regions map looks different but it may just be a change in colour scheme.
• Book 2, Chapter 5 – the chapter on temporal patterns has been greatly expanded and design patterns are now available at data.arr.org.au (although the website is currently unavailable).  The example in Book 2, Chapter 5.10 shows the use of design temporal patterns in RORB modelling.
• Book 2, Chapter 7 now covers continuous rainfall simulation (mainly just a change in Chapter numbering).
• Book 3, Chapter 3.12.  This is a new chapter: RFFE Implementation and Limitations, which includes a discussion on the likely accuracy of the RFFE and additional checking to be undertaken when using the tool.
• Book 4, Catchment Simulation.  There was no content in this book in the Dec 2015 version of ARR.  Now a draft of the whole book is available (Disclosure – I’m the lead author of Chapter 2).
• Book 5, Flood Hydrograph Estimation, has been extensively revised.  There are new chapters on: catchment representation (Book 5, Chapter 2); flood routing principles (Book 5, Chapter 5); and flood hydrograph modelling approaches (Book 5, Chapter 6).
• The losses Chapter (Book 5, Chapter 3) has been revised.  There are new methods to select losses for design flood estimation (Chapter 3.5).  The loss regions have changed (Figure 5.3.16).  There are now only 4 regions and new prediction equations are available for each region.  Median IL and CL values for much of Australia are provided in Figures 5.3.18 and 5.3.19.
• Book 6, Flood Hydraulics, was well advanced in the previous draft but there are several updates.  The chapters on Rock Chutes and Rock Riprap have been removed and a new chapter on safety design criteria has been added (Book 6, Chapter 7).  This was previously in Book 9.
• Book 7, Application of Catchment Modelling Systems, is now available as a complete draft.  There was no content in the December 2015 version of ARR.  This Book includes information relevant for hydrologic models, RORB, RAFTS, URBS and WBMN.
• Book 8, Estimation of Very Rare to Extreme Floods was well advanced in the Dec 2015 draft ARR and a quick review suggests there have been few changes.
• Book 9, Runoff in Urban Areas was not included in the earlier draft.   It was available as a separate PDF but is now integrated into ARR.  All of Book 9 is now available except Chapter 6 ‘Modelling Approaches’.
•  The safety design criteria information from the earlier version of Book 9 has been moved to Book 6.

One issue that is not specifically addressed is the continued use of the urban rational method.  It is not included in the urban book (Book 9) and the subtext is that there are better approaches. However, the urban rational method is widely used in practice and is recommended by some authorities e.g. Melbourne Water (see their hydrologic and hydraulic design guidelines here).

A general issue is that authority guidelines and standards will need to be updated to relate to the new Australian Rainfall and Runoff.  For example the Austroads Guide to Road Design, Part 5: Drainage – General and Hydrology Considerations, refers to the 1987 version of Australian Rainfall and Runoff.  There is a similar issue with the stormwater drainage code, AS/NZS3500.3.

# Teaching hydrology

I’m reading Richard Dawkins’ book An appetite for wonder.  In it, he writes about the tutorial system at Oxford.  As a senior undergraduate, his weekly tutorial assignments included:

• Reading a PhD thesis and writing something similar to an examiner’s report, reviewing the history of the field in which the thesis was written, proposing follow up research and discussing the theoretical and philosophical issues raised by the thesis.
• Becoming, close-to, a world expert in some topic by reading theses, papers, and books and writing an essay on a topic selected by his tutor.

Then there was a one hour one-on-one session with a tutor, discussing and defending his essay.

Image doing something like that with undergraduates in hydrology.  In one week, a student could become close to the world expert in something like the rational method, its history, applications, limitations and requirements for further research.  Or, they could cover hydrologic routing, or some aspect of flood frequency analysis.

At a university, where I taught, our students were taking four units in disparate fields, working part-time, and taking tutorials with 20 or more colleagues.  Hydrology was covered in, perhaps 2 units in a few lectures and a few tutorials.  The teaching was commendably broad but so shallow in comparison with that described by Dawkins.

The common feedback from students was that they did not get sufficient feedback.  How would it be if they had an hour, one-on-one with a tutor to defend and discuss their written work, every week?

Teaching engineering hydrology is a particular challenge.  At a minimum you would like graduates to be able to apply the current methods, work through the steps and come up with a defensible design.  But it would be nice if they could do more.  Ideally, graduates should be able to criticise the current methods, understand the limitations, know when they should be used and when other approaches are better and be able to identify where research is required.  I remember setting an exam question once where I asked students to follow a particular procedure and come up with an answer.  They generally did well.  Next I asked the circumstances when the procedure should be used and when it was not appropriate.  The answers were poor.  Clearly my teaching was lacking.  They had learned to follow standard procedures but not much about when they should be used or when they wouldn’t work.

There are a lot of new tools available with the new version of Australian Rainfall and Runoff.  We need to help the next crop of hydrologists to learn and use these new methods, but we also need to be clear about their limitations.  There is some good material on approaches and limitations in ARR Book 1, Section 3, and in the ‘limitations’ tab of regional flood frequency tool.   The limitations of the recommended approaches to flood frequency analysis are also important.

### References

Dawkins, R. (2001) Evolution in biology tutoring? In: Palfreyman, D. (Ed) The Oxford Tutorial: ‘Thanks you taught me how to think’.  Oxford Centre for Higher Education Policy Studies.  Second edition (link).  Dawkins’ essays starts on page 36.

Dawkins, R. (2013) An appetite for wonder: the making of a scientist.  Bantam Press.  www.richarddawkins.net/afw (link).