Category Archives: Hydrology

Flood frequency plots using ggplot

This post provides a recipe for making plots like the one below using ggplot2 in R.  Although it looks simple, there are a few tricky aspects:

  • Superscripts in y-axis labels
  • Probability scale on x-axis
  • Labelling points on the x-axis that are different to the plotted values i.e. we are plotting the normal quantile values but labelling them as percentages
  • Adding a title to the legend
  • Adding labels to the legend
  • Positioning the legend on the plot
  • Choosing colours for the lines
  • Using commas as a thousand separator.



Code is available as a gist, which also shows how to:

  • Enter data using the tribble function, which is convenient for small data sets
  • Change the format of data to one observation per row using the tidyr::gather function.
  • Use a log scale on the y-axis.

Links for more information:

Climate change and flood investigations

One surprising finding from the review of the state of hydrologic practice in Victoria, is that climate change impacts on flooding are not being widely considered. Only half the studies reviewed (10 of 20), mention climate change.  Similar findings are reported in other work that shows some Victorian flood managers are not keeping up with their national and international colleagues in considering the additional flood risk predicted with a change in climate.

There is already evidence that rainfall intensity for short duration storms is increasing, which could lead to more frequent and larger flash floods.  This is a particular issue in towns and cities because small urban catchments are especially vulnerable.

In the corporate world, consideration of climate change is being taken seriously.   The recent Hutley opinion found that many climate change risks “would be regarded by a Court as being foreseeable at the present time” and that Australian company directors “who fail to consider ‘climate change risks’ now, could be found liable for breaching their duty of care and diligence in the future”.

The Task Force on Climate Related Financial Disclosures (TCFD), chaired by Michael Bloomberg, has recently released recommendations on how companies should report on climate change risks.  This includes the need to report on risks of “Increased severity of extreme weather events such as cyclones and floods” and “Changes in precipitation patterns and extreme weather variability”.

In the Australian flood scene, the latest Handbook 7Managing the floodplain: a guide to best practice in flood risk management in Australia – provides advice on assessing and reporting on climate change risk.  But the accompanying project brief template and guide, describe climate change aspects of a flood investigation as optional.  The latest version of Australian Rainfall and Runoff provides recommended approaches to assessing climate change impacts on flooding but recent research  argues these methods are too conservative.

On a positive note for Victoria, the Floodplain Management Strategy does encourage consideration of climate change (Policy 9A):

Flood studies prepared with government financial assistance will consider a range of floods of different probabilities, and the rarer flood events will be used to help determine the location’s sensitivity to climate change. Further climate change scenarios may be considered where this sensitivity is significant.



Figure 1: Flooding in Creswick 4 Aug 2010 (link to source)

Flood investigations lead on to decisions about land use zoning and design of mitigation works.  Are climate change risks to these measures foreseeable at the present time?  If so, then they should be considered and reported on.

Clearly this is an area where knowledge and ideas are changing rapidly. Practising hydrologists need to keep up with latest methods, and managers and boards of floodplain management authorities need to be aware of the latest thinking on governance, risk management, and disclosure.

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.

Actual ET and productivity

I was reading a post over at Dynamic Ecology presenting an appreciation of Michael Rosenzwieg, a Professor of Ecology and Evolutionary Biology at the University of Arizona. What caught my eye was his most cited paper which is on the correlation between AET (actual evapotranspiration and productivity).  Here is the abstract:

Actual evapotranspiration (AET) is shown to be a highly significant predictor of the net annual above-ground productivity in mature terrestrial plant communities. Communities included ranged from deserts and tundra to tropical forests. It is hypothesized that the relationship of AET to productivity is due to the fact that AET measures the simultaneous availability of water and solar energy, the most important rate-limiting resources in photosynthesis.

As a hydrologist I knew about actual evapotranspiration (evaporation plus transpiration) but hadn’t paid attention to the link with productivity.  To an ecologist, productivity refers to the rate of biomass production through photosynthesis –  where inorganic molecules, like water and carbon dioxide, are converted to organic material.  Productivity can be measured as mass per unit area per unit time e.g. g m-2 d-1.

In Australia, Actual evapotranspiration is mapped by the Bureau of Meteorology (Figure 1).  There are high values along the coast north of Brisbane, Cape York and ‘The Top End‘.  If Rosenzeig’s correlations hold, these areas are the most ecologically productive in Australia.  In Victoria the highest AET is around Warrnambool, Gippsland and particularly, a small area on the east coast near Mallacoota.  Many of the areas with highest AET are heavily forested.


