- 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:

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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 7 – *Managing 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.

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.

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- 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.

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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.

Rosenzweig quantified the relationship between AET and productivity:

Where:

- 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).

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From Wikipedia:

If a certain event did not occur in a sample with

nsubjects, 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.

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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**

**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).

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.

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 (Maribyrnong) |
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 |

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(hours) (equation 1)

where *A* is measured in km^{2}.

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:

(equation 2)

Where:

- is the flood of years average recurrence interval.
- is the runoff coefficient for a particular average recurrence interval.
- is the rainfall intensity which is a function of (time of concentration) and .
- is the catchment area.

For a catchment with a stream gauge, where flood frequency analysis can be undertaken, this will provide the values on the left hand side of equation 2. We also know the catchment area (). If 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 .

(equation 3)

This was the approach used in ARR1987. A large number of gauges were selected and values calculated. Ultimately 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 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 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.

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In addition to the IL/CL model discussed in the previous post, RORB can be run using an initial loss / runoff coefficient model, where the runoff coefficient specifies the proportion of rainfall lost in each time step after the initial loss is satisfied. This reason these different loss models are of interest is that the new version of Australian Rainfall and Runoff is recommending that the IL/CL model is used in place of the runoff coefficient model (Book 5, Section 3.3.1). In some areas, modelling approaches will need to change and this will have implications for flood estimates.

The runoff coefficient loss model is selected as shown in Figure 1.

The user inputs the runoff coefficient, *C*, for a pervious surface. For an impervious surface, there is no opportunity to specify the runoff coefficient which is hard-wired in RORB as 0.9. For mixed sub-areas, the runoff coefficient is scaled, the equations from the RORB manual are:

Where *C _{i}* is the runoff coefficient for the i

Example: For a fraction impervious, and

The initial loss is calculated as as a weighted average of the pervious and impervious initial losses as shown in the previous post. The impervious initial loss is always set to zero in RORB.

Let’s do the calculations for a 100% impervious surface. RORB will set and . Using the 6 hour, 1% rainfall as before, the rainfall excess hyetograph is shown in Figure 2.

Example calculation:

As explained in the previous post, the rainfall between 1.5 hour and 2 hour is 19.4 mm. With a runoff coefficient of 0.9, the rainfall excess will be: 0.9 x 19.4 = 17.5 mm.

The rainfall excess hydrograph from a 10 km^{2} impervious sub-area can be calculated from the rainfall excess hyetograph using the method described in the previous post. The peak flow corresponding to the 17.5 mm rainfall peak is 97.2 m^{3}s^{-1} (see the previous post for sample calculations).

The key point is that we have changed the peak flow from an impervious surface, just by changing the loss model. With the IL/CL model, both initial and continuing loss for a 100% impervious surface are hard-wired to zero. The peak runoff was 107.8 m^{3}s^{-1}. For the runoff coefficient model, initial loss is hard-wired to zero, but the runoff coefficient is hard-wired to 0.9, i.e. we have some loss from the impervious surface. This changes the hydrograph as shown in Figure 3.

The value of the runoff coefficient for an impervious surface is noted in the RORB manual:

The impervious area runoff coefficient

Cis set by the program to 0.9, reflecting the fact that losses occur even on nominally impervious surfaces in urban areas._{imp}

This is reasonable, but inconsistent with the treatment of the continuing loss when the IL/CL loss model is used. In this case, *CL* is hard-wired to zero so there are no losses from impervious surfaces; a feature of RORB for modellers to be aware of.

Also note Equation 3.6 above. This suggests that if the user inputs a runoff coefficient larger than the impervious coefficient (i.e. larger than 0.9) then a value of 0.9 will be used. This isn’t actually implemented. If a runoff coefficient of 1 is input, there is a direct conversion of rainfall to runoff i.e. there is no loss. It is even possible to input runoff coefficients greater than 1.

Equation 3.6 may just be the result of a typo. Some experimenting suggests the behaviour in the model is represented the combination of equation 3.5, above and the following in place of equation 3.6:

That is, the runoff coefficient for an impervious surface is 0.9 unless the runoff coefficient input by the user is larger than 0.9.

Calculations are available via a gist.

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Usually, losses are reduced for impervious compared to pervious surfaces and RORB sets both initial and continuing loss to zero if a surface is 100% impervious.

As an example, consider the 6 hour 1% rainfall for Melbourne, which is 83.4 mm. If we use the ARR1987 temporal pattern (see the previous post), the hyetograph is as shown in Figure 1.

Example calculation:

In the ARR1987 temporal pattern, the time period between 1.5 and 2 hours has 23.3 percent of the rain. The total rainfall is 83.4 mm so the rain in this period is 83.4 x 23.3% = 19.43 mm which is consistent with Figure 1.

The corresponding rainfall excess hydrograph, for an area of 10 km^{2}, which is 100% impervious, is shown in Figure 2 (Note that Areal Reduction Factors have not been used).

Example calculation:

The instantaneous flow at a 2 hours will be

To explain factors at the start of the equation, 1/3.6 is for unit conversion, 1/0.5 is because the temporal pattern has a 0.5 hour time step. The RORB output matches the calculations (Figure 3).

By default, RORB will show the rainfall excess hyetograph above the calculated hydrograph but this is based on the initial and continuing loss as provided by the user. In this case, I’ve specified *IL* = 10 mm and *CL* = 2 mm/h for the pervious areas. These losses, and the hyetograph, are misleading where a sub-catchment has some impervious component. In this case, for a 100% impervious sub-area, both IL and CL are set to zero by the program. It would be best not to display the misleading hyetograph, which can be turned off as shown in Figure 4.

If a sub-area is a combination of both impervious and pervious surfaces, this must be specified to RORB as a Fraction Impervious (*F _{i}*). The initial and continuing losses are scaled based on this fraction.

Where and are the initial and continuing losses for pervious areas as input by the user.

For example, if the pervious value of *IL* is set to 10 mm and *CL* to 2 mm/h, then for a sub-area with a Fraction Impervious value of 60%, the initial and continuing losses will be:

The continuing loss is 0.8 mm/h which is 0.4 mm per 30 min time step.

Running the model with these parameters results in a rainfall excess hydrograph as shown in Figure 5. Note that the start of the rise of the hydrograph is delayed because of the initial loss. The peak is reduced by a small amount (from 108 cumec to 106 cumec because of the continuing loss).

Example calculation, flow peak:

For a real catchment with a 60% fraction impervious, we would expect some early runoff from the impervious surfaces that would provide flow directly into the urban drainage system. RORB doesn’t model this process, which may not matter, depending on the application, but as modellers we need to be aware of this limitation.

Calculations are available as a gist.

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‘Rainfall excess’ is the rainfall left over after the initial and continuing loss are removed. Rainfall excess hydrographs are used in the runoff-routing program RORB. The RORB manual (Section 3.3.4) describes them as follows:

In catchment studies, the program calculates hyetographs for all sub-areas. After deducting losses, it converts the hyetograph ordinates to ‘hydrographs’ of rainfall-excess on the sub-areas, in m

^{3}/s, and interprets the average ‘discharge’ during a time increments as an instantaneous discharge at the end of the time increment.

Lets look at an example. I’m using the methods from the 1987 version of Australian Rainfall and Runoff so I can compare results with calculations in RORB.

I’m working on a catchment in Gippsland where the 1% AEP 6 hour rainfall is of interest. Rainfall IFD data is available from the Bureau of Meteorology via this link.

For the site of interest, the 1% (100-year), 6 hour rainfall depth is 90.9 mm.

Temporal patterns are available in Australian Rainfall and Runoff Volume 2, Table 3.2. Gippsland is in zone 1 and ARI is > 30 years so we need the bottom row from the table below. This shows the percentage of the rainfall depth in each 30 min time period

Applying the temporal pattern to the design rainfall depth results in the following hyetograph.

Calculate the rainfall excess hyetograph by removing the initial loss and continuing loss. For this example,

- IL = 10 mm and
- CL = 2 mm/h.

Note that the continuing loss is 2 mm/h and the time step of the hyetograph is 0.5 h so 1 mm is lost per time step.

The rainfall excess hyetograph is shown in Figure 2.

The procedure to convert a rainfall excess hyetograph to a rainfall excess hydrograph is explained in the quote at the start of the blog. We need to:

- Multiply the rainfall excess by the catchment area (converts rainfall to a volume)
- Divide by the time step (to calculate volume per unit time)
- Ensure flow is allocated to the correct time step – the rainfall during a time step produces the instantaneous flow at the end of the time step
- Ensure the units are correct – calculated flow is is m
^{3}/s, rainfall is in mm and catchment area is in km^{2}.

There is also a discussion of this in ARR2016 Book 5, Chapter 6.4.3.1.

Example calculation: in this case, the sub-catchment area is 78.7 km^{2}. The rainfall in the 3rd time step, between 1 hour and 1.5 hour, is 8.9 mm so the flow at the end of this time step will be:

The rainfall excess hydrograph is shown in Figure 3.

Figure 4 shows the rainfall excess hydrograph as calculated by RORB. The answers look close and I’ve confirmed this by looking at the calculated values.

Calculations are available as a gist.

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