Figure 1: Average annual areal actual evapotranspiration (link to source)

Rosenzweig quantified the relationship between AET and productivity:

\mathrm{log_{10}NAAP} = (1.66 \pm 0.27) \mathrm{log_{10}AET} - (1.66 \pm 0.01)


  • NAAP is the net annual above-ground productivity in grams per square meter.
  • AET is annual actual evapotranspiration in mm.

The 95% confidence intervals for the slope and intercept are provided.

Rosenzweig’s paper was published in 1968 and the relationship between AET and productivity is better understood now (e.g. Jasechko, S. et al., 2013).  But the simple relationship between AET and productivity does provide an interesting perspective on the Australian landscape.


Michael L. Rosenzweig  (1968) Net Primary Productivity of Terrestrial Communities: Prediction from Climatological Data,” The American Naturalist 102, no. 923 (Jan. – Feb., 1968): 67-74. DOI: 10.1086/282523 (link).

Jasechko, S., Sharp, Z., Gibson, J., Birks, S., Yi, Y. and Fawcett, P. (2013) Terrestrial water fluxes dominated by transpiration.  Nature 496(7445):347-350 (link).



Flood frequency and the rule of 3

There is a ‘rule of three‘ in statistics that provides a rapid method for working out the confidence interval for flood occurrence.

From Wikipedia:

If a certain event did not occur in a sample with n subjects, the interval from 0 to 3/n is a 95% confidence interval for the rate of occurrences in the population.

For example, if a levee hasn’t been overtopped since it was built 100 years ago, then it can be concluded with 95% confidence that overtopping will occur in fewer than 1 year in 33 (3/100).  Alternatively the 95% confidence interval for the Annual Exceedance Probability of the flood that would cause overtopping is between 0 and 3/100 (3%).  Of course you may be able to get a better estimate of the confidence interval if you have other data such as a flow record, information on water levels and the height of the levee.

The rule of 3 provides a reasonable estimate for n greater 30.

Old flood levels on railway bridge plans

Historic flood levels are an important input to flood studies to help with hydraulic model calibration and to improve the precision of flood frequency analysis.  A collection of old bridge plans from Victorian Railways (Victoria, Australia) is available that includes information on flood levels from some of the largest floods from the late 19th and early 20th Century.  From newspaper reports, we know these were big floods, but information on levels is hard to find which means the observations noted on these plans are a valuable resource.

A link to a PDF file with 409 plans is here:

I’ve pulled out some examples in the following figures.

Mitchell River, 1870 flood level at Bairnsdale


Figure 1: Mitchell River at Bairnsdale, 1870 flood level (link to plan)

Nicholson River, 1893 flood level at the Nicholson Bridge

The 1893 flood was a very significant event in this area, resulting in a change in course in the nearby Tambo River which isolated a wharf which was important for river traffic at the time (Erskine et al., 1990).


Figure 2: Nicholson River at Bairnsdale, 1983 flood level (link to plan)

I’ve looked through all the plans and found flood levels for about 15 waterways as noted in the following table. Not all of these will be useful but there are a few gems such as the 1934 flood levels in Orbost which include velocity estimates.

Table 1: Waterways and page numbers in the PDF file of plans
Waterway Location Page number in PDF Comment
Back Ck Taradale 381 “Max” flood level
Goulburn River Toolamba 400 1870 flood
Jacksons Ck Sunbury 361 1916 flood
Maribrynong River 168 ‘Adopted’ flood level
Mitchell River Bairnsdale 178 1870 flood
Moonee Ponds Ck Jacana 221 ‘Adopted’ flood level
Moorabool River 223
MurrayRiver Albury 265 Flood sometime prior to 1882
Murray River Tocumwal 388 Flood prior to 1892
Nicholson River Nicholson 275 1893 flood
Saltwater River
Footscray 334 1906 flood
Snowy River Orbost 301 1934 flood
Stoney Ck 351
Tambo River Bruthen 371
Thomson River Sale 317 Flood prior to Nov 1874
Yarra River Richmond 310 1863 flood
Woady Yallock Ck 405 1909 flood
Wombat Ck 407

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)


  • 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.


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.